74C05 Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials)
We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.
This paper focuses on
rate-independent damage in elastic bodies. Since the driving energy is nonconvex,
solutions may have jumps as a function of time, and in this situation it is known that the classical concept
of energetic solutions for rate-independent systems
may fail to accurately describe the
behavior of the system at jumps.
Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling
and approximate the model by its viscous regularization.
In fact, the analysis of the latter PDE system presents
remarkable difficulties, due to its highly nonlinear character.
We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with
careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable.
Hence, for the viscous problem we conclude the existence of weak solutions satisfying a
suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis.
The latter leads to the notion of (weak) parameterized
solution to our rate-independent system,
which encompasses the influence of viscosity in the description of the jump regime.
Global higher integrability of minimizers of variational problems with mixed boundary conditions
(2012)
We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order $p+\epsilon$ for a uniform $\epsilon >0$. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.
This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
We consider rate-independent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a generalized Prandtl-Ishlinskii operator.
This paper deals with error estimates for space-time
discretizations of 3D model for isothermal stress-induced
transformations in shape-memory materials. After recalling
existence and uniqueness results, a fully-discrete approximation is presented and an explicit space-time
convergence rate of order is derived.
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.
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.
Quasistatic small-strain plasticity in the limit of small hardening and its numerical approximation
(2011)
The quasistatic rate-independent evolution of the Prager-Ziegler-type model of linearized plasticity with hardening is shown to converge to the rate-independent evolution of the Prandtl-Reuss elastic/perfectly plastic model. Based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of Gamma-convergence for rate-independent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A two-sided energy inequality is a posteriori verified to document experimental convergence rates.
This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.
This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.
This paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.
A general abstract approximation scheme for rate-independent processes in the
energetic formulation is proposed and its convergence is proved under various
rather mild data qualifications. The abstract theory is illustrated on several
examples: plasticity with isotropic hardening, damage, debonding,
magnetostriction, and two models of martensitic 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.
We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and
Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the
remaining faces. Under natural smoothness assumptions on the data we obtain
$u\in L^\infty((0,T);H^{3/2-\delta}(\Omega))$ for the displacements and
$z\in L^\infty((0,T);H^{1/2-\delta}(\Omega))$ for the internal variables.
The proof is based on a difference quotient technique and a reflection argument.
In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general
three-dimensional system. The plastic behavior is modeled using the classical
three-dimensional von Mises plasticity model. It turns out that this single-yield model leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model,
given by a hysteresis operator of Prandtl-Ishlinskii type whose density
function can be determined explicitly. This result indicates that the use
of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity
is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness
of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.