Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, polyconvex functions which are always s.w.l.s. are usually considered. For isotropic as well as for transversely isotropic and orthotropic materials constitutive functions that are polyconvex already exist. The main goal of this contribution is to provide a new method for the construction of polyconvex hyperelastic models for more general anisotropy classes. The fundamental idea is the introduction of positive definite second-order structural tensors G=HH^T encoding the anisotropies of the underlying crystal. These tensors can be viewed as a push-forward of a cartesian metric of a fictitious reference configuration to the real reference configuration. Here the driving transformations H in the push-forward operation are mappings of the cartesian base vectors of the fictitious configuration onto crystallographic motivated base vectors. Restrictions of this approach are based on the polyconvexity condition as well as on the usage of second-order structural tensors and pointed out in detail.     

The study of equivalent material parameters in a hyperelastic model

We study a hyperelastic model of some biological soft tissues with emphasis on the problem of its matching with the material parameters acquired by experimental mechanical tests. First, we study the polyconvexity property of the hyperelastic model. Then, we explore the notion of equivalent sets of material parameters. We perform a numerical study of the regions of equivalent material parameters characterizing the curves predicted by the hyperelastic model that are close, within a prefixed tolerance, to those given by the experimental data. In the numerical study we use the quadratic variation and the Hausdorff distance. The study suggests that a qualitative knowledge of shape and volume of the regions of equivalent material parameters can provide both a criterion for the optimal match between the model with the experimental data and an indication on the reducibility of the number of parameters used in the model.     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998     

Distortional hardening plasticity model for paperboard

A distortional hardening elasto-plastic model at finite strains suitable for modeling of orthotropic materials is presented. As a prototype material, paperboard is considered. An in-plane model is established. The model developed is motivated from non-proportional loading tests on paperboard where the paperboard is pre-strained in one direction and then loaded in the perpendicular direction. A softening effect is revealed in the pre-strained samples. The observed experimental findings cannot be accurately predicted by current models for paperboard. To be able to model the softening effects, a yield surface based on multiple hardening variables is introduced. It is shown that the model parameters can be obtained from simple uniaxial experiments. The model is implemented in a finite element framework which is used to illustrate the behavior of the model at some specific loading situations and is compared with strain fields obtained from Digital Image Correlation experiments.     

Fractional Piola identity and polyconvexity in fractional spaces

In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler–Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal energy. Contrary to local problems in nonlinear elasticity, this existence result is compatible with solutions presenting discontinuities at points and along hypersurfaces.     

On conditions for polyconvexity

We give an example of a smooth function , which is not polyconvex and which has the property that its restriction to any ball of radius one can be extended to a smooth polyconvex function . In particular, it implies that there exists no `local condition' which is necessary and sufficient for polyconvexity of functions , where , . We also briefly discuss connections with quasiconvexity.

     

On the singular set for Lipschitzian critical points of polyconvex functionals

Partial regularity is proved for Lipschitzian critical points of polyconvex functionals provided ‖DuL^∞_‖ is small enough. In particular, the singular set for a Lipschitzian critical point has Hausdorff dimension strictly less than n when ‖DuL^∞_‖ is small enough. Model problems treated include