We define a class of deformations in W 1,p (Ω, R n), p > n−1, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality Det = det (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W 1,p , and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

In this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea to unify the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of creation of new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies the sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one a.e. deformations is also one-to-one a.e. We then use these results to obtain existence of minimizers of variational models that incorporate the elastic energy and this created surface energy, taking into account the orientation-preserving and the non-interpenetration conditions.

We study fracture and delamination of a thin stiff film bonded on a rigid substrate through a thin compliant bonding layer. Starting from the three-dimensional system, upon a scaling hypothesis, we provide an asymptotic analysis of the three-dimensional variational fracture problem as the thickness goes to zero, using Γ-convergence. We deduce a two-dimensional limit model consisting of a brittle membrane on a brittle elastic foundation. The fracture sets are naturally discriminated between transverse cracks in the film (curves in 2D) and debonded surfaces (two-dimensional planar regions). We introduce the vectorial plane-elasticity case, applying the rigorous results established for scalar displacement fields, in order to numerically investigate the typical cracking scenarios encountered in applications. To this end, we formulate a reduced-dimension, rate-independent, irreversible evolution law for transverse fracture and debonding of thin film systems. Finally, we propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional. We provide an illustration of the capabilities of the formulation exploring complex crack patterns in one and two dimensions, showing a qualitative comparison with geometrically involved real life examples.

Let B (0, R 0 ) ⊂ R 3 denote a three-dimensional spherical droplet of radius R 0 > 0, centered at the origin. Let S 0 denote the set of symmetric, traceless 3 × 3 matrices i.e.where M 3×3 is the set of 3 × 3 matrices. The corresponding matrix norm is defined to be [12]and we will use the Einstein summation convention throughout the paper.We work with the Landau-de Gennes theory for nematic liquid crystals [7] whereby a liquid crystal configuration is described by a macroscopic order parameter, known as the Q-tensor order parameter. Mathematically, the Landau-de Gennes Q-tensor order parameter is a symmetric, traceless 3 × 3 matrix belonging to the space S 0 in (1). The liquid crystal energy is given by the Landau-de Gennes energy functional and the associated energy density is a nonlinear function of Q and its spatial derivatives [7,17]. We work with the simplest form of the Landau-de Gennes energy functional that allows for a first-order nematic-isotropic phase transition and spatial inhomogeneities as shown below [12] -Here, L > 0 is a small material-dependent elastic constant, |∇Q| 2 = Q ij,k Q ij,k ( note that Q ij,k = ∂Q ij ∂x k ) with i, j, k = 1 . . . 3 is an elastic energy density and f B : S 0 → R is the bulk energy density. For our purposes, we take f B to be a quartic polynomial in the Q-tensor invariants as shown below -