According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group of invertible - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets. Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, this method allows us to prove the existence and uniqueness of a subsonic solution of the secular equation, a problem which remains unsolved in almost all generalized linear theories of elastic materials. Since the results are suitable to be used for numerical implementations, we propose two numerical algorithms which are viable for any elastic material. Explicit numerical calculations are made for aluminium-epoxy in the context of the Cosserat model. Since the novel form of the secular equation for isotropic elastic material has not been explicitly derived elsewhere, we establish it in this paper, too. © 2022 Elsevier B.V.

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present significant aspects related to the relaxed micromorphic model realization with the finite element method (FEM). The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation H1(B) × H1(B) and a nodal-edge formulation H1(B) × H(curl , B) , where the latter employs the Nédélec space. In this framework, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the computational convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length parameter. © 2022, The Author(s).

In this paper, a coherent boundary value problem to model metamaterials' behaviour based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary conditions, thus disclosing the possibility of exploring the scattering patterns of finite-size metamaterial specimens. Thanks to the simplified model's structure (few frequency- and angle-independent parameters), we are able to unveil the scattering metamaterial's response for a wide range of frequencies and angles of propagation of the incident wave. These results are an important stepping stone towards the conception of more complex large-scale meta-structures that can control elastic waves and recover energy. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'. © 2022 The Author(s).

We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies W: GL +(n) → R with an additive volumetric-isochoric split, i.e. W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF),which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of “least” rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλminis quasiconvex. In addition, we demonstrate that under affine boundary conditions, Wmagic+(F) allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis. © 2022, The Author(s).

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function W. We will demonstrate these methods using the planar isotropic rank-one convex function Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin,where λmax≥ λmin are the singular values of F, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies W: GL +(2) → R with an additive volumetric-isochoric split of the form W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF)with a concave volumetric part. This example is therefore of particular interest with regard to Morrey’s open question whether or not rank-one convexity implies quasiconvexity in the planar case. © 2022, The Author(s).

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [H1]3×3, [H(curl)] 3 and H(symCurl) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [H(curl)] 3 in place of the classical [H1]3×3. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H(symCurl). The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory. © 2022, The Author(s).

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups. © 2021 Elsevier Ltd

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [H1]3 for the displacement and [H(curl)]3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length Lc and the correlation to the classical Cauchy continuum theory. © 2022 Elsevier B.V.

Deformation microstructure is studied for a 1D-shear problem in geometrically nonlinear Cosserat elasticity. Microstructure solutions are described analytically and numerically for zero characteristic length scale. © The Author(s) 2022.

We consider the classical Mindlin–Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field P. In effect, P is now only required to match the tangential derivative of the classical displacement u which is known at the Dirichlet part of the boundary. We derive the full boundary condition, in adding the missing Neumann condition on the Dirichlet part. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part. Finally, we show that compared to classical Dirichlet conditions on u and P, the new boundary condition modifies the interpretation of the constitutive parameters. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

In this paper we show that an enriched continuum model of the micromorphic type (Relaxed Micromorphic Model) can be used to model metamaterials’ response in view of their use for meta-structural design. We focus on the fact that the reduced model's structure, coupled with the introduction of well-posed interface conditions, allows us to easily test different combinations of metamaterials’ and classical-materials bricks, so that we can eventually end-up with the conception of a meta-structure acting as a mechanical diode for low/medium frequencies and as a total screen for higher frequencies. Thanks to the reduced model's structure, we are also able to optimize this meta-structure so that the diode-behaviour is enhanced for both “pressure” and “shear” incident waves and for all possible angles of incidence. © 2022

In this paper, we present a unit cell showing a band-gap in the lower acoustic domain. The corresponding metamaterial is made up of a periodic arrangement of one unit cell. We rigorously show that the relaxed micromorphic model can be used for metamaterials’ design at large scales as soon as sufficiently large specimens are considered. We manufacture the metamaterial via metal etching procedures applied to a titanium plate so as to show that its production for realistic applications is viable. Experimental tests are also carried out confirming that the metamaterials’ response is in good agreement with the theoretical design. In order to show that our micromorphic model opens unprecedented possibilities in metastructural design, we conceive a finite-size structure that is able to focus elastic energy in a confined region, thus enabling its possible subsequent use for optimizing complex structures. Indeed, thanks to the introduction of a well-posed set of micromorphic boundary conditions, we can combine different metamaterials and classical Cauchy materials in such a way that the elastic energy produced by a source of vibrations is focused in specific collection points. The design of this structure would have not been otherwise possible (via e.g., direct simulations), due to the large dimensions of the metastructure, counting hundreds of unit cells. © 2022 Elsevier Ltd

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order O(h5) in the shell thickness h. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the O(h5) and O(h3) case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya (Int. J. Solids Struct. 37(39):5517–5528, 2000) which will be further strengthened. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H1, such that standard nodal H1-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H1 and H(curl) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates. © 2021, The Author(s).

We consider the regularity question of solutions for the dynamic initial-boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave-type equation for the displacement with a generalized Maxwell-type wave equation for the microdistortion. Naturally, solutions are found in H1 for the displacement u and H(Curl) for the microdistortion P. Using energy estimates for difference quotients, we improve this regularity. We show (Formula presented.) –regularity for the displacement field, (Formula presented.) –regularity for the microdistortion tensor P and that (Formula presented.) is H1–regular if the data are sufficiently smooth. © 2021 John Wiley & Sons, Ltd.

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, because slender specimens are, in general, described as stiffer. © The Author(s) 2021.

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. © 2021, The Author(s).

We consider the cylindrical bending problem for an infinite plate as modeled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner specimens are comparatively stiffer. We provide the analytical solution for each case and exhibits the predicted bending stiffness. The relaxed micromorphic continuum shows bounded bending stiffness for arbitrary thin specimens, while classical micromorphic continuum or gradient elasticity as well as Cosserat models (Neff et al. in Acta Mechanica 211(3–4):237–249, 2010) exhibit unphysical unbounded bending stiffness for arbitrary thin specimens. This finding highlights the advantage of using the relaxed micromorphic model, which has a definite limit stiffness for small samples and which aids in identifying the relevant material parameters. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin–Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness μ∗. This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

In this paper, we establish well-posed boundary and interface conditions for the relaxed micromorphic model that are able to unveil the scattering response of fully finite-size metamaterial samples. The resulting relaxed micromorphic boundary value problem is implemented in finite-element simulations describing the scattering of a square metamaterial sample whose side counts nine unit cells. The results are validated against a direct finite-element simulation encoding all the details of the underlying metamaterial’s microstructure. The relaxed micromorphic model can recover the scattering metamaterial’s behavior for a wide range of frequencies and for all possible angles of incidence, thus showing that it is suitable to describe dynamic anisotropy. Finally, thanks to the model’s computational performances, we can design a metastructure combining metamaterials and classical materials in such a way that it acts as a protection device while providing energy focusing in specific collection points. These results open important perspectives for the short-term design of sustainable structures that can control elastic waves and recover energy. © The Author(s) 2021.

In this paper, we notice a property of the extension operator from the space of tangential traces of H(curl; Ω) in the context of the linear relaxed micromorphic model, a theory that is recently used to describe the behavior of some metamaterials showing unorthodox behaviors with respect to elastic wave propagation. We show that the new property is important for existence results of strong solution for non-homogeneous boundary condition in both the dynamic and the static case. © 2020 John Wiley & Sons, Ltd.

While the design of always new metamaterials with exotic static and dynamic properties is attracting deep attention in the last decades, little effort is made to explore their interactions with other materials. This prevents the conception of (meta-)structures that can enhance metamaterials’ unusual behaviors and that can be employed in real engineering applications. In this paper, we give a first answer to this challenging problem by showing that the relaxed micromorphic model with zero static characteristic length can be usefully applied to describe the refractive properties of simple meta-structures for extended frequency ranges and for any direction of propagation of the incident wave. Thanks to the simplified model’s structure, we are able to efficiently explore different configurations and to show that a given meta-structure can drastically change its overall refractive behavior when varying the elastic properties of specific meta-structural elements. In some cases, changing the stiffness of a homogeneous material which is in contact with a metamaterial’s slab, reverses the structure’s refractive behavior by switching it from an acoustic screen (total reflection) into an acoustic absorber (total transmission). The present paper clearly indicates that, while the study and enhancement of the intrinsic metamaterials’ properties is certainly of great importance, it is even more challenging to enable the conception of meta-structures that can eventually boost the use of metamaterials in real-case applications. © Copyright © 2021 Rizzi, Collet, Demore, Eidel, Neff and Madeo.

