Quantitative Aspects of the Rigidity of Branching Microstructures in Shape Memory Alloys via H-Measures

Simon, Theresa M.

doi:10.1137/18m1220017

Cited by 6 publications

(1 citation statement)

References 16 publications

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“…Moreover, building on the first (more qualitative) step of investigating exactly stress-free structures, further quantitative properties of the resulting material patterns and the associated energies are studied in the literature. For instance, this includes the scaling and relaxation behaviour of the associated energies [8,[31][32][33][34][35][36][37][38][39][40][41][42] as well as the stability and fine-scale properties of these patterns [43][44][45][46]. We refer to [47] and [1] for a survey of these results.…”

Section: Relation To the Literaturementioning

confidence: 99%

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

Rüland

Simon

2023

J Elast

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We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components.

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Section: Relation To the Literaturementioning

confidence: 99%

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

Rüland

Simon

2023

J Elast

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Energy scaling laws for microstructures: from helimagnets to martensites

Ginster,

Zwicknagl

2023

Calc. Var.

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We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients $$({\pm }\,\alpha ,{\pm }\,\beta )$$ ( ± α , ± β ) , regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio $$\alpha /\beta $$ α / β and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

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Energy barriers for boundary nucleation in a two-well model without gauge invariances

Tribuzio,

Zemas

2024

Calc. Var.

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We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation (Knüpfer in Proc R Soc A Math Phys Eng Sci 467(2127): 695-717, 2011) for all rank-1 directions, but the ones normal to the confining hyperplane, where the scaling is as in a three-gradient problem in full space, resulting in a lower energy barrier (Rüland et al. J Nonlinear Sci 33(2): 25 2023).

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Quantitative Aspects of the Rigidity of Branching Microstructures in Shape Memory Alloys via H-Measures

Cited by 6 publications

References 16 publications

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

Energy scaling laws for microstructures: from helimagnets to martensites

Energy barriers for boundary nucleation in a two-well model without gauge invariances

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