Nabil Mr

Génie Industriel

Cmp Augusto

Vincent Cr

Stéphane Pr2

Pierre Badel

Ma

Balbo Flavien

Jean-François Bassereau

Batton-Hubert Mireille

Appliquees O Mathematiques

Maître-Assistant Roustant

O Informatique

Professeur Boissier

J C Formes

Professeur Genie Pinoli

N Industriel

Maitre Absi

Mohsen Nakhaei

ABSI

Keywords: . . . . . . . . . . . . . implementing . . . . . . . 5.4 FEM solver validation test cases . . . . . . . . . . . . . . . . . . Continuum micromechanics, Homogenization, Large strain elasticity, Soft tissue, Constitutive modeling, Nonlinear nite element method

solving the problem . . . . . .

5.3.4

Finite element discretization . . . .

I am also grateful to the rest of the wonderful committee members, Prof.

Christian Hellmich, Dr. Daniel Georg, Prof. Pierre Badel, and Prof. Giuseppe

Vairo, for their insightful comments and suggestions to this thesis work. I have been greatly inspired by the discussion with them.

I would like to thank my lab mates at École des Mines de Saint-Étienne and also at Boston University, too many to list here, who have been excellent sources of inspiration and knowledge.

And lastly, I would like to thank my family for all of their supports, without their unconditional love, encouragement, and spiritual support, this would not have been possible.

Thank you all.

List of Figures

1.1

The hierarchical structure of arteries. (A) macroscopic artery of a mouse, taken from [START_REF] Joseph T Keyes | Adaptation of a planar microbiaxial optomechanical device for the tubular biaxial microstructural and macroscopic characterization of small vascular tissues[END_REF]; (B) the three layers of arterial wall, the adventitia tunica (A), the tunica media (M) and the intima tunica (I) from [START_REF] Paul | Mechanics of vascular smooth muscle[END_REF], by means of an electron micrograph; (C) the adventitia is made of an arrangement of collagen bundles (red)

and elastin bers (green) taken from [START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF], by means of multiphoton microscope; (D) the media is made of an arrangement of lamellae sheets and interlamellar space [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]; (E) the collagen bundles are made of an arrangement of collagen brils, as taken from Elastin bers of rat aortic adventitia, magnication ×5, 500

2.9

Numerical algorithm for the multi-scale homogenization scheme. . .

3.1

Multiphoton microscopic imaging of bers in the adventitia [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF].

(A) collagen bundles; (B) elastin bers. . . . . . . . . . . . . . . .

3.2

Elastin ber and broblast (F) in aortic adventitia of rat, magnication ×1, 700 [START_REF] Tatsuo | Scanning electron microscopic studies of tissue elastin components exposed by a koh-collagenase or simple koh digestion method[END_REF]. . . . . . . . . . . . . . . . . . . . . . .

3.3

Orientation of a ber in the arterial wall. θ represents the ber inclination with respect to the axial direction of artery, and φ its direction in the radial-circumferential plane. . . . . . . . . . . . . .

Micromechanical RVEs representing the adventitia. (A) RVE

representing the adventitia tissue scale; (B) RVE representing the collagen bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Cross section of a collagen bundle of a mouse adventitia [START_REF] Jerey R Tonniges | Collagen bril ultrastructure in mice lacking discoidin domain receptor 1[END_REF]. . . .

3.6

Representative straight form and crimped form of collagen bundle and corresponding angles. . . . . . . . . . . . . . . . . . . . . . . .

3.7

Macroscopic mechanical response of adventitia, (a) human coronary artery according to [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF] and (b) human thoracic aorta according to [START_REF] Weisbecker | Layer-specic damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening[END_REF] . The free parameter θ b-ad and R 2 are reported for each tissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . experiment carried out by [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] for whole carotid artery sample, taking into account the media as a linear elastic material with dierent Young's moduli E med , 0.1, 0.5 and 1 MPa. . . . . . . . .

3.11

Comparison of the ber kinematics with the experiments carried out by [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] for whole carotid artery sample. . . . . . . . . . . . . .

4.4

Orientation of a ber in the arterial wall. θ represents the ber inclination with respect to the axial direction of artery, and φ its direction in the radial-circumferential plane. . . . . . . . . . . . . .

4.5

Macroscopic mechanical response of the media human coronary artery according to [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF], (a) θ c-med = 85

5.5

The adventitia RVE is remotely subjected to traction forces at its remote boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

The inner radius-pressure curve observed in mesh convergence study of a cylinder having a thickness of H =0.01mm. . . . . . . .

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie LIST OF FIGURES

5.7

Comparing the inner radius-pressure curve from FEM and analytical solution of a thin wall cylinder with dierent wall thicknesses (H =0.01mm, H =0.025mm, and H =0.05mm),

considering ve elements across the thicknesses. . . . . . . . . . . .

5.8

Schematic of ination test for an artery. . . . . . . . . . . . . . . .

5.9

Ination of an artery regarded as a thick-walled cylinder made of adventitia, elastic lamellae and interlamelar space. The initial inner radius of R i = 0.17 mm, and an outer radius of R o = 0.22 mm. . .

General Introduction

Arteries are part of the cardiovascular system which are responsible for circulating blood as a uid which contains the oxygen and nutrients all over the body. The mechanics of arteries is an important topic as the artery is subjected to stresses occurring over a cyclic load (heartbeat), and it is a complex multiphase material containing active elements involved in large deformation. At present, there is still a pressing need to understand better how the mechanical properties of the artery are related to its structure, which can undergo millions of cycled distensions and still maintain a functional blood circulation. Therefore, there has been important eorts in mechanics to provide better insights regarding arterial structure-function relationships, the pathophysiology, and treatment of arterial diseases [START_REF] Jay | Mechanics of the arterial wall: review and directions[END_REF].

The mechanical properties of arteries originate from physical mechanisms (such as ber rearrangement and elasticity) and mechanobiological eects (such as cell mechanosensing mediating tissue remodeling), which occur within the microstructure, and these aspects are signicantly altered in cardiovascular diseases [START_REF] Jay | Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms[END_REF].

The signicant role of the microstructure morphology and composition on arterial response motivated us to develop a detailed multi-scale model of the arterial wall. In this study, we developed a model, which accounts for the universal patterns across dierent scales in the two mechanically signicant layers of arteries, namely the adventitia and the media, and introduce the dierent constituents making up the tissue at these dierent scales as phases into representative volume elements (RVE). Each phase volume fraction and morphology, as well as its ability to undergo load-driven rotations, were identied from literature and post-processing of multiphoton microscopic images [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The framework of nite strain continuum micromechanics [START_REF] Morin | The ber reorientation problem revisited in the context of eshelbian micromechanics: theory and computations[END_REF] was employed in an incremental approach to compute the stress, strain, and phase reorientations, at each load increment. Within the adventitia, two scales were modeled to account for both the bril decrimping (through the progressive bril reorientation at the lowest scale) and bundle realignment (at the macroscopic scale). Within the media, a stack of lamellae and interlamellar space was modeled, and the progressive reorientation of the collagen and elastin bers were accounted for, all reorientation mechanisms being the source of the nonlinear response of the tissue.

We organized this thesis as follows:

• Chapter 1 presents background on arterial mechanics and structure as well as an introduction of the constitutive models. We described the structure of dierent layers of an artery at dierent scales, namely: tissue level (macroscopic), ber level (microstructure), and bril level (nanostructure).

The main focus was on structure-mechanics relation and load-induced microstructural changes as well as the elementary constituents. The chapter ends with an introduction to the constitutive models of arterial tissue and an overview of the thesis.

• Chapter 2 describes the theory of continuum micromechanics under large deformation.

The thermodynamic formulation of the local hypoelastic constitutive relation was developed, and the homogenization techniques were explained. We construct algorithms for applying dierent homogeneous boundary conditions, namely strain rate and stress-driven load on the boundary of RVE. Then, the method was extended to a multi-scale approach for the hierarchical structure of heterogeneous materials.

• Chapter 3 explains the application of the developed framework to model the arterial adventitia layer. The physical parameters of the model were determined through image processing and literature review.

Then, we validated the model for the adventitia layer against the experimental data of the individual layer. A sensitivity analysis was performed to all parameters of the model which exhibit variations in the literature, and we studied their impact on mechanical response and microstructure evolution. Next, the ability of the model to capture the mechanical response of the whole artery as well as ber evolution were investigated considering the arterial media layer as isotropic linear elastic.

• The extension of the model to the arterial media layer is described in Chapter 4. The physical parameters of the model were measured using image processing, and some were determined from the literature. The experimental data of the media layer was employed to validate the arterial media layer individually. Then, we performed a sensitivity analysis to study the impact of the parameters on the model. Finally, we combined the two micromechanical model of the arterial media and adventitia layers to perform a uniaxial simulation of the whole artery and validate the model against the experimental results both in terms of mechanical response and microstructure evolution.

• The nite element implementation of the model and its application to the structural model of arteries are presented in Chapter 5. An updated Lagrangian nite element formulation was employed due to the incremental nature of the developed multi-scale constitutive model. First, we validated the nite element solver for a thin wall cylinder. Then, an internal pressure simulation was performed on the artery to study the mechanical response and microstructure evolution over time. The pathogenesis of many arterial diseases implies dysfunction and loss of the organization of the complex microstructure and components [START_REF] Zenon | The eect of age on the unfolding of elastin lamellae and collagen bers with stretch in human carotid arteries[END_REF][START_REF] Kim | The relationship between mechanical properties of carotid artery and coronary artery disease[END_REF][START_REF] Michael | Mechanical factors in arterial aging: a clinical perspective[END_REF][START_REF] Johnson | Vascular calcication: pathobiological mechanisms and clinical implications[END_REF][START_REF] Lanzer | Medial vascular calcication revisited: review and perspectives[END_REF].

Introduction Générale

As prominent examples, structural changes due to aging, atherosclerosis, and aneurysm result in change of arterial mechanical properties [START_REF] Sylvain Marchais | Arterial compliance and blood pressure[END_REF][START_REF] Zenon | The eect of age on the unfolding of elastin lamellae and collagen bers with stretch in human carotid arteries[END_REF].

Multiscale Structure of arteries

Arterial tissues exhibit a vast diversity and variability among organs and species [START_REF] Sarah | Determinants of mechanical properties in the developing ovine thoracic aorta[END_REF]. Still, the fundamental hierarchical organization of elastic arterial tissues has remained largely unchanged across most arterial tissues [START_REF] Jessica | Vascular extracellular matrix and arterial mechanics[END_REF][START_REF] Faury | Functionstructure relationship of elastic arteries in evolution: from microbrils to elastin and elastic bres[END_REF], and can be described by means of the following morphological patterns:

• The macrostructure, with a characteristic length of 100 µm to 1 mm, features the three concentric layers of elastic arteries, namely the intima, the media, and the adventitia [START_REF] Jessica | Vascular extracellular matrix and arterial mechanics[END_REF], see Fig. 1.1(B).

• Zooming out pieces of any of these three layers reveals that all layers host various biological cells: broblasts in the adventitia (with a characteristic length of 20-30 µm), smooth muscle cells (SMC) in the media (with a characteristic length of 50 µm) [START_REF] Paul | Mechanics of vascular smooth muscle[END_REF][START_REF] Chen | Microstructural constitutive model of active coronary media[END_REF], and endothelial cells in the intima (with a characteristic length of 50 µm).

• The entire domain outside these cells is called the extracellular matrix.

The extracellular matrix is made of structural macromolecules, among which collagen and elastin are arranged hierarchically in networks of brils and bers.

The extracellular matrix appears as a composite with a characterisitic size of 50-200 µm, see Fig. 1.1(C). Within this extracellular space at this scale, dierent families of bers (mainly collagen and elastin bers) can be distinguished. Note that in the adventitia layer the collagen bers are arranged in the form of collagen bundles (see Fig. 1.1(C) in red).

• Finally, the elementary constituents can be distinguished, namely the collagen brils, the elastin, and the ground substance.

The precise organization of each layer will be detailed in subsequent sections. [START_REF] Romo | In vitro analysis of localized aneurysm rupture[END_REF][START_REF] Luo | A multi-modular tensegrity model of an actin stress ber[END_REF][START_REF] Cavinato | Biaxial loading of arterial tissues with 3d in situ observations of adventitia brous microstructure: A method coupling multi-photon confocal microscopy and bulge ination test[END_REF], and tension-ination tests [START_REF] Jtc Schrauwen | A method for the quantication of the pressure dependent 3d collagen conguration in the arterial adventitia[END_REF][START_REF] Genovese | An improved panoramic digital image correlation method for vascular strain analysis and material characterization[END_REF]. They all reveal a highly non linear mechanical response of arterial tissue, as exemplied in Fig. 1.2, [START_REF] Bergel | The static elastic properties of the arterial wall[END_REF][START_REF] Boone | Mechanical properties of arteries[END_REF]. More precisely, at low applied stresses, the arterial tissue can be easily stretched, resulting in a very low stretch-stress slope. When the applied stress is increased, the tissue progressively stiens, resulting in an increasing stretch-stress slope, whose microstructural origin is discussed in next section. Another interesting feature of the mechanical behavior of arteries is its anisotropy, although no clear conclusion could be drawn from the dierent papers [START_REF] Cavinato | Experimental characterization of adventitial collagen ber kinematics using second harmonic generation imaging microscopy: Similarities and dierences across arteries, species and testing conditions[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]]: it appears that the global anisotropy of arteries depends on the tissue location and on the species. It is widely accepted that the elastin ber network bears the load when the latter is small [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF][START_REF] Sarah | Determinants of mechanical properties in the developing ovine thoracic aorta[END_REF]. At larger loads, the progressive recruitment of collagen bundles in the adventitia allows them to bear the load and prevent the arteries from overdistension. Changes in the arterial morphology induce variations in the arterial stiness [START_REF] Haskett | Microstructural and biomechanical alterations of the human aorta as a function of age and location[END_REF].

I II III

Mohsen Nakhaei Thèse en Mécanique et Ingénierie

Role of the brous networks in the adventitia morphology

Two main constituents are known to play an important role in adventitia mechanics, namely the collagen bundles and the elastin bers, which can be tracked by means of multiphoton microscopy, see Fig. 1.1(C) [START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF][START_REF] Zoumi | Imaging coronary artery microstructure using second-harmonic and two-photon uorescence microscopy[END_REF][START_REF] Jtc Schrauwen | A method for the quantication of the pressure dependent 3d collagen conguration in the arterial adventitia[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF].

The latter technique indeed reveals the local orientations of collagen and elastin bers. First, several studies showed the very limited radial inclination of the ber networks within the adventitia [START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF][START_REF] Jtc Schrauwen | A method for the quantication of the pressure dependent 3d collagen conguration in the arterial adventitia[END_REF]. We therefore choose to neglect it also in the sequel. Second, while elastin bers are mainly oriented in the longitudinal direction, the orientation of collagen bundles in the unloaded conguration is more complex to determine due to crimping. Several contributions state that collagen bundles are preferentially aligned along the longitudinal directions [START_REF] Tsamis | Elastin and collagen bre microstructure of the human aorta in ageing and disease: a review[END_REF][START_REF] Chiquet | Collagen xii: protecting bone and muscle integrity by organizing collagen brils[END_REF][START_REF] Antonio D'amore | Characterization of the complete ber network topology of planar brous tissues and scaolds[END_REF],

while others opt for a diagonal orientation (in the longitudinal-circumferential plane) [START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF].

Load-induced morphological changes

The adventitia is characterized by important load-induced morphological changes.

The latter have been the topic of several studies, see e.g. [START_REF] Jtc Schrauwen | A method for the quantication of the pressure dependent 3d collagen conguration in the arterial adventitia[END_REF][START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. They underline the impressive rearrangement capabilities of the collagen bundles:

while highly crimped and apparently disorganised in the unloaded conguration [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF][START_REF] Koert P Dingemans | Extracellular matrix of the human aortic media: an ultrastructural histochemical and immunohistochemical study of the adult aortic media[END_REF], the collagen bundles are progressively recruited, i.e. they unfold, straighten, and progressively align towards the load direction [START_REF] Se Greenwald | Experimental investigation of the distribution of residual strains in the artery wall[END_REF][START_REF] Peter B Canham | Eect of pressure on circumferential order of adventitial collagen in human brain arteries[END_REF]. These features can be observed in uniaxial tension [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF], biaxial tension [START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF], and bulge-ination tests [START_REF] Cavinato | Biaxial loading of arterial tissues with 3d in situ observations of adventitia brous microstructure: A method coupling multi-photon confocal microscopy and bulge ination test[END_REF], although the biaxial character of the applied load limits the reorientation eects [START_REF] Cavinato | Experimental characterization of adventitial collagen ber kinematics using second harmonic generation imaging microscopy: Similarities and dierences across arteries, species and testing conditions[END_REF]. These two reorientation processes of collagen bundles can also be quantied [START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF][START_REF] Jtc Schrauwen | A method for the quantication of the pressure dependent 3d collagen conguration in the arterial adventitia[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] Zeinali-Davarani | Contribution of collagen ber undulation to regional biomechanical properties along porcine thoracic aorta[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF]; two theories are mainly debated:

(i) the reorientation of collagen bundles follows an ane kinematics, i.e. the apparent rotation of the bers follows from the stretching of the tissue, while the bers are rmly embedded in the matrix; (ii) the reorientation kinematics is larger than the ane kinematics and bers have the ability to shear the surrounding Mohsen Nakhaei Thèse en Mécanique et Ingénierie 1.3. THE TUNICA MEDIA 18 matrix to rotate. While the rst theory is a good approximation for restricted deformation (such as the physiological deformation) [START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF], the non-ane kinematics cannot be neglected for larger deformation ranges [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. Conversely, elastin bers exhibit only limited load-induced orientation changes [START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF][START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF].

Finally, the preferential longitudinal orientation of the adventitial ber networks explains the observed anisotropy of the adventitial tissue, namely: the adventitia is stier in the longitudinal direction than in the circumferential one [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF][START_REF] Weisbecker | Layer-specic damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening[END_REF].

1.3 The tunica media They have large, round fenestrations, which serve as anchorages for SMCs [START_REF] Koert P Dingemans | Extracellular matrix of the human aortic media: an ultrastructural histochemical and immunohistochemical study of the adult aortic media[END_REF].

Circumferentially-oriented, wavy collagen bundles run parallel to the elastic lamellae [START_REF] John | Transmural organization of the arterial media. the lamellar unit revisited[END_REF][START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF][START_REF] Lucas H Timmins | Structural inhomogeneity and ber orientation in the inner arterial media[END_REF][START_REF] Roy | Experimental characterization of the distribution of collagen ber recruitment[END_REF][START_REF] Michael R Hill | A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the artery wall[END_REF]. Between these elastic lamellae, the interlamellar space is made of SMCs. SMCs have an elongated shape, about 100-200µm of length and 5-10µm width [START_REF] Jay | Cardiovascular solid mechanics: cells, tissues, and organs[END_REF], which display a radial tilting of about 20

• and are mainly circumferentially oriented [START_REF] Chen | Microstructural constitutive model of active coronary media[END_REF][START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. Along the SMC, type III collagen brils contribute to the SMC cohesion and anchorage on the lamellae [START_REF] John | Transmural organization of the arterial media. the lamellar unit revisited[END_REF]. Thick radial elastin struts provide cohesion by bridging the lamellae layers together [START_REF] Koert P Dingemans | Extracellular matrix of the human aortic media: an ultrastructural histochemical and immunohistochemical study of the adult aortic media[END_REF][START_REF] Lucas H Timmins | Structural inhomogeneity and ber orientation in the inner arterial media[END_REF][START_REF] Ryan G Koch | A custom image-based analysis tool for quantifying elastin and collagen micro-architecture in the wall of the human aorta from multi-photon microscopy[END_REF]. Finally, thin elastin radial bers connect the SMC to both lamellae [START_REF] Raspanti | The extracellular matrix of the human aortic wall: ultrastructural observations by feg-sem and by tapping-mode afm[END_REF][START_REF] Koert P Dingemans | Extracellular matrix of the human aortic media: an ultrastructural histochemical and immunohistochemical study of the adult aortic media[END_REF].

Load-induced morphological changes

Several studies showed that lamellae of the media tend to be crimped in the unloaded conguration, while the load application leads to their progressive uncrimping, see e.g. [START_REF] Wolinsky | A lamellar unit of aortic medial structure and function in mammals[END_REF][START_REF] Yu | Transmural variation in elastin ber orientation distribution in the arterial wall[END_REF].

The SMCs also undergo a load-induced reorientation [START_REF] Chen | Microstructural constitutive model of active coronary media[END_REF]. Still, reorientations occurring in the media are less pronounced than the ones occurring in the adventitia [START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. Besides, the preferred circumferential orientation of the medial ber networks explains why the media is Mohsen Nakhaei Thèse en Mécanique et Ingénierie stier in the circumferential direction than in the longitudinal one [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF][START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF][START_REF] Weisbecker | Layer-specic damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening[END_REF].

The tunica intima

The tunica intima is the inner layer of the arterial wall. The surface close to the lumen is covered by endothelial cells. From a mechanical point of view, the endothelial cells are sensitive to shear stresses applied by the blood ow on the arterial wall. The endothelial cells respond to this mechanical signal through the reorganization of their actin lament structure [START_REF] Ookawa | Changes in the microstructure of cultured porcine aortic endothelial cells in the early stage after applying a uid-imposed shear stress[END_REF]. When subjected to high shear stresses, the endothelial cells are stretched along the blood ow direction, and the actin laments are observed around the cell nucleus [START_REF] Ookawa | Changes in the microstructure of cultured porcine aortic endothelial cells in the early stage after applying a uid-imposed shear stress[END_REF]. Furthermore, the endothelial cells active Nitric oxide (NO) pathway as an endothelium-derived relaxing factor which aects on SMCs [START_REF] Richard | Vascular nitric oxide: formation and function[END_REF].

