Local minimizers and planar interfaces in a phase-transition model with interfacial energy

Ball, J. M.; Crooks, Elaine

doi:10.1007/s00526-010-0349-8

Cited by 31 publications

(33 citation statements)

References 36 publications

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“…Thus, apparently for any of the accepted notions of local minimizer, infinite hysteresis is predicted. This dominance of interfacial energy at small scales, which overstabilizes linear deformations, also occurs when gradient models of interfacial energy are combined with the bulk energies studied here, as shown in [12]. Consider a frame-indifferent energy density W τ ∈ C 2 (M 3×3 + ), continuous in τ and satisfying W τ (A) → ∞ as det A → 0, and having positive-definite linearized elasticity tensor at I.…”

Section: Perspective On Metastability and Hysteresismentioning

confidence: 65%

Incompatible Sets of Gradients and Metastability

Ball

James

2015

Arch Rational Mech Anal

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We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an L 1 neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.

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Section: Perspective On Metastability and Hysteresismentioning

confidence: 65%

Incompatible Sets of Gradients and Metastability

Ball

James

2015

Arch Rational Mech Anal

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“…One possibility to incorporate this restriction is to let the energy depend (on parts) of the second gradient of the deformation. The canonical way to do so is to let the energy density be a convex function of the second gradient (cf., e.g., [6,9,41,48,54]). Yet, here we propose a different approach inspired by the notion of polyconvexity due to Ball [4].…”

Section: Gradient Polyconvexitymentioning

confidence: 99%

“…The contribution of the higher gradient is usually associated to interfacial energies, as in e.g. [6,9,41,48,54] which work with an energy functional of the type J(y) = Ω (w(∇y(x)) + γ|∇ 2 y(x)| d )dx, (1.9) for some γ > 0 and d > 1. Now, if d > n, any deformation of finite energy will satisfy L(F ) ≤ 0 with L from (1.6) and depending only on the energy bound by Sobolev embedding.…”

Section: Introductionmentioning

confidence: 99%

A note on locking materials and gradient polyconvexity

Benešová

Kružík

Schlömerkemper

2018

Math. Models Methods Appl. Sci.

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We use gradient Young measures generated by Lipschitz maps to define a relaxation of integral functionals which are allowed to attain the value +∞ and can model ideal locking in elasticity as defined by Prager in 1957. Furthermore, we show the existence of minimizers for variational problems for elastic materials with energy densities that can be expressed in terms of a function being continuous in the deformation gradient and convex in the gradient of the cofactor (and possibly also the gradient of the determinant) of the corresponding deformation gradient. We call the related energy functional gradient polyconvex. Thus, instead of considering second derivatives of the deformation gradient as in second-grade materials, only a weaker higher integrability is imposed. Although the second-order gradient of the deformation is not included in our model, gradient polyconvex functionals allow for an implicit uniform positive lower bound on the determinant of the deformation gradient on the closure of the domain representing the elastic body. Consequently, the material does not allow for extreme local compression.

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“…As is also well-known, these microstructures can develop with arbitrary fineness unless the diffuse interfaces between sub-regions of a single variant are penalized. This is done by inclusion of a dependence on ∇ 0 χ for regularization [Ball andCrooks, 2011, Rudraraju et al, 2014]. This ensures physically meaningful solutions and mathematical well-posedness.…”

Section: Free Energy Density Functionsmentioning

confidence: 99%

A variational treatment of material configurations with application to interface motion and microstructural evolution

Teichert

Rudraraju

Garikipati

2017

Journal of the Mechanics and Physics of Solids

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We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. These descriptions also lead to parameterizations of the free energies for the two cases, from which variational treatments can be developed and equilibrium conditions obtained. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with * Corresponding Author arXiv:1608.05355v2 [cond-mat.soft] 18 Nov 2016 respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (Arch. Rat. Mech. Anal., 11, 1962, 385-414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. The coupled system of equations governs the configurational change in crystal structure, and elastic deformation driven by elastic, Eshelbian, and configurational stresses. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

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Local minimizers and planar interfaces in a phase-transition model with interfacial energy

Cited by 31 publications

References 36 publications

Incompatible Sets of Gradients and Metastability

Incompatible Sets of Gradients and Metastability

A note on locking materials and gradient polyconvexity

A variational treatment of material configurations with application to interface motion and microstructural evolution

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