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Damage of nonlinearly elastic materials at small strain : existence and regularity results
Literaturverz. S. 31 In this paper an existence result for energetic solutions of rate-independent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [Mielke-Roubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ? W^1,r (Omega) with r>d for Omega ? R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.
Global existence for rate-independent gradient plasticity at finite strain
We provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables.
A metric approach to a class fo doubly nonlinear evolution euations and applications
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.
Thermally driven phase transformation in shape-memory alloys
This paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.
Complete damage evolution based on energies and stresses
The rate-independent damage model recently developed in Bouchitté, Mielke, Roubícek ``A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions
BV solutions and viscosity approximations of rate-independent systems
In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
Weak-convergence methods for Hamiltonian multiscale problems
We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients
Differential, energetic, and metric formulations for rate-independent processes
We consider different solution concepts for rate-independent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banach-space setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rate-independent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces.
Dispersive stability of infinite dimensional Hamiltonian systems on lattices
We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein-Gordon chain. If the nonlinearity is of degree higher than 4, then small localized initial data decay like in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different $ell^p$ spaces
Reverse approximation of energetic solutions to rate-independent processes
Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.