A dense elastin layer, called the internal elastic lamina (see Fig. 1.4), separates the endothelial cells and the tunica media. Scanning electron microscopy and transmission electron microscopy studies revealed that elastin bers are mostly longitudinally oriented in internal elastic lamina, while, in the media, the elastin bers are mostly circumferentially oriented to resist the circumferential stress generated by blood pulsation [START_REF] Farand | Structure of large arteries: orientation of elastin in rabbit aortic internal elastic lamina and in the elastic lamellae of aortic media[END_REF].

Mohsen Nakhaei Thèse en Mécanique et Ingénierie 1.5 The elementary constituents The Achilles tendon of rats under tensile test.

[10] 100 -800 Bovine Achilles tendon brils under atomic force microscopy hydrated collagen brils [START_REF] Joost | Micromechanical testing of individual collagen brils[END_REF] 350 -650 hydrated collagen brils of Rat tail tendon tensile test [START_REF] Kato | Mechanical properties of collagen bres: a comparison of reconstituted and rat tail tendon bres[END_REF] 410 -1310 hydrated collagen brils of Dermis of sea cucumber tensile test [START_REF] Zhilei L Shen | Stress-strain experiments on individual collagen brils[END_REF] 430 hydrated collagen brils in a bovine Achilles tendon tensile test [START_REF] Sasaki | Stress-strain curve and young's modulus of a collagen molecule as determined by the x-ray diraction technique[END_REF] 1000 -3900 hydrated collagen brils of Bovine Achilles tendon bending experiment by atomic force microscopy [START_REF] Yang | Mechanical properties of native and cross-linked type i collagen brils[END_REF] 1700 -2700 air-dried collagen brils of Rat tail tendon tensile test [START_REF] Kato | Mechanical properties of collagen bres: a comparison of reconstituted and rat tail tendon bres[END_REF] 2000 -7000 Bovine Achilles tendon brils under atomic force microscopy an-hydrated collagen brils [START_REF] Joost | Micromechanical testing of individual collagen brils[END_REF] 3750 -11500 hydrated collagen brils of Rat tail tendon atomic force microscopy [START_REF] Marco Pe Wenger | Mechanical properties of collagen brils[END_REF] Table 1.1: Overview of the range of values for the Young's modulus of collagen bril and ber found in the literature

The mechanical properties of type III collagen are dierent from the ones of type I collagen. According to our knowledge, the only investigation reporting mechanical properties for type III collagen brils relates to human placenta tissue.

More precisely, atomic force microscopy on collagen brils type I and type III from human placenta was performed by [START_REF] Asgari | In vitro brillogenesis of tropocollagen type iii in collagen type i aects its relative brillar topology and mechanics[END_REF] in the hydrated state in a phosphate buered saline solution. They reported Young's modulus for type I collagen brils ranging from 600 kPa to 1.5 MPa and for type III collagen brils ranging from 100 kPa to 300 kPa [START_REF] Asgari | In vitro brillogenesis of tropocollagen type iii in collagen type i aects its relative brillar topology and mechanics[END_REF].

Elastin bers

Another main elementary component of arterial tissue is elastin (see Fig. 1.5), which provides its elasticity to arterial tissue; elastin also ensures that arteries can undergo several million deformation cycles without appreciable fatigue [START_REF] Gundiah | Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests[END_REF].

Regarding the mechanical characterization of elastin, the elastic modulus of intact puried elastic bers from aortas was reported to be 0.3 MPa [START_REF] Burton | Relation of structure to function of the tissues of the wall of blood vessels[END_REF][START_REF] George | Elastic tissue. i. description of a method for the isolation of elastic tissue[END_REF], while the water-swollen, single elastin ber taken from a bovine ligamentum nuchae [START_REF] Aaron | Elastin as a random-network elastomer: A mechanical and optical analysis of single elastin bers[END_REF] Mohsen Nakhaei Thèse en Mécanique et Ingénierie the dierence in Young's moduli could be explained by the method for elastin extraction [START_REF] Gundiah | Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests[END_REF], the autoclaving method, as used in Aaron and Gosline (1981), providing higher stiness values than alkali-treated elastin, as proposed by [START_REF] Burton | Relation of structure to function of the tissues of the wall of blood vessels[END_REF], see Table 1.2 for more details. a dense network of actin-myosin laments in their cytoplasm. Thanks to their properties, smooth muscle cells are responsible for regulating the rapid stress development in the artery as well as for sustaining the imposed stress [START_REF] Paul H Ratz | Regulation of smooth muscle calcium sensitivity: Kcl as a calcium-sensitizing stimulus[END_REF][START_REF] Paul | Mechanics of vascular smooth muscle[END_REF].

SMCs are oriented circumferentially with an inclination of about 20 • in the radial direction [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. The tensile stiness of smooth muscle cells harvested from rat thoracic aortas ranges from 10 kPa in the relaxed condition to 100 kPa in the contracted state [START_REF] Nagayama | Mechanical anisotropy of rat aortic smooth muscle cells decreases with their contraction[END_REF]. In other study the Young's modulus of young and old monkeys were measured by [START_REF] Zhu | Temporal analysis of vascular smooth muscle cell elasticity and adhesion reveals oscillation waveforms that dier with aging[END_REF] as 4 kPa for young samples and 25 kPa for old samples.

Other constituents

These previously described main constituents are embedded in a soft ground substance, containing water, other cells, and proteins such as proteoglycans and glycosaminoglycans. Some mechanical properties of arteries such as viscoelasticity, permeability, and homeostasis have been shown to be related to these molecules [START_REF] Thomas | Cell biology of arterial proteoglycans[END_REF].

State of the art on arterial constitutive models

The art of describing the physical behavior of materials through mathematical formulation according to the physical concepts and experimental observation is called constitutive modeling. A powerful constitutive model plays an important role in the predictivity of structural models. Computational models of biological tissue along with the experimental work accelerated our analysis about dierent organs to improve and speed up the clinical approaches. Recently, the development of numerical simulations increased dramatically to simulate dierent pathologies in order to explain, investigate, and predict the behavior of a pathology. These numerical models assist clinical trials in a rapid and low-cost way, which would reduce the required for expensive experiments [START_REF] Samani | Elastic moduli of normal and pathological human breast tissues: an inversion-technique-based investigation of 169 samples[END_REF][START_REF] Walpole | Multiscale computational models of complex biological systems[END_REF][START_REF] Kitano | Computational systems biology[END_REF][START_REF] Jay | A constrained mixture model for growth and remodeling of soft tissues[END_REF][START_REF] Thomas | Computational pathology: Challenges and promises for tissue analysis[END_REF]. However, these experiments remain an essential key to validate the model's assumptions.

Finally, the validated model can be used to address questions that cannot be solved experimentally at this moment or treat the pathology in a patient-specic manner. Additionally, these models may enable the investigation in a clinical setting when a physician decides whether or not a drug or a surgical procedure should be administered to a patient [START_REF] Nicholas Hg Holford | Simulation of clinical trials[END_REF].

The main focus here is to develop arterial constitutive models, which are able to predict the elastic mechanical behavior of the tissue, although arterial tissue also exhibits inelastic behavior such as viscoelasticity, damage as well as remodeling and growth.

Hyperelastic models

Over the last decades, researchers have developed diverse models employing the framework of hyperelasticity to predict the mechanical behavior of arteries and to relate tissue structure to its mechanical functions [START_REF] Holzapfel | Nonlinear solid mechanics ii[END_REF][START_REF] Stylianopoulos | Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls[END_REF][START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF][START_REF] Chen | Microstructural constitutive model of active coronary media[END_REF][START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF]. These models are computationally ecient and precisely reproduce the macroscopic mechanical responses. However, most models remain phenomenological and do not address physical mechanisms and mechanobiological eects occurring within the microstructure of arteries or link these to the observed macroscopic mechanical responses [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen bre orientations[END_REF][START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] Li | A discrete bre dispersion method for excluding bres under compression in the modelling of brous tissues[END_REF].

Hyperelastic material models are based upon a potential function called strain energy density ψ that physically represents the mechanical energy stored in the body per unit volume of mass. In order to satisfy the objectivity requirement the strain energy density is written as a function of the right Cauchy-green deformation tensor ψ(C C C), (whereby C C C = F F F T F F F and F F F is the deformation gradient).

The thermodynamical requirements for a hyperelastic material where free energy is equal to strain energy yield the relationship between stress and strain.

Moreover, the hyperelastic materials exhibit zero dissipation under deformation, where the dissipation can be derived through Clausisus-Duhem expression [118] as,

D = π π π : 1 2 ∂C C C ∂t - ∂ψ ∂t (C C C(t)) = 0 (1.1)
with π π π, the Piola-Kirchho stress tensor reads as, π π π = 2 ∂ψ(C C C) ∂C C C . In order to use the hyperelastic models for brous soft tissues, in which the bers are embedded in a soft ground substance; the strain energy density function is usually decomposed into two parts, the contribution of the matrix and the contributions of the bers.

Where the matrix strain energy represents the isotropic behavior ψ iso and bers strain energy is associated with anisotropic response ψ ani . The bers in the tissue introduced by planar discrete ber families (n=2) with direction e 1 0 and e 2 0 in the reference conguration. Thus, in general, the strain energy function for the hyperelastic model of the arterial wall with two collagen families can be written as follows,

ψ(C C C, e 1 0 , e 2 0 ) = ψ iso (C C C) + ψ ani (C C C, e 1 0 , e 2 0 ). (1.2)
Based on invariant theory, a polynomial invariant basis is dened for the isotropic part ψ iso (C C C) as I iso = {I 1 , I 2 , I 3 } and for anisotropic part ψ ani (C C C, e 1 0 , e 2 0 ) the invariant basis is I ani = {I 1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 , I 8 }, which leads to an isotropic function for the strain energy density. Usually for sake of simplicity and in order to reduce the material parameters one can write the strain energy density in terms of some of its invariant basis [START_REF] Holzapfel | Nonlinear solid mechanics ii[END_REF], as:

ψ(C C C, e 1 0 , e 2 0 ) = ψ iso (I 1 ) + ψ ani (I 4 , I 6 ), (1.3)
where,

I 1 = tr C C C I 2 = 1 2 (tr C C C) 2 + tr (C C C) 2 I 3 = det C C C I 4 = C C C : e 1 0 ⊗ e 1 0 I 5 = C C C 2 : e 1 0 ⊗ e 1 0 I 6 = C C C : e 2 0 ⊗ e 2 0 I 7 = C C C 2 : e 2 0 ⊗ e 2 0 I 8 = (e 1 0 • e 2 0 )e 2 0 • C C C 2 e 2 0 (1.4)
Then the corresponding second Piola-Kirchho stress tensor is,

π π π = 2f M ∂ψ iso (I 1 ) ∂C C C + 2f f ∂ψ ani (I 4 , I 6 ) ∂C C C = 2f M ∂ψ iso ∂I 1 ∂I 1 ∂C C C + 2f f ∂ψ ani ∂I 4 ∂I 4 ∂C C C + ∂ψ ani ∂I 5 ∂I 5 ∂C C C , (1.5)
where f M is the volume fraction of matrix and f f is the volume fraction of collagen bers.

Multi-scale approaches

In order to include more structural features into the hyperelastic constitutive models a multi-scale approach was developed by [START_REF] Maceri | An insight on multiscale tendon modeling in muscletendon integrated behavior[END_REF][START_REF] Weisbecker | Constitutive modelling of arteries considering bre recruitment and three-dimensional bre distribution[END_REF] to represent collagen

LIMITS OF THE EXISTING MODELS AND OVERVIEW OF THE THESIS 26

bers in the soft tissue. For example in [START_REF] Maceri | An insight on multiscale tendon modeling in muscletendon integrated behavior[END_REF], the mechanical response of crimped collagen bers was described by an equivalent tangent ber modulus E f (λ 4 , λ 6 )

where λ 4 = (I 4 )

1 2 and λ 6 = (I 6 )

1 2 are ber stretches, accounting for collagen ber geometrical crimp (micro-scale) and introducing a worm-like chain model molecular response at the (nano-scale). Thus the strain energy density of bers in Eq. (1.2) is expressed as follows [START_REF] Marino | Constitutive Modeling of Soft Tissues[END_REF][START_REF] Marino | Multiscale elastic models of collagen bio-structures: from cross-linked molecules to soft tissues[END_REF][START_REF] Bianchi | An integrated computational approach for aortic mechanics including geometric, histological and chemico-physical data[END_REF]:

ψ ani (C C C, e 1 0 , e 2 0 ) = ψ ani (I 4 , I 6 ) = 1+ 1 2 [(λ 4 -1)+(λ 6 -1)] 1 1+(ξ-1) 1 E f (η)dηdξ. (1.6)

Limits of the existing models and overview of the thesis

In most hyperelastic models, the contribution of elastin bers is assumed as isotropic and combined with the matrix [START_REF] Holzapfel | Nonlinear solid mechanics ii[END_REF][START_REF] Stylianopoulos | Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls[END_REF][START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF][START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF][START_REF] Marino | Constitutive Modeling of Soft Tissues[END_REF][START_REF] Marino | Multiscale elastic models of collagen bio-structures: from cross-linked molecules to soft tissues[END_REF][START_REF] Bianchi | An integrated computational approach for aortic mechanics including geometric, histological and chemico-physical data[END_REF] although experiments have shown the anisotropic contribution of elastin bers in the mechanical response of the tissue [START_REF] Marino | Constitutive Modeling of Soft Tissues[END_REF]. Furthermore, one can only compute the macroscopic stress and deformation through hyperelastic constitutive models in which the bers are continuously embedded in their surrounding matrix and the bers follow the macroscopic deformation (ane deformation). However, experimental observations reveal that by increasing the mechanical load bers are capable to generate shear stresses to rotate faster than the matrix (non-ane deformation) [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. This exponential strain energy density function [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen bre orientations[END_REF] was recently extended to exclude the eect of collagen bers under compression [START_REF] Li | An exponential constitutive model excluding bres under compression: Application to extensionination of a residually stressed carotid artery[END_REF] and can reproduce the experimental results with accuracy but the material parameters do not have a direct relationship with microstructural features such as collagen ber waviness, intramuscular cross-link, and presence of other bers and molecules [START_REF] Marino | Constitutive Modeling of Soft Tissues[END_REF]. Furthermore, the development of multiscale models allow to include the undulation of collagen bers and taking into account the anisotropy of collagen brils in dierent scales [START_REF] Nierenberger | Mécanique multiéchelles des parois vasculaires : expérimentation, imagerie, modélisation[END_REF].

Recently, beyond developing the continuum models, discrete network models have also been implemented to describe microstructural details of the soft tissue [START_REF] Witthoft | A discrete mesoscopic particle model of the mechanics of a multi-constituent arterial wall[END_REF][START_REF] Mauri | A discrete network model to represent the deformation behavior of human amnion[END_REF] to include truly microstructural constituents and physical phenomena to bridge with the tissue biology in the constitutive models.

As we discussed in this chapter, the mechanical properties of arteries come from the physical and mechanobiological mechanisms, which occur within the Mohsen Nakhaei Thèse en Mécanique et Ingénierie • Developing layer-specic representative volume element (RVE):

The stacks of images from Krasny experiments [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] were used to reconstruct the RVEs for adventitia and media layers. Image analysis was performed on these images to determine the required physical quantities such as volume fractions and orientation of each constituent. However, the literature was also reviewed to obtain some other physical quantities to build up the three main layers of arterial structure; namely, adventitia, lamellae and interlamellar space. Within the adventitia, two scales were considered so as to account for both the bril decrimping (through the progressive bril reorientation at the lowest scale) and bundle realignment (at the macroscopic scale). Within the media, a stack of lamellae and interlamellar space was considered.

• Multi-scale modeling of the hierarchical structure of artery: The RVEs considered as a ber-reinforced soft composite, where the hypoelastic constitutive laws were taken into account to model the constituents within the RVEs. Due to the large deformation, an extension of Eshelby's matrix-inclusion problem was used to compute analytical expressions for the strain rate and spin concentration tensors. These concentration tensors were used in a multi-scale approach to compute averaged strain rate and averaged spin tensors over the phases through the hierarchical structure of Mohsen Nakhaei Thèse en Mécanique et Ingénierie

LIMITS OF THE EXISTING MODELS AND OVERVIEW OF THE THESIS 28

the artery from the applied macroscopic strain rate. The computed averaged strain rate at each phase provides access to the microscopic stress, while the averaged spin tensor determines the load-induced phase rotation.

• Integrating the homogenized models in a structural model of the artery: In order to account for the interaction of these RVEs, we implemented the micromechanical framework in a nite element formulation. We considered an arterial segment with a simplied geometry of a hollow cylinder. The multi-scale constitutive model was considered in a home-made nite element code to account for adventitia, lamellae and interlamellar space across the thickness of the arterial wall.

Résumé

La plupart des matériaux rencontrés dans notre vie, allant des matériaux manufacturés aux matériaux naturels, peuvent être considérés comme homogènes à une échelle dite macroscopique, mais ont une microstructure hétérogène. Ils sont constitués de diérents constituants présentant généralement une microstructure complexe, voir par exemple la microstructure de la couche la plus externe du tissu artériel présentée Fig. 2.1. Il est important de noter que la disposition des constituants composant la microstructure joue un rôle important dans la compréhension et la prédiction du comportement macroscopique des matériaux hétérogènes. A cet égard, la micromécanique des milieux continus est un cadre possible pour prendre en compte la morphologie de la microstructure an de développer un modèle mathématique permettant de prédire le comportement global de tels matériaux [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF][START_REF] Suquet | Eective properties of nonlinear composites[END_REF]158,[START_REF] Zaoui | Continuum micromechanics: survey[END_REF][START_REF] Sevostianov | On some controversial issues in eective eld approaches to the problem of the overall elastic properties[END_REF]. Ce chapitre présente les outils théoriques de micromécanique des milieux continus, étendus au cadre des grandes transformations. L'approche est basée sur une loi de comportement hypoélastique, et sur le calcul de la vorticité de chaque phase qui induit une rotation progressive.

Introduction

Most of the materials encountered in our life, ranging from manufactured to natural materials, can be considered as homogeneous at a so-called macroscopic scale, but have a heterogeneous microstructure. They are built up by dierent constituents usually exhibiting a complex microstructure, see for instance the microstructure of the most outer layer of arterial tissue shown on Fig. 2.1.

It is important to notice that the arrangement of the constituents making up the microstructure play a signicant role in understanding and controlling the macroscopic behavior of heterogeneous materials. In this respect, continuum micromechanics is a possible framework to take into account the microstructure morphology to develop a mathematical model that predicts the overall behavior of such materials [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF][START_REF] Suquet | Eective properties of nonlinear composites[END_REF]158,[START_REF] Zaoui | Continuum micromechanics: survey[END_REF][START_REF] Sevostianov | On some controversial issues in eective eld approaches to the problem of the overall elastic properties[END_REF].

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

Representative volume element and assumptions of scale separation

In continuum micromechanics, the rst step consists in identifying the patterns of the material's microstructure that govern the mechanical response of the material.

The complexity of many material's microstructures prevents from accounting for each and every detail: as a remedy, a statistical approach is

d l L (2.1)
• The characteristic length of the heterogeneities (d) must be small enough as compared to the characteristic length of the RVE (l), that is d l. This ensures that the homogenized response remains independent of the precise geometry of the RVE, i.e. the RVE is really representative.

• The characteristic length of the RVE must also be small enough in comparison to the characteristic length of the structure (L) and of the loading to which the structure is subjected that is l L, such that the macroscopic elds remain continuous.

Once the dierent RVEs have been dened under the requirement of scale separation, the geometrical and physical characteristics of their phases must be identied. These characteristics include the phase geometrical shape (and possibly its orientation), as well as its volume fraction and its constitutive relation, these characteristics being in most cases of statistical nature. in terms of a velocity eld v(x) as follows:

Momentum balance and kinematic compatibility

v(x) = D D D • x ∀x ∈ ∂Ω, (2.2)
whereby D D D is the macroscopic strain rate tensor, and x is the microscopic position vector in the current conguration. In addition, compatibility inside the RVE yields the microscopic strain rate eld d d d(x) as:

d d d(x) = 1 2 grad v(x) + [grad v(x)] T ∀x ∈ Ω, (2.3)
Thus, the kinematically compatible microscopic strain rates d d d(x) inside the RVE, subjected to the homogeneous strain rate D D D, fulll the Hashin strain rate average rule:

D D D = 1 |Ω| Ω d d d(x)dV = d d d , = 1 2|Ω| ∂Ω v(x) ⊗ n(x) + v(x) ⊗ n(x) dS, (2.4)
whereby we made use of the Green-Ostrogradski theorem to evaluate the rate of deformation at the boundary, • denotes the spatial average operator, n(x) is the unit normal outward-pointing vector at point x of the boundary of RVE, and ⊗

is the dyadic product.

