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This IMA Volume in Mathematics and its Applications SHOCK INDUCED TRANSITIONS AND PHASE STRUCTURES IN GENERAL MEDIA is based on the proceedings of a workshop that was an integral part of the 1990-91 IMA program on "Phase Transitions and Free Boundaries." The workshop focused on the thermodynamics and mechanics of dynamic phase transitions that are mainly inertially driven and brought together physicists, metallurgists, mathematicians, engineers, and molecular dynamicists with interests in these problems. Financial support of the National Science Foundation made the meeting pos­ sible. We are grateful to J .E. Dunn, Roger Fosdick, and Marshall Slemrod for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, .Jr. PREFACE When a body is subjected to a strong shock the material may suffer severe local structural changes. Rapid solidification, liquification, or vaporization can oc­ cur, and, moreover, complex structural heterogeneity is often left in the wake of the passing wave. Thus, inertially driven shock waves raise fundamental questions involving experiment, theory, and mathematics which bear on phase stability and metastability, as well as on reaction kinetics and appropriate measures of phase structure.

Inhaltsverzeichnis

Frontmatter

Nucleation, Kinetics and Admissibility Criteria for Propagating Phase Boundaries

Abstract
This paper reviews our recent studies on the nucleation and kinetics of propagating phase boundaries in an elastic bar and relates them to various admissibility criteria. First, we discuss how the field equations and jump conditions of the quasi-static theory of such a bar must be supplemented with additional constitutive information pertaining to the initiation and evolution of phase boundaries. The kinetic relation relates the driving traction f at a phase boundary to the phase boundary velocity ; thus f = φ (), where φ is a materially-determined function. The nucleation criterion specifies a critical value of f at an incipient phase boundary. We then incorporate inertial effects, and we find in the context of the Riemann problem that, as long as phase boundary velocities are subsonic, the theory again needs — and has room for — a nucleation criterion and a kinetic relation. Finally, we describe the sense in which each of three widely studied admissibility criteria for phase boundaries is equivalent to a specific kinetic relation of the form f = φ () for a particular choice of φ A kinetic relation based on thermal activation theory is also discussed.
Rohan Abeyaratne, James K. Knowles

On a Combustion-Like Model for Plastic Strain Localization

Abstract
A model is presented of the localization of plastic strain into an adiabatic shear band during rapid plastic shear. The model is compared with a well-known model of a confined thermal explosion, and some open problems are discussed.
Timothy J. Burns

Shear Waves and Phase Transformations

Abstract
Shock wave studies of phase transformations generally involve normal impact of plates, or other means for generating strong compression waves. For these waves, the high pressures and the resulting increases in temperature can cause a phase transformation. The shear stresses, while not zero, play a minor role in driving the transformation. However, in martensitic transformations and in twinning, shear stresses are understood to be important. This paper presents some recent developments in stress wave experiments that make it possible to study material response under large shear stresses and large shear strains. Two configurations are considered — both involving pressure-shear impact of parallel plates. In one, a symmetric impact configuration, the impacting plates are identical and the wave profiles generated in the target plate are used to investigate its dynamic response. In the other, a thin specimen plate is sandwiched between two hard plates so that the dynamic response of the specimen can be measured under nominally homogeneous states of stress and deformation.
Rodney J. Clifton

The Riemann Problem for Systems of Conservation Laws of Mixed Type

Without Abstract
Haitao Fan, Marshall Slemrod

On the Evolutionary Condition for Stationary Plane Waves in Inert and Reactive Substances

Abstract
Shock and detonation waves in phase-transforming substances often exhibit splitting instability. Application of the criterion of evolutionarity shows the conditions under which such instabilities can occur.
G. Richard Fowles

Dynamic Effects in Gradient Theory for Fluid Mixtures

Abstract
We propose a new method to study motions of mixtures in fluid interfaces. We extend the equations of equilibrium in interfaces and the results associated with traveling waves for van der Waals like fluids [21]. The Maxwell rule is extended to interfaces of fluid mixtures out of equilibrium. Formula like the Clapeyron relation are obtained for isothermal layers.
Henri Gouin

Continuum Limits of Discrete Gases II

Abstract
During his IMA lecture Greenberg presented a variety of new results for quasilinear hyperbolic conservation laws. The theme of that lecture was the existence of special systems of conservation laws which were capable of supporting time periodic solutions. Some of the results presented there were obtained jointly with Rascle [1] and will appear elsewhere and the final part of the lecture dealt with the model we discuss here. Our interest is in the continuum limits of flows associated with a discrete interacting particle system. This model was considered previously by Greenberg in [2] and [3]. Here we provide a thorough analysis of the weak or distributional limits of one particle distribution function associated with this gas and show that in certain cases the moments of the limit flow corresponding to density, momentum and pressure are weak solutions of the one–dimensional gas dynamics equations with a p = Sρ 3 equation of state. One of the solutions obtained via this limiting procedure is time periodic and has certain shock waves which would normally be regarded as inadmissible because the entropy S decreases across them. Anomalous solutions of the type described above are not necessarily the norm for this model. We also present an example where the limit flows consist of decaying N waves and all shocks are of the type that would normally be regarded as admissible, that is the entropy increases across them.
J. M. Greenberg, J. Hack