Let Ω ⊂ R3 be an open and bounded set with Lipschitz boundary and outward unit normal ν. For 1 < p< ∞ we establish an improved version of the generalized Lp-Korn inequality for incompatible tensor fields P in the new Banach space W01,p,r(devsymCurl;Ω,R3×3)={P∈Lp(Ω;R3×3)∣devsymCurlP∈Lr(Ω;R3×3),devsym(P×ν)=0on∂Ω}where r∈[1,∞),1r≤1p+13,r>1ifp=32.Specifically, there exists a constant c= c(p, Ω , r) > 0 such that the inequality ‖P‖Lp(Ω,R3×3)≤c(‖symP‖Lp(Ω,R3×3)+‖devsymCurlP‖Lr(Ω,R3×3))holds for all tensor fields P∈W01,p,r(devsymCurl;Ω,R3×3). Here, devX:=X-13tr(X)1 denotes the deviatoric (trace-free) part of a 3 × 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset Γ ⊂ ∂Ω. If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space KS,dSC which is determined by the conditions symP=0 and devsymCurlP=0. In that case one can replace ‖devsymCurlP‖Lr(Ω,R3×3) by ‖devsymCurlP‖W-1,p(Ω,R3×3). The new Lp-estimate implies a classical Korn’s inequality with weak boundary conditions by choosing P= D u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing P=A∈so(3). The proof relies on a representation of the third derivatives D 3P in terms of D2devsymCurlP combined with the Lions lemma and the Nečas estimate. We also discuss applications of the new inequality to the relaxed micromorphic model, to Cosserat models with the weakest form of the curvature energy, to gradient plasticity with plastic spin and to incompatible linear elasticity. © 2021, The Author(s).

For n≥ 3 and 1 < p< ∞, we prove an Lp-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields P: Ω → Rn×n having p-integrable generalized Curl n and generalized vanishing tangential trace Pτl=0 on ∂Ω , denoting by {τl}l=1,…,n-1 a moving tangent frame on ∂Ω. More precisely, there exists a constant c= c(n, p, Ω) such that ‖P‖Lp(Ω,Rn×n)≤c(‖devnsymP‖Lp(Ω,Rn×n)+‖CurlnP‖Lp(Ω,Rn×n(n-1)2)),where the generalized Curl n is given by (CurlnP)ijk:=∂iPkj-∂jPki and [InlineEquation not available: see fulltext.] denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product. © 2021, The Author(s).

For 1 < p < ∞ we provean Lp-version of the generalized trace-free Korn inequality for incompatible tensor fields P in W01,p (Curl;Ω,ℝ3×3). More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain. Then there exists a constant c > 0 such that (Formula Presented) such that holds for all tensor fields P ∈ W01,p(Curl;Ω,ℝ3×3), i.e., for all P ∈ W1,p(Curl;Ω,ℝ3×3) with vanishing tangential trace P × ν = 0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω and dev P :=P - 1/3 tr(P)· 1 denotes the deviatoric (trace-free) part of. We also show the norm equivalence (Formula Presented) for tensor fields P ∈ W1,p(Curl;Ω,ℝ3×3). These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset Γ ⊆ ∂Ω of the boundary. Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

For n ≥ 2 and 1 &lt; p &lt; ∞ we prove an Lp-version of the generalized Korn-type inequality for incompatible, p-integrable tensor fields P : Ω →Rn×n having p-integrable generalized Curl and generalized vanishing tangential trace Pτl = 0 on ∂Ω, denoting by {τl}l=1,...,n−1 a moving tangent frame on ∂Ω, more precisely we have: (Equation presented), where the generalized Curl is given by (Equation presented). © 2021 Elsevier Masson SAS. All rights reserved.

For 1 &lt; p &lt; ∞, we prove an Lp-version of the generalized Korn inequality for incompatible tensor fields P in (Formula presented.). More precisely, let (Formula presented.) be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) &gt; 0 such that (Formula presented.) holds for all tensor fields (Formula presented.), that is, for all (Formula presented.) with vanishing tangential trace (Formula presented.) on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible (Formula presented.), this recovers an Lp-version of the classical Korn's first inequality and for skew-symmetric (Formula presented.) an Lp-version of the Poincaré inequality. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C2-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping. © 2021 The Author(s).

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)=μ2∥F∥2detF+f(detF); such an energy is rank-one convex if and only if the function f is convex. © 2021, The Author(s).

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor inc(εp):=Curl[(Curl εp)T], where εp is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor inc(εp) and it contains isotropic hardening based on the rate of the plastic strain tensor (Formula presented.). We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion (Formula presented.) to their symmetric counterpart (Formula presented.). In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings. © The Author(s) 2019.

It has recently been shown that for a Cauchy stress response induced by a strictly rank-one convex hyperelastic energy potential, a homogeneous Cauchy stress tensor field cannot correspond to a non-homogeneous deformation if the deformation gradient has discrete values, i.e. if the deformation is piecewise affine linear and satisfies the Hadamard jump condition. In this note, we expand upon these results and show that they do not hold for arbitrary deformations by explicitly giving an example of a strictly rank-one convex energy and a non-homogeneous deformation such that the induced Cauchy stress tensor is constant. In the planar case, our example is related to another previous result concerning criteria for generalized convexity properties of conformally invariant energy functions, which we extend to the case of strict rank-one convexity. © 2019 Elsevier Ltd

It is well-known that many three-dimensional chiral material models become non-chiral when reduced to two dimensions. Chiral properties of the two-dimensional model can then be restored by adding appropriate two-dimensional chiral terms. In this paper we show how to construct a three-dimensional chiral energy function which can achieve two-dimensional chirality induced already by a chiral three-dimensional model. The key ingredient to this approach is the consideration of a nonlinear chiral energy containing only rotational parts. After formulating an appropriate energy functional, we study the equations of motion and find explicit soliton solutions displaying two-dimensional chiral properties. © 2019

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 Neff et al. (J. Elast., https://doi.org/10.1007/s10659-019-09752-w, 2019, in this volume). 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. © 2019, Springer Nature B.V.

In this paper, we explore the use of micromorphic-type interface conditions for the modeling of a finite-sized metamaterial. We show how finite-domain boundary value problems can be approached in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized tractions, as well as additional conditions on the micro-distortion tensor and on the double-traction. The case of a metamaterial slab of finite width is presented, its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to a direct finite-element simulation encoding all details of the selected microstructure. The reflection and transmission coefficients obtained via the two methods are presented as a function of the frequency and of the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies going from the long-wave limit to frequencies beyond the first band-gap and for angles of incidence ranging from normal to near-parallel incidence. The present paper sets the basis for a new viewpoint on finite-size metamaterial modeling enabling the exploration of meta-structures at large scales. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann’s principle on the overall representation of anisotropy. We explain our idea of “maximal” stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper (d’Agostino et al. in J. Elast. 2019. Accepted in this volume) to describe the wave propagation including band-gaps in the same tetragonal metamaterial. © 2019, Springer Nature B.V.

We reconsider anti-plane shear deformations based on prior work of Knowles and relate the existence of anti-plane shear deformations to fundamental constitutive concepts of elasticity theory like polyconvexity, rank-one convexity and tension–compression symmetry. In addition, we provide finite element simulations to visualize our theoretical findings. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

We present the complete set of constitutive relations and field equations for the linear thermoelastic relaxed micromorphic continuum and investigate its variants for wave propagation. It is found that the additional thermal effects give rise to new waves and generate couplings with longitudinal waves which are not existing in the relaxed micromorphic continuum without thermal effects. However, transverse waves go unaffected by the thermal properties. Thermal effects do not create any band gap in the dispersion curves of the model with three curvature parameters. The dispersion curves have been computed numerically for a particular model and compared with those presented in earlier studies. © 2019, © 2019 Taylor & Francis Group, LLC.

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 ϵ SL(2) and all R ϵ 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. Copyright © Royal Society of Edinburgh 2019.

We provide a faithful translation of Hans Richter’s important 1949 paper ‘Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formänderungen’ from its original German version into English, complemented by an introduction summarizing Richter’s achievements. © The Author(s) 2020.

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order O(h5) 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. © 2020, Springer Nature B.V.

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(h5) terms. The form of the energy allows us to show the coercivity for terms up to order O(h5) and the convexity of the energy. Secondly, we consider only that part of the energy including O(h3) 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(h5) the conditions on the thickness h are those considered in the modelling process and they are independent of the constitutive parameter, in the O(h3) -case the coercivity is proven under some more restrictive conditions on the thickness h. © 2020, Springer Nature B.V.

The Legendre-Hadamard necessary condition for energy minimizers is derived in the framework of Cosserat elasticity theory. © 2020 The Author, 2020.

We consider conformally invariant energies W on the group GL+(2) of 2 × 2 -matrices with positive determinant, i.e., W:GL+(2)→R such that W(AFB)=W(F)for allA,B∈{aR∈GL+(2)|a∈(0,∞),R∈SO(2)},where SO(2) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation W(F)=h(λ1λ2) of W in terms of the singular values λ1, λ2 of F, 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 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 W1,p-quasiconvex envelope on the domain GL+(n) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on GL+(2). © 2020, The Author(s).

In this paper, we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curlp giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge invariance conditions, i.e., only true state variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn’s inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain εp and standard isotropic hardening. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

We provide a faithful translation of Hans Richter’s important 1948 paper “Das isotrope Elastizitätsgesetz” from its original German version into English. Our introduction summarizes Richter’s achievements. © The Author(s) 2019.