Momentum balance and stress average rule -Hill's lemma

The rate of deformation existing on the boundary of the RVE results in the existence of traction forces t(x) on the boundary of RVE which must be in equilibrium, such that:

∂Ω t(x)dS = ∂Ω σ σ σ(x) • n(x)dS = 0 (2.5)
whereby use of Cauchy's theorem, t(x) = σ σ σ(x) • n(x), has been made. Once again, applying the Green-Ostrogradski theorem to Eq. (2.5) leads to the expected equilibrium equation at each point of the RVE, reading as:

div σ σ σ(x) = 0 ∀x ∈ Ω. (2.6)
Then, one can compute the external power of traction force P ext on the RVE, by applying Cauchy's theorem and accounting for the Hashin's boundary condition, leading to:

P ext = ∂Ω t(x) • v(x)dS = ∂Ω [σ σ σ(x) • n(x)] • [D D D • x]dS = Ω σ σ σ(x)dV : D D D.
(2.7)

Mohsen Nakhaei Thèse en Mécanique et Ingénierie

THERMODYNAMIC FOUNDATION OF THE MICROSCOPIC HYPO-ELASTIC CONSTITUTIVE RELATION 34

The position-independent volume integral of microscopic Cauchy stresses in the external power induces the existence of a macroscopic Cauchy stress dened by:

Σ Σ Σ = 1 |Ω| Ω σ σ σ(x)dV = σ σ σ , (2.8)
in the form of a stress average rule. The external power on the RVE is therefore a function of the macroscopic stress and strain rate,

P ext = Σ Σ Σ : D D D, (2.9)
Furthermore, substituting the stress average rule (2.8) into the principle of virtual power [START_REF] Germain | The method of virtual power in continuum mechanics. part 2: Microstructure[END_REF] leads to:

P ext = -P int = 1 |Ω| Ω σ σ σ(x) : d d d(x)dV = σ σ σ : d d d , (2.10)
where P int is the power of internal forces. Equations (2.9) and (2.10) reveal the Hill's lemma: the average power generated by the microscopic stresses on the microscopic strain rates is equal to the power generated by the macroscopic stress on the macroscopic strain rate reading as,

Σ Σ Σ : D D D = σ σ σ : d d d = 1 |Ω| Ω σ σ σ(x) : d d d(x)dV = σ σ σ : d d d .
(2.11)

Thermodynamic foundation of the microscopic hypo-elastic constitutive relation

The constitutive relations have to satisfy both rst and second principles of thermodynamics, which can be combined into the Clausius-Duhem inequality

[118], expressing the system's dissipation D, as:

D = σ σ σ : d d d -ρ Dψ Dt + s DT Dt - q T • grad T ≥ 0, (2.12)
whereby σ σ σ and d d d are respectively the Cauchy stress and strain rate tensors, ρ is the mass density, T is the absolute temperature, and s is the (microscopic) entropy per unit mass. Furthermore, the (microscopic) Helmholtz free energy per unit mass, ψ, is chosen here as the thermodynamic potential, and q is the outward heat ux vector at the boundary of the structure. The operator

D(•)
Dt is the material derivative of quantity (•). For an elastic (i.e. reversible) transformation, the dissipation remains equal to zero, D = 0.

D 1 = σ σ σ : d d d -ρ Dψ Dt + s DT Dt , D 2 = - q T • grad T. (2.13)
It is usually assumed that each dissipation should be positive, and consequently, for a reversible transformation, one has D 1 = 0. As proposed by Rajagopal [START_REF] Srinivasa | On a class of non-dissipative materials that are not hyperelastic[END_REF][START_REF] Srinivasa | A gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials[END_REF], we choose to work with the Gibbs thermodynamic potential G instead of the Helmholtz free energy ψ [START_REF] Morin | Non-ane ber kinematics in arterial mechanics: a continuum micromechanical investigation[END_REF]. The relation between the two potentials results from a partial Legendre transformation on the variable σ σ σ, as dened by:

ψ(d d d, T ) = ∂G(σ σ σ, T ) ∂σ σ σ : σ σ σ -G(σ σ σ, T ). (2.14)
As a consequence, the intrinsic dissipation reads as:

D 1 = σ σ σ : d d d -σ σ σ : ρ ∂ 2 G(σ σ σ, T ) ∂σ σ σ∂σ σ σ : Dσ σ σ Dt -ρ DG(σ σ σ, T ) DT DT Dt + ρs DT Dt = 0. (2.15)
This latter expression of the intrinsic dissipation shows that temperature T and the Cauchy stress σ σ σ are natural arguments of the Gibbs thermodynamic potential.

As a consequence, one gets the following state laws:

s = - ∂G(σ σ σ, T ) ∂T , d d d = ρ ∂ 2 G(σ σ σ, T ) ∂σ σ σ∂σ σ σ : Dσ σ σ Dt .
(2.16)

To ensure the objectivity of the second state law (i.e. the strain rate should not depend on the motion of the observer), one adds a gyroscopic term [START_REF] Morin | Non-ane ber kinematics in arterial mechanics: a continuum micromechanical investigation[END_REF][START_REF] Srinivasa | A gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials[END_REF],

dened as:

σ σ σ : ρ ∂ 2 G(σ σ σ, T ) ∂σ σ σ∂σ σ σ : (-σ σ σ • Ω Ω Ω + σ σ σ • Ω Ω Ω) = 0, (2.17)
where Ω Ω Ω is the spin skew-symmetric tensor (Ω Ω Ω -1 = Ω Ω Ω T and tr(Ω Ω Ω) = 0). This yields a thermodynamically reversible and objective hypo-elastic constitutive law in the form of: Choosing Ω Ω Ω as the skew-symmetric part of the microscopic velocity gradient, i.e.

d d d = ρ ∂ 2 G(σ σ σ, T ) ∂σ σ σ∂σ σ σ : Dσ σ σ Dt + Ω Ω Ω • σ σ σ -σ σ σ • Ω Ω Ω .
Ω Ω Ω = ω ω ω = 1 2

grad v(x) -[grad v(x)] T , (2.19)
where grad is the Eulerian gradient and T the transpose operator. Then, a Gibbs potential in the form of G = σ σ σ : C -1 : σ σ σ, gives the microscopic constitutive law in a hypo-elastic formulation as follows:

σ σ σ ∇ (x) = C(x) : d d d(x), (2.20)
where σ σ σ ∇ (x) is the (microscopic) stress rate in its objective form, C(x) is the hypoelastic stiness tensor, and d d d(x) is the strain rate. The choice of Ω Ω Ω = ω ω ω, yields the objective stress rate as the Jaumann one, reading as:

σ σ σ ∇ (x) = σ σ σ(x) + σ σ σ(x) • ω ω ω(x) -ω ω ω(x) • σ σ σ(x), (2.21)
where σ σ σ(x) =

d d d(x) = A(x) : D D D(x) ω ω ω(x) = R(x) : D D D(x), (2.22)
where A(x) is the fourth-order strain rate concentration tensor and R(x) is the fourth-order spin concentration tensor. The next section is devoted to estimating expressions for these concentration tensors.

Mori-Tanaka estimation of strain rate and spin concentration tensors from auxiliary Eshelby problems

Dierent approaches have been proposed to estimate the concentration relations.

One set of methods relies on the use of the variational principles, which allow to bound the eective properties [START_REF] Hashin | Analysis of composite materialsa survey[END_REF][START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF]. Depending on the assumptions taken on Mohsen Nakhaei Thèse en Mécanique et Ingénierie

MORI-TANAKA ESTIMATION OF STRAIN RATE AND SPIN

CONCENTRATION TENSORS FROM AUXILIARY ESHELBY PROBLEMS 37 the morphology of the RVE, the bounds are more or less restrictive. The other approach consists in the use of a mean eld approach. The framework in which our model is developed is based on the latter approach, which is detailed in the following sections.

Eshelby problem as a rst estimate of the concentration tensors

In the framework of homogenization, concentration tensors allow to determine the microscopic elds in the phases as functions of the applied macroscopic elds.

As a rst simple case, an Eshelby matrix-inclusion type problem [START_REF] John D Eshelby | The determination of the elastic eld of an ellipsoidal inclusion, and related problems[END_REF] is used to estimate the concentration tensors. However, we consider not only the strain rate but also the rotation of the inclusion or spin, which provides the basis for a large strain micromechanics theory.

The Eshelby inclusion problem considers an innite material with stiness C 0 , in which a small ellipsoidal volume Ω I is subjected to a uniform eld of eigenstrain rate η η η, while the rest of the material is free from any eigenstrain rate and no loading is applied on the material, see Fig. 2.3(A). Due to the linearity of the problem and to the shape of the inclusion, Eshelby showed that this eigenstrain rate induces a uniform velocity gradient eld in the inclusion as:

gradv(x) = L Esh : η η η ∀x ∈ Ω I , (2.23)
where L Esh is the Eshelby strain rate-to-velocity gradient tensor, depending on the stiness of the matrix C 0 , and on the shape and the orientation of the ellipsoidal inclusion. The orientation of the ellipsoidal inclusion is dened by an orthonormal basis n i ( i = 1, 2, 3), which coincides with the principal axes of the ellipsoidal inclusion.

The aspect ratios representing the shape of the inclusion are ω 1 = a 3 a 1

, ω 2 = a 3 a 2

,

and

ω 3 = a 1 a 2
, where 2a i (i = 1, 2, 3) are the lengths of the principal axes of the ellipsoidal inclusion, such that: a 1 ≤ a 2 ≤ a 3 .

In the present work, we focus on innitely long cylindrical inclusions (see Fig. 2.4), i.e. a 3 → ∞, and a 2 = a 1 . In this case, the non zero components of L Esh are expressed, in the n i basis, in terms of the Poisson's ratio of the isotropic matrix ν 0 [START_REF] Budiansky | Void growth and collapse in viscous solids[END_REF][START_REF] Morin | Non-ane ber kinematics in arterial mechanics: a continuum micromechanical investigation[END_REF] as:

L Esh 1111 = L Esh 2222 = 4ν 0 -5 8(ν 0 -1) L Esh 1122 = L Esh 2211 = - 4ν 0 -5 8(ν 0 -1) L Esh 1212 = L Esh 1221 = L Esh 2112 = L Esh 2112 = 4ν 0 -3 8(ν 0 -1) L Esh 1133 = L Esh 2233 = -ν 0 2(ν 0 -1) L Esh 3113 = L Esh 3223 = L Esh 3131 = L Esh 3232 = 1 2 .
(2.24)

We will also consider lamellar structures, and introduce inclusions in the form of oblate spheroids i.e. a 3 → ∞ and a 2 → ∞. In this case, the non zero L Esh components for an innitely thin plate embedded in an isotropic matrix with

Poisson's ratio ν 0 , read, in the n i basis, as:

L Esh 1111 = 1 L Esh 2112 = L Esh 2211 = 1 -2ν 0 2(1 -ν 0 ) L Esh 3113 = L Esh 3311 = 1 L Esh 1212 = - 1 -2ν 0 2(1 -ν 0 ) L Esh 1313 = ν 0 1 -ν 0 . (2.25)
More details on derivations of the Eshelby strain rate-to-velocity gradient tensor are provided in the Appendix A.

From the uniform velocity gradient in the inclusion expressed by Eq. (2.23), one can compute the uniform strain rate tensor and the spin tensor within the inclusion respectively as:

d d d(x) = 1 2 gradv(x) + [gradv(x)] T ∀x ∈ Ω I , (2.26
)

ω ω ω(x) = 1 2 gradv(x) -[gradv(x)] T ∀x ∈ Ω I , (2.27)
these are linearly related to the eigenstrain rate η η η in the domain Ω I , through the Eshelby strain rate-to-velocity gradient tensor,

d d d(x) = 1 2 L Esh + L Esh T : η η η = S Esh : η η η ∀x ∈ Ω I , (2.28
)

ω ω ω(x) = 1 2 L Esh -L Esh T : η η η = R Esh : η η η ∀x ∈ Ω I , (2.29)
where S Esh = 1 2 L Esh + L Esh T and R Esh = 1 2 L Esh -L Esh T are respectively fourth-order Eshelby concentration tensor for strain rate and spin in the Eshelby inhomogeneity problem, see Fig.

2.3(A).

This rst Eshelby problem can then be extended to the following problem: an ellipsoidal inclusion, with stiness C I and occupying the volume Ω I , is surrounded by an innite matrix with stiness C 0 ; the material is subjected to a macroscopic strain rate D D D 0 , at its remote boundary, see Fig. 2.

3(B).

The superposition principle is used and this second problem is decomposed as the sum of two problems: rst a homogeneous innite material, with stiness C 0 , is subjected to the boundary conditions v(x) = D D D 0 • x ; the second subproblem is the previously described inhomogeneity Eshelby problem, whereby the eigenstrain eld η η η is expressed in terms of the stiness tensors as:

η η η = -C -1 0 : [I + (C I -C 0 ) : P I ] -1 : (C I -C 0 ) : D D D 0 , (2.30)
where P I = S Esh : C 0 -1 is the Hill tensor, and I is the fourth-order identity tensor, with components

I ijkl = 1 2 (δ ik δ jl + δ il δ jk ), δ ij being the Kronecker delta, i.e. δ ij = 1, if i = j,
d d d I = [I + (C I -C 0 ) : P I ] -1 : D D D 0 = A ∞ : D D D 0 , (2.31) ω ω ω I = -R Esh : C -1 0 : [I + (C I -C 0 ) : P I ] -1 : (C I -C 0 ) : D D D 0 = R ∞ : D D D 0 , (2.32)
where A ∞ and R ∞ are respectively the expressions of the concentration tensors for strain rate and spin in the Eshelby problem. While the strain rate and spin Mohsen Nakhaei Thèse en Mécanique et Ingénierie The strain rate becomes equal to the prescribed strain rate far from the inclusion but close to the inclusion, strong variations in strain rate and spin are encountered.

Mori-Tanaka estimate for ber-reinforced composites

In this section, we consider heterogeneous materials, whose microstructure can be described by N inclusion phases embedded in a surrounding matrix M , and subjected to a uniform strain rate D D D at the boundary of the RVE (see Fig. 2.4). In this case, the strain rate and spin elds are not anymore uniform in the inclusions due to their mutual interactions. As a remedy, a mean eld approach is used, and the previously described Eshelby problem is taken as an auxiliary problem to evaluate the mean value of the strain rate and spin elds within the inclusion. In this auxiliary problem, the auxiliary matrix has the same mechanical properties as the matrix of the considered material, and the inhomogeneity has the same shape and mechanical properties as the inclusion for which the mechanical response needs to be determined. Equations (2.31) and (2.32) respectively provide the mean strain rate and mean spin in the inclusion. This result is substituted into the strain rate average rule Eq. (2.4) and leads to: strain rate is equal to the mean strain rate in the matrix phase of the considered problem, and its expression reads as:

D D D = N r=1 f r d d d r + f M d d d M , (2.33
D D D 0 = d d d M = N r=1 f r [ I + (C r -C 0 ) : P r ] -1 + f M I -1 : D D D.
(2.34)

As a consequence, the mean strain rate and spin in each inclusion phase are expressed as follows:

d d d r = [ I + (C r -C 0 ) : P r ] -1 : N r=1 f r [I + (C r -C 0 ) : P r ] -1 + f M I -1 : D D D = A r : D D D, (2.35
)

ω ω ω r = R ∞ : C 0 -1 : [I + (C r -C 0 ) : P r ] -1 : (C r -C 0 ) : N r=1 f r [I + (C r -C 0 ) : P r ] -1 + f M I -1 : D D D = R r : D D D. (2.36)
From Eqs. (2.35) and (2.36), the fourth-order strain rate and spin concentration tensors, A r , A M , and R r respectively, can be identied as:

A r = A ∞ r : N i=1 f i A ∞ i + f M I -1
∀r ∈ {1, 2, 3, ..., N },

(2.37)

A M = N i=1 f i A ∞ i + f M I -1 , (2.38) R r = R ∞ r : N i=1 f i A ∞ i + f M I -1 ∀r ∈ {1, 2, 3, ..., N }, (2.39)
where A ∞ r and R ∞ r are respectively the strain rate and spin concentration tensors of the auxiliary Eshelby problem, dened by Eqs. (2.31) and (2.32). This homogenization scheme is called the Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with mistting inclusions[END_REF]5].

Algorithmic treatment of ber kinematics 2.6.1 Motivation for time discretization

The homogenization scheme under large deformation is highly nonlinear due to the conformational changes of the microstructure. Therefore, an incremental approach is implemented to solve the governing equations: the load history is divided into a nite number of steps and an innitesimal part of the load is applied at each increment. Thus, the deformation and evolution of the microstructure can be approximated to be linear at each increment. Then one can solve the linearized equations and keep track of all the increments to get the solution of the nonlinear problem. We use an updated Lagrangian description [START_REF] Sn Atluri | Rotations in computational solid mechanics[END_REF], which requires the knowledge of the updated conguration at each increment: the current conguration is used as a reference conguration to compute the elds at the next increment.

Due to the complex dependence of the concentration tensors on the inclusion orientations, a forward Euler scheme is employed for time discretization, i.e. for a quantity a, one has:

ȧt = a t+dt -a t dt (2.40)
where ȧ is the time derivative of the quantity a, with a t the value of the quantity at time t and a t+dt the value of the quantity after a small time increment dt.

Dσ σ σ r Dt t = σ σ σ t r = C t r : d d d t r -σ σ σ t r • ω ω ω t r + ω ω ω t r • σ σ σ t r , (2.42)
i.e. for each phase:

σ σ σ t r = C t r : d d d t r -σ σ σ t r • ω ω ω t r + ω ω ω t r • σ σ σ t r , ∀r ∈ {1, 2, 3, ..., N }. σ σ σ t M = C t M : d d d t M .
(2.43)

Accounting for Eq. ( 2.41), one can compute the local constitutive equation in terms of the prescribed strain rate at the boundary, as follows:

σ σ σ t r = C t r : A t r : D D D t -σ σ σ t r • (R t r : D D D t ) + (R t r : D D D t ) • σ σ σ t r σ σ σ t M = C t M : A t M : D D D t .
(2.44)

It is interesting to write (2.44) using the index notation, which gives,

σr ij = C r ijkl A r lkmn -σ r ik R r kjmn + R r ikmn σ r kj D nm = C r eq ijmn D nm , σM ij = C M ijkl A M lkmn D nm = C M eq ijmn D nm , (2.45)
whereby all the variables are evaluated at time t and fourth-order equivalent "stiness" tensors C eq r for phase inclusion r and C eq M for matrix phase are dened, using the index notation, as:

C r eq ijmn = C r ijkl A r lkmn -σ r ik R r kjmn + R r ikmn σ r kj , C M eq ijmn = C M ijkl A M lkmn .
(2.46)

for the current stress state at time t.

a cart =     sinθcosφ cosθcosφ -sinφ sinθsinφ cosθsinφ cosφ cosθ -sinθ 0     • a sph , (2.48)
where a cart and a sph represent the same vector expressed respectively in the cartesian and spherical coordinate system, θ is the co-latitudinal angle with 0 ≤ θ ≤ π and φ is the longitudinal angle with 0 ≤ φ < 2π (see Fig.

Σ Σ Σ t+dt = N r=1 f r σ σ σ t+dt r + f M σ σ σ t+dt M = N r=1 f r σ σ σ t r + N r=1 f r C eq t r : D D D t dt + f M σ σ σ t M + f M C eq t M : D D D t dt, = Σ Σ Σ t + C t hom : D D D t dt, ∀r ∈ {1, 2, 3, ..., N }, (2.51)
whereby the homogenized "stiness" tensor C t hom is dened as:

C t hom = N r=1 f r C eq t r + f M C eq t M ,
∀r ∈ {1, 2, 3, ..., N }.

(2.52) A summary of the numerical scheme is presented in Fig. 2.6.

v(x) = D D D t • x D t d t M = A t M : D t d t r = A t r : D t Σ t+dt = σ σ σ t+dt ω t r = R t r (θ t , φ t ) : D t θ t+dt , φ t+dt
σ σ σ ∇ t M = C t M : d d d t M σ σ σ ∇ t r = C t r : d d d t r

Extension to stress-driven load cases

One can consider a most complex case where traction forces are applied on the boundary of the RVE. The knowledge of the material conformation at the initial time is used in equation (2.52) to compute the homogenized stiness. Then, the homogenized stiness allows to calculate the strain rate in a forward Euler scheme as follows, knowing the macroscopic stress at all times,

D D D t = C t hom -1 : Σ Σ Σ t+dt -Σ Σ Σ t dt (2.53)
Then, the previously described numerical scheme is adopted and allows to evaluate the macroscopic stress increment generated by this strain rate. Finally, this macroscopic increment of Cauchy stress is compared with the prescribed one.

If there is a dierence between the two values the calculation is repeated with the new homogenized stiness in order to determine the correct strain rate. On the other hand, if the dierence between the two macroscopic Cauchy stresses is negligible, we compute the macroscopic deformation gradient as follows,

F F F t+dt = D D D t dt + I I I • F F F t , (2.54)
where I I I is the second-order identity tensor and the macroscopic deformation gradient at initial instant is assumed to be the identity matrix (F F F t=0 = I I I). This completes the computational scheme for applied traction forces on the boundary over a small time increment dt. A summary of the numerical scheme for imposed stress conditions is illustrated in In the present study, a multi-scale approach is introduced by considering a lower scale RVE to model the heterogeneous structure of the material phase (see v(x,

t) = D D D t • x D t d up t M = A up t M : D t d up t I = A up t I : D t Σ t+dt = σ σ σ up t+dt ω up t I = R up t I (θ up t I , φ up t I ) : D t θ up t+dt , φ up t+dt
Upper scale morphology evolution

σ up t+dt M σ up t+dt I = σ σ σ low t+dt ω low t r,I = R low t r,I (θ low t r,I , φ low t r,I ) : d d d up t I θ low t+dt , φ low t+dt

Conclusion

In this chapter, we developed the framework of nite strain continuum micromechanics.

