Theory of diffusionless phase transitions

James, Richard D.; Kinderlehrer, David

doi:10.1007/bfb0024935

Cited by 58 publications

(22 citation statements)

References 43 publications

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“…A recently developed geometrically nonlinear theory predicts the martensitic microstructure by energy minimization, see for example [9,10,11,13,15,20,21,24,27,34] and the references therein. In this theory, the free energy density of the crystal below the transformation temperature is minimized on several energy wells representing the martensitic variants.…”

Section: Introductionmentioning

confidence: 99%

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Bhattacharya

Luskin

1999

Archive for Rational Mechanics and Analysis

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Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222) . The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank-one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers.

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Section: Introductionmentioning

confidence: 99%

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Bhattacharya

Luskin

1999

Archive for Rational Mechanics and Analysis

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“…Nonconvex variational problems often arise in the modeling of the equilibria of crystals or other ordered states [2][3][4][5][6][7][8][9], [11][12][13][14][15][16][17][18][19][20], For instance, the free energy for a solid crystal which has symmetry-related (martensitic) variants will have multiple, distinct energy wells. These variational problems may fail to attain a minimum value for any admissible deformation.…”

Section: Introductionmentioning

confidence: 99%

“…Rather, the deformation gradients of minimizing sequences can have oscillations which do not converge strongly enough to evaluate nonlinear integrals of the deformation gradient such as the bulk energy functional. Nevertheless, the solution to these variational problems can be described in terms of an appropriate mathematical description of microstructure such as the Young measure [2][3][4][5], [15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…A continuum theory to describe the equilibria of crystals such as CuZn, CuAINi, NiAl, and InTl which have symmetry-related variants has been recently developed [2][3][4][5][6][7][8][9], [11][12][13][14][15][16][17][18][19][20]. A corresponding theory of microstructure using the concept of the Young measure, or parametrized measure, has also been recently developed to describe solutions to the variational problems given by the above continuum theory [2][3][4][5], [15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…A corresponding theory of microstructure using the concept of the Young measure, or parametrized measure, has also been recently developed to describe solutions to the variational problems given by the above continuum theory [2][3][4][5], [15][16][17][18][19][20]. This theory of microstructure gives a calculus for the computation of macroscopic properties of the crystals.…”

Section: Introductionmentioning

confidence: 99%

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Optimal-Order Error Estimates for the Finite Element Approximation of the Solution of a Nonconvex Variational Problem

Collins¹,

Luskin²

1991

Mathematics of Computation

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Abstract. Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational problems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gradient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.

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A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

Friesecke

James

Müller

2002

Comm Pure Appl Math

632

1,023

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The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a -limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → R n , U ⊂ R n . We show that the L 2 -distance of ∇v from a single rotation matrix is bounded by a multiple of the L 2 -distance from the group SO(n) of all rotations.

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Theory of diffusionless phase transitions

Cited by 58 publications

References 43 publications

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Optimal-Order Error Estimates for the Finite Element Approximation of the Solution of a Nonconvex Variational Problem

A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

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