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We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \begin{align} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [\sigma] = \mathbb{H}^{\rm ZJ}(\sigma).D, \qquad D = {\rm sym} {\rm D} v\,, \end{align} operating on the Cauchy stress $\sigma$, the Eulerian strain rate $D$ and the spatial velocity $v$ with the novel \enquote{corotational stability postulate} (CSP)\begin{equation} \Bigl\langle \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma], D \Bigr\rangle > 0 \qquad \forall \, D\in{\rm Sym}(3)\setminus\{0\} \end{equation} to show that for a given isotropic Cauchy-elastic constitutive law $B \mapsto \sigma(B)$ in terms of the left Cauchy-Green tensor $B = F F^T$, the induced fourth-order tangent stiffness tensor $\mathbb{H}^{\rm ZJ}(\sigma)$ is positive definite if and only if for $\widehat{\sigma}(\log B):=\sigma(B)$, the strong monotonicity condition (TSTS-M$^{++}$) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M^{++}) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor $\mathbb{H}^{\rm ZJ}(\sigma)$. The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions.

We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for \begin{equation*} \frac{{\rm D}^{\circ}}{{\rm D} t}[B] = \mathbb{A}^{\circ}(B).D \end{equation*} that $\mathbb{A}^{\circ}(B)$ is positive definite. Here, $B = F \, F^T$ is the left Cauchy-Green tensor, $\frac{{\rm D}^{\circ}}{{\rm D}t}$ is a specific objective corotational rate, $D = {\rm sym} \, {\rm D} v$ is the Eulerian stretching and $\mathbb{A}^{\circ}(B)$ is the corresponding induced fourth order tangent 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 $\frac{{\rm D}^{\circ}}{{\rm D} t}$ we determine several conditions characterizing positivity. Among them an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers (2004). We also give a geometrical motivation for invertibility and positivity 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.

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress $\sigma$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma] = \frac{{\rm d}}{{\rm d}{t}}[\sigma] - W \, \sigma + \sigma \, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} \varphi$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, we show that \begin{equation} \label{eqCPSdef} \begin{alignedat}{2} \forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}: ~ \langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehat\sigma(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat \sigma(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\rm Sym}^{++}_4(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M$^+$) for $B \mapsto \sigma(B) = \widehat \sigma(\log B)$, i.e. \begin{equation} \langle \widehat\sigma(\log B_1)-\widehat\sigma(\log B_2),{\log B_1-\log B_2} \rangle> 0 \qquad \forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case.

The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the non-uniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider a remarkable possibility in this nonlinear scenario: Does symmetric loading lead only to symmetric deformations or also to asymmetric deformations? If so, what can we say about monotonicity for these homogeneous solutions, a property which is less restrictive than the energetic stability criteria of homogeneous solutions for Rivlin's cube problem. For the Neo-Hooke type materials we establish what properties the volumetric function $h$ depending on ${\rm det}\, F$ must have to ensure the existence of a unique radial solution (i.e. the cube must continue to remain a cube) for any magnitude of radial stress acting on the cube. The function $h$ proposed by Ciarlet and Geymonat satisfies these conditions. However, discontinuous equilibrium trajectories may occur, characterized by abruptly appearing non-symmetric deformations with increasing load, and a cube can instantaneously become a parallelepiped. Up to the load value for which the bifurcation in the radial solution is realized local monotonicity holds true. However, after exceeding this value, monotonicity no longer occurs on homogeneous deformations which, in turn, preserve the cube shape.

The classification of all fourth-order anisotropic tensor classes for classical linear elasticity is well known. In this article, we review the related problem of explicitly computing the dimension and the expressions of the elements belonging to these classes, and we extend this computation to fourth-order elasticity tensors acting on non-symmetric matrices. These tensors naturally appear in generalized continuum models. Based on tensor symmetrization, we provide the most general forms of these tensors for orthotropic, transversely isotropic, cubic, and isotropic materials. We present a self-contained discussion and provide detailed calculations for simple examples.

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 \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+2\log\lambda_{\rm min}\,, \] where $\lambda_{\rm max}\geq\lambda_{\rm min}$ 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 $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] 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.

