A conservative formulation for plasticity

In this paper we propose a fully conservative form for the continuum equations governing rate-dependent and rate-independent plastic flow in metals. The conservation laws are valid for discontinuous as well as smooth solutions. In the rate-dependent case, the evolution equations are in divergence form, with the plastic strain being passively convected and augmented by source terms. In the rate-independent case, the conservation laws involve a Lagrange multiplier that is determined by a set of constraints; we show that Riemann problems for this system admit scale-invariant solutions.     

A variational principle for doubly nonlinear evolution

The weighted energy-dissipation principle stands as a novel variational tool for the study of dissipative evolution and has already been applied to rate-independent systems and gradient flows. We provide here an example of its application to a specific yet critical doubly nonlinear equation featuring a super-quadratic dissipation.     

On the Quasistatic Limit of Dynamic Evolutions for a Peeling Test in Dimension One

The aim of this paper is to study the quasistatic limit of a one-dimensional model of dynamic debonding. We start from a dynamic problem that strongly couples the wave equation in a time-dependent domain with Griffith’s criterion for the evolution of the domain. Passing to the limit as inertia tends to zero, we find that the limit evolution satisfies a stability condition; however, the activation rule in Griffith’s (quasistatic) criterion does not hold in general, thus the limit evolution is not rate-independent.     

The effect of precipitates on the evolution of a martensitic phase boundary

In this article, we study the role of defects in the quasistatic evolution of a martensitic phase boundary. The formulation of the model gives rise to a nonlocal free boundary problem, for which we present an implicit finite-time discretization. For an approximate model, assuming a nearly flat interface, we show that the phase boundary exhibits a sick-slip behavior in the presence of a heterogeneous environment, thus leading to a transition from viscous kinetics to rate-independent behavior. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Model for the Texture Evolution in Polycrystalline Materials

We study a polycrystalline material that undergoes a rigid-plastic rate-independent evolution. Of our particular interest is the creation of texture and the resulting qualitative change in the macroscopic material response: from isotropic to anisotropic. Texture is represented by a crystal orientation distribution function, which associates every spatial point with a probability measure on the set of rotations. We discuss an energetic formulation for the texture evolution in the context of generalized standard materials. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

The Gilbert equation with dry-friction-type damping

A modified Gilbert equation for micromagnetics is considered, obtained by augmenting the standard viscous-like dissipation with a rate-independent term. We prove existence of a weak solution both with and without viscous dissipation. By scaling time we show that, if the applied field varies very slowly, then gyromagnetic effects and viscous dissipation become negligible. In the infinitesimally-slow-loading limit, the system thus becomes fully rate-independent.     

Damage and delamination processes in thermo-viscoelastic materials

This contribution addresses several models for rate-independent damage and delamination processes in thermo-viscoelastic materials. In the spirit of continuum damage mechanics, both degradation phenomena are modeled by means of internal variables, governed by a rate-independent flow rule. The latter is coupled in a highly nonlinear way with the heat equation and the momentum balance for the displacements. We present a suitable weak formulation for this class of models, and discuss existence and approximation results in the framework of variational convergence. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical study of a bacteria–fish model with level of infection structure

In this paper, we propose a mathematical model to study a bacteria–fish system, based upon the interactions between Clostridium botulinum and tilapia, Oerochromis mossambicus. The fish population is divided into susceptible and infected, and the infected fish population is considered structured by the level of infection. The model is thus a system with the infected fish equation being an evolution equation, while those corresponding to the susceptible fish and bacteria in water are ordinary differential equations. The model is firstly transformed into a system with distributed delay for susceptible fish and bacteria and, further, under some assumptions, into a system with discrete delay. The study of this system gives us some results concerning the existence, uniqueness, positivity and boundedness of solutions; we also discuss the existence and stability of its equilibrium points, including conditions for the appearance of Hopf bifurcation. The theoretical results are illustrated by some numerical simulations.     

An evidential opinion dynamics model based on heterogeneous social influential power

This paper introduces an evidential opinion dynamics model combing Dempster–Shafer evidence theory to explore the opinion evolution. Our model is an improvement of the Continuous Opinions and Discrete Actions (CODA) model. The process of people updating their opinions is regarded as a decision making process. The unavoidable uncertainty of the opinion evolution is handled by Dempster–Shafer evidence theory. Thereby, a new opinion group, i.e.the neutrals, is introduced into the system. Simulations show the neutral group plays a significant part in the opinion evolution. An opinion ice-breaking process at the early stage of the opinion evolution is observed. It is found that the consensus is not always reached then clusters emerge instead, which depends on the proportions of supporters, neutrals and opponents. The individual’s influential power is taken into account. With the positive feedback mechanism of people’s influential power, the obtained results are in accordance with people’s daily cognition such as the Authority Effect and the Matthew Effect. The final influential power distribution of all individuals in our model presents power-law characteristic.     

A first regularity result for the Armstrong–Frederick cyclic hardening plasticity model with Cosserat effects

The purpose of this article is to prove the Hölder continuity up to the boundary of the displacement vector and the microrotation matrix for the quasistatic, rate-independent Armstrong–Frederick cyclic hardening plasticity model with Cosserat effects. This model is of non-monotone and non-associated type. In the case of two space dimensions we use the hole-filling technique of Widman and Morrey's Dirichlet growth theorem.     

Homogenization for rate-independent systems

This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with hardening. Using the recently developed methods of two-scale convergence, periodic unfolding and the new introduced one, periodic folding, we show that the homogenized problem can be represented as a two-scale limit which is again an energetic formulation, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit

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 tend to 0. An application to three-dimensional elastic-plastic systems with hardening is given.     

A Formulation of Additive Finite Anisotropic Thermo-Plasticity in Logarithmic Lagrangean Strain-Entropy Space

A phenomenological model of rate-independent thermo-plasticity at finite strains is discussed. The formulation is based on an additive decomposition of the strain measure into an elastic and plastic part as proposed by Green and Naghdi. A constitutive model in the logarithmic Lagrangean strain-entropy space is developed capable of modelling isotropic elastic and anisotropic plastic material behaviour. The staggered solution scheme for coupled thermo-mechanical problems employs an isentropic phase for the deformation and an iso-geometrical phase for the thermal field. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermodynamics and analysis of rate-independent adhesive contact at small strains

We address a model for adhesive unilateral frictionless Signorini-type contact between bodies of heat-conductive viscoelastic material, in the linear Kelvin-Voigt rheology, undergoing thermal expansion. The flow rule for debonding the adhesion is considered rate-independent and unidirectional, and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned.