Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2010"
TL;DR: The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers.
Abstract: Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual-primal FETI methods as well as related inexact dual-primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI-DP approach and the behavior of FETI-DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual-primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems.
114 citations
TL;DR: In this paper, an existence result for energetic solutions of rate-independent damage processes and the temporal regularity of the solution are discussed, and a new technique for the construction of joint recovery sequences is presented.
Abstract: This paper discusses an existence result for energetic solutions of rate-independent damage processes and the temporal regularity of the solution. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [16] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W 1,r (Ω) with r > d for Ω C ℝ d , we can handle the case r > 1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Holder-continuity of solutions with respect to time.
112 citations
TL;DR: In this article, the authors revisited this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century, and discussed subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period.
Abstract: The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation, and related bifurcations. c
97 citations
TL;DR: In this article, a procedure for second-order computational homogenization of heterogeneous materials is derived from the unit cell homogenisation, in which an appropriate representation of the micro-displacement field is assumed as the superposition of a local macroscopic displacement field, expressed in a polynomial form related to the macro-disposition field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities.
Abstract: A procedure for second-order computational homogenization of heterogeneous materials is derived from the unit cell homogenization, in which an appropriate representation of the micro-displacement field is assumed as the superposition of a local macroscopic displacement field, expressed in a polynomial form related to the macro-displacement field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities. This second contribution is represented as the superposition of two unknown functions each of which related to the first-order and to the second-order strain, respectively. This kinematical micro-macro framework guarantees that the micro-displacement field is continuous across the interfaces between adjacent unit cells and implies a computationally efficient procedure that applies in two steps. The first step corresponds to the standard homogenization, while the second step is based on the results of the first step and completes the second-order homogenization. Two multi-phase composites, a three-phase and a laminated composite, are analysed in the examples to assess the reliability of the homogenization techniques. The computational homogenization is carried out by a FE analysis of the unit cell; the overall elastic moduli and the characteristic lengths of the second-order equivalent continuum model are obtained. Finally, the simple shear of a constrained heterogeneous two-dimensional strip made up of the composites considered is analysed by considering a heterogeneous continuum and a homogenized second-order continuum; the corresponding results are compared and discussed in order to identify the validity limits of the proposed technique.
80 citations
TL;DR: In this paper, the viscous correction terms for the isothermal quantum Euler system of Degond, Gallego, and Mataric were derived by using a Chapman-Enskog expansion up to order 1.
Abstract: The aim of this paper is to compute viscous correction terms for the isothermal quantum Euler system of Degond, Gallego, M ehats (SIAM Multiscale Model Simul.,6, 2007). We derive this model by using a Chapman-Enskog expansion up to order 1. In a last part, we con- sider a situation where the ow is nearly irrotational in order to get a simplified model.
72 citations
TL;DR: In this paper, a mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented and theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved.
Abstract: The mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented. Weak setup of the problems based on mechanical variational principles is studied. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. Some properties of the spectrum of the problems under consideration are established. The studies are performed applying the functional analysis techniques. Finally, the Rayleigh principle for eigenfrequencies is constructed.
65 citations
TL;DR: In this paper, a thermodynamically consistent phenomenological model for the anisotropic Mullins effect in filled elastomers is presented, which takes into account both softening and permanent set.
Abstract: In this paper a thermodynamically consistent phenomenological model for the anisotropic Mullins effect in filled elastomers is presented. The model takes into account anisotropic softening as well as permanent set. The formulation is based on an anisotropic three-dimensional softening criterion and a scalar damage function both formulated in terms of the principal stretches. The damage function describes the difference in stresses between the primary loading curve and unloading curve in uniaxial tension tests and is evaluated from experimental data. The predictive capabilities of the proposed model are examined in comparison to experimental data available in literature as well as to own experimental results on CR rubber presented in the paper. Good agreement with these experiments is observed. In particular, the characteristic S-shape of the stress softening curves is accurately captured.
