Mathematical models of martensitic microstructure

Ball, J. M.

doi:10.1016/j.msea.2003.11.055

Cited by 45 publications

(47 citation statements)

References 40 publications

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“…In this sense, the optical micrographs of habit planes, macrotwins, or more complex microstructures in this alloy have been used several times to confirm the theoretical predictions of the mathematical theory of martensitic microstructures [10,11], which explains the formation of the observed microstructures by means of energy minimization and kinematic compatibility.…”

Section: Introductionmentioning

confidence: 92%

Mobile Interfacial Microstructures in Single Crystals of Cu–Al–Ni Shape Memory Alloy

Seiner

2015

Shap. Mem. Superelasticity

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This paper summarizes the main properties of the microstructures formed during reverse (austenite ? martensite) transitions in single crystals of the CuAl-Ni shape memory alloy, and discusses the relation between these properties and the mechanical stabilization effect. It is shown that all experimentally observed interfacial microstructures (X-and k-interfaces and their nonclassical equivalents) are not local minimizers of the quasistatic energy, and their formation is probably governed by requirements on mobility and dissipation. This conclusion is supported by finite elements models, and acoustic emission measurements.

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Section: Introductionmentioning

confidence: 92%

Mobile Interfacial Microstructures in Single Crystals of Cu–Al–Ni Shape Memory Alloy

Seiner

2015

Shap. Mem. Superelasticity

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“…Solid-solid phase transitions are a classical model problem in the variational study of pattern formation in solids, both in the context of the theory of relaxation and in the study of singularly perturbed problems. Their study has led on the one side to many important abstract developments in the calculus of variations, on the other side to a mathematical explanation of the physical behavior of shape-memory alloys and other materials with peculiar properties [4,5,9,47,3,41,34]. The basic model is a vectorial, nonconvex variational problem, where the integrand depends on the gradient of the deformation field.…”

Section: Introductionmentioning

confidence: 99%

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

et al. 2020

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We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the shape of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for which we have matching upper and lower bounds.

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“…Note that (6) implies that detν x = det U 1 for any zero-energy microstructure. (See [9] for a description of gradient Young measures in the context of the nonlinear elasticity model for martensite.) In the case of cubic symmetry, and the absence of boundary conditions on ∂Ω, there always exist such zero-energy microstructures.…”

Section: Microstructure Of Polycrystalsmentioning

confidence: 99%

Geometry of polycrystals and microstructure

Ball

Carstensen

2015

MATEC Web of Conferences

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Abstract. We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-totetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.

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Mathematical models of martensitic microstructure

Cited by 45 publications

References 40 publications

Mobile Interfacial Microstructures in Single Crystals of Cu–Al–Ni Shape Memory Alloy

Mobile Interfacial Microstructures in Single Crystals of Cu–Al–Ni Shape Memory Alloy

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

Geometry of polycrystals and microstructure

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