As the RVEs undergo large morphological changes, a

Mori-Tanaka scheme extended to inclusion reorientation is adopted to compute analytical expressions for the strain rate and spin concentration tensors. Based on homogenization theory, these concentration tensors are then used to compute the averaged strain rate and spin tensor in each phase of RVE from the applied macroscopic strain rate on the boundary.

The phases in the RVE are represented as innitely long cylinders embedded in a matrix, where Mori-Tanaka scheme denes the interaction between dierent phases in which a local hypoelastic constitutive model is dened. The computed spin tensor for each phase determines the load-induced rotation of the phase. This approach will ensure the non-ane deformation of the phases in the RVE.

Furthermore, the method is extended to a multi-scale homogenization scheme to be capable of modeling the hierarchical structure of artery.

3 | Multi-scale modelling of the arterial adventitia

Introduction

In order to develop a microstructurally motivated constitutive model for the artery, the multi-scale micromechanical approach which was explained in Chapter 2 was rst applied to the arterial adventitia layer. This chapter presents the steps required to construct the arterial adventitia model with specic multi-scale RVEs from the multiphoton image stacks of the tissue. Then, the uniaxial tensile simulation was performed on the adventitia model to investigate the role of the collagen and elastin ber networks morphology changes and contribution of these networks in the mechanical response. Finally, these results were compared against the experimental results and observations to evaluate the capacity of the micromechanical model to predict the experimental evidences.

Micromechanical representation of the adventitia 3.2.1 Modelling the adventitia at the scale of a few hundreds of micrometers

As detailed in Chapter 1 and illustrated in Fig. 3.1, the microstructure of the adventitia is very complex and cannot be resolved in full details. As a remedy, we Mohsen Nakhaei Thèse en Mécanique et Ingénierie The morphology of the adventitia layer motivates the choice of a Mori-Tanaka scheme to account for all interactions between the dierent phases.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie The proposed multi-scale approach for the mechanical modeling of the arterial adventitia introduces several physical and morphological parameters whose determination is discussed in the present section. It is important to note that, among all parameters, the vast majority are actually universal parameters, since their values will remain xed, based on experimental observations. First, as reviewed in Chapter 1, all adventitial ber networks have a negligible transmural angle, and therefore all φ angles are set to π 2 .

Mechanical properties of elementary constituents

Our adventitia model introduces three distinct elementary constituents namely:

(i) type I collagen bers, (ii) elastin bers, and (iii) ground substance matrix.

We assign to these three constituents a hypoelastic constitutive behavior, as introduced in the Section 2.4 of Chapter 2. Despite the vast literature discussing the possible anisotropy of these constituents, we assume isotropic linear elastic properties for all constituents, i.e. the stiness tensor of phase r, C r , is dened as :

C r = 3k r J + 2µ r K, (3.1)
whereby k r and µ r are respectively the bulk and shear moduli of phase r, which can be expressed as functions of the Young's modulus E r and of the Poisson's ratio ν r by k r = Er 3(1-2νr) and µ r = Er 2(1+νr) . Furthermore, J is the volumetric part of the fourth-order identity tensor I with components J ijkl = 1 3 δ ij δ kl , and K is its deviatoric part, K = I -J. The components of the fourth-order identity tensor I read as I ijkl = 1 2 (δ ik δ jl + δ il δ jk ). δ ij stands for the Kronecker delta.

The literature review of Section 1.5 in Chapter 1 was guided our choice for the isotropic properties of collagen bers, elastin bers, and ground substance, which are summarized in Table 3.1. For all constituents, the Poisson's ratio is taken equal to 0.34 [START_REF] Vass | Review of universal rules governing bone composition, organization, and elasticity across organizational hierarchies[END_REF][START_REF] Cusack | Determination of the elastic constants of collagen by brillouin light scattering[END_REF].

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie Elastin ber 1 Tensile test on a water-swollen bovine ligamentum [START_REF] Aaron | Elastin as a random-network elastomer: A mechanical and optical analysis of single elastin bers[END_REF] Ground substance 0.1 Resistance of single cell to elastic deformation [4,[START_REF] Stricker | Mechanics of the f-actin cytoskeleton[END_REF][START_REF] Shelly R Peyton | Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion[END_REF][START_REF] Buxboim | Matrix elasticity, cytoskeletal forces and physics of the nucleus: how deeply do cells `feel'outside and in?[END_REF] Table 3.1: Mechanical parameters of the adventitita micromechanical model.

Volume fraction of elementary constituents

Amount of collagen brils in the collagen bundles Dierent studies report cross-sectional views of collagen bundles, imaged by means of transmission electron microscopy, see Fig. 3.5. Automatic segmentation of these microscopy images (using the Otsu method [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF] to determine the thresholding grey value) allows to access the volume fraction of these brils, for dierent tissues, which we report in Table 3.2.

Figure 3.5: Cross section of a collagen bundle of a mouse adventitia [START_REF] Jerey R Tonniges | Collagen bril ultrastructure in mice lacking discoidin domain receptor 1[END_REF].

Specimen Imaging method Volume fraction Reference

Mouse blood vessel adventitia electron microscopy 63% [START_REF] Gregorio | Structural biology of the bres of the collagenous and elastic systems[END_REF] Mouse thoracic aorta adventitia electron microscopy 51% [START_REF] Jerey R Tonniges | Collagen bril ultrastructure in mice lacking discoidin domain receptor 1[END_REF] Rat inferior vena cava adventitia two-photon uorescent microscopy 31% [START_REF] Han | Fluid in the tissue channels of vascular adventitia investigated by afm and tem[END_REF] Rat aorta adventitia transmission electron microscope 30% [START_REF] Berillis | The role of collagen in the aorta's structure[END_REF] Table 3.

2: Collagen bril volume fraction within a collagen bundle

This literature survey motivates our choice of a volume fraction of collagen brils of 30 %. An analysis is performed to understand the eect of a variation of this volume fraction on the overall result, see section 3.5 of this chapter.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

DETERMINATION OF THE PARAMETERS OF THE MODEL 59

Amount of collagen bundles in the adventitia

The stacks of adventitia images of rabbit carotid arteries taken by Witold Krasny [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] were analyzed with ImageJ to separate the two recorded channels: second harmonic generation (collagen bundles signals) and autouorescence (elastin bers signals). Then, the Otsu thresholding technique [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF] is performed to extract binary images. Since the fraction of collagen is not uniform through the thickness of the arterial adventitia, the fraction is computed from taking the average through the image stacks of each sample. Finally, we set the fraction of collagen bundles to 30%, corresponding to the average of the fractions computed for the nine tested samples, which is in agreement with other studies [START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF][START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF].

Specimen Imaging method Volume fraction Reference

Porcine coronary adventitia multiphoton microscopy 33% [START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF] Porcine coronary adventitia multiphoton microscopy 33% [START_REF] Chen | A validated 3d microstructure-based constitutive model of coronary artery adventitia[END_REF] Rabbit carotid adventitia multiphoton microscopy 30% ± 10% Image processing from [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] Table 3.3: Reported values for the volume fraction of collagen bundles in the adventitia

Amount of elastin bers in the adventitia

In the literature, the amount of elastin bers has been evaluated in dierent tissue samples by a commercial image analysis software such as SIGMA SCAN PRO 5 [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF] or ImageJ [START_REF] Garcia | Right coronary artery becomes stier with increase in elastin and collagen in right ventricular hypertrophy[END_REF]. We also processed the dierent stacks of images resulting from uniaxial tensile tests on rabbit carotid arteries performed by Witold Krasny [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The microscopic images of the adventitia tissue were analyzed with ImageJ and the Otsu thresholding technique [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF] performed on the autouorescence channel allows to access the elastin volume fraction. All values are reported in Table 3.4. The last line of Table 3.4 corresponds to our own evaluation of the elastin fraction. We noticed a great variation in the elastin fraction among the dierent samples. Given the higher values reported in the literature, we nally decided to adopt a volume fraction of 20%. The sensitivity of the model with respect to this parameter will be discussed later.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

DETERMINATION OF THE PARAMETERS OF THE MODEL 60

Specimen

Imaging method Volume fraction Reference

Porcine coronary adventitia multiphoton microscopy 25.7% [START_REF] Jerey R Tonniges | Collagen bril ultrastructure in mice lacking discoidin domain receptor 1[END_REF] Porcine coronary adventitia multiphoton microscopy 22% [START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF] Rabbit carotid adventitia multiphoton microscopy 15% ± 10% Image processing from [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] Porcine thoracic adventitia light microscopy 5% [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF] Table 3.4: Reported values for the volume fraction of elastin bers in the adventitia

Orientation of the dierent ber networks

Orientation of the collagen brils in the collagen bundles

The ber waviness (or its inverse, the ber straightness) is an often-reported parameter for collagen bundles. It is dened as the ratio of the total length of the collagen bundle on the end-to-end distance of the crimped collage bundle. The ber straightness P is directly related to the orientation of the crimps, through the following relation: cos(θ) = P , whereby θ is the angle between the main bundle axis and the local crimp orientation, see Fig. 3.6. Table 3.5 reports the values collected from the literature on dierent tissue samples. We here adopt the average value of all values reported in Table 3.5. Once again, a sensitivity analysis on this parameter will be performed in a later section. Porcine coronary adventitia 0.85

31 • [22]
Porcine coronary adventitia 0.82

Orientation of the collagen bundles

The orientation of bers was analyzed using orientationJ, a series of ImageJ plugins for directional image analysis [START_REF] Püspöki | Transforms and operators for directional bioimage analysis: a survey[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF], which was employed previously to compute the global orientation and waviness of collagen bundles in the axial-circumferential plane. Through this method, they extracted a global orientation at around 45 • for the collagen bundles of the adventitia layer in the relax condition [START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF]. However, since this parameter is dicult to estimate from our images, and since the results are very sensitive to this parameter, we decided to determine this parameter by curve tting.

Orientation of the elastin bers

Microscopic observations [START_REF] Chen | The layered structure of coronary adventitia under mechanical load[END_REF][START_REF] Chen | Biaxial deformation of collagen and elastin bers in coronary adventitia[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] show that the elastin bers are mainly oriented along the axial direction, i.e. θ el-ad = 0.

The summary of model parameters and constants is presented in Table 3

f b-ad = 30% f el-ad = 20% f m-ad = 1 -i f i lower scale f f -b = 30%

Results

In this section, we validate the micromechanical model by comparing our model predictions with the macroscopic mechanical response obtained experimentally on each layer of human coronary arteries according to experiments from [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF] and of human thoracic aorta according to [START_REF] Weisbecker | Layer-specic damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening[END_REF]. This validation was performed individually for the adventitia layer. The parameters of the model were chosen based on Section 3.3.2 and summarized in Table 3.6. The only free parameter of the model, the global angle of collagen bundles in the adventitia θ b-ad , was calibrated by minimizing the gap between experimental and computational stress-strain curves using a genetic algorithm. Furthermore, as a rst estimate, a linear elastic representation of the media was considered along with the adventitia micromechanical model in order to evaluate the model by comparing the mechanical response and kinematics of the bers with experiments performed by [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] on rabbit carotid artery.

Tensile test on the arterial adventitia

Uniaxial tensions were simulated based on the micromechanical model of the adventitia in order to compare the predicted mechanical response with the experimental results according to [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF] (see Fig. 3.7(a)) and also according to [START_REF] Weisbecker | Layer-specic damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening[END_REF] (see Fig. We compared the bundle rotations with the ane kinematic model, where the θ angle evolution is given by the following equation:

θ = tan -1 λ circ λ long tan Θ (3.2)
whereby λ circ and λ long respectively stand for the measured circumferential and longitudinal stretches and Θ is the initial bundle orientation.

(a) (b) The model is also able to predict the bril decrimping, where brils tend to align with the bundle direction, as shown on Fig. 3.9 (although the predicted decrimping remains very limited here). It is interesting to notice that the decrimping does not occur from the beginning of the load application, which is in agreement with experimental observations [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. In the present model, the non linearity of the mechanical response arises from the combined reorientation of the collagen bundles and brils, while the elastin bers exhibit very limited reorientations (again in good agreement with experiments).

Tensile test on artery

Finally, the global response of the arterial wall was computed by averaging across the whole thickness, pooling together the adventitia and the media in order to compare the results of the model with the experiment performed on a carotid artery strip [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The arterial structure is stratied. Therefore, when loading the structure in directions parallel to the layers, the average response is equal to the Voigt bound, and reads as: Accounting for the media as a linear material is a good rst approximation, since it allows to retrieve the correct initial slope of the curve (which would be Mohsen Nakhaei Thèse en Mécanique et Ingénierie

Discussion

The multiscale micromechanical approach has been used to model the nonlinear and anisotropic mechanical responses of adventitia including directly phenomena occurring at the microscopic scale such as collagen ber recruitment and non-ane ber rotations. The objective was to propose a model for the adventitia layer built on the minimum of microstructural information, which is necessary to obtain reliable predictions of macroscopic mechanical behavior. To this aim, the main orientation of collagen and elastin bers was obtained from image analysis of the adventitia images and considered in our representative volume element. Note that modeling the main ber orientation instead of the ber networks was also used in the well-established soft tissue models [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen bre orientations[END_REF][START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] Holzapfel | Nonlinear solid mechanics ii[END_REF][START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF].

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie ane ber rotation at low stresses, where the bers have the same deformation as matrix, (ii) non-ane ber rotation at higher stresses, where the bers can generate shear stresses to rotate more than the matrix, which is in good agreement with experimental observations [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF]. Furthermore, the brils in the lower scale RVE tend to align with the ber direction (bril decrimping, see Fig. 3.9), where the decrimping does not happen right after the load application in agreement with the experimental observations [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF].

Anisotropy

In the model, at the lower scale, the matrix phase and collagen brils are isotropic materials, while the homogenized RVE (collagen bundle at upper scale) behaves anisotropically. Therefore, at the upper scale, the collagen bundles are anisotropic, while the matrix is isotropic and nally the homogenized response of the adventitia behaves anisotropically. The shape and the orientation of the bers and brils phase within the RVEs dictate the anisotropy behavior of the model. This hierarchical structure and preferred orientation of constituents have been often seen in the biological tissues, where the micromechanical framework has been applied before to model this anisotropic behavior in bone [START_REF] Fritsch | universal'microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity[END_REF][START_REF] Fritsch | Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength[END_REF][START_REF] Fritsch | Mechanical behavior of hydroxyapatite biomaterials: an experimentally validated micromechanical model for elasticity and strength[END_REF].

Sensitivity analysis on load increment size

The developed multiscale model is highly non linear, which requires the development of an incremental algorithm, as developed in Chapter 2. Besides, the implicit dependence of the dierent concentration operators on the orientations Mohsen Nakhaei Thèse en Mécanique et Ingénierie 3.5. DISCUSSION 68 of the inclusions were solved with an explicit scheme for the discretization of the equations. We therefore propose a sensitivity analysis to verify the convergence of the results for dierent increment sizes. The results are reported in Fig. 3.12, in which the variation of mechanical response was not signicant. Hence, increments of 0.2 kPa were chosen to perform the simulations.

Sensitivity analysis on the collagen orientation and fraction at both scales

We extended our sensitivity analysis to all parameters of the models whose values exhibited variations in the literature, namely: the volume fractions of collagen brils and bundles and their orientations. For each parameter, we here show its impact on the response while maintaining all other parameters constant and equal to the value indicated above in Table 3.6.

The most inuential parameter is the orientation of the collagen bundles: the variation of only 4 • in the bundle orientation produces in fact the similar scattering in the mechanical response as the variation of 15

• in the bril orientation (compare Fig. 3.13(a) and Fig. 3.14(a)). Increasing the angle between collagen bundles and direction of the load (i.e. in circumferential tension 90 -θ b-ad ) leads to increasing the time it takes to stien the material, since it takes more time for the ber to reorient and to align with the load direction. As seen in Fig. 3.13(b), by increasing the angle between collagen bundles and direction of the load, the ber is aligned with the direction of the load at higher stretches. While by increasing the angle between collagen brils and collagen bundles, collagen brils reorient less (see

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie Increasing the volume fraction of collagen bundles leads to decreasing the time needed to stien the material. In this condition, we have less volume fraction of ground substance and more interaction between bundles which leads to faster reorientation with the load direction.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

Contribution of the matrix stiness

We here test how the matrix stiness aects the mechanical response of the adventitia. To this aim, the matrix Young's modulus at both scales was varied in the range kPa, and the eect is shown in the Fig. 3.17

Conclusion

The micromechanical modeling was employed in a multi-scale approach to address the non-ane deformation of bers at large deformation as well as taking into account the anisotropy of collagen and elastin bers.

To the best of our knowledge, the developed constitutive model is the rst to address the nonlinearity of the adventitia tissue through the collagen bundle reorientation and the anisotropy through the preferred orientation of the bers as it has been observed experimentally [START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The modeling of these microstructural phenomena and access to the microstructural deformation and stresses is crucial for the constitutive model to be capable of accounting for the growth and remodeling, damage and modeling of other pathologies.

The model built on the minimum microstructural information, which can directly be measured from images and predictions of the macroscopic and the microscopic mechanical behavior of adventitia. It is interesting to note that our model well captures the macroscopic mechanical response of the adventitia, with only one tted parameter, all the others being taken from the microstructural analysis. As the results demonstrate, the model predicts the contribution of collagen and elastin to mechanical load as well as the evolution of micromechanical parameters such as microstructural stresses and microstructural evolution.

In order to have a better understanding of the contribution of the arterial media layer into nonlinearity and anisotropy of the mechanical response of the artery, it is required to build a detailed, microstructural model. This is the aim of the next chapter.

Micromechanical representation of media

The media layer is made of lamellar units, each unit being made of lamellae, consisting of oblate spheroids containing dense networks of elastin and collagen (see Fig. 4.1), as well as of interlamellar space in between where smooth muscle cells are living (see Fig. 4.2). The mechanical properties of these layers arise from the interplay of their constituents. We here employ the previously introduced theoretical framework (see chapter 2) to propose a micromechanical model of the media, which whose microstructure was described in details in chapter 1.

At the scale of a few hundreds of micrometers

The modeling of the media layer at the scale of a few hundreds of micrometers

At the scale of a few micrometers

Modeling the elastic lamellae

The elastic lamellae are made of dense networks of elastin bers, and a few percent of collagen bers type I and III. In the elastic lamellae micromechanical model, these bers are modeled as innitely long cylinders embedded in a soft ground substance, which is a suitable candidate for the Mori-Tanaka homogenization scheme (see Fig. 4.3 (C)) [START_REF] Mori | Average stress in matrix and average elastic energy of materials with mistting inclusions[END_REF]. The collagen ber phase is labeled by the subscript "c -la", while the elastin phase is labeled by the subscript "el -la", and the matrix phase by "m -la". Accordingly, the respective volume fractions of each of these phases read as f c-la , f el-la , and f m-la , with their sum being equal to 1.

Modeling the interlamellar space

The interlamellar space contains SMCs, elastin, and collagen bers type I and III. Similarly, in the interlamellar space micromechanical model, the collagen and elastin bers are modeled as innitely long cylinders embedded in a soft ground Mohsen Nakhaei Thèse en Mécanique et Ingénierie substance containing the SMCs, which is a suitable candidate for the Mori-Tanaka homogenization scheme (see Fig. 4.3 (B)) [START_REF] Mori | Average stress in matrix and average elastic energy of materials with mistting inclusions[END_REF]. The collagen ber phase is labeled by the subscript "c -in", while the elastin phase is labeled by the subscript "el -in", and the matrix phase by "m -in". Accordingly, the respective volume fractions of each of these phases read as f c-in , f el-in , and f m-in , with their sum being equal to 1.

Determination of the parameters of the model 4.3.1 Mechanical properties of elementary constituents

The micromechanical models for both lamellae and interlamellar space introduce three distinct elementary constituents, namely: (i) type I and type III collagen bers, (ii) elastin bers, and (iii) ground substance matrix. The mechanical parameters for the elementary constituents of the media layer, i.e., of elastin and collagen bers as well as of the ground substance are the same as those of the adventitia layer model, which are summarized in Table 3.1. In particular, for sake of simplicity, we choose to consider the same mechanical properties for all collagen bers (i.e. type I and type III). Regarding the mechanical properties of the interlamellar space matrix, the mechanical properties of SMCs have been Mohsen Nakhaei Thèse en Mécanique et Ingénierie reviewed in Chapter 1, and a Young's modulus of 50 kPa was chosen for the ground substance of the interlamellar space.

