On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure

In this work we are interested in one dimensional motion of a body capable of phase transitions. Such a problem is often described by the usual equations of elasticity with non-monotone stress-strain relations [1]. This system of conservation laws is elliptic in the interval of values of the strain where the stress function is monotone decreasing, and is hyperbolic otherwise. This interval is usually referred to as the unstable region. For such systems of mixed type, it has been shown in [2] that a constant state in the elliptic region is unstable in the sense that any local disturbance of a constant state, taken as initial data for the numerical solution, gives rise to a sequence of approximate solutions converging to a non-trivial solution of the Cauchy problem with the constant initial data. This non-trivial solution may be a distributional solution (also called weak solution), or, more generally, a measure-valued solution (abbreviated m-v solution), a concept first introduced by DiPerna [3] (see Definition 2.1). It was also shown that a new type of Riemann waves may appear as (measure-valued) solutions of Riemann problems with data inside the elliptic region. These waves are called oscillation waves, and are associated with persistent oscillations in the numerical solutions of the approximation scheme. To approach the solution of such a 2 × 2 system of conservation laws, we employ a scheme in which each numerical solution approximates the solution of a 3 × 3 system describing the one dimensional motion of a rate-type viscoelastic body (see [4]–[8]) with a Maxwell type viscosity, which is inversely proportional to the grid spacing of the numerical scheme and hence reduces to the original 2 × 2 system as the grid spacing tends to zero. This scheme, based on the method of characteristics proposed in [7], has the mathematical advantage of allowing an energy estimate, which is crucial in the proof of the existence of measure-valued solutions.

/pdf/oscillation-waves-in-riemann-problems-for-phase-transitions-4dbu691vlt.pdf

Oscillation waves in Riemann problems for phase transitions

The standard method of characteristics is applied to a hyperbolic system of partial differential equations describing the motion of a viscoelastic body. The equilibrium curve can be of the van der Waals type such that the system may describe phase transition phenomena. For isolated body problems, sufficient conditions are given on the time integration steps of the first and second numerical approximations such that the total energy of the numerical solution be a nonincreasing function of the number of time integration steps. This property is required since the total energy of the exact solution is a nonincreasing function of time as required by the second law of thermodynamics. Numerical examples of nonisolated body problems when the violations of the above conditions lead to numerical instabilities are presented elsewhere (see Introduction).



Die standard-Charakteristenmethode wird auf ein hyperbolisches System von partiellen Differentialgleichungen, die die Bewegung eines viskoelastischen Korpers beschreiben, angewandt. die Gleichgewichtskurve kann vom van der Waals-Typ sein, so das das System Phasenubergangsphanomene beschreiben kann. Fur Probleme isolierter Korper werden notwendige Bedingungen uber die Stufen der Zeitintegration der ersten und zweiten numerischen Approximationen gegeben, so das die Gesamtenergie der numerischen Losung eine nicht wachsende Funktion der Zahl der zeitlichen Intergrationsstufen ist. Diese Eigenschaft wird verlangt, weil die Gesamtenergie der exakten Losung eine nicht wachsende Funktion der Zeit ist, wie es der zweite Hauptsatz der Thermodynamik erfordert. Numerische Beispiele fur Problem isolierter Korper, wenn die Storungen obiger Bedingungen zu numerischen Instabilitaten fuhren, werden woanders vorgestellt (siehe Einleitung).

On the Method of Characteristics in Rate‐Type Viscoelasticity with Non‐Monotone Equilibrium Curve

Solids undergoing martensitic phase transformations are currently a subject of intense interest in mechanics. In spite of recent progress in understanding the absolute stability of elastic phases under applied loads, the presence of metastable configurations remains a major puzzle. In this overview we presented the simplest possible discussion of nucleation and growth phenomena in the framework of the dynamical theory of elastic rods. We argue that the resolution of an apparent nonuniqueness at the continuum level requires “de-homogenization” of the main system of equations and the detailed description of the processes at micro scale.

Nucleation and Growth in Elastodynamics

An initial and boundary value problem for a rate type elastic-visco-plastic material is considered. An Euler method as well as internal and external approximation techniques are used in order to obtain the numerical solution of the problem. The error is estimated over a finite time interval. In the viscoelastic case the error is estimated over an infinite time interval provided that the time step is less than a critical value. A one dimensional numerical example is successfully integrated using this numerical approach and a finite element technique.



Es wird ein Anfangswert- und ein Randwertproblem fur ein als elasto-visko-plastisch geltendes Material betrachtet. Eine Eulersche Methode sowie innere und ausere Approximationsverfahren werden benutzt, um die numerische Losung des Problems zu erhalten. Der Fehler wird uber ein endliches Zeitintervall abgeschatzt. Im viskoelastischen Fall wird der Fehler uber ein unendliches Zeitintervall abgeschatzt, vorausgesetzt, der Zeitschritt ist kleiner als ein kritischer Wert. Ein eindimensionales numerisches Beispiel wird erfolgreich integriert, wobei dieser numerische Ansatz und eine Finite-Elemente-Methode benutzt werden.

Error Estimates of an Euler Method for a Quasistatic Elastic-Visco-Plastic Problem

This paper deals with the standard method of characteristics for a system of partial differential equations encountered in viscoelasticity (and also in viscoplasticity, electric transmission lines, etc.). The second law of thermodynamics implies the existence of a nonlinear functional called the total energy, which for the exact solutions of the considered problems is a nonincreasing function of time. A sufficient condition on the integration step is given such that the total energy of the numerical solution be also a non-increasing function of the number of time integration steps. An example is given showing that, if this condition is violated then the numerical solution diverges in time.



Die vorliegende Arbeit beschaftigt sich mit Standard-Charakteristikenmethoden fur ein in der Viskoelastizitat (und auch in Viskoplastizitat, elektrischen Ubertragungsleitungen usw.) auftretendes partielles Differentialgleichungssystem. Der zweite Hauptsatz der Thermodynamik garantiert die Existenz eines Gesamt-Energie genannten nichtlinearen Funktionals, das fur die exakten Losungen des betrachteten Problems eine nicht-wachsende Funktion der Zeit ist. In einem Integrationsschritt wird eine hinreichende Bedingung dafur angegeben, das die Gesamt-Energie der numerischen Losung ebenfalls eine nichtwachsende Funktion der Anzahl der Zeit-Integrationsschritte ist. Es wird ein Beispiel angegeben, das zeigt, das die numerische Losung mit der Zeit divergiert, falls diese Bedingung verletzt ist.

I. Suliciu

Papers

On the Method of Characteristics in Rate‐Type Viscoelasticity