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In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^3)$ model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted., Comment: arXiv admin note: text overlap with arXiv:2010.14308, arXiv:2003.08594

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size effects., Archive of applied mechanics; Vol. 93. 2023, Issue1, pp 5-21

International audience; We study the existence of solutions arising from the modelling of elastic materials using generalized theories of continua. In view of some evidence from physics of metamaterials, we focus our effort on two recent nonstandard relaxed micromorphic models including novel micro-inertia terms. These novel micro-inertia terms are needed to better capture the band-gap response. The existence proof is based on the Banach fixed-point theorem.

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size-effects., Comment: 16 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2104.11322

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type$W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$W(F)=μ2∥F∥2detF+f(detF); such an energy is rank-one convex if and only if the function$f$fis convex.

We solve the St.Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical stiffness singularity for a vanishing rod diameter, since slender specimens are in general described as stiffer., 41 pages, 29 figures

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^{5})$ terms. The form of the energy allows us to show the coercivity for terms up to order $O(h^{5})$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^{3})$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^{5})$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^{3})$ -case the coercivity is proven under some more restrictive conditions on the thickness $h$ .

We considerconformally invariantenergiesWon the group$${{\,\mathrm{GL}\,}}^{\!+}(2)$$GL+(2)of$$2\times 2$$2×2-matrices with positive determinant, i.e.,$$W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}$$W:GL+(2)→Rsuch that$$\begin{aligned} W(A\, F\, B) = W(F) \quad \text {for all }\; A,B\in \{a\, R\in {{\,\mathrm{GL}\,}}^{\!+}(2) \,|\,a\in (0,\infty ),\; R\in {{\,\mathrm{SO}\,}}(2)\}, \end{aligned}$$W(AFB)=W(F)for allA,B∈{aR∈GL+(2)|a∈(0,∞),R∈SO(2)},where$${{\,\mathrm{SO}\,}}(2)$$SO(2)denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute)quasiconvex envelopeof these functions. Our results, which are based on the representation$$W(F)=h\bigl (\frac{\lambda _1}{\lambda _2}\bigr )$$W(F)=h(λ1λ2)ofWin terms of the singular values$$\lambda _1,\lambda _2$$λ1,λ2ofF, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion$${\mathbb {K}}:=\frac{1}{2}\frac{\Vert F \Vert ^2}{\det F}$$K:=12‖F‖2detF. Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the$$W^{1,p}$$W1,p-quasiconvex envelope on the domain$${{\,\mathrm{GL}\,}}^{\!+}(n)$$GL+(n)which, in particular, ensure that a stricter version ofDacorogna’s formulais applicable to conformally invariant energies on$${{\,\mathrm{GL}\,}}^{\!+}(2)$$GL+(2).

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order $O(h^{5})$ in the shell thickness $h$ . The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is decisive for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess.

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

International audience; In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micro-morphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial the meso-and micro-coefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of angular momentum are formulated at an infinitesimal cube considering the total stress as linear and quadratic approximation of a spatial Taylor series expansion.

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order [Formula: see text] in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending–curvature tensor, with coefficients depending on the initial curvature of the shell.

International audience; In this paper we venture a new look at the linear isotropic indeterminate couple stress model in the general framework of second gradient elasticity and we propose a new alternative formulation which obeys Cauchy-Boltzmann's axiom of the symmetry of the force stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility to switch from a {4th order} (gradient elastic) problem to a 2nd order micromorphic model are also discussed with a view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational type boundary conditions are used in the partial integration, the classical anti-symmetric nonlocal force stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence--free second order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ a novel integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the complete, consistent traction boundary conditions for both models.

