Professeur Invité Liu Xiaobing

Prof Fahmi

Prof Zaoui Ali

Four

Keywords: Polymer systems, Molecular dynamics, Continuum mechanics, Mechanical properties, Interface, Strain-induced mechanisms

Polymers are important raw materials in the actual industrial production due to their peculiar properties. They offer a lot of new extraordinary opportunities in a wide range of engineering applications including 3D printing, waste-water treatment, civil engineering, biomedical engineering, actuators and energy storage. The present PhD dissertation deals with the multiscale modeling of the mechanical behavior of polymer systems using the molecular-level description via molecular dynamics and the continuum-level mechanics.

In a first part of the thesis, molecular dynamics simulations are employed to study the physical and mechanical properties of polyethylene (PE) and carbon-black filled vulcanized cispolyisoprene (cis-PI) systems. We first show the determinant role of the structural composition (i.e. length and number of chains) and temperature on the crystallinity in PE systems obtained by two general measurement methods (bond orientation and global order parameters). We especially highlight the strong influence of the latter factors on the biaxial large-strain stressstrain response till failure, the orientation crystallization provoking a significant directional effect. We then point out that the branch chain composition has a great effect on the shape memory capacities of branched PE systems. Finally, the vulcanization and nanofiller reinforcement mechanisms in cis-PI are investigated at the nanoscale. Besides, we quantify the mechanical response upon stretching-retraction loading sequences and the related nonlinear and inelastic features are studied. The interactions between fullerenes and cis-PI atoms and the aggregation configuration of cis-PI atoms around fullerenes are analyzed in details by introducing local number density and potential mean force through radial distribution function.

In a second part of the present thesis, we address the problem of the continuum-based representation of various polymer systems, including PE, polyvinylidene fluoride (PVDF) and entangled cis-PI systems. We first develop a multi-scale model for strain-induced morphological anisotropy in semi-crystalline PE formulated within a continuum-based micromechanical framework. The crystallographic shear in the crystalline lamellae and the molecular alignment/relaxation of the amorphous phase are two underlying inelastic processes integrated in the constitutive representation. The cavitation damage accumulation related to the progressive nucleation and anisotropic growth of nano-sized cavities in the amorphous phase is ii also integrated. The coupling between the different phases is performed by means of a multiscale homogenization-based approach in which the interfacial interaction is considered. The model output is compared to tensile experimental observations of PE containing a broad range of crystallinities obtained under monotonic and oligo-cyclic tensile loading sequences upon large-strain plastic deformation. We also examine the damage behavior upon stretching and stretching-retraction-recovery sequences at different strain rates and temperatures. We then enrich the continuum-based model to semi-crystalline PVDF by connecting the macroscopic response to strain-induced α → β phase transition. Some fundamental parameters taken from molecular dynamics simulations are transferred to the continuum scale. By completing the scale bridging method by the identification of the other model parameters, the model outputs are found in good agreement with the existing tensile experiments of a PVDF upon large-strain plastic deformation. Finally, we examine the biaxial capacities of a hyperelastic constitutive model thanks to molecular dynamics data of entangled cis-PI systems obtained for different deformation modes.

iii

Résumé

La présente thèse de doctorat porte sur la modélisation multi-échelle du comportement mécanique de systèmes polymères en utilisant la description au niveau moléculaire via la dynamique moléculaire et à l'échelle du continuum via la mécanique des milieux continus.

Dans une première partie, des simulations de dynamique moléculaire sont utilisées pour étudier les propriétés physiques et mécaniques de systèmes en polyéthylène (PE) et en cis-polyisoprène (cis-PI) vulcanisés et renforcés en noir de carbone. Nous montrons d'abord le rôle déterminant de la composition structurelle (c'est-à-dire la longueur et le nombre de chaînes) et de la température sur la cristallinité de systèmes en PE, obtenue par deux méthodes de mesure générales (à savoir les paramètres d'orientation des liaisons et d'ordre global). Nous mettons particulièrement en évidence la forte influence de ces derniers facteurs sur la réponse mécanique biaxiale en grandes déformations et ce, jusqu'à la rupture, la cristallisation de l'orientation provoquant un effet directionnel significatif. Nous montrons ensuite que la composition de la chaîne de ramification a un effet important sur les capacités de mémoire de forme des systèmes en PE ramifiés. Enfin, les mécanismes de vulcanisation et de renforcement par des nanocharges dans le cis-PI sont étudiés à l'échelle nanométrique. En outre, nous quantifions la réponse mécanique lors de séquences d'étirement-rétraction et les caractéristiques non linéaires et inélastiques associées sont étudiées. Les interactions entre les fullerènes et les atomes de cis-PI et la configuration d'agrégation des atomes de cis-PI autour des fullerènes sont analysées en détail en introduisant la densité de nombre local et la force moyenne potentielle par le biais de la fonction de distribution radiale.

Dans une deuxième partie, nous abordons le problème de la représentation constitutive de divers systèmes polymères, notamment le PE, le polyfluorure de vinylidène (PVDF) et le cis-PI enchevêtrés. Nous développons d'abord un modèle multi-échelle pour représenter l'anisotropie morphologique induite par la déformation dans le PE semi-cristallin, formulé dans un cadre micromécanique. Le cisaillement cristallographique dans les lamelles cristallines et l'alignement/relaxation moléculaire de la phase amorphe sont deux processus inélastiques sousjacents intégrés dans la représentation constitutive. L'accumulation de l'endommagement par cavitation liée à la nucléation progressive et à la croissance anisotrope de cavités de taille iv nanométrique dans la phase amorphe est également intégrée. Le couplage entre les différentes phases est effectué au moyen d'une approche basée sur l'homogénéisation multi-échelle dans laquelle l'interaction interfaciale est prise en compte. Les résultats du modèle sont comparés aux observations expérimentales en traction de PE contenant une large gamme de cristallinités, obtenues sous des séquences de chargement en traction monotones et oligo-cycliques lors de grandes déformations plastiques. Nous examinons également le comportement endommageable lors de séquences d'étirement et d'étirement-rétraction-récupération à différentes vitesses de déformation et températures. Nous enrichissons ensuite le modèle constitutif à l'application du PVDF semi-cristallin en reliant la réponse macroscopique à la transition de phase α → β induite par la déformation. Certains paramètres fondamentaux issus de simulations de dynamique moléculaire sont transférés à l'échelle du continuum. En complétant la méthode de transfert d'échelle par l'identification des autres paramètres du modèle, les résultats de ce dernier sont trouvés en bon accord avec les expériences de traction existantes d'un PVDF lors d'une grande déformation plastique. Enfin, nous examinons les capacités biaxiales d'un modèle constitutif hyperélastique grâce aux données de dynamique moléculaire de systèmes cis-PI enchevêtrés obtenues pour différents modes de déformation.

General introduction

Polymer systems have a large range of applications, in sectors like oil industry, automobile, aeronautic, robotic, biomechanics, 3D printing, waste-water treatment, civil engineering, actuators and energy storage, in which they may be subjected to large strains under in-service or manufacturing process. These materials present an extremely large variety of molecular architectures and structural factors (length and number of chains, crystallinity, vulcanization, nanofiller reinforcement…) that leads to an extremely large variety of mechanical responses from the plastic response characteristics of thermoplastics to the nonlinear elastic response characteristics of elastomers. Besides, the external mechanical loading conditions (strain rate, temperature, non-monotonicity, multiaxiality…) affect significantly both the deformation mechanisms (molecular orientation, cavitation damage, phase transformation…) and the largestrain mechanical behavior. As a meaningful prerequisite of advanced applications of polymer systems, predictive multiscale modeling is needed. This PhD dissertation deals with the multiscale modeling of the large-strain mechanical behavior of polymer systems using the molecular-level description via molecular dynamics and the continuum-level mechanics.

Various polymers are investigated in the present work namely, semi-crystalline polyethylene (PE), semi-crystalline polyvinylidene fluoride (PVDF) and carbon-black filled vulcanized cispolyisoprene (cis-PI).

The first part of this PhD dissertation containing Chapters I, II and III is exclusively related to molecular dynamics simulations that are employed to study the physical and mechanical properties of PE and cis-PI systems. In Chapter I, the atomistic approach is employed to study the crystallinity and mechanical properties of PE systems. The influence of the structural composition (i.e. length and number of chains) and temperature on the crystallinity in PE systems is firstly reported by means of two general measurement methods, namely bond orientation parameter and global order parameter. For the different designed PE systems, the large-strain stress-strain response till failure is then examined under different conditions in terms of mechanical loading modes and stretching temperatures. We especially highlight the strong influence of the structural composition and loading conditions on the biaxial large-strain General introduction 2 stress-strain response till failure, the orientation crystallization provoking a significant directional effect. In Chapter 2, through the molecular dynamics simulations, we bring some insights on the influence of branched chain composition and temperature on the large-strain mechanical behavior and the shape memory performance of the same material system. Chapter 3 is dedicated to carbon-black filled vulcanized cis-PI systems. We report the effects of vulcanization and nanofiller reinforcement mechanisms on the physical and mechanical properties of the different designed cis-PI systems. The mechanical response is quantified upon stretching-retraction loading sequences and the related nonlinear and inelastic properties are studied. The local interactions at the nanoscale are analyzed in details by introducing local number density and potential mean force through radial distribution function.

In the second part of this PhD dissertation containing Chapters IV to VII we address the problem of the continuum-based representation of various polymer systems, including PE, PVDF and entangled cis-PI systems. In Chapter IV, we examine the ability of a multi-scale homogenization-based constitutive model to capture the PE large-strain tensile response variation with crystal concentration. The model is applied on a high-density polyethylene with 0.72 crystal content, a low-density polyethylene with 0.3 crystal content and an ultra-lowdensity polyethylene with 0.15 crystal content. The results of our simulations are criticized by analyzing the model ability to capture a series of experimental observations under monotonic, loading-unloading and oligo-cyclic stretching over a large strain range. The effect of the amorphous phase fraction on the microstructure evolution is finally discussed thanks to the model. In Chapter V, the continuum-based model is further developed to consider the cavitation damage effects in the material anisotropy during finite plastic deformation of polyethylene. By this way, a more physically consistent description is formulated in continuum terms to describe the complex coupling existing between microstructure (amorphous and crystalline phases), micromechanical deformation processes including cavitation damage and large-strain flow behavior. The efficiency of the proposed plastic-damage model to capture material response in terms of finite deformation stress-strain behavior and inelastic volumetric strain is critically discussed by comparing simulations to experimental results including strain rate and stretching temperature effects. The effects of crystal plasticity and amorphous cavitation damage on the macroscopic and internal responses are observed thanks to the model under stretching and stretching-retraction conditions. In Chapter VI, we enrich the continuum-based model to render it capable to replicate the mechanical response of PVDF along with the α → β phase transition caused by the straining in the final aim to clarify the connection between the mesostructural evolution and the benefices on the electroactive response. Besides, molecular dynamics simulations are employed to examine the respective mechanical properties of α-PVDF, β-PVDF and amorphous phase. The elastic constants of the distinguishing phases and the onset of α → β phase transition, when α-PVDF is drawn, are then used as direct inputs into the continuumbased model. The intermolecular constants of the amorphous phase are also obtained thanks to molecular dynamics data. The other parameters of the continuum-based model are determined using the stress-strain experimental data of a PVDF deformed over a large-strain range. The effect of the progressive variation in the mesostructural properties caused by the straining on the piezoelectric properties is finally observed. In Chapter VII, we address the problem of the multiscale prediction of the multiaxial hyperelastic behavior of rubbers. We attempt to verify the predictive ability of a hyperelastic constitutive model based on the theoretical framework of continuum mechanics while connecting the continuum-level mechanics to the molecularlevel description by means of molecular dynamics method.

A general conclusion closes this PhD dissertation with the main results obtained.

Ⅰ.1. Introduction

Due to their peculiar properties, polyethylene became important raw materials in the actual industrial production and offer many new extraordinary opportunities in a wide range of engineering applications including 3D printing, wastewater treatment, biomedical engineering, actuators and energy storage.

As a meaningful prerequisite of advanced applications of polyethylene, the formulation of continuum-based constitutive models is needed (van Dommelen et al., 2003;Ayoub et al., 2010Ayoub et al., , 2011;;Abdul-Hameed et al., 2014;Jiang et al., 2019). In the most sophisticated developments, they are developed within a multiscale framework using statistical mechanics and an adequate transition scale scheme. A representative example is the introduction of the crystallization micro-mechanism at the chain-scale and the prediction of its effect on the response at the macroscale (Guo et al., 2018(Guo et al., , 2020)). A rigorous constitutive representation of polyethylene requires the most physically realistic introduction of the structural and loading parameters that govern the macroscopic response. A deeper understanding of the separate and synergistic effects of key parameters is still crucially needed to propose physically based models avoiding (or at least limiting) the arbitrary inclusion of heuristic model constants and providing accurate information for the polyethylene design.

In the meantime, even if the qualitative understanding of the deformation mechanisms in polyethylene is substantial (Schrauwen et al., 2004a(Schrauwen et al., , 2004b;;[START_REF] Hong | A model treating tensile deformation of semicrystalline polymers: quasi-static stress-strain relationship and viscous stress determined for a sample of polyethylene[END_REF]Addiego et al., 2006;Pawlak et al., 2007;Bartczak et al., 2010;[START_REF] Cheng | Chain entanglements and mechanical behavior of high-density polyethylene[END_REF][START_REF] Xiong | Micro/macro-stress relationship and local stress distribution in polyethylene spherulites upon uniaxial stretching in the small strain domain[END_REF], the assessment of their effects on the macroscopic response is difficult to quantify. Number of experiments have been devoted to the study of the relationship between structure and properties in polyethylene. For instance, small angle X-ray diffraction, small-angle neutron scattering, and proton magnetic resonance have been used to characterize the structure. Ayoub et al. (2010) and Schrauwen et al. (2004a) have realized tensile experiments on polyethylene and analyzed the four typical regimes before fracture of the large-strain stress-strain curves (elastic deformation, yield strength and strain-softening in the small-strain domain and strain-hardening in the moderate/large-strain domain), and further explored the crystallinity and crystal region. The Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 8 influence of lamellar thickness and orientation of crystalline regions on the mechanical behavior of polyethylene was analyzed by [START_REF] Hong | A model treating tensile deformation of semicrystalline polymers: quasi-static stress-strain relationship and viscous stress determined for a sample of polyethylene[END_REF], [START_REF] Cheng | Chain entanglements and mechanical behavior of high-density polyethylene[END_REF] and [START_REF] Xiong | Micro/macro-stress relationship and local stress distribution in polyethylene spherulites upon uniaxial stretching in the small strain domain[END_REF].

They investigated the influence of structural factors such as crystal region, amorphous region, crystallinity, molecular weight and entanglement on the different strain regimes of the macroscopic response.

Due to the objective limitations of the current experimental techniques, it is nonetheless difficult to highlight the separate effects of the structural features on the macroscopic response, to measure the material properties at the molecular level and to quantify the various deformation mechanisms that may take place at different scales, at any location in the loaded specimen and at any time upon the mechanical loading. With the continuous improvement of computer level, molecular simulation has become an indispensable tool for studying the properties of complex material systems by including structural parameters and interactions at relevant scales. Indeed, nowadays such a method is a powerful tool to study local interactions and structure properties relationship of polymers from the nanoscale to the macroscale (Brown et al., 1991;Zhang et al., 2001;[START_REF] Simoes | Molecular deformation mechanisms and mechanical properties of polymers simulated by molecular dynamics[END_REF][START_REF] Li | Monte Carlo simulation of uniaxial tension of an amorphous polyethylene-like polymer glass[END_REF][START_REF] Makke | Mechanical testing of glassy and rubbery polymers in numerical simulations: Role of boundary conditions in tensile stress experiments[END_REF][START_REF] Hossain | Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene[END_REF][START_REF] Sanmartin | Following the crystallization process of polyethylene single chain by molecular dynamics: The role of lateral chain defects[END_REF][START_REF] Anoukou | Molecular dynamics study of the polymer clay nanocomposites (PCNs): Elastic constants and basal spacing predictions[END_REF][START_REF] Anoukou | Structural and thermodynamics properties of organo-modified montmorillonite clay[END_REF]Liao et al., 2018;[START_REF] Hagita | Structure formation of a quenched single polyethylene chain with different force fields in united atom molecular dynamics simulations[END_REF]Hagita et al., , 2019)). [START_REF] Li | Monte Carlo simulation of uniaxial tension of an amorphous polyethylene-like polymer glass[END_REF] used the Monte Carlo (MC) method to explore the uniaxial stretching of polyethylene under glassy state at different strain rates and temperatures. The simulation results were highly consistent with those obtained from experiment. [START_REF] Makke | Mechanical testing of glassy and rubbery polymers in numerical simulations: Role of boundary conditions in tensile stress experiments[END_REF] carried out uniaxial and multiaxial tensile simulations on polymers under glassy and rubbery states at different strain rates by the MC method and obtained stress-strain behavior in agreement with experiment. This article discusses the effects of different loading methods on the properties of non-homogeneous glassy and rubbery polymers, including the boundary conditions of loading and the direction of loading, but does not specifically analyze the structural composition of the non-homogeneous polymers. [START_REF] Simoes | Molecular deformation mechanisms and mechanical properties of polymers simulated by molecular dynamics[END_REF] also used the MC method to simulate the tensile deformation of amorphous polymers and investigated the mechanical properties and the deformation mechanisms upon the tensile process. Brown and Clarke (1991) qualitatively obtained the deformation behavior of polyethylene-like amorphous polymers in uniaxial tensile simulations at higher strain rates than experimental ones. [START_REF] Hossain | Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene[END_REF] adopted united-atom (UA) Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 9 method to model the amorphous polyethylene system in the simulation in order to investigate the deformation mechanisms and the influence of temperature, chain number and chain length on the tensile deformation. They found that the elastic deformation and yielding phases are primarily inter-molecular chain interactions, while the strain-reinforced phase is primarily intramolecular interaction. [START_REF] Hagita | Structure formation of a quenched single polyethylene chain with different force fields in united atom molecular dynamics simulations[END_REF] used the Molecular Dynamics (MD) approach to investigate the evolution of single polyethylene molecular structure with eight different force fields under quenching processes. They found that different force fields caused different degrees of chain folding, resulting in different degrees of crystallinity. Hagita et al. (2019) studied the crystallization properties of polyethylene with different molecular structures, namely linear and ring polymers. They found that the linear polyethylene has a slower crystallization rate than cyclic polyethylene. Zhang et al. (2001) and [START_REF] Sanmartin | Following the crystallization process of polyethylene single chain by molecular dynamics: The role of lateral chain defects[END_REF] studied the effect of side chains on the crystallization of single-chain polyethylene. Their results showed that both crystallization temperature and crystallization parameters decrease as the number of branches increases, which corresponds to the conclusion obtained from experiment. Liao et al. (2018) studied the influence of composition of chain branching (Methyl-, Ethyl-and Butyl-groups) on the mechanical properties of branched and linear polyethylene.

In this chapter, MD simulation is employed to study crystallization and mechanical properties of various semi-crystalline polyethylene systems. The large-strain stress-strain response is examined under different conditions in terms of mechanical loading modes and stretching temperatures. This chapter attempts to analyze the inelastic deformation mechanism of multichain semi-crystalline polyethylene using the concept of strain decomposition to investigate the relationship between shear strain and cavitation damage in inelastic deformation.

Ⅰ.2. Computational details

Using the explicit atom model for polymer simulations requires large number of atoms, complex force fields and small time-step that should be about 1/10 of the vibrational period of the C-H bond (the smallest vibrational period in most organics). This means such simulations require high performing computer and long simulation time, which is suitable for studying the intramolecular interactions of effects. However, the aim of our work is to study the interactions Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 10 between each linear chain molecules, including slipping, unentanglement, de-crosslinking etc., and therefore the united atom model is more suitable for this study. In the united atom model, each Methyl-, Ethyl-and Butyl-groups (i.e. the CH, CH2, CH3 monomer) will be considered as a single bead (Fig. Ⅰ.1). By this way, united-atom model requires only one-third of atoms in the explicit atom model. In addition, there is only one kind of bead, which results in fewer interaction between beads and therefore does not need smaller time steps and different integration schemes such as in the explicit atom model, and therefore shorter computational time [START_REF] Chen | A comparison of united atom, explicit atom, and coarse-grained simulation models for poly(ethylene oxide)[END_REF]. The MC self-avoiding random walks is employed here to create the different initial structures.

During the chain generation process, a face-centered cubic (FCC) lattice superimposed on the simulation cell is used, with nearest neighbor distance of 1.53 Å [START_REF] Binder | Monte Carlo and Molecular Dynamics Simulations in Polymer Science[END_REF][START_REF] Shepherd | Multiscale modeling of the deformation of semicrystalline polymers[END_REF][START_REF] Hossain | Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene[END_REF][START_REF] Vu-Bac | Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations[END_REF]. In this work, we create eight structures with different number and length of chain to describe the effect of structural composition on the material properties, as shown in Table Ⅰ.1. The simulation cell is orthogonal and the initial length is determined based on a representative density of 0.95 g/cm 3 for high-density polyethylene (HDPE). A snapshot of the initial state of the polyethylene molecular structure is provided in The DREIDING force field (Mayo et al., 1990) was used in this chapter. This force field has four components: bond stretching, bond angle bending, bond angle torsion and non-bonded interaction, which can be expressed as:

𝐸 𝑣𝑎𝑙 = 𝐸 𝑏𝑜𝑛𝑑 + 𝐸 𝑎𝑛𝑔𝑙𝑒 + 𝐸 𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 + 𝐸 𝑛𝑏 (Ⅰ.1)

The energy expressions and parameters for each component are displayed in Table Ⅰ.2. The large-scale molecular dynamics code LAMMPS was used. Before deforming the simulation cells, we carry out energy minimization of the initial structure, followed immediately by structural relaxation step. The minimize step relaxes any high-energy configurations that crested during the MC self-avoiding random walks. The simulation cells are subjected to a fivestage relaxation process as shown in Table Ⅰ.3. The Nosé-Hoover Thermostat and Barostats were applied to control temperature and pressure. This simulation uses periodic boundary conditions. Periodic boundary conditions lead to greater stresses, but we study the overall trend of the different structures, so their effects can be ignored. The energy evolution is constantly monitored throughout the relaxation process to ensure that the simulation cells remain in equilibrium at the end of each stage, as shown in Fig. Ⅰ.2. In order to see the degree of crystallization of each simulation cell, we analyse chain orientation and global order of simulation cell.

Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene * During relaxation step, the pressure in X, Y and Z directions was kept constant at 1 atm, but independently.

Fig. Ⅰ.2.

Total energy evolution during the relaxation step (end temperature is 100 K).

i) Chain orientation 𝜒 𝑛 , 𝜒 𝑙

Here, the length of the trans-segment is the number of consecutive trans-bonds with dihedral angles |φ|< π/3; which is referred to as "stem length" in the polymer-crystallization field (Fujiwara et al., 1997[START_REF] Fujiwara | Structure formation of a single polymer chain. I. Growth of trans domains[END_REF]. Two crystallization parameters, namely 𝑛 𝑖,𝑡𝑟𝑎𝑛𝑠 and 𝑙 𝑖,𝑡𝑟𝑎𝑛𝑠 , are used in this chapter: 𝑛 𝑖,𝑡𝑟𝑎𝑛𝑠 is the number of the trans-segments in which the monomer i is involved in the formation of the trans-segment, and takes a value ranged from 0 to 4 (Fig. Ⅰ.3), and 𝑙 𝑖,𝑡𝑟𝑎𝑛𝑠 is the length of the trans-segments whose formation involves the monomer i. In general, the crystallinity χ is determined as the number of crystalline atoms divided by the total number of atoms. In this work, the value of χ can be estimated if we deem that an atom

Time (ns)

Step warm

Step 1

Step 2

Step 3

Step 4 ii) Global order (𝑟 𝑐 , 𝑃 2,𝑖 )

The Hermans orientation factor was used to compute the order parameter, which is used to determine the global crystallinity parameters, as follows (Fujiwara et al., 1997[START_REF] Fujiwara | Structure formation of a single polymer chain. I. Growth of trans domains[END_REF][START_REF] Yi | Molecular simulation of bundle-like crystal nucleation from neicosane melts[END_REF]Lavine et al., 2003;Kim et al., 2014;Nicholson et al., 2016;Hagita et al., 2019):

𝑃 2,𝑖 = 3 2 〈(𝑒 𝑖 • 𝑒 𝑗 ) 2 〉 - 1 2 (Ⅰ.2)
Here, i and j index different atoms, all atoms j within a cutoff distance 𝑟 𝑐 of atom i are included in the set for calculating the average value, where 𝜎 is the van der Waals diameter (Fig. Ⅰ.3).

The atom i with 𝑃 2,𝑖 > 𝑃 2,𝑡ℎ are designated as crystalline. In addition, two crystal beads that are within a distance 𝑟 𝑡ℎ are considered to be in the same nucleus. Values of (𝑟 𝑐 , 𝑃 2,𝑡ℎ ) of (2.5σ, 0.40) and (1.5σ, 0.52) are adopted to calculate crystallinity of each atom [START_REF] Yi | Molecular simulation of bundle-like crystal nucleation from neicosane melts[END_REF]Nicholson et al., 2016).

