Generalized Models and Non-classical Approaches in Complex Materials 1

The asymptotic homogenization method is applied to a family of boundary value problems for linear piezoelectric heterogeneous media with periodic and rapidly oscillating coefficients.We consider a two-phase fibrous composite consisting of identical circular cylinders perfectly bonded in a matrix. Both constituents are piezoelectric 622 hexagonal crystal and the periodic distribution of the fibers follows a rectangular array. Closed-form expressions are obtained for the effective coefficients, based on the solution of local problems using potential methods of a complex variable. An analytical procedure to study the spatial heterogeneity of the strain and electric fields is described. Analytical expressions for the computation of these fields are given for specific local problems. Examples are presented for fiber-reinforced and porous matrix including comparisons with fast Fourier transform (FFT) numerical results.

The spectra of SH guided waves in an isotropic continuously layered plate with arbitrary profile of the limiting slowness ŝ(y) across the plate are explicitly analyzed for high frequencies ω in the framework of “adiabatic” approximation. Dispersion equations and their solutions are analytically found for free, clamped or free-clamped faces of the plate. The positions of horizontal asymptotes for dispersion branches are determined by extreme points y _i of the dependence ŝ(y) including also inflection points and “linear” (non-extreme) min/max points. In the vicinity of all asymptotic levels, apart from the upper one, spectra form specific ladder-like patterns. Explicit asymptotics of dispersion curves are derived for a series of particular local dependencies ŝ(y) in the vicinity of points y _i .

The Quantum Fluid Dynamics (QFD) representation has its foundations in the works of Madelung (1929), De Broglie (1930 - 1950) and Bohm (1950 - 1970). It is an interpretation of quantum mechanics with the goal to find classically identifiable dynamical variables at the sub-particle level. The approach leads to two conservation laws, one for "mass" and one for "momentum", similar to those in hydrodynamics for a compressible fluid with a particular constitutive law. The QFD equations are a set of nonlinear partial differential equations. This paper extends the QFD formalism of quantum mechanics to the Nonlinear Schrödinger and the Gross-Pitaevskii equation.

This paper deals with the problem of local buckling caused by uniaxial stretching of an infinite plate with a circular hole or with circular inclusion made of another material. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible. This paper also shows the difference between them when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. Computational models show that instability modes are different both when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. The case when plate and inclusion have the same modulus of elasticity, but different Poisson’s ratio is investigated too. It is also discussed here the case when a plate with inclusion is under biaxial tension. For each ratio of the modulus of elasticity of plate versus inclusion it’s obtained the range of the load parameters for which the loss of stability is impossible.

The paper presents an extension to viscoelastic composites of a former developed numerical homogenization procedure which was used for elastic and piezoelectric material systems. It is based on an unit cell model using the finite element method. In the paper a brief description of the basic equations and the homogenization algorithm with specific attention to the numerical model is given. The investigated composites consist of a viscoelastic matrix with unidirectional embedded cylindrical elastic fibers. Hence the homogenized behavior of the composite is also viscoelastic. Consequently the effective coefficients are time-dependent. The geometrical shape of the unit cell is rhombic which allows to analyze a wide range of nonstandard unidirectional fiber distributions. Otherwise it includes the special cases for square and hexagonal fiber arrangements which can be used for comparisons with other solutions. Here results are compared with an analytical homogenization method. Furthermore the influences of rhombic angle and fiber volume fraction on effective coefficients are investigated. In addition two limit cases are considered. One is with air as inclusions which is equivalent to a porous media and the other is the pure matrix without fibers.

This chapter deals with the modeling and design of inner resonance media, i.e. media that present a local resonance which has an impact on the overall dynamic behaviour. The aim of this chapter is to provide a synthetic picture of the inner resonance phenomena by means of the asymptotic homogenization method (Sanchez-Palencia, 1980). The analysis is based on the comparative study of a few canonical realistic composite media. This approach discloses the common principle and the specific features of different inner resonance situations and points out their consequences on the effective behavior. Some general design rules enabling to reach such a specific dynamic regime with a desired effect are also highlighted. The paper successively addresses different materialshaving different behaviours and inner structures as elastic composites, reticulated media, permeable rigid and elastic media,undergoing phenomena governed either by momentum transfer or/and mass transfer,in which the inner resonance mechanisms can be highly or weakly dissipative,in situation of inner resonance or inner anti-resonance.The results related to different physical behaviours show that inner resonance requires a highly contrasted microstructure. It constrains the resonant constituent to respond in a forced regime imposed by the non resonant constituent. Then, the effective constitutive law is determined by this latter while the resonating constituent acts as an atypical source term in the macroscopic balance equation. It is established that inner resonance governed by momentum (resp. mass) balance yields unconventional mass (resp. bulk modulus). Furthermore, inner-resonance in media characterized by hyperbolic or parabolic dynamic equations can be handled in a similar manner, leading however to strongly distinct effective features.