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function W:GL+(n) → ℝwhich is equal to the classical Hencky strain energy WH(F) = μ||devnlog U||2 + κ 2 [tr(log U)]2 = μ||log U||2 + Λ 2 [tr(log U)]2 in a neighborhood of the identity matrix ; here, GL+(n) denotes the set of n×n-matrices with positive determinant, F GL+(n) denotes the deformation gradient, U = FT F is the corresponding stretch tensor, log U is the principal matrix logarithm of U, tr is the trace operator, ||X|| is the Frobenius matrix norm and devnX is the deviatoric part of X ℝn×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 WVL in the so-called Valanis-Landel form WVL(F) =∑i=1nw(λ i) with w:(0,∞) → ℝ, where λ1,...,λn denote the singular values of F. © 2019 World Scientific Publishing Company.

In this contribution we discuss the interest of using enriched continuum models of the micromorphic type for the description of dispersive phenomena in metamaterials. Dispersion is defined as that phenomenon according to which the speed of propagation of elastic waves is not a constant, but depends on the wavelength of the traveling wave. In practice, all materials exhibit dispersion if one considers waves with sufficiently small wavelengths, since all materials have a discrete structure when going down at a suitably small scale. Given the discrete substructure of matter, it is easy to understand that the material properties vary when varying the scale at which the material itself is observed. It is hence not astonishing that the speed of propagation of waves changes as well when considering waves with smaller wavelengths. In an effort directed toward the modeling of dispersion in materials with architectured microstructures (metamaterials), different linear-elastic, isotropic, micromorphic models are introduced, and their peculiar dispersive behaviors are discussed by means of the analysis of the associated dispersion curves. The role of different micro-inertias related to both independent and constrained motions of the microstructure is also analyzed. A special focus is given to those metamaterials which have the unusual characteristic of being able to stop the propagation of mechanical waves and which are usually called band-gap metamaterials. We show that, in the considered linear-elastic, isotropic case, the relaxed micromorphic model, recently introduced by the authors, is the only enriched model simultaneously allowing for the description of non-localities and multiple band-gaps in mechanical metamaterials. © Springer Nature Switzerland AG 2019. All rights reserved.

Truesdell's empirical inequalities are considered essential in various fields of nonlinear elasticity. However, they are often used merely as a sufficient criterion for semi-invertibility of the isotropic stress strain-relation, even though weaker and much less restricting constitutive requirements like the strict Baker–Ericksen inequalities are available for this purpose. We elaborate the relations between such constitutive conditions, including a weakened version of the empirical inequalities, and their connection to bi-coaxiality and related matrix properties. In particular, we discuss a number of issues arising from the seemingly ubiquitous use of the phrase “X,Y have the same eigenvectors” when referring to commuting symmetric tensors X,Y. © 2019 Elsevier Ltd

We determine the optimal orthogonal matrices R ∈ O(n) which minimize the symmetrized Euclidean distance W : O(n) → R, W (R ; D):= | | sym(RD - 1)| | 2 , where 1 denotes the identity matrix and sym(X) = 1/2 (X + XT ) is the symmetric part of X, for a given positive definite diagonal matrix D = diag(d1, . . ., dn) with distinct entries d1 &gt; d2 &gt; ⋯ &gt; dn &gt; 0. The number of critical points depends on D and can grow faster than exponential in n. In the process, we prove and use a novel result of independent interest: every real matrix whose square is symmetric can be expressed as a block-diagonal matrix composed of blocks of size at most two by a suitable orthonormal change of basis. © 2019 Society for Industrial and Applied Mathematics

In this paper we set up the full two-dimensional plane wave solution for scattering from an interface separating a classical Cauchy medium from a relaxed micromorphic medium. Both media are assumed to be isotropic and semi-infinite to ease the semi-analytical implementation of the associated boundary value problem. Generalized macroscopic boundary conditions are presented (continuity of macroscopic displacement, continuity of generalized tractions and, eventually, additional conditions involving purely microstructural constraints), which allow for the effective description of the scattering properties of an interface between a homogeneous solid and a mechanical metamaterial. The associated “generalized energy flux” is introduced so as to quantify the energy which is transmitted at the interface via a simple scalar, macroscopic quantity. Two cases are considered in which the left homogeneous medium is “stiffer” and “softer” than the right metamaterial and the transmission coefficient is obtained as a function of the frequency and of the direction of propagation of the incident wave. We show that the contrast of the macroscopic stiffnesses of the two media, together with the type of boundary conditions, strongly influence the onset of Stoneley (or evanescent)waves at the interface. This allows for the tailoring of the scattering properties of the interface at both low and high frequencies, ranging from zones of complete transmission to zones of zero transmission well beyond the band-gap region. © 2019

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. © The Author(s) 2019.

We consider the problem to determine the optimal rotations (Formula presented.) which minimize (Formula presented.) for a given diagonal matrix (Formula presented.) with positive entries (Formula presented.). The objective function W is the reduced form of the Cosserat shear-stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy (Formula presented.), determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block-diagonal matrix with blocks of size at most two. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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 presented.) 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. © The Author(s) 2019.

In this paper, we show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced by a corresponding generalised continuum of the relaxed micromorphic type. Specifically, we compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties: (i) the relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic continuum. We show that, while the Cauchy theory is able to describe the overall behaviour of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency band-gap and at frequencies intersecting the optical branches. In addition, we observe a computational time reduction associated with the use of the relaxed micromorphic continuum, compared to the sensible computational time needed to perform a transient computation in a micro-structured domain. © 2018 Elsevier Ltd

In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell's so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably. © 2018 Elsevier Ltd

We study the fully nonlinear dynamical Cosserat micropolar elasticity problem in three dimensions with various energy functionals dependent on the microrotation [Formula presented] and the deformation gradient tensor [Formula presented]. We derive a set of coupled nonlinear equations of motion from first principles by varying the complete energy functional. We obtain a double sine–Gordon equation and construct soliton solutions. We show how the solutions can determine the overall deformational behaviours and discuss the relations between wave numbers and wave velocities thereby identifying parameter values where the soliton solution does not exist. © 2018 The Authors

Constitutive laws in terms of the logarithmic strain tensor logU, i.e. the principal matrix logarithm of the stretch tensor U=FTF 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 ‖logU‖2, i.e. an energy function W:GL+(n)→R of the form W(F)=Ψ(‖logU‖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 ‖devnlogU‖2, where devnlogU is the deviatoric part of logU. 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(‖devnlogU‖2) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F↦Ψ(‖devnlogU‖2)+Wvol(detF) cannot be rank-one convex for any function Wvol:(0,∞)→R if Ψ is strictly monotone. © 2018 Elsevier Ltd

We consider a dislocation-based rate-independent model of single crystal gradient plasticity with isotropic or linear kinematic hardening. The model is weakly formulated through the so-called primal form of theflowrule as a variational inequality for which a result of existence and uniqueness is obtained using the functional analytical framework developed by Han-Reddy. © The Author, 2017.

In isotropic finite elasticity, unlike in linear elastic theory, a homogeneous Cauchy stress may be induced by non-homogeneous strains. To illustrate this, we identify compatible non-homogeneous three-dimensional deformations producing a homogeneous Cauchy stress on a cuboid geometry, and provide an example of an isotropic hyperelastic material that is not rank-one convex, and for which the homogeneous stress and associated non-homogeneous strains are given explicitly on a domain similar to those analysed. © 2016, © The Author(s) 2016.

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the bandgap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure. © Springer Science+Business Media Dordrecht 2017.

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. © 2018 Elsevier Masson SAS

In the present paper, the material parameters of the isotropic relaxed micromorphic model derived for a specific metamaterial in a previous contribution are used to model its transmission properties. Specifically, the reflection and transmission coefficients at an interface between a homogeneous solid and the chosen metamaterial are analyzed by using both the relaxed micromorphic model and a direct FEM implementation of the detailed microstructure. The obtained results show excellent agreement between the transmission spectra derived via our enriched continuum model and those issued by the direct FEM simulation. Such excellent agreement validates the indirect measure of the material parameters and opens the way towards an efficient metastructural design. © The Author(s) 2017.

For homogeneous higher-gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain ε = sym Grad[u], the new invariance condition is equivalent to the isotropy of the linear formulation. Therefore, it may also be used in higher-gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple-stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity. © The Author(s) 2016.

We consider the recently introduced microcurl model which is a variant of strain gradient plasticity in which the curl of the plastic distortion is coupled to an additional micromorphic-type field. For both single crystal and polycrystal cases, we formulate the model and show its well-posedness in the rate-independent case provided some local hardening (isotropic or linear kinematic) is taken into account. To this end, we use the functional analytical framework developed by Han-Reddy. We also compare the model to the relaxed micromorphic model as well as to a dislocation-based gradient plasticity model. © 2017 Elsevier Ltd

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by (Formula presented.)where (Formula presented.) is the (infinitesimal) shear modulus, (Formula presented.) is the (infinitesimal) bulk modulus, k and (Formula presented.) are additional dimensionless material parameters, (Formula presented.) is the right stretch tensor corresponding to the deformation gradient(Formula presented.), (Formula presented.) denotes the principal matrix logarithm on the set of positive definite symmetric matrices, (Formula presented.) and (Formula presented.) are the deviatoric part and the Frobenius matrix norm of an (Formula presented.)-matrix (Formula presented.), respectively, and (Formula presented.) denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging “eversion of elastic tubes” problem. © 2017 Springer-Verlag GmbH Germany, part of Springer Nature

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,tand 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. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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 themicrostructure 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. © Springer Nature Singapore Pte Ltd. 2017.