46 citations
TL;DR: In this article, a fractional approach to describe the diffusion process in fractal media is proposed, where the continuity and constitutive equations are derived by means of local fractional calculus, and the problem is formulated both in the steady-state regime and in the transient regime.
Abstract: In this paper, a fractional approach to describe the diffusion process in fractal media is put forward. After introducing anomalous diffusion quantities, the continuity and constitutive equations are derived by means of local fractional calculus, and the problem is formulated both in the steady-state regime and in the transient regime. Eventually, a simple heat conduction problem in the steady-state regime is solved analytically.
46 citations
TL;DR: In this article, the in-plane motion of elastic strings on tree-like network, observed from the 'leaves' is considered, and the inverse problem of recovering not only the physical properties, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges.
Abstract: We consider the in-plane motion of elastic strings on tree-like network, observed from the 'leaves' We investigate the inverse problem of recovering not only the physical properties, i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges To this end we use the Boundary Control method for wave equations on graphs established in [4, 7]. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim
40 citations
TL;DR: In this paper, the concept of representative directions is proposed to generalize one-dimensional material models for uniaxial tension to complete three-dimensional constitutive models for the finite element method.
Abstract: The concept of representative directions is intended to generalize one-dimensional material models for uniaxial tension to complete three-dimensional constitutive models for the finite element method. The concept is applicable to any model which is able to describe uniaxial loadings, even to those for inelastic material behavior without knowing the free energy. The typical characteristics of the respected material class are generalized in a remarkable similarity to the input model. The algorithm has already been implemented into the finite element systems ABAQUS and MSC.MARC considering several methods to increase the numerical efficiency. The implementation enables finite element simulations of inhomogeneous stress conditions within technical components, though the input model predicts uniaxial material behavior only.
40 citations
TL;DR: In this paper, a constitutive material model for filling-reinforced rubber is proposed to study the process dependence of the dynamic modulus, bimodal DMA tests and transient multistep tests.
Abstract: Filler-reinforced rubber shows many interesting nonlinear effects under cyclic deformations. As a result of the dynamic loading, a change in the materials' microstructure and hence in the dynamic behaviour of the elastomer is observed. In this context, the frequency-, amplitude-, temperature-, and the preload-dependence are well-known effects. Additionally, pronounced thermomechanical couplings are observed, e.g., heat build-up phenomena. Mechanical coupling effects can be demonstrated by studying the transient dynamic behaviour of the Payne-effect (amplitude dependence). Using the technique of dynamical mechanical analysis (DMA) the mentioned effects can be investigated in a very comfortable way. To study the process dependence of the dynamic modulus, bimodal DMA tests and transient multistep tests have been carried out. The non-trivial postprocessing of the bimodal measurements is shortly explained in the paper. The approach of finite nonlinear viscoelasticity with additional internal variables provides an excellent basis for constitutive material modelling. The thermodynamical consistency of the developed constitutive model is demonstrated. This offers the possibility to represent thermomechanical coupling effects like the dissipative heat build-up. A series of numerical results of FEM simulations under more complicated transient loading histories, computed with the developed and implemented material model, are presented.
TL;DR: In this paper, the problem of the steady laminar flow of an incompressible viscous electrically conducting fluid over a rotating disk in the presence of a uniform transverse magnetic field is extended to the case of partial slip.
Abstract: The present paper is devoted to the solution of the steady laminar flow of an incompressible viscous electrically conducting fluid over a rotating disk in the presence of a uniform transverse magnetic field. Classical von Karman problem of a rotating disk is extended to the case where the disk surface admits partial slip. Using von Karman similarity transformation the nonlinear equations of motion are reduced to a boundary value problem whose solution is then obtained in terms of a series of exponentially decaying functions for the full range of slip coefficients. The exact numerical method is found to improve as the strength of the magnetic field and the strength of the applied slip are increased. The effects of the magnetic field together with the slip on the physically significant relevant parameters, such as the wall shears, the torque, and the vertical suction are clarified. Purely explicit analytical expressions for the solution of magnetohydrodynamic equations to support the numerically evaluated solutions are also obtained via the homotopy analysis method.
