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We use gradient Young measures generated by Lipschitz maps to define a relaxation of integral functionals which are allowed to attain the value + ∞ and can model ideal locking in elasticity as defined by Prager in 1957. Furthermore, we show the existence of minimizers for variational problems for elastic materials with energy densities that can be expressed in terms of a function being continuous in the deformation gradient and convex in the gradient of the cofactor (and possibly also the gradient of the determinant) of the corresponding deformation gradient. We call the related energy functional gradient polyconvex. Thus, instead of considering second derivatives of the deformation gradient as in second-grade materials, only a weaker higher integrability is imposed. Although the second-order gradient of the deformation is not included in our model, gradient polyconvex functionals allow for an implicit uniform positive lower bound on the determinant of the deformation gradient on the closure of the domain representing the elastic body. Consequently, the material does not allow for extreme local compression. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement. [ABSTRACT FROM AUTHOR]

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort. [ABSTRACT FROM AUTHOR]

The paper deals with a scalar conservation law in an arbitrary dimension d with a discontinuous flux. The flux is supposed to be a discontinuous function in the spatial variable x and in an unknown function u. Under some additional hypothesis on the structure of possible discontinuities, we formulate an appropriate notion of entropy solution and establish its existence and uniqueness. The framework for proving the existence and uniqueness of entropy weak solutions is provided by the studies on entropy measure-valued solutions and may be viewed as a corollary of the uniqueness theorem for entropy measure-valued solutions. [ABSTRACT FROM AUTHOR]

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures. [ABSTRACT FROM AUTHOR]