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 of switching from a fourth-order (gradient elastic) problem to a second-order micromorphic model are also discussed with the 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 integration by parts, 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 an integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the orthogonal boundary conditions for both models. © SAGE Publications.

We consider the Cosserat shell approach under finite rotations. The Cosserat shell features an additional, in principle independent orthogonal frame. In this setting we establish a novel curvature tensor which we call the shell dislocation density tensor. For this variant, we derive the equations and in a hyperelastic contextwe showexistence of minimizers under generic convexity assumptions on the elastic energies in terms of nonlinear strain measures. The correspondence between our formulation and proposals in the literature is established. © Springer Nature Singapore Pte Ltd. 2017.

Modeling two-dimensional chiral materials is a challenging problem in continuum mechanics because three-dimensional theories reduced to isotropic two-dimensional problems become nonchiral. Various approaches have been suggested to overcome this problem. We propose a new approach to this problem by formulating an intrinsically two-dimensional model which does not require references to a higher dimensional one. We are able to model planar chiral materials starting from a geometrically nonlinear Cosserat-type elasticity theory. Our results are in agreement with previously derived equations of motion but can contain additional terms due to our nonlinear approach. Plane wave solutions are briefly discussed within this model. © 2017 Mathematical Sciences Publishers.

We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy. © 2016 The Authors

In this note, we show that the Cauchy stress tensor (Formula presented.) in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This means that (Formula presented.) implies (Formula presented.) under strict rank-one convexity. As a consequence of this seemingly unnoticed observation, it follows that rank-one convexity and a homogeneous Cauchy stress imply that the left Cauchy-Green strain is homogeneous, as is shown in Mihai and Neff (Int. J. Non-Linear Mech., 2016, to appear). © 2016 Springer Science+Business Media Dordrecht

In this note, we extend integrability conditions for the symmetric stretch tensor U in the polar decomposition of the deformation gradient ∇φ=F=RU to the nonsymmetric case. In doing so, we recover integrability conditions for the first Cosserat deformation tensor. Let (Formula presented.). Then, (Formula presented.), giving a connection between the first Cosserat deformation tensor U¯ and the second Cosserat tensor K. (Here, Anti denotes an isomorphism between R3 × 3 and So(3):={A∈R3×3×3|A.u∈so(3)∀u∈R3}). The formula shows that it is not possible to prescribe U¯ and K independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy), we show existence of minimizers in Sobolev spaces. © 2016, Springer International Publishing.

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure. © 2017 Springer Science+Business Media Dordrecht

We consider the two logarithmic strain measures ωiso = ‖devn logU‖ and ωvol = |tr(log U)|, which are isotropic invariants of the Hencky strain tensor log U = log(FTF), and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group GL(n). Here, F is the deformation gradient, U = √FT F is the right Biot-stretch tensor, log denotes the principal matrix logarithm, ‖. ‖ is the Frobenius matrix norm, tr is the trace operator and devn X = X – 1/n tr(X) · 1 is the n-dimensional deviator of X ∈ Rn×n. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇u, which is the symmetric part of the displacement gradient ∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potential in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor R, where F = RU is the polar decomposition of F. © Springer Nature Singapore Pte Ltd. 2017.

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies W(F) = Wˆ (FTF) = Wˆ (C) which are convex with respect to the right Cauchy-Green tensor C= FTF, where F denotes the gradient of deformation. Examples of such energies exhibiting a blow up for det F→ 0 are given. © 2016, Springer Science+Business Media Dordrecht.

In this paper, the role of gradient micro-inertia terms η//∇u,t//2and free micro-inertia terms η//P,t//2is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term η//∇u,t//2alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term η//P,t//2alone describes the full complex behaviour of bandgap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia η//∇u,t//2, in the sense of Cartan-Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia η//∇u,t//2on more classical enriched models such as the Mindlin-Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin-Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials. © 2017 The Author(s) Published by the Royal Society. All rights reserved.

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. © Copyright Royal Society of Edinburgh 2017.

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. © 2017 The Author(s) Published by the Royal Society. All rights reserved.

In this paper we propose an anisotropic extension of the isotropic exponentiated Hencky energy, based on logarithmic strain invariants. Unlike other elastic formulations, the isotropic exponentiated Hencky elastic energy has been derived solely on differential geometric grounds, involving the geodesic distance of the deformation gradient (Formula presented.) to the group of rotations. We formally extend this approach towards anisotropy by defining additional anisotropic logarithmic strain invariants with the help of suitable structural tensors and consider our findings for selected case studies. © 2017 Springer-Verlag GmbH Germany

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field F(colon equals)∇ϕ:Ω→ GL +(n) and the microrotation field R:Ω→ SO (n), the shear-stretch energy is necessarily of the form Wμ,μc(R;F)(colon equals)μ sym (RTF-1)2+μc skew (RTF-1)2.We aim at the derivation of closed form expressions for the minimizers of Wμ,μc(R;F) in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension n=3, as a function of F∈ GL +(3). In a previous contribution (Part I), we have first shown that, for all n≥2, the full range of weights μ>0 and μc≥0 can be reduced to either a classical or a non-classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in SO(2) and proved their global optimality. In the present contribution (Part II), we characterize the non-classical optimal rotations in dimension n=3. After a lift of the minimization problem to the unit quaternions, the Euler-Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates rpolar μ,μc±(F)∈ SO (3) which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of SO(3). Geometrically, our proposed optimal Cosserat rotations rpolar μ,μc±(F) act in the plane of maximal stretch. Our previously obtained explicit formulae for planar optimal Cosserat rotations in SO(2) reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear-stretch energy and criteria for the existence of non-classical optimal rotations. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F:=∇ϕ:Ω→GL+(n) and the microrotation field R:Ω→SO(n)the shear–stretch energy is necessarily of the form Wμμc(R;F):=μ||sym(RTF-1)||2 μc||skew(RTF-1)||2 where μ&gt;0 is the Lamé shear modulus and μc≥0 is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations argminR∈SO(n) Wμ,μc(R;F) as a function of F∈ GL+(n) and weights μ&gt;0 and μc≥0. For n≥2, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases:(μ,μc)=(1,1) and (μ,μc)=(1,0). In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range μ&gt;μc 0 in dimension n=2. Currently, optimality results for n 3 are out of reach for us, but we contribute explicit representations for n=2 which we name rpolarμ,μc ±(F) ∈ SO(2) and which arise for n=3 by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations rpolarμ,μc ± (Fγ) for simple planar shear. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The rotation polar (F) ∈ SO (3) arises as the unique orthogonal factor of the right polar decomposition F=polar(F)U of a given invertible matrix F∈ GL +(3). In the context of nonlinear elasticity Grioli (Boll Un Math Ital 2:252–255, 1940) discovered a geometric variational characterization of polar (F) as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli’s variational approach with weights (material parameters) μ&gt; 0 and μc≥ 0 (Grioli: μ= μc). The energy subject to minimization coincides with the Cosserat shear–stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field F: = ∇ φ: Ω → GL +(3) and the microrotation field R: Ω → SO (3). The corresponding set of non-classical energy-minimizing rotations rpolarμ,μc±(F):=arg minR∈SO(3){Wμ,μc(R;F):=μ||sym(RTF-1)||2+μc||skew(RTF-1)||2}represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping φnano which models an idealized nanoindentation and compare the corresponding optimal rotation patterns rpolar1,0±(Fnano) with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored. © 2017, Springer International Publishing AG.

We prove the . sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors . x,y∈Rn whose elementary symmetric polynomials satisfy . ek(x)≤ek(y) (for . 1≤k<n) and . en(x)=en(y), the inequality . ∑i(logxi)2≤∑i(logyi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function . f:M⊆Cn→R with . f(z)=∑i(logzi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of . z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI. © 2016 Elsevier Inc.

In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micromorphic 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. © 2017 Elsevier Ltd

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. © 2016 Elsevier Masson SAS. All rights reserved.

We discuss in detail existing isotropic elasto-plastic models based on 6-dimensional flow rules for the positive definite plastic metric tensor Cp = FTp Fp and highlight their properties and interconnections. We show that seemingly different models are equivalent in the isotropic case. © Springer International Publishing Switzerland 2016.

