M Miniaci

Matteo Mazzotti

Ada Amendola

Fernando Fraternali

Effect of prestress on phononic band gaps induced by inertial amplification

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Phononic crystals and elastic metamaterials have proved to be powerful platforms to achieve a plethora of unconventional vibrational behaviours deriving from their peculiar dispersion diagrams, characterized by frequency band gaps (BGs) and pass-bands, i.e. frequency regions where the propagation of waves is inhibited or supported, respectively [START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF] . This allowed to achieve frequency selective wave filtering 2 , guiding [START_REF] Zhu | Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces[END_REF][START_REF] Mousavi | Topologically protected elastic waves in phononic metamaterials[END_REF][START_REF] Miniaci | Experimental observation of topologically protected helical edge modes in patterned elastic plates[END_REF] and splitting [START_REF] Su | Elastic metasurfaces for splitting sv-and p-waves in elastic solids[END_REF][START_REF] Miniaci | Valley-based splitting of topologically protected helical waves in elastic plates[END_REF] , stimulating the conception of pioneering applications, such as ultra-sensitive devices [START_REF] Miniaci | Proof of concept for an ultrasensitive technique to detect and localize sources of elastic nonlinearity using phononic crystals[END_REF] , large scale metamaterials for vibration shielding [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Fraternali | Innovative seismic isolation devices based on lattice materials: A review[END_REF] , subwavelength imaging [START_REF] Moleron | Acoustic metamaterial for subwavelength edge detection[END_REF] , elastic wave cloaking / lensing [START_REF] Zhang | Broadband acoustic cloak for ultrasound waves[END_REF][START_REF] Misseroni | Cymatics for the cloaking of flexural vibrations in a structured plate[END_REF][START_REF] Bigoni | Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization[END_REF] , etc.

In this context, an important issue is that phononic crystal and elastic metamaterial operational frequencies are typically fixed once the structure has been designed and fabricated. In contrast, in the majority of practical applications, including the ones previously mentioned, it may be desirable to dynamically or adaptively tune the BGs (or some of the dispersion branches) in terms of frequency, also after the structure has been fabricated. To overcome this limitation, periodic systems with adaptive elastic properties have recently been proposed. For instance, BG tuning has been achieved by means of piezoelectric materials [START_REF] Casadei | Broadband vibration control through periodic arrays of resonant shunts: experimental investigation on plates[END_REF][START_REF] Bergamini | Phononic crystal with adaptive connectivity[END_REF][START_REF] Kherraz | Controlling bragg gaps induced by electric boundary conditions in phononic piezoelectric plates[END_REF] , temperature variation [START_REF] Jim | Thermal tuning of phononic bandstructure in ferroelectric ceramic/epoxy phononic crystal[END_REF][START_REF] Cheng | Temperature effects on the band gaps of lamb waves in a one-dimensional phononic-crystal plate (l)[END_REF][START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF][START_REF] Wu | Effect of thermal stresses on frequency band structures of elastic metamaterial plates[END_REF] , magneto-and light-based approaches [START_REF] Robillard | Tunable magnetoelastic phononic crystals[END_REF][START_REF] Matar | Band gap tunability of magneto-elastic phononic crystal[END_REF][START_REF] Guo | Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses[END_REF][START_REF] Zhang | Tunability of band structures in a two-dimensional magnetostrictive phononic crystal plate with stress and magnetic loadings[END_REF][START_REF] Gliozzi | Tunable photo-responsive elastic metamaterials[END_REF] , as well as by the application of external mechanical loads [START_REF] Bordiga | Prestress tuning of negative refraction and wave channeling from flexural sources[END_REF] . Among them, the latter approach can be easily implemented by imposing controlled displacements into specific portions of the structure.

In this context, Bigoni et al. [START_REF] Bigoni | Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics[END_REF] proposed for the first time the prestress as a practical way to reversibly alter the dispersion diagram of a periodic structure, including the possibility of shifting the BG position. The feasibility of the approach was confirmed formulating a theoretical model for an orthotropic, prestressed (compressible) elastic layer vibrating on an elastic half space and assuming long-wave asymptotics for the solution. Gei et al. [START_REF] Gei | Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects[END_REF] relaxed the hypothesis of perfect periodicity and investigated the effect of the prestress in quasi-periodic structures in the case of flexural vibrations. Amendola et al. [START_REF] Amendola | Tuning frequency band gaps of tensegrity mass-spring chains with local and global prestress[END_REF] studied the band structure of tensegrity mass-spring chains, and the possibility to tune the dispersion relation of such systems by suitably varying local and global prestress variables, given their remarkable softening / stiffening response under axial or compressive loading [START_REF] Fraternali | On the mechanical modeling of the extreme softening/stiffening response of axially loaded tensegrity prisms[END_REF][START_REF] Amendola | Experimental investigation of the softening-stiffening response of tensegrity prisms under compressive loading[END_REF] .

Periodic elastomeric structures, thanks to their capacity of repetitively undergoing large strain deformations in a fully reversible manner, brought to BG nucleation / annihilation mechanisms triggered by the application of external loads radically changing the unit cell geometry [START_REF] Babaee | Harnessing deformation to switch on and off the propagation of sound[END_REF] . Deformations in the linear and nonlinear regimes, as well as diverse geometrical topologies have been explored [START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF][START_REF] Bertoldi | Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations[END_REF][START_REF] Wang | Mechanically tunable phononic band gaps in three-dimensional periodic elastomeric structures[END_REF][START_REF] Rudykh | Transforming wave propagation in layered media via instability-induced interfacial wrinkling[END_REF] .

Finally, it has been observed that also mechanical instabilities induced, for instance, by the application of external loads, may alter the propagation of elastic waves. Slesarenko et al. induced instabilities in soft composite materials achieving a significant decrease of the group velocity (up to going negative) of the transverse waves under specific micropolar conditions [START_REF] Slesarenko | Foreshadowing elastic instabilities by negative group velocity in soft composites[END_REF] .

However, the majority of the aforementioned investigations are limited to the context of (i) Bragg-or (ii) ordinary resonance-induced BGs, whereas in the present paper, we report about the effects of an applied prestress on (iii) periodic structures exhibiting BGs induced by inertial amplification mechanisms. While in (i) the wave scattering derives from assemblages of periodic unit cells, requiring thus the wavelength of the incoming wave to be comparable to the structural periodicity [START_REF] Romero-García | Multi-resonant scatterers in sonic crystals: Locally multi-resonant acoustic metamaterial[END_REF] , and in (ii) the dynamic behaviour is mainly governed by the eigenfrequencies of resonators included in the structure [START_REF] Liu | Locally Resonant Sonic Materials[END_REF] , in (iii) large inertial forces are generated by amplifying the motion of a mass, which in turn increases the inertia of the overall system and lowers its resonance frequency, allowing thus for sub-wavelength and broadband BGs, while keeping the structure lightweight [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Acar | Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures[END_REF] .