Ⅰ.3. Results and discussion

Ⅰ.3.1. Glass transition temperature

The glass transition temperature 𝑇 𝑔 is a fundamental physical feature of polymers. At temperatures below 𝑇 𝑔 , the material is in a rigid "glassy state". When the temperature is higher than 𝑇 𝑔 , the amorphous molecular chains begin to move, and the material becomes rubber-like Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 14 in its softness and flexibility. In the present work, 𝑇 𝑔 was determined according to the change in the slope of the specific volume-temperature curve [START_REF] Takeuchi | Molecular dynamics simulation of local chain motion in bulk amorphous polymers. II. Dynamics at glass transition[END_REF][START_REF] Han | Glass transition temperatures of polymers from molecular dynamics simulations[END_REF][START_REF] Gee | The role of the torsional potential in relaxation dynamics: A molecular dynamics study of polyethylene[END_REF]Capaldi et al., 2004). The system was equilibrated at 500 K followed by quenching from 500 K to 100 K at a rate of 100 K/ns. Fig. Ⅰ.4 shows the specific volumetemperature curves for different systems. First, a linear fit is performed at both ends of the curve to obtain two linear fit lines. Thus, the temperature at the intersection of these two first points is defined as 𝑇 𝑔 . The obtained values are in agreement with previous theoretical studies [START_REF] Takeuchi | Molecular dynamics simulation of local chain motion in bulk amorphous polymers. II. Dynamics at glass transition[END_REF][START_REF] Han | Glass transition temperatures of polymers from molecular dynamics simulations[END_REF][START_REF] Gee | The role of the torsional potential in relaxation dynamics: A molecular dynamics study of polyethylene[END_REF]Capaldi et al., 2004aCapaldi et al., , 2004b)), as well as typical experimental values [START_REF] Gaur | The glass transition temperature of polyethylene[END_REF]Brandrup et al., 1999;Jiang et al., 2019). The systems will be deformed at four temperatures: 100 K that falls into the glassy state, 250 K and 300 K that are within the transition temperature range (between glassy and rubbery states) and, 400 K that falls into the rubbery state. In the first part, the stress increases almost linearly with strain increasing, the slope of the linear part corresponds to the elastic modulus. At the end of the elastic region, the material reaches the yield point. At this time, in addition to the elastic deformation, some plastic deformations also occur. Then, the stress decreases suggesting a strain-softening process after the yield point.

Further deformation of polyethylene causes a drastic strain hardening and a final stress drop. In this stage, the stiffness decreases catastrophically and the system can be considered in a failure state. Similar behaviors were obtained in previous works (Yoon et al., 1993;Kremer et al., 2001;Faller et al., 2002;Fukunaga et al., 2002). The macro-response may be connected to the structure evolution with the applied strains. of polyethylene at a temperature of 300 K. Firstly, the more chains there are, the smoother the curve is within the natural stretching rate, but the overall trend is the same, so that the properties of the material can be accurately obtained even with a small number of chains.

Above the natural draw rate, the friction between the beads of the united atom model is altered by the presence of hydrogen atoms due to the united atom model ignoring the presence of For simulated cells of the same chain length, the larger the system, i.e. the higher the number of chains, the slightly higher the ductility of the cell, which is also reflected in the response to volume strain, where the steep increase in volume strain is delayed, i.e. the necking is delayed

For long chain length structure, the material has higher elastic stiffness, yield strength and breaking elongation. The material with shorter length chain exhibits more mechanical behavior similar to crystals.

This proves that when the chains are shorter, the polymer cannot be fully entangled and therefore the stretch ability is smaller. a directional effect. At a strain of approximately 0.15, the yield strength along the Y direction is higher than that of the two other stretch directions, which means highest elastic stiffness.

Along the X and Z directions, the initial elastic response is followed by a rollover to yield and no strain-softening is observed. This is a result of the crystal orientation. Indeed, because the polymer chains are parallel in the Y-axis, the folded chains are mainly opened instead of slipping when strain occurs.

The inelastic deformation mechanisms can be analyzed during the course of the uniaxial stretching using the strain decomposition concept [START_REF] Zaïri | Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer[END_REF]. The elastic volumetric strain  𝑒𝑙 , shearing strain  𝑠ℎ𝑒𝑎𝑟 and cavitation strain  𝑐𝑎𝑣 , making up the complete strain, are additively split and plotted in Fig. Ⅰ.9. This figure allows a distinction to be made between the dilatational mechanisms, giving rise to a volume increase, and the shear yielding, occurring at constant volume. shearing strain is negative. The latter behavior is due to some tiny gaps in the structure that still exist in the equilibrium state. In the initial deformation stage, due to the slippage or opening of the chains, these gaps become smaller. At larger strain region, the shearing strain progressively increases and informs about progressive plasticity/fragmentation mechanisms of the crystalline domains. The cavitation strain takes place as the plastic deformation proceed. This most direct indicator of the progressive voiding mechanism informs about the cavities between crystalline domains that become larger and larger until the final failure (Addiego et al., 2006;Pawlak et al., 2007;Bartczak et al., 2010). The voiding mechanism enters in competition with the shear yielding. The latter is the predominant deformation mechanism after yielding but the cavitation strain takes over the shearing strain at larger strains. The competition between these two inelastic mechanisms is clearly directional dependent. The anisotropic response at the macroscale is therefore related to both the initial anisotropy in crystal orientation and the anisotropic deformation micro-mechanisms. The effect of the stretching temperature on the uniaxial stress-strain behavior is reported in Fig. The polyethylene response is known to be sensitive to the mechanical loading mode [START_REF] Zaïri | Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer[END_REF]Hachour et al., 2014). The equibiaxial stress-strain response is provided in Fig. Ⅰ.12 and can be compared to the uniaxial stress-strain response in Fig. Ⅰ.8(b). The illustrative example concerns Structure 2 deformed at 100 K. For the uniaxial stretching, as explained above, the stress-strain curve has four distinct regimes before failure (manifested by the catastrophic stiffness decrease), along with a directional effect due to the crystal orientation.

Different loading axis combinations for equibiaxial stretching show a great influence on the anisotropic stress-strain response. The equibiaxial stretchability before reaching the peak stress is significantly lower than that observed under uniaxial loading. For stretching axis combinations, the stress-strain curves of the same axis are also different. Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 26 be also observed that the increase in temperature leads to the decrease of the two envelopes.

Ⅰ.4. Conclusion

MD simulations were carried out to investigate the crystallization of multi-chain polyethylene structures and the mechanical behavior under uniaxial and biaxial stretching loading. The simulations were conducted for different semi-crystalline polyethylene structures under various loading temperatures representing different polymer states (100 K: glassy state, 250 K and 300 K: transition state and 400 K: rubbery state). From the main results, the following conclusions can be drawn:

1) When side lengths change independently, polymer crystallinity in each direction is different.

Structure composition and temperature have a great effect on the crystallinity parameters (i.e.

crystallinity based on bond orientation and global order parameters). The higher the temperature, the lower the crystallinity of linear polyethylene and the longer the chain length, the lower the crystallinity.

2) The stress-strain curves obtained during the course of the uniaxial stretching show four distinct regimes before failure and the same structure has similar behavior under different stretching temperatures. Structure composition and temperature have a great effect on the different mechanical properties, such as elastic stiffness, plastic flow and failure. Besides, the strain decomposition concept allows to clearly highlight the main deformation mechanisms split into shear yielding and dilatational mechanisms. A directional effect is pointed out both on the uniaxial response and the underlying deformation micro-mechanisms.

3) The biaxial strain ratio has a significant effect on the different mechanical properties, especially the yield strength and the strain-hardening ability. Data are collected to construct yield and failure envelopes under a wide range of biaxial strain ratios and temperatures.

Ⅱ.1. Introduction

Polyethylene (PE) is becoming increasingly important in industries due to its performant mechanical properties and moldability. In order to expand the versatility of its use, it is important to have comprehensive features and corresponding mechanisms. In fact, different molecular structures result in different properties of polyethylene [START_REF] Omar | Effect of molecular structures on dynamic compression properties of polyethylene[END_REF][START_REF] Vicente-Alique | Study of the effect of the molecular architecture of the components on the melt rheological properties of polyethylene blends[END_REF].

The different properties of polyethylene are based primarily on density and the degree of molecular branching. When the single chains of polyethylene are linear or approximately linear, as high-density polyethylene (HDPE), the molecules stack and pack well during the crystallization process. When the linear chains are longer, the material is called ultra-high molecular weight polyethylene (UHMWPE) and stacks and packs molecules much better. In contrast, low-density polyethylene (LDPE), such as linear low-density polyethylene (LLDPE)

and cross-linked polyethylene (PEX) (Liao et al., 2018), have molecules that are loose and do not stack and pack well, whereas LDPE is tough and flexible. Which is especially important.

The effect of these different molecular structures on properties has attracted extensive research interest. Let us note here that the most important is the deformation mechanism of the mechanical properties [START_REF] Ramachandran | Persistence length of short-chain branched polyethylene[END_REF]Yan et al., 2021). Many experiments have been devoted to the study of the relationship between structure and properties in polymers, such as small-angle X-ray diffraction, small-angle neutron scattering and proton magnetic resonance.

With the continuous improvement of computer computing power, molecular simulation has become an important tool for studying the characteristics of complex material systems. Brown and Clarke (1991) qualitatively obtained the deformation behavior of polyethylene-like amorphous polymers in uniaxial tensile simulations at higher strain rates than experimental ones. The autohrs previous studies (Yan et al., 2021) adopted united-atom (UA) model to model the semi-crystalline polyethylene system in the simulation in order to investigate the deformation mechanisms and the influence of temperature, chain number and chain length on the deformation mechanisms. They found that the stress-strain curve of polyethylene has four distinct regimes: elastic, yield, softening and hardening. It is also found that the elastic regime Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 34 and yield are mainly intermolecular chain interactions, while the strain-hardening state is mainly intramolecular interactions. Liao, Huang and Meng (2018) used molecular dynamics to study the mechanical properties of polyethylene with short branch chain. The results show that the longer length of chain branching induces lower density and higher yield of branched polyethylene. In addition, more branched chains bring higher density and lower yield of branched polyethylene. However, these studies are quite limited. For example, the effect of long branched chains and a large number of branched chains on the properties of polyethylene has not been better studied. There have been many previous studies on shape memory property [START_REF] Shi | Study on the effect of transition temperature on shape memory behavior in polyurethane based on molecular dynamics simulation[END_REF][START_REF] Shi | The difference in molecular orientation and interphase structure of SiO2/shape memory polyurethane in original, programmed and recovered states during shape memory process[END_REF]. In Shi's study, the molecule dynamic (MD) simulations were used to characterize the property of shape memory polyurethane (SMPU) and silicon dioxide/shape memory polyurethane (SiO2/SMPU) composite from the atomistic scale. The shape memory property of polyurethane is largely dependent on molecular self-aggregation and thermalinduced molecular diffusion. They suggested set the melting point of soft segment as the target temperature for bring SMPU to the optimal self-recovery capacity.

The purpose of this study is to investigate the uniaxial tensile deformation and shape memory properties of branched polyethylene in molecular dynamics simulations using UA model. In this chapter, we investigate the stress-strain response of branched polyethylene for different branched chain lengths, number of branched chains and temperatures. The stress-strain behavior of polyethylene deformation and its dependence on strain rate and temperature are investigated and compared with previous theoretical and experimental results. To investigate the shape memory properties of branched polyethylene, we applied temperature-induced shape memory behavior on different structures after stretching at different stretching temperatures and performed uniaxial stretching on the structures after shape memory behavior to compare the mechanical properties before and after shape memory behavior.

Ⅱ.2. Computational details

Ⅱ.2.1. Molecular structures of polyethylene chains

In this work, we adopted the UA model to model the branched polyethylene systems. In the UA model, each Methyl-, Ethyl-and Butyl-groups (i.e. the CH, CH2, CH3 monomer) [START_REF] Yi | Molecular simulation of bundle-like crystal nucleation from neicosane melts[END_REF] Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 35 Capaldi et al., 2004;David et al., 2016;Yan et al., 2021) will be considered as a single bead (Fig. Ⅱ.1(a)). By this way, the number of atoms included in the simulation is reduced which will results in fewer interaction between atoms, and consequently the calculation time becomes shorter. The Monte Carlo self-avoiding random walks method is employed here to build the different branched polyethylene structures. During the chain generation process, a face-centered cubic (FCC) lattice superimposed on the simulation cell is used, with nearest neighbor distance of 1.53 Å [START_REF] Yi | Molecular simulation of bundle-like crystal nucleation from neicosane melts[END_REF]Lavine et al. 2003;Capaldi et al., 2004;[START_REF] Vu-Bac | Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations[END_REF]David et al., 2016;Yan et al., 2021). Three different branched polyethylene molecules will be built to study the influence of the different branched chain lengths, number of branched chains and temperatures on the mechanical properties. The length and unmber of main and branched chain in three structures is given in Table Ⅱ.1. These include a large number of short branch chains (Structure 1), a small number of short branch chains (Structure 2) and long branched chains (Structure 3).

Ⅱ.2.2. Force field

The DREIDING force field (Mayo et al., 1990;Kim et al., 2014;Yuji et al., 2016;[START_REF] Bowman | Free volume and internal structural evolution during creep in model amorphous polyethylene by Molecular Dynamics simulations[END_REF], special for the UA model, was used in this work. This potential describes the energy Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 36 compensation for deviations from the equilibrium position as a function of bond expansion and bond angle bending during molecular motion (Kremer et al., 2001;Faller et al., 2002;Fukunaga et al., 2002;Hagita et al., 2019). It also includes related functions that describe the change in energy when a chemical bond is rotated, and the interactions between non-bonded particles. It can be expressed as:

𝐸 𝑣𝑎𝑙 = 𝐸 𝑏𝑜𝑛𝑑 + 𝐸 𝑎𝑛𝑔𝑙𝑒 + 𝐸 𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 + 𝐸 𝑛𝑏 (Ⅱ.1)
The functional expression and respective parameters for each part of the force field for the UA model of polyethylene system are listed in Table Ⅱ.2. to approximate bulk-like behavior. In this section, to obtain the mechanical properties of branched low-density polyethylene, we will first perform uniaxial deformation, NPT ensemble was used in this step, that is, in the stretching direction, the polyethylene systems were stretched at the strain rate of 10 10 s -1 , and the face perpendicular to the direction of stretching, free shrinkage. The stretch simulations were performed for three different structures for a total of 20000 or 11000 total united atoms. Four temperatures were examined: 100 K, 250 K,300 K and 400 K, as shown in Table Ⅱ.4. These temperatures were purposely chosen to select a lowest one (100 K), which corresponds to a temperature just below and above the glass transition temperature (250 K and 300 K), and a temperature above the glass transition temperature (400 K). Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 38 of branched polyethylene. As the temperature rises, the Young's modulus of the material becomes larger, and the softening is also evident after yielding point. For the small strain level (10% strain), we can see that, branched polyethylene with short branch chain (Structure 1) have higher strength than one with long branched chain (Structure 3). This is because the short chain is advantageous to the crystallization of the material, which results in a higher degree of crystallinity. For lager stain level (200 % strain), it is obvious that the longer branch chain can stand greater stretching force due to entanglement of long chain.

Ⅱ.3.2. Shape memory behavior and results

Smart polymers are materials that can adapt their shape and physical properties by responding to external stimuli. Shape memory polymers (SMPs) can recover their original shape from a temporarily deformed state upon responding to an external stimulus such as temperature, light, chemical, electrical or electromagnetic field.

Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 39

Ⅱ.3.2.1. Loading and unloading

In order to make the simulation about the shape memory behavior of branched polyethylene, the material is first stretched until the strain is 100%, and then the stretch is removed. Since this is viscoelastic-plastic material, we know that when the stress returns to zero, the shape of the material will not be completely restored, and the strain that is not restored is inelastic strain, which includes plastic deformation and the resulting voids. that for branched polyethylene materials, when the stress returns to zero, the recovered elastic strain is about 0.2 and the inelastic one is about 0.8, while the inelastic strain for branched polyethylene is less than 0.8.

Ⅱ.3.2.2. Shape memory behavior

The shape memory of polyethylene was achieved according to four steps after unloading: cooling down, shape fixation, warm up and shape memory, as summarized in Table Ⅱ.6. Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 40 First, the material, after unloading stress, is cooled down to 100 K (low temperature), and stress relaxation is performed at low temperature, then the relaxed structure is warmed up to 450 K (higher than 𝑇 𝑔 ). Finally, shape memory simulation is performed at high temperature. is always maintained at 0 during the shape memory process. The recovery R will be defined as:

R = (1 - 𝜀 𝜀 max ) × 100 (Ⅱ.2)
where 𝜀 and 𝜀 max are, respectively, the remanent strain stored in the material and its maximum value. The shape memory response is shown in Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 41 branched chains is larger, the strain recovery is smaller, and the shape memory performance became weaker. Moreover, it is also clear that for the same number of branched chains, when the branched chain length is short, the strain recovery is higher.

Ⅱ.3.3. Shape memory effect

The evolution of mechanical properties caused by shape memory behavior will be studied in this part. The structures previously obtained will undergo the two-relaxation step presented in Table Ⅱ.7. This is to obtain the stable structure at different temperatures followed by being stretched at strain rate of 10 10 s -1 and various temperatures. The stress-strain curve of branched polyethylene before and after shape memory behavior at 100 K were shown in

Initial structure 1 Initial structure 2 Initial structure 3 After shape memery structure 1 After shape memery structure 2 After shape memery structure 3

Hardening regime

Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from molecular dynamics 42 find that the shape memory behavior has a greater influence on the mechanical properties of branched chain, especially the hardening regime. It is because the shape memory behavior leads to the re-polymerization and crystallization of the chain, and the the branched chain facilitate the re-polymerization. From Fig. Ⅱ.5(b), it is evident that the stress hardening is increased for all structures after the shape memory behavior.

Ⅱ.4. Conclusion

Molecular dynamics simulations were used here in order to study the mechanical properties and shape memory behavior of branched low-density polyethylene in uniaxial stretching. We have found that the stress-strain variations obtained during the progress of uniaxial stretching show four distinct regimes before failure, and the same structure has similar behavior under different temperatures. Structure composition and temperature have a great effect on the different mechanical properties, such as elastic stiffness, plastic flow and failure. Moreover, the response of the memory behavior indicates that the material can restore the original shape well, the lower the temperature, the higher the recovery rate. The material through the shape memory behavior, has better mechanical properties.

Ⅲ.1. Introduction

Natural rubber is an elastomeric polymer with high molecular weight and whose main component is cis-Polyisoprene (cis-PI). It is widely used in industry due to its scalable mechanical properties and low masses [START_REF] Sheiko | Architectural code for rubber elasticity: from supersoft to superfirm materials[END_REF]. It has nonetheless the disadvantages of low elasticity, low tensile strength and low abrasion resistance, which limit its applications. Therefore, it is important to think about improving its intrinsic properties by proceeding with other methods [START_REF] Sethulekshmi | Recent developments in natural rubber nanocomposites containing graphene derivatives and its hybrids[END_REF].

The most popular method to improve the mechanical and thermomechanical properties of natural rubber is to vulcanize it [START_REF] Akiba | Vulcanization and crosslinking in elastomers[END_REF]Mark et al., 2013). The process of vulcanization is to heat the rubber and sulfur together to form crosslinks between the single chains of rubber. Considerable experimental studies have been devoted to elaboration and characterization of crosslinked rubbers. To name a few, [START_REF] Zhao | Influence of crosslink density on mechanical properties of natural rubber vulcanizates[END_REF], [START_REF] Chaikumpollert | Mechanical properties and cross-linking structure of cross-linked natural rubber[END_REF], [START_REF] Kashihara | Effects of the degree of crosslinking and test rate on the tensile properties of a crosslinked polyacrylic pressure-sensitive adhesive and vulcanized rubber[END_REF] and [START_REF] Kim | Correlation between the crosslink characteristics and mechanical properties of natural rubber compound via accelerators and reinforcement[END_REF] used experimental means to study the crosslinking process and the effect of crosslinking on properties and especially the tensile strength. However, only few theoretical studies were developped so far. Note for example continuum-based (Guo et al., 2018) and coarse-grained (Yagyu et al., 2018) models that were developed to predict the tensile strength of crosslinked rubbers.

Another approach enhancing durability and strength consists to add nanofillers into the rubber, the most widely used being carbon black (CB) (Yazdani et al., 2018;[START_REF] Gao | Surface modification of nano-sized carbon black for reinforcement of rubber[END_REF][START_REF] Jian | Creep performance of CNT-based nanocomposites: a parametric study[END_REF]. The structure of CB is composed of CB particles aggregated by van der Waals forces.

Therefore, each CB structure is different, depending on the particle size, aggregation size, etc.

Among them, the most frequently used materials are fullerenes in zero dimension, carbon nanotubes (CNT) in one dimension, and graphene in two dimensions. Researchers have developed a great interest in studying natural rubber-based nanocomposites containing CB.

Various properties of NR composites that can be enhanced by filling the NR matrix with CB, including but not limited to: graphene, CNT or fullerenes (C60). Yazdani et al. (2018) studied the effect of CNT on polyethylene (PE). They found that CNT increases the tensile strength of LDPE-CNT and HDPE-CNT composites in the studied concentration range lying between 0.79 and 2.63 wt%. [START_REF] Jian | Creep performance of CNT-based nanocomposites: a parametric study[END_REF] found that the scale, weight fraction and good dispersion state of the nanofiller of nanocomposites have a great influence on the resistance to timedependent deformation. In addition, the experiments of [START_REF] Gao | Surface modification of nano-sized carbon black for reinforcement of rubber[END_REF] showed that when the pre-treatment temperature is lower than 450 ℃, the 300% modulus of rubber increases with the pre-treatment temperature increase; when the pre-treatment temperature is higher than 450 ℃, the conclusion is the opposite.

Based on today's advanced computers, there is an increasing interest in using molecular dynamics simulations (MD) to study the properties of complex material systems (Yan et al., 2021). However, so far, only a few previous studies have dealed with MD applied on rubber filled with carbon nanoparticles. [START_REF] Li | A molecular dynamics simulation study on enhancement of mechanical and tribological properties of polymer composites by introduction of graphene[END_REF] used MD to simulate the properties of graphene-filled polymers and found that after adding graphene reinforcements, the Young's modulus of the polymer composites increases by about 150%, the shear modulus increases by 27.6%, and the hardness increases by 27.6%. In addition, the average coefficient of friction and wear rate of the polymer composites were reduced by about 35% and 48%, respectively. In the same context, [START_REF] He | Thermal conductivity of carbon nanotube/natural rubber composite from molecular dynamics simulations[END_REF], [START_REF] Kwon | A large increase in the thermal conductivity of carbon nanotube/polymer composites produced by percolation phenomena[END_REF] and [START_REF] Gong | Mesoscopic modeling of heat transfer in carbon nanotube multiphase polymer composites[END_REF] demonstrated the MD ability to quantitatively predict the thermal conductivity of polymers.

Based on the aforementioned research, this chapter quantitatively studies the effects of vulcanization and nanofiller on the physical and mechanical properties of rubber by means of MD simulation. In addition, the packing on the surface of the filler is also analyzed. For this purpose, natural rubber-CB composites were investigated, where the rubber melt consists of 30 single chains (C170H274) containing 34 monomers (C5H8), and C60, whose amount is in the range between 0 and 26.42 phr (per hundred rubber). By creating aggregates, we can study the effect of the number of fullerenes on the mechanical and thermodynamic properties of rubber-based materials.

Ⅲ.2. Structure and potential

The explicit atom model (Kang et al., 2019;Sharma et al., 2020;Zhang et al., 2021) adopted Here, we adopt the polymer consistent force field (PCFF) (Sun et al., 1994) as the basis of the potential function, which belongs to the consistent force fields family with the same functional form as other members such as CFF91, CFF, and COMPASS. It involves a Lennard-Jones 9-6 potential for the intermolecular and intramolecular dispersion-repulsion interactions. Specific stretching, bending and torsion terms describe 1-2, 1-3 and 1-4 interactions, respectively.

Electrostatic charges are determined from pair contributions. In this force field, the carbon atoms in the cis-PI can be classified into 3 types according to the way the atoms are grouped, described in details in references (Sun et al., 1994;Li et al., 2016) and skipped here for brevity.

The total force field is the sum of the bonded force field U bonded and the non-bonded force field U no-bonded . The functional form can be illustrated in Eq. (Ⅲ.1)-Eq.(Ⅲ.12).