In this paper, we substantiate the claim implicitly made in previous works that the relaxed micromorphic model is the only linear, isotropic, reversibly elastic, nonlocal generalized continuum model able to describe complete band-gaps on a phenomenological level. To this end, we recapitulate the response of the standard Mindlin–Eringen micromorphic model with the full micro-distortion gradient ∇P, the relaxed micromorphic model depending only on the Curl P of the micro-distortion P, and a variant of the standard micromorphic model, in which the curvature depends only on the divergence Div P of the micro distortion. The Div-model has size-effects, but the dispersion analysis for plane waves shows the incapability of that model to even produce a partial band gap. Combining the curvature to depend quadratically on Div P and Curl P shows that such a model is similar to the standard Mindlin–Eringen model, which can eventually show only a partial band gap. © 2016

Let Ω ⊂ ℝn, n ≥ 2, be a bounded Lipschitz domain and 1 < q < ∞. We prove the inequality ∥T∥Lq(Ω) ≤ CDD (∥ dev T∥Lq(Ω) + ∥ Div T∥Lq(Ω)) being valid for tensor fields T : Ω → ℝnxn with a normal boundary condition on some open and non-empty part Γν of the boundary ∂Ω. Here dev T = T - 1/n tr (T) · Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove ∥T∥L2(Ω) ≤ CDSC (∥ dev sym T∥L2(Ω) + ∥ Curl T∥L2(Ω)) if n ≥ 3, ∥T∥L2(Ω) ≤ CDSDC (∥ dev sym T∥L2(Ω) + ∥ dev Curl T∥L2(Ω)) if n = 3, being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γτ of ∂Ω. Here, sym T = 1/2 (T + TT) denotes the symmetric part of T and Curl is the rotation row-wise. © EDP Sciences, SMAI 2015.

In this paper, we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulus μc, still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic length Lc measuring the non-locality of the phononic crystal is of the order of 1/3 of the diameter of its fluidfilled holes. © 2016 The Author(s).

We consider the two logarithmic strain measures (Formula presented.), which are isotropic invariants of the Hencky strain tensor log U, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group GL (n). Here, F is the deformation gradient, U=FTF is the right Biot-stretch tensor, log denotes the principal matrix logarithm, ‖ · ‖ is the Frobenius matrix norm, tr is the trace operator and devnX=X-1ntr(X)·1 is the n-dimensional deviator of X∈ Rn × n. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε= sym ∇ u, which is the symmetric part of the displacement gradient ∇ u, and reveals a close geometric relation between the classical quadratic isotropic energy potential (Formula presented.) in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy (Formula presented.), where μ is the shear modulus and κ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor R, where F= RU is the polar decomposition of F. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity. © 2016, Springer-Verlag Berlin Heidelberg.

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 FW(F)=W(logU) defined in terms of logarithmic strain logU, where U=FTF, happens to be everywhere rank-one convex as a function of F, the new function FW(F)=W(logU-logUp) need not remain rank-one convex at some given plastic stretch Up (viz. EploglogUp). This is in complete contrast to multiplicative plasticity (and infinitesimal plasticity) in which FW(FFp-1) remains rank-one convex at every plastic distortion Fp if FW(F) is rank-one convex (usymu-εp 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. © 2016 Elsevier Ltd.

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parameterization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated. © American Institute of Mathematical Sciences.

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general six-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements (GFEs), which leads to an objective discrete model which naturally allows arbitrarily large rotations. GFEs of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton’s method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkling of thin elastic sheets in shear. © 2016, Springer-Verlag Berlin Heidelberg.

In a series of papers which are either published [Hadjesfandiari, A., Dargush, G. F., 2011a. Couple stress theory for solids. Int. J. Solids Struct. 48 (18), 2496-2510; Hadjesfandiari, A., Dargush, G. F., 2013. Fundamental solutions for isotropic size-dependent couple stress elasticity. Int. J. Solids Struct. 50 (9), 1253-1265.] or available as preprints [Hadjesfandiari, A., Dargush, G. F., 2010. Polar continuum mechanics. Preprint arXiv:1009.3252; Hadjesfandiari, A. R., Dargush, G. F., 2011b. Couple stress theory for solids. Int. J. Solids Struct. 48, 2496-2510; Hadjesfandiari, A. R., 2013. On the skew-symmetric character of the couple-stress tensor. Preprint arXiv:1303.3569; Hadjesfandiari, A. R., Dargush, G. F., 2015a. Evolution of generalized couple-stress continuum theories: a critical analysis. Preprint arXiv:1501.03112; Hadjesfandiari, A. R., Dargush, G. F., 2015b. Foundations of consistent couple stress theory. Preprint arXiv:1509.06299] 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 do not consider that the set of boundary conditions intervening in the virtual work principle is not unique, their statement is not tenable and leads to some misunderstandings in the indeterminate couple stress model. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and we show that it still yields a well-posed boundary value problem. © 2015 Elsevier Ltd. All rights reserved.

We consider the Cosserat continuum in its finite strain setting and discuss the dislocation density tensor as a possible alternative curvature strain measure in three-dimensional Cosserat models and in Cosserat shell models. We establish a close relationship (one-to-one correspondence) between the new shell dislocation density tensor and the bending-curvature tensor of 6-parameter shells. © Springer Science+Business Media Singapore 2016.

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors a,b ∈ R+ n so that. Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for a,b ∈ R+ n and k1, . . . , kn ∈ [0,∞). © Element, Zagreb.

We discuss a completely forgotten work of the geologist GF Becker on the ideal isotropic nonlinear stress-strain function (Am J Sci 1893; 46: 337-356). Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker's work, combining his approach with the current framework of the theory of nonlinear elasticity. Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker's deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation, which in today's notation can be written as TBiot= 2 G · dev3 log (U) + K · tr [ log (U) ] · 11 = 2 G · log (U) + Λ · tr [ log (U) ] · 11, where TBiot is the Biot stress tensor, log(U) is the principal matrix logarithm of the right Biot stretch tensor U = √FTF, tr X = ∑i=13 Xi,i denotes the trace and dev3 X = X - (1/3) tr (X) · 11 denotes the deviatoric part of a matrix X ∈ ℝ3 × 3. Here, G is the shear modulus and K is the bulk modulus. For Poisson's number ν = 0 the formulation is hyperelastic and the corresponding strain energy WBeckerν= 0 (U) = 2 G [ < U, log (U) - 11 &gt; + 3 ] has the form of the maximum entropy function. © The Author(s) 2016.

In 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. © 2016 Elsevier Ltd

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine-Gordon type equation which is known to have soliton solutions. © 2015 Elsevier B.V.

This work presents a hyperviscoelastic model, based on the Hencky-logarithmic strain tensor, to model the response of a tire derived material (TDM) undergoing moderately large deformations. The TDM is a composite made by cold forging a mix of rubber fibers and grains, obtained by grinding scrap tires, and polyurethane binder. The mechanical properties are highly influenced by the presence of voids associated with the granular composition and low tensile strength due to the weak connection at the grain-matrix interface. For these reasons, TDM use is restricted to applications involving a limited range of deformations. Experimental tests show that a central feature of the response is connected to highly nonlinear behavior of the material under volumetric deformation which conventional hyperelastic models fail in predicting. The strain energy function presented here is a variant of the exponentiated Hencky strain energy, which for moderate strains is as good as the quadratic Hencky model and in the large strain region improves several important features from a mathematical point of view. The proposed form of the exponentiated Hencky energy possesses a set of parameters uniquely determined in the infinitesimal strain regime and an orthogonal set of parameters to determine the nonlinear response. The hyperelastic model is additionally incorporated in a finite deformation viscoelasticity framework that accounts for the two main dissipation mechanisms in TDMs, one at the microscale level and one at the macroscale level. The new model is capable of predicting different deformation modes in a certain range of frequency and amplitude with a unique set of parameters with most of them having a clear physical meaning. This translates into an important advantage with respect to overcoming the difficulties related to finding a unique set of optimal material parameters as are usually encountered fitting the polynomial forms of strain energies. Moreover, by comparing the predictions from the proposed constitutive model with experimental data we conclude that the new constitutive model gives accurate prediction. © 2016 by ASME.

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented) based on the Hencky-logarithmic (true, natural) strain tensor log U. Here, μ &gt; 0 is the infinitesimal shear modulus, κ = 2μ+3λ/3 &gt; 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k are additional dimensionless material parameters, F = ∇ϕ is the gradient of deformation, U = √FT F is the right stretch tensor, and devn log U = log U –1/ntr(log U) · 1 is the deviatoric part of the strain tensor log U. Based on the multiplicative decomposition Fn = Fe Fp, we couple these energies with some isotropic elasto-plastic flow rules Fpd/dt[Fp −1] ∈ −∂χ (dev3 ∑e) defined in the plastic distortion Fp, where ∂χ is the subdifferential of the indicator function χ of the convex elastic domain εe (∑e, 1/3σy 2) in the mixed-variant ∑e-stress space, ∑e = Fe T DFe Wiso (Fe), and Wiso (Fe) represents the isochoric part of the energy. While WeH may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of WeH on the one hand and the elastic domain in ∑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. © Springer-Verlag Berlin Heidelberg 2015.

The purpose of this article is to prove the Hölder continuity up to the boundary of the displacement vector and the microrotation matrix for the quasistatic, rate-independent Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects. This model is of non-monotone and non-associated type. In the case of two space dimensions we use the hole-filling technique of Widman and Morrey's Dirichlet growth theorem. © 2014 Elsevier Inc.