If compared to the cases of Bragg scattering and ordinary local resonant metamaterials [START_REF] Mazzotti | Modeling bloch waves in prestressed phononic crystal plates[END_REF] , we show that the inertial amplification allows for a more remarkable curve shift, modulated through large but fully reversible compression(stretch) of the unit cell, eventually triggering significant enlargement(reduction) of a specific band gap width. We also observe a selective up(down)-shift of some dispersion branches, leading to negative group velocities over specific wavenumber ranges. In addition, a non-monotone trend of the lower limit of the first BG under the same type of external applied prestrain is found and explained through an analytical model, which shows that this behaviour derives from the different unit cell effective mass and stiffness variations as the prestrain level increases. Examining the deformed geometries consequent the application of the prestress highlights how the remarkable band shift is due to the hinge-like behaviour of some regions of the unit cell, typical of the inertial amplification mechanism and responsible for a large deformation state, i.e. change of the unit cell geometry. We consider the static deformation induced by the prestress to be in the linear elastic regime so to have a complete reversibility of the phenomena (tunability). The analysis is performed in terms of small amplitude motions superimposed on a deformed state once the desired load has been applied.

The paper is organized as follows: in section II, the 2-step Updated Lagrangian scheme, including (i) a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load and (ii) the Floquet-Bloch decomposition applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration, is presented. Section III provides evidence for the dispersion band alteration induced by the application of the prestress in a periodic structure exhibiting BG induced by inertial amplification mechanism.

Parametric dispersion curves for different values of prestress are presented and compared to the original structure (i.e., without the application of any prestress). Afterwards, a 2-step nonlinear transient numerical simulation confirms the BG shift induced by the prestress in a finite structure, proving its ability to switch from a pass-to a stop-band behaviour over the same frequency range. The prestress is easily implemented by applying prescribed displacement at specific edges of the structure prior to the wave propagation. The full tunability of the structure is guaranteed by the possibility of readily applying and removing the imposed deformations. Finally, section IV summarizes the main results of the present research and provides future perspectives, shedding light on the possibility of investigating the effect of an additional state of prestress, internal, cross-linking the fields of metamaterials and tensegrity structures.

II. FLOQUET-BLOCH ANALYSIS OF A PRESTRESSED PHONONIC CRYSTAL

In this work, the band structure diagrams are computed using a Floquet-Bloch finite element method formulated within an Updated Lagrangian scheme. As schematically presented in Fig. 1, the procedure consists of two main computational steps, namely (i) a nonlinear static analysis involving large strains and displacements, and (ii) a so called small-on-large dynamic analysis, in which small vibrations are superimposed on the statically deformed unit cell. The main computational aspects of the two steps are outlined in the following.

A. Static analysis

Following the application of a static volume load f V 0 , or surface load f S0 , the unit cell Ω 0 identified by the position vector x and represented by the lattice vector r 0 = {r 01 , 0} T and its reciprocal vector g 0 = {g 01 , 0} T = {r 01 /2π, 0} T , undergoes a displacement u 0 that results in a change of configuration from the undeformed state C 0 to the static deformed state C (see Fig. 1). The deformed domain and matrix of the lattice vectors in the (deformed) configuration C are identified by Ω and r = {r 1 , 0} T , respectively. The relation between r 0 and r can be expressed as r = F L r 0 , where F L defines the affine component of the deformation gradient such that F L = F -1 F P , being F = ∇ x u 0 + I the deformation gradient, F P a periodic non-affine deformation [START_REF] Zhang | Soft phononic crystals with deformation-independent band gaps[END_REF] and ∇ x = {∂/∂x 1 , ∂/∂x 2 } T . The equilibrium equations with respect to the undeformed configuration C 0 can be derived from the variational statement:

Ω (S(x) : δE(x) -f V 0 • δu 0 ) da = ∂Ω f s0 • δu 0 ds, (1)
subjected to the Dirichlet boundary conditions:

u 0 (x + r 0 ) = u 0 (x), (2)
in which E = 1 2 (F T )F -I is the Green-Lagrange strain tensor and S = D 0 : E the second Piola-Kirchhoff stress tensor. The tensor of tangential elastic moduli, D 0 , is expressed by

D 0 = 4∂ 2 ψ/(∂C∂C), being C = F T F the right
Cauchy-Green deformation tensor and ψ the elastic energy density. The material of the unit cell in C 0 is specified by the density ρ 0 while, assuming a hyperelastic material behavior described by the Murnaghan's model [START_REF] Murnaghan | Finite deformations of an elastic solid[END_REF][START_REF] Pau | Nonlinear guided wave propagation in prestressed plates[END_REF][START_REF] Dubuc | The effect of applied stress on the phase and group velocity of guided waves in anisotropic plates[END_REF][START_REF] Dubuc | Higher order longitudinal guided wave modes in axially stressed seven-wire strands[END_REF] , the elastic energy density can be defined as:

ψ = 1 2 (λ + 2µ) I 2 1 (E) -2µI 2 (E) + 1 3 (l + 2m) I 3 1 (E) -2mI 1 (E)I 2 (E) + nI 3 (E), (3)
in which λ and µ denote the first and second Lamé parameters, respectively, (l, m, n) the third order Murnaghan parameters, and I 1 (E), I 2 (E) and I 3 (E) the first, second and third invariants of the Green-Lagrange strain tensor, respectively.

The application of a standard Galerkin approach to Eq. ( 1) results in the generalized system of equations:

Γ T 0 K(Q 0 )Γ 0 Q 0 (X) = P 0 (X), (4)
where K(Q 0 ) is the static stiffness matrix, P 0 the global vector of nodal forces, Q 0 the global vector of independent nodal displacements and Γ 0 the mapping operator resulting from Eq. ( 2) and realizing the condition

U 0 = Γ 0 Q 0 .
U 0 indicates the full vector of nodal displacements. In this work, the solution of Eq. ( 4) is carried out using Comsol Multiphysics 5.3 50 .

Following the Updated Lagrangian scheme, once the displacements Q 0 are obtained, the reference configuration is updated from C 0 to C by calculating the corresponding nodal coordinates x = x 0 + Γ 0 Q 0 (x 0 ). The updated material properties in C are given by ρ = ρ 0 (det F) -1 and D ijkl = (detF) -1 F iI F jJ F kK F lL (D 0 ) IJKL , while the Cauchy stress tensor is obtained from the relation σ = (det F) -1 FSF T . The geometry of the unit cell in the configuration C is then re-meshed and used as the basis for the linear dynamic analysis described in the next section.

