Configurational Forces in Continuous Theories of Elastic Ferroelectrics

Kalpakides, V.K.; Arvanitakis, Antonios I.

doi:10.1007/978-90-481-3447-2_21

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…A similar analysis was performed by Kalpakides and Arvanitakis [14,15] in the context of electro-elasticity by reproducing successfully the microstructure of two-dimensional ferroelectric materials after using two level set functions. It becomes evident that as in the sharp interface theory, the level set formulation provides us with the potential to determine only the driving force within the transition layer.…”

Section: Material-inhomogeneity Forcesmentioning

confidence: 78%

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The concept of material forces in phase transition problems within the level-set framework

Arvanitakis

Kalpakides

2011

Arch Appl Mech

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The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine-Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.

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Section: Material-inhomogeneity Forcesmentioning

confidence: 78%

“…The force terms on the right-hand side of Eq. (14) are material forces, where with f int we denote the material internal forces and with f inh the material-inhomogeneity forces defined by the relations…”

Section: Sharp Interface Theorymentioning

confidence: 99%

The concept of material forces in phase transition problems within the level-set framework

Arvanitakis

Kalpakides

2011

Arch Appl Mech

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“…The velocity in oxidation problems modeled by Rao et al [2000], Rao and Hughes [2000], Garikipati and Rao [2001] is based on material composition. Finally, we note that Kalpakides and Arvanitakis [2009] used a velocity based on configurational forces to model ferroelastic materials, although it is arrived at differently than in the present work.…”

Section: Introductionmentioning

confidence: 98%

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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An implicit representation of phase interface motion with internal variables

Arvanitakis

2018

Arch Appl Mech

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Configurational Forces in Continuous Theories of Elastic Ferroelectrics

Cited by 4 publications

References 10 publications

The concept of material forces in phase transition problems within the level-set framework

The concept of material forces in phase transition problems within the level-set framework

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

An implicit representation of phase interface motion with internal variables

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