We describe ellipticity domains for the isochoric elastic energy F →||devn logU||2 = ||log √ FTF (det F)1/n||2 = 1/4||log C (detC)1/n || 2 for n=2, 3, where C=FTF for F ε GL+(n). Here, devn logU =logU - (1/n) tr(logU) 1 is the deviatoric part of the logarithmic strain tensor logU. For n=2, we identify the maximal ellipticity domain, whereas for n=3, we show that the energy is Legendre- Hadamard (LH) elliptic in the set E3(Wiso H , LH,U, 23 ) := U PSym(3)| dev3 logU2 ≤ 23 , which is similar to the von Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy WH(F)=μ dev3 logU2 + (k/2)[tr(logU)]2, U = √ FTF with μ&gt;0 andk &gt; 23 μ, previously obtained by Bruhns et al.

In this note we show that the relaxed linear micromorphic model recently proposed by the authors can be suitably used to describe the presence of band-gaps in metamaterials with microstructures in which strong contrasts of the mechanical properties are present (e.g. phononic crystals and lattice structures). This relaxed micromorphic model only has 6 constitutive parameters instead of 18 parameters needed in Mindlin- and Eringen-type classical micromorphic models. We show that the onset of band-gaps is related to a unique constitutive parameter, the Cosserat couple modulus μ<inf>c</inf> which starts to account for band-gaps when reaching a suitable threshold value. The limited number of parameters of our model, as well as the specific effect of some of them on wave propagation can be seen as an important step towards indirect measurement campaigns. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium system as a minimization problem. Applying the direct methods of the calculus of variations we show the existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on the weak lower semi-continuity of the total energy functional. Use is made of the dislocation density tensor $\overline{\boldsymbol{K}}= \overline{\boldsymbol{R}}^{T}\operatorname{Curl}\overline{\boldsymbol{R}}$ as a suitable Cosserat curvature measure. © 2015, Springer Science+Business Media Dordrecht.

Assume n≥. Consider the elementary symmetric polynomials e<inf>k</inf>(y<inf>1</inf>,y<inf>2</inf>,…,y<inf>n</inf>) and denote by E<inf>0</inf>,E<inf>1</inf>,…,E<inf>n</inf> the elementary symmetric polynomials in reverse order (Formula presented.). Let, moreover, S be a non empty subset of {0,1,…,n−1}. We investigate necessary and sufficient conditions on the function f:I→R, where I⊂R is an interval, such that the inequality f(a<inf>1</inf>)+f(a<inf>2</inf>)+⋯+f(a<inf>n</inf>)≤f(b<inf>1</inf>)+f(b<inf>2</inf>)+⋯+f(b<inf>n</inf>) holds for all a=(a<inf>1</inf>,a<inf>2</inf>,…,a<inf>n</inf>)∈In and b=(b<inf>1</inf>,b<inf>2</inf>,…,b<inf>n</inf>)∈In satisfying E<inf>k</inf>(a)<E<inf>k</inf>(b) for k∈S and E<inf>k</inf>(a)=E<inf>k</inf>(b) for k∈{0,1,…,n−1}∖S. As a corollary, we obtain our inequality (∗) if 2≤n≤4, f(x)=log2x and S={1,…,n−1}, which is the sum of squared logarithms inequality previously known for 2≤n≤3. © 2015, Pompe and Neff; licensee Springer.

For a bounded domain Ω⊂R3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γt≠θ of Γ, we prove the existence of some c&gt;0, such that(0.1)c{norm of matrix}T{norm of matrix}L2(Ω,R3×3)≤{norm of matrix}symT{norm of matrix}L2(Ω,R3×3)+{norm of matrix}CurlT{norm of matrix}L2(Ω,R3×3) holds for all tensor fields in H(Curl;Ω), i.e., for all square-integrable tensor fields T:Ω→R3×3 with square-integrable generalized rotation CurlT:Ω→R3×3, having vanishing restricted tangential trace on Γt. If Γt=θ, (0.1) still holds at least for simply connected Ω and for all tensor fields T∈H(Curl;Ω) which are L2(Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise.For compatible tensor fields T=;v, (0.1) reduces to a non-standard variant of the well known Korn's first inequality in R3, namelyc{norm of matrix};v{norm of matrix}L2(Ω,R3×3)≤{norm of matrix}sym;v{norm of matrix}L2(Ω,R3×3) for all vector fields v∈H1(Ω,R3), for which ;vn, n=1, . . ., 3, are normal at Γt. On the other hand, identifying vector fields v∈H1(Ω,R3) (having the proper boundary conditions) with skew-symmetric tensor fields T, (0.1) turns to Poincaré's inequality since2c{norm of matrix}v{norm of matrix}L2(Ω,R3)=c{norm of matrix}T{norm of matrix}L2(Ω,R3×3)≤{norm of matrix}CurlT{norm of matrix}L2(Ω,R3×3)≤2{norm of matrix};v{norm of matrix}L2(Ω,R3). Therefore, (0.1) may be viewed as a natural common generalization of Korn's first and Poincaré's inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through CurlT. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate. © 2014.

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance toward constitutive relations in nonlinear elasticity. Focusing on primary matrix functions on the set of symmetric matrices, we discuss and compare different criteria for monotonicity. The demonstrated results are particularly applicable to computations involving the true-stress–true-strain monotonicity condition, a constitutive inequality recently introduced in an Arch. Appl. Mech. article by C.S. Jog and K.D. Patil. We also clarify a statement by Jog and Patil from the same article which could be misinterpreted. © 2015, Springer-Verlag Berlin Heidelberg.

In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencicy energies defined in the first part of this series, i.e. W-eH(F)={mu/k e(k)parallel to dev(n) log U parallel to(2)+kappa/2 (k) over cap e((k) over cap (log det U)]2) if det F > 0, +infinity if det F <= 0, where F = del phi is the gradient of deformation, U = root(FF)-F-T is the right stretch tensor and dev(n) log U is the deviatoric part of the strain tensor log U. The main result in this paper is that in plane elastostatics, i.e. for n=2, the energies of this family are polyconvex for k >= 1/4, (k) over cap >= 1/8, extending a previous result which proves polyconvexity for k >= 1/3, (k) over cap >= 1/8. This leads immediately to an extension of the existence result. (C) 2015 Elsevier Ltd. All rights reserved.

We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ&gt;0 is the infinitesimal shear modulus, (Formula Presented.) is the infinitesimal bulk modulus with λ the first Lamé constant, k, are additional dimensionless material parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and [InlineEquation not available: see fulltext.] is the n-dimensional deviatoric part of the strain tensor logU. For small elastic strains, WeH approximates the classical quadratic Hencky strain energy (Formula Presented.) which is not everywhere rank-one convex. In plane elastostatics, i.e., n=2, we prove the everywhere rank-one convexity of the proposed family WeH, for (Formula Presented.). Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n=2,3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family WeH is not preserved in dimension n=3 and that the energies (Formula Presented.) are also not rank-one convex. © 2015, Springer Science+Business Media Dordrecht.

We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with λ the first Lamé constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) 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 (Formula Presented.) 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 (Formula Presented.) are polyconvex for (Formula Presented.) 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. © 2015, Springer Basel.

We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch, and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. © The Author(s) 2014.

We consider the equilibrium problem in the relaxed linear model of micromorphic elastic materials. The basic kinematical fields of this extended continuum model are the displacement u ε R3 and the non-symmetric micro-distortion density tensor P ε R3×3. In this relaxed theory, a symmetric force-stress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor Curl P. However, the relaxed model is able to fully describe rotations of the microstructure and to predict non-polar size-effects. In contrast to classical linear micromorphic models, we allow the usual elasticity tensors to become positive-semidefinite. We prove that, nevertheless, the equilibrium problem has a unique weak solution in a suitable Hilbert space. The mathematical framework also settles the question of which boundary conditions to take for the micro-distortion. Similarities and differences between linear micromorphic elasticity and dislocation gauge theory are discussed and pointed out. © The Author, 2015.

In the present paper we consider a 2D pantographic structure composed of two orthogonal families of Euler beams.Pantographic rectangular ‘long’ waveguides are considered in which imposed boundary displacements can induce the onset of travelling (possibly non-linear) waves. We performed numerical simulations concerning a set of dynamically interesting cases. The system undergoes large rotations, which may involve geometrical non-linearities, possibly opening a path to appealing phenomena such as the propagation of solitary waves. Boundary conditions dramatically influence the transmission of the considered waves at discontinuity surfaces. The theoretical study of this kind of objects looks critical, as the concept of pantographic 2D sheets seems to have promising possible applications in a number of fields, e.g. acoustic filters, vascular prostheses, and aeronautic/aerospace panels. © 2015, Estonian Academy Publishers. All rights reserved.

In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013), Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus μ<inf>c</inf>, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology. © 2013, Springer-Verlag Berlin Heidelberg.

In this work we establish the well-posedness for infinitesimal dislocation based gradient viscoplasticity with isotropic hardening for general subdifferential plastic flows. We assume an additive split of the displacement gradient into non-symmetric elastic distortion and non-symmetric plastic distortion. The thermodynamic potential is augmented with a term taking the dislocation density tensor Curlp into account. The constitutive equations in the models we study are assumed to be of self-controlling type. Based on the self-controlling property the existence of solutions of quasi-static initial-boundary value problems under consideration is shown using a time-discretization technique and a monotone operator method. © 2015 Elsevier Ltd.

