Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.
We propose a model for phase transformations that are driven by changes in the temperature. We consider the temperature as a prescribed prescribed quantity like an applied load. The model is based on the energetic formulation for rate-independent systems and thus allows for finite-strain elasticity. Time-dependent Dirichlet boundary conditions can be treated by decomposing the deformation as a composition of a given deformation satisfying the time-dependent boundary conditions and a part coinciding with the identity on the Dirichlet boundary.
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.
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.
We show that the coupled balance equations for a large class of dissipative materials
can be cast in the form of GENERIC (General Equations for Non-Equilibrium
Reversible Irreversible Coupling). In dissipative solids, also called generalized standard
materials, the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows
for an efficient derivation of thermodynamically consistent coupled field equations,
while revealing additional underlying physical structures, like the role of the free energy
as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects.
Applications to large and small-strain thermoplasticity is given. Moreover, for the
quasistatic case, where the deformation can be statically eliminated, we derive a generalized
gradient structure for the internal variable and the temperature with a reduced
entropy as driving functional.
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
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.
We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems
with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces.
The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.
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.
A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.
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.
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.
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.
We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic $\lambda$-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
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.
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.
We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
We consider two particular systems of this type, namely, a
diffusion-reaction bipolar energy transport system, and a
drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitly calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.
We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon-densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.