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International audience; A non-affine microsphere model for rubberlike materials is proposed, based on a local minimization of the network free energy under a maximal advance path constraint. It accounts for any chain weight distribution and for damage such as Mullins softening observed in filled rubber materials. The non-affine equal-force model is compared to the common affine model and a hybrid equal-force model from the literature, when considering the isotropic hyperelastic behavior without damage of rubber materials presenting chains of various lengths. The non-affine model shows an improved deformability compared to the affine model limited by the maximal extension of the shorter chains and a significantly softer behavior. Possible damage is introduced by increasing the chain lengths according to the submitted maximal chain traction force. Each chain are impacted independently resulting in a directional softening that introduces the evolution of the stress-free configuration that needs to be assessed over the loadings. The model was successfully tested on the cyclic uniaxial tension stretch-stress responses of carbon-black filled styrene butadiene rubbers that were well fitted with three parameters only.

International audience; This work presents a general methodology to analyze three-dimensional Freedericksz transitions in twisted-nematic liquid crystal (LC) bilayers. Using two equivalent coupled electromechanical variational formulations, the problem is treated as a bifurcation instability triggered by an externally applied electric field. Specifically, we consider LC bilayer materials anchored between two bounding plates and subjected to an electric field across the bilayer thickness. The plates are also twisted by an overall angle leading to different orientations of the directors in each layer. We first evaluate the corresponding ground state of the director field, and subsequently, we analyze the bifurcation problem by using a combined analytical-numerical method leading to a one-dimensional finite element discretization of the resulting stiffness matrix of the system. An analytical solution for the zero-twist bilayer is also obtained. The developed methodology is used to study the effect of the volume fraction of the constituents forming the bilayer upon the resulting critical electric field and corresponding eigenmodes. We find that by assembling a relatively thin 5CB layer with a thicker 7E layer, one can obtain periodic Freedericksz transitions (i.e. local modes) even for a zero-twist LC bilayer. We also show that when a 5CB material is assembled together with another electrically similar LC, such as a PCH12, the combined LC bilayer can exhibit an even lower Freedericksz transition than a LC of the same thickness consisting of any of the two constituents alone.

Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t , width a and length l , assuming t ≪ a ≪ l . It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Two popular theories can be recovered as limiting cases, namely Kirchhoff’s rod model for small bending and twisting strains, | κ i | ≪ t ∕ a 2 , and Sadowsky’s inextensible ribbon model for | κ i | ≫ t ∕ a 2 ; we point out that Sadowsky’s inextensible model may be a poor approximation even for ribbons having a very thin cross-section (say, with t ∕ a as small as 0 . 02 ). By way of illustration, the one-dimensional model is applied (i) to the lateral-torsional instability of a ribbon, showing good agreement with both experiments and finite-element shell simulations, and (ii) to the stability of a twisted ribbon subjected to a tensile force. The non-convexity of the one-dimensional model is discussed; it is addressed by a convexification argument.

International audience; This paper addresses localization of the deformation due to buckling that occurs immediately following the onset of bifurcation in the axisymmetric buckling of a perfect spherical elastic shell subject to external pressure. The localization process is so abrupt that the buckling mode of the classical eigenvalue analysis, which undulates over the entire shell, becomes modified immediately after bifurcation transitioning to an isolated dimple surrounded by an unbuckled expanse of the shell. The paper begins by revisiting earlier attempts to analyze the initial post-buckling behavior of the spherical shell, illustrating their severely limited range of validity. The unsuccessful attempts are followed by an approximate Rayleigh-Ritz solution which captures the essence of the localization process. The approximate solution reveals the pathway that begins at bifurcation from the classical mode shape to the localized dimple buckle. The second part of the paper presents an exact asymptotic expansion of the initial post-buckling behavior which accounts for localization and which further exposes the analytic details of the abruptness of the transition.

International audience; This work studies experimentally and numerically the post-bifurcation response of a magnetorheological elastomer (MRE) film bonded to a soft non-magnetic (passive) substrate. The film-substrate system is subjected to a combination of an axial mechanical pre-compression and a transverse magnetic field. The non-trivial interaction of the two fields leads to a decrease of the critical magnetic field with applied pre-compression, while the observed wrinkling patterns evolve into crinkles, a bifurcation mode that is defined by the accompanied curvature localization and strong shearing of the side faces of the wrinkled geometry. Using a magneto-elastic variational formulation in a two-dimensional finite element numerical setting, we find that the crinkling is an intrinsic feature of magnetoelasticity and its presence is directly associated with the repulsive magnetic forces of the neighboring wrinkled-crinkled faces. As a result, the presence of the magnetic field prohibits the formation of creases and folds. In an effort to obtain a good quantitative agreement between the numerical and the experimental results, we also introduce an approximate way to model the friction of the lateral film-substrate faces. This analysis reveals the strong effects of friction upon the magneto-mechanical wrinkling modes.