TL;DR: In this article, the authors used the recurrence plot technique to identify oscillations, rotations, and transient chaotic vibrations for relatively short time series composed of only few cycles, and used RQA to distinguish different types of motion.
Abstract: We investigated dynamic responses of a parametric pendulum obtained experimentally. Using the recurrence plot technique designed to analyze experimental time series we have distinguished different types of motion. This method, supplemented by recurrence quantification analysis (RQA), has been used to identify oscillations, rotations, and transient chaotic vibrations for relatively short time series composed of only few cycles.
TL;DR: In this paper, the problem of trapped mode frequencies in a three-dimensional infinite channel is considered in the framework of linear water-waves, where the body has a rough surface characterized by a small parameter e > 0 while the distance of the body to the water surface is also of order e. Under a certain symmetry assumption, the accumulation effect for trapped modes frequencies is established.
Abstract: The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter e > 0 while the distance of the body to the water surface is also of order e. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely it is proved that, for any given d > 0 and integer N > 0, there exists a number e (d, N) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case e ∈ ( 0, e (d, N)). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.
TL;DR: In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered, and it is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum.
Abstract: A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.
TL;DR: In this article, boundary control problems governed by a system of semilinear parabolic PDEs with pointwise control constraints are considered, and the existence of an optimal solution is shown.
Abstract: In this work, boundary control problems governed by a system of semilinear parabolic PDEs with pointwise control constraints are considered. This class of problems is related to applications in the chemical catalysis. After discussing existence and uniqueness of the state equation with both linear and nonlinear boundary conditions, the existence of an optimal solution is shown. Necessary and sufficient optimality conditions are derived to deal with numerical examples, which conclude the paper.
TL;DR: In this article, the influence of the temperature history on the Mullins effect, its recovery behavior and the rate dependence is experimentally investigated using NR/BR (NR: natural rubber, BR: polybutadiene rubber) rubber blend.
Abstract: The influence of the temperature history on the Mullins effect, its recovery behaviour and the rate dependence is experimentally investigated using NR/BR (NR: natural rubber, BR: polybutadiene rubber) rubber blend. The crystallization which occurs in rubber during long term storage below the melting temperature has been taken into account to interpret the experimental data. To study the influence of low temperatures and large deformations on the Mullins effect, cyclic strain-controlled processes are applied under different temperatures. The softened specimens are subjected to a sequence of heating, cooling, and conditioning processes in order to study the influence of the temperature history on healing, melting, and recrystallization. The results indicate the existence of a threshold temperature: if the specimen temperature is larger than this threshold, a nearly complete recovery of the material occurs within finite time, while any temperature below this limit will be too small for healing. The temperature dependence of both the healing and the Mullins effect in rubber with different degrees of crystallinity is resolved by considering the melting and recrystallization rates. The rate dependence of the blend is investigated under different temperatures via monotonic and cyclic tension tests at different strain rates and relaxation tests. The experimental data suggests a decrease in the strain rate sensitivity at higher temperatures.
TL;DR: In this article, the electro-mechanical behavior of lead zirconate titanate PIC151 (PI Ceramics) samples is studied by a joint experimental and numerical investigation.
Abstract: The electro-mechanical behavior of ferroelectric materials is studied by a joint experimental and numerical investigation. The experiments were performed on lead zirconate titanate PIC151 (PI Ceramics) samples, and the numerical simulations were done by a 2D Finite Element implementation of a continuum phase field model. To limit computational costs, the material is idealized as a single crystal. The phase field parameters for the material are determined through experimental parameter identification. Numerical results are presented to show the domain structure evolution within the crystal. In comparison to experimental results, calculations are carried out to disclose mesoscopic properties of the material under various poling scenarios, the dependence of the coercive field on the electric loading frequency, and the improvement of the material properties by stress-assisted poling procedures.