B. Dynamic Analysis Using the Floquet-Bloch Decomposition

Following the small-on-large analysis approach [START_REF] Pau | Nonlinear guided wave propagation in prestressed plates[END_REF][START_REF] Mazzotti | Guided waves dispersion analysis for prestressed viscoelastic waveguides by means of the safe method[END_REF][START_REF]Harnessing instability-induced pattern transformation to design tunable phononic crystals[END_REF][START_REF] Mazzotti | Band structure analysis of leaky bloch waves in 2d phononic crystal plates[END_REF][START_REF] Mazzotti | Wave dispersion in thin-walled orthotropic waveguides using the first order shear deformation theory[END_REF] , in which C is the new reference configuration, the position vector for the unit cell in the dynamic deformed configuration C (Fig. 1) is approximated as x ≈ x while, from the application of the Floquet-Bloch theorem, any small harmonic perturbation u(x) can be expressed as [START_REF] Collet | Floquet-bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF] :

u(x) = ũ(x)exp(ikx)exp(-iωt), (5)
in which ũ(x) is a Ω-periodic displacement amplitude, t denotes the time, ω the angular frequency, and k ∈ Λ the Bloch wavenumber, being Λ the reciprocal unit cell defined in C by the reciprocal lattice vector g = F -T L g 0 .

By defining the k-shifted gradient of a generic Ω-periodic vector field φ(x) as:

∇ k φ(x) = ∇ x φ(x) + ik φ(x) ⊗ (r r -1 ), (6)
the solution of the elastodynamic problem for free vibrations of the unit cell in C subjected to an initial stress σ 0 can be obtained from the variational statement:

-ω 2 Ω ρ(x)δ ũ * (x) • ũ(x)dΩ + Ω δẽ * k (x, ϑ) : D(x) : ẽk (x)dΩ + Ω σ 0 (x) : (∇ k δ ũ * (x)) T ∇ k ũ(x) dΩ = 0, (7)
subjected to the Dirichlet boundary condition:

ũ(x + r) = ũ(x) on ∂Ω, (8)
in which (•) * stands for the conjugate of a complex vector or tensor field, ẽk (

x) = 1 2 [∇ k ũ(x) + (∇ k ũ(x)) T ] follows
from Eq. ( 6) and denotes the linearized Green-Lagrange strain tensor.

The finite element discretization of Eqs. ( 7) and ( 8) proceeds by first generating a new mesh for the deformed geometry of the unit cell in C and then applying a Galerkin approach [START_REF] Mazzotti | Ultrasonic leaky guided waves in fluid-coupled generic waveguides: hybrid finite-boundary element dispersion analysis and experimental validation[END_REF] . As a results, the following generalized linear eigenvalue problem is obtained:

Γ T k 2 K 3 + ik K 2 -K T 2 + K 1 -ω 2 M Γ Q(ω) = 0, (9)
in which Γ is a mapping operator implementing the Dirichlet boundary condition in Eq. ( 8), such that Ũ = Γ Q, where Ũ is the global vector of nodal displacement amplitudes and Q a subvector of Ũ collecting only its independent components. The expressions of K 3 , K 2 , K 1 and M are given in the Appendix A, for the sake of brevity.

The band diagrams of the phononic structure can be computed from the eigenvalue problem in Eq. ( 9) (i) by extracting the Bloch wavenumbers k(ω) for any fixed frequency ω, or (ii) by computing the natural frequencies ω(k) of the system for any fixed Bloch wavenumber k. Since the focus of the present research does not require the knowledge of the spatial attenuation, the latter approach has been used, resulting computationally more convenient, as it does not require the transformation of the system into the state space.

III. RESULTS

To study the effect of an external mechanical load on a periodic structure exhibiting BGs induced by inertial amplification, the numerical method presented in the previous section is here applied to the representative unit cell reported in Fig. 2A. The unit cell consists of an epoxy matrix exhibiting elongated cross-like holes, and it was proposed for the first time by Acar and Yilmaz [START_REF] Acar | Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures[END_REF] . Its geometrical and material parameters are reported in Tables I, II. As discussed in Section II, the first step for the band structure calculation is to extract the deformation induced in the unit cell by an initial state of stress / strain applied to the structure. To do this, two different sets of loading conditions are applied to the structure in the form of a normal displacement u 0 (x 0 ) • n 0 (x 0 ) prescribed to the x 02 -parallel faces of the unit cell (highlighted in purple in Fig. 2A). The out-of-plane degrees of freedom of the unit cell are blocked, so to prevent any possible bending deformation during the application of the loading conditions. The first loading set induces a state of compression and varies from 0 (no prestrained condition) to -360 µm (maximum compression condition) with a step of ∆u 0 (x 0 ) • n 0 (x 0 ) = -20 µm, while the second set induces a tensile state and varies from 0 (no prestrained condition) to +130 µm (maximum traction condition) with a step of ∆u 0 (x 0 ) • n 0 (x 0 ) = +10 µm.

For the sake of brevity, only the deformation of the unit cell under the conditions of maximum compression / traction are reported in Figs. 2B,C, respectively. Examining the induced deformation, it is possible to infer that the stress, and thus the deformation, is mainly localized into the hinge-like regions, responsible for the activation of the inertial amplification mechanism [START_REF] Acar | Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures[END_REF] . The analysis also shows that the maximum Von Mises stress level reached in the structure is of 33.1 and 21.2 MPa, for the two types of loading condition, respectively. Given these values of maximum stress, and applying the von Mises yield criterion [START_REF] Mises | Mechanik der festen körper im plastisch-deformablen zustand[END_REF] , the minimum mono-axial tensile strength required for our material to have a safety factor ≥ 1 is σ min = 33.1 • (3) = 56 MPa (if we consider a state of pure shear solicitation), which is an acceptable value for epoxy [START_REF] Tsiafis | Experimental-analytical procedures to determine the performance of epoxy resins under various mechanical loadings[END_REF] . Considering that the aforementioned value is obtained in the most strict condition of pure shear (which is not the case in the system under consideration), an elastic behaviour of the material, and thus of the band diagram, over the full range [-360, +130] µm, modulated by the intensity of the applied prestrain, is guaranteed, allowing for a full reversibility of the undeformed configuration (original dispersion diagram), once the load is removed. For the sake of completeness, since the small on large theory is considered here, the dynamic component of the stress is negligible with respect to the static prestress. For this reason it can be assumed that the safety factor does not change between the static and dynamic configuration.