U (Ⅲ.1)

U bonded = U b + U θ + U φ + U ℵ + U b-b ′ + U b-θ + U b-φ + U θ-φ + U θ-θ ′ + U θ-θ-φ (Ⅲ.2) U b = k 2 (b -b 0 ) 2 + k 3 (b -b 0 ) 3 + k 4 (b -b 0 ) 4 (Ⅲ.3) U θ = k 2 (θ -θ 0 ) 2 + k 3 (θ -θ 0 ) 3 + k 4 (θ -θ 0 ) 4 (Ⅲ.4) U φ = k 1 (1 -cos 𝜙) + k 2 (1 -cos 2𝜙) + k 3 (1 -cos 3𝜙) (Ⅲ.5) U ℵ = k 2 𝜒 2 (Ⅲ.6) U b-b ′ = k(b -b 0 )(b ′ -b ′ 0 ) (Ⅲ.7) U b-φ = (b -b 0 )[k 1 cos 𝜙 + k 2 cos 2𝜙 + k 3 cos 3𝜙] (Ⅲ.8) U θ-φ = (θ -θ 0 )[k 1 cos 𝜙 + k 2 cos 2𝜙 + k 3 cos 3𝜙] (Ⅲ.9) U θ-θ ′ = k(θ -θ 0 )(θ ′ -θ ′ 0 ) (Ⅲ.10) U θ-θ-φ = k(θ -θ 0 )(θ ′ -θ ′ 0 ) cos 𝜙 (Ⅲ.11) U b-θ = k(b -b 0 )(θ -θ 0 ) (Ⅲ.12)
Ⅲ.3. Simulation details

Ⅲ.3.1. Vulcanization of natural rubber

To obtain vulcanized rubber, elemental sulfur (-S-S-, as the most common vulcanizing agent) (Fig. III.1(c)) [START_REF] Engelmann | Computer simulation of thermal conductivity in vulcanized polyisoprene at variable strain and temperature[END_REF] is added to the cis-PI melt and, by heating at 600 K, crosslinking bonds are formed, as shown in Fig. III.1(d). PCFF force field is also applied to C-S bonds, and the sulfur crosslinking bonds are supposed only generated in the C atom of the double bond. In this process, we control the degree of crosslinking of natural rubber by adding different amounts of elemental sulfur. The degree of crosslinking was defined by volume

U bonded
fraction. The vulcanization degree is displayed in Table III.1, where 0 indicates unvulcanized natural rubber.

Table Ⅲ.1. Crosslink density.

Crosslink degree (%) 0, 0.3, 1.16, 2.3

Ⅲ.3.2. Addition of fullerenes

In order to add fullerene (Fig III.2. The interaction between the CB atoms and between the CB atoms and the atoms of the rubber are also determined by the PCFF force field. Previous studies [START_REF] Xiong | Atomistic investigation of the interfacial mechanical characteristics of carbon nanotube reinforced epoxy composite[END_REF][START_REF] Choi | A multiscale mechanical model for the effective interphase of SWNT/epoxy nanocomposite[END_REF][START_REF] Singh | Effect of temperature on elastic properties of CNT-polyethylene nanocomposite and its interface using MD simulations[END_REF]Jiang et al., 2019;[START_REF] Jian | Understanding the effect of functionalization in CNT-epoxy nanocomposite from molecular level[END_REF][START_REF] Liu | Thermal conductivity of aligned CNT-polyethylene nanocomposites and correlation with the interfacial thermal resistance[END_REF] have proven that PCFF is well suited for CBpolymer composite systems, and the results they provide are reliable, compared to the theoretical and experimental values. The fraction of nanofillers was defined by nanofiller mass fraction in phr.

Table Ⅲ.2. Nanofiller amount.

Nanofiller fraction (phr) 0.97, 2. 94, 7.83, 11.74, 26.42

Ⅲ.3.3. Simulation step

The initial rubber-fullerene composites structure will be subjected to a four-stages relaxation step to relax any high-energy configurations. Initially, the simulation run for 2×10 6 timesteps (Δt=0.5 fs) using NVT ensemble at 600 K, then the structure is relaxed for 2×10 6 timesteps (Δt=0.5 fs) using NPT ensemble at 600 K. In this process, the crosslinking reaction is performed in the simulation cell, and in order to obtain a random hinge structure, crosslinking is performed every 100,000 steps until the sulfur is completely crosslinked. When the crosslinking is Chapter Ⅲ. Physical and mechanical properties of filled vulcanized rubber by molecular dynamics simulations 52 completed, the vulcanized structure will continue to relax at 600 K with NPT ensemble. The next relaxation cooled the structure down to the room temperature, 297 K, for 2×10 6 timesteps (Δt=0.5 fs) followed by further relaxation of 2×10 6 timesteps (Δt=0.5 fs) at room temperature.

Ⅲ shows that the physical density of vulcanized natural rubber increases almost linearly with the crosslink volume fraction increase. These results coincide with the experimental value of the literature (0.91 g/cm3 (Fujiwara et al.,1997)) and the previous simulation results (0.825 g/cm 3 [START_REF] Harmandaris | Detailed molecular dynamics simulation of the self-diffusion of n-alkane and cis-1,4 polyisoprene oligomer melts[END_REF], 0.84 g/cm 3 [START_REF] Pandey | Multiscale modeling of polyisoprene on graphite[END_REF], 0.854 g/cm 3 [START_REF] Khuntawee | Molecular dynamics study of natural rubber-fullerene composites: connecting microscopic properties to macroscopic behavior[END_REF] and 0.885 g/cm 3 [START_REF] Sharma | Validation of force fields of rubber through glass-transition temperature calculation by microsecond atomic-scale molecular dynamics simulation[END_REF]). The coefficient of thermal expansion is defined as the fractional increase in volume per unit rise in temperature:

α 𝑣 = 𝜕𝑉 𝜕𝑇 • 1 𝑉
. It is plotted in Fig. III.2(b). It can be observed that it decreases with the crosslink volume fraction increase. In the unvulcanized case, the thermal expansion value is found to be 4.81×10 -4 K -1 , which is quite close to experimental values of the literature (5.97×10 -4 K -1 [START_REF] Kawahara | Thermal properties and crystallization behavior of highly deproteinized natural rubber[END_REF] and6 Mark et al., 2007)). The specific heat capacity is calculated by

.1×10 - 4 K -1 (
𝐶 𝑣 = 𝑑𝐸 𝑡𝑜𝑡𝑎𝑙 𝑁𝐾𝑇 2 , in which
𝐸 𝑡𝑜𝑡𝑎𝑙 is the total energy, N is the number of atoms, K is the Boltzmann constant and T is the temperature. In this work, the 𝐶 𝑣 value was found to be in the range between 1.36 J•g -1 •K -1 and 2.3 J•g -1 •K -1 and increases with increasing the crosslink volume fraction, as shown in Fig.

III.2(c

). These findings are in good agreement with experimental data (1.87 J•g -1 •K -1 [START_REF] Sapkota | Influence of clay modification on curing kinetics of natural rubber nanocomposites[END_REF]), and preceding simulation result (1.75 J•g -1 •K -1 [START_REF] Mandal | Variations in specific heat and microstructure in natural rubber filled with different fillers as studied by differential scanning calorimetry[END_REF]). In this part, we study the mechanical response of the system under a stretching at a strain rate of 10 10 s -1 . All stress and strain quantities are defined in their true form: Cauchy stress vs.

Ⅲ.4.1.2. Mechanical properties

Hencky strain. Note that due to the small size of the simulated cell, there are fluctuations in the stress-strain curve, a phenomenon that exists in many studies (Yazdani et al., 2018;Payal et al., 2019;Zhang et al., 2021). The Savitzky-Golay filter (Larive et al., 2013) was used to reduce the magnitude of the fluctuations and finally to obtain the trend of the stress. the stress evolution as a function of the strain. Firstly, it can be observed that the stress-strain response presents a yield event characteristic of thermoplastic-type materials although natural rubber is an elastomeric medium. Actually, its special structure (crosslinking, long chains, etc.)

renders the simulation at molecular level not accurate regarding their mechanical properties till today [START_REF] Sen | Viscoelastic properties of polymer melts from equilibrium molecular dynamics simulations[END_REF][START_REF] Sapkota | Influence of clay modification on curing kinetics of natural rubber nanocomposites[END_REF]Larive et al., 2013;[START_REF] Mandal | Variations in specific heat and microstructure in natural rubber filled with different fillers as studied by differential scanning calorimetry[END_REF]. Secondly, the crosslinking influence is not significant below a strain of 1 and the three curves are superimposed. Using the Neo-Hookean law to fit the elastomer response, C1 is displayed in Table III.3. It can be seen that the difference is very small. When the strain is greater than 1, crosslinked bonds come into play and strain-hardening occurs. For C60 less than 5 phr, the filler has essentially no effect on the density; while at greater than 5 phr, the density increases, similar to the results of a previous study [START_REF] Jian | Creep performance of CNT-based nanocomposites: a parametric study[END_REF]. However, the coefficient of thermal expansion is more complicated, which should be caused by the aggregation of the filler in the simulation cell. Specific heat capacity, on the other hand, is the Chapter Ⅲ. Physical and mechanical properties of filled vulcanized rubber by molecular dynamics simulations 57 opposite of density, which varies greatly in small quantities and does not increase excessively when the amount of filler becomes larger. All simulation results in this section match previous studies (Fujiwara et al., 1997;[START_REF] Kawahara | Thermal properties and crystallization behavior of highly deproteinized natural rubber[END_REF][START_REF] Harmandaris | Detailed molecular dynamics simulation of the self-diffusion of n-alkane and cis-1,4 polyisoprene oligomer melts[END_REF]Mark et al., 2007;[START_REF] Sapkota | Influence of clay modification on curing kinetics of natural rubber nanocomposites[END_REF][START_REF] Mandal | Variations in specific heat and microstructure in natural rubber filled with different fillers as studied by differential scanning calorimetry[END_REF][START_REF] Pandey | Multiscale modeling of polyisoprene on graphite[END_REF][START_REF] Sharma | Validation of force fields of rubber through glass-transition temperature calculation by microsecond atomic-scale molecular dynamics simulation[END_REF][START_REF] Khuntawee | Molecular dynamics study of natural rubber-fullerene composites: connecting microscopic properties to macroscopic behavior[END_REF], where the experimental data are presented in Section III.4.1.

W = C 1 (𝐼 1 -3) with 𝐼 1 the first strain invariant.

Ⅲ.4.2.2. Configuration of the interface

To explain the nanofiller-induced changes on the macroscopic properties of natural rubber, we analyse the interactions between fullerenes and cis-PI at different concentrations and the where 𝜌 = 𝑁 𝑉 ⁄ is the number density in which N is the total number of the atoms in system and V is the volume of the system; K is the Boltzmann constant; T is the temperature. In the centre of the contour there is a schematic of a fullerene, where the interior of the fullerene and the surface immediately adjacent to the fullerene are pure white, which is an indication that within this range no stacking of the atoms occurs and shows that neither the initial position of the plateau period is enhanced, i.e. the resulting pile thickness widens, but all converge to the global density as the distance increases. In contrast, when the C60 fraction is 26.42 phr, there is no plateau period, because when the C60 fraction is larger, some fillers are closer together, as in

Ⅲ.4.2.3. Mechanical properties

The simulation unit is then uniaxially stretched and unloaded with a strain rate of 10 9 s -1 and 10 10 s -1 . Stress-strain curves are shown in Eyring-like model [START_REF] Pavlov | Filler reinforcement in cross-linked elastomer nanocomposites: insights from fully atomistic molecular dynamics simulation[END_REF] was used to derive the moduli at experimental dimensions, as shown in Fig. III.11(c). We can find that the calculated results then converge to the previous study [START_REF] Ogden | Non-linear Elastic Deformations[END_REF], and the value of the elastic modulus is larger when the filler concentration is more important. When the strain is less than one, there is a clear difference between the three curves. We have used Neo-Hookean fitting stress-strain curves, and C1 is given in Table III.4. The difference is greater than 30%, indicating that the nanofillers can improve the initial elastic properties of the material, and the slope of the stress is basically similar at this time in the large strain stage, so it indicates that crosslinking will control the strain-hardening. During unloading, as the C60 content increases, the more strain is recovered.

Ⅲ.5. Conclusion

The object of this study is the main component of natural rubber, cis-polyisoprene(cis-PI),

which is vulcanized by adding element sulphur to cis-PI and vulcanizing at high temperatures to obtain vulcanized rubber, and then adding different amounts of fullerenes (C60) to the vulcanized rubber at high temperatures to obtain CB filled vulcanised rubber. The macroscopic physical and thermodynamic properties of vulcanized rubber and CB filled vulcanised rubber were studied to obtain the effect of vulcanisation and nanofiller on natural rubber. The higher the degree of vulcanization, the formation of a network between the molecular chains, resulting in higher density, lower thermal expansion and higher specific heat capacity. Vulcanization therefore also leads to an increased ability of the rubber to withstand stress at large strains.

Fillers, on the other hand, have the same effect on density and specific heat capacity as vulcanization, but there is no clear trend towards an effect on the coefficient of thermal expansion. The filler also leads to an increase in the constant parameter (C1) of the rubber at small strains, about 30% and an increase in the recovery strain. The attraction of fullerenes to cis-PI atoms and the aggregation configuration of cis-PI atoms around fullerenes were analysed by radial distribution function (RDF), introducing local number density and potential mean force (PMF).

Ⅳ.1. Introduction

Semi-crystalline polyethylene systems have a large range of applications, in sectors like oil industry, automobile, aeronautic, robotic, biomechanics and civil engineering, in which they may be subjected to large strains under in-service or manufacturing process. These materials present an extremely large variety of molecular architectures with a crystallinity index ranged from about seventy percent down to a few percent only, passing the morphology from wellorganized crystalline lamellae to discrete crystalline lamellae randomly dispersed within an amorphous matrix [START_REF] Peacock | Handbook of Polyethylene: Structures, Properties, and Applications[END_REF]. The broad morphology range leads basically to an extremely large variety of responses from the plastic response characteristics of thermoplastics to the nonlinear elastic response characteristics of elastomers (Ayoub et al., 2011;Abdul-Hameed et al., 2014) with a profound influence of the amorphous and crystalline features such as density of stress transmitters in the amorphous layers and crystal dimensions [START_REF] Hillmansen | The effect of strain rate, temperature, and molecular mass on the tensile deformation of polyethylene[END_REF]Argon et al., 2005;Kazmierczak et al., 2005;Bartczak and Galeski, 2010). The development of continuum-based models is a prerequisite for designing and performance predicting polyethylene-based products whether in the form of fibers, films or massive parts.

Modeling response of semi-crystalline polyethylene systems is to relate to their complex organization hierarchy from the nano-sized lamellar structure to the macroscopic scale. Purely phenomenological approaches using theories such as viscoelasticity [START_REF] Zhang | Nonlinear mechanical response of high density polyethylene. Part II: Uniaxial constitutive modeling[END_REF] or viscoplasticity [START_REF] Drozdov | Constitutive equations of finite viscoplasticity of semicrystalline polymers[END_REF][START_REF] Colak | Modeling viscoelastic and viscoplastic behavior of high density polyethylene (HDPE)[END_REF][START_REF] Zaïri | Numerical modelling of elastic-viscoplastic equal channel angular extrusion process of a polymer[END_REF][START_REF] Khan | Amorphous and semicrystalline solid polymers: Experimental and modeling studies of their inelastic deformation behaviors[END_REF][START_REF] Hadj Hamouda | Viscoplastic behaviour of a medium density polyethylene (MDPE): Constitutive equations based on double nonlinear deformation model[END_REF]Dusunceli and Colak, 2008;Drozdov et al., 2013) have the advantage of simplicity. Nonetheless, these models provide only a mathematical description of the different aspects of the material response, generally under lowstrain and without obvious connection to the actual microstructure and its deformation-induced evolution. Over the years, continuum-based models have been developed with the concern to introduce microstructural specificities as precisely as possible in the aim to provide a deep understanding of the separate and synergic effects of key microstructural parameters governing the macroscopic response. The link with the microstructure strongly depends on the approach used for the constitutive representation and the observation scale from which the model starts.

Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 69

In the most physically consistent models, the microstructure approximation of semi-crystalline polymer systems is generally based on composite-type representations.

Following the pioneering work of [START_REF] Haward | The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics[END_REF], continuum-based models have been proposed to constitutively describe the large-strain material behavior of semi-crystalline polymers by the combination of resistances representing intermolecular and molecular network micro-mechanisms (Ayoub et al., 2010(Ayoub et al., , 2011;;Abdul-Hameed et al., 2014;Makki et al., 2017;Sepiani et al., 2018;Chen et al., 2019;Deplancke et al., 2019;Qi et al., 2019;Bernard et al., 2020). The latter micro-mechanism describes the resistance to deformation of the amorphous molecular structure to simulate the molecular orientation/relaxation process during the strainhardening stage of the stress-strain response (Boyce et al., 2000). In the above-cited models, a simple averaging homogenization is used and only for the intermolecular resistance which does not allow to represent the crystallinity effect on the deformation-induced orientation. The active interaction between crystalline and amorphous domains is a first-order factor which requires to be taken into account using more sophisticated micromechanics-based approaches. For instance, micromechanics-based models using the Eshelby inclusion theory were proposed through the matrix-inclusion constitutive representation of the semi-crystalline polymer system (Bédoui et al., 2006;Gueguen et al., 2010;Anoukou et al., 2014;Hachour et al., 2014). These models were restricted to the elastic stiffness and yield strength predictions, and more recently extended to the small-strain post-yielding behavior (Mesbah et al., 2021). Multi-scale homogenizationbased constitutive models have been also developed by considering at the mesoscopic scale an aggregate of two-phase layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers. Using this concept, the elastic-viscoplastic deformation behavior of high-density polyethylene was predicted at small strain levels (Nikolov and Doghri, 2000;Nikolov et al., 2002) and moderate strain levels by simulating the texture evolution (Lee et al., 1993a(Lee et al., , 1993b;;van Dommelen et al., 2003;Agoras and Ponte Castaneda, 2012;Uchida and Tada, 2013;Mirkhalaf et al., 2019). The models are generally identified using loading modes in which the crystal plasticity is a first-order phenomenon, such as compression, channel die compression and shear. While the local events involved in yielding due to tensile straining are not connected with clear crystal shearing but with the phenomena occurring in the amorphous phase and the Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 70 relation to crystal thickness is not expected, rather to crystallinity degree, read amorphicity (Seguela et al., 1998a(Seguela et al., , 1998b;;[START_REF] Bartczak | Influence of molecular parameters on high-strain deformation of polyethylene in the plane-strain compression. Part I. Stress-strain behavior[END_REF]Bartczak and Galeski, 2010;Rozanski and Galeski, 2013). The latter is an influential and decisive parameter in tensile yielding, the strength and consistency of the amorphous phase being thus first-order factors (Mesbah et al., 2021).

The with a high elastic modulus in the chain direction while the amorphous layer consists of randomly oriented entangled chains and is thus supposed to be an isotropic medium.

Ⅳ.2.2. Constitutive equations

In what follows, the three-dimensional constitutive equations are described for the different physical sources governing the macro-response. The model satisfies the continuum mechanics rules within the large inelastic deformation kinematics. In the latter framework, the deformation

Ⅳ.2.2.1. Elastic-viscoplastic-viscohyperelastic polyethylene amorphous network

We assume that the behavior of the amorphous domain is the sum of an elastic-viscoplastic resistance that accounts for intermolecular interactions and a viscohyperelastic resistance that accounts for molecular network stretching and orientation process, the two physically distinct sources acting in parallel. The amorphous Cauchy stress tensor ( )

_ inter _ inter _ inter 1 ln e a a a a J = σ K V (Ⅳ.2)
where

_ inter _ inter det e a a J = F
is the elastic intermolecular volume change, ( )

_ inter ln e a V
is the Hencky elastic strain and a K is the fourth-order elastic stiffness tensor of the isotropic amorphous layers given by: 66 _ 000 000 000

0 0 0 0 0 00 0 2 2 2 2 2 2 0 0 0 0 0 0 0 a ij K                      =    +  +    +    (Ⅳ.3)
in which  and  are the amorphous Lame's constants.

From the rheological point of view, the molecular network resistance is constituted by a nonlinear spring in series with a viscous dashpot (Fig. Ⅳ.2). The molecular network Cauchy stress _ net a σ is given by the Arruda and Boyce (1993) eight-chain model of rubber elasticity introducing the extensibility limit of amorphous chains as a deformation measure: ( )

( ) 2 _ net _ net _ net _ net _ net _ n t 1 e 1 3
coth 1 =- x x
x which can be treated by a Padé approximation:

( ) ( ) ( )

1 2 2 31 x x x x -  - - .
The time derivative of the deformation gradient tensor writes are the amorphous intermolecular and network shear strain rates, respectively, given by (Boyce et al., 2000):

_ inter _ inter ,0 = exp 1 a aa G ks          --       σ and _ net _ net _ net 1 = 1 a v a in a C      -  σ (Ⅳ.8)
The left formula of Eq. (Ⅳ.8) captures barriers to molecular chain segment rotation in the amorphous layers. The term ,0 a  is the pre-exponential factor, G  is the activation energy (which must be overcome for inelasticity begins), s is the shear strength controlling the amorphous chain segment rotation:

1 0 _ inter nn n a s ss n   - = (Ⅳ.9)
where 0 s is the initial shear strength, n is the intermolecular hardening coefficient and  is a factor introduced to take into account the crystal-dependency of concentration of stress transmitters in the amorphous layers.

The right formula of Eq. (Ⅳ.8) captures molecular relaxation process in the amorphous layers.

The term _ net in a  must be slightly higher than one at the beginning of the loading process to avoid singularity and ensure stability of the numerical algorithm. It is given by:

( ) 12 _ net _ net 1 = trace 3 in a in a   +   B (Ⅳ.10) in which _ net _ net _ net T a in in aa in = FF B
is the inelastic left Cauchy-Green tensor and  = 0.01 is the added perturbation coefficient ensuring numerical stability.

The term v C in Eq. (Ⅳ.8) is the amorphous network viscosity coefficient given by:

exp v Q CD R  =-  (Ⅳ.11)
in which D and Q are two molecular relaxation parameters and R is the universal gas constant.

Ⅳ.2.2.2. Elastic-viscoplastic polyethylene crystals

The contribution of the polyethylene crystals may be represented by an elastic spring in series with a viscoplastic dashpot capturing barriers to crystallographic shear in the crystalline phase

c ij K           =            [GPa] (Ⅳ.13)
The plastic deformation of polyethylene crystals occurs by crystallographic slip that is captured by the following relation to account for the s N distinct slip systems:

1 s N in cc    = =  LR , ( ) 1 = 2       +  R n m m n (Ⅳ.14)
where  R is the nonsymmetric Schmid tensor with  n and  m are the "slip plane" normal vector and the "slip direction" vector in this plane, respectively, and ( )

1 ,0 = sign m cc g        (Ⅳ.15)
where ,0 c  is a reference strain rate, m is the strain rate sensitivity parameter and g  is the shear strength of the slip system  listed in Table Ⅳ.1. The term ( ) sign x is the signum function defined as:

( )

10 sign = 0 0 11 x xx x -   =     (Ⅳ.16)
The crystalline inelastic deformation gradient in c F at time

1 nn t t t + = +  is calculated
Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 77 incrementally using the previous one at time n t (van Dommelen et al., 2003):

( ) ( ) 1_ in in in c n c inc c n tt + = F F F (Ⅳ.17) in which _ in c inc

F

is the incremental inelastic deformation gradient for the time increment t  which can be treated by a Padé approximation:

( ) ( ) 1 _ 22 in in in c inc c c tt -  - +  F I L I L (Ⅳ.18)
The crystalline elastic deformation gradient is then extracted from the multiplicative deformation decomposition:

1 e in cc c - = F F F .
The definition of the mechanical coupling of the crystalline lamella with its corresponding amorphous layer remains unchanged from that initially proposed by Lee et al. (1993aLee et al. ( , 1993b) ) and van Dommelen et al. (2003). Therefore, only a summary of the concept is addressed in the next subsection since the details can be found in the paper abovementioned.

Ⅳ.2.3. Volume-averaging

The mechanical coupling between the deformation modes in amorphous and crystalline domains is obtained using as ensemble-volume averaged homogenization procedure a volumeweighted average of the respective phases.

The local Cauchy stress tensor i I σ of the composite inclusion i is given by: ,0 ,0

= i i i i i I c c a a ff + σ σ σ (Ⅳ.
i cc i c ii a a c c fJ f f J f J = + and ,0 ,0 ,0 i aa i a ii a a c c fJ f f J f J = + (Ⅳ.20)
where ,0 i c f and ,0 i a f are the corresponding initial volume fractions.

The macroscopic Cauchy stress tensor σ in the semi-crystalline polymer is obtained by the volume-averaging of the I N inclusions in the aggregate:

1 = I N ii II i f =  σσ , ,0 ,0 1 1 I ii II i I N jj I II j fJ f N fJ = =  = (Ⅳ.21)
The local inclusion-averaged deformation gradient tensor i I F is given by: ,0 ,0

= i i i i i I c c a a ff + F F F (Ⅳ.22)
The latter is equal to the imposed macroscopic deformation of the aggregate of the Although the Taylor assumption may be questionable, it has the advantage of being easy to implement compared to more sophisticated micromechanical approaches based on an Eshelby formulation (Mesbah et al., 2021), in particular, regarding the complex coupling existing between microstructure, finite-strain, time-dependency, non-linearity and external mechanical loading. Note that the resulting model, applied here to a broad range of crystallinities, involves a relatively low number of parameters with all a physical meaning. The compatibility condition on the deformation gradients is introduced at the amorphous-crystalline interfaces: vector at the amorphous-crystalline interface in the reference configuration5 . Additionally, the stress equilibrium condition at the interface between the crystalline lamella and the amorphous layer writes:

0 0 0 i i i i i i c a I  =  =  F F F x x x (Ⅳ.
i i i i i i c a I  =  =  σ σ σ n n n (Ⅳ.24)
where i n is the unit normal vector at the amorphous-crystalline interface in the current configuration.