TL;DR: In this paper, a suction/injection controlled free convective motion of a viscous incompressible fluid between two periodically heated infinite vertical parallel plates is presented and the temperature and velocity fields are separated into steady and periodic parts and the resulting second order ordinary differential equations solved to obtain the solution to the problem.
Abstract: A suction/injection controlled free convective motion of a viscous incompressible fluid between two periodically heated infinite vertical parallel plates is presented. The temperature and velocity fields are separated into steady and periodic parts and the resulting second order ordinary differential equations solved to obtain the solution to the problem. The influence of each governing parameter is discussed with the aid of contour maps. The significant result from this study is that temperature is higher near the plate with injection while velocity is more enhanced near the plate with suction. It is also interesting to note that for large values of suction, the nature of the flow is that of constant heating of the plates, while for large values of Strouhal number, the flow behaves as if there was no suction/injection at the plates (Wang, [11]).
TL;DR: In this paper, a boundary element method (BEM) implementation for SH harmonic waves in a class of inhomogeneous anisotropic media is presented, where the inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density.
Abstract: The author presents a Boundary Element Method (BEM) implementation for SH harmonic waves in a class of inhomogeneous anisotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The implementation is based on a closed form fundamental solution for SH waves derived by Daros [C. H. Daros, A fundamental solution for SH‐waves in class of inhomogeneous anisotropic media, Int. J. Eng. Science 46 (2008) 809–817]. He shows numerical results obtained by the traditional boundary integral equation. Moreover, the non‐hypersingular traction based BEM is also implemented, allowing the modelling of cracks in inhomogeneous anisotropic media. The author obtains numerical results for the stress intensity factors (SIF) which are compared to previous published results.
TL;DR: In this article, a family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order, where a k-th order reconstruction requires the first k - 1 time derivatives of the velocity.
Abstract: It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion. Moreover, the reconstruction of the body's motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order, where a k-th order reconstruction requires the first k - 1 time derivatives of the velocity. Such reconstruction formulas could be used whenever the velocity field is accessible. As an example the formulas are applied to the rotation update in a momentum preserving time stepping scheme for the dynamic Euler equations.
TL;DR: In this paper, the results of Altenbach and Eremeyev were used to obtain bounds on the constitutive parameter α t from the six-parameter shell theory model, and the results showed that the drilling stiffness α t affects the FEM results.
Abstract: In this paper we discuss the constitutive relations for micropolar plates recently obtained by Altenbach and Eremeyev. We pay particular attention to their relation for the resultant drilling stress couple and compare it with that used so far in the statically and kinematically exact nonlinear six-parameter shell theory. Using the results of Altenbach and Eremeyev, we present bounds on values of the constitutive parameter α t from the six-parameter shell theory model. Some representative numerical simulations show how the drilling stiffness α t affects the FEM results.
TL;DR: In this paper, some analytical and numerical aspects of time-dependent models with internal variables are discussed and the focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials.
Abstract: In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.
TL;DR: In this article, necessary and sufficient conditions for a given system of second-order ODEs to be of Lagrangian form with additional dissipative forces were derived and shown to be independent.
Abstract: In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not independent and prove a stronger result accordingly.
TL;DR: In this article, a viscoelastic elastomer is examined with respect to its thermo-mechanical behavior and uniaxial tension tests and relaxation tests are performed at different constant temperatures up to the glass transition region.
Abstract: In this contribution a viscoelastic elastomer is examined with respect to its thermo-mechanical behaviour. Therefore uniaxial tension tests and relaxation tests are performed at different constant temperatures up to the glass transition region. Hence an experimental data pool is provided. On the theoretical side a finite viscoelastic and incompressible material model is used and enhanced by temperature dependency in order to model the experimentally observed effects. The material parameters are strategically identified by means of biologic evolution strategies. Thereby it turned out that the investigated material cannot be modelled as a thermo-rheologically simple material. Quite the contrary, a new ansatz is chosen.