Once the static analysis performed for the aforementioned sets of prestrain, the deformed geometries are assigned as the input unit cells to calculate the dispersion diagrams exploiting the Floquet-Bloch theory (see section II). The out-of-plane displacement of the structure is kept blocked also in this phase, in order to limit the dispersion analysis to waves belonging to the x 1 -x 2 plane. The band structures are computed considering the unit cell to infinitely duplicate in a periodic linear array, and assuming the epoxy in its linear elastic regime (the hypothesis of small displacements is now applied). The unit cell domain is meshed by means of 8-node hexagonal elements of maximum size L F E = 0.2 mm, which is found to provide accurate eigensolutions up to the frequency of interest [START_REF] Marchi | A dispersion compensation procedure to extend pulse-echo defects location to irregular waveguides[END_REF] . It should be noted that, since the applied prestrain applied to each face of the cell induces isotropic deformation in the x 1 -x 2 plane in both compression and traction cases, the deformation gradient F and its affine component F L are diagonal and, as a consequence, the orientation of the reciprocal lattice vectors g 1 in the deformed configurations does not change with respect to that in the undeformed configuration. This implies that the orientation of the Bloch wavevector also remains unchanged between the undeformed and deformed configurations (see Fig. 1D). Therefore, the resulting eigenvalue problem (K -ω 2 M)u = 0 is solved by varying the non-dimensional wavevector k * along the irreducible path [Γ -X], with Γ ≡ (0, 0) and X ≡ π/a, 0), being a = 2 • for increasing values of the prestrain (see Fig. 4A). However, although the global width of the first BG experiences a limited variation, both its upper and lower bounds undergo a remarkable frequency shift (they both almost triple their frequency). On the contrary, the lower bound of the second BG is rather stable in frequency for increasing values of the prestrain, while its upper limit experiences a considerable frequency up-shift. This is responsible for an overall enlargement of the BG width (almost doubling with respect to the unprestressed case). Finally, it is possible to notice that the lower bound of the second BG experiences a sort of inflection point when no displacement is imposed. The different behaviours of the two BGs highlights a selective nature of the prestress in altering the dispersion diagram.

The aforementioned effects partially apply also for the case of unit cell subjected to an external traction pre-loading, i.e. both curve shifting and group velocity inversion are observed (Fig. 3B). However, in this case, group velocity inversion involves more bands (the third and fourth ones) and occurs at lower reduced wavenumber values (k * π 2a ), as highlighted by the black arrows in Fig. 3B (refer to Fig. S1B and Fig. S1C for a direct comparison of the dispersion curves singularly plotted for the 0 µm and +130 µm prestrain cases). This implies that, differently from the compression case, when an external state of traction is induced in the pre-loading phase, only few dispersion curves clearly shift towards higher frequencies (the first, the second, the fifth and sixth bands) over the full range of k * (from 0 to π/a), whereas some others (the third and the fourth ones) exhibit both a down-and an up-shift over a wide range of k * . As a consequence, while the first BG is almost kept unaltered in terms of frequency width and slightly shifts towards higher frequencies (see Fig. 4B), the second BG experiences a remarkable width decrease (up to 3 times less the original BG).

At this point, it is worth to point out here the different dynamic behaviour of inertially amplified elastic metamaterials with respect to the ordinary ones when subjected to an external state of prestress that emerges from the present research. Focusing the attention on the case of prestrain inducing tensile deformation in the structure, in ordinary PCs the BGs tend to decrease its frequency regime, being this effect mainly driven by the geometrical changes of the unit cell rather then by the effective stiffness and mass alteration introduced by the prestress (see for instance Fig. 3A of ref. [START_REF] Mazzotti | Modeling bloch waves in prestressed phononic crystal plates[END_REF] and relevant literature on ordinary PCs presented in section I). On the contrary, in the case of inertially amplified elastic metamaterials, where the displacement mechanism is used to amplify the effective inertia of small masses 60 , the state of external solicitation represents an important means to control the effective stiffness / effective mass ratio of the structure. This additional degree of freedom may lead to both an increase or decrease of the lower limit of the first BG frequency regime, under the same type of applied prestrain (see for instance Figs. 3B and4B, where the lower limit of the first BG increases although a prestress condition inducing tensile solicitation in the structure is applied).

To gain further mechanical insights about this peculiar behaviour, a thorough explanation on the change of the effective stiffness / effective mass ratio of the structure as a function of the applied prestrain is provided with the help of the model reported in Fig. 5, and proposed for the first time by the research group of Yilmaz [START_REF] Taniker | Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges[END_REF][START_REF] Yuksel | Realization of an ultrawide stop band in a 2-d elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF][START_REF] Yuksel | Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms[END_REF][START_REF] Taniker | Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands[END_REF] . Figure 5A shows the rigid link equivalent model, which assumes pin joints at the middle points of the flexural hinges. Lumped parameters m, m a and k ef f are reported along with the principal geometrical parameters ϑ (initial link angle) and displacements (δ x and δ y ) consequent the application of a prestrain inducing tensile solicitation in the system. Already in the undeformed configuration, an angle ϑ exists between the equivalent rigid links (lines in red) and the ground.

f p = k ef f m ef f + m /(2π) = k ef f m a (cot 2 (ϑ) + 1)/4 + m /(2π) (10)
where m a (cot 2 (ϑ) + 1)/4 defines the effective mass of the system.

When a tensile prestress is applied, ϑ decreases to ϑ u , and the effective mass increases. If the stiffness k ef f is left unaltered, from Eq. ( 10) it clearly emerges that the lower limit of the first BG would decrease due to increase of the effective mass. However, if the effective stiffness k ef f increases more than the effective mass m ef f (the edge mass m does not depend on ϑ), then the lower limit of the first BG may also increase (this reasoning also applies to the case of prestrain inducing a compression state of solicitation and it may lead to analogous considerations).

To verify the correctness of the numerical approach proposed in this paper, a comparison of the analytical values of f p deriving from Eq. ( 10) and the numerical solutions is performed and reported in Figs. 5B-D. Figure 5B reports the parametric plot of the dispersion curves of the unit cell when an external prestrain ([0 -40] µm range) is applied inducing a tensile state in the structure. The [100 -220] Hz frequency range is considered. The same polarization reported in Fig. 3 applies. The values of the frequency of the longitudinal branch responsible for the lower limit of the first BG are reported at the X high symmetry point in correspondence of the black arrows. Left panel of Fig. 5C reports the direct comparison of the analytical solution (triangular markers connected by the black line) deriving from Eq. ( 10) and the numerical results (magenta square markers). The values of k ef f introduced in the analytical model are directly calculated from the deformed numerical configuration according to standard homogenization procedures [START_REF] Timoshenko | Theory of elastic stability[END_REF][START_REF] Dirrenberger | Computational homogenization of architectured materials[END_REF] .

Right panel of Fig. 5C shows the variation of the effective stiffness ∆k ef f and of the effective mass ∆m ef f . This explains why the lower limit of the first BG increases, as shown in Figs. 3B and4B, even if a tensile prestress is applied to the structure. Indeed, the effective mass increase ∆m ef f (Fig. 5D) is far lower than the stiffness increase ∆k ef f (this is due to the chosen geometrical characteristics of the unit cell, including the out-of-plane dimension h = t 2 limiting the value of m a ). As a consequence, it is possible to conclude that in this case the BG alteration is mainly stiffness driven.

Finally, to clearly show the enhanced potential of inertial amplification PCs with respect to the ordinary ones when an external prestrain condition is applied, additional calculations by changing some of the geometrical and mechanical parameters of the unit cell are performed. Inspired by the analytical model reported in Fig. 5A, a configuration of the unit cell allowing for the lower limit of the first BG to exhibit a non-monotone trend has been found and reported in Fig. 6. It is possible to observe that the lower limit of the first BG now first decreases, i.e., the increase of the effective mass is dominant -the red curve is above the blue one at low values of prestrain (yellow rectangle) and then increases, i.e., the increase in effective stiffness is dominant -the red curve is below the blue one at higher values of prestrain (green rectangle). This has been obtained by changing some of the model parameters as follows: d 3 = 18.5 mm, t 3 = 18.5 mm, m a = 1 kg/m 3 and multiplying by a factor of 10 the Lamè and Murnaghan material parameters.