Volume fraction of elementary constituents Amount of elastin bers in the media

The multiphoton stacks of medial tissue from rabbit carotid arteries taken by Witold Krasny [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] were analyzed using the same procedure as for the adventitia, namely: rst, ImageJ allows to separate the two channels of the second harmonic generation (collagen signal) and autouorescence (displaying mainly the elastin bers, but also few collagen bers). Then, the Otsu thresholding technique [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF] allows to access the ber fraction by means of conversion to binary images. As a result the elastin bers volume fraction amounts to 25% as estimated by the average over the nine samples. This is in good agreement with an medial elastin volume fraction of 29% as measured in rat aorta by [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. Our estimated value is also in good agreement with the amount of elastin measured in small animals arteries, as reported in Table 4.1. Bigger species, like porcs, sheeps, or even humans, have a larger amount of elastin. Besides, [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF] stated that approximately Mohsen Nakhaei Thèse en Mécanique et Ingénierie

DETERMINATION OF THE PARAMETERS OF THE MODEL

78

71% of the total medial elastin belongs to the lamellae. Thus, if we split the total volume fraction of medial elastin between 50% lamellae and 50% interlamellar space such that 71% of elastin is present in the lamellae, we can compute an elastin volume fraction in the lamellae of around 35% and in the interlamellar space of around 15%.

Specimen Imaging method Volume fraction Reference

Human abdominal aorta photomicrograph 40% to 60% [START_REF] Tonar | Microscopic image analysis of elastin network in samples of normal, atherosclerotic and aneurysmatic abdominal aorta and its biomechanical implications[END_REF] Sheep carotid photomicrograph 40% to 50% [START_REF] Dehkordi | Scleroprotein and vascular smooth muscle cell distribution of common carotid media in sheep and goats[END_REF] Rat subrenal aorta photomicrograph 35% to 45% [START_REF] Kochova | Morphology and mechanical properties of the subrenal aorta in normotensive and hypertensive rats[END_REF] Porcine aorta photomicrograph 30% to 50%

[135]

Wistar rats aorta photomicrograph 20% to 25% [START_REF] Hayashi | Composition of connective tissues and morphometry of vascular smooth muscle in arterial wall of doca-salt hypertensive ratsin relation with arterial remodeling[END_REF] Table 4.1: Review of the volume fraction of elastin bers in the media

Amount of collagen bers in the media

The volume fraction of medial collagen bers is also evaluated using the second harmonic generation channel through ImageJ followed by the Otsu thresholding technique [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF]. As a result, the fraction of collagen ber in the media amounts to around 25% as given by the average over the nine samples tested by Witold Krasny [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF], which falls well in the range of all other measures reported in the literature, as summarized in Table 4.2.

It has been observed experimentally that the collagen bers are distributed between the lamellae and the interlamellar space [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF], but the exact proportions are not reported. For simplicity, we split the total volume fraction of medial collagen between 50% in the lamellae and 50% in the interlamellar space, such that the fraction of collagen ber in the lamellae and interlamellar space becomes equal to 25%.

Specimen Imaging method Volume fraction Reference

Sheep carotid photomicrograph 40% to 50% [START_REF] Dehkordi | Scleroprotein and vascular smooth muscle cell distribution of common carotid media in sheep and goats[END_REF] Wistar rats aorta photomicrograph 20% to 25% [START_REF] Hayashi | Composition of connective tissues and morphometry of vascular smooth muscle in arterial wall of doca-salt hypertensive ratsin relation with arterial remodeling[END_REF] Porcine aorta photomicrograph 15% to 22%

[135]

Human abdominal aorta photomicrograph 12% to 20% [START_REF] Tonar | Microscopic image analysis of elastin network in samples of normal, atherosclerotic and aneurysmatic abdominal aorta and its biomechanical implications[END_REF]

Specimen Imaging method Volume fraction Reference

Porcine abdominal aorta photomicrograph 65% [START_REF] Tonar | Orientation, anisotropy, clustering, and volume fraction of smooth muscle cells within the wall of porcine abdominal aorta[END_REF] Rat subrenal aorta photomicrograph 50% to 55% [START_REF] Kochova | Morphology and mechanical properties of the subrenal aorta in normotensive and hypertensive rats[END_REF] Human abdominal aorta photomicrograph 40% to 45% [START_REF] Tonar | Microscopic image analysis of elastin network in samples of normal, atherosclerotic and aneurysmatic abdominal aorta and its biomechanical implications[END_REF] Table 4. Dierent contributions [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF][START_REF] Chen | Microstructural constitutive model of active coronary media[END_REF][START_REF] Yu | Transmural variation in elastin ber orientation distribution in the arterial wall[END_REF] showed that the elastin bers in the media are mainly oriented along the circumferential direction both in the lamellae and the interlamellar space, i.e. θ el-med = 90 • . A small fraction of elastin (5%) was assigned a longitudinal orientation, θ el-med = 0 • to represent the interconnection of the elastin network. This circumferential orientation was also found when analyzing the multiphoton stacks of Witold Krasny [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. Besides, some portion of the elastin bers are tilted in the radial-circumferential plane, with an orientation parallel to the tilting of the SMC, i.e. φ el-in = φ el-la = 70 • [100], while some elastin bers networks display no radial tilting, i.e. φ el-in = φ el-la = 90 • . The existence of truss bers in the interlamellar space motivates us for the introduction of a small portion (5%) of elastin bers with a purely radial orientation, i.e.

φ el-in = 0 [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF].

Orientation of the collagen bers

The orientation of collagen bers was analyzed using OrientationJ, a series of ImageJ plugins for directional image analysis [START_REF] Püspöki | Transforms and operators for directional bioimage analysis: a survey[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF]. We extracted a main orientation for medial collagen bers θ c-med = 85 • , which is in good agreement with the results of [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. According to the literature [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF], these collagen bers have an inclination in the radial-circumferential plane but no study reported this angle. Thus, the inclination of collagen bers in the radial-circumferential plane of the media φ c-med is chosen as a free parameter. We consider that the collagen bers have the same initial orientation in the lamellar and interlamellar space.

A summary of model parameters and constants is given in table 4.4.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

f c-la = 25% f m-la = 1 -i f i Interlamellar space f c-in = 25% f m-in = 1 -i f i orientation Lamellae θ c-med = 85 • NA φ c-med = Free parameter NA Interlamellar space θ c-med = 85 • NA φ c-med = Free parameter NA Parameters Elastin 1 Elastin 2 Elastin 3 mechanical constant E el1 = 1 MPa E el2 = 1 MPa E el3 = 1 MPa ν el1 = 0.34 ν el2 = 0.34 ν el3 = 0.34 volume fraction Lamellae NA f el2-la = 30% f el3-la = 5%
Interlamellar space

f el1-in = 5% f el2-in = 5% f el3-in = 5% orientation Lamellae NA θ el2-med = 90 • θ el3-med = 0 • NA φ el2-la = 70 • φ el3-la = 90 •
Interlamellar space

θ el1-med = 90 • θ el2-med = 90 • θ el3-med = 0 • φ el1-in = 0 • φ el2-in = 70 • φ el3-in = 90 • Table 4
.4: Summary of the medial model parameters and constants.

Results

Tensile test on the arterial media

Uniaxial tensions were simulated with the micromechanical model of the media to compare the predicted mechanical response with the experimental results reported for human coronary arteries [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF] in the circumferential and longitudinal directions.

A genetic algorithm was employed to determine the collagen inclination φ c-med so as to t simultaneously the circumferential and longitudinal curves. (a)

θ c-med = 85 • R 2 = 0.71 (b) θ c-med = 85 • &70 • R 2 = 0.95

83

results of the model with the experiment performed on the whole carotid artery [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The overall stress of the artery is obtained from Eq. (3.3) by assuming the same strain rate in the adventitia and media layers (according to the arrangement of the layers with respect to the load directions).

The constants of the micromechanical model of the media and adventitia are the ones reported for the previous validations. Nevertheless, the free parameter of the adventitia model, i.e. the global orientation of collagen bundles θ b-ad and the free parameter of the media model φ c-med were determined by tting the predicted response to the experimental response using a genetic algorithm. The inclination of collagen bers in the radial-circumferential plane of the media for all the carotid artery samples is obtained as φ c-med = 50 The micromechanical approach was extended to model as well the medial layer of artery.

For the sake of simplicity two RVEs represent the media layer namely: elastic lamellae and interlamellar space. The minimum microstructural information, which is necessary to achieve reliable predictions of macroscopic mechanical behavior was taken into account to establish the RVEs. Then, the media layer was modeled as a lamellar structure of these two units, which has been observed experimentally [START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. The RVE of elastic lamellae consists of elastin and collagen bers and interlamellar space's RVE consists of elastin and collagen bers, and smooth muscle cells (SMCs). In the RVE of interlamellar space, SMCs was represented as a soft ground substance, since modeling the SMCs as inclusion (see primary media's RVEs in Fig. The results show that the model is capable of capturing the macroscale mechanical response and the microscale ber kinematics for the medial layer. The bers in the medial layer under load do not reorient largely as in the adventitial layer (compare Fig. 4.6 to Fig. 3.8 (b) from Chapter 3 for the adventitial ber reorientations), which is also in good agreement with experimental observations [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. This bers' interaction has been correctly predicted by the Mori-Tanaka assumption [START_REF] Mori | Average stress in matrix and average elastic energy of materials with mistting inclusions[END_REF], although Mori-Tanaka scheme only considers the modeling of ber-matrix interaction, in which bers can only feel each other through the matrix rather than the direct interaction between the network of bers.

Furthermore, considering all these interactions in a complex biological tissue through the implementation of nite element is computationally expensive [START_REF] Khang | Multi-scale modeling of the heart valve interstitial cell[END_REF].

Although for the sake of simplicity we consider one family of collagen bers, results show the ber dispersion can improve the prediction of the mechanical behavior of the medial layer (compare Fig. 4.5(a) and Fig. 4.5(b)). Note that we did not have access to the microstructural information of the tested medial samples on the human coronary artery by [START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF], and microstructural information was obtained from the rabbit carotid artery images. The mechanical responses of the whole artery obtained from the Voigt bound estimation Eq. (3.3), this is only valid when the load direction is parallel to the direction of the layers. In order to apply more complex load conditions such as an ination test on the artery, the implementation of the nite element method is necessary to account for the interaction of these RVEs.

Among the parameters introduced in the medial model, the dierent volume fractions vary in wide ranges, depending on the species or on the considered organ.

We therefore chose to perform a sensitivity analysis, so as to understand how these volume fractions impact the overall medial mechanical response. In this sensitivity analysis, we also include the inclination parameter which was determined by a genetic algorithm. For each parameter, we here show its impact on the uniaxial circumferential response of the media, while maintaining all other parameters constant and equal to the value indicated above in Table 4.4.

The investigation reveals that the most inuential parameter is the orientation of the collagen bers φ c-med : increasing the angle between the bers and the load directions (i.e. in circumferential tension 90-φ c-med ) leads to increasing the time it takes to stien the material, since it takes more time for the ber to reorient and to align with the load direction (see Fig. the lamellae and interlamellar space in the media. As lamellae contains higher fraction of elastin bers which make it stier as compared to the interlamellar space, these elastin bers as well resist against reorientation of collagen bers.

Increasing the fraction of lamellae leads to decreasing the time it takes to stien the material as well as decreasing the non-linear behavior of the material (see

Conclusion

In this chapter, the micromechanical approach was extended to model the were compared with the experimental results of the mouse renal artery [START_REF] Faury | Developmental adaptation of the mouse cardiovascular system to elastin haploinsuciency[END_REF].

Furthermore, the distribution of stress and ber orientations were investigated across the arterial wall.

Position of the problem

We consider an arterial segment with a simplied geometry of a hollow cylinder with length L , internal radius R i , and thickness H, and made of two concentric layers, namely the adventitia as the outer layer and the media as the inner one (see Fig. 5.1). The intima is not modeled due to its negligible role in the arterial mechanics. This initial conguration is considered as the unloaded conguration and the reference for stresses and strains is chosen on this conguration. The arterial segment is subjected to a pressure eld P im on the inner surface ∂Ω i , Mohsen Nakhaei Thèse en Mécanique et Ingénierie 5.2.

POSITION OF THE PROBLEM 92

while the outer surface ∂Ω o is let free of traction forces. Finally, a zero velocity is imposed on the two crosssectional surfaces ∂Ω end . The hypoelastic constitutive models developed in chapters 3 and 4 are chosen for modeling the response of the adventitial and medial layers respectively. This problem can be summarized by the following set of equations:

div (Σ Σ Σ(X, t)) = 0 ∀X ∈ Ω, (5.1)
Σ Σ Σ(X, t) • n(X, t) = P im (X, t) n(X, t) ∀X ∈ ∂Ω i , (5.2) Σ Σ Σ(X, t) • n(X, t) = 0 ∀X ∈ ∂Ω o , (5.3) V (X, t) • e z = V (X, t) • e θ = 0 ∀X ∈ ∂Ω end , (5.4)
Σ Σ Σ(X, t) = C(X, t) : D D D(X, t) ∀X ∈ Ω, (5.5)
whereby Σ Σ Σ, D D D, V , X and n are respectively the macroscopic Cauchy stress tensor, the macroscopic Eulerian strain rate, the macroscopic velocity vector, the macroscopic location vector, and the outer unit normal vector, all of them being evaluated at time t and location X; Ω, ∂Ω i , ∂Ω o , and ∂Ω end are respectively the cylinder volume, inner and outer surfaces, and crosssectional surfaces. Finally, the macroscopic eld of hypoelastic stiness is dened as:

C(X, t) =            C adv if X ∈ Ω adv C lam if X ∈ Ω lam C in if X ∈ Ω in
with subscripts adv, lam, and in denoting respectively the adventitia, the lamellar space, and the interlamellar space.

Mohsen Nakhaei Thèse en Mécanique et Ingénierie

Numerical strategy

Weak formulation

The ination problem is solved by a numerical scheme, which requires rewriting the previously described problem in a weak formulation.

To this aim, the equilibrium equation (5.1) is multiplied by a virtual velocity eld Ṽ and integrated over the arterial volume Ω. Then the divergence theorem is used, and the dierent boundary conditions need to be accounted for. In more details, one gets:

Ω Σ Σ Σ : D D D dΩ = ∂Ω (Σ Σ Σ • n) • Ṽ dS, = ∂Ω i (Σ Σ Σ • n) • Ṽ dS + ∂Ω o (Σ Σ Σ • n) • Ṽ dS + ∂Ω end (Σ Σ Σ • n) • Ṽ dS.
(5.6)

The second integral of the right-hand side is zero according to the boundary condition (5.3). The virtual velocity eld Ṽ belongs to the set of continuous and continuously dierentiable eld on Ω, which are kinematically admissible with the problem dened by Eqs. (5.1) to (5.5). Accordingly, the third equation of the right-hand side is also equal to zero. Therefore, Eq. (5.6) simplies as: ȧt = a t+dt -a t dt def = ∆a t .

Ω Σ Σ Σ : D D D dΩ = ∂Ω i (P im n) • Ṽ dS, Ṽ (X, t) • e z = Ṽ (X, t) • e θ = 0 ∀X ∈ ∂Ω end . ( 5
(5.8)

Leading, for the Cauchy stress eld, to:

Σ Σ Σ t+dt = Σ Σ Σ t + Σ Σ Σdt = Σ Σ Σ t + C t : D D D t dt, (5.9)
where equation (5.5) has been accounted for.

Considering that all mechanical congurations are known up to time t, one aims at computing the mechanical equilibrium at time t + dt. Incorporating Eq. (5.9) into Eq. (5.7), written for the conguration at time t + dt, leads to:

Ω t+dt D D D t : C t : D D D dΩ = ∂Ω i,t+dt P t im n t+dt • Ṽ dS + ∂Ω i,t+dt ∆P t im n t+dt • Ṽ dS - Ω t+dt Σ Σ Σ t : D D D dΩ, Ṽ (X, t + dt) • e z = Ṽ (X, t + dt) • e θ = 0 ∀X ∈ ∂Ω end , (5.10)
whereby the pressure eld P im has been discretized according to Eq. (5.8).

Simplifying assumptions for solving the problem

The weak formulation Eq. (5.10) is written on the conguration at time t + dt, which remains unknown, and on which there is no certainty that the stress at time t is in equilibrium with the internal pressure at time t (i.e. that the rst and last terms on the right-hand side cancel). Also, the stiness of the tissue and the hidden concentration tensors are evolving with the load application. To overcome these diculties, extra assumptions need to be made: (i) the stiness remains constant during the load increment, (ii) the geometry change during the interval dt is small enough to assume that the stress eld and the applied pressure at time t are in equilibrium on the conguration at time t + dt, and (iii) the update of the tissue stiness does not induce a further imbalance. In other words, we assume that the loading is discretized in suciently small steps, and therefore, Mohsen Nakhaei Thèse en Mécanique et Ingénierie equilibrium can be neglected. As a result, the equilibrium reduces to:

Ω t D D D t : C t : D D D dΩ = ∂Ω i,t ∆P t im n t • Ṽ dS, Ṽ (X, t) • e z = Ṽ (X, t) • e θ = 0 ∀X ∈ ∂Ω end .
(5.11)

Finite element discretization

In order to solve the spatial integral over the volume of the geometry at each time increment, a nite element approach is used. As it is shown in Fig. 5.1, both the geometry and the loading applied on the artery display a cylindrical symmetry.

This reduces the problem to an axisymmetric problem. The study is restricted to the surface corresponding to the intersection of the hollow cylinder with the plane (e r , e z ) dened in cylindrical coordinate system by a constant θ angle (see Fig. 5.2). The equilibrium equation (5.1) in cylindrical coordinates, reads as:

∂Σ rr ∂r + ∂Σ rz ∂z + Σ rr -Σ θθ r = 0 ∂Σ rz ∂r + ∂Σ zz ∂z + Σ rz r = 0, (5.12)
and the boundary conditions reduce to: where the deformation and evolution of the microstructure can be approximated to be linear at each increment. The homogenized fourth order stiness matrix C at each time increment was used to compute the element stiness matrix.

Σ Σ Σ • n = P im on ∂Ω i , Σ Σ Σ • n = 0 on ∂Ω o , Ṽ (X, t) • e z = 0 ∀X ∈ ∂Ω end .
Furthermore, the computed spin tensor at each increment was utilized to update the microstructure evolution θ and φ. A summary of the numerical scheme for the nite element implementation is illustrated in

FEM SOLVER VALIDATION TEST CASES 98

axial stress, which is predicted uniform through the wall thickness by our model, although it is not the case in the non-linear Abaqus code [START_REF] Smith | ABAQUS/Standard User's Manual[END_REF]. It seems that the above-mentioned error caused by our linearization approach as this uniform axial stress through the wall thickness is also observed when we solve the problem with the linearized Abaqus code [START_REF] Smith | ABAQUS/Standard User's Manual[END_REF] (see Fig.

Validation for a thin-wall cylinder by comparison with an equivalent problem on a RVE

As a second validation test case, we consider a thin-wall cylinder (with average radius R a = 0.28mm and thickness H = 0.01mm) subjected to an inner pressure P im on the inner surface, while all other surfaces are considered as traction free.

The cylinder is made up of the adventitial material as described in chapter 3 (see Fig. 5.5).

The analytical expression of the stress eld of such a problem reads as:

Σ Σ Σ =     Σ rr 0 0 0 Σ θθ 0 0 0 0     , (5.14)
with,

Σ θθ = P im R a H Σ rr = - P im 2 .
(5.15)

The stress is computed from Eq. step. Then, assuming a uniform stress eld in the thin wall cylinder, we compared the response of an RVE with the FEM implementation.

Thus, an ination simulation was performed using the nite element solver, then the computed inner radius-pressure curve was compared with the analytical solution. The FE simulation was performed with dierent numbers of elements (2, 5 and 10 elements) across the thickness of the cylinder, with four integration points per element. The constitutive relation being computed at the integration point, one RVE is assigned to each integration point. The results show that a convergence is reached with 5 elements across the thickness (see Fig. 5.6). Furthermore, the results show an excellent agreement between the nite element and analytical solutions ( 0.5% error with respect to the analytical solution).

Mohsen Nakhaei Thèse en Mécanique et Ingénierie However, by increasing the thickness, we do not obtain a uniform stress across the cylinder wall as it was assumed for the analytical solution, thus there is an increasing discrepancy between the analytical and FE results. The error is 0.9% for the cylinder with a thickness of H =0.025mm and is 1.3% for the cylinder with a thickness of H =0.05mm.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

Application to the arterial ination

Since the ination test is the closest test to the physiological conditions [START_REF] Jay D Humphrey | Computer-aided vascular experimentation: a new electromechanical test system[END_REF]88,[START_REF] Yu | Transmural variation in elastin ber orientation distribution in the arterial wall[END_REF], we here simulate the experiments performed on mice samples by Gilles Faury [START_REF] Faury | Developmental adaptation of the mouse cardiovascular system to elastin haploinsuciency[END_REF]. More precisely, the experiments were performed as follows: a segment of the mouse renal artery was quickly excised and placed in a physiological buer. Then, the artery was cleaned of connective tissue and fat before being mounted onto a pressure arteriograph, where the proximal and distal ends of the arterial segment were grasped with extra-ne-point no. 5 microforceps and held in position on the microcannula with a Lexan pinch clamp [START_REF] Faury | Developmental adaptation of the mouse cardiovascular system to elastin haploinsuciency[END_REF][START_REF] Faury | Relation between outer and luminal diameter in cannulated arteries[END_REF] (see Fig. Accordingly, the arterial wall is modeled as a thick wall cylinder, with axially constrained ends (see Fig. 5.9). The mechanical loading and boundary conditions to the ination problem read as follows: an increasing internal pressure P im is applied incrementally from 0 to 175 mmHg on the inner surface of the cylinder, while the outer surface is let traction free. Furthermore, a symmetry condition along the axial direction is considered at the two crosssections. The experiments reported an initial inner radius of R i = 0.17 mm, and an outer radius of R o = 0.22 mm, the wall thickness is divided into two equally thick sub-layers, the media and adventitia. The adventitia layer is modeled by employing the developed multi-scale model of Chapter 3. The media layer is divided into four equally thick sublayers, being alternatively an elastic lamellae and some interlamellar space (see Fig. 5.9). The mechanical properties of each of these layers are assigned according to Chapter 4.