Constitutive laws in terms of the logarithmic strain tensor log U , i.e. the principal matrix logarithm of the stretch tensor U = F T F corresponding to the deformation gradient F , have been a subject of interest in nonlinear elasticity theory for a long time. In particular, there have been multiple attempts to derive a viable constitutive law of nonlinear elasticity from an elastic energy potential which depends solely on the logarithmic strain measure ‖ log U ‖ 2 , i.e. an energy function W : GL + ( n ) → R of the form (1) W ( F ) = Ψ ( ‖ log U ‖ 2 ) with a suitable function Ψ : [ 0 , ∞ ) → R , where ‖ . ‖ denotes the Frobenius matrix norm and GL + ( n ) is the group of invertible matrices with positive determinant. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form (1) is Legendre–Hadamard elliptic. Similarly, we consider the related isochoric strain measure ‖ dev n log U ‖ 2 , where dev n log U is the deviatoric part of log U . Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n = 2 , we show that for n ≥ 3 , no strictly monotone function Ψ : [ 0 , ∞ ) → R exists such that F ↦ Ψ ( ‖ dev n log U ‖ 2 ) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F ↦ Ψ ( ‖ dev n log U ‖ 2 ) + W vol ( det F ) cannot be rank-one convex for any function W vol : ( 0 , ∞ ) → R if Ψ is strictly monotone.

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies $$F \mapsto W_{\rm eH}(F):= \widehat{W}_{\rm eH}(U) := \left\{ \begin{array}{lll} \frac{\mu}{k}\,e^{k\, \| {\rm dev}_n \log {U}\|^2}+\frac{\kappa}{2\, {\widehat{k}}}\,e^{\widehat{k}\,[ {\rm tr}(\log U)]^2}&\quad \text{if}& \det\, F > 0,\\ + \infty & \quad \text{if} & \det F \leq 0,\end{array} \right.$$ based on the Hencky-logarithmic (true, natural) strain tensor $${\log U}$$ . Here, $${\mu > 0}$$ is the infinitesimal shear modulus, $${\kappa=\frac{2 \mu+3\lambda}{3} > 0}$$ is the infinitesimal bulk modulus with λ the first Lame constant, $${k,\widehat{k}}$$ are additional dimensionless material parameters, $${F=\nabla \varphi}$$ is the gradient of deformation, $${U=\sqrt{F^T F}}$$ is the right stretch tensor, and dev n $${\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot{\mathbb{1}}}$$ is the deviatoric part of the strain tensor $${\log U}$$ . Based on the multiplicative decomposition $${F=F_e\, F_p}$$ , we couple these energies with some isotropic elasto-plastic flow rules $${F_p\,\frac{\rm d}{{\rm d t}}[F_p^{-1}]\in-\partial \chi({\rm dev}_3 \Sigma_{e})}$$ defined in the plastic distortion F p , where $${\partial \chi}$$ is the subdifferential of the indicator function $${\chi}$$ of the convex elastic domain $${\mathcal{E}_{\rm e}({\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\mathbf{y}}^2)}$$ in the mixed-variant $${\Sigma_{e}}$$ -stress space, $${\Sigma_{e}=F_e^T D_{F_e}W_{\rm iso}(F_e)}$$ , and $${W_{\rm iso}(F_e)}$$ represents the isochoric part of the energy. While $${W_{\rm eH}}$$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $${W_{\rm eH}}$$ on the one hand and the elastic domain in $${\Sigma_{e}}$$ -stress space on the other hand are closely related. Thus, the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true stress–true strain relation remains monotone, as observed in experiments.