When the crystalline lamella is sheared, the plastic deformation is frozen and the homogeneously plasticized crystals can be then regarded as rigid bodies connecting the amorphous chains and having the possibility to rotate during the stretching of amorphous chains, as conceptually realized in (Guo and Zaïri, 2020) for crystallized rubber systems. The texture evolution is related to the orientation of the amorphous-crystalline interfaces which can be updated at each increment.

T T i i i i i i i i i i i ii   - -  =  F F F = FF F x x n n xx n (Ⅳ.25) with ( ) 01 02 0 det T i i i i i i i  - = F F F F x
x n , the symbol  meaning the cross product of the vectors.

The model introduces a connection between the polyethylene response, the structure and the physics of the deformation modes of the crystalline and amorphous phases. The combination of amorphous and crystalline stiffness and plastic flow describes the small and moderate-strain response. For the large deformation behavior, two molecular orientation effects that take place as the strain-hardening stage proceeds are taken into account: Amorphous chains network

c p cw c t L f   = (Ⅳ.27)
where  is the density of the whole material (Table Ⅳ.2), c  =1.00 g/cm3 is the density of the crystalline phase and cw f is the crystal weight fraction measured by Differential Scanning Calorimetry (DSC) experiments (Table Ⅳ.3). ,0

f cw f H f H  =  (Ⅳ.29) in which ,0 f H 
is the melting enthalpy of a theoretically fully crystalline polyethylene Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 83 [START_REF] Wunderlich | Macromolecular Physics, Crystal Melting[END_REF] taking into account the melting temperature dependence for every crystal concentration (see Table IV.3).

Fig. Ⅳ.4.

Melting endotherms and cooling exotherms of the three polyethylene systems. The HD heat flow is typical of a high-crystalline polymer. The very broad LL and UL melting endotherms show non-uniform chemical composition distributions due to high co-unit concentrations (Ayoub et al., 2010).

Ⅳ.3.1.2. Large-plastic deformation

The intrinsic large-plastic deformation was obtained using video-controlled tensile mechanical tests. They were performed on an electromechanical Instron-5800 universal testing machine interfaced with a non-contact optical extensometer for the local strains measurement on dog bone shaped samples 8 of 28 mm gauge length, 8 mm width and 1 mm thickness; The gauge part presents a curvature radius of about 120 mm in order to localize the necking (if any) while minimize the stress triaxiality effect. The local true axial strain rate was controlled to remain strictly constant throughout the tests in the smallest sample cross-section even after necking by 8 Although no initial preferential orientation was expected, the samples were cut off in the same direction of 1 mm thick compression-molded sheets; The latter were molded from polyethylene pellets at 180°C for 10 min and slowly cooled to room temperature in order to avoid residual thermal stresses generated during cooling. • Single loading steps up to very large strains for model identification: the sample is stretched to a prescribed strain level of about 1.8 at different constant true strain rates: 0.01 /s, 0.005 /s, 0.001 /s, 0.0005 /s and 0.0001 /s.

• Single loading-unloading steps for model verification: the strain is ramped to a prescribed level at a constant true strain rate of 0.001 /s and then ramped down to zero stress under the same absolute strain rate. Three maximum applied strains are considered: 0.9, 1.2 and 1.5.

• Oligo-cyclic stretching for model verification: The previous sequence is performed repeatedly to determine the progressive stress-softening at maximum applied strain and the accumulation of plastic strain at zero stress with increasing cycle number.

The responses are plotted in the form of true (Cauchy) axial stress versus true (Hencky) axial strain.

Ⅳ.3.2. Model identification

The model parameters were identified to fit the constant true strain rate monotonic curves up to From the earlier studies of [START_REF] Peterlin | Plastic deformation of polymers with fibrous structure[END_REF], the prevailing picture of the microstructure evolution due to tensile straining of a semi-crystalline polymer has been stated (Seguela et al., 1998a(Seguela et al., , 1998b;;[START_REF] Bartczak | Influence of molecular parameters on high-strain deformation of polyethylene in the plane-strain compression. Part I. Stress-strain behavior[END_REF]Bartczak and Galeski, 2010;Rozanski and Galeski, 2013); The tensile yielding results from massive structural modifications accompanied by widely diverse deformation mechanisms in the two constitutive phases.

 = 0  = 0.5  = 1  = 2 (b)  = 0  = 0.5  = 1  = 2
It is determined by the stretching of chains in the amorphous phase occurring in the first place (involving interlamellar separation, inter-lamellar shear and lamellar rotation) and by the plastic deformation events in the crystalline lamellae followed by their fragmentation leading to the formation of a fibrillar microstructure at large strains.

x 1 The microstructure evolution can be also analyzed using the local axial stresses and stretches plotted in Figs. Ⅳ.7 and Ⅳ.8, respectively. In the early stage of deformation until yield point, the response is governed by the plastic deformation in the crystalline phase and by the intermolecular interactions in the amorphous phase; Their relative importance depends on the amorphicity. It is worth noticing that the interfacial interactions between crystalline and amorphous domains may form a region in the vicinity of crystalline lamella where the amorphous chain mobility is reduced (see Fig. Ⅳ.1). Due to the loss of chain segmental mobility, the glass transition temperature is expected to be severely modified leading to an enhancement of the intermolecular resistance of the amorphous layers. The latter resistance depends on the crystallinity. For the higher crystalline system, the small-strain tensile yielding, manifested by the initial elastic response followed by the rollover to yield, is predominantly associated with the crystallographic shear in the crystalline phase but also, in a certain extent, with the intermolecular barriers to chain-segment rotation in the minor amorphous phase.

x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 Chapter Ⅳ. Continuum-

Fig. Ⅳ.7.

Local axial stresses in the three polyethylene systems upon oligo-cyclic loading at a strain level of 1.5 and at a true strain rate of 0.001 /s.

In the plastic regime, the extension of chains in the amorphous phase progressively governs the response manifested by the progressive strain-hardening and finally the dramatic strainhardening at very large strains. For low-crystalline systems, the molecular orientation/relaxation of the amorphous phase dominates the material response in all strain range, the contribution of the discrete crystalline lamellae to the overall resistance to deformation being slight and they rather act as junction points in the amorphous matrix.Nonmonotonic conditions being one of the most severe tests for the verification of a constitutive model, its predictive capacities are discussed in the two next subsections using the model parameters previously identified in monotonic loading.

Ⅳ.3.3.2. Loading-unloading

The model predictions under a loading-unloading path are depicted in Inter.

Crystalline

Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 89 the model predicts the nonlinearity of the unloading path when the strain level is large. For more moderate strains, the model is unable to capture the nonlinear unloading path which leads difference between unloading and reloading paths), (iii) an accumulation of plastic strain at zero stress [START_REF] Drozdov | Mullins' effect in semicrystalline polymers[END_REF][START_REF] Wang | Network alteration theory on Mullins effect in semicrystalline polymers[END_REF]Makki et al., 2017;[START_REF] Guo | Characterization of the spherulitic deformation in equatorial region and cavitation in HDPE materials submitted to mixed-mode oligo-cyclic tensile loading[END_REF]. Their amounts depend on the amorphous phase fraction. The plastic strain evolution is plotted in response evolution with increasing cycle number (Makki et al., 2017). Nonetheless, our objective here is to limit the number of model parameters and to present the model plots for The quite good agreement with the experimental observations of the HD nonlinear cyclic response confirms the ability of the constitutive representation to describe high crystalline media. Fig. Ⅳ.12 compares the predicted and measured curves to present the strain level effect on different aspects of the HD cyclic response considering both one cycle with increasing maximum strain levels and oligo-cyclic loading with two strain levels. It can be observed that the model recaptures the same path as that corresponding to the monotonic one when the strain exceeds the maximum strain previously applied. The general trends of the oligo-cyclic stretching are adequately predicted by the model, in particular the increase in stress-softening magnitude with strain level.

Ⅳ.4. Conclusion

In this chapter, the highly nonlinear and rate-dependent mechanical response of semi-crystalline polyethylene systems covering a wide spectrum of the crystallinity was investigated within a

Max stress ratio

Cyclic number ε = 0.9 ε = 1.5

Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 94 multi-scale homogenization-based approach. The model ability to capture the large-strain response variation with the crystallinity was criticized. The model exhibited weaknesses in small-strain region but it was able to reproduce, within large-strain region, the rate-dependent monotonic response variation with crystallinity. The general trends of the high-crystalline material cyclic response were quantitatively well reproduced by the model while the deviation from experiments increased with amorphicity, which would require to take into account the amorphous network alteration.

In addition to molecular orientation effects that takes place as the deformation proceeds, the cavitation damage associated with progressive nucleation and growth of cavities in the amorphous layers will be presented in the next chapter.

Ⅴ.1. Introduction

Crystallographic texturing and strain-induced preferred orientation of the amorphous molecular network are the two main inelastic mechanisms at the origin of the finite-strain morphological anisotropy occurring in semi-crystalline polyethylene under in-service or manufacturing processing (Seguela et al., 1998;Argon et al., 2005;Kazmierczak et al., 2005;Bartczak et al., 2010;Rozanski et al., 2013). Due to 3D stress being developed in the thin amorphous layers embedded between broad and wide crystalline lamellae, the local inelastic events may also include nano-sized voids nucleated within the amorphous domain. Over the years, a substantial qualitative understanding of the cavitation damage in semi-crystalline polymers has been brought [START_REF] Parsons | Three-dimensional large-strain tensile deformation of neat and calcium carbonate-filled high-density polyethylene[END_REF][START_REF] Pawlak | Plastic deformation of crystalline polymers: The role of cavitation and crystal plasticity[END_REF]Pawlak et al., , 2007;;Addiego et al., 2006;[START_REF] Castagnet | Relative influence of microstructure and macroscopic triaxiality on cavitation damage in a semi-crystalline polymer[END_REF][START_REF] Boisot | Experimental investigations and modeling of volume change induced by void growth in polyamide 11[END_REF][START_REF] Detrez | Plasticity/damage coupling in semi-crystalline polymers prior to yielding: micromechanisms and damage law identification[END_REF][START_REF] Ponçot | True intrinsic mechanical behaviour of semicrystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF][START_REF] Laiarinandrasana | Three dimensional quantification of anisotropic void evolution in deformed semicrystalline polyamide 6[END_REF][START_REF] Ovalle | Nanocavitation mechanisms in deformed High Density PolyEthylene (HDPE) using synchrotron radiation NanoTomography[END_REF]Yan et al., 2021). The reader may also refer to a recent comprehensive review on the literature experimental observations [START_REF] Zhang | Quantification of strain-induced damage in semi-crystalline polymers: A review[END_REF]. The voiding damage process may experience anisotropic growth during deformation and enter in competition with the plastic shearing of the nano-sized crystalline lamellae. The relative plastic-damage extent depends on complex interactions existing between external loading conditions and features of the two distinguishing phases such as crystal dimensions, crystal degree, crystal form, amorphous strength/consistency and molecular structure.

Continuum-based constitutive modeling of the inelastic deformation behavior of semicrystalline polyethylene is a prerequisite for advanced engineering use while offering deep understanding of the separate and synergic effects of key microstructural and loading parameters governing the macroscopic response. The model should be as most as physically consistent by considering the above-mentioned microstructure specificities. Although it is a very active subject of research, quantitative modeling of the finite strain anisotropic plastic behavior including cavitation damage in the amorphous phase in connection to the crystallographic shear in the crystalline lamellae is still lacking, and that, for all existing semicrystalline polymer systems. Addressing this problematic is therefore of prime importance to better understand the interactions between the strain-induced morphological anisotropy and the cavitation damage. The microstructure-based models can be classified according to the approximation of the semi-crystalline polymer microstructure. The overall material response can be constitutively captured in continuum terms by the combination of different rheological resistances invoking mechanical analogs (using combinations of elements, usually linear elastic Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 102 springs, viscoplastic dashpots and nonlinear elastic springs) and representing physically distinct sources in the crystalline and amorphous domains (Dusunceli et al., 2008;Ayoub et al., 2010Ayoub et al., , 2011;;Drozdov et al., 2013;Abdul-Hameed et al., 2014;Makki et al., 2017;Sepiani et al., 2018;Chen et al., 2019;Deplancke et al., 2019;Qi et al., 2019;Bernard et al., 2020). These rheological models use a simplified composite-type representation and may integrate the crystalline and amorphous intermolecular micro-mechanisms but neglect both hierarchical semi-crystalline structure and internal interactions. Besides, the crystallographic texturing cannot be predicted because the constitutive representation remains rather macroscopic even if some microstructure ingredients are incorporated such as amorphous chains features and crystal fraction. Proper physically-based constitutive representation requires to take into account the multi-scale inelastic processes while the local interactions are properly considered. In this regard, continuum-based models considered the mesoscopic scale as the starting observation of the constitutive representation by employing multi-scale homogenization-based approaches.

Using the Eshelby inclusion theory, some of these models assume a matrix-inclusion microstructure approximation based on a composite-type description (Bédoui et al., 2006;Gueguen et al., 2010;Anoukou et al., 2014;Hachour et al., 2014;Mesbah et al., 2021), the inclusions (individually crystalline lamellae at low crystallinity or amorphous domains at high crystallinity) being randomly dispersed in the matrix material (rubbery amorphous matrix at low crystallinity or percolated crystalline matrix at high crystallinity). The theoretical frameworks are nonetheless restricted to strain levels from the elastic region to the small-strain post-yield region. As an alternative, the semi-crystalline material may be regarded as a mixture of crystalline and amorphous domains organized as an aggregate of two-phase inelastic layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers which are randomly oriented (Lee et al., 1993a(Lee et al., , 1993b;;Nikolov et al., 2000Nikolov et al., , 2002;;van Dommelen et al., 2003;Agoras et al., 2012;Uchida et al., 2013;Mirkalaf et al., 2019;Yan et al., 2021). These

Ⅴ.2. Model formulation

The present modeling satisfies the continuum mechanics rules within the large deformation kinematics and is based on a two-step homogenization process, i.e. microscopic/mesoscopic and then mesoscopic/macroscopic. The two-phase heterogeneous material exhibits a hierarchical organization of the regularly ordered (crystalline) phase from the nano-sized lamellar structure to the well-organized crystalline lamellae, in the form of micro-sized

Ⅴ.2.1. Microscopic scale

In a first step of the homogenization process, to achieve the transition from the smallest scale to the mesoscale, the microscopic scale of the material medium must be described by the local stresses in each phase, namely the crystalline Cauchy stress c σ and the amorphous Cauchy stress a σ . In accordance with continuum mechanics, both solid phases are considered as homogeneous and continuous media. The physical consistency of their respective constitutive formula depends on the description degree of the deformation modes. In what follows, the crystalline lamellae plasticity and the amorphous phase damage are successively described in connection to features such as finite-strain time-dependency and non-linearity.

Ⅴ.2.1.1. Plasticity in crystalline lamellae

The elastic-viscoplastic deformation of the crystalline lamellae can be defined via the ( )

1 = 2       +  R n m m n (Ⅴ.3)
where  R is the nonsymmetric Schmid tensor with is the resolved shear stress and g  is the shear strength of the slip system  .

Ⅴ.2.1.2. Damage in amorphous layers

Let us now consider the Cauchy stress in the amorphous layers. It may be constitutively where 1 a and 2 a are the semi-axes of the void, respectively, along axis 1 and along axis 2. The void is prolate if 213 a a a = and oblate if 2 1 3 a a a =. The terms 1 b and 2 b are the semi-axes of the volume element in which the ellipsoidal void is included. Note that the representative 11 The confocality of the ellipsoidal void implies that the semi-axes are related by the following condition: Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 107 material element is considered initially spherical. The initial void volume fraction is denoted

33 0 _ 0 0 1 void void f a b f =  
where 0 a and 0 b are the radii, respectively, of the void and of the volume element in the reference configuration.

Ⅴ.2.1.2.1. Intermolecular interactions

The intermolecular deformation gradient tensor _ inter a F may be obtained using the multiplicative decomposition concept of the deformation:

== F R U V R (Ⅴ.9)
The intermolecular Cauchy stress tensor is the local intermolecular shear strain rate governing the amorphous phase surrounding the voids that takes the Argon form and is expressed as:

_ inter _ inter ,0 = exp 1 a aa G ks          --       σ (Ⅴ.13)
where ,0 a  is the pre-exponential factor, G  is the activation energy (which must be overcome for inelasticity begins), k is the Boltzmann's constant,  is the absolute temperature and s

is the shear strength controlling the amorphous chain segment rotation:

( )

, _ inter a s a s h s s  =- (Ⅴ.14)
where h is the post-yield hardening slope prior to the strain-hardening stage, , as s is the steady- state shear strength and 0 s is the initial shear strength. The yield function _ inter a  may be defined using one of the Gurson-type models proposed to capture the inelastic response of porous polymers [START_REF] Lazzeri | Applications of a dilatational yielding model to rubbertoughened polymers[END_REF][START_REF] Steenbrink | Void growth in glassy polymers[END_REF][START_REF] Pijnenburg | Macroscopic yield in cavitated polymer blends[END_REF][START_REF] Pijnenburg | Successively refined models for crack tip plasticity in polymer blends[END_REF][START_REF] Guo | Modeling vapor pressure effects on void rupture and crack growth resistance[END_REF][START_REF] Guo | Continuum modeling of a porous solid with pressure-sensitive dilatant matrix[END_REF][START_REF] Jeong | A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices[END_REF]Seeling et al., 2002;[START_REF] Imanaka | Fracture toughness of rubbermodified epoxy adhesives: effect of plastic deformability of the matrix phase[END_REF][START_REF] Imanaka | Crack-growth behavior of epoxy adhesives modified with liquid rubber and cross-linked rubber particles under mode I loading[END_REF][START_REF] Zaïri | Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer[END_REF][START_REF] Zaïri | Modelling of the elastoviscoplastic damage behaviour of glassy polymers[END_REF]Zaïri et al., , 2011;;Challier at al., 2006;Liarinandrasana et al., 2009;[START_REF] Holopainen | Influence of damage on inhomogeneous deformation behavior of amorphous glassy polymers. Modeling and algorithmic implementation in a finite element setting[END_REF]. The Gurson model was either used as is or with changes more or less deep in order to; introduce polymer specificities for the solid ligaments between voids such as the matrix yielding pressure dependence [START_REF] Lazzeri | Applications of a dilatational yielding model to rubbertoughened polymers[END_REF][START_REF] Jeong | A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices[END_REF], the matrix initial yield strain effects (Sreenbrink et al., 1997;Pijnenbug, 2001) and the matrix plastic anisotropy due to molecular network orientation (Zaïri et al., 2011;Holopainea et al., 2014). Note that the anisotropic damage of material may be formulated within the continuum damage mechanics framework and described by a second-order tensor [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Voyiadjis | A theory of anisotropic healing and damage mechanics of materials[END_REF][START_REF] Ayadi | Anisotropic (continuum damage mechanics)based multi-mechanism model for semi-crystalline polymer[END_REF]. In a Gurson-type formulation, a scalar damage variable is conveniently used along with a shape factor making the void growth anisotropic.

The anisotropic expression of [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF] already employed by Zaïri et al. (2011) for rubber-toughened glassy polymers is here used: Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene

109 _ inter_h _ inter _ inter _ inter_h 2 _i 2 nter _ inte 2 r 2 2 cosh 1 0 a a a a void voi a a ww d C f q q f   = + + -- =      σ σ σ Y     (Ⅴ.15)
where _ inter_h _ inter :

aa  = σ
J is a generalized hydrostatic stress in which the tensors J and Y are associated to the void axis:

( )

2 1 1 2 3 3 2 2 2 12    =  +  + -  J
e e e e e e and 2 2 1 1 3 3

2 1 1 3 3 3 =  -  -  Y
e e e e e e (Ⅴ.16)

where ( )

1 2 3
,, e e e is an orthogonal basis with 2 e parallel to the axisymmetric axis of the void. The parameters C ,  ,  and 2  are given in Appendix V. A. The parameter w q is given by: ( ) ( )

0 exp 1 2 1 1 exp 2 = + - + w S qq S (Ⅴ.17)
where 0 q is a material constant that may be related to interaction effects between voids and ( )

ln Sw =
is the void shape parameter with 0 S its initial value. ( )

_ inter _ inter _ inter 1 2 T i aa i a n in n =+ D L L and
( )

_ inter _ inter _ inter 1 2 T i aa i a n in n =- W L L (Ⅴ.19)
The intermolecular inelastic flow is considered irrotational12 , i.e. is given by the differentiation of the strain energy density W with respect to the elastic deformation:

( )

_ net _ net e i e a a i i W J    =  σ with 1, 2, 3 i = (Ⅴ.22)
No summation on i is implied in this equation.

With the consideration of a strain energy density

W based on the first stretch invariant:

0 0 2 2 1 00 01 1 sin b e e a i i W W I R d d dR VI       =       (Ⅴ.23)
where the integration is carried out in the standard spherical coordinate system ( ) The local behavior of the amorphous network is governed by the Arruda and Boyce (Arruda et al., 1993) eight-chain model of rubber elasticity introducing the extensibility limit of amorphous chains as a deformation measure: ( ) ( ) ( )

,, R
1 ln cst 3 sinh h I W N N C     = + +   
1 2 2 31 -  - - x x x
x . Using the kinematically admissible deformation field defined by [START_REF] Hou | Cavitation in elastic and elastic-plastic solids[END_REF] for void growth in a hyperelastic solid, the motion of every material point of the representative material element is defined by the components in the deformed configuration:

( ) = _net 13 e i ii a x R X J   with 1, 2, 3 i = (Ⅴ.26)
No summation on i is implied in this equation. The components 1 X , 2 X and 3 X of the reference configuration are given in the standard spherical coordinate system by:

1 sin sin XR = , 2 sin cos XR = and 3 cos XR = (Ⅴ.27)
The term  is given as a function of the radial distance R from the origin of the representative material element as:

( ) ( )  = + -    13 3 0 _net 3 11 a b RJ R  (Ⅴ.28)
The local first stretch invariant 1 I may be expressed as [START_REF] Hou | Cavitation in elastic and elastic-plastic solids[END_REF]: The network inelastic strain rate tensor _ net in a D is given by the following flow rule: where  = 0.01 is a small perturbation coefficient ensuring numerical stability of the numerical algorithm at the beginning of the loading process and v C is the amorphous network viscosity coefficient.

( ) 2 2 2 2 2 2 2 2 1 1 1 1 2 2 3 t 3 2 3 2 4 _ ne 1 1 1 e e e a I I X X X J R    = + + + -             (Ⅴ.
_ net _ net _ net _ net 2 a in aa a   = σ D σ (Ⅴ.30) in which ( ) 2 _ net _ ne 1 t _ net 2 .
As indicated earlier for the intermolecular interactions, the rate kinematics are described by the network spatial velocity gradient tensor _ net

_ net _ net _ net _ net 1 e in a a a a a - = = + L F F L L : 1 _ net _ net _ net a a
-- = = + L F F F F D W (Ⅴ.34) with ( ) _ net _ net _ net 1 2 T i a in in aa n =+ D L L and
( )

_ net _ net _ net 1 2 T i a in in aa n =- W L L (Ⅴ.35)
By choosing again the inelastic flow irrotational, i.e.

Ⅴ.2.2. Mesoscopic scale: Two-phase layered composite inclusions

The homogenized amorphous layer stress is volume-averaged with the crystalline lamella stress to give the local inclusion-averaged stress i I σ of the composite inclusion i: The local inclusion-averaged deformation gradient tensor i I F of the composite inclusion i is given by 13 : ,0 ,0

= i i i i i I a a c c ff + σ σ σ (Ⅴ.
= i i i i i I a a c c ff + F F F (Ⅴ.39)
in which i a F and i c F are the local quantities previously defined. The stress equilibrium is considered at the amorphous-crystalline interface of the composite inclusion i using the following condition:

i i i i i i c a I  =  =  σ σ σ n n n (Ⅴ.40)
where i n is the unit normal vector at the interface in the current configuration. The compatibility of deformation is also considered:

0 0 0 i i i i i i c a I  =  =  F F F x x x (Ⅴ.

Ⅴ.2.3. Macroscopic scale: Aggregate

In a second step of the homogenization process, the transition to the macroscale is conducted by the volume-averaging of all the inclusions in the aggregate. In the present simulations, the aggregate contains I N =125 two-phase layered composite inclusions. The macroscopic Cauchy stress tensor σ is given:

1 = I N ii II i f =  σσ with ,0 ,0 1 I ii II i I N jj II j fJ f fJ =  = (Ⅴ.

