Searcharxiv⌕ Search

arXiv subjects

Robert J. Martin

Publications and source records attributed to Robert J. Martin.

At least 19 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 \[ Σ_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad Σ_f(Q^T\mathrm{diag}(λ_1,\dotsc,λ_n)\, Q) = Q^T\mathrm{diag}(f(λ_1,\dotsc,λ_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(λ_{π(1)},\dotsc,λ_{π(n)}) = f_{π(i)}(λ_1,\dotsc,λ_n) \] for any permutation $π\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(λ_1,\dotsc,λ_n)$ is the diagonal matrix with diagonal entries $λ_1,\dotsc,λ_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $Σ_f$ 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 $V\mapstoσ(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm Dσ(1\!\!\!\:1)$ in the natural state imply the strong Baker-Ericksen inequalities.

math.AP↗
arXiv2025

Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints

We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}φ^T \operatorname{D}φ}-I_2\rVert^2$ for planar mappings $φ\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}φ\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}φ,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}φ\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.

math.AP↗
arXiv2024

The corotational stability postulate: positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity

In isotropic nonlinear elasticity the corotational stability postulate (CSP) is the requirement that \begin{equation*} \langle\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}[σ] , D \rangle > 0 \quad \forall \ D \in \text{Sym}(3)\setminus \{0\} \, , \end{equation*} where $\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}$ is any corotational stress rate, $σ$ is the Cauchy stress and $D = \text{Sym} \, L$, condition $L= \dot{F} \, F^{-1}$ is the deformation rate tensor. For $\widehatσ(\log V) := σ(V)$ it is equivalent to the monotonicity (TSTS-M$^+$) \begin{equation*} \langle \widehatσ (\log V_1) - \widehatσ (\log V_2) , \log V_1 - \log V_2 \rangle > 0 \quad \forall \ V_1, V_2 \in \text{Sym}^{++}(3), \ V_1 \neq V_2 \, . \end{equation*} For hyperelasticity, (CSP) is in general independent of convexity of the mapping $F \mapsto \mathrm{W}(F)$ or $U \mapsto \widehat{\mathrm{W}}(U)$. Considering a family of diagonal, homogeneous deformations $t \mapsto F(t)$ one can, nevertheless, show that (CSP) implies positive incremental Cauchy stress moduli for this deformation family, including the incremental Young's modulus, the incremental equibiaxial modulus, the incremental planar tension modulus and the incremental bulk modulus. Aside, (CSP) is sufficient for the Baker-Ericksen and tension-extension inequality. Moreover, it implies local invertibility of the Cauchy stress-stretch relation. Together, this shows that (CSP) is a reasonable constitutive stability postulate in nonlinear elasticity, complementing local material stability viz. LH-ellipticity.

math.AP↗
arXiv2024

Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain

We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \begin{align} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [σ] = \mathbb{H}^{\rm ZJ}(σ).D, \qquad D = {\rm sym} {\rm D} v\,, \end{align} operating on the Cauchy stress $σ$, 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}[σ], 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 $B \mapsto σ(B)$ in terms of the left Cauchy-Green tensor $B = F F^T$, the induced fourth-order tangent stiffness tensor $\mathbb{H}^{\rm ZJ}(σ)$ is positive definite if and only if for $\widehatσ(\log B):=σ(B)$, 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 $\mathbb{H}^{\rm ZJ}(σ)$. 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.

math.AP↗
arXiv2024

A natural requirement for objective corotational rates -- on structure preserving corotational rates

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.

math.AP↗
arXiv2024

On the representation of fourth and higher order anisotropic elasticity tensors in generalized continuum models

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.

math-ph↗
arXiv2024

A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain

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 $σ$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ] = \frac{\rm d}{{\rm d}{t}}[σ] - W \, σ+ σ\, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} φ$, 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}[σ]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehatσ(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat σ(\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 $B \mapsto σ(B) = \widehat σ(\log B)$, i.e. \begin{equation} \langle \widehatσ(\log B_1)-\widehatσ(\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.

math.AP↗
arXiv2024

The Biot stress -- right stretch relation for the compressible Neo-Hooke-Ciarlet-Geymonat model and Rivlin's cube problem

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.

math.AP↗
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

Polyconvex anisotropic hyperelasticity with neural networks

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.

cond-mat.mtrl-sci↗
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

Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

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.

math.AP↗
arXiv2019

Inconsistency of uhyper and umat in Abaqus for compressible hyperelastic materials

In this article, we revisited Bažant'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.

math.NA↗
arXiv2019

A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity

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.

math.AP↗