arXiv subjects

Jendrik Voss

Publications and source records attributed to Jendrik Voss.

15 recordsLinked to original sources

arXiv2026

Monotonicity of isotropic tensor functions on the set of symmetric matrices: completing Rodney Hill's generalization of the Chandler Davis convexity theorem

Motivated by classical constitutive inequalities in isotropic nonlinear elasticity theory, we investigate the monotonicity of isotropic tensor functions of the form \[ \Sigma_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad \Sigma_f(Q^T\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)\, Q) = Q^T\mathrm{diag}(f(\lambda_1,\dotsc,\lambda_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(\lambda_{\pi(1)},\dotsc,\lambda_{\pi(n)}) = f_{\pi(i)}(\lambda_1,\dotsc,\lambda_n) \] for any permutation $\pi\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)$ is the diagonal matrix with diagonal entries $\lambda_1,\dotsc,\lambda_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $\Sigma_f$ on $\mathrm{Sym}(n)$. Our results generalize the Chandler Davis theorem for convex scalar isotropic functions and are obtained independently of Hill's original proof of this equivalence. We also discuss simple invertibility conditions for isotropic matrix functions. We conclude by showing that injectivity of the Cauchy stress $V\mapsto\sigma(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm D\sigma(1\!\!\!\:1)$ in the natural state imply the strong Baker-Ericksen inequalities.

math.AP↗

arXiv2025

Effective interface forces to model boundary effects in a finite-size metamaterial through the reduced relaxed micromorphic model

We use the reduced relaxed micromorphic model (RRMM) to capture the effective "bulk" dynamical response of finite size metamaterial specimens made out of a Labyrinthine unit cell. We show that for small finite-size specimens, boundary effects can play a major role, so that the RRMM needs an enrichment to capture the metamaterial's bulk response, as well as the boundary effects. A benchmark test is introduced to show that different metamaterial/ homogeneous material interfaces can drive completely different responses even if the bulk metamaterial remains the same. We show with no remaining doubts that the concept of "interface forces" must necessarily be introduced if one wants to model finite-size metamaterials in a homogenized framework.

physics.app-ph↗

arXiv2024

Yet another best approximation isotropic elasticity tensor in plane strain

For plane strain linear elasticity, given any anisotropic elasticity tensor $\mathbb{C}_{\rm aniso}$, we determine a best approximating isotropic counterpart $\mathbb{C}_{\rm iso}$. This is not done by using a distance measure on the space of positive definite elasticity tensors (Euclidean or logarithmic distance) but by considering two simple isotropic analytic solutions (center of dilatation and concentrated couple) and best fitting these radial solutions to the numerical anisotropic solution based on $\mathbb{C}_{\rm aniso}$. The numerical solution is done via a finite element calculation, and the fitting via a subsequent quadratic error minimization. Thus, we obtain the two Lam\'e-moduli $\mu$, $\lambda$ (or $\mu$ and the bulk-modulus $\kappa$) of $\mathbb{C}_{\rm aniso}$. We observe that our so-determined isotropic tensor $\mathbb{C}_{\rm iso}$ coincides with neither the best logarithmic fit of Norris nor the best Euclidean fit. Our result calls into question the very notion of a best-fit isotropic elasticity tensor to a given anisotropic material.

math.AP↗

arXiv2024

From frequency-dependent models to frequency-independent enriched continua for mechanical metamaterials

Mechanical metamaterials have recently gathered increasing attention for their uncommon mechanical responses enabling unprecedented applications for elastic wave control. To model the mechanical response of large metamaterials' samples made up of base unit cells, so-called homogenization or upscaling techniques come into play trying to establish an equivalent continuum model describing these macroscopic metamaterials' characteristics. A common approach is to assume a priori that the target continuum model is a classical linear Cauchy continuum featuring the macroscopic displacement as the only kinematical field. This implies that the parameters of such continuum models (density and/or elasticity tensors) must be considered to be frequency-dependent to capture the complex metamaterials' response in the frequency domain. These frequency-dependent models can be useful to describe some of the aforementioned macroscopic metamaterials' properties, yet, they suffer some drawbacks such as featuring negative masses and/or elastic coefficients in some frequency ranges. More than being counter-intuitive, this implies that the considered Cauchy continuum is not positive-definite for all the considered frequencies. In this paper, we present a procedure, based on the definition of extra kinematical variables (with respect to displacement alone) and the use of the inverse Fourier transform in time, to convert a frequency-dependent model into an enriched continuum model of the micromorphic type. All the parameters of the associated enriched model are constant (i.e., frequency-independent) and the model itself remains positive-definite for all the considered frequency ranges. The response of the frequency-dependent model and the associated micromorphic model coincide in the frequency domain, in particular when looking at the dispersion curves.

physics.app-ph↗

arXiv2022

Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach

We present an inertia-augmented relaxed micromorphic model that enriches the relaxed micromorphic model previously introduced by the authors via a term $\text{Curl}\dot{P}$ in the kinetic energy density. This enriched model allows us to obtain a good overall fitting of the dispersion curves while introducing the new possibility of describing modes with negative group velocity that are known to trigger negative refraction effects. The inertia-augmented model also allows for more freedom on the values of the asymptotes corresponding to the cut-offs. In the previous version of the relaxed micromorphic model, the asymptote of one curve (pressure or shear) is always bounded by the cut-off of the following curve of the same type. This constraint does not hold anymore in the enhanced version of the model. While the obtained curves' fitting is of good quality overall, a perfect quantitative agreement must still be reached for very small wavelengths that are close to the size of the unit cell.

physics.app-ph↗

arXiv2022

Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

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 \[ W_{\rm magic}^+(F)=\frac{λ_{\rm max}}{λ_{\rm min}}-\log\frac{λ_{\rm max}}{λ_{\rm min}}+\log\det F=\frac{λ_{\rm max}}{λ_{\rm min}}+2\logλ_{\rm min}\,, \] where $λ_{\rm max}\geqλ_{\rm min}$ 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.

math.AP↗

arXiv2021

Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions

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 \[ W_{\rm magic}^+(F)=\frac{λ_{\rm max}}{λ_{\rm min}}-\log\frac{λ_{\rm max}}{λ_{\rm min}}+\log\det F=\frac{λ_{\rm max}}{λ_{\rm min}}-2\logλ_{\rm min} \] 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.

math.AP↗

arXiv2020

A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets

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.

math.AP↗

arXiv2020

Rank-one convexity vs. ellipticity for isotropic functions

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.

math.AP↗

arXiv2020

Sharp rank-one convexity conditions in planar isotropic elasticity for the additive volumetric-isochoric split

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 $W(F)=\fracμ{2}\frac{\lVert F\rVert^2}{\det F}+f(\det F)$; such an energy is rank-one convex if and only if the function $f$ is convex.

math.AP↗

arXiv2019

Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

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.

math.AP↗

arXiv2018

The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity

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 $W(F)=h(\frac{λ_1}{λ_2})$ 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 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.

math.AP↗

arXiv2018

Do we need Truesdell's empirical inequalities? On the coaxiality of stress and stretch

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 $X,Y$.

math.AP↗

arXiv2018

Shear, pure and simple

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.

math.AP↗

arXiv2018

Again anti-plane shear

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

math.AP↗