Ⅴ.2.4. Microstructure evolution and model implementation

The present multi-scale homogenization-based approach considers assembled together into the same model, the synergic effects on the material anisotropy of key orientation-induced deformation modes taking place as the large-plastic deformation proceeds, namely crystallographic texture evolution, amorphous molecular chains orientation/relaxation and anisotropic voiding process. This subsection provides specific kinetics of the microstructure evolution along with some numerical aspects concerning the composite inclusions model solving process.

The time domain is subdivided into discrete time steps. For each discrete time level, the set of non-linear equations, comprising the interaction law, the local the interface conditions and the volume-averaging equations, is solved. Thereafter, both interface normals and cavitation damage are explicitly updated.

Ⅴ.2.4.1. Time-integration of inelastic flow

The inelastic component in c F at time 1 nn t t t + = +  is calculated incrementally by using the previous one at time n t (van Dommelen et al., 2003):

( ) ( ) 1_

Ⅴ.2.4.2. Morphological texture evolution

The morphological texture evolution is directly related to the orientation of the amorphouscrystalline interfaces which can be updated at each increment. Considering two independent vectors at time 0 t = , ( )

1 0 i  x and
( ) where ( )

2 0 i  x ,
F i n
t can be either ( )

F i an t , ( ) F i cn t or ( ) F i In t .
The normal vector i n at the amorphous-crystalline interface is then obtained by: Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 115

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 12 0 0 0 00 0 F F F = FF F T T i i i i i i n n n i i i i i ii nn n t t t t tt t   - -  =  x x n n xx n (Ⅴ.46) with ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 0 0 det 0 F F F F T i i i i i i i n n n n t t t t  - = x x n
, the symbol  meaning the cross product of the vectors.

Ⅴ.2.4.3. Damage evolution

The progressive cavitation damage accumulation gives rise to a volume fraction whose the kinetics void f may be expressed as an additive expression of a nucleation part n f and a growth part

g f : ( ) ( ) ( ) void n n g nn ff t t t f =+ (Ⅴ.47)
The nucleation phenomenon of voids is difficult to put in equation. In the Gurson-type models, this part introduces the stochastic nature of the process from a statistical form, either strain or stress induced. The mechanism is assumed to be controlled by the local intermolecular shear strain rate:

( ) ( ) ( ) 2 _ inter 1 _ inter 1 exp 2 2 + a n N a n n a n N N t f f t t s s    -   =-     (Ⅴ.48)
where N  is the strain at which nucleation takes place, N s is standard deviation and a f is the amorphous volume fraction defined above.

Considering the plastic incompressibility of the amorphous solid phase around growing voids, the change in void volume fraction due to void growth is proportional to the macroscopic intermolecular inelastic volumetric strain rate:

( ) ( ) ( ) ( ) +1 _ inter 1: I D g vo in n n a n id tt f t f =- (Ⅴ.49)
The void aspect ratio rate w is defined by:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 _ inter :: + AD in n n w n n n n a n w t w t t t t t t  =  -  n n n n (Ⅴ.50)
in which w  is a parameter and is a fourth-order tensor expressed as [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation[END_REF]:

1 13 : 2 void f -   =-  with 1 3 3 21 void void f f -  =+ - (Ⅴ.51)
where  is the deviatoric part of the fourth-order identify tensor and is expressed in Appendix V. B.

Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 116

Ⅴ.3. Results and discussion

In what follows, the model is applied to the intrinsic large-plastic deformation of high-density polyethylene represented by the stress-strain behavior along with the inelastic volumetric strain as the most direct indicator of the voiding mechanism nucleated within the amorphous phase.

Monotonic data are used to identify the model parameters while non-monotonic (stretchingretraction-recovery) data are used to verify the model predictability. The internal microstructure evolution is then visualized and discussed thanks to the model.

Ⅴ.3.1. Macroscopic response Ⅴ.3.1.1. Model inputs

The crystalline parameters are those used in Chapter Ⅳ using the deviatoric version of the constitutive model (Yan et al., 2021). Lamellar surfaces in polyethylene are of the {h0l}-type (i.e. the crystal lamellar growth direction is the crystallographic b-axis) where the angle between the chain direction and the normal of interface varies between 20° and 40° (Bassett et al., 1981). The initial angle between chain direction and normal of interface is set to be 35°.

The crystalline phase has therefore an orthorhombic behavior with a high elastic modulus in the chain direction. The values of the elastic stiffness components of the polyethylene crystal are taken from the literature (Tashiro et al., 1978):

c ij K           =            [GPa] (Ⅴ.52)
The shear strength of the slip system is also taken from the literature (Lee et al.,1993) and reported in Table V. 1. They are both introduced as direct inputs into the model. The other model constants are identified using the intrinsic tension data of the high-density polyethylene extracted from the papers of Addiego et al. (2006) and [START_REF] Ponçot | True intrinsic mechanical behaviour of semicrystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF]. The two key features of the macroscopic material response (true stress-strain and volumetric strain data) reported in Figs.

V. 2(a) and V. 3(a) were experimentally quantified upon monotonic stretching for different true strain rates (0.005 /s, 0.001 /s and 0.0005 /s) at room temperature by Addiego et al. (Addiego et al., 2006) and different stretching temperatures (room temperature, 40°C and 45°C) at a true strain rate of 0.001 /s by [START_REF] Ponçot | True intrinsic mechanical behaviour of semicrystalline and amorphous polymers: Influences of volume deformation and cavities shape[END_REF]. In the two papers, the data were measured Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 117 for the same material (containing 0.76 of crystals) and the same protocol under well-controlled conditions by means of a video-controlled system able to maintain constant true axial strain rate in the plastic instability of the tested specimen during stretching up to an assigned maximum value. The values of the identified model parameters are listed in Table V. 2. The procedure of identification uses a deterministic scheme that consists of successively identifying the different parts of the model. The identification of the cavitation damage parameters was the outcome of a calibration that provides the best fit of the monotonic inelastic volumetric data of Addiego et al. (2006) and Ponçot et al. (2014). Taking as initial values those of the deviatoric version of the constitutive model (Yan et al., 2021), the final values of the amorphous intermolecular and network parameters were then obtained through the adjustment of the best monotonic stressstrain response by means of trial and error.

Table V.1. Slip systems and associated shear strength (Lee et al., 1993a). as well as the volumetric strain under increasing tensile strain. The model is able to reproduce well the cavitation damage taking place as the plastic deformation proceeds at room temperature as well as the delay of the damage onset observed with the stretching temperature increase. The significant influence of stretching temperature on the critical strain, beyond which the cavitational behavior takes place, is well captured by the model. Note that the negative volumetric strain experimentally observed below the critical strain is the manifestation of a temperature-dependent material densification phenomenon that may be attributed to the compaction of amorphous polymer chains due to extensive shear yielding delaying the cavitation onset.

The shearing mechanism being assumed to be purely isochoric in our model, the negative volume before the critical strain cannot be captured. Nonetheless, the onset of the cavitationinduced positive inelastic volumetric strain as well as the change in the kinetics of the inelastic Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 118 (2006) and were obtained at room temperature under a true strain rate of 0.001 /s. The experiments were replicated by the model using the constants calibrated under monotonic stretching. The macroscopic strain is ramped to the level of 1 at the constant true strain rate of 0.001 /s, then it is ramped down to zero macroscopic stress under the same absolute strain rate and finally the macroscopic stress was maintained at a zero value to calculate the recovery in strain and in dilatation.

Fig. Ⅴ.7.

Oligo-cyclic response at a strain level of 0.9 and at a true strain rate of 0.001 /s: (a) stress-strain curve, (b) stress-softening evolution (normalized to the initial value), (c) plastic strain accumulation at zero stress (normalized to the initial value).

Ⅴ.3.2. Microstructure evolution

As a final point, the microstructure evolution in high-density polyethylene is discussed. The Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 127

The internal mechanics within the spherulitic microstructure of high-density polyethylene is another relevant feature experimentally observed to understand more precisely the mechanisms [START_REF] Xiong | Micro/macro-stress relationship and local stress distribution in polyethylene spherulites upon uniaxial stretching in the small strain domain[END_REF]. The shear yielding remains the predominant deformation mechanism and results from both the plastic yielding in the crystalline phase and the intermolecular barriers to chain-segment rotation in the amorphous phase. The role of the cavitation damage on the internal stresses and stretches is pointed out using the model in Figs. Ⅴ.11 and Ⅴ.12.

Ⅴ.4. Conclusion

As concluding remarks, the synergic effects of crystal texturing and anisotropic cavitation damage on the plastic deformation behavior of high-density polyethylene were integrated into a constitutive model formulated within a multi-scale homogenization-based approach. By comparing simulations and experimental data, the model ability to capture large-strain macroscopic stress and inelastic dilatation was shown. The proposed multi-scale approach provides a good approximation of the intrinsic response of high-density polyethylene and the basic underlying inelastic mechanisms by considering strain rate and stretching temperature effects. Besides, the role of cavitation damage in overall and local responses was observed to better understand its importance under stretching and stretching-retraction conditions.

Although quite sophisticated, the model needs improvements for a fully realistic description of the microstructure evolution through the consideration of the crystal fragmentation. (  )

12 1 2  =- =- Qf H (Ⅴ.A2)
The parameters 1  , 2  and 1  are expressed as follows: The non-zero terms of  are given in the local ellipsoidal coordinate system by [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation[END_REF]: Poly(vinylidene fluoride) (PVDF) is a well-known multifunctional piezoelectric semicrystalline polymer offering flexibility and electro-mechanical conversion capabilities for a wide range of sophisticated engineering applications in modern technologies and industries including tactile sensors, actuators, energy storage, transducers and non-volatile memories [START_REF] Chang | Piezoelectric nanofibers for energy scavenging applications[END_REF]. The piezoelectric properties are highly dependent on polymer crystalline structure. The α-phase and the β-phase are by far the most important crystalline structures [START_REF] Nalwa | Ferroelectric Polymers: Chemistry, Physics and Applications[END_REF]. The α-phase is non-polar, non-piezoelectric, nonpyrroelectric while the β-phase exhibits strong ferroelectric and piezoelectric properties. However, it is not easy to obtain PVDF consisting of entirely β-phase crystalline structure. It has been proved experimentally that the β crystalline form can be obtained by stretching PVDF initially in the α-form, the α → β phase transition benefiting to the electroactive properties [START_REF] Ye | Effect of electroactive phase transformation on electron structure and dielectric properties of uniaxial stretching poly(vinylidene fluoride) films[END_REF][START_REF] Sharma | Process induced electroactive β-polymorph in PVDF: Effect on dielectric and ferroelectric properties[END_REF]. The loading temperature has a significant effect both on the strain-induced α → β phase transition and the elementary plastic deformation mechanisms with a transition, according to the temperature range, from a cavitational behavior to a more roughly homogeneous deformation governed by shearing mechanisms [START_REF] Guo | Insitu synchrotron SAXS and WAXS investigations on deformation and α-β transformation of uniaxial stretched poly(vinylidene fluoride)[END_REF][START_REF] Osaka | The optical transparency and structural change of quenched poly(vinylidene fluoride) caused by cold-drawing[END_REF][START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF]. The advanced use of PVDF in engineering applications requires accurate and flexible predictive tools allowing to tailor and optimize the strain-induced microstructure evolution and its influence on the overall response. That is a challenging task due to the complex multiscale relationship between microstructure features (in terms crystal and amorphous properties, α → β phase transition and morphological anisotropy) and macroscopic response. Concurrent multiscale modeling techniques can be used. In this regard, local interactions and structural properties of PVDF may be investigated using atomistic simulations [START_REF] Pei | Elastic properties of poly(vinyldene fluoride) (PVDF) crystals: A density functional theory study[END_REF][START_REF] Erdtman | Simulation of α-and β-PVDF melting mechanisms[END_REF][START_REF] Satyanarayana | Molecular dynamics simulations of αto βpoly(vinylidene fluoride) phase change by stretching and poling[END_REF][START_REF] Sahihi | Interfacial interactions in a model composite material: Insights into α → β phase transition of the magnetite reinforced poly(vinylidene fluoride) systems by all-atom molecular dynamics simulation[END_REF]. Alternatively, quantitative predictive modeling of the structure-property relationship may be constitutively examined in continuum terms. Over the years, continuum-based constitutive models have been proposed for semi-crystalline polymers. Some of them were developed to capture the macroscopic response via rheological elements, combining linear/nonlinear elastic springs and viscoelastic/viscoplastic dashpots, representing the elementary micro-mechanisms and assembled within a simplified composite-type representation of the semi-crystalline material medium (Dusunceli and Colak, 2008;Ayoub et al., 2010Ayoub et al., , 2011;;[START_REF] Drozdov | Effect of temperature on the viscoelastic and viscoplastic behavior of polypropylene[END_REF][START_REF] Laiarinandrasana | Damage of semicrystalline polyamide 6 assessed by 3D X-ray tomography: From microstructural evolution to constitutive modeling[END_REF][START_REF] Drozdov | Mullins' effect in semicrystalline polymers: Experiments and modeling[END_REF]Drozdov et al., 2013;Abdul-Hameed et al., 2014;Makki et al., 2017;Sepiani et al., 2018;Chen et al., 2019;Deplancke et al., 2019;[START_REF] Mahjoubi | A micro-macro constitutive model for strain induced molecular ordering in biopolymers: Application to polylactide over a wide range of temperatures[END_REF][START_REF] Mahjoubi | Strain-induced phase transformation in poly(lactic acid) across the glass transition: Constitutive model and identification[END_REF]Qi et al., 2019;Bernard et al., 2020;[START_REF] Wang | Experimental and constitutive modelling studies of semicrystalline thermoplastics under solid-state stamp forming conditions[END_REF][START_REF] Satouri | A gradient enhanced constitutive framework for the investigation of ductile damage localization within semicrystalline polymers[END_REF]. In other more physically consistent constitutive models, the macroscopic response is evaluated within a micro-macro homogenization-based approach through Eshelbytype inclusion problems for elastic behavior (Bédoui et al., 2006;Gueguen et al., 2010;Anoukou et al., 2014), yield behavior (Hachour et al., 2014) and small-strain post-yield behavior (Mesbah et al., 2021;[START_REF] Mamache | Micromechanical modeling of the biaxial behavior of strain-induced crystallizable polyethylene terephthalate-clay nanocomposites[END_REF]; This theoretical framework uses a composite-type representation in a more suitable. description of the local interactions but the developments are generally limited to small-strain behavior. Other micro-macro homogenization-based models constitutively represent within the finite-strain framework the Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 139 organization hierarchy starting from the nano-sized lamellar structure, allowing thus access to the texture evolution in the semi-crystalline material medium and the underlying plastic anisotropy (Lee et al., 1993a(Lee et al., , 1993b;;Nikolov and Doghri, 2000;Nikolov et al., 2002;van Dommelen et al., 2003;[START_REF] Sedighiamiri | Micromechanical modeling of the deformation kinetics of semicrystalline polymers[END_REF]Agoras and Ponte Castaneda, 2012;[START_REF] Poluektov | Micromechanical modelling of short-term and long-term large-strain behaviour of polyethylene terephthalate[END_REF]Uchida and Tada, 2013;[START_REF] Popa | Formulation and implementation of a constitutive model for semicrystalline polymers[END_REF]Mirkhalaf et al., 2019;[START_REF] Okereke | Two-process constitutive model for semicrystalline polymers across a wide range of strain rates[END_REF]Yan et al., 2021). At the mesoscopic scale, the semi-crystalline polymer is regarded as an aggregate of two-phase layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers in which the internal interfacial interaction is explicit.

Chapter Ⅴ. A multi-
a I Q Q a a I Q a a I Q a a I QQ     = - - = - -               = - - = +         = + = +     == ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 3 2 3 1 3 1 3 22 12 ' ' 2 2 13 31 2 3 ' ' 2 2 23 32 1 3 8π 3 π 1 ˆˆ8π 3 4π 1 ˆˆ8π 3 4π a a a a a a a a a a a a I I a a I Q Q I I a a I Q Q I I a a I  - + - +   = = - + - +   = = - + - +  (Ⅴ.B1) with ( ) ( ) 13 12 2 1 3 1 2 2 3 23 1 3 1 1 3 3 1 3 2 2 1 2 2 3 12 22 1 2 1 2 2 2 32 2 22
-       = = - -          -  - = -- = = - = - = = = - - (
Each distinguishing layer may have each own rheological representation to capture, in addition to the crystallographic texturing, local inelastic events such as intermolecular interactions and orientation/relaxation process of the amorphous molecular network (Yan et al., 2021). In spite of the important number of studies dealing with the development of continuum-based models for semi-crystalline polymers, only a few were proposed for PVDF [START_REF] Odegard | Constitutive modeling of piezoelectric polymer composites[END_REF]Challier et al., 2006;[START_REF] Baudet | A two-phase model for the diffusomechanical behaviour of semicrystalline polymers in gaseous environment[END_REF]Laiarinandrasana et al., 2009;[START_REF] Erol | A microstructure-based approach to modeling electrostriction that accounts for variability in spatial locations of domains[END_REF][START_REF] Garcia-Gonzalez | Low temperature mechanical behaviour of PVDF: Cryogenic pre-treatment, quasi-static, cyclic and dynamic experimental testing and modelling[END_REF] and, to the best of our knowledge, no one considers the strain-induced structure changes such as α → β phase transition and anisotropy induced by the crystallographic texturing.

The formulation of coupled constitutive models would be useful in the design and optimization of multi-functional piezoelectric PVDF-based material media. It would allow a more accurate information of the evolution of the electroactive activity if the α → β phase transition along with the orientation of both the amorphous and crystalline phases are considered.

In the present chapter, a microstructure-based model is developed using a micro-macro transition approach within the continuum mechanics rules to replicate the α → β phase transition caused by the straining in the final aim to clarify the connection between the mesostructural evolution and the benefices on the electroactive response. The present chapter is organized as follows. The continuum-based constitutive model is formulated in Section Ⅵ.2. Section Ⅵ.3 provides the model inputs identified using results issued both from molecular dynamics simulation performed for this work and existing experiments. The effect of both material anisotropy and α → β phase transition on the piezoelectric properties is discussed. Concluding remarks are finally given in Section Ⅵ.4.

Ⅵ.2. Continuum-based model

In this section, a microstructure-based model is formulated in continuum terms to describe the finite-strain deformation processes in the amorphous molecular network and in the crystals while taking into account the strain-induced α → β phase transition. The intrinsic response is described within an elasto-viscoplastic-viscohyperelastic coupling. The superscripts e and in will refer to the elastic (linear elastic and hyperelastic) part and the inelastic (viscoplastic and viscous) part, respectively.

The following notation is used throughout the text. Tensors and vectors are denoted by normal boldfaced letters and italicized boldfaced letters, respectively, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative. The superscript T indicates the transpose quantity.

Ⅵ.2.1. Microstructure-based model

The continuum-based model is formulated within an ensemble-volume averaged homogenization approach in which the interfacial interaction is considered. As illustrated in Fig.

Ⅵ.2, the semi-crystalline PVDF structure is regarded at the mesoscopic scale as an aggregate of randomly oriented three-phase layered composite inclusions consisting in parallel amorphous layer, α-form lamella and β-form lamella. All the three phases are considered as homogeneous and continuous media in accordance with the continuum mechanics; A key quantity of the finite-strain kinematics framework is the deformation gradient tensor =   F xX mapping a material point from its initial position X to the current position x . The local quantity i I F of the composite inclusion i is obtained by a volume-weighted average of the three phases:

,0 ,0 ,0

= i i i i i i i I a a f f f     ++ F F F F (Ⅵ.1)
in which the subscripts a ,  ,  and I refer to the amorphous phase, the α-phase, the β-phase Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 141 and the inclusion, respectively.

Fig. Ⅵ.2.

Continuum model of PVDF based on an aggregate of randomly oriented threephase layered composite inclusions consisting in three parallel layers of amorphous phase, αphase and β-phase. Two distinct rheological representations are introduced to constitutively represent the amorphous and crystalline layers constructed, respectively, using elastoviscoplastic-viscohyperelastic and elasto-viscoplastic elements. The material constants of the different rheological elements were identified using a scale bridging method coupling molecular dynamics simulations and experimental data. While the aggregate is subjected to a macrostrain, the α → β phase transition in each individual inclusion is considered by varying the crystal degree with the local strain, f  varying from its initial value _0 f  to 0 and f  varying from 0 to its maximum value _ f   .

The terms i f  and i f  are the α-phase and β-phase volume fractions, ,0 i a f is the initial amorphous volume fraction and, i F  , i F  and i a F are the local deformation gradient tensors respecting the interfacial compatibility condition: . The two volume fractions are expressed here as follows:

0 0 0 0 i i i i i i i i aI  =  =  =  F F F F  x x x x (Ⅵ.
_0 i i i f f f =-    and _ ii ff  =   (Ⅵ.4)
where  is a parameter capturing the transformation degree (as described in Section Ⅵ.2.1.3).

The local Cauchy stress tensor i I σ of the composite inclusion i is obtained by a volumeweighted average of the three phases:

= i i i i i i i I a a f f f ++ σ σ σ σ     (Ⅵ.5)
where i σ  , i σ  and i a σ are the local Cauchy stress tensors respecting the interfacial equilibrium condition:

i i i i i i i i aI  =  =  =  σ σ σ σ  n n n n (Ⅵ.6)
in which i n is the unit normal vector in the current configuration given by: 01 02 0 01 02 0 ( ) ( ) ( )

T T i i i i i i i i i i i ii   - -  =  F F F = FF F x x n n xx n (Ⅵ.7)
1 2 2 31 x x x x -  - - .
The strain rates given, respectively, by the [START_REF] Argon | A theory for the low temperature plastic deformation of glassy polymers[END_REF] expression for the thermally activated process (Boyce et al., 2000) and the Bergstrom-Boyce (1998) expression for the thermalinduced relaxation process of the amorphous layers:

_ inter _ inter ,0 = exp 1 a aa G ks          --       σ and 1 _ net _ net _ net =1 in a v

Ⅵ.2.4. Strain-induced α → β phase transition

The random orientation of the inclusions in the aggregate before straining, along with the tilting Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 146 towards the maximum principal strain of the normal to the surface of the crystalline lamellae during plastic deformation, implies a non-uniform distribution of the α → β phase transition mechanism at the scale of the inclusions. The degrees of α-phase i f  and β-phase i f  in each inclusion i are governed by the following conditions on the α-phase micro-strain

i   : _0 i f f f  =      0 i c i c       and 0 i f f  =     0 i c i c       (Ⅵ.17)
where c  is the critical strain for the onset of α → β phase transition when α-PVDF is drawn,

f    0,1  and f    0,1 
are the macro-variables of α-phase and β-phase amounts and _0 f  is the initial α-phase amount.

The α-phase degree f  decreases progressively with the increase of the β-phase degree f  upon straining according to the following formulae:

_0 f f f =-    and _ ff  =   (Ⅵ.18) where _ f  
is the maximum β-phase amount (taken equal to _0 f  for a full transformation) and  is the total degree of transformation given by the following Avrami-type kinetics [START_REF] Doufas | Simulation of melt spinning including flowinduced crystallization: Part I. Model development and predictions[END_REF]: Eq. (Ⅵ.20) is an empirical formula emerged for the study of spherulitic growth in thermallyinduced crystallization. It was employed in previous studies of the literature in the context of strain-induced crystallization of poly(ethylene terephthalate) [START_REF] Ahzi | Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature[END_REF][START_REF] Makradi | A two-phase self-consistent model for the deformation and phase transformation behavior of polymers above the glass transition temperature: application to PET[END_REF] and poly(lactic acid) [START_REF] Mahjoubi | A micro-macro constitutive model for strain induced molecular ordering in biopolymers: Application to polylactide over a wide range of temperatures[END_REF][START_REF] Mahjoubi | Strain-induced phase transformation in poly(lactic acid) across the glass transition: Constitutive model and identification[END_REF] as a phenomenological description of the evolution of newly formed phase. The involved constants are here slightly modified in order to fit the observations in the PVDF model application. The Avrami-type kinetics is probably not Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 147 an optimal choice for the description of the micro-mechanism of α → β phase transition which requires more work to propose an expression through a thermodynamic reasoning, see e.g. (Guo and Zaïri, 2020).

( ) ( ) ( )
The Ⅵ.1. The ReaxFF reactive force field was used.

