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 nonlocal variational problems in L p , like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand, or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the Calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal Poincaré inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in L p in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, non-positive integrands may give rise to positive nonlocal functionals.
We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in Solid Mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincaré inequality. We cover Dirichlet, Neumann and mixed boundary conditions. The existence theory is set in the Lebesgue L p spaces and in the fractional Sobolev W s,p spaces, for 0 < s < 1 and 1 < p < ∞.for some 1 < p < ∞ and 0 ≤ α < n + p. For this special growth, we distinguish the weakly singular case 0 ≤ α < n and the strongly singular case n < α < n + p. When 0 ≤ α < n, the analysis of the lower semicontinuity is reduced to the recent study carried out by Elbau  and lies in the functional framework of Lebesgue L p spaces. The weak lower semicontinuity is proved in  to be equivalent to an interesting convexity property of the integrand w, of a different nature that those convexity properties equivalent to weak lower semicontinuity for local problems (see, e.g., [16, Ch. 8]); we will discuss this issue in Section 3 in our particular peridynamics framework. The coercivity for the Dirichlet problem was proved by Andreu et al.  in their study of nonlocal diffusion problems, and later used by [3,26] in the context of peridynamics. The coercivity for the Neumann and mixed problem was proved by Aksoylu & Mengesha  using a Poincaré-type inequality proved by Ponce  in his study of nonlocal characterizations of Sobolev spaces (see also ). As a matter of fact, we shall need some adaptations of those results to our context. At this point, we ought to mention that Dirichlet and mixed boundary value problems have a slightly different meaning than for local problems, one the reasons being that L p functions do not have traces of the boundary ∂Ω. In contrast, Dirichlet conditions in the context of peridynamics prescribe the value of the deformation in a set of positive measure.The lower semicontinuity in the case n < α < n + p is in fact trivial, since the functional framework is that of the fractional Sobolev spaces W s,p with s = α−n p , and weak convergence in W s,p implies (for a subsequence) convergence a.e. The coercivity, on the other hand, is a consequence of an improved Poincaré-type inequality in fractional Sobolev spaces recently proved in Hurri-Syrjänen & Vähäkangas . It is worth mentioning that the need of improved Poincaré-type inequalities is a result of the assumption that w(x, ·) vanishes for |x| large.The existence theory for the critical case α = n is also covered by reducing it to the case 0 ≤ α < n and to the functional framework of L p spaces. In doing that, we do not provide a full characterization of the lower semicontinuity, so that our conditions on w may not be optimal.Nonlocal variational problems, of which (1.1) is a particular case, h...
Fracture surfaces and the regularity of inverses for BV deformations the date of receipt and acceptance should be inserted later Abstract Motivated by nonlinear elasticity theory, we study deformations that are approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy E was used by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that E measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing that, we also obtain an SBV regularity result for the inverse.
Based on a previous work by the authors on the modelling of cavitation and fracture in nonlinear elasticity, we give an alternative proof of a recent result by Csörnyei, Hencl and Malý on the regularity of the inverse of homeomorphisms in the Sobolev space W 1;n 1 . With this aim, we show that the notion of fracture surface introduced by the authors in their model corresponds precisely to the original notion of cavity surface in the cavitation models of Müller and Spector (1995) and Conti and De Lellis (2003). We also find that the surface energy introduced in the model for cavitation and fracture is related to Lusin's condition (N) on the non-creation of matter.A fundamental question underlying this paper is whether Det Du D det Du necessarily implies that the deformation u opens no cavities. We show that this is not true unless Müller and Spector's condition INV for the non-interpenetration of matter is satisfied. Having thus provided an additional justification of its importance, we prove the stability of this condition with respect to weak convergence in the critical space W 1;n 1 . Combining this with the work by Conti and De Lellis, we obtain an existence theory for cavitation in this critical case.
In this paper we propose a nonlocal model of hyperelasticity obtained by substitution of the classical gradient by the Riesz fractional gradient. We show existence of solutions for those nonlocal models in Bessel fractional spaces under the main assumption of polyconvexity of the energy density. 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 hyperelastic energy.
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