We consider a family of isotropic volumetric–isochoric decoupled strain energies $$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm dev}_n{\rm log} {U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}({\rm log} U)]^2}&\text{if}& \det\, F > 0,\\+\infty &\text{if} &\det F\leq 0,\end{array}\right.$$ based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, \({\kappa=\frac{2\mu+3\lambda}{3} > 0}\) is the infinitesimal bulk modulus with \({\lambda}\) the first Lame constant, \({k,\hat{k}}\) are dimensionless parameters, \({F=\nabla \varphi}\) is the gradient of deformation, \({U=\sqrt{F^T F}}\) is the right stretch tensor and \({{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}\) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy $$\begin{array}{ll}F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U)&:={\mu}\,\|{\rm dev}_n{\rm log} U\|^2+\frac{\kappa}{2}\,[{\rm tr}({\rm log} U)]^2,\end{array}$$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family \({W_{_{\rm eH}}}\) are polyconvex for \({k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}\) , extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ which is equal to the classical Hencky strain energy \[ W_{\mathrm{H}}(F) = \mu\,\lVert\operatorname{dev}_n\log U\rVert^2+\frac{\kappa}{2}\,[\operatorname{tr}(\log U)]^2 = \mu\,\lVert\log U\rVert^2+\frac{\Lambda}{2}\,[\operatorname{tr}(\log U)]^2 \] in a neighborhood of the identity matrix; here, $\operatorname{GL^+}(n)$ denotes the set of $n\times n$-matrices with positive determinant, $F\in\operatorname{GL^+}(n)$ denotes the deformation gradient, $U=\sqrt{F^TF}$ is the corresponding stretch tensor, $\log U$ is the principal matrix logarithm of $U$, $\operatorname{tr}$ is the trace operator, $\lVert X\rVert$ is the Frobenius matrix norm and $\operatorname{dev}_n X$ is the deviatoric part of $X\in\mathbb{R}^{n\times n}$. The extension can also be chosen to be coercive, in which case Ball's classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions $W_{\mathrm{VL}}$ in the so-called Valanis-Landel form \[ W_{\mathrm{VL}}(F) = \sum_{i=1}^n w(\lambda_i) \] with $w\colon(0,\infty)\to\mathbb{R}$, where $\lambda_1,\dotsc,\lambda_n$ denote the singular values of $F$.

We consider the weighted isotropic relaxed micromorphic model and provide an in depth investigation of the characteristic dispersion curves when the constitutive parameters of the model are varied. The weighted relaxed micromorphic model generalizes the classical relaxed micromorphic model previously introduced by the authors, since it features the Cartan-Lie decomposition of the tensors P,t and Curl P in their dev sym , skew and spherical part. It is shown that the split of the tensor P,t in the micro-inertia provides an independent control of the cut-offs of the optic banches. This is crucial for the calibration of the relaxed micromorphic model on real band-gap metamaterials. Even if the physical interest of the introduction of the split of the tensor Curl P is less evident than in the previous case, we discuss in detail which is its effect on the dispersion curves. Finally, we also provide a complete parametric study involving all the constitutive parameters of the introduced model, so giving rise to an exhaustive panorama of dispersion curves for the relaxed micromorphic model.

In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of the microstructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials., Comment: arXiv admin note: text overlap with arXiv:1607.07385

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin-Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps. The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro- parameters are identified by numerical homogenization in static tests on the unit-cell level in [30]. The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch-Floquet analysis for two wave directions. We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial., Comment: To appear in Journal of Elasticity

For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

In this paper we consider the bending theory of Mindlin-type elastic plates. We study the spatial evolution of the solution of a non-standard problem. This problem is characterized by some constraint conditions in which the displacement and velocity at a given time are proportional with their respective initial values. The study is organized in two parts: in the first part we consider the materials with internal energy density positive definite and in the end we study the strongly elliptic materials. Using numerical analysis for some materials, we compare the different Saint-Venant’s type decay rates obtained.

In this paper we consider the initial–boundary value problem which describes the bending of Mindlin type thermoelastic plates with voids. In terms of some appropriate time-weighted surface power functions, we study the spatial behaviour of thermodynamic processes for a large class of thermoelastic materials with voids. Bounded domains and unbounded domains are considered.

This paper deals with the linear theory of micropolar solid–solid mixtures. First, some existence, uniqueness and continuous dependence results are derived. Then, the propagation of plane waves is studied. There are possible four types of waves: longitudinal displacement waves, longitudinal microrotation waves, transverse displacement waves and transverse microrotation waves. A detailed analysis is presented for the longitudinal displacement waves and the longitudinal microrotation waves.

International audience; In this paper we consider the Grioli–Koiter–Mindlin–Toupin linear isotropic indeterminate couple stress model. Our main aim is to show that, up to now, the boundary conditions have not been completely understood for this model. As it turns out, and to our own surprise, restricting the well known boundary conditions stemming from the strain gradient or second gradient models to the particular case of the indeterminate couple stress model, does not always reduce to the Grioli–Koiter–Mindlin–Toupin set of accepted boundary conditions. We present, therefore, a proof of the fact that when specific “mixed” kinematical and traction boundary conditions are assigned on the boundary, no “a priori” equivalence can be established between Mindlin's and our approach.