The goal of this study is to better understand the mathematical structure and ramifications of the second-order homogenization of low-frequency wave motion in periodic solids. To this end, multiple-scales asymptotic approach is applied to the scalar wave equation (describing anti-plane shear motion) in one and two spatial dimensions. In contrast to previous studies where the second-order homogenization has lead to the introduction of a single fourth-order derivative in the governing equation, present investigation demonstrates that such (asymptotic) approach results in a family of field equations uniting spatial, temporal, and mixed fourth-order derivatives – that jointly control incipient wave dispersion. Given the consequent freedom in selecting the affiliated lengthscale parameters, the notion of an optimal asymptotic model is next considered in a one-dimensional setting via its ability to capture the salient features of wave propagation within the first Brillouin zone, including the onset and magnitude of the phononic band gap. In the context of two-dimensional wave propagation, on the other hand, the asymptotic analysis is first established in a general setting, exposing the constant shear modulus as sufficient condition under which the second-order approximation of a bi-periodic elastic solid is both isotropic and limited to even-order derivatives. On adopting a chessboard-like periodic structure (with contrasts in both modulus and mass density) as a testbed for in-depth analytical treatment, it is next shown that the second-order approximation of germane wave motion is governed by a family fourth-order differential equations that: (i) entail exclusively even-order derivatives and homogenization coefficients that depend explicitly on the contrast in mass density; (ii) describe anisotropic wave dispersion characterized by the “ sin 4 θ + cos 4 θ ” term, and (iii) include the asymptotic model for a square lattice of circular inclusions as degenerate case. For completeness, the analysis is illustrated by a set of numerical results highlighting the effects of periodic structure on the long-wavelength material response in terms of wave dispersion, phononic band gap, and second-order anisotropy.

International audience; This work provides a rigorous analysis of the effective response, i.e., average magnetization and mag-netostriction, of magnetoelastic composites that are subjected to overall magnetic and mechanical loads. It clarifies the differences between a coupled magnetomechanical analysis in which one applies a Eulerian (current) magnetic field and an electroactive one where the Lagrangian (reference) electric field is usually applied. For this, we propose an augmented vector potential variational formulation to carry out numerical periodic homogenization studies of magnetoelastic solids at finite strains and magnetic fields. We show that the developed variational principle can be used for bottom-up design of microstructures with desired magne-tomechanical coupling by properly canceling out the macro-geometry and specimen shape effects. To achieve that we properly treat the average Maxwell stresses arising from the medium surrounding the magnetoelastic representative volume element (RVE) while at the same time we impose a uniform average Eulerian–and not Lagrangian–magnetic field. The developed variational principle is then used to study a large number of ideal as well as more realistic two-dimensional microstructures. We study the effect of particle volume fraction, particle distribution and particle shape and orientation upon the effective magnetoelastic response at finite strains. We consider also unstructured isotropic microstructures based on random adsorption algorithms and we carry out a convergence study of the representativity of the proposed unit cells. Finally, three-phase two-dimensional auxetic microstructures are analyzed. The first consists of a periodic distribution of voids and particle chains in a polymer matrix, while the second takes advantage of particle shape and chirality to produce negative and positive swelling by proper change of the chirality and the applied magnetic field.

International audience; The purpose of this paper is to experimentally validate a 1D probabilistic model of damage evolution in unidirectional SiC/SiC composites. The key point of this approach lies in the identification and validation at both local and macroscopic scales. Thus, in addition to macroscopic tensile tests, the evolution of microscopic damage mechanisms - in the form of matrix cracks and fiber breaks - is experimentally analyzed and quantified through in-situ scanning electron microscope and computed tomography tensile tests. A complete model, including both matrix cracking and fiber breaking, is proposed on the basis of existing modeling tools separately addressing these mechanisms. It is based on matrix and fiber failure probability laws and a stress redistribution assumption in the vicinity of matrix cracks or fiber breaks. The identification of interfacial parameters is conducted to fit the experimental characterization, and shows that conventional assumptions of 1D probabilistic models can adequately describe matrix cracking at both macro- and microscopic scales. However, it is necessary to enrich them to get a proper prediction of ultimate failure and fiber break density for Hi-Nicalon type S fiber-reinforced SiC/SiC minicomposites.

International audience; We develop a novel continuum description of lattice dynamics for a particle chain with nearest neighbor interactions. Our continuum model interpolates discrete solutions and allows one to deal adequately with singular and impact loadings. The resulting theory is local in space and nonlocal in time. It is characterized by a nontrivial hereditary structure which blends inertial and elastic forces. The proposed methodology can be used to build continuum approximations for lattice dynamics which incorporate successively finer scales. It can also serve as a seamless interface between discrete and continuum elasticity theories.