These results suggest that a deformation of the unit cell geometry induced by a compressive / tensile prestress state, already in the elastic regime, can lead to significant changes in the passband and BG behaviours of a periodic structure, especially if the BG nucleation mechanism is lead by IA.

B. Transient Calculation

The confirmation of the above mentioned BG tunability is here verified through a nonlinear transient numerical simulation of wave propagation conducted on a finite waveguide comprising 50 unit cells disposed in the x 1 -direction, as shown in Fig. 7A. The idea, here, is to pre-load some unit cells of the waveguide before exciting elastic waves at one of its edge, in order to locally and reversibly change the dispersion diagram and confirm the ability of the waveguide to switch from a pass-to a stop-band behaviour over a specific frequency range.

After having pre-loaded the array, so to uniformly reach the prestrain condition of -360 µm in 6 unit cells (Fig. 7B), elastic waves are excited at the left edge of the waveguide by means of an imposed displacement of 1 µm in the x 1 -direction (red arrow in Fig. 7A). Two input signals are considered: (i) a triangular-like excitation (top-left panel of Fig. 7C) and (ii) a Hanning modulated 11 sine cycles centered at 1400 Hz (top-right panel of Fig. 7C), exhibiting a rather broadband and narrowband frequency content, respectively (bottom panels of Fig. 7C). Such pulses have been chosen according to the band structure shifts reported in Fig. 3 and to highlight the tunable filtering capabilities of the designed waveguide under prestressed conditions. In both excitation cases, 20 ms long time transient explicit simulations have been performed in order to allow multiple wave reflections to take place at both the edge of the waveguide and of the prestressed unit cells.

In the case of excitation (i), time transient displacements in the x 1 -direction are recorded at the two acquisition points R1 and R2 (Fig. 7B), taken equidistant from the prestressed regions and chosen respectively before (B/f) and after (A/t) the prestressed portion of the waveguide. After acquisition, signals are Fourier transformed and compared to highlight the differences of the two responses in terms of frequency content. Figure 7D reports the displacement along the x 1 -direction at points R1 and R2 (top panel), as well as their energy content in the frequency domain (lower panel). The frequencies of the computed BGs as a function of the applied prestrain are also highlighted as a shaded region, where the color scale refers to the level of imposed displacement in the pre-loading phase. Examining the energy content it emerges that the Fourier transform of the signal acquired before the prestrained region (B/f -red line), where the diagram reported in Fig. S1B applies, presents component in the 1000 -1600 Hz frequency range, whereas the signal registered after the prestrained region (A/t -black line) has no frequency components in this frequency range. On the contrary, for frequencies above 1600 Hz, the amplitudes of the frequency contents are comparable. This is in agreement with the two different dispersion diagrams and clearly confirms the possibility of the waveguide to to switch from a pass-to a stop-band behaviour over the 1000 -1600 Hz frequency range by readily applying and removing the imposed deformations, respectively.

When the second type of excitation (i.e., the Hanning modulated 11 sine cycles centered at 1400 Hz) is applied to the left edge of the waveguide, the switch potential is even more evident. Indeed, reconstructing the full wave field displacements at specific time instants, it is clearly visible that the pulse is fully supported in the first portion of the waveguide (where no prestress is applied), whereas when the prestrained region begins the wave is strongly reflected back.

IV. CONCLUSIONS AND FUTURE PERSPECTIVES

In conclusion, in this work the effect of the application of an external prestress on the dispersion diagram has been investigated. The geometrical deformation of the unit cell consequent the action of an applied external load has been determined through a static geometrically nonlinear analysis, representing the first step of a so called Updated Lagrangian scheme of calculation. A Floquet-Bloch decomposition has then been applied (second step) to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration.

The effect of the prestress on the the original band structure (i.e., unit cell without any pre-loading phase) has been demonstrated through parametric Floquet-Bloch analysis and further confirmed by nonlinear transient numerical simulations, proving the capability of the structure to switch from a pass-to a stop-band behaviour in the same frequency range ([1000 -1600] Hz).

The results presented herein provide insights in the behaviour of band gaps induced by inertial amplification, and suggest new opportunities for real-time tunable wave manipulation.

Future investigations will concern the extension of the present study in the direction of a tensegrity-inspired redesign [START_REF] Skelton | Tensegrity systems[END_REF][START_REF] Fraternali | Mechanical modeling of superelastic tensegrity braces for earthquake-proof structures[END_REF] of the unit cell reported in Fig. 1. A self-similar 69 reorganization of the structural elements will allow the construction of a tensegrity architecture. The design will derive from the D-bar tensegrity systems [START_REF] Skelton | Tensegrity systems[END_REF][START_REF] Fraternali | Mechanical modeling of superelastic tensegrity braces for earthquake-proof structures[END_REF] , equipped with longitudinal and transverse cables. The idea will be to recursively divide the longitudinal span of the cell into segments of equal length and replace each new segment with a smaller scale unit with equal shape 67 . Self-equilibrated tensile forces in the cables and compressive forces in the struts will give rise to an internal state of prestress. Cables and struts will present an offset in the transverse direction in order to prevent material overlapping. The application of a self-equilibrated system of forces composed of tensile forces in the cables and compressive forces in the struts will give rise to an internal state of prestress, which can superimpose to the external state of prestress, analysed in the present work. Such initial states of stress will contribute to the geometric term of the stiffness matrix of the structure [START_REF] Skelton | Tensegrity systems[END_REF] , and allow for an extra degree of freedom for the optimal tuning of the dispersion relation of the system 31 . are subjected to a remarkable frequency shift (towards higher frequencies). On the contrary, the lower bound of the second BG (black dashed lines delimited by blue square markers) is rather stable in frequency, while its upper bound still experiences a strong shift towards higher frequencies. This is responsible for an important global BG width enlargement (twice the initial value). (B) In the case of prestrain inducing a state of traction in the unit cell, a similar behaviour is observed for the first BG, whereas an important width reduction can be deduced for the second BG (up to three times the initial value), as the initial imposed solicitation is increased to 130 µm. Applied Prestrain [um] Ampl.

[-] The variation of the normalized effective stiffness and of the normalized effective mass explaining the reason of the decrease / increase of the lower limit frequency of the first BG even if the same type of prestress (tension) is applied to the structure.

Yellow and green rectangles determine the mass and stiffness driven regions, respectively.