The response of the FE model in terms of internal pressure versus inner radius evolution is compared to the experimental ination results, as shown on Fig. 5.10 and the collagen ber and bril evolutions are also reported. Two dierent simulations have been performed: the rst one was run with the sets of parameters identical to the ones reported in Chapter 3 and 4 (see Table 5 curve. This may be due to a dierent microstructural composition or morphology.

To investigate this aspect, several simulations were performed to capture the mechanical properties of the mouse renal artery. These investigations illustrate that there are two ways to achieve better prediction of the mouse renal artery's mechanical response: (i) reducing the volume fraction of the collagen bers, and

(ii) increasing the level of hydration of the collagen bers.

E el1 = 1 MPa E el2 = 1 MPa E el3 = 1 MPa ν el1 = 0.34 ν el2 = 0.34 ν el3 = 0.34 volume fraction Lamellae NA f el2-la = 30% f el3-la = 5%
Interlamellar space literature, the media layer contains collagen type III bers, which have a lower elastic modulus as compared to the bers made of collagen type I. Asgari et al. [START_REF] Asgari | In vitro brillogenesis of tropocollagen type iii in collagen type i aects its relative brillar topology and mechanics[END_REF] showed that the elastic modulus of a tissue decreases exponentially as the ratio of Col-III:Col-I increases. For the sake of simplicity, we previously considered the same mechanical properties for the collagen bers in the adventitia and media layers (E = 250MPa). In the lower scale RVE of the adventitia model, the collagen brils have a Young's modulus of E=250 MPa, which leads to a modulus of the collagen bers of E = 50 MPa after homogenization (which accounts for their hydration degree). Consequently, we decided to assign the hydrated medial collagen bers the lattest value for their Young's modulus. Accordingly, the model is able to better capture the mechanical response of the mouse renal artery, when this change is added to the previously detailed modications in the parameters (see Fig. The collagen brils are oriented initially with an angle of 43 • with respect to the collagen bundle orientation (θ f -adv ) in the adventitia.

f el1-in = 5% f el2-in = 5% f el3-in = 5% orientation Lamellae NA θ el2-med = 90 • θ el3-med = 0 • NA φ el2-la = 70 • φ el3-la = 90 • Interlamellar space θ el1-med = 90 • θ el2-med = 90 • θ el3-med = 0 • φ el1-in = 0 • φ el2-in = 70 • φ el3-in = 90
The results demonstrate that increasing the internal pressure also induces an evolution of the brils orientation. Fig. 5.14 shows that the brils were reoriented toward the direction of the collagen bundles by increasing the internal pressure, i.e. they undergo decrimping. The region with maximum circumferential stress corresponds also to the region where the brils experienced the largest decrimping.

Mohsen Nakhaei Thèse en Mécanique et Ingénierie

Discussion

The multiscale micromechanical model was implemented in the nite element formulation to simulate the ination of an artery. For the sake of simplicity and in order to reduce the computational costs, several simplication assumptions have been made to solve the problem. The arterial geometry was simplied to a hollow cylinder and the initial conguration was considered as stress-free and strain-free. Due to the hypoelastic formulation of the problem, it is assumed that the stiness remains constant during the load increment for innitely small time steps dt. It is also assumed that the deformation during the time interval dt is small such that the stress eld and the applied pressure at time t are in equilibrium on the conguration at time t + dt, and that the update of the stiness tensor does not induce a further imbalance.

After validating the home-made nite element solver, the multi-scale Furthermore, the stress distribution through the arterial wall revealed good accordance with ber orientation, which has also been observed experimentally [START_REF] Yu | Transmural variation in elastin ber orientation distribution in the arterial wall[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The results show the capability of the model to capture the microstructural evolution as well as the macroscopic mechanical response of the artery over time from ber evolving congurations to structural modeling.

Conclusion

In conclusion, we have shown that the hypoelastic formulation in the framework of continuum micromechanics is capable of accurately predicts the microscale ber kinematics and macroscopic mechanical response of the artery over time from ber evolving conguration to structural modeling. This model could provide new insights into structural constitutive modeling of arteries to incorporate the patient-specic features from microstructure up-to arterial geometry of the patient, since the model permits the investigation of dierent parameters such as bers orientation and fraction on the macroscopic response of an artery, as well as damage occurring at the ber level. As a perspective towards patient-specic simulation, the present model can be integrated inside nite element commercial software to solve the complex geometry of the artery in 3D and without the simplications which have been made in this study.

General Conclusion

As we discussed, the physical mechanisms and microstructural evolution that occur within the microstructure of the artery dictate its mechanical properties. To investigate these microstructural mechanisms and their macroscopic mechanical consequences, we proposed a hypoelastic constitutive model in the framework of continuum micromechanics, taking into account universal patterns (representative volume element) of the hierarchical organization of arteries at dierent scales.

• For the sake of simplicity, the minimum of microstructural information, which is essential to obtain reliable predictions of macroscopic mechanical behavior of the artery. The main orientation of collagen and elastin bers, as well as their volume fractions, were obtained from image analysis of the arterial images and considered in our representative volume element (RVE). The representation of the ber network in the form of the principal orientation of bers was also used in some well-established constitutive models [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen bre orientations[END_REF][START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] Holzapfel | Nonlinear solid mechanics ii[END_REF][START_REF] Holzapfel | Determination of layer-specic mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling[END_REF].

• To the best of our knowledge, the developed constitutive model is the rst to address the nonlinearity of the arterial tissue through the collagen bundle reorientation and the anisotropy through the preferred orientation of the bers as it has been observed experimentally [START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF][START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. In our model, elastin ber, collagen bril, and soft ground matrix were modeled as hypoelastic material (hypoelastic modulus and the hypoelastic Poisson's ratio) and the only source of non-linearity is due to microstructural evaluation.

Hypoelastic formulation accounts for the large rotation and deformation such that at higher stresses the bers can generate shear stresses to rotate more than the matrix (non-ane deformation). The bers at upper scale RVE reoriented toward the load direction ( ber decrimping) and the brils in the lower scale RVE tend to be aligned with the ber direction (bril decrimping), while the bril decrimping does not occur right after applying the load. All these microstructural phenomena are in good agreement with experimental observations [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF].

• In the model, the matrix phase and bers are isotropic material, while the homogenized RVE behaves anisotropically. The anisotropy behavior of the model comes from the shape and the orientation of the bers within the RVEs. The latter has been also applied in the framework of continuum micromechanics to model the anisotropic behavior of bone [START_REF] Fritsch | universal'microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity[END_REF][START_REF] Fritsch | Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength[END_REF][START_REF] Fritsch | Mechanical behavior of hydroxyapatite biomaterials: an experimentally validated micromechanical model for elasticity and strength[END_REF].

• The ber interaction has been correctly predicted by the Mori-Tanaka assumption [START_REF] Mori | Average stress in matrix and average elastic energy of materials with mistting inclusions[END_REF]. In the media layer, the higher fraction of bers limits their rotation compared to the rotation of bers in the adventitia layer.

This observation is in good agreement with the experimental measurements [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Chow | Arterial extracellular matrix: a mechanobiological study of the contributions and interactions of elastin and collagen[END_REF]. Although Mori-Tanaka scheme only considers the modeling of ber-matrix interaction, in which bers can only interact between each other through the matrix rather than the direct interaction between the network of bers. However, considering all these interactions in a complex biological tissue through the implementation of the nite element method is computationally expensive [START_REF] Khang | Multi-scale modeling of the heart valve interstitial cell[END_REF].

• The multiscale micromechanical model was implemented in the nite element formulation to simulate the ination of an artery. For the sake of simplicity and for reducing the computational costs, several simplication • Although bers are oriented randomly in all directions in the tissue, the model considers only the main orientation family bers. The model also can account for a probability distribution function of ber orientation as it was tested during the implementation. However, for a statistically valid sampling size, we needed a large number of bers which made the computational cost very high.

• The biological components are known as complex heterogeneous and nonlinear materials, for instance, proteins are even modeled as a polymer chain [START_REF] Ikai | The world of nano-biomechanics[END_REF]. However, in the proposed model, the elementary constituents were modeled as isotropic linear elastic at lower scale RVEs. For the sake of simplicity and in order to account for the microstructural phenomena and to illustrate its eect on the macroscopic mechanical response, the constituents were assumed as isotropic linear elastic. However, some experimental observation supports this assumption. For instance, brils have been shown

to behave almost linearly at the lower scale in their deformation range [START_REF] Joost | Micromechanical testing of individual collagen brils[END_REF][START_REF] Ikai | The world of nano-biomechanics[END_REF].

Future work will permit to extend the proposed arterial constitutive model in contributing to:

• Growth and remodeling: as this micromechanical framework has become a reliable tool to model hard tissues (bone) [START_REF] Fritsch | universal'microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity[END_REF][START_REF] Fritsch | Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength[END_REF][START_REF] Morin | A multiscale poromicromechanical approach to wave propagation and attenuation in bone[END_REF][START_REF] Vass | Review of universal rules governing bone composition, organization, and elasticity across organizational hierarchies[END_REF], it is expected that it can also become a basic tool to study the contribution of dierent microstructural components in the mechanical behavior of the artery. This • Accounting for further microstructural mechanisms: The model is capable to include the pre-strain in the microstructure at the ber network level. Due to the existence of the pre-strain in the artery [START_REF] Cardamone | Origin of axial prestretch and residual stress in arteries[END_REF][START_REF] Chuong | Residual stress in arteries[END_REF], this could improve the model to have a better prediction of arterial structure mechanical response.

This can also be extended to incorporate the microstructural damage at the ber network level.

• Patient-specic simulation: the constitutive model could be integrated into the nite element commercial software to incorporate the patient-specic features from the microstructure up-to arterial geometry of the patient.

A | Eshelby auxiliary tensor

The Eshelby auxiliary tensor L is dened as L(x) = Ω I G(x, x )dV (x ), where G(x, x ) is the elastic Green's function. The elastic Green's function is the velocity eld v(x), solution of the momentum balance in which the elastic constitutive function and compatibility equation have been incorporated, i.e. the solution of C m ijkl u k,li + b j = 0, when the body force b is a delta function, i.e. b j (x) = δ(x -x )δ kj , and C m is the stiness of an innite homogeneous matrix [START_REF] John D Eshelby | The determination of the elastic eld of an ellipsoidal inclusion, and related problems[END_REF] The fourth-order Eshelby concentration tensors for strain rate and spin were introduced in the Eqs. (2.26) and (2.27) are respectively equal to [START_REF] John D Eshelby | The determination of the elastic eld of an ellipsoidal inclusion, and related problems[END_REF]:

S ijmn = 1 2 (L iklj + L jkli ), R ijmn = 1 2 (L iklj -L jkli ), (A.4)

B | Image processing

The stacks of images from Krasny experiments [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF] were analyzed to determine the volume fractions and orientation of collagen and elastin bers in the adventitia and media layers. In total, we analyzed 12 stacks of images, coming from four samples tested in the longitudinal, circumferential and diagonal directions respectively. In each case, the initial conguration which is reported is analyzed after few cycles of preconditioning in the load direction. These images were analyzed with ImageJ through the Otsu thresholding technique [START_REF] Otsu | A threshold selection method from gray level histograms[END_REF] to compute the volume fractions of elastin and collagen. The orientation of bers was analyzed using orientationJ, a series of ImageJ plugins for directional image analysis [START_REF] Püspöki | Transforms and operators for directional bioimage analysis: a survey[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF], as well as the 2D

fast Fourier transformation method proposed by [START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]. The results are shown for the adventitia in Arterial tissues are made of variously organized collagen and elastin networks and exhibit a highly nonlinear anisotropic behavior with the ability to sustain large reversible strains and to undergo a load-induced progressive morphological rearrangement of the microstructure. In the present study motivated by these specicities of arterial mechanics, we developed a detailed multi-scale model of the arterial wall. The model accounts for the universal patterns across dierent scales in the two mechanically signicant layers of arteries, namely the adventitia and the media. The framework of nite strain continuum micromechanics was employed in an incremental approach to compute stress, strain, and ber reorientations. The extensions of Eshelby's matrix-inclusion problem allowed for deriving analytical expressions for the concentration tensors. These concentration tensors relate the macroscopic strain rate tensor to the strain rate and spin tensors averaged over the constituents of the microstructure, to the macroscopically applied strain rate. In the model, the strain rate is related to the averaged stress in each constituent, and the spin tensor is related to the load-induced ber rotation. The proposed adventitial model accounts for both collagen decrimping and realignment. The media is modeled as a stack of lamellae and interlamellar space; both lamellae and interlamellar space being homogenized with relevant representative volume element (RVE). Furthermore, the multi-scale constitutive model was implemented in a nite element formulation to solve the structural model of the artery. The model was validated against dierent experimental data sets on arterial samples from dierent species (rabbit, human, mouse). The results show that the model is able to estimate the contribution of each component into the macroscopic response of the tissue for dierent loading and can predict both the macroscopic response and microscopic ber kinematics accurately. We submit that such model would help in predicting the evolution of the mechanical tissue response overtime during, for instance, remodeling and growth or damage.

Stéphane

  Avril and Dr. Claire Morin, for their continuous support and guidance during my Ph.D. They gave me the great opportunities to work on various interesting topics such as large deformation continuum micromechanics and nonlinear nite element methods. Their insightful comments were very inspiring and challenging, which enriched this thesis. I also gratefully acknowledge the opportunity of working at Multi-Scale Tissue Biomechanics Lab at Boston University. I would like to express my great appreciation to Prof. Katherine Yanhang Zhang for giving me the opportunity to obtain a deeper perspective of experimental tests and become familiar with developing dierent constitutive modelings.

. 4

 4 [139], by means of electron microscopy; (F1) & (F2) respectively show the collagen and elastin bers in the structure of lamellae sheets; (G) the structure of interlamellar space contains elastin, collagen bers and SMC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Non linear mechanical response under a uniaxial tensile test of a rabbit carotid artery. Collagen bers are initially crimped (I), then start decrimping (II), and are stretched at large stresses (III). . . . 16 1.3 The structure of rat aorta media [139]. (a) The transverse section of media, magnication ×1, 500. (b) Close view of elastic lamellae, magnication ×11, 000. . . . . . . . . . . . . . . . . . . . . . . . . 19 1Internal elastic lamina of rat aorta, magnication ×3, 000 [139]. . . 20 1.5

  Macroscopic mechanical response of adventitia at high stress. (b) Evolution of ber angles under uniaxial tension in the adventitia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Evolution of bril angle under uniaxial tension . . . . . . . . . . . Mohsen Nakhaei Thèse en Mécanique et Ingénierie LIST OF FIGURES 3.10 Comparison of the mechanical response of the model with the

3. 12

 12 Increment size of load step in simulation of uniaxial tension on arterial adventitia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 The impact of global orientation of collagen bundles on the mechanical response (a) and ber kinematics (b) of arterial adventitia 3.14 The impact of orientation of collagen brils on the mechanical response (a) and ber kinematics (b) of arterial adventitia . . . . . 3.15 The impact of collagen bundles volume fraction on the mechanical response (a) and ber kinematics (b) of arterial adventitia . . . . . 3.16 The impact of collagen brils volume fraction on the mechanical response (a) and ber kinematics (b) of arterial adventitia . . . . . 3.17 The impact of the matrix stiness on the mechanical response (a) and ber kinematics (b) of arterial adventitia . . . . . . . . . . . . 3.18 The impact of the elastin fraction on macroscopic mechanical response of adventitia. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Multiphoton microscopic imaging of bers in the media. (A) collagen bers; (B) elastin bers [82]. . . . . . . . . . . . . . . . . . 4.2 Scanning electron micrographs of a tangentially cut surface of the media in the rat aorta. The elastin brils and smooth muscle cells (M) covered the surface of the elastic lamellae, magnication ×3, 500 [140]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Micromechanical RVEs representing the media. (A) RVE representing the tissue scale; (B) RVE representing the interlamellar space; (C) RVE representing the lamellae. . . . . . . .

  . (b) θ c-med = 85 • &70 • . . angles under uniaxial tension in the media. . . . 4.7 Comparison of the modeled mechanical response and ber kinematics with the experiments [82] for dierent carotid artery samples. φ c-med = 50 • for all samples. . . . . . . . . . . . . . . . . 4.8 Primary micromechanical RVEs representing the media. (A) RVE representing the tissue scale; (B) RVE representing the interlamellar space; (C) RVE representing the lamellae; (D) RVE representing the elastic lamellae. . . . . . . . . . . . . . . . . . . . 4.9 Primary model macroscopic mechanical response of the media human coronary artery according to [63]. . . . . . . . . . . . . . . . 4.10 The impact of dierent parameters of the media model on the circumferential mechanical response (a) out of axial-circumferential plane inclination angle of medial collagen φ c-med (b) in axial-circumferential plane inclination angle of the medial collagen θ c-med (c) fraction of collagen in the media f c-med (d) fraction of elastin in the media f el-med (e) fraction of lamellae in the media f lam (f ) dierent proportions of collagen fraction in the medial lamellae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 (a) The geometry of the problem. (b) Crosssection of cylinder. . . 5.2 Imposed boundary condition on the geometry accounting for symmetry conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 (a) Radial displacement, (b) radial stress, (c) circumferential stress, and (d) axial stress of the nodes across the thickness of the cylinder as computed by our implemented FEM and by Abaqus for a linear elastic material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Axial stress of the nodes across the thickness of the cylinder as computed by linearized Abaqus code and developed FEM solver for a linear elastic material. . . . . . . . . . . . . . . . . . . . . . .

5. 10 97 Nomenclaturedissipation of the system D 1 intrinsic dissipation of the system D 2 thermal 2 characteristicψ

 1097122 Comparing FE model with ination experiment on the mouse renal artery[35]. FEM model (A) : θ b-adv = 39 • and φ c-med = 50 • . FEM model (B): θ b-adv = 40 • and φ c-med = 33 • . FEM model (C): θ f -b = 43 • , f c-la = f c-in = 5% and f f -b = 15%. . . . . . . . . . . 5.11 Comparing FE model with ination experiment on the mouse renal artery[35]. FEM model (D) : θ b-adv = 40 • , φ c-med = 32 • , θ f -b = 43 • , and f c-la = f c-in = f f -b = 15%. . . . . . . . . . . . . . . . . 5.12 (a) Radial displacement, (b) radial stress, (c) circumferential stress, and (d) axial stress at the center of the elements across the thickness of the artery. . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Fiber reorientation at the center of elements across the thickness of the artery at three internal pressures. (a) The orientation of bers in the axial-circumferential plane. (b) The orientation of bers in the radial-circumferential plane. . . . . . . . . . . . . . . . . . . . . 5.14 Fibril reorientation at the center of elements across the thickness of the artery at three internal pressures. . . . . . . . . . . . . . . . A.1 Ellipsoidal inclusion with principal axess of a, b, and c. . . . . . . List of Tables1.1 Overview of the range of values for the Young's modulus of collagen bril and ber found in the literature . . . . . . . . . . . . . . . . 21 1.2 Overview of the range of values for the Young's modulus of elastin ber found in the literature . . . . . . . . . . . . . . . . . . . . . . 22 2.1 Numerical algorithm for the constitutive model under imposed stress conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Mechanical parameters of the adventitita micromechanical model. . 58 3.2 Collagen bril volume fraction within a collagen bundle . . . . . . 58 3.3 Reported values for the volume fraction of collagen bundles in the adventitia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Reported values for the volume fraction of elastin bers in the adventitia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 The waviness and corresponding local angle of collagen bundles . . 61 3.6 The summary of arterial adventitia model parameters and constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Review of the volume fraction of elastin bers in the media . . . . 78 4.2 Review of the volume fraction of collagen bers type I and III in the media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Review of the volume fraction of SMCs in the media . . . . . . . . 79 4.4 Summary of the medial model parameters and constants. . . . . . 81 4.5 Obtained model parameters after calibration for each carotid artery sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1 Numerical algorithm for nite element implementation. . . . . . . strain rate concentration tensor A ∞ fourth-order eigenstrain rate-to-spin concentration tensor in the Eshelby inhomogeneity problema i half length of the principal axes of the ellipsoidal inclusion C fourth-order hypo-elastic stiness tensor C 0 fourth-order hypo-elastic stiness tensor of the matrix of the Eshelby problem C I fourth-order hypo-elastic stiness tensor of the inclusion of the Eshelby problem C hom homogenized fourth-order hypo-elastic stiness tensor C eq r fourth-order equivalent "stiness" tensor of phase r accounting for the objective rate D dissipation of the system d second-order microscopic Eulerian strain rate tensor D second-order macroscopic Eulerian strain rate tensor D 0 applied second-order macroscopic Eulerian strain rate tensor in the Eshelby problem length of the heterogeneities of the lower scale representative volume element d r second-order microscopic averaged Eulerian strain rate in phase r d M second-order microscopic averaged Eulerian strain rate in the matrix phase dt time increment E Young's modulus e r , e θ , e φ spherical base vectors f volume fraction f second-order microscopic deformation gradient F second-order macroscopic deformation gradient G Gibbs thermodynamic potential I fourth-order identity tensor I second-order identity tensor J volumetric part of the second-order identity tensor K deviatoric part of the second-order identity tensor k bulk modulus l characteristic length of the representative volume element l 2 characteristic length of a lower scale representative volume element L characteristic length of the arterial structure or of the applied loading L Esh fourth-order eigenstrain rate-to-velocity gradient concentration tensor in the Eshelby problem M denotes the matrix phase N Hill tensor q outward heat ux vector R fourth-order strain rate-to-spin concentration tensor R r fourth-order strain rate-to-spin concentration tensor of phase r R Esh fourth-order eigenstrain rate-to-spin concentration tensor in the Eshelby problem R ∞ fourth-order eigenstrain rate-to-spin concentration tensor in the Eshelby inhomogeneity problem s entropy per unit mass (also called specic entropy) S Esh fourth-order eigenstrain rate-to-strain rate concentration tensor in the Eshelby problem T Poisson's ratio of the matrix of the Eshelby problem ρ mass density σ microscopic second-order Cauchy stress tensor Σ macroscopic second-order Cauchy stress tensor φ The longitudinal angle between inclusion and the global coordinate system Helmholtz free energy per unit mass Ω second-order, skew symmetric, spin tensor ω of the lower scale RVE up ... of the upper scale RVE Subscripts Description b -ad ... of the collagen bundle phase in the adventitia RVE c -in ... of the collagen phase in the interlamellar space RVE c -la ... of the collagen phase in the lamellae RVE el -ad ... of the elastin phase in the adventitia RVE el -in ... of the elastin phase in the interlamellar space RVE el -la ... of the elastin phase in the lamellae RVE f -b ... of the brils in the collagen bundle RVE M ... of matrix phase m -ad ... of the matrix phase in the adventitia RVE m -b ... of the matrix phase in the collagen bundle RVE m -in ... of the matrix phase in the interlamellar space RVE of the matrix phase in the lamellae RVE M, I ... of the matrix phase of the lower scale RVE attached to the phase I of the upper scale RVE r ... of phase r r, I ... of phase r of the lower scale RVE attached to the phase I of the upper scale RVE

1. 1 . 15 Figure 1 . 1 :

 11511 Figure 1.1: The hierarchical structure of arteries. (A) macroscopic artery of a mouse, taken from [74]; (B) the three layers of arterial wall, the adventitia tunica (A), the tunica media (M) and the intima tunica (I) from [112], by means of an electron micrograph; (C) the adventitia is made of an arrangement of collagen bundles (red) and elastin bers (green) taken from [16], by means of multiphoton microscope; (D) the media is made of an arrangement of lamellae sheets and interlamellar space [100]; (E) the collagen bundles are made of an arrangement of collagen brils, as taken from [139], by means of electron microscopy; (F1) & (F2) respectively show the collagen and elastin bers in the structure of lamellae sheets; (G) the structure of interlamellar space contains elastin, collagen bers and SMC.