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 $W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split, i.e. \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F)\,, \] 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 \enquote{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 \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-2\log\lambda_{\rm min} \] is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $W_{\rm magic}^+(F)$ 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.

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.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\operatorname{GL}^+(2)$ of invertible $2\times2-\,$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 $\operatorname{GL}^+(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W:\operatorname{GL}^+\to\mathbb{R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

It is well known that a twice-differentiable real-valued function $W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ on the group $\operatorname{GL}^+(n)$ of invertible $n\times n-$matrices with positive determinant is rank-one convex if and only if it is Legendre-Hadamard elliptic. Many energy functions arising from interesting applications in isotropic nonlinear elasticity, however, are not necessarily twice differentiable everywhere on $\operatorname{GL}^+(n)$, especially at points with non-simple singular values. Here, we show that if an isotropic function $W$ on $\operatorname{GL}^+(n)$ is twice differentiable at each $F\in\operatorname{GL}^+(n)$ with simple singular values and Legendre-Hadamard elliptic at each such $F$, then $W$ is already rank-one convex under strongly reduced regularity assumptions. In particular, this generalization makes (local) ellipticity criteria accessible as criteria for (global) rank-one convexity to a wider class of elastic energy potentials expressed in terms of ordered singular values. Our results are also directly applicable to so-called conformally invariant energy functions. We also discuss a classical ellipticity criterion for the planar case by Knowles and Sternberg which has often been used in the literature as a criterion for global rank-one convexity and show that for this purpose, it is still applicable under weakened regularity assumptions.

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

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 $O(h^5)$ 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.

We provide a faithful translation of Hans Richter's important 1949 paper "Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Form\"anderungen" from its original German version into English, complemented by an introduction summarizing Richter's achievements.

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.

We provide a faithful translation of Hans Richter's important 1948 paper "Das isotrope Elastizit\"atsgesetz" from its original German version into English. Our introduction summarizes Richter's achievements.

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function $W\colon\operatorname{SL}(2)\to\mathbb{R}$ with $W(RF)=W(FR)=W(F)$ for all $F\in\operatorname{SL}(2)$ and all $R\in\operatorname{SO}(2)$, where $\operatorname{SL}(2)$ and $\operatorname{SO}(2)$ 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 $\operatorname{GL}^+(2)$ of $2\times2$-matrices with positive determinant, i.e. $W\colon\operatorname{GL}^+(2)\to\mathbb{R}$ such that \[W(AFB) = W(F) \qquad\text{for all }\; A,B\in\{aR\in\operatorname{GL}^+(2) \,|\, a\in(0,\infty)\,,\; R\in\operatorname{SO}(2)\}\,,\] where $\operatorname{SO}(2)$ denotes the special orthogonal group, and provide an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $W(F)=h(\frac{\lambda_1}{\lambda_2})$ of $W$ in terms of the singular values $\lambda_1,\lambda_2$ of $F$, are applied to a number of example energies in order to demonstrate the convenience of the eigenvalue-based expression compared to the more common representation in terms of the distortion $\mathbb{K}:=\frac12\frac{\lVert F\rVert^2}{\det F}$. Special cases of our results can be obtained from earlier works by Astala et al. and Yan.

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 $X,Y$.

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.

We reconsider anti-plane shear deformations of the form $\varphi(x)=(x_1,\,x_2,\,x_3+u(x_1,x_2))$ 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.

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

The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\operatorname{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation} W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $\Psi\colon[0,\infty)\to\mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function $\Psi$ such that $W$ of the form (1) is Legendre-Hadamard elliptic. Similarly, we consider the related isochoric strain measure $\lVert\operatorname{dev}_n\log U\rVert^2$, where $\operatorname{dev}_n \log U$ is the deviatoric part of $\log U$. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case $n=2$, we show that for $n\geq3$, no strictly monotone function $\Psi\colon[0,\infty)\to\mathbb{R}$ exists such that $F\mapsto \Psi(\lVert\operatorname{dev}_n\log U\rVert^2)$ is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form $F\mapsto \Psi(\lVert\operatorname{dev}_n\log U\rVert^2) + W_{\mathrm{vol}}(\det F)$ cannot be rank-one convex for any function $W_{\mathrm{vol}}\colon(0,\infty)\to\mathbb{R}$ if $\Psi$ is strictly monotone.