In an earlier work, Elliott et al. [2006a, Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. Journal of the Mechanics and Physics of Solids 54(1), 193–232], the authors used temperature-dependent atomic potentials and path-following bifurcation techniques to solve the nonlinear equilibrium equations and find the temperature-induced martensitic phase transformations in stress-free, perfect, equi-atomic binary B2 crystals. Using the same theoretical framework, the current work adds the influence of stress to study the model's stress-induced martensitic phase transformations. The imposition of a uniaxial Biot stress on the austenite (B2) crystal, lowers the symmetry of the problem, compared to the stress-free case, and leads to a large number of stable equilibrium paths. To determine which ones are possible reversible martensitic transformations, we use the (kinematic) concept of the maximal Ericksen–Pitteri neighborhood (max EPN) to select those equilibrium paths with lattice deformations that are closest, with respect to lattice-invariant shear, to the austenite phase and thus capable of a reversible transformation. It turns out that for our chosen parameters only one stable structure (distorted α IrV ) is found within the max EPN of the austenite in an appropriate stress window. The energy density of the corresponding configurations shows features of a stress-induced phase transformation between the higher symmetry austenite and lower symmetry martensite paths and suggests the existence of hysteretic stress–strain loops under isothermal load–unload conditions. Although the perfect crystal model developed in this work over-predicts many key material properties, such as the transformation stress and the Clausious–Clapeyron slope, when compared to real experimental values (based on actual polycrystalline specimens with defects), it is—to the authors' knowledge—the first atomistic model that has been demonstrated to capture all essential trends and behavior observed in shape memory alloys.

International audience; The present work is a detailed study of the connections between microstructural instabilities and their macroscopic manifestations -- as captured through the effective properties -- in finitely strained fiber- reinforced elastomers, subjected to finite, plane-strain deformations normal to the fiber direction. The work, which is a complement to a previous and analogous investigation by the same authors on porous elastomers, (Michel et. al. , 2007), uses the linear comparison, second-order homogenization (S.O.H.) technique, initially developed for random media, to study the onset of failure in periodic fiber-reinforced elastomers and to compare the results to more accurate finite element method (F.E.M.) calculations. The influence of different fiber distributions (random and periodic), initial fiber volume fraction, matrix constitutive law and fiber cross- section on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition, constraints to the principal solution due to fiber/matrix interface decohesion, matrix cavitation and fiber contact are also addressed. It is found that both microscopic and macroscopic instabilities can occur for periodic microstructures, due to a symmetry breaking in the periodic arrangement of the fibers. On the other hand, no instabilities are found for the case of random microstructures with circular section fibers, while only macroscopic instabilities are found for the case of elliptical section fibers, due to a symmetry breaking in their orientation.

International audience; The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.

This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.

International audience; When using a classical SHPB (split Hopkinson pressure bar) set-up, the useful measuring time is limited by the length of the bars, so that the maximum strain which can be measured in material testing applications is also limited. In this paper, a new method with no time limits is presented for measuring the force and displacement at any station on a bar from strain or velocity measurements performed at various places on the bar. The method takes the wave dispersion into account, as must inevitably be done when making long time measurements. It can be applied to one-dimensional and single-mode waves of all kinds propagating through a medium (flexural waves in beams, acoustic waves in wave guides, etc.). With bars of usual sizes, the measuring time can be up to 50 times longer than the time available with classical methods. An analysis of the sensitivity of the results to the accuracy of the experimental data and to the quality of the wave propagation modelling was also carried out. Experimental results are given which show the efficiency of the method.

A linear stability criterion for strain-rate sensitive solids and structures is proposed and validated with the help of two versions of Shanleys column, the first with two discrete supports and the second with a continuous distribution of supports. Linear stability transition is defined by the change in sign of the second derivative with respect to time of the columns angular position evaluated at the onset of perturbation. This criterion pertains to the initiation of instabilities but is not expected to provide information on their long term development. Two parameters influence linear stability: the dimensionless number T, defined as the ratio of the relaxation time of the viscous support to the characteristic loading time, and the perturbation size. It is found that the critical load of principal equilibria, defined for a straight column and a zero value of T, is the classical reduced modulus load, in agreement with existing stability criteria for rate-independent models based on maximum dissipation. For arbitrary values of T, two critical loads are identified at the linear stability transition. The first is named the rate-dependent tangent modulus load and is valid for perturbations sufficiently small to prevent initial unloading. That load coincides with the classical tangent modulus load for T tending to zero and is, surprisingly, a decreasing function of that dimensionless number. The second critical load is termed the rate-dependent reduced modulus load, and is applicable to columns that are partly unloaded at the onset of perturbation. This critical load approaches the classical reduced modulus load in the limit of T tending to zero, is a decreasing function of T, and depends on the imperfection size. Similar results are found for the second model, with a new insight on the role of the unloading zone extent in determining the critical load in the singular limit of T equal to zero. The proposed stability criterion is validated by comparing its predictions with the outcome of nonlinear perturbation analyses and of imperfection sensitivity studies. It is shown, in particular, that an imperfection evolves according to our stability predictions as long as the relative difference between the irreversible displacements of the supports can be disregarded. A generalization of the proposed linear stability criterion to viscoplastic continua is finally sketched.

Partial damage is considered in this paper. It is defined as a transformation of mechanical properties between two materials along a surface, in elasticity or anelasticity. The transformation is controlled by a generalized energy criterion of the Griffith type. The equilibrium state of the structure, the total dissipation with increasing loading and the associated rate boundary value problem are studied within the framework of generalized standard materials. A global formulation of the rate boundary value problem allows us to characterize the actual equilibrium state assumed to be known. Finally, the condition of stability and the possibility of bifurcation are given, as are applications to simple composite structures.

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method. The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.