Another construction of BV solutions to rate-independent systems

We study one kind of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order ε and then taking the limit ε → 0. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke, Rossi and Savare

BV solutions constructed by epsilon-neighborhood method

We study a certain class of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order $\varepsilon$ and then taking the limit $\varepsilon \to 0$. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke, Rossi and Savar\'e

A rate-independent gradient system in damage coupled with plasticity via structured strains

This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni [FRC10], including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers (cf., e.g., [Mie05, Mie11b])

A rate-independent model for the isothermal quasi-static evolution of shape-memory materials

This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.Comment: 33 pages, 3 figure

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

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

Globally stable quasistatic evolution in plasticity with softening

We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.Comment: 43 page

Weak solutions to rate-independent systems: Existence and regularity

Weak solutions for rate-independent systems has been considered by many authors recently. In this thesis, I shall give a careful explanation (benefits and drawback) of energetic solutions (proposed by Mielke and Theil in 1999) and BV solutions constructed by vanishing viscosity (proposed by Mielke, Rossi and Savare in 2012). In the case of convex energy functional, then classical results show that energetic solutions is unique and Lipschitz continuous. However, in the case energy functional is not convex, there is very few results about regularity of energetic solutions. In this thesis, I prove the SBV and piecewise C^1 regularity for energetic solution without requiring the convexity of energy functional. Another topic of this thesis is about another construction of BV solutions via epsilon-neighborhood method