The balance of material momentum is applied to the motion of an ideal, incompressible fluid with special emphasis on water waves. To this end, the fluid flow is represented by its material or Lagrangian description. A variational approach using Hamilton’s principle is employed, with the incompressibility condition incorporated into the Lagrangian by means of a Lagrange multiplier. The balance of material momentum is obtained in its standard form known from nonlinear elasticity, however with the peculiarity that the dynamic Eshelby stress becomes hydrostatic and its divergence reduces to the (negative) gradient of an “Eshelby pressure”. The balance is applied to Gerstner’s nonlinear theory of water waves.

We exploit Clemmow’s complex plane-wave representation of electromagnetic fields to construct globally exact solutions of Maxwell’s equations in a piecewise homogeneous dispersive conducting medium containing a plane interface that can sustain (possibly dissipative) field-induced surface electric currents. Families of solutions, parametrised by the complex rotation group SO(3, ℂ), are constructed from the roots of complex polynomials with coefficients determined by constitutive properties of the medium and a particular interface admittance tensor. Such solutions include coupled TE and TM-type surface polariton and Brewster modes and offer a means to analyse analytically their physical properties given the constitutive characteristics of bulk meta-materials containing fabricated meta-surface interfaces.

The paper deals with continuous models of elasto-plastic materials with microstructural defects such as dislocations and disclinations. The basic assumptions concern the existence of plastic distortion and so-called plastic connection with metric property and the existence of the free energy function. This is dependent on the Cauchy-Green strain tensor, and its gradient with respect to the plastically deformed anholonomic configuration, and on the dislocation and disclination densities. The defect densities are defined in terms of the incompatibility of the plastic distortion and non-integrability of the plastic connection. The evolution of plastic distortion and disclination tensor has been postulated under the appropriate viscoplastic and dissipative type equations, which are compatible with the principle of the free energy imbalance. The associated small distortion model is provided. The present model and the previous ones have been also compared.

The present work deals with the estimation of the linear viscoelastic effective properties for composites with periodic structure and rectangular cross-section fibers, using the two-scale asymptotic homogenization method (AHM). As a particular case, the effective properties for a layered medium with transversely isotropic properties are obtained. Two times the homogenization method, in different directions, according to the geometrical configuration of the composite material is applied for deriving the analytical expressions of the viscoelastic effective properties for a composite material with rectangular cross-section fibers, periodically distributed along one axis. In addition to that, models with different creep kernels, in particular, the Rabotnov’s kernel are analyzed. Finally, the numerical computation of the effective viscoelastic properties is developed for the analysis of the results. Moreover, a numerical algorithm using FEM is developed in the present work. Comparisons with other approaches are given as a validation of the present model.

A theory of plastic bending of single crystal beam with two active slip systems accounting for continuously distributed excess dislocations is proposed. If the resistance to dislocation motion is negligibly small, then excess dislocations pile up against the beam’s neutral line, leaving two small layers near the lateral faces dislocation-free. The threshold value at the onset of plastic yielding, the dislocation density, as well as the moment-curvature curve are found. If the energy dissipation is taken into account, excess dislocations at the beginning of plastic yielding occupy two thin layers, leaving the zone near the middle line as well as two layers near the beam’s lateral faces dislocation-free. The threshold bending moment at the dislocation nucleation and the hardening rate are higher than those in the case of zero dissipation.

The aim of this review is to give an overview of techniques and methods used in the modeling of acoustic and elastic metamaterials. Acoustic and elastic metamaterials are man-made materials which present exotic properties capable to modify and drive wave propagation. In particular in this work we will focus on locally resonant microstructures. Such metamaterials are based on local resonances of the internal structure, the dimensions of which are much smaller than the wavelengths of the waves under analysis. We will consider the seminal papers in the fields to grasp the most important ideas used to develop locally resonant metamaterials, such as homogenization techniques and optimization topology. Finally, we will discuss some interesting application to clarify the aforementioned methods.

