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

Conti, Sergio; Diermeier, Johannes; Melching, David; Zwicknagl, Barbara

doi:10.1051/cocv/2020020

Cited by 19 publications

(14 citation statements)

References 50 publications

(98 reference statements)

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“…Recalling ( 22) one obtains (20) together with ( 21), (23) for the bottom and top area and |f j (h j )| → 0 as j → ∞. This concludes the proof of strong convergence.…”

Section: Outline Of the Main Argumentssupporting

confidence: 69%

“…In their model one partial derivative is constrained to take only two values, leading to the characteristic non-convexity. Their work has been meanwhile generalized to different volume fractions [19,59], to vectorial settings in the context of linearized elasticity [3,10,11,23,36,38,39,49,52] and to geometrically nonlinear formulations [12,50]. These vectorial generalizations have confirmed that the Kohn-Müller scalar model indeed captures the correct scaling of the energy.…”

Section: Introductionmentioning

confidence: 90%

“…In the case of equal volume fractions, θ = 1 2 , the present model reduces to the one originally studied by Kohn and Müller. A variant of I in which the Dirichlet boundary condition is replaced by the elastic energy of austenite outside R Lx ,Ly has been studied in [19], a vectorial extension in [23], resulting in a rich phase diagram. The limit ε ∼ θ 2 → 0 of a similar model was addressed in [21].…”

Section: Introductionmentioning

confidence: 99%

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

Conti

Diermeier

Koser

et al. 2021

J Elast

Self Cite

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We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as $\varepsilon ^{2/3}$ ε 2 / 3 as well as the one where the energy scales as $\varepsilon ^{1/2}$ ε 1 / 2 .

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“…Recalling ( 22) one obtains (20) together with ( 21), (23) for the bottom and top area and |f j (h j )| → 0 as j → ∞. This concludes the proof of strong convergence.…”

Section: Outline Of the Main Argumentssupporting

confidence: 69%

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Conti

Diermeier

Koser

et al. 2021

J Elast

Self Cite

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“…Our result should be viewed in the context of scaling laws in the calculus of variations in general and more specifically in the modelling of shape-memory alloys and related phase transformation problems (see [37] and [45] for surveys on this). In the context of the modelling of shape-memory alloys, scaling laws, providing some insights on the possible behaviour of energy minimizers, have been deduced in various settings [3,5,6,8,9,14,15,20,[34][35][36][38][39][40][41][42]55]. For certain models, in subsequent steps, even finer properties (such as for instance almost periodicity results) have been derived [13].…”

Section: Relation To the Literaturementioning

confidence: 99%

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

Rüland

Tribuzio

2021

Arch Rational Mech Anal

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In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185–207, 1997), Chipot (Numer Math 83(3):325–352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.

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“…Particularly, see [21] for a simplified scalar model in SBV addressing the low volume-fraction of one phase, and dealing with the problem of internal jumps. (We also refer to [33] for some extensions to a vectorial model in the geometrically linear setting, and to [24] for a corresponding scaling law in the case of a martensitic nucleus embedded in an austenitic matrix.) 3.4.…”

Section: Characterization Of Admissible Limiting Triplesmentioning

confidence: 99%

Two-well linearization for solid-solid phase transitions

Davoli,

Friedrich

2020

Preprint

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In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV ). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus not being detected by the limiting deformations. By Γ-convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [32] for a model with anisotropic second-order perturbation.

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

Cited by 19 publications

References 50 publications

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

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

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

Two-well linearization for solid-solid phase transitions

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