Figure 1 . 2 :

 12 Figure 1.2: Non linear mechanical response under a uniaxial tensile test of a rabbit carotid artery. Collagen bers are initially crimped (I), then start decrimping (II), and are stretched at large stresses (III).

1. 2 . THE TUNICA ADVENTITIA 17 1. 2

 2172 The tunica adventitiaThis paragraph focuses on the description of the extracellular matrix of the tunica adventitia from a mechanical point of view.

1. 3 . 1

 31 The media morphologyThe tunica media is made of an arrangement of several medial lamellar units, see Fig.1.1(D)[START_REF] John | Transmural organization of the arterial media. the lamellar unit revisited[END_REF][START_REF] O'connell | The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3d confocal and electron microscopy imaging[END_REF]. Each lamellar unit consists in a row of overlapping SMCs surrounded on the lower and upper sides by two concentric elastic lamellae (see Fig.1.3(a)).More precisely, the elastic lamellae are made of a dense network of circumferentially-oriented elastic bers[START_REF] Farand | Structure of large arteries: orientation of elastin in rabbit aortic internal elastic lamina and in the elastic lamellae of aortic media[END_REF][START_REF] Lucas H Timmins | Structural inhomogeneity and ber orientation in the inner arterial media[END_REF] (see Fig. 1.3(b)).

Figure 1 . 3 :

 13 Figure 1.3: The structure of rat aorta media [139]. (a) The transverse section of media, magnication ×1, 500. (b) Close view of elastic lamellae, magnication ×11, 000.

1. 5 . THE ELEMENTARY CONSTITUENTS 20 Figure 1 . 4 :

 52014 Figure 1.4: Internal elastic lamina of rat aorta, magnication ×3, 000 [139].

1. 5 .

 5 THE ELEMENTARY CONSTITUENTS 22 has a Young's modulus up to 1.2 MPa. According to Gundiah et al. (2007),

Figure 1 . 5 :

 15 Figure 1.5: Elastin bers of rat aortic adventitia, magnication ×5, 500 [139].

1. 7 .

 7 LIMITS OF THE EXISTING MODELS AND OVERVIEW OF THE THESIS 27 microstructure, and these aspects are signicantly altered in cardiovascular diseases [67]. Arterial tissue is a complex multiphase material containing active components such as cells and structural macromolecules. These components undergo a load-induced progressive morphological rearrangement such that tissue exhibits highly nonlinear anisotropic behavior with the ability to sustain large reversible strains. The signicant role of the microstructure morphology and composition on arterial response motivated us to develop a detailed multiscale model of the arterial wall, which could provide new insights into the arterial structure-function relationships. To this aim, we propose a model in the framework of continuum micromechanics under large strains, taking into account universal patterns (representative volume element) of the hierarchical organization of arteries at dierent scales to relate microstructural mechanisms to their macroscopic mechanical consequences. This aim was reached by achieving the following steps:

2. 2 . 31 50µmFigure 2 . 1 :

 23121 Figure 2.1: Microstructure of the most outer layer (called adventitia) of a rabbit carotid artery, imaged under a multiphoton microscope [82].

Figure 2 . 2 :

 22 Figure 2.2: Representative volume element of the arterial adventitia layer.

( 2

 2 .18) Mohsen Nakhaei Thèse en Mécanique et Ingénierie 2.5. MORI-TANAKA ESTIMATION OF STRAIN RATE AND SPIN CONCENTRATION TENSORS FROM AUXILIARY ESHELBY PROBLEMS 36

  Figure 2.3: Inhomogeneity and Eshelby inclusion problems.

2. 5 .

 5 MORI-TANAKA ESTIMATION OF STRAIN RATE AND SPIN CONCENTRATION TENSORS FROM AUXILIARY ESHELBY PROBLEMS 40 are uniform in the inclusion, the strain rate and spin are not uniform outside the inclusion.

  Figure 2.4: Representative Volume Element of heterogeneous materials subjected to homogeneous strain rate as the velocity vector at the boundary.

  Finally, since the material derivative is equivalent to a partial derivative with respect to time in the Lagrangian formulation (xed reference frame) D(•) Dt = ∂(•) ∂t , using a forward Euler scheme for the time derivative of the Cauchy stress σt = σ t+dt -σ t dt , one can compute the updated microscopic Cauchy stress in the form of, σ σ σ t+dt r = σ σ σ t r + C eq t r : D D D t dt, ∀r ∈ {1, 2, 3, ..., N }. σ σ σ t+dt M = σ σ σ t M + C eq t M : D D D t dt.

(2. 47 ) 2 . 6 . 4

 47264 Fiber reorientation and stress average rule Since the inclusions are not necessarily aligned with the global base frame, the previously introduced base frame n i is replaced by a spherical base frame (e r , e θ , Mohsen Nakhaei Thèse en Mécanique et Ingénierie 2.6. ALGORITHMIC TREATMENT OF FIBER KINEMATICS 44 e φ ), and a global Cartesian base frame (e 1 , e 2 , e 3 ) is introduced and attached to the matrix. The relation between the two base frames reads as:

  2.5).Furthermore, in the presence of inclusions in the material which can reorient e r Second Eulerian angle φ

Figure 2 . 5 :)

 25 Figure 2.5: Denition of the two Euler angles θ and φ.

Figure 2 . 6 :

 26 Figure 2.6: Numerical algorithm of homogenization scheme.

5 .

 5 t+dt from Eq. (2.51) c) Compare macroscopic Cauchy stresses (computed and prescribed) and compute the residual:R R R = Σ Σ Σ t+dt -Σ Σ Σ t+dt d) Check the tolerance: ||R R R|| ≤ tol, exit the algorithm, otherwise go back to step 3 and apply the residual R R R instead of ∆Σ Σ Σ 4. Update the homogenized behavior of the material C t+dt hom Compute the macroscopic deformation gradient F F F t+dt from Eq.(2.54)

Figure 2 . 7 :

 27 Figure 2.7: Representation of the dierent RVEs in a multi-scale homogenization scheme.

Fig. 2 .

 2 Fig. 2.8). The material properties of the heterogeneous inclusion in the upper scale is estimated by homogenization over the RVE of the lower scale. The orientation of heterogeneities in the lower RVE is taken into account by two Euler angles θ low and φ low , dening an orthonormal spherical basis (e low r , e low θ , e low φ ) with respect to

Figure 2 . 8 : 59 )

 2859 Figure 2.8:Representative Volume Elements for heterogeneous materials exhibiting a hierarchical microstructure.

(2. 60 )

 60 where f low r,I and f low M,I are respectively the volume fractions of phase r and of the matrix in the lower scale RVE, A low r,I and A low M,I are the strain rate concentration tensors into the phase r and the matrix phase in the lower scale RVE, and C low r,I and C low M,I are the fourth-order stiness tensors of the phase r and of the matrix in the lower scale RVE, all of them being attached to the phase I of the upper RVE. C eq low r,I and C eq low M,I follow from Eq. (2.46), incorporating into this equation the concentration tensors A low r,I , A low M,I , and R low r,I , as well as the local stress tensor σ σ σ low r,I . C I is the homogenized stiness tensor over the lower scale RVE, which allows to Mohsen Nakhaei Thèse en Mécanique et Ingénierie 2.9. CONCLUSION 50 compute the eective (homogenized) Cauchy stress tensor over the lower scale RVE. A summary of the numerical scheme is illustrated in Fig. 2.9.

Figure 2 . 9 :

 29 Figure 2.9: Numerical algorithm for the multi-scale homogenization scheme.

Contents 3 . 1 53 3. 2 . 1 53 3. 2 . 2 56 3. 3 57 3. 3 . 1 62 3. 4 . 1 62 3. 4 . 2 67 3. 5 . 3 67 3. 5 . 4 72 3

 31532153225635731624162426753675472 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Micromechanical representation of the adventitia . . Modelling the adventitia at the scale of a few hundreds of micrometers . . . . . . . . . . . . . . . . . . . . . . . Modelling the collagen bundles of adventitia at the scale of a few micrometers . . . . . . . . . . . . . . . . . . . . Determination of the parameters of the model . . . . Mechanical properties of elementary constituents . . . . 57 3.3.2 Volume fraction of elementary constituents . . . . . . . 58 3.3.3 Orientation of the dierent ber networks . . . . . . . . 60 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensile test on the arterial adventitia . . . . . . . . . . . Tensile test on artery . . . . . . . . . . . . . . . . . . . 64 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.1 Non-linearity . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5.2 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity analysis on load increment size . . . . . . . . Sensitivity analysis on the collagen orientation and fraction at both scales . . . . . . . . . . . . . . . . . . . 68 3.5.5 Contribution of the matrix stiness . . . . . . . . . . . . 70 3.5.6 Contribution of elastin in the mechanical response . . . 71 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . un modèle de comportement de l'artère motivé par la microstructure, l'approche micromécanique multi-échelle qui a été expliquée au chapitre 2 a d'abord été appliquée à l'adventice, qui est la couche externe de l'artère. Ce chapitre présente les étapes nécessaires pour construire le modèle de l'adventice avec des volumes élémentaires représentatifs spéciques à plusieurs échelles élaborés à partir d'observations microstructurales du tissu. Ensuite, la simulation de la traction uniaxiale a été réalisée sur le modèle d'adventice pour étudier le rôle des changements de morphologie des réseaux de bres de collagène et d'élastine et la contribution de ces réseaux à la réponse mécanique. Enn, ces résultats ont été comparés aux résultats expérimentaux et aux observations an d'évaluer la capacité du modèle micromécanique à prédire les observations expérimentales.

3. 2 .

 2 MICROMECHANICAL REPRESENTATION OF THE ADVENTITIA 54 here use the previously introduced theoretical framework to propose a multi-scale model of the adventitia.

Figure 3 . 1 : 55 Figure 3 . 2 : 3 Figure 3 . 3 :

 315532333 Figure 3.1: Multiphoton microscopic imaging of bers in the adventitia [82]. (A) collagen bundles; (B) elastin bers.

3. 2 . 56 3. 2 . 2

 25622 MICROMECHANICAL REPRESENTATION OF THE ADVENTITIA Modelling the collagen bundles of adventitia at the scale of a few micrometersThe collagen bundles modeling needs to be further rened, since collagen bundles are made, at a lower scale, of a staggered arrangement of collagen brils. As reviewed in Chapter 1, the recruitment process of collagen bers has a signicant impact on the mechanical response of arteries and therefore needs to be accounted for. Following the multi-scale homogenization procedure described in Section 2.8 of Chapter 2, we introduce a lower scale RVE, with a characteristic length of 5-10 µm, i.e. satisfying the scale separation conditions, and hosting heterogeneities in the form of innitely long cylindrical inclusions representing the collagen brils, which have a characteristic size of 30100 nm [139]. The remaining volume of the RVE is made of a ground substance matrix. The brils and matrix are respectively labeled by subscripts "f -b" and "m -b", and their respective volume fractions read as f f -b and f m-b = 1-f f -b . We assume an helicoidal crimping: the collagen bers are isotropically distributed in the transverse plane, i.e., φ ∈ [0 -2π]. Once again, a Mori-Tanaka scheme is chosen to account for bers-matrix interactions. The resulting multi-scale micromechanical model representing the arterial adventitia layer is illustrated in Fig. 3.4.

Figure 3 . 4 : 57 3. 3

 34573 Figure 3.4: Micromechanical RVEs representing the adventitia. (A) RVE representing the adventitia tissue scale; (B) RVE representing the collagen bundles.

Figure 3 . 6 :

 36 Figure 3.6: Representative straight form and crimped form of collagen bundle and corresponding angles.

  θ b-ad = Free parameter θ el-ad = 0 • NA φ b-ad = 90 • φ el-ad = 90 • NA lower scale θ f -b = 35 • NA NA φ b-ad = 90 • NA NA

  Fig.3.8(b)). They are in good agreement with experimental observations, namely the bundles tend to align with the direction of tension at large stretches. Note that ber alignment is not completely circumferential or longitudinal at the end of the simulation, as also observed by Witold Krasny during his experiments[START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF][START_REF] Krasny | Kinematics of collagen bers in carotid arteries under tension-ination loading[END_REF].

Figure 3 . 7 :

 37 Figure 3.7: Macroscopic mechanical response of adventitia, (a) human coronary artery according to [63] and (b) human thoracic aorta according to [147] . The free parameter θ b-ad and R 2 are reported for each tissue.

Figure 3 . 8 : 64 Figure 3 . 9 :

 386439 Figure 3.8: (a) Macroscopic mechanical response of adventitia at high stress. (b) Evolution of ber angles under uniaxial tension in the adventitia.

D 45 •

 45 D D artery = D D D adv = D D D med (3.3) σ σ σ artery = f adv σ σ σ adv + f med σ σ σ med (3.4) As a rst estimate the mechanical response of the media layer considered as a linear elastic with Young's modulus of E med . Parameters of the adventitial micromechanical model are chosen identical to the previous validation case. Nevertheless, the only free parameter of the model, i.e. the global orientation Mohsen Nakhaei Thèse en Mécanique et Ingénierie 3.4. RESULTS 65 of collagen bundles θ b-ad , was chosen as a free parameter and determined by calibrating the modeled response against experiments using a genetic algorithms. Choosing θ b-ad = 68 • for sample preconditioned longitudinally and θ b-ad = 31 • for sample preconditioned circumferentially lead to the best tting with R 2 = 0.67. These values of θ b-ad were also in good agreement with the morphology of the tissue. Fiber tracking was performed on few stacks of images recorded by Witold Krasny [82], and it was interesting to notice that the preconditionning of the sample induces a preferred global orientation of the collagen bundles: on these processed stacks, a global orientation of 60 • ±10 • was obtained in the longitudinal direction, whereas the circumferential sample exhibited a global orientation of 35 • ±10 • . This justies the use of two dierent values for the two loading directions. Interestingly also, the global orientation was found to be further away from the angle as compared to the tests performed on the adventitia alone on porcine arteries. The mechanical responses of the arterial model, taking into account the media as a linear elastic with dierent Young's modulus (E med ) of 0.1, 0.5 and 1 MPa are illustrated in Fig. 3.10.

Figure 3 . 10 :

 310 Figure 3.10: Comparison of the mechanical response of the model with the experiment carried out by [82] for whole carotid artery sample, taking into account the media as a linear elastic material with dierent Young's moduli E med , 0.1, 0.5 and 1 MPa.

  no media would have been accounted for). However, developing a multiscale model for the media would permit to gain in non linearity and anisotropy of the global mechanical response of the tissue, as will be shown in the next chapter.

Figure 3 . 11 :

 311 Figure 3.11: Comparison of the ber kinematics with the experiments carried out by [82] for whole carotid artery sample.

  Fig. 3.8 (b)), where the two regimes of ber reorientation were observed; (i)

Figure 3 . 12 :

 312 Figure 3.12: Increment size of load step in simulation of uniaxial tension on arterial adventitia

3. 5 . DISCUSSION 69 Fig. 3 .

 5693 Fig. 3.14(b)).

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: The impact of global orientation of collagen bundles on the mechanical response (a) and ber kinematics (b) of arterial adventitia

Figure 3 . 15 :Figure 3 . 16 :

 315316 Figure 3.15: The impact of collagen bundles volume fraction on the mechanical response (a) and ber kinematics (b) of arterial adventitia

Figure 3 . 17 :

 317 Figure 3.17: The impact of the matrix stiness on the mechanical response (a) and ber kinematics (b) of arterial adventitia

Figure 3 . 18 :

 318 Figure 3.18: The impact of the elastin fraction on macroscopic mechanical response of adventitia.

  élémentaires représentatifs spéciques, construits à partir des piles d'images multiphotoniques du tissu. La validation de notre modèle repose sur la simulation d'un essai de traction uniaxiale et sur la comparaison avec les données expérimentales correspondantes.4.1 IntroductionIn this chapter, we extend the micromechanical model to the arterial media layer, to take into account the nonlinear and anisotropic contribution of this layer in the mechanical response of the artery. Similarly to the development of the micromechanical model of the adventitia layer, this chapter is devoted to discussing the steps required to construct the micromechanical model of the media with specic RVEs built from the multiphoton image stacks of the tissue. The validation of our model relies on the simulation of a uniaxial tension test and on the comparison with the corresponding experimental data.

75 Figure 4 . 1 :

 7541 Figure 4.1: Multiphoton microscopic imaging of bers in the media. (A) collagen bers; (B) elastin bers [82].

4. 3 . 76 Figure 4 . 2 :

 37642 Figure 4.2: Scanning electron micrographs of a tangentially cut surface of the media in the rat aorta. The elastin brils and smooth muscle cells (M) covered the surface of the elastic lamellae, magnication ×3, 500 [140].

4. 3 . 77 Figure 4 . 3 :

 37743 Figure 4.3: Micromechanical RVEs representing the media. (A) RVE representing the tissue scale; (B) RVE representing the interlamellar space; (C) RVE representing the lamellae.

3 : 3 Figure 4 . 4 :

 3344 Figure 4.4: Orientation of a ber in the arterial wall. θ represents the ber inclination with respect to the axial direction of artery, and φ its direction in the radial-circumferential plane.

  MPa E m-la = 0.01 & E m-in = 0.05 MPa ν b = 0.34 νm = 0.34 volume fraction Lamellae

  Finally, an orientation of φ c-med = 53 • for human coronary artery leads to the best agreement between model predictions and experimental results (see Fig. 4.5 (a)). Although the model could predict correctly the mechanical response of the media in the circumferential direction, the prediction of the mechanical response in longitudinal direction remains linear. As the microstructure of the specimen is not reported for the human coronary artery, further investigation was performed on the model to have a better estimation of the mechanical response in both circumferential and longitudinal directions. This investigation shows that the collagen dispersion toward longitudinal direction provides a nonlinear mechanical response in the longitudinal direction. Finally, the collagen ber dispersion with equal volume Mohsen Nakhaei Thèse en Mécanique et Ingénierie 4.4. RESULTS 82 fraction (12.5%) in θ c-med = 85 • and θ c-med = 70 • leads to a better prediction of the mechanical response of the human coronary artery (see Fig. 4.5 (b)). Note that even for a sample of the rabbit carotid artery was observed dispersion and inclination of collagen bers around the main orientation (θ c-med = 80 • ).

Figure 4 . 5 :

 45 Figure 4.5: Macroscopic mechanical response of the media human coronary artery according to [63], (a) θ c-med = 85 • . (b) θ c-med = 85 • &70 • .

Figure 4 . 6 :

 46 Figure 4.6: Evolution of ber angles under uniaxial tension in the media.

Figure 4 . 7 :

 47 Figure 4.7: Comparison of the modeled mechanical response and ber kinematics with the experiments [82] for dierent carotid artery samples. φ c-med = 50 • for all samples.