In nonlinear theories the axiom of equipresence requires all the effects of the same order to be taken account. In this paper the mathematical modelling of deformation waves in media is analysed involving nonlinear and dispersive effects together with accompanying phenomena caused by thermal or electrical fields. The modelling is based on principles of generalized continuum mechanics developed by G.A. Maugin. The analysis demonstrates the richness of models in describing the physical effects in media with complex properties.

Within the theory of continuous distributions of defects in materials, this paper advocates a point of view based on the geometry of differential forms. Applications to smectic liquid crystals and to multi-walled nanotube composites serve to show how this dual mathematical approach fits perfectly with the intended physical reality. Moreover, the weak formulation of the theory in terms of de Rham currents delivers the description of discrete isolated dislocations as a generalization of the smooth theory.

We discuss the particular class of strain-gradient elastic material models which we called the reduced or degenerated strain-gradient elasticity. For this class the strain energy density depends on functions which have different differential properties in different spatial directions. As an example of such media we consider the continual models of pantographic beam lattices and smectic and columnar liquid crystals.

The method of virtual power, put forward by Paul Germain and celebrated by Gérard A. Maugin, is used (and abused) in the present work in combination with continuum thermodynamics concepts in order to develop generalized continuum, phase field, higher order temperature and diffusion theories. The systematic and effective character of the method is illustrated in the case of gradient and micromorphic plasticity models. It is then tentatively applied to the introduction of temperature and concentration gradient effects in diffusion theories leading to generalized heat and mass diffusion equations.

The influence of stresses and strains on a chemical reaction rate and a chemical reaction front velocity is studied basing on the concept of the chemical affinity tensor. The notion of forbidden zones formed by strains or stresses at which the reaction cannot go is discussed. Examples of forbidden zones are constructed.

The problem of propagation of acoustic waves in the presence of a submerged elastic object of cylindrical or spherical shape is formulated here in a unified fashion. With this general formalism is associated a synthetic procedure of resolution of the continuity conditions for the fields at the interfaces. This procedure, based on the operator concept, leads in a simple and direct fashion to the generalized Debye series for the exact solution of wave propagation in the case of separable geometries with the acoustic source either inside or ouside to the scatterer. For cylindrical and spherical geometries, this result, which we term the "Generalized Debye Series Theory" (GDST), is exploited for various applications and appears as a complementary contribution to the Resonance Scattering Theory (RST) as established by H. Überall et al.

We consider gyrocontinuum, whose each point-body is an infinitesimal rigid body containing inside an axisymmetric rotor, attached to the body but freely rotating about its axis. Point-bodies of the medium may perform independent translations and rotations of general kind. The proper rotation of their rotors does not cause stresses in the medium. We consider the case of infinitesimal density of inertia tensor both of rotor and carrying body and large proper rotation velocity of the rotor, resulting together in a finite dynamic spin. Rotor inside each point body does not interact with anything but its carrying point body, i.e. its existence only contributes into the kinetic energy but not to the strain energy.We suppose that this medium does not react to the gradient of turn of the carrying bodies, therefore we call it “reduced”. This yields in zero couple stresses. For simplicity we consider the elastic energy of the medium to be isotropic. This is a medium similar to the reduced Cosserat medium but with the kinetic moment consisting of a gyroscopic term. An example of such an artificially made medium could be a medium consisting of interacting light spheres with light but fast rotating rotors inside them. We consider linear motion of the carrying spheres and investigate harmonic waves in this continuum. We see that, similar to isotropic reduced Cosserat medium, longitudinal wave is non-dispersional, and shear-rotational wave has dispersion and one its branch has a band gap. The band gap depend on the dynamic spin of point bodies and can be controlled via it. Note that all the shear harmonic waves in this medium are not plane waves but have polarization, if the direction of propagation is not orthogonal to the rotor axes.

The DNA denaturation, the double-stranded DNA unwinding process, is a vital process for cells. The percentage of the denaturation is used as an index of organism complexity and it is the base of the DNA hybridisation technique, which provides a great deal of information. It can be detected by observing the increase in the ability of a DNA solution to absorb ultraviolet light at a wavelength of 260 nm. Based on experimental data, we found a mathematical model capable of predicting the behaviour of a general DNA, given the melting temperature T _m .