Note that the traditional force fields for polymers, such as AMBER, CHARMM, CVFF, UFF, CFF, COMPASS, and so on, are not adaptive here because the bond breaking/formation are not taken into account. Indeed, ReaxFF eschews explicit bonds in favor of bond orders calculated from interatomic distances that are updated every molecular dynamics simulation step, which allows for continuous bond breaking/formation during the simulation. After preparing all the structures, the molecular dynamics simulations were performed. Each

Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 148 simulation involves three steps. In the first step, a geometry optimization was performed on the model to obtain an energy minimized structure. In the second step, a NPT dynamic equilibration was running for 10000 steps at 298 K. The evolution in total energy and density during the equilibrium process was analyzed (see Appendix Ⅵ.A) for both crystal forms. In the equilibrium phase, both energy and density have been equilibrated. The β-PVDF density is found to be slightly higher than the α-PVDF density, which is similar to the 1.92 g/cm 3 for pure α-PVDF and 1.97 g/cm 3 for pure β-PVDF [START_REF] Hasegawa | Crystal structures of three crystalline forms of poly(vinylidene fluoride)[END_REF][START_REF] Ameduri | From vinylidene fluoride (VDF) to the applications of VDF-containing polymers and copolymers: Recent developments and future trends[END_REF]. Based on the information provided by the atoms contributions to radial distribution function (RDF), the partial RDF of bonded atom for both crystal forms was analyzed (see Appendix VI.A) and it was confirmed that the behavior is found close to that found in earlier studies [START_REF] Hasegawa | Crystal structures of three crystalline forms of poly(vinylidene fluoride)[END_REF][START_REF] Wang | Structure and piezoelectric properties of poly(vinylidene fluoride) studied by density functional theory[END_REF][START_REF] Zhu | Microscopic characterization of polarization fatigue in ferroelectric vinylidene fluoride and trifluoroethylene copolymer films[END_REF][START_REF] Li | Density functional theory simulation of PVDF transition in electric field polarization[END_REF]. They are reported in Table Ⅵ.2 and Ⅵ.3, respectively, and compared to theoretical results of the literature [START_REF] Pei | Elastic properties of poly(vinyldene fluoride) (PVDF) crystals: A density functional theory study[END_REF] obtained by density-functional theory. We only retained the 9 most important components of the elastic stiffness tensors:

_11 C  , _ 22 C  , _ 33 C  , _ 44 C  , _ 55 C  , _ 66 C  , _12 C  , _13 C  , _ 23 C  (Ⅵ.21) for α-PVDF and _11 C  , _ 22 C  , _ 33 C  , _ 44 C  , _ 55 C  , _ 66 C  , _12 C  , _13 C  , _ 23 C  (Ⅵ.22)
for β-PVDF since the remaining elastic coefficients are very weak and can be neglected.

Ⅵ.3.1.3. Onset of α → β phase transition

As the manifestation of the α → β phase transition caused by the straining, the α-PVDF exhibits a stress drop followed by a resumption in Fig. Ⅵ.4(a). That results in two distinct processes of linear elasticity, one from 0 to a strain level of 0.05, and a second one from a strain level of 0.08. The elastic constant _ 33 C  calculated in the second stage is found to be 310 GPa which is very close to the β-PVDF elastic constant _ 33 C  . The structure of α-PVDF in the second stage becomes the same than the β-PVDF structure. The screenshots of the stretching process inserted

Ⅵ.3.2. Molecular dynamics analysis of the amorphous phase

Molecular dynamics simulations are employed to examine the properties of the amorphous PVDF phase. It is modeled using the explicit atom model (Kang et al., 2019;Sharma and Roy, 2020;Zhang et al., 2021). The monomer atomic formula of amorphous PVDF is C2H2F2 as shown in Fig. Ⅵ.3(b). Each single chain is represented by 50 monomers. Twenty chains are then heated and melted to form a PVDF melt.

Ⅵ.3.2.1. Simulation details and structural properties

The polymer consistent force field (PCFF) is adapted as the basis of the potential function (Sun et al., 1994). The PCFF force-field is one member of the ab initio consistent force-field family (composed of CFF91, CFF, pcff, and COMPASS force-field) with the same functional form. It involves a Lennard-Jones 9-6 potential for the intermolecular and intramolecular dispersionrepulsion interactions. Specific stretching, bending and torsion terms describe 1-2, 1-3 and 1-4 Ⅵ.3(b). The parameter of these interactions are described in details in (Sun et al., 1994;Li et al., 2016) and skipped here for brevity.

The initial structure is studied for four relaxation steps to relax any high-energy configurations.

Initially, the simulation ran for 2×10 5 timesteps (Δt=1 fs) using NVT ensemble at 600 K, then the structure is relaxed for 5×10 5 timesteps (Δt=1 fs) using NPT ensemble at 600 K. The next relaxation cooled the structure down to the desired temperature at 1 K/ps cooling rate (Δt=1 fs)

followed by further relaxation of 3×10 6 timesteps (Δt=1 fs) at the desired temperature. The energy history of the models was monitored through this process to ensure that final structure was stable. (see Appendix VI.A).

Ⅵ.3.2.2. Elastic and yield constants

After the stable structure was obtained, uniaxial tension was performed at different straining temperatures and strain rates and, the stress-strain curves are plotted in fluctuations in the response are due to the small size of the simulated cell as also observed in other studies for different systems (Yazdani et al., 2018;Payal et al., 2019;Zhang et al., 2021) and they were smoothed via the Savitzky-Golay filter (Larive and Sweedler, 2013) to obtain the stress trends. It can be seen that the latter trends exhibit a clear dependence on the loading parameters; The lower the strain rate (the higher the loading temperature), the lowest the stiffness and the yield stress.

In order to identify the equilibrium response, the system was uniaxially stretched at a constant strain rate of 10 9 s -1 for 10000 steps (ε=0.1) followed by equilibration for 200000 steps (Δt=1 fs) with the axial dimension kept fixed to stabilize the energy in the system and to provoke the stress relaxation. This process is iterated until the desired strain is obtained. The relaxed stressstrain response is plotted in The amorphous stiffness modulus can be obtained from the initial slope of stress-strain curve.

The small-strain region of the relaxed response was also used to evaluate the amorphous yield strength as shown in Fig. Ⅵ.7(a). Different graphical methods may be utilized to identify the stress at yield onset from a monotonic stress-strain curve according to its shape (Ward and

Ⅵ.3.3. Continuum-based simulation

The continuum-based model was entirely coded in Fortran language and compared to the homogeneous material data taken from the work of [START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF] (see also Defebvin In the following subsection, a robust hybrid deterministic calibration procedure of the continuum-based model is detailed by using molecular dynamics and experimental data.

Ⅵ.3.3.1. Model parameters

The elastic constants of the α phase, the β phase and the amorphous phase obtained separately The intermolecular yield parameters of the amorphous phase are obtained by using simultaneously the yield point extracted from the molecular dynamics results and an inverse identification of the experimental yield point at 10 -3 s -1 . A very low strain rate of 10 -8 s -1 is used to plot the quasi-static yield stress computed from the molecular dynamics simulations. The amorphous yield stress is plotted against the logarithmic strain rate in  using the yield stress evolution with strain rate (constructed by taking the quasi-static data (at 10 -8 s -1 ) from the MD curve and the data at 10 -3 s -1 by inverse identification using the experimental curve) and (b) identification of the hardening coefficient n using the MD data in terms of logarithmic stress vs. logarithmic inelastic strain. To complete the identification of the viscoplastic intermolecular features of the amorphous layers, the hardening coefficient n was determined from the slope of the molecular dynamics e  =-0.28 C/m 2 . The overall piezoelectric strain coefficient is plotted as a function of the applied strain in Fig. Ⅵ.11(a). The simulation consists to an isothermal straining at the temperature of 360 K up to a pre-determined strain level under a constant strain rate of 10 -3 s -1 and to extract at this strain level the values of the electroelastic moduli at room temperature. As soon as the α → β phase transition begins, the piezoelectric strain coefficient increases with the increasing applied strain following a gradual rate decrease until a maximum strain level showing a good correlation with the experimental data of [START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF] (see also

  = +            ( ) ( )

Ⅶ.1. Introduction

A reliable design of rubber systems can only be achieved by proposing accurate structuremechanics relationships accounting for the multiaxial character of the mechanical loading. The prediction problem of the mechanical behavior for this class of materials may be approached using two ways of investigation, namely, molecular dynamics and continuum-based modeling.

Molecular dynamics is a very interesting tool to analyze and predict the mechanics of polymers under biaxial monotonic loading. Indeed, if constitutive modeling has attracted attention of many researchers, connecting the molecular-level description to the continuum-level mechanics under monotonic and multiaxial loading is less documented (Uddin andJu, 2016, 2017;[START_REF] Chaube | Multiscale analysis of large-strain deformation behaviour of random cross-linked elastomers[END_REF][START_REF] Chen | Molecular simulation-guided and physics-informed mechanistic modeling of multifunctional polymers[END_REF][START_REF] Yuan | Molecular simulation guided constitutive modeling of filled rubber: Bridging structural parameters to constitutive equations[END_REF].

The hyperelastic model proposed by Arruda and Boyce (1993) constitutively represents the rubber network by eight long-chain molecules. This so-called eight-chain model has found many applications. It was for example combined to viscous and plastic components to describe the inelastic features of a wide range of polymer systems such as unfilled rubbers and filled rubbers, amorphous and semi-crystalline thermoplastics; the reader may refer to the literature background in the following references (Yan et al., 2021[START_REF] Yan | A multi-scale plastic-damage model for straininduced morphological anisotropy in semi-crystalline polyethylene[END_REF]. Although the eight-chain model has several advantages, especially a flexible and explicit formulation, its main weakness relies in its incapacities to predict rubber biaxial mechanics [START_REF] Boyce | Constitutive models of rubber elasticity: A review[END_REF]. Indeed, several studies showed that models based solely on the first strain invariant have poor multiaxial predictive capacities whatever the mathematical expression [START_REF] Horgan | Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility[END_REF][START_REF] Wineman | Some results for generalized neo-Hookean elastic materials[END_REF][START_REF] Horgan | The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials[END_REF][START_REF] Puglisi | Multi-scale modelling of rubber-like materials and soft tissues: An appraisal[END_REF][START_REF] Destrade | Methodical fitting for mathematical models of rubber-like materials[END_REF], the presence of the second strain invariant appearing as necessary.

Several extensions of the eight-chain model were proposed in order to enhance its predictive capacities under biaxial loading [START_REF] Boyce | Constitutive models of rubber elasticity: A review[END_REF]Meissner andMatejka, 2003, 2004;[START_REF] Miroshnychenko | Heuristic search for a predictive strain-energy function in nonlinear elasticity[END_REF][START_REF] Bechir | A three-dimensional network model for rubber elasticity: The effect of local entanglements constraints[END_REF][START_REF] Kroon | An 8-chain model for rubber-like materials accounting for non-affine chain deformations and topological constraints[END_REF][START_REF] Dal | An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects[END_REF][START_REF] Anssari-Benam | A generalised neo-Hookean strain energy function for application to the finite deformation of elastomers[END_REF]. Among these extensions, [START_REF] Kroon | An 8-chain model for rubber-like materials accounting for non-affine chain deformations and topological constraints[END_REF] proposed modifications using a reasoning from the micromechanics of the chain molecules by considering the chain deforms non-affinely while the interactions with neighboring chains are accounted for through the adding of the topological constraint effects imposed on the lateral motions. The Kroon model contains three supplementary parameters.

Our purpose in this chapter is to examine the validity of the Kroon network theory by connecting the continuum-level mechanics to the molecular-level description. The latter The stress state is obtained from the differentiation of strain energy with respect to deformation.

The Kroon strain energy can be expressed at the macro-scale as a function of the three stretch invariants: The first term in Eq. (Ⅶ.2) describes the molecular network stretching process while the last term corresponds to the volumetric part in which p is the Lagrange multiplier. The two other terms were added by [START_REF] Kroon | An 8-chain model for rubber-like materials accounting for non-affine chain deformations and topological constraints[END_REF]. The power-law term originates from the consideration of the non-affine polymer chain deformation and the third term introduces the 2 I -dependence coming from the tube-constraints. The model contains five material constants to be identified: n , c c , nc c ,  and con c . The three last material constants are the supplementary ones compared to the original Arruda-Boyce eight-chain strain energy (Arruda and Boyce, 1993). In the limit case nc c →, 1   and con 0 c = , the interactions between polymer chains are ignored and the Kroon strain energy becomes the original Arruda-Boyce form based on an affine description of the deformation. Moreover, it degenerates to the more classical Neo-Hookean model for extremely long-chain molecules, i.e. n → . A fully affine deformation is obtained when nc 1 =  , otherwise it is an unknown in Eq. (Ⅶ.2) and can be determined via a Newton method from the following condition: The deformation gradient tensor F can be written on the form of the principal directions: where  is the applied stretch and n  [-0.5, 1.0] is a biaxiality exponent varying from -0.5 for uniaxial tension to 1.0 for equibiaxial tension. where  is the depth of potential well,  is the finite distance at which the inter-particle potential is zero, and r is the distance between the particles for Lennard-Jones potentials. The parameter of these interactions are described in Table Ⅶ.1.

Ⅶ.3.2. Mechanical response

The model system was then deformed under a uniaxial tensile loading at a constant strain rate of 10 10 s -1 . This deformation condition was implemented in LAMMPS by decoupling the boundary in the loading direction from the NPT equations of motion. The stretch contribution of each individual chain defined as the ratio of end-to-end vectors after and before deformation is plotted in Fig. VII.5(a). Fig. VII.5(b) presents stress-strain curves in the three main directions. It can be observed that the material response exhibits no preferential direction. The rate-dependency of the uniaxial stress-strain response is illustrated in Fig. VII.6(a) by considering three decades of strain rates of 10 9 s -1 , 10 10 s -1 and 10 11 s -1 . The material response exhibits a clear rate dependence. The overall response of the network is dependent on the loading path since the chains experience different extent of deformation. The system was biaxially stretch at different strain ratios 1:1, 1:0.2 and 1:0.1 at a strain rate of 10 10 s -1 and the results are reported in Fig. VII.6(b). x:y=1:1

x:y=1:0.2

x:y=1:0.1

Ⅶ.4. Kroon model predictions

Our purpose now is to examine the validity of the Kroon model discussed above to predict the biaxial behavior. The model parameters were calibrated using the uniaxial response.

Ⅶ.5. Conclusion

A series of model systems were constructed by molecular dynamics simulation method. The relaxed-stress was quantified under different biaxial strain ratios. Checking the validity of the Kroon model for the model system, we have found that it gives good estimates of the behavior x:y=1:1

x:y=1:0.2

x:y=1:0.1

Chapter Ⅶ: Kroon network theory verification for rubber biaxial mechanics via a multi-scale modeling 180 for different biaxial loading paths. Nonetheless, the molecular dynamics data requires a deepener analysis by extending the variety of model systems for pure elastomers (i.e. no additive fillers) as well as filled elastomeric systems.

General conclusions and perspectives

This PhD dissertation concerned the multiscale modeling of the large-strain mechanical behavior of various polymer systems using two different, but complementary, approaches, namely molecular dynamics simulations and constitutive modeling within the continuum mechanics rules. The first approach provides the molecular-level description whereas the second approach provides the continuum-level mechanics.

Molecular dynamics simulations were firstly carried out to investigate the structure-property relation of semi-crystalline polyethylene (PE) systems. The simulations were conducted for different material systems under various loading temperatures representing different states (100 K: glassy state, 250 K and 300 K: transition state and 400 K: rubbery state). When side lengths change independently, polymer crystallinity in each direction was different. It was found that structure composition and temperature have a great effect on the crystallinity parameters (i.e. crystallinity based on bond orientation and global order parameters). The higher the temperature, the lower the crystallinity and the longer the chain length, the lower the crystallinity. The stress-strain curves obtained during the course of the uniaxial stretching showed four distinct regimes before failure with a strong effect of both the structure composition and the stretching temperature. Besides, the strain decomposition concept allowed to clearly highlight the main deformation mechanisms split into shear yielding and dilatational mechanisms. A directional effect was pointed out both on the uniaxial response and the underlying deformation micro-mechanisms. A significant effect of the biaxial strain ratio was pointed out on the different mechanical properties, especially the yield strength and the strainhardening ability. Data were collected to construct yield and failure envelopes under a wide range of biaxial strain ratios and temperatures. The response of the memory behavior indicated that PE with low crystallinity can restore the original shape well, the lower the temperature, the higher the recovery rate. In addition, it was evidenced that the material through the shape memory behavior has better mechanical properties. Then, we investigated the structureproperty relation of carbon-black filled vulcanized cis-polyisoprene (cis-PI) systems. The latter were vulcanized by adding element sulphur to cis-PI and vulcanizing at high temperatures to obtain vulcanized rubber, and then adding different amounts of fullerenes (C60) to the vulcanized rubber at high temperatures to obtain carbon-black (CB) filled vulcanized rubber.

The macroscopic physical and thermodynamic properties of vulcanized rubber and CB filled vulcanized rubber were studied to obtain the effect of vulcanization and nanofiller on natural General conclusion and perspectives 184 rubber. The higher the degree of vulcanization, the formation of a network between the molecular chains, resulting in higher density, lower thermal expansion and higher specific heat capacity. It was pointed out that vulcanization leads to an increased ability of the rubber to withstand stress at large strains. Moreover, it was showed that fillers have the same effect on density and specific heat capacity as vulcanization, but there is no clear trend towards an effect This work led to several aspects that can be treated in the future for both sides concerning the molecular dynamics simulations and the development of continuum-based models. We need to deepen the understanding of the reinforcement mechanisms of different polymer systems using a wider range of nanofillers (nanotube, graphene, …). On the continuum terms side, although quite sophisticated, the constitutive model needs improvements for a fully realistic description of the microstructure evolution and the underlying large-strain-induced anisotropy. In particular, the fragmentation mechanism operating in the crystalline lamellae and leading to the formation of a fibrillar microstructure deserves to be included.

Abstract

As a meaningful prerequisite of advanced applications of polymer systems, predictive multiscale modeling is needed. This thesis deals with the multiscale modeling of the large-strain mechanical behavior of polymer systems using both molecular dynamics and continuum-level mechanics. The key role of the structural factors (length and number of chains, crystallinity, cavitation damage, phase transformation, vulcanization and nanofiller reinforcement) and the loading conditions (strain rate, temperature, non-monotonicity and biaxiality) on the largestrain mechanical behavior is highlighted for various polymers, including semi-crystalline polyethylene, semi-crystalline polyvinylidene fluoride and carbon-black filled vulcanized cispolyisoprene.

Keywords

Résumé

En vue de leurs applications toujours plus innovantes, la modélisation multi-échelle prédictive des systèmes polymères est un prérequis nécessaire. Cette thèse porte sur la modélisation multiéchelle du comportement mécanique en grandes déformations de cette classe de matériaux en utilisant la description au niveau moléculaire via la dynamique moléculaire et à l'échelle du continuum via la mécanique des milieux continus. Le rôle clé des facteurs structuraux (tels que longueur et nombre de chaînes, cristallinité, endommagement par cavitation, transformation de phase, vulcanisation et renforcement des nanocharges) et des conditions de chargement (telles que vitesse de déformation, température, non monotonicité et biaxialité) sur le comportement mécanique en grandes déformations est mis en évidence pour divers polymères, notamment le polyéthylène semi-cristallin, le fluorure de polyvinylidène semi-cristallin et le cis-polyisoprène vulcanisé et chargé en noir de carbone.

Mots clés : Systèmes polymères ; Dynamique moléculaire ; Mécanique des milieux continus ; Propriétés mécaniques; Interface ; Mécanismes de déformation.

Mots clés :

 clés Systèmes polymères ; Dynamique moléculaire ; Mécanique des milieux continus ; Propriétés mécaniques; Interface ; Mécanismes de déformation.

  Fig. Ⅰ.1. Polyethylene models: (a) Explicit-atom and united-atom models, (b) multi-chain molecular structure.

Fig

  Fig. Ⅰ.1 taking the Structure 1 as an illustrative example.

  Fig. Ⅰ.3. Schematic images depicting the judgment of crystalline: (a) Chain orientation and (b) global order.

Fig

  Fig. Ⅰ.4. Volume evolution as a function of temperature used to extract the glass transition temperature: (a) Structure 1, (b) Structure 2 and (c) Structure 3.

  Fig. Ⅰ.7. (a) Uniaxial stress-strain response of the Structure 1 deformed along Y direction at a stretching temperature of 100 K. (b) External and internal screenshots of the simulated cell at different loading strains, with the external screenshot showing each marble and the internal structure showing the cavitation of the structure.

  Necking Chapter Ⅰ. Crystallization and mechanical behavior of semi-crystalline polyethylene 20 hydrogen atoms, which affects the overall performance when multiple chains create necking and voids (part of the increased volume strain, Fig. Ⅰ.8), according to Stokes' law, which relates the coefficient of friction to the hydrodynamic radius.

  Fig. Ⅰ.8(c) shows an obvious breakpoint for Structure 3 (short length chain).

  Fig. Ⅰ.8. Structural composition effect on the uniaxial stress-strain response and volume strain at a stretching temperature of 300 K for different structures.

  Fig. Ⅰ.9. Direction effect on the uniaxial stress-strain response along with deformation mechanisms of Structure 1 at a stretching temperature of 100 K: (a) along X direction, (b) along Y direction and (c) along Z direction.The elastic volumetric strain plays an important role within the small strain region, whereas the

  Fig. Ⅰ.10. Temperature effect on the uniaxial stress-strain response of Structure 1 deformed along different directions: (a) 250 K, (b) 300 K and (c) 400 K.

  Fig. Ⅰ.12. Stress-strain response of Structure 2 at a stretching temperature of 100 K: (a) XY equibiaxial stretching, (b) XZ equibiaxial stretching, (c) YZ equibiaxial stretching.

  Fig. Ⅰ.13. Biaxial stress-strain response of Structure 1: (a), (b) different biaxial strain ratios X:Y at a stretching temperature of 100 K, (c) different temperatures for a biaxial strain ratio 0.5:1, (d) different temperatures for a biaxial strain ratio 1:0.5.

  Fig. Ⅰ.14. Temperature effect on the biaxial response: (a) yield strength and (b) failure point.

  Fig. Ⅱ.1. Polyethylene models for: (a) all-atom and (b) United-atom models and (c) single branched polyethylene chain.

Fig. Ⅱ. 2 .

 2 Fig. Ⅱ.2. Stress-strain variation for: Structure 1 at 100 K (a) and three structures at 100 K (b).

Fig. Ⅱ. 3

 3 Fig. Ⅱ.3 displays the stress versus temperature at 10% strain (a) and 200% strain (b) for different structures.For the small strain level (10% strain), we can see that, branched polyethylene with

Fig. Ⅱ. 3 .

 3 Fig. Ⅱ.3. Stress versus temperature at 10% strain (a) and 200% strain (b) for different structures.

Fig. Ⅱ. 4 .

 4 Fig. Ⅱ.4. Response to shape memory behavior for: Stress-strain variation (a); Recovery rate evolution (b); and Temperature and structure response of recovery rate (c).

  Fig. Ⅱ.4(c) in which it can be seen that the higher the stretching temperature, the lower the final strain recovery. From Fig. Ⅱ.4(c), we can see that for the structures with the branched chain of the same length, when the number of

Fig. Ⅱ. 5 .

 5 Fig. Ⅱ.5. Stress-strain relationship at: 100 K for structure 1 (a), and stress versus temperature (b) at 200% strain before and after shape memory.

  Fig. Ⅱ.5(a) and the stress at 200% strain of different structure for different temperature were shown in Fig. Ⅱ.5(b). Through the study of the stress-strain curve, we can 0

  here for intial cis-PI model is generally accepted. The monomer atomic formula is C5H8 as shown in Fig. III.1(a). Each cis-PI single chain is a polymeric chain consisting of 34 monomers. The thirty chains are then heated and melted to form a cis-PI melt (Fig. III.1(b)).

Fig. Ⅲ. 1 .

 1 Fig. Ⅲ.1. Structures used in this study: (a) cis-PI chains, (b) natural rubber simulation cell, (c) elemental sulfur, (d) vulcanized natural rubber, (e) fullerene molecules (C60) and (f) filled vulcanized natural rubber.

  and mechanical properties of filled vulcanized rubber by molecular dynamics simulations 50 as indicated by the different colors in Fig. III.1. The parameters of these interactions are

  . III.1(e)) to the simulation cell, we first create a certain number of cylindrical indentations in the simulation cell, where 1 indentation is for 1 added fullerene, 3 indentations for 3 added fullerenes, 4 indentations for 8 or 12 added fullerenes, and 9 indentations for 27 added fullerenes, while the diameter of the cylindrical indentation is slightly larger than the diameter of the fullerene. Then, we add a certain amount of fullerene to the simulation cell, as shown in Fig.III.1(f), and in Table

  Fig. Ⅲ.2. Effect of crosslink volume fraction on (a) density, (b) thermal expansion and (c) specific heat capacity.

Fig. Ⅲ. 3 .

 3 Fig. Ⅲ.3. Stress-strain curves of simulated units with different crosslinking degrees upon stretching process.

Fig

Fig. Ⅲ. 4 .

 4 Fig. Ⅲ.4. Stress-strain curves of simulation cell upon (a) stretching-retraction and (b) cyclic loading sequences.

Fig

  Fig. III.7 shows the effect of nanofillers fraction on the macroscopic and thermal properties.

Fig. Ⅲ. 7 .

 7 Fig. Ⅲ.7. Effect of C60 amount on (a) density, (b) thermal expansion and (c) specific heat capacity.

  interfacial build-up due to the attraction, and the nanofiller aggregation due to the mutual attraction between fullerenes. The pair distribution functions (C60-cis-PI, C60-H(cis-PI), C60-C(cis-PI), C60-S(cis-PI), C60-C60) were calculated and local number density ( 𝜌 𝑙𝑜𝑐𝑎𝑙 ) and potential mean forces (PMF) were obtained using the following equations:ρ 𝑙𝑜𝑐𝑎𝑙 = ρ • 𝑔(𝑟)Physical and mechanical properties of filled vulcanized rubber by molecular dynamics simulations 58 PMF = -𝐾𝑇 ln(𝑔(𝑟)) (Ⅲ.14)

  C60 is involved in the calculations, the centre of mass of C60 is used for all calculations, as shown in Fig. III.8. The local aggregation density in the cis-PI chains around fullerenes can be used to study the probabilistic configuration of polymers around fullerenes. The peak values of PMF were used to justify preferences of C60 molecules for aggregation and cis-PI chain adsorption (Khuntawee et al., 2019).

