Rank-one convexity and polyconvexity of Hencky-type energies

We investigate a family of isotropic volumetric-isochorically decoupled strain energies based on the Hencky-logarithmic (true, natural) strain tensor log U. The main result of this note is that for n = 2 the considered energies are rank-one convex for suitable values of two material parameters. We also conjecture that there are values of the material parameters such that the corresponding energies are polyconvex. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Loss of polyconvexity by homogenization: a new example

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with P-growth, where p ≥ 2 can be fixed arbitrarily.     

On the construction of anisotropic polyconvex energy densities

In this contribution we propose a general framework for the construction of polyconvex energies for arbitrary anisotropy classes. The main idea is the introduction of an anisotropic metric reflecting the material symmetry of the underlying crystal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of polyconvex,anisotropic free‐energy functions

The existence of minimizers of some variational principles in finite elasticity is based on the concept of quasiconvexity, introduced by Morrey [6]. This integral inequality is rather complicated to handle. Thus, the sufficient condition for quasiconvexity, the polyconvexity condition in the sense of Ball [1], is a more important concept for practical applications, see also Ciarlet [4] and Dacorogna [5]. In the case of isotropy there exist some models which satisfy this condition. Furthermore, there does not exist a systematic treatment of anisotropic, polyconvex free‐energies in the literature. In the present work we discuss some aspects of the formulation of polyconvex, anisotropic free‐energy functions in the framework of the invariant formulation of anisotropic constitutive equations and focus on transverse isotropy.     

Polyconvex energies and cavitation

We study the existence of singular minimizers in the class of radial deformations for polyconvex energies that grow linearly with respect to the Jacobian.     

Modeling dissipative effects in anisotropic materials by means of evolving generalized structural tensors

In this work, an anisotropic dissipative model is proposed as an extension of a recently presented polyconvex anisotropic strain-energy function for fiber-reinforced materials. This thermodynamically consistent model is able to describe different softening phenomena and includes residual deformations. The so-called preconditioning behavior of a soft biological tissue sample is considered as a numerical example. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rank-one convexity implies polyconvexity for isotropic,objective and isochoric elastic energies in the two-dimensional case

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on GL⁺(2) is already polyconvex on 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. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyconvex energy densities with applications to anisotropic thin shells

This work shows the capability to simulate anisotropic thin shells using polyconvex energy densities by analyzing numerical examples. The variational framework is based on the enhanced assumed strain formulation. The iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three–dimensional constitutive equations. We consider an additive structure of the energy decoupled in an isotropic part for a matrix material and the superposition of transversely isotropic parts for embedded fiber families and focus on polyconvex strain energy functions. In two representative examples we document anisotropy effects. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Interface Problem in Solid Mechanics with a Linear Elastic and a Hyperelastic Material

The three dimensional interface problem is considered with the homogeneous Lamé system in an unbounded exterior domain and some quasistatic nonlinear elastic material behavior in a bounded interior Lipschitz domain. The nonlinear material is of the Mooney-Rivlin type of polyconvex materials. We give a weak formulation of the interface problem based on minimizing the energy, and rewrite it in terms of boundary integral operators. Then, we prove existence of solutions.     

The two-well problem in three dimensions

We study properties of generalized convex hulls of the set with . If K contains no rank-1 connection we show that the quasiconvex hull of K is trivial if H belongs to a certain (large) neighbourhood of the identity. We also show that the polyconvex hull of K can be nontrivial if H is sufficiently far from the identity, while the (functional) rank-1 convex hull is always trivial. If the second well is replaced by a point then the polyconvex hull is trivial provided that there are no rank-1 connections. Received: March 25, 1999 / Accepted: April 23, 1999     

Convergence of total variation of curvature measures

It is known that the curvature measures of parallel ɛ-neighbourhoods of a set with positive reach or a polyconvex set converge vaguely if ɛ tends to zero to the curvature measures of the set itself. We show that in the case of a set with positive reach, the total variations of the curvature measures converge as well, whereas in the case of a polyconvex set this is no more true in general. Supported by MSM 113200007 and GAČR 201/06/0302. Author’s address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic     

EXISTENCE THEOREM AND FINITE ELEMENT METHOD FOR STATIC PROBLEMS OF A CLASS OF NONLINEAR HYPERELASTIC SHELLS

In this paper, various boundary value problems of hyperelastic shells are considered. It is assumed that the storede-nergy function W(x, F) of the material,of which the shell is made, satisfies polyconvex conditions proposed by Ball~([2]).Existence of minimum points of the total energy of the shell in suitably chosen function spaces, and in suitably chosen finite element spaces is proved. Convergence of the finite element solutions is proved under certain regular conditions on the minimum points and some additional assumptions on W(x, F). A Gradient type computing scheme for solving the finite element solutions is given, and global convergent result is obtained.     

An example of a C1,1 polyconvex function with no differentiable convex representative

We construct a C^1,1 polyconvex function W such that there exists a fixed 2×2 matrix Y with the property that all convex representatives of W have at least two distinct subgradients (and are hence not differentiable) at the point (Y,detY), showing in particular that a polyconvex function can be smoother than any of its convex representatives. To cite this article: J. Bevan, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Polyconvex potentials,invertible deformations,and thermodynamically consistent formulation of the nonlinear elasticity equations

It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.     

Numerical Calculation of Fiber Orientation in Three-Dimensional Arterial Walls

In this contribution an approach for the fiber reorientation in three-dimensional arterial walls is presented. In detail the load-bearing capacity of the tissue is increased by re orienting the fibers with respect to the principal stresses, cf. [1]. The improved fiber reorientation algorithm is combined with the polyconvex nonlinear anisotropic material model presented in [3]. The results of a three-dimensional finite element simulation, where the reorientation approach is applied to a short segment of a patient-specific arterial geometry, are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A SimpleModel for Anisotropic Damage with Applications to Soft Tissues

Arteries are reinforced by helically arranged collagen fibers and posses orthotropic elastic properties. In this paper a polyconvex anisotropic energy is used in order to guarantee the existence of minimizers for the purely elastic boundary value problem. Anisotropic discontinuous damage effects, which are induced by decreasing stiffness of particular fibers, are observed in a certain range of overexpansion. A simple thermodynamical consistent anisotropic damage model is constructed, basing on the assumption that damage mainly takes place in the fiber directions as a result of breakage of collagen cross‐links.Finally a cycled overexpansion of a test material from an artery is analyzed in order to show the main characteristics of the proposed model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bauer's maximum principle and hulls of sets

We use the Bauer maximum principle for quasiconvex, polyconvex and rank-one convex functions to derive Krein-Milman-type theorems for compact sets in . Further we show that in general the set of quasiconvex extreme points is not invariant under transposition and it is different from the set of rank-one convex extreme points. Finally, a set in with different polyconvex, quasiconvex and rank-one convex hulls is constructed. Received September 14, 1999 / Accepted January 14, 2000 /Published online July 20, 2000     

A remark on polyconvex envelopes of radially symmetric functions in dimension 2 × 2

We study polyconvex envelopes of a class of functions related to the function of Kohn and Strang introduced in [4]. We present an example of a function of this class for which the polyconvex envelope may be computed explicitly and we also point out some general features of the problem.     

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.