Kinetic relations and the propagation of phase boundaries in solids

Abeyaratne, Rohan; Knowles, Jonathan C.

doi:10.1007/bf00375400

Cited by 353 publications

(360 citation statements)

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“…Nonclassical solutions have the distinctive feature of being dynamically driven by small-scale effects such as diffusion, dispersion, and other high-order phenomena. Their selection requires an additional jump relation, called a kinetic relation, and introduced in the context of phase transition dynamics by Slemrod [35,36,13], Truskinovsky [37,38], Abeyaratne and Knowles [1,2], LeFloch [23], and Shearer [33,34], and developed in the more general context of nonlinear hyperbolic systems of conservation laws by LeFloch and collaborators [15]- [17], [3]- [5], [28]- [30], and [27]. See [24] for a review.…”

Section: State Of the Artmentioning

confidence: 99%

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Boutin¹,

Chalons²,

Lagoutìère³

et al. 2008

Interfaces Free Bound.

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We propose a new numerical approach to computing nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keeps nonclassical shock waves as sharp interfaces, unlike standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux functions.

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Section: State Of the Artmentioning

confidence: 99%

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Boutin¹,

Chalons²,

Lagoutìère³

et al. 2008

Interfaces Free Bound.

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show abstract

“…The problem of the velocity of the phase-transition front is well understood and extensively discussed in the case of displacive martensitic transformations in solids [6]- [10]. It is shown that the conventional theory, when supplemented by the appropriate kinetic relation, does indeed lead to a well-posed problem.…”

Section: Discussionmentioning

confidence: 99%

“…The notion of a kinetic relation is introduced by Abeyaratne and Knowles [6] following ideas from materials science. They have demonstrated its applicability in the thermoelastic setting and impact problems [7]- [9].…”

Section: Introductionmentioning

confidence: 99%

On the velocity of a moving phase boundary in solids

Berezovski

Maugin

2005

Acta Mechanica

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A general relationship between the driving force and the velocity of a moving phase boundary in thermoelastic solids is established on the basis of non-equilibrium jump relations at the discontinuity. The non-equilibrium jump relations are formulated in terms of contact quantities and local equilibrium fields. The contact quantities are introduced following ideas of the thermodynamics of discrete systems. It is shown that under certain simplifications the derived relationship can be reduced to a known kinetic relation.

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“…Remarkably, the velocity V depends only on the ends states through the thermodynamic driving force defined by the end states γ ± [17]. Further resulting kinetic relation V(f ) is monotone, passes smoothly through the origin and takes values only in the first and third quadrants (i.e.…”

Section: (C) Homogeneous Materialsmentioning

confidence: 99%

Length scales and pinning of interfaces

Tan

Bhattacharya

2016

Phil. Trans. R. Soc. A.

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One contribution of 11 to a theme issue 'Trends and challenges in the mechanics of complex materials' . The pinning of interfaces and free discontinuities by defects and heterogeneities plays an important role in a variety of phenomena, including grain growth, martensitic phase transitions, ferroelectricity, dislocations and fracture. We explore the role of length scale on the pinning of interfaces and show that the width of the interface relative to the length scale of the heterogeneity can have a profound effect on the pinning behaviour, and ultimately on hysteresis. When the heterogeneity is large, the pinning is strong and can lead to stick-slip behaviour as predicted by various models in the literature. However, when the heterogeneity is small, we find that the interface may not be pinned in a significant manner. This shows that a potential route to making materials with low hysteresis is to introduce heterogeneities at a length scale that is small compared with the width of the phase boundary. Finally, the intermediate setting where the length scale of the heterogeneity is comparable to that of the interface width is characterized by complex interactions, thereby giving rise to a non-monotone relationship between the relative heterogeneity size and the critical depinning stress.

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Kinetic relations and the propagation of phase boundaries in solids

Abstract: This paper investigates the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: the first

Cited by 353 publications

References 38 publications

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

On the velocity of a moving phase boundary in solids

Length scales and pinning of interfaces

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