We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy W on the special linear group SL ( 2 ) implies the polyconvexity of W.

International audience; n this paper we propose to study wave propagation, transmission and reflection in band-gap mechanical metamaterials via the relaxed micromorphic model. To do so, guided by a suitable variational procedure, we start deriving the jump duality conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, relaxed micromorphic media. Jump conditions to be imposed at surfaces of discontinuity embedded in Cauchy and Mindlin continua are also presented as a result of the application of a similar variational procedure. The introduced theoretical framework subsequently allows the transparent set-up of different types of micro-macro connections granting the description of both (i) internal connexions at material discontinuity surfaces embedded in the considered continua and, as a particular case, (ii) possible connections between different (Cauchy, Mindlin or relaxed micromorphic) continua. The established theoretical framework is general enough to be used for the description of a wealth of different physical situations and can be used as reference for further studies involving the need of suitably connecting different continua in view of (meta-)structural design. In the second part of the paper, we focus our attention on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side in order to perform explicit numerical simulations of wave reflection and transmission. This particular choice is descriptive of a specific physical situation in which a classical material is connected to a phononic crystal. The reflective properties of this particular interface are numerically investigated for different types of possible micro-macro connections, so explicitly showing the effect of different boundary conditions on the phenomena of reflection and transmission. Finally, the case of the connection between a Cauchy continuum and a Mindlin one is presented as a numerical study, so showing that band-gap description is not possible for such continua, in strong contrast with the relaxed micromorphic case.

In a series of papers which are either published [A.R. Hadjesfandiari and G.F. Dargush, Couple stress theory for solids, Int. J. Solids Struct. 48, 2496-2510, 2011; A.R. Hadjesfandiari and G.F. Dargush, Fundamental solutions for isotropic size-dependent couple stress elasticity, Int. J. Solids Struct. 50, 1253-1265, 2013] or available as preprints Hadjesfandiari and Dargush have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must always be skew-symmetric. Since they use an incomplete set of boundary conditions in their virtual work principle their statement contains unrecoverable errors. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and still yields a well-posed boundary value problem., arXiv admin note: text overlap with arXiv:1504.00868

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy F ↦ W ( F ) = W ^ ( log U ) defined in terms of logarithmic strain log U , where U = F T F , happens to be everywhere rank-one convex as a function of F, the new function F ↦ W ˜ ( F ) = W ^ ( log U − log U p ) need not remain rank-one convex at some given plastic stretch Up (viz. E p log ≔ log U p ). This is in complete contrast to multiplicative plasticity (and infinitesimal plasticity) in which F ↦ W ( F F p − 1 ) remains rank-one convex at every plastic distortion Fp if F ↦ W ( F ) is rank-one convex ( ∇ u ↦ ∥ sym ∇ u − e p ∥ 2 remains convex). We show this disturbing feature of the additive logarithmic plasticity model with the help of a recently introduced family of exponentiated Hencky energies.

In this paper we consider the bending theory of Mindlin type thermoelastic plates with voids. We study the temporal behaviour of the solution of the boundary-initial value problem of this theory. Assuming that the internal energy density is positive definite, relations describing the asymptotic behaviour of the Cesaro means of various parts of total energy are established. An extension of the results to a large class of thermoelastic materials with voids is given.

This paper deals with the theory of mixtures which have as constituents two micropolar incompressible fluids. First, using a specific algorithm, a Galerkin type representation of solution is given for the linearized two-dimensional dynamical problem. Then, the steady-state vibration problem is considered and uniqueness theorems are established for both bounded and unbounded domains. Finally, the Galerkin type representation is used to construct the fundamental solution for two-dimensional steady-state vibration problem.

The Cesaro means of various parts of the total energy are introduced in the context of the linear theory of micromorphic thermopiezoelectricity. Then, using some Lagrange identities, the relations describing the asymptotic behavior of the Cesaro means are established.

