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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.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group$\\textrm {GL}^{\\!+}(2)$of invertible$2\\times 2$- - 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$\\textrm {GL}^{\\!+}(2)$as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function$W\\colon \\textrm {GL}^{\\!+}(2)\\to \\mathbb {R}$which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

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

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 ( A F B ) = W ( F ) for all A , B ∈ { a R ∈ 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 ( λ 1 λ 2 ) of W in terms of the singular values λ 1 , λ 2 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 : = 1 2 ‖ F ‖ 2 det F . Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the W 1 , p -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 ) .

We investigate a family of isotropic volumetric-isochoric decoupled strain energies based on the Hencky-logarithmic (true, natural) strain tensor log U , where μ >0 is the infinitesimal shear modulus, is the infinitesimal bulk modulus with λ the first Lamé constant, are additional dimensionless material parameters, F =∇ φ is the gradient of deformation, is the right stretch tensor and is the n -dimensional deviatoric part of the strain tensor log U . For small elastic strains, W eH approximates the classical quadratic Hencky strain energy 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 W eH , for and . 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 W eH is not preserved in dimension n =3 and that the energies are also not rank-one convex.

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.

In this paper, we linearise the recently introduced geometrically nonlinear constrained Cosserat-shell model. In the framework of the linear constrained Cosserat-shell model, we provide a comparison of our linear models with the classical linear Koiter shell model and the “best” first-order shell model. For all proposed linear models, we show existence and uniqueness based on a Korn’s inequality for surfaces.

We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for$$\\begin{aligned} \\frac{\\textrm{D}^{\\circ }}{\\textrm{D}t}[B] = \\mathbb {A}^{\\circ }(B).D \\end{aligned}$$D ∘ D t [ B ] = A ∘ ( B ) . D that the characteristic stiffness tensor$$\\mathbb {A}^{\\circ }(B)$$A ∘ ( B ) is positive-definite. Here,$$B = F \\, F^T$$B = F F T is the left Cauchy–Green tensor,$$\\frac{\\textrm{D}^{\\circ }}{\\textrm{D}t}$$D ∘ D t is a specific objective corotational rate,$$D = {{\\,\\textrm{sym}\\,}}\\, D_\\xi v$$D = sym D ξ v is the Eulerian stretching and$$\\mathbb {A}^{\\circ }(B)$$A ∘ ( B ) is the corresponding induced characteristic fourth-order stiffness tensor. Well-known corotational rates like the Zaremba–Jaumann rate, the Green–Naghdi rate and the logarithmic rate belong to this family of “positive” corotational rates. For general objective corotational rates$$\\frac{\\textrm{D}^{\\circ }}{\\textrm{D}t}$$D ∘ D t , we determine several conditions characterizing positivity. Among them is an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers [84]. We also give a geometrical motivation for invertibility and positivity of$$ \\mathbb {A}^{\\circ }(B)$$A ∘ ( B ) and highlight the structure-preserving properties of corotational rates that distinguish them from more general objective stress rates. Applications of this novel concept are indicated.

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

We show how to explicitly compute the homogenised curvature energy appearing in the isotropic$$\\Gamma $$Γ -limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimisation problem on$$\\Omega \\subset \\mathbb {R}^3$$Ω ⊂ R 3 for a Cosserat energy based on the second-order dislocation density tensor$$\\alpha :=\\overline{R} ^T \\textrm{Curl}\\overline{R} ın \\mathbb {R}^{3\\times 3}$$α : = R ¯ T Curl R ¯ ∈ R 3 × 3 ,$$\\overline{R}ın \\textrm{SO}(3)$$R ¯ ∈ SO ( 3 ) is used.