Poly-, Quasi- and Rank-One Convexity in Applied Mechanics

These lectures are largely based on two previous survey articles (2001), (2002), and cover a selection of open problems with some new remarks and updates. But they also give an introduction to the convexity conditions that are the objects of study of this course.

We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using T-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices.

Hyperelastic material behavior can be preferably described by using polyconvex energies, since the existence of minimizers is then guaranteed, if, in addition, the coercivity condition is satisfied. We give an overview of the construction of polyconvex energies for the description of non-trivial anisotropy classes, namely the triclinic, monoclinic, rhombic, tetragonal, trigonal and cubic symmetry groups, as well as transverse isotropy. The anisotropy of the material is described by invariants in terms of the right Cauchy-Green tensor and a specific second-order and a fourth-order structural tensor, respectively. To show the capability of the proposed polyconvex energies to simulate real anisotropic material behavior we focus on fittings of fourth-order elasticity tensors near the reference state to experimental data of different anisotropic materials.

In this contribution a general framework for the construction of polyconvex anisotropic strain energy functions, which a priori satisfy the condition of a stress-free reference configuration, is given. In order to show the applicability of polyconvex functions, two application fields are discussed. First, a comparative analysis of several polyconvex functions is provided, where the models are adjusted to experiments of soft biological tissues from arterial walls. Second, thin-shell simulations, where polyconvex material models are used, show a strong influence of anisotropy when comparing isotropic shells with anisotropic ones.

The article presents a variational theory of sharp phase interfaces bearing a deformation dependent energy. The theory involves both the standard and Eshelby stresses. The constitutive theory is outlined including the symmetry considerations and some particular cases. The existence of phase equilibria is proved based on appropriate convexity properties of the interfacial energy. Some generalization of the convexity properties is given and a relationship established to the semiellipticity condition from the theory of parametric integrals over rectifiable currents.

We review in these notes some of our recent work on modelling the mechanical response of nematic elastomers, both under static and dynamic loadings. Our aim is to compare theoretical results based on mathematical analysis and numerical simulations with the available experimental evidence, in order to examine critically the various accomplishments, and some interesting problems that remain open.

These notes consist of two main parts. In the first are described some recent results on the practical adaptation of polyconvexity criteria to models of nonlinearly elastic isotropic and transversely isotropic solids. The second part is concerned with the application of strong ellipticity in three dimensions to the construction of two-dimensional models for the combined bending and stretching of thin plates.

I consider the Γ-limit to a three-dimensional Cosserat model as the aspect ratio h > 0 of a flat domain tends to zero. The bulk model involves already exact rotations as a second independent field intended to describe the rotations of the lattice in defective elastic crystals. The Γ-limit based on the natural scaling consists of a membrane like energy and a. transverse shear energy both scaling with h, augmented by a curvature energy due to the Cosserat bulk, also scaling with h. A technical difficulty is to establish equi-coercivity of the sequence of functional as the aspect ratio h tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption. While the three-dimensional problem is well-posed for the Cosserat couple modulus μ_c ≥0, equi-coercivity needs a. strictly positive μ_c > 0. Then the Γ-limit model determines the midsorfaee deformation m ∈ H ^1,2 (ω, ℝ³). For the true defective crystal case, however, μ_c=0 is appropriate. Without equi-coercivity, we obtain first an estimate of the Γ-lim in and Γ-lim sup which can be strengthened to the Γ-convergence result. The Reissner-Mindlin model is “almost” the linearization of the Γ-limit for μ_c=0.