We propose a mathematical model for propagation of the long nonlinearly elastic longitudinal strain waves in a bar, which contains nanoscale structural inclusions. The model is governed by a nonlinear doubly dispersive equation (DDE) with respect to the one unknown longitudinal strain function. We obtained the travelling wave solutions to DDE, and, in particular, the strain solitary wave solution, which was shown to be significantly affected by parameters of the inclusions. Moreover we found some critical inaccuracies, committed in papers by others in the derivation of a constitutive equation for the long strain waves in a microstructured medium, revised them, and showed an importance of improvements for correct estimation of wave parameters.

The paper is concerned with the linear theory of piezoelectricity for isotropic chiral Cosserat elastic solids. The behavior of chiral bodies is of interest for the investigation of auxetic materials, carbon nanotubes, bones, honeycomb structures, as well as composites with inclusions. First, we establish the basic equations which govern the behavior of thin plates. It is shown that, in contrast with the theory of achiral plates, the stretching and flexure cannot be treated independently of each other. Then, we present a uniqueness result with no definiteness assumption on elastic constitutive coefficients. A reciprocity theorem is also established. Then, we present the conditions on the constitutive coefficients which guarantee that the energy of the system is positive definite and we give a continuous dependence result. In the case of stationary theory we derive a uniqueness result for the Neumann problem. Finally, the effects of a concentrated charge density in an unbounded plate are investigated.

In previous papers the concepts of caloric and entropic temperatures were outlined and illustrated in a few examples (ideal gas, ideal systems with two or three energy levels, solids with defects or dislocations). In equilibrium states, all degrees of freedom are at the same temperature, but out equilibrium they have different non-equilibrium temperatures. In this work, using a systematic methodology of classical irreversible thermodynamics, we take into consideration an undeformable medium in which the contributions of microscopic phenomena to the macroscopic specific internal energy U can be described by introducing two internal variables and one hidden variable. Internal variables are measurable (from the thermal point of view, they exchange directly heat with the system acting as thermometer) but not controllable, whereas (in our proposal) hidden variables are also not controllable but, in addition, they do not exchange directly heat with the thermometer, but only with other variables. The aim of this paper is to explore the difference between internal and hidden variables and to establish connections and relations among their corresponding non-equilibrium temperatures and the equilibrium temperature of the medium under consideration.

In this paper a generalized discrete elastica model including bending, normal and shear interactions is developed. Nonlinear static analysis of the discrete model is accomplished, its buckling and post-buckling behavior are thoroughly studied. It is revealed that based on what finite strain theory is used, the discrete model yields a generalized (extensible) Engesser elastica, or a generalized (extensible) Haringx elastica. The local continuum counterparts of these models are also obtained. Then nonlocal models are developed from the introduced flexural, extensible, shearable discrete systems using a continualization technique. Analytical and numerical solutions are given for the discrete and nonlocal models, and it is shown that the scale effects of the discrete models are well captured by the continualized nonlocal models.

In this chapter, a dynamic finite-strain plate theory for incompressible hyperelastic materials is deduced. Starting from nonlinear elasticity, we present the three-dimensional (3D) governing system through a variational approach. By series expansion of the independent variables about the bottom surface, we deduce a 2D vector dynamic plate system, which preserves the local momentum-balance structure. Then we propose appropriate position and traction boundary conditions. The 2D plate equation guarantees that each term in the variation of the generalized potential energy functional attains the required asymptotic order. We also consider the associated weak formulations of the plate model, which can be applied to different types of practical edge conditions.

We investigate a nonlinear, one-dimensional problem of thermoelectroelasticity with thermal relaxation and in quasi-electrostatics. The system of basic equations is a restriction to one spatial dimension of that proposed earlier in Abou-Dina et al (2017). This model is based on the introduction of the heat flow vector as an additional state variable, thus leading to a Cattaneo-type evolution equation. It includes several non-linear couplings and may be useful in studying problems of elastic dielectrics at low temperatures, as well as in problems of high-temperature heat conduction in dielectric solids subjected to strong high-frequency laser beams. For the present investigation, however, only a few nonlinearities have been retained in the equations for conciseness. A numerical solution is presented for the system of nonlinear equations using an iterative, quasilinearization scheme by finite differences. The numerical results are discussed.