Nonequilibrium Molecular Dynamics

Abstract
The development of nonequilibrium molecular dynamics is described, with emphasis on massively-parallel simulations involving the motion of millions, soon to be billions, of atoms. Corresponding continuum simulations are also discussed.
William G. Hoover

Nonlinear Stability and Instability of Overcompressive Shock Waves

Abstract
Overcompressive shock waves arise in nonlinear elasticity, magnetohydrodynamics, multiphase flows and other physical situations. The inviscid models are non-strictly hyperbolic and there have been controversies on the admissibility of overcompressive shocks. We study the nonlinear stability of these waves for the viscous models and show that there are two distinct types. The first type is a combination of classical shock waves and is nonlinear stable with respect to the strength of dissipations. For sufficiently small dissipation a fixed perturbation will give rise to waves of distinct speeds. The second type is stable, but not uniformly with respect to the strength of dissipations. For sufficiently small dissipation a fixed perturbation will give rise to waves of distinct speeds.
Tai-Ping Liu

The Dissipation Topography Associated with Solutions to a Riemann Problem Involving Elastic Materials Undergoing Phase Transitions

Abstract
Riemann problems are examined for elastic solids in an isothermal framework for the case in which the material constitutive response allows for the possibility of change of type in the equations of motion. Solutions are examined in terms of a contact locus: stress vs. velocity on x = 0. A family of solutions will often exist for those values of initial data which permit change of type. Various criteria for resolving uniqueness involve additional assumptions upon the mechanical energy dissipation rate (entropy generation rate). In order to understand the efficacy of these criteria, the dissipation rate topography associated with the contact locus is examined. Conclusions regarding those solutions selected by the entropy rate admissibility criterion are obtained.
Thomas J. Pence

Kinks versus Shocks

Abstract
Localized phase transitions as well as shock waves can often be modeled by material discontinuities satisfying Rankine-Hugoniot (RH) jump conditions. The use of Maxwell, Gibbs-Thompson, Hertz-Knudsen, and similar (supplementary to RH) relations in the theory of dynamic phase changes suggest that the classical system of jump conditions is at least incomplete in the case of phase transitions. While the propagation of a shock wave is completely determined by the conservations laws, the boundary conditions of the problem and the condition that the entropy increases in the process, the same is not true for the propagation of phase boundaries. Additional condition must be added to the RH conditions in order to provide sufficient data for the unique determination of the transformation process. The necessity was tacitly assumed by those who attacked the calculation of the phase boundary velocity without even trying to determine this parameter from the conservation laws and boundary conditions alone.
In order to be able to point out the contrast between shock waves and phase boundaries we treat both of them on the basis of the same assumption that the process takes place over a zone of finite width and consider a rather general model of the internal structure of the interface with special emphasis on the interplay between dispersion and dissipation effects. Our extended model of the continuum, capable of describing a “thick” interface, incorporates a weak form of nonlocality together with a number of dissipative mechanisms. Analysis of a model-type solution of the structure problem clarifies the distinction between supersonic (shock) and subsonic(kink) discontinuities and provides explicit examples of additional jump relations in the case of kinks (which simulate subsonic phase boundaries).
It is emphasized, that an extended system of jump relations for kinks may depend on the ratios of internal scales of length introduced by a more detailed description. An original theory, which provides discontinuous solutions, must therefore be complemented by these nondimensional parameters, even though internal scales by themselves are considered to be zero in this theory.
L. Truskinovsky

Shear Strain Localization in Plastic Deformations

Abstract
Shear instabilities in the form of shear bands are often observed during high speed, plastic deformations of metals. According to one theory, their formation is attributed to effective strain-softening response, which results at high strain rates as the net outcome of the influence of thermal softening on the, normally, strain-hardening response of metals. In order to test the core instability mechanism set forth by such theories, we consider one-dimensional shear deformations of a material exhibiting strain softening and strain-rate sensitivity. The deformation is caused by either prescribed tractions or prescribed velocities. As it turns out, for moderate amounts of strain softening, strain-rate sensitivity exerts a dissipative effect and stabilizes the motion. However, once a threshold is exceeded, the response becomes unstable and shear strain localization can occur.
Athanasios E. Tzavaras
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