K 3 = e Ωe N T (x) H T D(x)H + H T 0 Σ 0 (x)H 0 N(x)da, (13)
where Ω e denotes the domain of the e-th finite element of the mesh, e (•) the standard direct stiffness assembling procedure, N(x) the matrix of shape functions for the e-th element, and Σ 0 (x) is a block-diagonal matrix of the form:

Σ 0 (x) =   σ 0 (x) 0 0 σ 0 (x)   , (15)
while the different compatibility operators are expressed as:

B = ∂ ∂x 1     1 0 0 0 0 1     + ∂ ∂x 2     0 0 0 1 1 0     , (16)
B 0 = ∂ ∂x 1        1 0 0 0 0 1 0 0        + ∂ ∂x 2        0 0 1 0 0 0 0 1        (17)
H =     1 0 0 0 0 1     , H 0 =        1 0 0 0 0 1 0 0        . ( 18
) APPENDIX B
In this Appendix a direct comparison of the dispersion curves plotted for the specific cases of -360 µm, 0 µm and +130 µm prestrain loads are reported as singular diagrams in Fig. S1.

3 i=1 d i + d 4

 34 the lattice parameter.

Figure 3

 3 Figure 3 reports parametric plots of the dispersion diagrams of the unit cell as a function of the external prestrain intensity inducing compression (Fig.3A) or traction (Fig.3B) states in the structure. The dispersion curves are color-coded on the base of the level of prestrain applied at the boundaries of the unit cell in the pre-loading phase.Specifically, the color bar of Fig.3Avaries gradually from -360 µm (blue: maximum compression state) to 0 (green: unprestrained condition), respectively. Analysing the band diagrams, it is possible to observe that increasing the compressive state in the unit cell induces (i) a general up-shift (curves shading into dark blue) of the dispersion branches, as well as (ii) group velocity inversion in the third band when k * gets close to the high symmetry point X, as indicated by the black arrow. The band inversion is more evident comparing the dispersion curves singularly plotted for the -360 µm and 0 µm prestrain cases, as reported in Fig.S1Aand Fig.S1B. Further inspection of the the dispersion diagram allows to infer that the unprestrained unit cell (green curves) allows for the opening of two BGs in the [0 -3000] Hz frequency range, going from 170 to 470 Hz and 703 to 1180 Hz, respectively (see Fig.4Aat 0 µm imposed displacement). The upper and lower bounds for both the first and second BGs shift in frequency

FIG. 1 .

 1 FIG. 1. Graphic representation of the Updated Lagrangian scheme. (A) The undeformed configuration C0, i.e. the initial unit cell (delimited by dashed lines) used to calculate the displacement and stress fields introduced by the external mechanical load. (B) The static deformed configuration C, resulting from the application of the external mechanical load. (C) The dynamic configuration C undergoing a harmonic motion. (D) The reference systems in the direct and reciprocal spaces.

FIG. 2 .FIG. 3 .FIG. 4 .

 234 FIG. 2. Schematic representation of the undeformed unit cell and its deformed conditions under isotropic compression / traction. (A) Schematic representation of the unit cell exhibiting inertial amplification mechanism, positioned at x0 with respect to the original reference systems x0i, with i = 1,2,3. The structure is made of epoxy and it was proposed for the first time by Yilmaz et al. in ref. 43 . (B, C) Deformed configuration of the unit cell under isotropic u0(x0) • n0(n0) = 130µm traction(-360µm compression). The colors denote the Von Mises stress in MPa. A maximum stress of 21.2 (28.6 33.1) MPa is observed. Deformations are in 1:1 scale.

FIG. 5 .

 5 FIG. 5. Change of the effective stiffness / effective mass ratio of the structure as a function of the applied prestrain: numerical -analytical comparison. (A) Schematic representation of the rigid link equivalent model of the examined PC with inertial amplification mechanism 62 . Lumped parameters m, ma and k ef f are reported along with the principal geometrical parameters ϑ (initial link angle) and displacements (δx and δy) consequent the application of a prestrain inducing tensile solicitation. (B) Parametric plot of the the real part of the reduced wavenumber k * along the Γ -X irreducible path as a function of the applied prestrain inducing a tensile state in the structure in the [100 -220] Hz frequency range. The same polarization reported in Fig. 3 applies. The values of the frequency of the longitudinal branch responsible for the lower limit of the first BG are reported at the X high symmetry point in correspondence of the black arrows. (C) Comparison of the analytical solution (triangular markers connected by the black line) deriving from Eq. 1 and the numerical values (magenta square markers) is reported in the left panel. On the right panel, the variation of the effective stiffness ∆k ef f and of the effective mass ∆m ef f explaining the reason of the lower frequency increase of the BG even if a tensile prestress is applied to the structure. Indeed, the effective mass increase is far lower than the stiffness increase. (D) Magnification of the change of the effective mass showing its increase.

FIG. 6 .

 6 FIG. 6. Additional degree of tunability of inertially amplified elastic metamaterials under external prestrain solicitation state. (Left panel) Lower limit frequency of the first BG as a function of the applied prestrain. (Right panel)

FIG. 7 .N

 7 FIG. 7. Switch from a pass-to a stop-band behaviour of the waveguide. (A) Schematic representation of the finite structure implemented for the nonlinear transient simulation. The model comprises 50 unit cells arranged in the x01-direction. (B) Before exciting the propagation of elastic waves, an external load is applied over 6 internal unit cells of the array, so to locally induce the uniform prestrained condition of -360 µm. (C) Triangular-like (left panel) and 11 sine cycles centered at 1400 Hz Hanning modulated (right panel) excitations. (D) Signals recorded before (B/f) and after (A/t) the 6 prestressed unit cells (top panel) and their Fourier Transform (low panel), for the triangular-like excitation signal. The Fourier content clearly shows how the introduction of the prestress alters the frequency response function of the structure. The tunable BG is reported as a color-coded rectangle. The green region of the rectangle is the BG corresponding to the unprestrained unit cells (up to 1180 Hz), whereas the one fading to dark blue corresponds to the BG extension (up to 1680 Hz) induced by the introduction of the external compressive prestrain (up to -360 µm: dark blue). The comparison of the two signals proves that the same waveguide is capable to support or to inhibit waves having the same frequency content. (E) Snapshots at different time steps, for the narrowband excitation signal, showing how the unprestrained waveguide is capable of supporting the propagation of the wave, whereas the stressed region reflect back the majority of the waveform.

TABLE I .

 I Material constants for the epoxy[START_REF] Mazzotti | Modeling bloch waves in prestressed phononic crystal plates[END_REF] .

	Epoxy	1540	2.59	1.34	-18.94 -13.36	-9.81

ρ0 [kg/m 3 ] λ [GPa] µ [GPa] l [GPa] m [GPa] n [GPa]

TABLE II .

 II Geometrical parameters of the unit cell presented in Fig. 2A. All the given parameters are in [mm].

	t1	t2	t3	t4	d1	d2	d3	d4
	3.2	0.4	3.2	0.4	5.2	2.0	20.0	4.0
	A. Band structure analysis

Strain [ m] X k* 0 Γ X Re(k*)

Acknowledgements MiM has received funding from the European Union's Horizon 2020 FET Open ("BOHEME") under grant agreement No. 863179.

Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.


Jongun Moon

Olivier Bouaziz
email: olivier.bouaziz@univ-lorraine.fr

Hyoung Seop Kim
email: hskim@postech.ac.kr

Yuri Estrin

Twinning Engineering of a CoCrFeMnNi High-Entropy Alloy Inspired by Twinning-Induced Plasticity Steels

Keywords: high-entropy alloys, twinning, recovery, strain hardening, ductility

Deformation-induced twinning has been a notable example of overcoming the strength/ductility trade-off dilemma as a strengthening mechanism. By borrowing this concept from the area of TWIP steels, we designed a thermomechanical treatment for a CoCrFeMnNi high-entropy alloy to improve its mechanical characteristics. We used pre-straining at 77 K to introduce deformation-induced twins in the microstructure of the alloy, and then recovered it by annealing at 773 K, while avoiding recrystallization. The deformation-induced twins generated by pre-straining at 77 K were retained after this heat treatment, whilst partial recovery of dislocations occurred. As a result, the room-temperature mechanical properties of the alloy, including its strain hardening ability, were improved substantially.

Deformation-induced twins (DTs) are known to act as strong obstacles to moving dislocations [1,2]. Sustained deformation twinning in high manganese twinning-induced plasticity (TWIP) steels gives rise to a continual reduction of the mean free path of dislocations, thus enabling strain hardening and promoting high ductility [3][4][5]. This profuse twinning promoted by a favourable magnitude of the stacking fault energy (SFE) enables the TWIP steels to possess substantial strain hardening at room temperature (RT) [1,2].

Moreover, deformation-induced twins in TWIP steels were found to be thermally stable up to ~898 K [6]. Building on this thermal stability of DTs, some ten years ago a metallurgical route was proposed to improve the yield strength of TWIP steels [7]. It encompasses the following steps: First, pre-straining is applied at RT to generate deformation-induced twins. This is followed by annealing at 773 K with the aim to reduce the dislocation density by static recovery, while sustaining the DTs due to their thermal stability being higher than that of the dislocations. Through this 'twinning engineering', a favourable combination of high yield strength and enhanced uniform elongation of TWIP steels was achieved [7].

An exciting development in materials engineering has been the advent of a novel class of materials, which comprise multiple principal elements in a high concentration and are referred to as high-entropy alloys (HEAs) [8][9][10]. They were developed in the 1970s, but came to the fore in the last decade due to their exceptional mechanical characteristics [11].

Specifically, face-centered cubic (FCC) HEAs exhibit outstanding mechanical properties in the temperature range from cryogenic [12][13][14] to ambient [15][16][17][18] temperatures. The mechanisms underlying these properties are associated with the deformation-induced twinning [12,15] and/or phase transformations [13,14]. Among the FCC-based HEAs, the now classical Cantor alloy CoCrFeMnNi exhibits deformation-induced twinning at 77 K, resulting in outstanding tensile properties along with a high fracture toughness [12]. Using this feature, Sun et al. [19,20] demonstrated that the yield strength of the CoCrFeMnNi HEA at RT is enhanced by pre-straining at 77 K. No annealing heat treatment was involved in their work.

In the present communication, we propose a way to improve the tensile properties of the Cantor alloy even further. The novelty of this work consists in translating the strategy based on the thermomechanical route (pre-straining and recovery), which has been so successful for TWIP steels, to a high-entropy alloy. The results of a systematic investigation of the mechanical properties and microstructure evolution of an equiatomic CoCrFeMnNi alloy at RT after pre-straining at 77 K followed by an annealing treatment are reported below.

An ingot of a HEA Co20Cr20Fe20Mn20Ni20 (atomic percent, at%) was fabricated using vacuum induction melting in argon atmosphere. The ingot was homogenized at 1,373 K for 6 h in argon atmosphere, followed by water quenching. The surface of the homogenized ingot was mechanically milled in order to reduce surface roughness and remove oxides. After this preparation, cold rolling with a thickness reduction from 7 mm to 1.5 mm was performed. The cold-rolled plate was annealed at 1073 K for 1 h in argon atmosphere to achieve full recrystallization [21], followed by water quenching.

From this plate, dog-bone shaped tensile specimens with a gauge length of 5 mm and a width of 2.5 mm were cut. All tensile tests were conducted on an Instron 1361 machine. The deformation curves for the annealed alloy measured at RT and 77 K are presented in Fig. 1(a).

In addition to continuous tests to failure, the annealed alloy was deformed at 77 K to different strains, 20%, 35%, and 50%, as shown in Fig. 1(b). After straining at 77 K, the specimens were heat treated at 773 K for 1 h in argon atmosphere, followed by water quenching.

Henceforth, the specimens with different processing histories are referred to as P20, P35, P50, P20 + HT, P35 + HT, and P50 + HT, according to the strain to which they were deformed at 77 K and the recovery heat treatment. The processing histories and nomenclature used in the present work are summarized in Table 1. Before the tensile tests, the specimens were polished using SiC papers up to 1200-grit. All tensile tests, including pre-straining at 77 K, were performed at a strain rate of 10 -3 s -1 . An extensometer was attached to the gauge section of each specimen to measure the engineering strain during the tests.

Table 1. The Co20Cr20Fe20Mn20Ni20 alloy with the different processing histories.

Processing histories Naming

Pre-straining to a true strain of 20% at 77 K P20

Pre-straining to a true strain of 35% at 77 K P35

Pre-straining to a true strain of 50% at 77 K P50

Pre-straining to a true strain of 20% at 77 K + recovery heat treatment at 773 K for 1 h P20 + HT Pre-straining to a true strain of 35% at 77 K + recovery heat treatment at 773 K for 1 h P35 + HT Pre-straining to a true strain of 50% at 77 K + recovery heat treatment at 773 K for 1 h P50 + HT Scanning electron microscopy (SEM) with electron backscatter diffraction (EBSD) was performed using a JEOL 7800F device and TSL/OIM software. EBSD measurements were conducted with a step size of 50 nm. The surface of the alloy for SEM/EBSD characterization was polished up to 1200-grit SiC paper, followed by electro-polishing in an etchant solution of 8% perchloric acid. HT), and 1% (P50 + HT) with growing pre-strain. The combination of the yield strength and the uniform elongation after the heat treatment appears to be particularly good for the P20 + HT and P35 + HT alloys -similar to the results for TWIP steels [7]. This demonstrates that the thermomechanical processing strategy adopted from TWIP research also works for HEAs.

As anticipated, the DTs retained after this processing enhance the yield strength of the alloy.