In order to discuss the transmission of forces and moments in cellular materials, a computational scheme by use of rod-beam element is adopted to represent open-celled skeletal materials with random configuration. The target domain is obtained by so-called the Voronoi tessellation technique, in which we consider the line segments of the polyhedra as the substantial beam-like members, and these members are interconnected with each other at the corners. Finite element analyses by rod-beam elements are then carried out to examine the characteristics of the complex structures. We discuss the transition from microscopic deformation in member beams to macroscopic response of such structures.

Electro-active polymers (EAPs) are a comparatively new class of smart materials that can change their properties and undergo large deformations as a result of an external electric excitation. These characteristics make them promising candidates in a wide range of applications, for example in sensor and actuator technology. As the experimental testing is both expensive and time consuming, simulation methods are developed in order to predict the material behavior. These simulations are based on well established energy formulation that are amended by additional coupling terms, often times in form of an invariant description. While the form of the purely mechanical and purely electric invariant quantities does not vary among the contributions of the electro-mechanical community, two different formulations for the coupling invariant can be found. In this contribution we demonstrate the influence of the selected coupling invariant on the material response. Therefore a thermo-electromechanically coupled constitutive model is derived based on the frequently used total energy approach.We devise the relevant constitutive equations starting from the basic laws of thermodynamics. Two distinctively different non-homogeneous boundary value problems are solved analytically in order to demonstrate the influence of the selected coupling invariant.

The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p₀ to reach ‘by chance’ a site p for the first time? Further important are recurrence and transience features of a random walk: A random walker starting at p₀ will he ever reach site p (ever return to p₀)? How often a site is visited? Here we investigate Markovian random walks generated by fractional (Laplacian) generator matrices L$$ \frac{\alpha }{2} $$ 2 (0 < $$ \alpha $$ ≤2) where L stands for ‘simple’ Laplacian matrices. This walk we refer to as ‘Fractional Random Walk’ (FRW). In contrast to classical Pólya type walks where only local steps to next neighbor sites are possible, the FRW allows nonlocal long-range moves where a remarkably rich dynamics and new features arise. We analyze recurrence and transience features of the FRW on infinite d-dimensional simple cubic lattices. We deduce by means of lattice Green’s function (probability generating functions) the mean residence times (MRT) of the walker at preselected sites. For the infinite 1D lattice (infinite ring) we obtain for the transient regime (0 < $$ \alpha $$ < 1) closed form expressions for these characteristics. The lattice Green’s function on infinite lattices existing in the transient regime fulfills Riesz potential asymptotics being a landmark of anomalous diffusion, i.e. random motion (Lévy flights) where the step lengths are drawn from a Lévy $$ \alpha $$-stable distribution.

The aim of this paper is a review on recently found new aspects in the theory of micropolar media. For this purpose the necessary theoretical framework for a micropolar continuum is initially presented. Here the standard macroscopic equations for mass, linear and angular momentum, and energy are extended in two ways. First, the aspect of coupling linear and angular rotational kinetic energies is emphasized. Second, the equations are complemented by a recently proposed kinetic equation for the moment of inertia tensors containing a production term. We then continue to explore the possibilities of this new term for the case of micropolar media encountering a change of moment of inertia during a thermomechanical process. Particular emphasis is put on the general form of the production of moment of inertia for a transversally isotropic medium and its potential to describe, for example, structural changes from a transversally isotropic state to an isotropic one. In order to be able to comprehend and to study the influence of the various material parameters the production term is interpreted mesoscopically and various other examples are solved in closed form. Moreover, in context with the presented example problems it will also become clear that the traditional Lagrangian way of describing the motion of solids might sometimes no longer be adequate and must then be replaced by a Eulerian approach.

State space and entropy rate of a discrete non-equilibrium system are shortly considered including internal variables and the contact temperature. The concept of internal variables in the context of non-equilibrium thermodynamics of a closed discrete system is discussed. The difference between internal variables and degrees of freedom are repeated, and different types of their evolution equations are mentioned in connection with Gérard A. Maugin’s numerous papers on applications of internal variables. The non-equilibrium contact temperature is recognized as an internal variable and its evolution equation is presented.

In the classical continuum mechanics several quantities related to angular velocity of rotation are introduced. Examples include vorticity vector, twirl tensors and logarithmic spin. Furthermore the corresponding rotation tensors can be defined to capture the orientation of triads. All of these quantities are measures of accompanying rotational motion and can be related to the deformation and velocity gradient. Such relationships are crucial for constitutive modeling of material behavior. The aim of this contribution is to recall classical definitions of rotation-like quantities and to present several new relationships between them.