4 . 8 )

 48 have shown not to have an important impact on the mechanical response of the medial layer due to its low stiness compared to the other constituents. Furthermore, in the primary micromechanical media's RVEs, lamellae modeled with two scales; at the upper scale, elastic lamellae and collagen bers embedded in a soft matrix, and the lower scale was represented the dense networks of elastin bers in the elastic lamellae (see Fig. 4.8). These arrangements had almost doubled the parameters but could not improve the prediction of the model (see Fig. 4.9, R 2 = 0.88).

Figure 4 . 8 :

 48 Figure 4.8: Primary micromechanical RVEs representing the media. (A) RVE representing the tissue scale; (B) RVE representing the interlamellar space; (C) RVE representing the lamellae; (D) RVE representing the elastic lamellae.

Figure 4 . 9 :

 49 Figure 4.9: Primary model macroscopic mechanical response of the media human coronary artery according to [63].

  4.10(a)). In contrast to φ c-med , the angle θ c-med does not have an impact on the results as collagen bers are aligned almost circumferentially and therefore are already aligned with the load direction (see Fig. 4.10(b)).

Figure 4 . 10 :

 410 Figure 4.10: The impact of dierent parameters of the media model on the circumferential mechanical response (a) out of axial-circumferential plane inclination angle of medial collagen φ c-med (b) in axial-circumferential plane inclination angle of the medial collagen θ c-med (c) fraction of collagen in the media f c-med (d) fraction of elastin in the media f el-med (e) fraction of lamellae in the media f lam (f ) dierent proportions of collagen fraction in the medial lamellae.

Fig. 4 .

 4 Fig. 4.10(e)). Further investigations show increasing or decreasing the volume fraction of collagen bers in the lamellae dose not impact the mechanical responses of the model (see Fig. 4.10(f )).

  medial layer. The developed hypoelastic constitutive model in the framework of continuum micromechanics correctly predicted the nonlinear and anisotropic mechanical responses of arteries at the elastic regime. The multiscale layer-specic Mohsen Nakhaei Thèse en Mécanique et Ingénierie 4.6. CONCLUSION 89 model includes directly phenomena occurring at the microscopic scale, such as bers interaction through the matrix, collagen ber recruitment, and non-ane ber rotations, and takes into account universal patterns of the hierarchical organization of arteries at dierent scales. The results illustrate the capability of the model to capture the macroscopic mechanical response of the tissue as well as micromechanical evolution. We hope that this microstructure-based model will bring improvements in the prediction of damage and strength of arterial tissue, since failure of the dierent constituents can be predicted and upscaled to predict the rupture of the tissue. 5.1. INTRODUCTION 91 en couches, à savoir : lamelles, espace interlamellaire et adventice. En raison de la nature incrémentale du modèle constitutif matériel, une formulation lagrangienne actualisée des éléments nis a été utilisée pour prendre en compte les RVE aux points intégraux de chaque élément et mettre à jour la morphologie de la microstructure à chaque incrément. Après validation du modèle par rapport aux solutions semi-analytiques, le modèle d'éléments nis est utilisé pour simuler un test d'ination artérielle. La pression interne en fonction du rayon interne a été calculée pendant la simulation et les résultats ont été comparés aux résultats expérimentaux de l'artère rénale de la souris [35]. De plus, la distribution des contraintes et l'orientation des bres ont été étudiées sur la paroi artérielle. 5.1 Introduction In this chapter, a home-made nite element code was written by taking into account the multi-scale arterial constitutive model for the layer-specic structure of the artery; namely: lamellae, interlamellar space, and adventitia. Due to the incremental nature of the material constitutive model, an updated Lagrangian nite element formulation was employed to take in to account the RVEs at integral points of each element and update the morphology of the microstructure at each increment. After validating the model against the semi-analytical solutions, the nite element model is used to simulate an arterial ination test. The internal pressure versus inner radius was calculated during the simulation and the results

Figure 5

 5 Figure 5.1: (a) The geometry of the problem. (b) Crosssection of cylinder.

  between two successive congurations can be neglected and iterations between local enforcement of the constitutive relation and the global

Figure 5 . 2 :

 52 Figure 5.2: Imposed boundary condition on the geometry accounting for symmetry conditions.

5 .

 5 a) For every element i = 1 to n b) Compute the coordinates of nodes c) Initialize the stiness matrices K K K and K K Ke and right hand sides b and b e to zero (i) Loop over the Gauss points (ii) Compute the Gauss weights wi (iii) Compute the shape functions (iv) Compute the Jacobian (v) Evaluate the derivatives of the shape functions with respect to the global coordinates (vi) Compute the stiness matrix K K Ke and right hand side b e for the element d) Assemble into the global stifness matrix K K K and right hand side b and solve for K K KV = b e) Compute deformation gradient F F F t+dt and Cauchy stress Σ Σ Σ t+dt 4. Update the homogenized behavior of the material C t+dt hom Update the microstructure morphology θ and φ using spin tensor ω ω ω 6. Update the geometry based on deformation gradient Table 5.1: Numerical algorithm for nite element implementation. 5.4 FEM solver validation test cases 5.4.1 Validation with a commercial FEM code: Linear elasticity As a rst step, we considered a linear elastic material with a Young's modulus E = 1 MPa and a Poisson's ratio ν = 0.34 for the arterial wall. The cylinder has an inner radius R i = 0.275 mm and a thickness H = 0.1 mm. 20 quadrilateral elements are taken into account through the thickness of the cylinder. An internal pressure P im = 225 mmHg is imposed on the inner surface of the cylinder. The results of the implemented FEM model are compared with the same model in an FEM commercial software (Abaqus). As seen on Fig. 5.3, both the nal displacement eld and the nal stress eld are correctly predicted by our nite element implementation. A maximum error of 3% has been observed with respect to the Abaqus results. The biggest error in the prediction occurs for the normal Mohsen Nakhaei Thèse en Mécanique et Ingénierie 5.4.

Figure 5 . 3 :Figure 5 . 4 :

 5354 Figure 5.3: (a) Radial displacement, (b) radial stress, (c) circumferential stress, and (d) axial stress of the nodes across the thickness of the cylinder as computed by our implemented FEM and by Abaqus for a linear elastic material.

( 5 .Figure 5 . 5 :

 555 Figure 5.5: The adventitia RVE is remotely subjected to traction forces at its remote boundary.

5. 4 .FEM SOLVER VALIDATION TEST CASES 101 RaFigure 5 . 6 :

 410156 Figure 5.6: The inner radius-pressure curve observed in mesh convergence study of a cylinder having a thickness of H =0.01mm.

5. 5 . 102 RaFigure 5 . 7 :

 510257 Figure 5.7: Comparing the inner radius-pressure curve from FEM and analytical solution of a thin wall cylinder with dierent wall thicknesses (H =0.01mm, H =0.025mm, and H =0.05mm), considering ve elements across the thicknesses.

  5.8). The experiments were performed at 37• C in a bath lled with physiological buer. The setup was placed under an inverted microscope connected to a computerized system Mohsen NakhaeiThèse en Mécanique et Ingénierie 5.5. APPLICATION TO THE ARTERIAL INFLATION 103 Pim ro ri

Figure 5 . 8 :

 58 Figure 5.8: Schematic of ination test for an artery.

Figure 5 . 9 :

 59 Figure 5.9: Ination of an artery regarded as a thick-walled cylinder made of adventitia, elastic lamellae and interlamelar space. The initial inner radius of R i = 0.17 mm, and an outer radius of R o = 0.22 mm.

First, a

  simulation (labelled as FEM model C) was performed where the fraction of collagen bers was reduced in both layers: media and adventitia layers, f c-la = f c-in = 5% and f f -b = 15%. Furthermore, the degree of adventitial collagen bril crimping was increased and set as θ f -b = 43 • . The results illustrated that the FEM model (C) provides a better estimation of the mechanical response of the mouse renal artery (see Fig. 5.10, model C: R 2 = 0.71). The alternative solution to obtain a better prediction of the mechanical response of the unknown microstructure mouse renal artery is increasing the level of hydration of collagen bers, in particular in the media layer. According to the E b = 250 MPa E el = 1 MPa Em = 0.01 MPa ν b = 0.34 νe = 0.34 νm = 0.34 volume fraction upper scalef b-ad = 30% f el-ad = 20% f m-ad = 1i f i lower scale f f -b = 30%NA NA orientation upper scale θ b-ad = Free parameterθ el-ad = 0 • NA φ b-ad = 90 • φ el-ad = 90 • MPa E m-la = 0.01 & E m-in = 0.05 MPa ν b = 0.34 νm = 0.34volume fraction Lamellaef c-la = 25% f m-la = 1i f i Interlamellar space f c-in = 25% f m-in = 1i f iorientationLamellae θ c-med = 85 • NA φ c-med = Free parameter NA Interlamellar space θ c-med = 85 • NA φ c-med = Free parameter

5. 5 . 106 Figure 5 . 10 :

 5106510 Figure 5.10: Comparing FE model with ination experiment on the mouse renal artery[35]. FEM model (A) : θ b-adv = 39 • and φ c-med = 50 • . FEM model (B): θ b-adv = 40 • and φ c-med = 33 • . FEM model (C): θ f -b = 43 • , f c-la = f c-in = 5% and f f -b = 15%.

107 R 2

 1072 Fig.5.12). From the inner radius R i = 0.17mm, the rst 15 elements represent the media layer, and the other 15 elements belong to the adventitia layer. The media layer is divided into four equally thick sublayers, being alternatively an

Figure 5 .Figure 5 .Figure 5 . 13 :

 55513 Figure 5.11: Comparing FE model with ination experiment on the mouse renal artery[35]. FEM model (D) : θ b-adv = 40 • , φ c-med = 32 • , θ f -b = 43 • , and f c-la = f c-in = f f -b = 15%.

5. 6 . DISCUSSION 110 Figure 5 . 14 :

 6110514 Figure 5.14: Fibril reorientation at the center of elements across the thickness of the artery at three internal pressures.

  constitutive model described and validated in Chapter 3 and Chapter 4 for the human coronary artery was implemented in the nite element solver. The variation of pressure-radius was calculated during the internal pressure simulation and compared with the experimental results of the mouse renal artery[START_REF] Faury | Developmental adaptation of the mouse cardiovascular system to elastin haploinsuciency[END_REF]. The result for the pressure-radius variation exhibited the same trend as the experiment, although the response for the human coronary artery, i.e. with the free parameters being set to θ b-adv = 39 • and φ c-med = 50 • was stier as compared to the mouse Mohsen Nakhaei Thèse en Mécanique et Ingénierie 5.7. CONCLUSION 111 renal artery. Since we did not have access to microstructural images for these tests, we decided to vary the free parameters (i.e. we chose θ b-adv = 40 • and φ c-med = 32 • ) as well as the volume fraction and hydration of the collagen bers. It was shown to provide a good estimation of the mechanical response of the artery.

114 •

 114 assumptions have been made to solve the problem. The results show the capability of the model to capture the microstructural evolution as well as the macroscopic mechanical response of the artery over time from ber evolving congurations to structural modeling. An agreement was observed between the simulation and experiment both for bers reorientations and macroscopic mechanical response of the artery.The modeling of these microstructural phenomena and access to the microstructural deformation and stresses is crucial for the constitutive model to be capable of accounting for the growth and remodeling, damage and modeling of other pathologies. The proposed model has also some limitations: The complexity of tissue microstructures cannot be modelled in every detail. Thus, constituents with known physical quantities (shape, volume fraction and Young's modulus) are identied. These physical quantities introduced as physical parameters in the model for each constituent such as shape, volume fraction, orientation, and homogeneous mechanical properties. Determining these parameters is a challenging part of building the constitutive model, although most of them were shown to be constant from sample to sample.

  will be benecial in predicting the evolution of the tissue mechanical response overtime during pathologies inducing remodeling and damage of Mohsen Nakhaei Thèse en Mécanique et Ingénierie General Conclusion 115 the extracellular matrix [122, 105, 142]. This will imply coupling the current version of the model with biological considerations in the near future.

( 3 Figure A. 1 :I ab c 2 I cc = 0 (A. 2 )

 31202 Figure A.1: Ellipsoidal inclusion with principal axess of a, b, and c.

  Modélisation mécanique multi-échelle des tuniques de la paroi artérielle à partir de l'évolution des réseaux de fibres organiques Résumé:Les tissus artériels sont constitués de réseaux de collagène et d'élastine diversement organisés et présentent un comportement anisotrope hautement non linéaire avec la capacité de supporter de grandes déformations réversibles. Ces dernières s'accompagnent d'un réarrangement morphologique progressif de la microstructure induit par le chargement. Dans cette thèse, le rôle important de la microstructure sur la réponse artérielle macroscopique nous a motivé à développer un modèle multi-échelle détaillé de la paroi artérielle. Le modèle tient compte des motifs universels à diérentes échelles dans les deux couches mécaniquement importantes des artères, à savoir l'adventice et la media.Le cadre de la micromécanique des milieux continus a été utilisé dans une approche incrémentale pour calculer la contrainte, la déformation et les réorientations de bres. Les extensions du problème d'inclusion de la matrice d'Eshelby permettent d'obtenir des expressions analytiques pour les tenseurs de concentration, qui relient le tenseur de vitesse de déformation macroscopique à la vitesse de déformation et les tenseurs de spin moyennés sur les phases, à la vitesse de déformation appliquée macroscopiquement. Dans le modèle, la vitesse de déformation donne accès à la contrainte moyenne dans chaque phase, et le tenseur de spin permet de déterminer la rotation de la bre induite par la charge. Le modèle adventitiel proposé tient compte à la fois de la déformation et du réalignement du collagène. La media est constituée d'un empilement de lamelles et d'un espace interlamellaire; les lamelles et l'espace interlamellaire sont tous deux homogénéisés à partir de VER. De plus, le modèle de comportement multi-échelle a été implémenté dans une formulation éléments nis, an de réaliser des calculs de structure sur l'artère. Le modèle a été validé par rapport à diérents ensembles de données expérimentales sur des échantillons artériels de diérentes espèces (lapin, humain, souris).Les résultats montrent que le modèle est capable d'estimer la contribution de chaque composant dans la réponse macroscopique du tissu pour diérents chargements et peut prédire avec précision à la fois la réponse macroscopique et la cinématique microscopique des bres. Nous pensons qu'un tel modèle pourrait aider à prédire l'évolution de la réponse mécanique du tissu au l du temps, par exemple lors d'un remodelage ou d'une lésion. Mots-clefs : Micromécanique du continu, Homogénéisation, Elasticité à grande déformation, Tissus mous, Modélisation constitutive, Méthode des éléments nis non linéaires. LAYER-SPECIFIC MULTISCALE MECHANICAL MODELING OF ARTERIAL STRUCTURES WITH EVOLVING FIBER CONFIGURATIONS Abstract:

  Representative Volume Element of heterogeneous materials subjected to homogeneous strain rate as the velocity vector at the boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Denition of the two Euler angles θ and φ. . . . . . . . . . . . . .

								LIST OF FIGURES
		2.4
		2.5
		2.6	Numerical algorithm of homogenization scheme. . . . . . . . . . . .
		2.7	Representation	of	the	dierent	RVEs	in	a	multi-scale
			homogenization scheme. . . . . . . . . . . . . . . . . . . . . . . . .
		2.8	Representative Volume Elements for heterogeneous materials
										[139]. . 22
	2.1	Microstructure of the most outer layer (called adventitia) of a
		rabbit carotid artery, imaged under a multiphoton microscope [82].	31
	2.2	Representative volume element of the arterial adventitia layer. . . . 31
	2.3	Inhomogeneity and Eshelby inclusion problems.	. . . . . . . . . . 37

exhibiting a hierarchical microstructure. . . . . . . . . . . . . . . .

  • L'extension du modèle à la média est décrite au chapitre 4.

Là

encore, les paramètres physiques du modèle ont été mesurés à l'aide du traitement d'images, et certains ont été déterminés à partir de la littérature.

Les données expérimentales de la média ont été utilisées pour valider individuellement la réponse mécanique de cette couche du tissu. Ensuite, nous avons eectué une analyse de sensibilité pour étudier l'impact des paramètres sur le modèle. Enn, nous avons combiné les deux modèles micromécaniques de la média et de l'adventice pour eectuer une simulation uniaxiale de l'artère entière et valider le modèle par rapport aux résultats expérimentaux tant en termes de réponse mécanique que d'évolution de la microstructure.

  and zero otherwise.

	Substitution of Eq. (2.30) into Eqs. (2.28) and (2.29) gives the linear relation
	between the prescribed strain rate at the boundary of RVE and the uniform strain
	rate in the inclusion, reading as:

Table 2 .

 2 1. 1. Prescribed macroscopic Cauchy stress Σ Σ Σ t and Σ Σ Σ t+dt at time t and t + dt 2. Initialize the homogenized behavior of the material C t hom at time t according to Eq. (2.52) 3. Perform an iteration loop to applying the Cauchy stress ∆Σ Σ Σ = Σ Σ Σ t+dt -Σ Σ Σ t a) Compute macroscopic strain rate D D D t at time t from the linear approximation Eq. (2.53) b) For all constituents perform down-scaling, local constitutive behavior and up-scaling (i) Compute microscopic strain rate d d d t and spin ω ω ω t at time t from Eq. (2.41) (ii) Compute Cauchy microscopic stress σ σ σ t+dt from Eq. (2.47) (iii) For all bers compute updated orientation angles θ t+dt , φ t+dt from from Eq. (2.50) (iv) Compute updated homogenized behavior of the material C t hom from Eq. (2.52) (v) Compute macroscopic Cauchy stress Σ

	Σ Σ

Table 2 .

 2 1: Numerical algorithm for the constitutive model under imposed stress impacts its mechanical behavior. It is then possible to model this hierarchical structure in a multi-scale approach by considering a lower scale RVE, which takes into account the heterogeneity of that phase (see Fig. 2.7). This lower RVE has a characteristic length l 2 , with l 2 ≤ d. Again, the RVE representing the phase must follow the scale separation rule, that is d 2 l 2 , with d 2 as the characteristic length of the lower scale heterogeneities, to build up the multi-scale homogenization scheme (see Fig. 2.7).

	conditions
	Mohsen Nakhaei	Thèse en Mécanique et Ingénierie

Table 3 .

 3 5: The waviness and corresponding local angle of collagen bundles

	35 •	[19]

  .6.

	Parameters	Collagen	Elastin	Matrix
	mechanical constant	E b = 250 MPa	E el = 1 MPa	Em = 0.01 MPa
			ν b = 0.34	νe = 0.34	νm = 0.34
	volume fraction	upper scale

Table 3 .

 3

6: The summary of arterial adventitia model parameters and constants Mohsen Nakhaei Thèse en Mécanique et Ingénierie 3.4. RESULTS 62

Table 4 .

 4 2: Review of the volume fraction of collagen bers type I and III in the media Amount of SMCs in the media Although this fraction is not needed in our model, we here report values for the volume fraction of SMCs in the media, for a future extension of the model, explicitely accounting for the SMCs. The volume fraction of SMCs has been assessed by a stereological method on histological sections of the medial porcine abdominal aorta by[START_REF] Tonar | Orientation, anisotropy, clustering, and volume fraction of smooth muscle cells within the wall of porcine abdominal aorta[END_REF]. They reported a volume fraction of around 65% for the SMCs. Other studies, as summarized in Table4.3, report smaller fractions of SMC for other species and arteries.

Table 4 .

 4

	Sample	Preconditioning θ b-ad	R 2
	Sample 1	circ long	30 67.5	0.82
	Sample 2	circ long	37 66	0.87
	Sample 3	circ long	31.5 71	0.75
	Sample 4	circ long	29.5 68.5	0.78

• . The global orientation of adventitial collagen bundles θ b-ad , obtained after calibration, is reported in Table

4

.5 for each carotid artery sample. The results also show the eect of preconditioning, which also observed from experimental tests

[START_REF] Krasny | A comprehensive study of layer-specic morphological changes in the microstructure of carotid arteries under uniaxial load[END_REF]

as discussed in Chapter 3. Good agreements are found between the modeled response and the experiments, both in terms of macroscopic stress-stretch behavior and in terms of rotations of microstructural components. Results are shown in Fig. 4.7. 5: Obtained model parameters after calibration for each carotid artery sample.

Mohsen Nakhaei

Thèse en Mécanique et Ingénierie

  As in the previous chapters, an explicit forward Euler scheme is used, i.e. any time derivative is evaluated as follows:

		5.3. NUMERICAL STRATEGY	94
	5.3.2 Temporal discretization
	Due to the non-linear nature of the problem (resulting from the hypoelastic
	constitutive formulation), a temporal discretization is introduced; all variables
	are evaluated at a series of time instants t represented by a constant time interval
	dt.
			.7)
	Mohsen Nakhaei	Thèse en Mécanique et Ingénierie

Table 5

 5 Initialize Σ Σ Σ t , F F F t , and the homogenized behavior of the material C t hom at time t from Eq. (2.52) 3.Perform an iteration loop to compute deformation gradient F F F t+dt and Cauchy stress Σ Σ Σ t+dt

	.1.

Table 5 .

 5 2: Summary of the model parameters and constants.

	Mohsen Nakhaei	Thèse en Mécanique et Ingénierie

  Table C.1 and for media in Table C.2. Note that the orientation values are reported in these tables are the main orientations that were obtained from 2D fast Fourier transformation.

Acknowledgements

First of all, I would like to express my great gratitude to my advisors, Prof.