At the same time, the ability of the dislocation density to increase from a low base established by the recovery equips the material with a significant SHR capability, which ensures good tensile ductility [Fig. 4(b)]. Effectively, the SHR of the heat-treated alloy is higher than that of the initially annealed alloy when the cumulative strain accounting for the pre-strain is considered. This becomes evident when the true stress-strain curves at RT are juxtaposed so that a shift by the amount of the pre-strain at 77 K is made [Fig. The yield strength ( ) of the present alloy after the different processing routes can be described as follows:

,

where is the lattice friction and , , and denote the contributions associated with grain boundaries, retained DTs, and dislocations, respectively. Since the variability of the grain size among the samples with different processing histories is insignificant, the change in before and after pre-straining can be neglected. Thus, the increment of the yield strength of the alloy after the thermomechanical processing can be written as:

∆ ∆ ∆ . ( 2
)
Therefore, the enhancement of may be associated with the retained DTs and the dislocation density affected by the thermomechanical processing. We can surmise that the dislocation-related effect will enter the yield strength directly, via the Taylor relation

∆ / , (3)
where is the texture-dependent Taylor factor, is the shear modulus, is the magnitude of the dislocation Burgers vector, is the dislocation density acquired after the thermomechanical treatment, and is a numerical constant. By contrast, the effect of the retained twins is indirect: it comes to bearing through enhanced accumulation of the dislocation density due to the presence of the twins, leading to a higher strain hardening rate.

Indeed, the mean free path of dislocations ( ) can be expressed by the following equation [23]:

, ( 4
)
where and are the average grain size and the twin spacing, respectively. The retained DTs reduce the mean free path of mobile dislocations, leading to a faster rate of dislocation density accumulation, according to the equation [24]:

-" #, (5)
where k2 is a dynamic dislocation recovery coefficient. Obviously, a smaller effective mean free path at the beginning of the RT deformation owing to low-temperature pre-straining, and a further decrease of due to additional twinning during RT straining, lead to a higher SHR than in the annealed material.

Figure 5 shows a comparison of the yield strength of CoCrFeMnNi with respect to the grain size. The yield strength of the alloy reported previously follows the classical Hall-Petch relation [25]:

∆ " $ / , (6)
where , " , and are the lattice friction (125 MPa [26]), the Hall-Petch strengthening coefficient (494 MPa•μm -1/2 [26]), and the average grain size, respectively. Some data reported for the CoCrFeMnNi alloy slightly deviate from the Hall-Petch trend, and this may be associated with the high dislocation density and/or DTs induced by pre-straining [20] and high-pressure torsion (HPT) [28]. By contrast, the pre-strained and annealed material investigated in the present work outperforms the conventional grain boundary-strengthened alloys quite remarkably. As seen in Fig. 4(c), the yield strength of the pre-strained and recovered specimens exceeds the flow stress of the annealed specimens attained at the same strain levels. Since the heat treatment causes partial recovery of dislocations (Fig. 2), it can be surmised that the enhancement of the yield strength observed is associated with the retained boundaries of DTs. Ref. [26].

Although the present thermomechanical processing enhances the yield strength and SHR of the alloy, its tensile elongation is not improved sufficiently (Fig. 4). The latter observation can be attributed to partial recovery of dislocations as a result of the heat treatment (Figs. 2(b,e,h)). Despite the partial recovery, the annealed material still contains a considerable dislocation density. This leads to accumulation of mobile dislocations at localized obstacles, which imparts negatively on the ductility of the alloy. A more in-depth study is needed for finding an optimum condition of recovery heat treatment providing an improvement of both strength and ductility characteristics of the alloy. In such a recovery heat treatment, recrystallization is to be avoided and the DT density maintained, along with a sizeable reduction of the dislocation density.

There are reasons to believe that the success of the twinning engineering approach as applied to the equiatomic HEA CoCrFeMnNi is not fortuitious and can be utilized for other HEAs as well. Indeed, the propensity to deformation twinning that was so crucial in this approach is predicated on the low magnitude of their stacking-fault energy. [29].This feature is quite common to multicomponent HEAs with an equiatomic or nearly equiatomic ratio. It has been reported that they may exhibit local chemical ordering and/or clustering, which influence intrinsic and extrinsic SFEs [29,30]. A broad range of compositional space in HEAs facilitates alloy design to tune their composition for lowering the SFE intrinsically through local chemical fluctuations, thus promoting deformation twinning [29]. Hence, it can be anticipated that the processing strategy proposed here can be successfully emulated for other high-entropy alloy systems.

In summary, the mechanical properties and microstructural evolution of the equiatomic HEA Co20Cr20Fe20Mn20Ni20 subjected to a specially designed thermomechanical treatment, viz. pre-straining at 77 K and annealing at 773 K, were investigated systematically, inspired by the metallurgical route developed earlier for TWIP steels. The following conclusions can be drawn:

• The deformation-induced twins introduced by deformation at 77 K are retained during heat treatment at 773 K, while part of dislocations is recovered.

• The remnant deformation-induced twins give rise to an increase of the yield strength and the ultimate tensile strength of the alloy to 1,284 MPa and 1,297 MPa, respectively, with a substantial strain hardening rate at room temperature. These excellent mechanical properties are attributed to a reduction in the mean free path of dislocations by the deformation-induced twins formed by pre-straining at 77 K.

• The ductility of the alloy improves because the remnant deformation-induced twins act to raise the accumulation of dislocations. However, the increase of the tensile elongation attained is moderate, as a sizeable dislocation density survives the recovery heat treatment, which limits the strain hardening capability of the alloy.

• We believe that the two-step thermomechanical treatment strategy developed here for a particular high-entropy alloy, Co20Cr20Fe20Mn20Ni20, can be further fine-tuned to increase the gain in properties. We also expect that this processing concept will be applicable to a broader range of HEAs.

Figure 1 .

 1 Figure 1. (a) Engineering stress-strain curves at room temperature and 77 K. (b) The true

Figure 2

 2 Figure2shows the kernel average misorientation (KAM) maps and its distribution for

Figure 2 .

 2 Figure 2. EBSD KAM maps and the corresponding distribution plots for the pre-strained

Figure 3 .

 3 Figure 3. EBSD image quality maps of (a) annealed, (b) P20 + HT, (c) P35 + HT, and (d) P50

  4(c)]. Figure 4(d) shows the SHR as a function of true stress, revealing a better trade-off between the stress and the SHR of the alloy recovered by the heat treatment compared to the annealed alloy. It should also be noted that at a fixed SHR level (set at 3,000 MPa) the true flow stress increases with the prestrain, starting from 484 MPa for the annealed alloy and attaining the values of 833 MPa, 1,040 MPa, and 1,301 MPa for P20 +HT, P35 + HT, and P50 + HT, respectively [Fig. 4(d))].

Figure 4 .

 4 Figure 4. (a) Engineering and (b) true stress-strain curves of the annealed, P20 + HT, P35 +

Figure 5 .

 5 Figure 5. Room-temperature yield strength versus grain size of the present and the previously

Acknowledgments

This work was supported by Creative Materials Discovery Program through the National Research Foundation of Korea (NRF) funded by Ministry of Science and ICT (2016M3D1A1023384).

Declarations of Interest

None.