Fig. Ⅲ. 8 .

 8 Fig. Ⅲ.8. A schematic diagram describing the distance between the center of mass of the C60 molecule and the explicit atoms of cis-PI.

r

  Fig. Ⅲ.9. Local number density contour map of cis-PI around C60 interface for cis-PI-C60 composite at 1 phr.

Fig

  Fig. Ⅲ.10(c), where the first peak of 26.42 phr is 9.75 Å, i.e. the surface distance between two C60 is 2.25 Å. At this point, the closer C60 form a cluster. And as can be seen from the RDF, when the fraction is 26.42 phr, the rdf exhibits an amorphous state, different from the small content. The first minimum of the PMF is the minimum of the curve in Fig.Ⅲ.10(b), i.e. the attraction between C60 and cis-PI is more attractive than the attraction between cis-PI.

  Fig. III.11. By calculating the slope of the initial elastic phase of the stress-strain curve in Fig. III.11, we obtained the initial modulus of natural 0

Fig. Ⅲ. 11 .

 11 Fig. Ⅲ.11. Stress-strain curves (a) upon stretching and (b) stretching-retraction loading sequences and (c) Young's modulus.

  Fig. Ⅳ.1.Two-scale view of the semi-crystalline structure; At the mesoscopic scale, the crystalline lamellae may be arranged in highly regular stacking to form spherulitic morphology for high-crystalline media or they may be randomly distributed in the amorphous matrix for low-crystalline media; Both phases are coupled in a complex manner.

Fig. Ⅳ. 2 .

 2 Fig. Ⅳ.2. Constitutive representation considering the semi-crystalline medium as an aggregate of randomly distributed two-phase layered composite inclusions. Each inclusion is constituted by a stiff crystalline lamella with an internal elasti-viscoplastic stress and a rubbery amorphous layer with an internal stress including elastic-viscoplastic intermolecular and viscohyperelastic network stresses.

  map a material point from its initial position X in the reference configuration to its actual position x in the deformed configuration. The inelastic deformation mechanisms within each phase are considered to be distinct and are described using the multiplicative decomposition concept of the deformation. The introduction of an intermediate configuration during a spontaneous elastic unloading allows the multiplicative deformation decomposition into elastic and inelastic parts: e in = F F F in which the elastic and inelastic components are depicted by superscripts e and in, respectively. The two tensors can be further decomposed into stretch (right U or left V ) and rotation R movements using the polar decomposition: The subscripts c and a are used to differentiate the crystalline and amorphous phases, respectively.

aσ

  is thus split into Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and point of view, the intermolecular resistance is represented by an elastic spring in series with a viscous dashpot (Fig. Ⅳ.2). The intermolecular Cauchy stress _ inter a σ is constitutively coordinated with the corresponding strain by the following relationship:

  average density of amorphous chains per unit volume, k is the Boltzmann's constant and  is the absolute temperature) and the average number of molecular units N between entanglements (similar to average length of amorphous chains). The term I is the identity tensor, _ net _ net det e a a J = F is the elastic network volume Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 74 change, _ inter e a  is the elastic stretch on each chain in the amorphous layers: elastic left Cauchy-Green tensor and 1 -is the inverse Langevin function ( )

  gradient tensor of the spatial velocity described by: is the elastic spatial velocity gradient tensor and

F

  Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 76 (Fig. Ⅳ.2). The crystalline stress tensor c τ is constitutively coordinated with the strain as: is the elastic volume change, c σ is the crystalline Cauchy stress tensor, -Lagrange strain tensor and c K denotes the anisotropic fourth- order elastic stiffness tensor of the orthorhombic crystalline phase of polyethylene taken from the literature(Tashiro et al., 1978):



  is the crystalline shear strain rate constitutively coordinated to the resolved shear stress : c   =τR via a power law:

  19) in which i c σ and i a σ are the local stresses previously defined and, i c f and i a f are the crystalline and amorphous volume fractions, respectively, expressed as: Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 78 ,0

  the latter Taylor assumption, compatibility of deformation is satisfied everywhere by considering the local deformation to be related to the global one in an affine sense. It is obvious that the underlying assumptions to relate local and global responses impact the physical consistency and predictability of the model. A realistic modeling allows to propose a framework avoiding (or at least limiting) the arbitrary inclusion of heuristic parameters.

x

  denotes an arbitrary vector in the plane of the interface spanned by the unit vectors 01 i e and 02 i e of the local orthogonal basis ( ) in which 0 03 ii = ne denotes the unit normal Chapter Ⅳ. Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 79

  Fig. Ⅳ.3. AFM images at different magnifications of the three polyethylene systems showingisotropically distributed stacks of well-defined crystalline lamellae (spherulitic and lamellar morphologies) in the HD and LL systems and fuzzy crystallites akin to fringed micelles (bundle-like crystals embedded in the rubbery amorphous matrix) in the UL system(Ayoub et al., 2010).

  the crystallinity characteristics at room temperature. The specific melting enthalpy f H  was measured by DSC at a scanning rate of 10°C/min under nitrogen flow with a Q100-apparatus from TA Instruments, which has been indium-calibrated. Melting endotherms and the crystallizing exotherms are provided in Fig. Ⅳ.4; The extremums correspond to

  Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 84 regulating the cross-head speed of the testing machine. A series of mechanical tests were carried out to obtain data on the main features of the large-strain plastic deformation at room temperature. Different experiments have been undertaken in the plastic instability:

  Fig. Ⅳ.5. Stress-strain curves of the three polyethylene systems at different true strain rates (symbols: experimental data, solid lines: constitutive model).

FigⅣ

  Fig. Ⅳ.5 presents the identification resultsfor the different local true axial strain rates: 0.01 /s, 0.005 /s, 0.001 /s, 0.0005 /s and 0.0001 /s. The solid lines represent the model simulations while the symbols represent the experimental data. The model appears to be relatively effective to capture the influence of the crystallinity on various features of the nonlinear mechanical response along with the nonlinear rate-dependency. Moreover, it is able to reproduce the crystal content effect on the gradual polyethylene response transition from the thermoplastic-like behavior of the HD system to the elastomeric-like behavior of the UL system. However, the low-strain response is captured rather poorly by the model with a divergence that increases with the crystallinity decrease. That can be mainly the result of the elastic properties of the polyethylene crystal taken from the literature and introduced as direct inputs into the model for the three polyethylene systems. Despite the latter, the model adequately reproduces, as the crystal content decreases, the decrease in plastic features (yield strength and strain-hardening ability) along with the decrease in rate-dependency.

Fig. Ⅳ. 6 .

 6 Fig. Ⅳ.6. Model predictions of pole figures at different macroscopic strain levels

  based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity 87 crystalline lamellae towards the maximum principal stretch direction. The (010) poles spread around the transverse direction (Fig. Ⅳ.6(a)) while the inclusion interface normals are migrating towards the transverse direction (Fig. Ⅳ.6(b)).

  Fig. Ⅳ.9 by comparing them with the experimental observations of the three polyethylene systems. A global view atthese plots shows that the unloading path is strain dependent. A good agreement appears between model simulations and experimental data of the HD system. The plastic strain at zero stress is very well predicted by the model. For the LL system, it is satisfactory to observe that

Fig. Ⅳ. 8 .

 8 Fig. Ⅳ.8. Local axial stretches in the three polyethylene systems upon oligo-cyclic loading at a strain level of 1.5 and at a true strain rate of 0.001 /s.

  Fig. Ⅳ.9.Stress-strain curves of the three polyethylene systems upon loading-unloading at different strain levels and at a true strain rate of 0.001 /s (dashed lines: experimental data, solid lines: constitutive model).

  Fig. Ⅳ.12. Stress-strain curves of the HD system upon (a) loading-unloading-reloading, (b)oligo-cyclic loading (dashed lines: experimental data, solid lines: constitutive model), (c) stress-softening at different strain levels and at a true strain rate of 0.001 /s (dashed lines: experimental data, symbols: constitutive model).

  models are used to capture both pre-yielding and post-yielding while taking into account crystal elasto-plasticity, crystallographic texturing and amorphous molecular network preferred orientation during finite plastic deformation. All these deformation mechanisms are purely isochoric. To date, no model takes into account the multi-scale inelastic processes at the origin of the strain-induced anisotropy in semi-crystalline polymers in relation to the inelastic dilatation mechanism occurring in the amorphous phase. The development of such a modeling would bring a more accurate information of the intrinsic mechanical behavior and a deeper understanding of the interactions between the above-mentioned inelastic isochoric mechanisms Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 103 and the cavitation damage. In a previous work (Yan et al., 2021), a multi-scale homogenization-based model was developed and applied to the large-plastic deformation of semi-crystalline polyethylene covering a wide spectrum of the crystallinity allowing to demonstrate the determinant role of the strength and consistency of the amorphous phase on the tensile yielding. The aim of the present chapter is to further develop the model to consider the cavitation damage effects in the material anisotropy during finite plastic deformation of polyethylene. By this way, a more physically consistent description is formulated in continuum terms to describe the complex coupling existing between microstructure (amorphous and crystalline phases), micromechanical deformation processes including cavitation damage and large-strain flow behavior. The efficiency of the proposed plastic-damage model to capture material response in terms of finite deformation stress-strain behavior and inelastic volumetric strain is critically discussed by comparing simulations to experimental results including strain rate and stretching temperature effects. The effects of crystal plasticity and amorphous cavitation damage on the macroscopic and internal responses are observed thanks to the model under stretching and stretching-retraction conditions. The present chapter is organized as follows. The main elements of the fully three-dimensional model that accounts for the role of cavitation damage are described within the continuum-based micromechanical framework in Section Ⅴ.2 Section Ⅴ.3 presents comparisons between model simulations and experimental data as well as microstructure evolution predictions. Concluding remarks are finally given in Section Ⅴ.4. The following notation is used throughout the text. Tensors and vectors are denoted by normal boldfaced letters and italicized boldfaced letters, respectively, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative. The superscript T indicates the transpose quantity. The symbol  denotes tensor product.

  Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 104 interconnecting spherulites in the case of high crystallinity polyethylene. The crystalline lamellae are interspaced by a certain amount of randomly oriented entangled chains, namely the amorphous phase. Starting point for the derivation of the multi-scale plastic-damage model is the constitutive representation of the semi-crystalline polymer system. Fig. Ⅴ.1 shows the representative material element of the lamellar structure that is regarded at the mesoscopic scale as an aggregate of two-phase layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers, involving respectively, crystal plasticity and internal cavitation as main deformation modes.

Fig. Ⅴ. 1 .

 1 Fig. Ⅴ.1. Semi-crystalline polymer seen as an aggregate of randomly distributed two-phase layered composite inclusions (stiff crystalline lamella/rubbery amorphous layer) considering crystal plasticity and amorphous cavitation as elementary inelastic mechanisms. The internal cavitation in the amorphous layer is introduced thanks to an ellipsoid unit cell homogenized for both the elastic-viscoplastic intermolecular resistance and the viscohyperelastic network resistance.

KF

  introduction of an intermediate configuration during a spontaneous elastic unloading allowing the multiplicative decomposition of the crystalline deformation gradient tensor c is the fourth-order elastic stiffness tensor of the crystalline phase, is the elastic volume change of the crystalline lamellae.The elementary inelastic mechanisms operating in the polyethylene crystals are introduced as volume preserving. The inelastic spatial velocity gradient tensor in c



  n and  m are the "slip plane" normal vector and the "slip direction" vector in this plane, respectively (Fig. Ⅴ.1), and c   is the crystalline shear strain rate given by: Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semiis the reference strain rate, m is the strain rate sensitivity parameter,

  represented by considering two additive micro-mechanisms acting in parallel: resistance to alignment and orientation of the amorphous chains at large strains considering changes in configurational entropy and molecular relaxation process during the strain-hardening stage of the stress-strain response. The two physically distinct sources require different constitutive theories considering the cavitation damage mechanism giving rise to a volume increase. The amorphous phase surrounding the voids is made of an elastic-viscoplastic medium for the intermolecular interactions and of a viscohyperelastic medium for the molecular network alignment and orientation process. The ellipsoid unit cell illustrated in Fig. Ⅴ.1 introduces the void volume fraction void f and the void aspect ratio w as the basic microstructural features of the voided medium 11 :

  where c denotes the focal distance.

  elastic strain and a is the fourth-order elastic stiffness tensor of the isotropic amorphous layers whose the components may be expressed using the two Lame's constants  and  .The intermolecular inelastic strain rate tensor _ inter in a D at the upper scale (that of the porous amorphous phase) is given by the differentiation of the yield function _ inter a scalar parameter obtained from the necessary condition ensuring a coherence in the definition of the energy dissipation at the two scales involved in the amorphous layer. The equivalence between the inelastic work rate _ inter _ inter : in aa σD of the porous amorphous phase and the local dissipation ( ) _ inter _ inter 1 aa f -σ  in the amorphous phase surrounding the voids leads Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semi-

   and 02   . The term W is the local strain energy density, 1

  molecular network properties are introduced: the average number of molecular units N between entanglements and the amorphous network hardening modulus hc C n k = in which c n is the average density of amorphous chains per unit volume. The term 1 -is the inverse of the Langevin function ( ) ( ) may be approximated by Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semi-

  multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 112

f

  are the corresponding initial volume fractions and, c J and a J are the corresponding volume changes. Chapter Ⅴ. A multi-scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene 113

x

  denotes an arbitrary vector in the plane of the interface spanned by the unit vectors unit normal vector at the interface in the reference configuration.

F

  void volume fraction.13 In virtue of the Taylor assumption, the local deformation is related to the global one in an affine sense. In other words, the local inclusion-averaged deformation gradient tensor i I of the composite inclusion i is taken equal to the imposed macroscopic deformation gradient 𝐅 ̅ of the aggregate of the I N inclusions: i I = FF .

F

  time increment and I is the identity tensor.The elastic component e c is then extracted from the multiplicative deformation decomposition: . The time-integration of the amorphous deformation is carried out using the formulae (V. 20) and (V. 36).

  in the crystalline-amorphous interface, they become at time t :

  Figs. V. 2(b) and V. 3(b) show the model-experiments comparison. It can be seen that the model captures in a satisfactory manner the general trends of the elastic-plastic stress-strain response

  Fig. Ⅴ.2. Stress and volumetric responses under stretching for different strain rates: (a) experiments taken from Addiego et al. (2006) and (b) model simulations. volumetric strain are correctly reproduced by the model. In other words, the nucleation and growth process of voids are modeled during plastic deformation. The quite well reproduction of the kinetics of volumetric strain during the course of tensile tests shows that the model can

  Fig. Ⅴ.4. Stress and volumetric responses under a stretching-retraction-recovery sequence: (a) experiments taken from Addiego et al. (2006) and (b) model simulations.

Fig. Ⅴ. 5 .

 5 Fig. Ⅴ.5. Pole figures of the stress at the end of the retraction. At the end of the retraction, individual crystalline lamellae may be still stressed as observed in the pole figures provided in Fig. Ⅴ.5 and the recovery process is carried out at zero-stress creep of the volume-averaged stress of all the inclusions in the aggregate. Fig. Ⅴ.4(b) presents the model predictions of the macroscopic material response under the stretching-retractionrecovery path. It can be observed that the model results are satisfactory when compared with the experimental measurements during both the retraction segment and the zero-stress creep

  Fig. Ⅴ.8. Pole figures of the crystal orientation at different macroscopic strain levels.

  figures highlights the competition between this isochoric plastic mechanism and the voiding mechanism in the amorphous phase. The internal intermolecular and network stresses are accumulated in the amorphous phase in different extents, the former one being the most affected by the presence of voids. As illustrated by Fig.Ⅴ.12, the progressively accumulated damage modifies also the local stretches, especially upon the non-monotonic sequence.

  eccentricities of the inner and outer ellipsoidal shapes. Since the inner and outer ellipsoids are confocal, these eccentricities are related to f and S by:

  Fig. Ⅵ.1. Typical tensile experimental result of a PVDF (containing 0.70 of crystals initially in the α-form) strained up to very large strains in terms of stress as a function of strain along with X-ray plots showing both strain-induced texturing and α → β phase transition; strain rate of 10 -3 s -1 and temperature of 363 K (Defebvin et al., 2016).

  Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 138

Fig

  Fig. Ⅵ.1 provides an illustrative experimental example extracted from the work of Defebvin et al. (2016) of a PVDF stress-strain curve up to very large strains along with the microstructural evolution during straining showed by X-ray plots. The mechanical response is accompanied by the progressive micro-mechanism of α → β phase transition until complete transformation and the internal microstructure ordering in terms of crystallographic texturing and amorphous molecular network orientation.

  The strain-induced morphological anisotropy is considered in the continuum-based model while the physics of the deformation modes of the different phases is described by formulating an elasto-viscoplasticviscohyperelastic coupling. Molecular dynamics simulations are employed to examine the respective mechanical properties of α-PVDF, β-PVDF and amorphous phase. The elastic constants of the distinguishing phases and the onset of α → β phase transition, when α-PVDF is drawn, are then used as direct inputs into the continuum-based model. The intermolecular constants of the amorphous phase are also obtained thanks to molecular dynamics data. The other parameters of the continuum-based model are determined using the stress-strain experimental data of a PVDF deformed over a large-strain range. The effect of the progressive Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 140 variation in the mesostructural properties caused by the straining on the piezoelectric properties is finally observed.

x=

  is an arbitrary vector in the plane of the interface spanned by the unit vectors 01 ne the unit normal vector at the interface in the reference configuration (Fig. Ⅵ.2).

  Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 142 Considering a set of I N =125 inclusions in the aggregate, the macroscopic Cauchy stress tensor σ in the semi-crystalline polymer is obtained by the averaging homogenization: initial inclusion volume fraction and i I J is the inclusion volume change. While the aggregate is subjected to a macrostrain, the α → β phase transition is considered in each individual inclusion i by varying the α-phase degree i f  from its initial value _0 i f  to 0 and the β -phase degree i f  from 0 to its maximum value _ i f  

FC

  . The symbol  means the cross product of the vectors.The monitoring of the orientation of the interfaces allows simulating the texture evolution and thus the strain-induced anisotropy. In what follows, the constitutive equations are specified in order to describe the different micro-mechanisms governing the macroscopic response. Fig. Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 143 Ⅵ.2 provides a rheological representation for each phase, analog of their resistance to deformation. Both α-form and β-form crystals are described by a linear elastic spring in series with a viscoplastic slider. The amorphous phase is described by a hyperelastic spring in series with a nonlinear viscous damper, all in parallel with a linear elastic spring in series with a viscoplastic damper. Ⅵ.2.2. Elasto-viscoplasticity in α-PVDF and in β-PVDF Elasto-viscoplastic deformation occurs both in α-form and β-form crystals. The general elastic stress-strain relationship in α-PVDF and in β-PVDF is given, respectively, by: and C  are the fourth-order elastic stiffness tensors, τ  and  τ are the second Piola-Kirchhoff stress tensors and, e E  and e E  are the elastic Green-Lagrange strain tensors. The stress tensors τ  and  τ are related to their corresponding Cauchy stress tensors σ  and σ elastic volume changes and, e F  and e F  are the elastic deformation gradient tensors coming from the multiplicative decomposition: which F  and F  are the deformation gradients and, in F  and in F  are the inelastic parts. The time derivative of the latter are in in in are the reference strain rates, m  and m  are the strain rate sensitivity Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large--viscoplastic-viscohyperelasticity in amorphous PVDF The intermolecular interactions and the molecular orientation/relaxation process occurring in the PVDF amorphous domain is captured by considering the elasto-viscoplasticviscohyperelastic deformation. The amorphous stress a σ is additively decomposed into an elasto-viscoplastic intermolecular stress and an Arruda-Boyce eight-chain viscohyperelastic network stress: left Cauchy-Green tensor, I is the identity tensor and 1 -is the inverse Langevin function which can be treated by a Padé approximation:

h

  of Eq. (Ⅵ.16) captures barriers to molecular chain segment rotation in the amorphous layers. The term ,0 a  is the pre-exponential constant, G  is the activation energy barrier to molecular mobility, k is the Boltzmann's constant,  is the absolute temperature and s is the shear strength controlling the amorphous chain segment rotation: is the hardening rate given by the following power-law: and n is the intermolecular hardening coefficient. The right formula of Eq. (Ⅵ.16) captures molecular relaxation process in the amorphous layers. The term tensor and  = 0.01 is an added perturbation coefficient avoiding singularity of the right formula at the beginning of loading process.

  which  is the applied strain rate, ref  is the reference strain rate, av  is the Avrami exponent and av K is the phase transformation rate given here in the case of PVDF by the empirical form defined as follows: Nu represents the number density of nuclei.

  Fig. Ⅵ.3(a) displays packing structure in a unit cell of the two crystalline structures. The optimized cell parameters are listed in Table Ⅵ.1. The ReaxFF reactive force field was used.

  Fig. Ⅵ.3. Molecular dynamics model of the three constitutive phases in PVDF: (a) Side and top view of packing crystalline structure in a unit cell for α-phase and β-phase, (b) Single chain and supercell of the amorphous phase structure (Gray, white, and blue spheres represent carbon, hydrogen and fluorine atoms, respectively).Ⅵ.3.1.2. Elastic constants

  Fig. Ⅵ.4. Stress as a function of strain issued from molecular dynamics simulation at the temperature of 298 K for (a) α-PVDF and (b) β-PVDF.

  Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 151 interactions, respectively. Electrostatic charges are determined from pair contributions. In this force field, the carbon atoms in the PVDF can be classified into three types according to the way the atoms are grouped, as shown by the different colors in Fig.

  Fig. Ⅵ.5. Temperature effect of the α-PVDF response along with onset of α → β phase transition issued from molecular dynamics simulation: (a) Stress as a function of strain and (b) critical strain for α → β phase transition.

Fig

  Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 152

Fig

  Fig. Ⅵ.6. Stress as a function of strain issued from molecular dynamics simulation for the amorphous PVDF: (a) Temperature effect (at the strain rate of 10 10 s -1 ) and (b) strain rate effect (at the temperature of 363 K).

  Fig. Ⅵ.7. Amorphous PVDF response at the temperature of 363 K: (a) Stress as a function of strain at different strain rates issued from continuum-based simulation in comparison to the relaxed result issued from molecular dynamics (MD) calculations and (b) transversal strain as a function of longitudinal strain issued from MD calculations. Due to the presence of fluctuations, the yield point determination follows the criterion presented in Fig. Ⅵ.7(a); it was obtained as the intersecting point of two lines (shown in dashed lines in Fig. Ⅵ.7(a)), one follows the linear elastic domain and the other one corresponds to the slope of the post-yield hardening domain. To complete the characterization of the elastic response of the amorphous phase (assumed to be isotropic), the amorphous Poisson's ratio was obtained from the slope of the transversal strain-longitudinal strain curve issued from molecular dynamics calculations as shown in Fig. Ⅵ.7(b).

  Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 154 (2015)) of a semi-crystalline PVDF strained up to very large strains. The authors examined the temperature-dependent tensile deformation and the strain-induced α → β phase transition in PVDF (containing 0.70 of crystals initially in the α-form) by means of in situ SAXS/WAXS experiments. The phase transformation occurred in the whole temperature range between glass transition temperature and melting temperature. Nonetheless, brittle or non-homogeneous ductile behaviors were not exploited and only homogeneous material data were employed for the model identification. The data were obtained at the strain rate of 10 -3 s -1 and the temperature of 360 K over a large-strain range. The experimental data are reported in Fig.Ⅵ.1 in which are also reported X-ray plots at different strain levels showing a complete α → β phase transition at the maximum strain reached.

  by molecular dynamics are introduced as direct inputs into the continuum-based model. The identification of the inelastic and α → β phase transition mechanism parameters of the model was carried out using the large tensile deformation data of[START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF]. The values of the identified parameters for each part of the model are listed in TableⅥ.4.In accordance with the actual material, the initial microstructure is modeled with composite inclusions including fully α-form crystals. The critical strain for α → β phase transition taken from the molecular dynamics results (Fig.Ⅵ.5(b)) are directly used as constraints at the scale of the inclusions. That constitutes a fundamental aspect of this scale bridging method transferring molecular dynamics results to constitutive modeling. The moment when the maximum β-phase degree is reached being known from the X-ray plots (see Fig.Ⅵ.1), the kinetics of the α → β phase transition is designed using the experimental stress-strain data of[START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF]. The crystallographic shear parameters are assumed identical for the αform and the β-form. The pre-exponential factors are taken identical for all the phases, to the structure of the constitutive representation of the amorphous layers, a deterministic scheme is used to identify separately the parameters of the viscoplastic intermolecular part and the viscohyperelastic molecular network part.Chapter Ⅵ. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range 155

  Fig. Ⅵ.8. Identification of the amorphous viscoplastic intermolecular features using molecular dynamics (MD) data: (a) identification of the initial shear strength 0 s and the activation energy G using the yield stress evolution with strain rate (constructed by taking the quasi-static data (at 10 -8 s -1 ) from the MD curve and the data at 10 -3 s -1 by inverse identification using the experimental curve) and (b) identification of the hardening coefficient

  data ofFig. Ⅵ.8(b) in terms of logarithmic stress-logarithmic inelastic strain by reformulating the left formula of Eq. (Ⅵ.16) as follows: stress in the post-yield region at small/moderate strains.The parameter identification exercise ends with the viscohyperelastic molecular network parameters of the amorphous phase, were identified using a standard optimization procedure providing the best fit of the strain-hardening level at large strains. Fig. Ⅵ.7(a) provides the strain rate effect on the individual amorphous post-yield response computed from the continuum-based simulations. Interestingly, a reasonable agreement 0

ⅥC.

