Nonconvex Energy Density Research Articles

Analysis of a dynamic frictional contact problem for hyperviscoelastic material with non-convex energy density

Using the time approximation method we obtain the existence of a weak solution for the dynamic contact problem with damping and a non-convex stored elastic energy function. On the contact boundary we assume the normal compliance law and the generalization of the Coulomb friction law which allows for non-monotone dependence of the friction force on the tangential velocity. The existence result is accompanied by two numerical examples, one of them showing lack of uniqueness for the numerical solution.

Young-measure quasi-static damage evolution: The nonconvex and the brittle cases

A rate-independent model for incomplete damage is considered, with nonconvex energy density, mixed boundary condition, and nonzero external load, both for non-brittle and brittle materials. An existence result for a Young measure quasi-static evolution is proved.

Quasistatic evolution for a phase‐transition model: a Young measure approach

AbstractA quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the use of suitable regularity arguments in order to prove an existence result for a generalized notion of evolution (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Rate-independent phase transitions in elastic materials: A Young-measure approach

A quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the usage of suitable regularity arguments in order to prove an existence result for a notion of evolution presenting some improvements with respect to the one defined in [13], for infinitely many phases.

On the numerical relaxation of single‐slip plasticity in finite strains

AbstractThe modeling of the elastoplastic behaviour of single crystals with infinite latent hardening leads to a nonconvex energy density, whose minimization produces fine structures. The computation of the quasiconvex envelope of the energy density is faced in this case with huge numerical difficulties caused by the clusters of local minima. By exploiting the structure of the problem, we present a fast and efficient numerical relaxation algorithm as alternative to global optimization techniques usually adopted in literature which are computationally more expensive. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

In the theory of solid-solid phase transitions the deformation Open image in new window of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study

Young-Measure approximations for elastodynamics with non-monotone stress-strain relations

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density φ. Their time-evolution leads to a nonlinear wave equation utt =d ivS(Du) with the non-monotone stress-strain relation S = Dφ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young- measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young-measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and microstructures of a nonlinearly vibrating string. Mathematics Subject Classification. 35G25, 47J35, 65P25.

Finite Element Analysis of Microstructure for the Cubic to Tetragonal Transformation

Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry-related, rotationally invariant energy wells. We give estimates for the numerical approximation of a first-order laminate for such martensitic crystals. We give bounds for the L2 convergence of directional derivatives in the "twin" plane, for the L2 convergence of the deformation, for the weak convergence of the deformation gradient, for the convergence of the microstructure, and for the convergence of nonlinear integrals of the deformation gradient.