In this article, we revisited Ba\v{z}ant's comments on the implementation of hyperelastic material models in commercial finite element software. We would like to clarify that our assertions only apply if the material models are implemented as hypoelastic, i.e. by incremental stress updates, in common interfaces (including, in particular, umat in Abaqus). This assumption was not made sufficiently clear in the article. If, on the other hand, the stress calculations are implemented using the umat interface with absolute (or "total") stress updates, as is also assumed in the uhyper interface, there is no difference in the internal processes or the results between the umat and the uhyper implementation. This applies to highly compressible formulations as well, where the Kirchhoff and Cauchy stress tensors are clearly distinguished., Comment: The article does not sufficiently distinguish between the behaviour of Abaqus for incremental and absolute stress updates

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfrac{\mu}{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n \log\boldsymbol{U}\right\rVert^2} + \dfrac{\kappa}{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr} (\log\boldsymbol{U})]^2 }\,, \] where $\mu>0$ is the (infinitesimal) shear modulus, $\kappa>0$ is the (infinitesimal) bulk modulus, $k$ and $\hat{k}$ are additional dimensionless material parameters, $\boldsymbol{U}=\sqrt{\boldsymbol{F}^T\boldsymbol{F}}$ and $\boldsymbol{V}=\sqrt{\boldsymbol{F}\boldsymbol{F}^T}$ are the right and left stretch tensor corresponding to the deformation gradient $\boldsymbol{F}$, $\log$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $\mbox{dev}_n \boldsymbol{X} = \boldsymbol{X}-\frac{\mbox{tr} \boldsymbol{X}}{n}\boldsymbol{1}$ and $\lVert \boldsymbol{X} \rVert = \sqrt{\mbox{tr}\boldsymbol{X}^T\boldsymbol{X}}$ are the deviatoric part and the Frobenius matrix norm of an $n\times n$-matrix $\boldsymbol{X}$, respectively, and $\mbox{tr}$ 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.

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 $\mathrm{SL}(2)$ implies the polyconvexity of $W$.

We describe ellipticity domains for the isochoric elastic energy $ F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2 $ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W_{_{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg\{U\in{\rm PSym}(3) \;\Big|\, \|{\rm dev}_3\log U\|^2\leq \frac{2}{3}\bigg\}$, 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 $ W_{_{\rm H}}(F)=\mu \,\|{\rm dev}_3\log U\|^2+ \frac{\kappa}{2}\,[{\rm tr} (\log U)]^2 $, $U=\sqrt{F^TF}$ with $\mu>0$ and $\kappa>\frac{2}{3}\, \mu$, previously obtained by Bruhns et al.

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on $\mathrm{GL}^+(2)$ is already polyconvex on $\mathrm{GL}^+(2)$. Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. 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.

We consider the two logarithmic strain measures\[\omega_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad \omega_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]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 $\mathrm{GL}(n)$. Here, $F$ is the deformation gradient, $U=\sqrt{F^TF}$ is the right Biot-stretch tensor, $\log$ denotes the principal matrix logarithm, $\|.\|$ is the Frobenius matrix norm, $\mathrm{tr}$ is the trace operator and $\mathrm{dev}_n X$ is the $n$-dimensional deviator of $X\in\mathbb{R}^{n\times n}$. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $\varepsilon=\mathrm{sym}\nabla u$, which is the symmetric part of the displacement gradient $\nabla u$, and reveals a close geometric relation between the classical quadratic isotropic energy potential \[\mu\,\|\mathrm{dev}_n\mathrm{sym}\nabla u\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\mathrm{sym}\nabla u)]^2=\mu\,\|\mathrm{dev}_n\varepsilon\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\varepsilon)]^2\]in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy\[\mu\,\|\mathrm{dev}_n\log U\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\log U)]^2=\mu\,\omega_{\rm iso}^2+\frac\kappa2\,\omega_{\rm vol}^2\,,\]where $\mu$ is the shear modulus and $\kappa$ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $R$, where $F=R\,U$ 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.