The unitary polar factor Q = U-p in the polar decomposition of Z = U-p H is the minimizer over unitary matrices Q for both \\Log(Q*Z)\\(2) and its Hermitian part \\sym(*)(Log(Q*Z))\\(2) over both R and C for any given invertible matrix Z is an element of C-nxn and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any n and for the Frobenius matrix norm for n <= 3. The result shows that the unitary polar factor is the nearest orthogonal matrix to Z not only in the normwise sense but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all z is an element of C\{0}min(v is an element of(-pi,pi]) vertical bar Log(C)(e(-i upsilon)z)|(2) = vertical bar log vertical bar z vertical bar vertical bar(2), min(upsilon is an element of(-pi,pi]) vertical bar Re Log(C)(e(-iv)z)|(2) = vertical bar log vertical bar z vertical bar vertical bar(2).

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n) of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires the Riemannian metric to be right-O(n)-invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters.In deriving the result, geodesics on GL(n) have to be parameterized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved. © 2014.

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models. © 2013 Springer-Verlag Berlin Heidelberg.

We show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P: Ω ⊂ ℝ3 → ℝ3× 3 $$ \Vert DuP+(DuP)^T\Vert _q+\Vert u \Vert _q\geq c \Vert Du\Vert_q\quad \hbox{for $u\in W_0^{1,q}(\Omega;\mathbb{R}^3), \quad 1 \le q \le \infty$} $$ is in general false, even if P ∈ SO(3), while the Legendre-Hadamard condition and ellipticity on ℂn for the quadratic form |Du P+(DuP)T|2 is satisfied. Thus Gårding's inequality may be violated for formally positive quadratic forms. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

This paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6-parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations essentially using the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze the particular cases of isotropic shells, orthotropic shells and composite shells separately. © The Author(s) 2013.

We present some Poincaré-type inequalities for quadratic matrix fields with applications e.g. in gradient plasticity or fluid dynamics. In particular, applications to the pseudostress-velocity formulation of the stationary Stokes problem and to infinitesimal gradient plasticity are discussed. © 2013.

For a bounded domain Ω in RN with Lipschitz boundary Γ = ∂Ω and a relatively open and non-empty subset Γt of Γ, we prove the existence of a positive constant c such that inequality c||T ||L 2(Ω,RN×N) ≤ ||sym T ||L 2(Ω,RN ×N) + || Curl T ||L 2(Ω,RN ×N(N −1)/2) holds for all tensor fields T ∈ H(Curl; Γt,Ω,RN ×N), this is, for all T : Ω → RN ×N which are square-integrable and possess a row-wise square-integrable rotation tensor field Curl T : Ω → RN ×N(N −1)/2 and vanishing row-wise tangential trace on Γt. For compatible tensor fields T = ∇v with v ∈ H1(Ω,RN) having vanishing tangential Neumann trace on Γt the inequality reduces to a non-standard variant of the first Korn inequality since Curl T = 0, while for skew-symmetric tensor fields T the Poincaré inequality is recovered. If Γt = ∅, our estimate still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω,RN ×N) which are L2(Ω,RN ×N)-perpendicular to so(N), i.e., to all skew-symmetric constant tensors. © Springer Science+Business Media Dordrecht 2014.

In this paper we rediscover Grioli's important work on the optimality of the orthogonal factor in the polar decomposition in an euclidean distance framework. We also draw attention to recently obtained generalizations of this optimality property in a geodesic distance framework. © 2014 Elsevier Ltd. All rights reserved.

Let n≥2. We prove a condition on f ∈ C2(ℝ+,ℝ) for the convexity of f o det on ℙ ym(n), namely that f o det is convex on ℙ ym(n) if and only if f''(s)+ n-1/ns · f'(s) ≥ 0 and f'(s)≤ 0 ∀ s ∈ ℝ+.This generalizes the observation that C → - ln det C is convex as a function of C. © The Author(s) 2013.

In the framework of the geometrically nonlinear 6-parameter resultant shell theory we give a characterization of the shells without drilling rotations. These are shells for which the strain energy function W is invariant under the superposition of drilling rotations, i.e. W is insensible to the arbitrary local rotations about the third director d3. For this type of shells we show that the strain energy density W can be represented as a function of certain combinations of the shell deformation gradient F and the surface gradient of d3, namely W(FTF,FT d3,FTGrads d3). For the case of isotropic shells we present explicit forms of the strain energy function W having this property. © 2014 Elsevier B.V. All rights reserved.

We propose an extension of the cyclic hardening plasticity model formulated by Armstrong and Frederick which includes micropolar effects. Our micropolar extension establishes coercivity of the model which is otherwise not present. We study then existence of solutions to the quasistatic, rate-independent Armstrong-Frederick model with Cosserat effects which is, however, still of non-monotone, non-associated type. In order to do this, we need to relax the pointwise definition of the flow rule into a suitable weak energy-type inequality. It is shown that the limit in the Yosida approximation process satisfies this new solution concept. The limit functions have a better regularity than previously known in the literature, where the original Armstrong-Frederick model has been studied. © 2014 Elsevier Inc.

We show that the unitary factor Up in the polar decomposition of a nonsingular matrix Z=UpH is a minimizer for both∥-Log(Q *Z)∥-and∥-sym*(Log(Q *Z))∥- over the unitary matrices QεU(n) for any given invertible matrix ZεCn n×, for any unitarily invariant norm and any n. We prove that Up is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization. © 2014 Published by Elsevier Inc.

In this paper we show the existence of global minimizers for the geometrically non-linear equations of elastic plates, in the framework of the general 6-parameter shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field and the rotation tensor field (representing in total 6 independent scalar kinematic variables). For isotropic plates, we prove the existence theorem by applying the direct methods of the calculus of variations. Then, we generalize our existence result to the case of anisotropic plates. © 2012 Springer Science+Business Media B.V.

Let [InlineEquation not available: see fulltext.] be such that [InlineEquation not available: see fulltext.] and [Equation not available: see fulltext.] Then [Equation not available: see fulltext.] This can also be stated in terms of real positive definite [InlineEquation not available: see fulltext.]-matrices [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.]: If their determinants are equal, [InlineEquation not available: see fulltext.], then [Equation not available: see fulltext.] where log is the principal matrix logarithm and [InlineEquation not available: see fulltext.] denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated. MSC: 26D05, 26D07. © 2013 Bîrsan et al.; licensee Springer.

Let Ω⊂ℝN be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ∂Ω. We show that the solution to the linear first-order system:, vanishes if G∈L1(Ω;R(N×N)×N) and ζ∈W1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with G∈L1∩L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that: is a norm if P∈L∞(Ω;R3×3) with CurlP∈Lp(Ω;R3×3), CurlP-1∈Lq(Ω;R3×3) for some p, q&gt;1 with 1/p+1/q=1 as well as detP≥c+&gt;0. We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let Φ∈H1(Ω;R3), Ω⊂R3, satisfy sym(∇;Φ⊤∇;Ψ)=0 for some Ψ∈W1,∞(Ω;R3)∩H2(Ω;R3) with det∇;Ψ≥c+&gt;0. Then there exists a constant translation vector a∈R3 and a constant skew-symmetric matrix A∈so(3), such that Φ=AΨ+a. © 2013.

Let Ω ⊂ ℝN be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ∂Ω. We show that the solution to the linear first-order system (Formula is Presented) is unique if (Formula is Presented) and (Formula is Presented). As a consequence, we prove (Formula is Presented) to be a norm for (Formula is Presented) for some p, q &gt; 1 with 1/p + 1/q = 1 as well as det (Formula is Presented). We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let (Formula is Presented) satisfy sym (Formula is Presented) for some (Formula is Presented). Then, there exist a constant translation vector (Formula is Presented). © 2013 Springer Basel.

For a bounded domain Ω ⊂ R N with connected Lipschitz boundary, we prove the existence of some c &gt; 0, such that c ∥P∥ L 2(Ω,R N× N) ≤∥sym P∥ L 2(Ω,R N× N) + ∥Curl P∥ L 2(Ω,R N×(N-1)N/2) holds for all square-integrable tensor fields P: Ω→R N× N, having square-integrable generalized "rotation" Curl P: Ω→R N× (N-1)N/2 and vanishing tangential trace on ψ,Ω, where both operations are to be understood row-wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in R N. For compatible tensor fields P, that is, P = ∇ v, the latter estimate reduces to a non-standard variant of Korn's first inequality in R N, namely c∥ ∇ v∥ L 2(Ω,R N× N) ≤ ∥ sym ∇v∥ L 2(Ω,R N× N) for all vector fields v ∈ H 1(Ω, R N), for which ∇ v n,n = 1, ... ,N, are normal at ψΩ. Copyright © 2012 John Wiley & Sons, Ltd.

