Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

Conti, Sergio; Diermeier, Johannes; Koser, Melanie; Zwicknagl, Barbara

doi:10.1007/s10659-021-09862-4

Cited by 6 publications

(1 citation statement)

References 52 publications

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“…The lower bounds are more subtle -their proofs use rigidity theorems and/or convexity of the relaxed energy -but they make no use of the Euler-Lagrange equations that characterize critical points of our functional. It is, in fact, difficult to use the stationarity or minimality of elastic plus surface energy; however minimality has been used successfully in [12,13].…”

Section: Related Workmentioning

confidence: 99%

An Energy Minimization Approach to Twinning with Variable Volume Fraction

Conti

Kohn

Misiats

2022

J Elast

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In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call “twinning with variable volume fraction.” Two 1996 papers by Chopra et al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. The upper bounds are ansatz-based, providing full details about some (nearly) optimal microstructures. The lower bounds are ansatz-free, so they explain why no other arrangement of the two phases could be significantly better.

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Section: Related Workmentioning

confidence: 99%

An Energy Minimization Approach to Twinning with Variable Volume Fraction

Conti

Kohn

Misiats

2022

J Elast

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The Tapering Length of Needles in Martensite/Martensite Macrotwins

Conti

Zwicknagl

2023

Arch Rational Mech Anal

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We study needle formation at martensite/martensite macro interfaces in shape-memory alloys. We characterize the scaling of the energy in terms of the needle tapering length and the transformation strain, both in geometrically linear and in finite elasticity. We find that linearized elasticity is unable to predict the value of the tapering length, as the energy tends to zero with needle length tending to infinity. Finite elasticity shows that the optimal tapering length is inversely proportional to the order parameter, in agreement with previous numerical simulations. The upper bound in the scaling law is obtained by explicit constructions. The lower bound is obtained using rigidity arguments, and as an important intermediate step we show that the Friesecke–James–Müller geometric rigidity estimate holds with a uniform constant for uniformly Lipschitz domains.

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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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Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

Cited by 6 publications

References 52 publications

An Energy Minimization Approach to Twinning with Variable Volume Fraction

An Energy Minimization Approach to Twinning with Variable Volume Fraction

The Tapering Length of Needles in Martensite/Martensite Macrotwins

Energy scaling laws for microstructures: from helimagnets to martensites

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