  Fig. Ⅵ. 9. Continuum-based simulation of the PVDF response (containing 0.70 of crystals initially in the α-form): (a) Stress as a function of strain (solid line: model result, triangular symbols: experimental data) and (b) strain-induced α → β phase transition; strain rate of 10 -3s -1 and temperature of 363 K.Ⅵ.3.3.2. Simulations vs. experiments

  Fig. Ⅵ.11(a). The evolution of the piezoelectric strain coefficient interacts closely with the micro-mechanism of α → β phase transition as highlighted by the effect of the maximum βphase amount whereas slight effects are found on the overall mechanical response (Fig. Ⅵ.11(b)). This clarifies the relation between large-strain mesostructural properties and piezoelectric activity evolution. An illustrative example of the coupled electromechanical behavior is provided in Fig. Ⅵ.12(a). The simulations are performed under different applied electric fields while stretching-retraction sequences are applied at two pre-determined strain levels. Although only the β-phase is polarized, the internal stresses of the two other phases are also affected by the external electric loading as illustrated in Fig. Ⅵ.12(b) in reason of the mechanical coupling between the different phases.

  Fig. Ⅶ.1. Multiscale modeling rubber network connecting the molecular-level description and the continuum-level mechanics using a rubber network theory adapted from Kroon (2011) including non-affine deformation of the molecular network and topological constraint effects.Ⅶ.2. Kroon model for rubber biaxial mechanics

e

  denote basis vectors in a Cartesian coordinate system. Generally, to simplify the calculation, material incompressibility is assumed (i.e.2 3 1 IJ == ) and the deformation gradient tensor F may take the following form:

  The rubber network structure is treated as a collection of chains using the coarse-grained model considering each methyl group (i.e. the C5H8 monomer, Fig. Ⅶ.2(a)) as single beads with a force field that determines the interactions between beads. A single polymer chain consists of beads connected by potential energy determined as the sum of energies for bonded and non-Construction of molecular-level model systems: (a) Chemical structure (monomers) and (b) a full-atomic model and coarse-grained model.The term bonded U includes specific stretching and bending terms to describe 1-2 and 1spring constants for bond stretching and angular motion and, 0 r and 0  are the equilibrium bond length and angle, respectively.The non-bonded (van der Walls) interactions are approximated by a Lennard-Jones-12-6 potential:Coarse-grainedChapter Ⅶ: Kroon network theory verification for rubber biaxial mechanics via a multi-

  Fig. Ⅶ.3. Relaxation of the molecular structure: (a) Total energy evolution during the equilibrium process and (b) screenshot of equilibrated structure in VMD. The model consists of 120 chains with a long chain length of 400 beads (48000 beads in total) to simulate the entangled chain structure (Fig. VII.2(b)). The relaxation of the molecular structure involves four different steps. Initially, the simulation ran for 10 6 timesteps (∆t = 1 fs)using NVT (constant number of atoms, volume, and temperature) dynamics at 600 K followed by relaxation for different timesteps (∆t = 1 fs) for getting enough relax structures at 600 K.The next relaxation relaxes the new molecular structure at 600 K, NPT (constant number of

  Fig. Ⅶ.5. Local and global responses of the molecular model under uniaxial stretching: (a) Stretch ratio of individual chains at a stretch ratio of 1 and (b) stress-strain curves along the three main directions.

  Fig. Ⅶ.7. Stress relaxation after (a) uniaxial stretching at different strain levels and (b) biaxial stretching at different biaxial strain ratios. In order to identify the equilibrium response of the molecular model, the boxes were maintained in the deformed state for different prescribed strain levels in order to provoke the stress relaxation. A sufficient holding time was used to reach the stabilized relaxed-response. At each relaxation, all information about the extended polymer in the box were recorded such as velocity, position and virial stress of every atom. The uniaxial relaxed-stress is plotted in Fig. VII.7(a). The relaxed-stress was quantified for different biaxial strain ratios and the results are reported in Fig. VII.7(b).

  Fig. Ⅶ.8. Molecular dynamics data vs. continuum-based model: (a) Uniaxial stress-strain response, (b) biaxial stress-strain curve in x-direction, (c) biaxial stress-strain curve in ydirection for different biaxial strain ratios; Symbols with standard deviation: dynamics molecular data, Solid lines: Kroon continuum-based model (n=13, cc=2.2, cnc=0.15, α=8 and ccon=1), Dashed lines: Arruda-Boyce continuum-based model (n =13, cc=2, cnc→, α =2.2 and ccon=0).

  on the coefficient of thermal expansion. The reinforcement mechanisms were studied at the macroscale through the mechanical behavior and at the nanoscale by analyzing the local interactions. The attraction of fullerenes to cis-PI atoms and the aggregation configuration of cis-PI atoms around fullerenes were analyzed by radial distribution function, introducing local number density and potential mean force.Besides, the finite-strain mechanical response and deformation processes of various polymer systems were analyzed by means of microstructure-based models formulated in continuum terms. The highly nonlinear and rate-dependent mechanical response of semi-crystalline polyethylene systems covering a wide spectrum of the crystallinity was firstly investigated within a multi-scale homogenization-based approach. The model ability to capture the largestrain response variation with the crystallinity was criticized. The model exhibited weaknesses in small-strain region but it was able to reproduce, within large-strain region, the rate-dependent monotonic response variation with crystallinity. The general trends of the high-crystalline material cyclic response were quantitatively well reproduced by the model while the deviation from experiments increased with amorphicity, which would require to take into account the amorphous network alteration. In addition to molecular orientation effects that takes place as the deformation proceeds, the cavitation damage associated with progressive nucleation and growth of cavities in the amorphous layers was presented. The synergic effects of crystal texturing and anisotropic cavitation damage on the plastic deformation behavior of high-density polyethylene were integrated into a constitutive model formulated within a multi-scale homogenization-based approach. By comparing simulations and experimental data, the model ability to capture large-strain macroscopic stress and inelastic dilatation was shown. The proposed multi-scale approach provides a good approximation of the intrinsic response of highdensity polyethylene and the basic underlying inelastic mechanisms by considering strain rate and stretching temperature effects. Besides, the role of cavitation damage in overall and local responses was observed to better understand its importance under stretching and stretchingretraction conditions. We have then investigated the α → β phase transition in semi-crystalline polyvinylidene fluoride (PVDF) within a hierarchical multi-scale analysis coupling molecular dynamics simulations in atomistic and microstructure-based modeling in a continuum term. We have presented a continuum-based model formulated within a micromechanical framework to connect the PVDF elastic-viscoplastic-viscohyperelastic deformation behavior to the α → β phase transition caused by the straining. The identification of the developed micromechanics model was performed thanks to a rigorous deterministic scheme using relevant results issued from molecular dynamics simulations performed in this work and available experiments.Quantitative comparisons showed that the model was able to correctly capture the large-strain stress-strain response of a PVDF under uniaxial straining accounting for the strain-induced α → β phase transition. Besides, the model was used to clarify the influence of the microstructure evolution, in terms of α → β phase transition and internal ordering, on the piezoelectric activity.The continuum-based model contains very few parameters and allows to better understand the relation between key PVDF microstructure features (strain-induced crystallographic texturing and α → β phase transition) and mechanical response. Although it appears as a valuable tool for the reliable design and optimization of multi-functional piezoelectric PVDF-based material media, a complete verification of the model capabilities under more complex mechanical loading conditions remains an important issue for further investigations. Finally, we check the validity of a hyperelastic model to provide good estimates of the behavior of a rubber model system under different biaxial loading paths by means of the molecular dynamics simulation method.

Table Ⅰ

 Ⅰ

	Structure ID	Chain length	Number of chains	Number of total beads
	Structure 1	1000	20	20000
	Structure 1-i		80	80000
	Structure 1-ii		160	160000
	Structure 2	1000/100	10/100	20000
	Structure 2-i		80/800	160000
	Structure 3	100	200	20000
	Structure 3-i		800	80000
	Structure 3-ii		1600	160000

.1. Specific parameters of the three initial structures.

Table Ⅰ .

 Ⅰ 2. Functional form of force field and potential parameters used for MD calculations.

	Interaction	Form		Parameters
	Bond Length Bond Angle	𝐸 𝑏𝑜𝑛𝑑 = 𝐸 𝑎𝑛𝑔𝑙𝑒 =	1 2 1 𝐾 𝑏 (𝑟 -𝑟 0 ) 2 2 𝐾 𝜃 (𝜃 -𝜃 0 ) 2	𝐾 b = 700 kcal mol ⁄ 𝐾 𝜃 = 100 kcal mol rad 2 , 𝑟 0 = 1.53Å ⁄ ⁄ , 𝜃 0 = 109.1

∘

Dihedral Angle 𝐸 𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 = ∑ 𝐶 𝑖 (cos 𝜙) 𝑖

Table Ⅰ .

 Ⅰ 3. Parameter setting for five steps during the relaxation.

	Process		Ensemble	Temperature (K) Time step (fs)	Steps	Time (ns)
		Warm	NVT	0-500	1	1000000	1
		1	NVT	500	1	15000000	15
	Relaxation	2	NPT (aniso)*	500	0.5	20000000	10
		3	NPT (aniso)*	500-𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	14000000	7
		4	NPT (aniso)*	𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	24000000	12

Table Ⅱ .1. Three

 Ⅱ branched polyethylene systems.

		Chain length	Chain number
	Main chain	1000	10
	Branch structure 1	10	100 (on each chain)
	Branch structure 2	10	10 (on each chain)
	Branch structure 3	100	10 (on each chain)

Table Ⅱ .2. Functional

 Ⅱ form of force field and potential parameters used for MD calculations.

	Interaction	Form			Parameters
	Bond Length Bond Angle	𝐸 𝑏𝑜𝑛𝑑 = 𝐸 𝑎𝑛𝑔𝑙𝑒 =	1 2 1 𝐾 𝑏 (𝑟 -𝑟 0 ) 2 2 𝐾 𝜃 (𝜃 -𝜃 0 ) 2 4		𝐾 b = 700 kcal mol ⁄ 𝐾 𝜃 = 120 kcal mol rad 2 , 𝑟 0 = 1.53Å ⁄ ⁄ , 𝜃 0 = 109.3 ∘
	Dihedral Angle 𝐸 𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 = ∑ 𝐶 𝑖 (cos 𝜙) 𝑖 𝑖=0 Non-bonded 𝐸 𝑛𝑜𝑛-𝑏𝑜𝑛𝑑 = 4𝜀 [( 𝜎 𝑟 ) 12 -( 𝜎 𝑟	6 )	] , r ≤ 𝑟 𝑐	𝐶 0 = 1.736, 𝐶 1 = -4.49, 𝐶 2 = 0.776, 𝐶 3= 6.99 , 𝐶 4= 0 (kcal mol ⁄ ) σ = 4.01Å, ε = 0.112 kcal mol ⁄
	Ⅱ.3

. Results and discussion Ⅱ.3.1. Uniaxial stretching Table Ⅱ.3. Parameter setting for four steps during the relaxation.

	Chapter Ⅱ. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene from
			molecular dynamics
	temperature and pressure. Periodic boundary conditions were employed in all three directions
	Process	Ensemble	Temperature (K) Timestep (fs)	Steps	Time (ns)
	1	NVT	500	1	2000000	2
	2	NPT (aniso)	500	0.5	2000000	1
	3	NPT (aniso)	500-𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	500000	0.25
	4	NPT (aniso)	𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	3000000	1.5
	The large-scale molecular dynamics code LAMMPS was used here to minimize the energy of
	the initial structure followed by an equilibration sequence before mechanical loading of
	polyethylene. The minimize relaxes any high energy configurations that crested during the
	Monte Carlo self-avoiding random walks. The structures after minimize step of each
	polyethylene model were subjected to a four-stages equilibration for final equilibrium state, as
	shown in Table Ⅱ.3. The Nosé-Hoover Thermostat and Barostats were applied to control
			37

Table Ⅱ

 Ⅱ

		.4. Parameter setting during the deformation.
	Process	Ensemble	Temperature	Time step (fs)	Strain rate
	Tensile	NPT	𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	1	10 10 s -1

Table Ⅱ

 Ⅱ

	.5 gives the Young's modulus in the elastic regime. It can be seen that Structure 1 and
	Structure 2, with the same length and different number of branched chains, have similar Young's
	modulus within the error range [±5]. Structure 3 has a lower Young's modulus than others,
	indicating that the elastic modulus decreases when the branched chains are longer, what means
	that short branch chains promote crystallization of the molecule. Fig. Ⅱ.2 (b) shows the stress-
	strain curves of the three structures at a temperature of 100 K. For short branch structures,
	Structure 1 which contains larger number of branch chains has lower hardening strength than
	Structure 2, but the resistance to fracture is higher (higher fracture strain). For long branched
	structures (Structure 3), the hardening strength is stronger than that of short branch chains
	during the initial hardening phase, but the resistance to fracture is lower and the fracture strain
	is smaller.
	Table Ⅱ.5. Young's modulus of three structure at 100 K.
	Structure	Structure 1	Structure 2	Structure 3
	Young's modulus (MPa)	1900	1905	1847

Table Ⅱ .6. Four shape memory steps.

 Ⅱ

	step	Ensemble	Temperature (K)
	Cool down	NVT	T deformation → 100 K
	Stress relaxes	NPT	100 K
	Warm up	NPT	100 K → 450 K
	Shape memory	NPT	450 K
	Fig. Ⅱ.4(a) displays the stress evolution of branched polyethylene, which shows that the stress

Table Ⅱ .7. Parameter

 Ⅱ setting for four steps during the relaxation.

	Process	Ensemble	Temperature (K) Timestep (fs)	Steps
	1	NPT (aniso)	450-𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	1250 steps/K
	2	NPT (aniso)	𝑇 𝑑𝑒𝑠𝑖𝑟𝑒𝑑	0.5	3000000

Table Ⅲ

 Ⅲ

	Crosslink degree	0.3	1.61	2.3
	C 1 *	32.367	33.6514	39.5154

.3. C1 for different crosslink degrees.

* The Neo-Hookean model is the simplest form among polynomial models. Its strain energy density function is presented as:

  The initial moduli of natural rubber were obtained as 478.7 MPa for crosslink degree=0.3, 899.2 MPa for crosslink degree=1.16 and 1332.8 MPa for crosslink degree=2.3 at a strain rate of 10 10 s -1 , and 252.2 MPa for crosslink degree=0.3, 458.3 MPa for crosslink degree=1.16 and 699.5 MPa for crosslink degree=2.3 at a strain rate of 10 9 s -1 . A linearized

	rubber.
	2.94 phr
	7.83 phr
	11.74 phr
	26.42 phr

Table III .

 III 4. C1 for different nanofiller fractions.

	Nanofiller fraction	0	11.74	26.42
	C1 39.5154	44.6596	58.2745

Strain rate (s -1 )

	Chapter Ⅲ. Physical and mechanical properties of filled vulcanized rubber by molecular dynamics
	simulations
	(c)
			0.97 phr
			2.94 phr
			11.74 phr
			26.42 phr
	10 -2	10 2	10 6	10 10
	62

Table Ⅳ

 Ⅳ

	Slip system	g 	(MPa)
	(100)[001]		8
	(010)[001]		20
	{110}[001]		20
	(100)[010]	13.3
	(010)[100]		20
	{110}<11 ̅ 0>	17.6

.1 Slip systems and associated shear strength

(Lee et al., 1993a)

.

Table Ⅳ .3

 Ⅳ Physical characteristics of the three polyethylene systems: long period

										L ; mean
										p
	crystalline thickness	c t ; melting enthalpy of a perfect polyethylene crystal	H 	f	,0	; crystal
	weight fraction	f ; crystal volume fraction	f .
			cw					c
	Material	p L (nm)	c t (nm)	H 	f	,0	(J/g)	cw f (%)	c f (%)
	Experimental method	SAXS	SAXS				DSC		DSC	DSC
	UL	13	2			~ 290.3 17.4 ± 0.3 15.1 ± 0.2
	LL	14	4.2			~ 299.3 33.3 ± 0.8	30 ± 0.7
	HD	26.5	19.2			~ 296.5 75.5 ± 0.8 72.4 ± 0.8
	The crystal and amorphous volume fractions,

Table Ⅳ

 Ⅳ

		.4 HD model parameters.
	Crystalline	Reference strain rate Strain rate sensitivity	𝛾̇𝑐 ,0 m	10 -6 (s -1 ) 0.1355
	Amorphous	First Lame's constant	λ	100 (GPa)
		Second Lame's constant	μ	200 (MPa)
		Reference strain rate	𝛾ȧ ,0	10 -6 (s -1 )
		Activation energy	ΔG	4.03×10 -20 (J)
		Initial shear strength	s0	14 (MPa)
		Intermolecular hardening	n	193
		Interaction coefficient		19.6
		Hardening modulus	Ch	1.2 (MPa)
		Chain length	N	26
		Molecular relaxation	D	7.8×10 28 (MPa -1 s -1 )
		Molecular relaxation	Q/R	25.5×10 3 (K)

Table Ⅳ .

 Ⅳ 5 UL and LL model parameters (all other parameters are listed in Table4).

	Material	λ (GPa)	μ(MPa)		Ch (MPa)	N
	UL	1	2	0.28	0.24	42
	LL	16.6	33	1.4	0.8	42

Table V .

 V 2. Identified model parameters.

	Crystalline	Reference strain rate		c	,0	(s -1 )	10 -6
		Strain rate sensitivity	m		0.1355
	Amorphous	First Lame's constant	λ	(MPa)	1938 (25 ℃)	1615 (40 ℃)	1454 (45 ℃)
		Second Lame's constant	μ	(MPa)	425 (25 ℃)	354 (40 ℃)	319 (45 ℃)
		Reference strain rate		a	,0	(s -1 )	10 -6 (s -1 )
		Activation energy	G 	(J)	7.44×10 -20
		Initial shear strength	s0	(MPa)	38 (25 ℃)	29 (40 ℃)	27 (45 ℃)
		Intermolecular hardening	ss	(MPa)	37 (25 ℃)	28 (40 ℃)	25 (45 ℃)
		Interaction coefficient	h		60 (25 ℃)	20 (40 ℃)	10 (45 ℃)
		Network hardening	Cv	(MPa -1 s -1 )	2.2×10 -8
		Network viscosity	Ch	(MPa)	2.8 (25 ℃)	0.75 (40 ℃)	0.7 (45 ℃)
		Entanglement density	N		26
	Damage	Initial void volume fraction	f0		0.001
		Initial void shape parameter	S0		0
		Aspect ratio parameter	αω		4
		Interaction parameter	q0		4 (25 ℃)	1 (40 ℃)	0.1 (45 ℃)
		Standard deviation	sN		0.95 (25 ℃)	0.35 (40 ℃)	0.25 (45 ℃)
		Mean nucleating strain	γN		0.2(25 ℃)	0.65 (40 ℃)	0.86 (45 ℃)

  scale plastic-damage model for strain-induced morphological anisotropy in semicrystalline polyethylene

	2 '' 2 2 22 1 21 11 22 33 11 ˆ4π 4π π 22 π 22 a a a a I a I I a I QQ	1 1 a a
	3 11 3 3 2 3 ' 44 3 ˆ4π ' 33 1-π 2 2 4π a a a I	3	(		2 1	2	2	)	12 aa
	' 55 ˆ1-	( 11 ) 2 3 2 2 ' 2 3 66 3 1-4π 4π a a	3	(	2 1		3	2	)	1 3 a a
	'' 12 21 ˆˆ4 1
		129

Table Ⅵ .

 Ⅵ 1. Optimized cell parameters of nine crystalline forms of PVDF. The α, β and γ are angles between cell vectors a and b, b and c and, a and c, respectively.

		a (Å)	b (Å)	c (Å)	α (°)	β (°)	γ (°)
	α-phase	4.45868	8.44829	4.54098	90	90	90
	β-phase	7.85031	4.94145	1.92177	90	90	90

Table Ⅵ .

 Ⅵ 2. Elastic constants of α-PVDF calculated in this work by molecular dynamics simulation in comparison with other theoretical results of the literature obtained by densityfunctional theory (values in GPa). Table Ⅵ.3. Elastic constants of β-PVDF calculated in this work by molecular dynamics simulation in comparison with other theoretical results of the literature obtained by densityfunctional theory (values in GPa).

	This work	_11 17.169 15.916 145.25 C  _ 22 C  _ 33 C 	_ 44 1.34 C 	_ 55 3.877 C 	_ 66 3.47 C 	_12 10.203 3.003 C  _13 C 	_ 23 1.14 C 
	Pei and Zeng (2011)	16.10	15.70	148.1	0.5	3.9	6.5	6.2	8	0.8
	This work	_11 16.089 20.554 304.64 C  _ 22 C  _ 33 C 	_ 44 3.41 C 	_ 55 5.078 C 	_ 66 6.778 11.486 11.766 21.266 C  _12 C  _13 C  _ 23 C 
	Pei and Zeng (2011)	19.83	24.50 287.33	1.17	0.33	1.67	4.33	0.83	3.33

Table Ⅵ .

 Ⅵ 4. Continuum-based model parameters.

	Crystalline	Initial α-phase amount	f 	_0	0.7
		Reference strain rate	,0  = ,0   	10 -6 (s -1 )
		Strain rate sensitivity	m  = m 	0.1355
	Amorphous	Young's modulus	a E	600 (MPa)
		Poisson's ratio		a	0.4
		Reference strain rate		a	,0	10 -6 (s -1 )
		Activation energy	G 	7.44×10 -20 (J)
		Initial shear strength	0 s	19 (MPa)
		Intermolecular hardening	n	2.8
		Hardening modulus	h C	2.6 (MPa)
		Chain length	N	25
		Viscosity coefficient	v C	2.42×10 -6 (MPa -1 s -1 )
	Phase transformation	Reference strain rate		ref	0.001 (s -1 )
		Maximum β-phase amount	_ f  	0.7
		Avrami exponent		av	2.2
		Number density of nuclei	Nu	3.88×10 9

Table Ⅶ .

 Ⅶ 1. Potential parameters used for coarse-grained calculations.

:

  Polymer systems; Molecular dynamics; Continuum mechanics; Mechanical properties; Interface; Strain-induced mechanisms.

This chapter is based on the following paper: Yan, Z., Zaoui, A.,Zaïri, F.,

Crystallization and mechanicalbehavior of semi-crystalline polyethylene. Physica Scripta 96, 125729.

This chapter is based on the following paper: Yan, Z., Zaoui, A., Zaïri, F., 2022. Uniaxial stretching and shape memory behavior of branched amorphous polyethylene. Chinese Journal of Physics 79, 6-12.

This chapter is based on the following paper: Yan, Z., Zaoui, A., Zaïri, F.. Physical and mechanical properties of filled vulcanized rubber by molecular dynamics simulations. Under review.

Lamellar surfaces in polyethylene are of the {h0l}-type where the angle between the chain direction and the normal of interface varies between 20° and 40°(Bassett and Hodge, 1981). The initial angle between chain direction and normal of interface is set to be 35°.

This chapter is based on the following paper: Yan, Z.,[START_REF] Yan | A multi-scale plastic-damage model for straininduced morphological anisotropy in semi-crystalline polyethylene[END_REF]. A multi-scale plasticdamage model for strain-induced morphological anisotropy in semi-crystalline polyethylene. International Journal of Non-Linear Mechanics 147, 104238.

The multiplicative decomposition of the deformation gradient implies that the unloading process (relating the deformed configuration with the relaxed configuration) is not uniquely defined. Indeed, an arbitrary rigid body rotation of the relaxed configuration still leaves the relaxed configuration stress free. One way to make unique the unloaded configuration, with no loss in generality, is to assume the inelastic flow irrotational(Gurtin, et al.,2005).

This chapter is based on the following paper: Yan, Z.,Zaïri, F., Zaoui, A., 2023. Multiscale modeling of the strain-induced α → β phase transition in semi-crystalline poly(vinylidene fluoride) over a large-strain range. Mechanics of Materials, accepted.

This chapter is based on the following paper: Yan, Z., Zaïri, F., Zaoui, A.. Kroon network theory verification for rubber biaxial mechanics via a multi-scale modeling. Under review.

Acknowledgements

 in terms of the variables f and S . The parameters  ,  and C are given by:

)