We prove a Korn-type inequality in H(Curl; Ω, ℝ 3×3) for tensor fields P mapping Ω to ℝ 3×3. More precisely, let Ω ⊂ ℝ 3 be a bounded domain with connected Lipschitz boundary ∂Ω. Then there exists a constant c &gt; 0 such that, for all tensor fields P ∈ H(Curl; Ω, ℝ 3×3), i. e., all P ∈ H(Curl; Ω, ℝ 3×3) with vanishing tangential trace on ∂Ω. Here the rotation and tangential trace are defined row-wise. For compatible P of form P = ∇v, Curl P = 0, where v ∈ H 1(Ω, ℝ 3) is a vector field with components v n for which ∇v n are normal at ∂Ω, estimates (0. 1) is reduced to a non standard variant of Korn's first inequality:, For skew-symmetric P (with sym P = 0), estimates (0. 1) generates a nonstandard version of Poincaré's inequality. Therefore, the estimate is a generalization of two classical inequalities of Poincaré and Korn. Bibliography: 24 titles. © 2012 Springer Science+Business Media, Inc.

We prove existence of solutions to a non-monotone and non-gradient type quasi-static model of poroplasticity with Cosserat effects. It is shown that this model possesses global in time solutions, where the inelastic constitutive equation is satisfied in the sense of Young measures. The methods of proof are a monotone approximation, energy estimates, the fundamental theorem on Young measures, and a passage to the limit with the monotone approximation. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Well-posedness for infinitesimal dislocation based gradient viscoplasticity with linear kinematic hardening is shown using a time-discretization technique for the rate-dependent model and methods of convex analysis.

We prove a Korn-type inequality in H̊(Curl;Ω,R{double-struck}3×3) for tensor fields P mapping Ω to R{double-struck}3×3. More precisely, let Ω⊂R{double-struck}3 be a bounded domain with connected Lipschitz boundary ∂ Ω. Then, there exists a constant c&gt; 0 such that (1) c||P||L2(Ω,R{double-struck}3×3)≤||symP||L2(Ω,R{double-struck}3×3)+||CurlP||L2(Ω,R{double-struck}3×3) holds for all tensor fields P∈H̊(Curl;Ω,R{double-struck}3×3), i.e., all P∈H(Curl;Ω,R{double-struck}3×3) with vanishing tangential trace on ∂ Ω. Here, rotation and tangential traces are defined row-wise. For compatible P, i.e., P=∇;v and thus Curl. P=0, where v∈H1(Ω,R{double-struck}3) are vector fields having components vn, for which ∇;vn are normal at ∂ Ω, the presented estimate (1) reduces to a non-standard variant of Korn's first inequality, i.e., c||∇;v||L2(Ω,R{double-struck}3×3)≤||sym∇;v||L2(Ω,R{double-struck}3×3). On the other hand, for skew-symmetric P, i.e., sym. P=0, (1) reduces to a non-standard version of Poincaré's estimate. Therefore, since (1) admits the classical boundary conditions our result is a common generalization of these two classical estimates, namely Poincaré's resp. Korn's first inequality. © 2011 Académie des sciences.

An appropriate continuum theory to predict the behavior of flexible magnetic or electrically polarized materials undergoing large deformations is explained. The formulation treats the angular momentum as an explicit complementary principle including net-couples from magnetic resp. electric fields. As a consequence non-symmetric Cauchy stresses are mandatory for equilibrium, which is unlike in classical theories. However, the micropolar model is in accordance with classical phenomenological modeling parameters but with the feature to cover large deformations and non-classical types of loading. The formulation considers rotational degrees of freedom to appear in the kinematical equations as exact rotations in SO(3). This is a source of nonlinearity in the model but allows easily for large deformation as well as for net-couples. A simple example is the torque of a compass needle to explain the effect of materials with remanent magnetization within a magnetic field. The twisting moment becomes a maximum for remanent magnetization being perpendicular to an outer magnetic field. It vanishes if both fields are parallel. We investigate magnetic structures using finite element simulations. The development of active materials on the micro-level is in the focus. © 2011 SPIE.

We present a Cosserat-based 3D-1D dimensional reduction for a viscoelastic finite strain model. The numerical resolution of the reduced coupled minimization/evolution problem is based on a splitting method. We start by approximating the minimization problem using the finite element method with P1 Lagrange elements. The solution of this problem is used in the time-incremental formulation of the evolution problem. © SAGE Publications 2011.

We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor εP:= sym (P-1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plasticity or in problems with eigenstresses. The tensor field P induces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P-T = ∇ψ is a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on P is highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates. © EDP Sciences, SMAI, 2010.

We consider a specific case of unidirectional reinforced material under applied tensile load. The reinforcement of the material is inclined with 45° to the direction of the tensile resultant. Different approaches are discussed: one experiment and three computational models. Two models use the classical Cauchy continuum theory whereas the third computational model is based on a Cosserat continuum. It is well known that test specimen with inclination between unidirectional reinforcement and tensile direction show, besides Poissons effect, additional deformation perpendicular to the load direction. The classical transversely isotropic continuum theory predicts this deformation as typical S-shape. In the Cosserat continuum the orientation of the inner structure is incorporated. Thus, structural parameters influence the deformation. With the proposed geometrically non-linear Cosserat model classical and non-classical behaviour can be modelled. In the non-classical case, the transverse deformation is not described by one S-shape but by multiple S-shaped modes. The additional rotational parameters in the Cosserat continuum are responsible for the non-classical behaviour which is due to non-symmetric strain. © 2010 Springer-Verlag.

Existence and uniqueness for infinitesimal dislocation based rate-independent gradient plasticity with linear isotropic hardening and plastic spin are shown using convex analysis and variational inequality methods. The dissipation potential is extended non-uniquely from symmetric plastic rates to non-symmetric plastic rates and three qualitatively different formats for the dissipation potential are distinguished. © 2010 The Author(s).

We investigate the weakest possible constitutive assumptions on the curvature energy in linear Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is μL2 c ∥dev sym ∇axl A∥2 where axl A is the axial vector of the skewsymmetric microrotation A ∈ so(3) and dev is the orthogonal projection on the Lie-algebra sl(3) of trace free matrices. The proposed Cosserat parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed that unphysical stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four. One Cosserat coupling parameter μc &gt; 0 and only one length scale parameter L c &gt; 0. Use is made of a new coercive inequality for conformal Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the microrotations need to be specified; thus avoiding boundary layer effects. Since the curvature energy is the weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus μc &gt; 0 remains a material parameter independent of the sample size which is impossible for stronger curvature expressions. © 2010 SAGE Publications.

We review the continuous and fully-discrete problem setting of an elasto-plastic Cosserat model, which is an enhanced continuum model with independent rotational degrees of freedom. The continuous model is defined by a variational inequality, and for the discrete model a corresponding non-smooth nonlinear variational problem is derived which can be solved with a generalized Newton method. We present numerical results for two representative geometries, where we test the convergence for different finite element discretizations, we study the parameter dependency, and we demonstrate the efficiency of the parallel solution method for very large problem sizes. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, for the description of hyperelastic materials poly-convex functions which are always s.w.l.s. should be preferably used. A variety of isotropic and anisotropic polyconvex energies, in particular for the triclinic, monoclinic, rhombic and transversely isotropic symmetry groups, already exist. In this contribution we propose a new approach for the description of trigonal, tetragonal and cubic hyperelastic materials in the framework of polyconvexity. The anisotropy of the material is described by invariants in terms of the right Cauchy Green tensor and a specific fourth-order structural tensor. In order to show the adaptability of the introduced polyconvex energies for the approximation of real anisotropic material behavior we focus on the fitting of a trigonal fourth-order tangent moduli near the reference state to experimental data.

A minimization problem modeling geometrically exact generalized continua of micromorphic type is considered. The solution consists of two fields, the elastic deformation φ{symbol} of a given body and a tensorial field P which can model different additional features needed for a more reliable description of solids. For the solution of this minimization problem, a staggered algorithm is introduced which decouples the original problem into two separate problems. In each of these subproblems, one of the variables, φ{symbol} or P, respectively, is kept fixed and the subproblem is solved for the remaining variables, i.e., P or φ{symbol} respectively. Each of the problems is discretized with finite elements. Numerical results are presented for a cubic and a cylindrical geometry. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We describe a principle of bounded stiffness and show that bounded stiffness in torsion and bending implies a reduction of the curvature energy in linear isotropic Cosserat models leading to the so-called conformal curvature case μ Lc2/2 ∥ rm dev ∇ axl ̄ A 2∥ where ̄ A ∈ so (3) is the Cosserat microrotation. Imposing bounded stiffness greatly facilitates the Cosserat parameter identification and allows a well-posed, stable determination of the one remaining length scale parameter L c and the Cosserat couple modulus μ c . © 2009 Springer-Verlag.

The linear Reissner-Mindlin membrane-bending plate model is the rigourous γ-limit for zero thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we use the natural nonlinear scaling for the displacements u and the linear scaling for the infinitesimal microrotations Ā ∈ so(3). We also provide formal calculations for other combinations of scalings by retrieving other plate models previously proposed in the literature by formal asymptotic methods as corresponding γ-limits. No boundary conditions on the microrotations are prescribed. © 2010 World Scientific Publishing Company.