As dictated by modern statistical physics, the second law is to be replaced by the fluctuation theorem (FT) on very small length and/or time scales. This means that the deterministic continuum thermomechanics must be generalized to a stochastic theory allowing randomly spontaneous violations of the Clausius-Duhem inequality to take place anywhere in the material domain. This paper outlines a formulation of stochastic continuum thermomechanics, where the entropy evolves as a submartingale while the dissipation function is consistent with the FT. A summary is then given of the behavior of an atomic fluid in Couette flow, studied using a combination of kinetic theory, hydrodynamic theory, and molecular dynamics. Overall, the developed framework may be applied in many fields involving fluid flow and heat conduction on very small spatial scales.

The algorithm is developed to model two-dimensional dynamic processes in a nonlocal square lattice on the basis of the shift operators. The governing discrete equations are obtained for local and nonlocal models. Their dispersion analysis reveals important differences in the dispersion curve and in the sign of the group velocity caused by nonlocality. The continuum limit allows to examine possible auxetic behavior of the material described by the nonlocal discrete model.

We propose a new class of models for electrorheological fluids. While the standard constitutive relations for electrorheological fluids are based on the assumption that the stress is a function of the symmetric part of the velocity gradient and the intensity of the electric field, we formulate constitutive relations in an implicit way. The stress, the symmetric part of the velocity gradient and the intensity of the electric field are linked via a tensorial implicit equation. The potential benefit of the new class of models is investigated by the analysis of a simple shear flow in a transverse electric field.

The goal of this paper is to link the geometric variables of strain gradient continuum with electromagnetic fields. For that purpose, we derive the electromagnetic wave equation within a Riemann-Cartan continuum where curvature and torsion are present. We also show that the gravitational and electromagnetic fields are respectively identified as geometric objects of such a continuum, namely the curvature $$ \Re_{\alpha \beta \lambda }^{\gamma } $$ for gravitation which is a classical result, and the torsion $$ \aleph_{\alpha \beta }^{\gamma } $$ as source of electromagnetism.

One of the reasons why the finite element method became the most used technique in Computational Solid Mechanics is its versatility to deal with bodies having a curved shape. In this case method’s isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard elements in the case of polygonal or polyhedral domains. However isoparametric finite elements helplessly require the manipulation of rational functions and the use of numerical integration. This can be a brain teaser in many cases, especially if the problem at hand is non linear. We consider a simple alternative to deal with boundary conditions commonly encountered in practical applications, that bypasses these drawbacks, without eroding the quality of the finite-element model. More particularly we mean prescribed displacements or forces in the case of solids. Our technique is based only on polynomial algebra and can do without curved elements. Although it can be applied to countless types of problems in Continuum Mechanics, it is illustrated here in the computation of small deformations of elastic solids.

An asymptotic two-dimensional formulation for the potential energy of a thin magnetoelastic plate is obtained from that for a three-dimensional magnetoelastic body subjected to conservative tractions and an applied magnetic field.

Ionic polymer-metal composites consist in a thin film of electro-active polymers (Nafion® for example) sandwiched between two metallic electrodes. They can be used as sensors or actuators. The polymer is saturated with water, which causes a complete dissociation and the release of small cations. The strip undergoes large bending motions when it is submitted to an orthogonal electric field and vice versa. We used a continuous medium approach and a coarse grain model; the system is depicted as a deformable porous medium in which flows an ionic solution.We write microscale balance laws and thermodynamic relations for each phase, then for the complete material using an average technique. Entropy production, then constitutive equations are deduced: a Kelvin-Voigt stress-strain relation, generalized Fourier’s and Darcy’s laws and a Nernst-Planck equation.We applied this model to a cantilever electro-active polymer strip undergoing a continuous potential difference (static case); a shear force may be applied to the free end to prevent its displacement. Applied forces and deflection are calculated using a beam model in large displacements. The results obtained are in good agreement with data published in the literature.

The Cahn-Hilliard and Ginzburg-Landau (Allen-Cahn) equations are derived from the second law. The intuitive approach of separation of full divergences is supported by a more rigorous method, based on Liu procedure and a constitutive entropy flux. Thermodynamic considerations eliminate the necessity of variational techniques and explain the role of functional derivatives.