This paper deals with the Eringen’s theory for binary mixtures between elastic micropolar solids and incompressible micropolar fluids (Eringen in J Appl Phys 94:4184–4190, 2003). Using the weighted energy method, an uniqueness result in the case of unbounded domains for small displacement of the solid and for non-slow flow of fluid is presented.

In this paper we study the steady vibrations problem in linear theory of isothermal micropolar solid–fluid mixture. With the help of fundamental solution we establish representations of Somigliana type. Then, using the potentials of single layer and double layer, we reduce the boundary value problems to singular integral equations for which the Fredholm’s theorems are valid. Existence and uniqueness results for interior and exterior problems are presented.

In this paper we study the linear thermal theory of the micropolar solid-fluid mixtures. We consider a mixture consisting of an isotropic micropolar solid and a compressible fluid. First, we establish estimates of Saint–Venant type for bounded bodies in terms of two appropriate time–weighted surface power functions. An alternative estimate of Phragmen–Lindelof type for unbounded bodies is also presented. Further, for unbounded bodies we are able to establish an estimate which proves that the measure associated with the solution decays faster than an exponential of a second degree polynomial, provided an appropriate class of mixtures is considered. This estimate shows that at large distance from the support of the external given data, the spatial decay of processes is influenced only by the thermal effect and by the viscosities of the fluid.

The isothermal theory of binary micropolar solid–fluid mixture is considered in this paper. For the dynamical problem, a Galerkin type representation of the solution is established. Then, a fundamental solution is given for the three-dimensional partial differential system which describes the steady vibrations. Also, some basic properties of the fundamental solution and a direct application to the point load problem are presented.

In this paper we present inhomogeneous plane wave solutions within the context of linear theory of poroelastic materials. We consider the class of strongly elliptic homogeneous poroelastic materials with a center of symmetry. To construct the solutions in concern, we use the “directional-ellipse” method, which reduces the propagation problem to a secular equation for wave speeds and gives further on the algebraic system which furnishes the slowness and amplitude bivectors. Explicit expressions for all possible inhomogeneous plane waves are presented in the case of isotropic poroelastic materials. These solutions are further used to study the Rayleigh waves in an isotropic poroelastic half-space. The explicit equation for the Rayleigh surface wave speed is established.

This note deals with the isothermal linear theory of porous viscoelastic mixtures. Questions of uniqueness and continuous dependence for solutions of various classes of initial boundary value problems in mixtures consisting of two constituents: a porous elastic solid and a porous Kelvin–Voigt material are studied. The Lagrange identity and Logarithmic convexity methods are used to establish uniqueness and continuous dependence results, with no definiteness assumptions upon the internal energy.

In the present paper, we investigate a model for propagating progressive waves associated with the voids within the framework of a linear theory of porous media. Owing to the use of lighter materials in modern buildings and noise concerns in the environment, such models for progressive waves are of much interest to the building industry. The analysis of such waves is also of interest in acoustic microscopy where the identification of material defects is of paramount importance to the industry and medicine. Our analysis is based on the strong ellipticity of the poroelastic materials. We illustrate the model of progressive wave propagation for isotropic and transversely isotropic porous materials. We also study the propagation of harmonic plane waves in porous materials including the thermal effect.

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.

The present paper deals with the study of the amplitude of the steady-state vibrations in a right finite cylinder made of an isotropic Kelvin-Voigt material. Some exponential decay estimates, similar to those of Saint-Venant type, are obtained for appropriate cross-sectional area measures associated with the amplitude of the steady-state vibrations. It is proved that due to dissipative effects, the estimates in question hold for every value of the frequency of vibrations and for arbitrary values of the elastic coefficients. The results are extended to a semi-infinite cylinder and some alternatives of Phragmen-Lindelof type are established.

The aim of this paper is to study the asymptotic partition of the energy associated with the solution of the initial-boundary value problem who describes the behavior of binary homogeneous micropolar mixtures of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid. Some Lagrange-Brun identities are established. Using the Cesaro means of various parts of total energy, the relations that describe the asymptotic behavior of mean energies are established.

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.