TL;DR: In this article, a constitutive model for the numerical simulation of rubber behavior in a wide frequency range is presented, which combines Simo's viscoelastic model and a pseudo-elastic approach.
Abstract: A constitutive model for the numerical simulation of rubber behavior in a wide frequency range is presented. The combination between the well known Simo's viscoelastic model and a pseudo-elastic approach enables for the modeling of inelastic effects at low frequencies, such as nonlinear elasticity, hysteretic behavior, and damage (Mullins effect). The constitutive formulation is derived in detail with an aim of the finite element implementation. Because mechanical response at high frequencies is usually characterized by complex modulus, the developed viscoelastic damage model is extended to high-frequency analysis. A key idea is the decomposition of the deformation gradient into linear and nonlinear part. The nonlinear part is associated with inelastic deformations established at low frequencies, while the linear contribution plays an important role for dynamic analysis at high frequencies. As a result, the steady-state response of rubber at a certain static deformation is evaluated and consequently leads to the numerical solution for complex modulus. The computational efficiency of the proposed model can be seen from a good agreement with experimental data.
TL;DR: In this paper, a method in terms of parametric variable representations via the plastic work is proposed to study two-way plastic flow, which may give rise to any given shape of strain recovery loops.
Abstract: Finite elastoplastic J 2 -flow models with combined hardening are found to exhibit strain recovery effects. For uniaxial deformation of bars, these models produce plastic flow with the axial stress growing beyond the initial yield stress and then generate reverse plastic flow with the axial stress reducing to zero. A novel method in terms of parametric variable representations via the plastic work is proposed to study such two-way plastic flow. By means of this new method, it is shown that the foregoing two-way plastic flow may give rise to any given shape of strain recovery loops, and, furthermore, explicit models are constructed with strain recovery loops formed by two-way plastic flow. In conjunction with the parametric variable representation method proposed, these findings suggest the possibility of applying straightforward, classical elastoplasticity models to characterizing pseudoelastic behaviour of shape memory materials.
TL;DR: In this article, the asymptotic fields of mixed-mode frictional cohesive cracks in quasi-brittle materials have been derived after reformatting the cohesive-law in a special but universal polynomial containing fractional or integer powers.
Abstract: This paper discusses the asymptotic fields ahead of mixed mode frictional cohesive cracks in quasi-brittle materials. These fields have been derived after reformatting the cohesive-law in a special but universal polynomial containing fractional or integer powers. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions for a traction-free crack. The coefficients of the expansions however depend nonlinearly on the softening law and the boundary conditions. As expected, the asymptotic field of a frictional cohesive crack reduces to that of a frictionless cohesive crack with normal cohesive separation when the friction coefficient becomes zero. It is also shown that the asymptotic field of a frictionless cohesive crack can be used to a non-cohesive crack opened by crack face loading, after including the singular terms corresponding to the eigenvalue 1/2. Furthermore, the coefficients are given explicitly for two special softening laws that are commonly used in practice. The leading terms of the true displacement asymptotic field are also derived explicitly. These are especially useful as the enrichment functions at the tip of a mixed mode cohesive crack for accurate simulation of crack growth in quasi-brittle materials using the extended finite element (XFEM).
TL;DR: In this paper, the authors considered a contact problem in linear thermoelastic diffusion theory and proved the existence of a weak solution by the use of Sobolev's basic space energy arguments.
Abstract: We consider a one-dimensional contact problem in linear thermoelastic diffusion theory. The coupled system of equations consists of a hyperbolic equation and two parabolic equations. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. The existence of a weak solution is proved by the use of Sobolev’s basic space energy arguments. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity.
TL;DR: HAL as discussed by the authors is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. On the phase transitions in deformable solids Victor Eremeyev, Franz Fischer