Nucleation Barriers for the Cubic‐to‐Tetragonal Phase Transformation

Knüpfer, Hans; Kohn, Robert V.; Otto, Félix

doi:10.1002/cpa.21448

Cited by 50 publications

(69 citation statements)

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“…11]). Using a similar setting as in [], we note that the energy can be in general expressed in the form

\begin{matrix} true \\ right {scriptE}_{elast} false[u false] = \int_{R^{3}} \underset{i = 0, 1, 2, 3}{trueprefixmin} \{0true ω^{(i)} + \sum_{α, β, γ, δ} {double-struckC}_{α β γ δ}^{(i)} {(() e false(u false) - e^{(i)})}_{α β} {(() e false(u false) - e^{(i)})}_{γ δ}\}, \end{matrix}

where the elastic rank‐4 tensors

C^{false(i false)}

are the elastic moduli of the austenite and the three martensite phases and where the constants

ω^{false(i false)}

are the corresponding energy densities of martensite and austenite. We assume that the first two variants of martensite have the same energy, i.e.…”

Section: Model and Statement Of Resultsmentioning

confidence: 99%

“…We show that in the setting described above, within a geometrically linear description and in the presence of an interfacial energy, the minimal energy scales like

V^{\frac{11}{13}}

in terms of the volume of the new phase if the volume is sufficiently large. In particular, the scaling of the energy is much higher than in the case of a nucleus that consists of three martensite variants as considered by Kohn and the two authors in []. From the physical perspective, the main difference is the lack of self‐accommodation in the setting considered in this paper which leads to a different structure (and higher energy) for minimizing configurations.…”

Section: Introductionmentioning

confidence: 83%

“…From the physical perspective, the main difference is the lack of self‐accommodation in the setting considered in this paper which leads to a different structure (and higher energy) for minimizing configurations. In this sense, Theorem (and its counterpart [, Theorem 1]) is a quantitative version of the result by Battacharya on self‐accommodation of martensite for the cubic‐to‐tetragonal phase transformation in the geometrically linear theory [, Chapter 9]. We note that the energetics for the nucleation at corners for the cubic‐to‐tetragonal transformation have been considered by Bella and Goldman in [] and by Ball and Koumatos in [].…”

Section: Introductionmentioning

confidence: 88%

“…In particular, the local interfacial part of the energy is naturally expressed in terms of physical variables while the elastic energy ‐ which is nonlocal as a function of the phases ‐ is naturally expressed in Fourier space. Similarly as in [], the identification of the two characteristic length scales R , L (related to optimal diameter and thickness of the configurations) of the problem is essential both in the proof of the upper bound as well as in the proof of the lower bound. In the proof, we use one‐dimensional (and hence anisotropic) mollification kernels on the length scale R and discrete derivatives on the length scale

L \geq R

.…”

Section: Introductionmentioning

confidence: 99%

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Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

Knüpfer

Otto

2018

Z Angew Math Mech

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The austenite‐to‐martensite phase transformation is characterized by the creation and growth of small nuclei of the new martensitic phase. Within a geometrically linear description and including an interfacial energy, we show that the minimal energy scales like V1113 in the volume V of the nucleus for V≫1 when primarily only two martensite variants are present. This complements the findings in [Knüpfer, Kohn, Otto, CPAM 2012] where it has been shown that the minimal energy of the transient nuclei has the lower scaling V911 if all three martensite variants are present. The higher scaling of the minimal energy is related to a failure of self‐accommodation. The proof requires new analytical ideas and tools.

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“…11]). Using a similar setting as in [], we note that the energy can be in general expressed in the form

\begin{matrix} true \\ right {scriptE}_{elast} false[u false] = \int_{R^{3}} \underset{i = 0, 1, 2, 3}{trueprefixmin} \{0true ω^{(i)} + \sum_{α, β, γ, δ} {double-struckC}_{α β γ δ}^{(i)} {(() e false(u false) - e^{(i)})}_{α β} {(() e false(u false) - e^{(i)})}_{γ δ}\}, \end{matrix}

where the elastic rank‐4 tensors

C^{false(i false)}

are the elastic moduli of the austenite and the three martensite phases and where the constants

ω^{false(i false)}

are the corresponding energy densities of martensite and austenite. We assume that the first two variants of martensite have the same energy, i.e.…”

Section: Model and Statement Of Resultsmentioning

confidence: 99%

“…We show that in the setting described above, within a geometrically linear description and in the presence of an interfacial energy, the minimal energy scales like

V^{\frac{11}{13}}

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 88%

L \geq R

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

Knüpfer

Otto

2018

Z Angew Math Mech

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“…These studies suggest a transition between uniform structures and the formation of microstructures. Some of the results have been extended also to the vectorvalued case (see [8,9,10,24,7,6,32,33]). Similar phenomena have been found for a variety of variational models, with applications including pattern formation in ferromagnets (see [15,13,39,34]), in type-1-superconductors (see [14,12,17]), and in thin compressed films (see [4,30,31,5,3]).…”

Section: Introductionmentioning

confidence: 99%

Deformation concentration for martensitic microstructures in the limit of low volume fraction

Conti¹,

Diermeier²,

Zwicknagl³

2017

Calc. Var.

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We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of Γ-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for SBV p functions whose jump sets have a prescribed orientation.

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Optimal Fine‐Scale Structures in Compliance Minimization for a Shear Load

Kohn

Wirth

2015

Comm Pure Appl Math

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We return to a classic problem of structural optimization whose solution requires microstructure. It is well‐known that perimeter penalization assures the existence of an optimal design. We are interested in the regime where the perimeter penalization is weak; i.e., in the effect of perimeter as a selection mechanism in structural optimization. To explore this topic in a simple yet challenging example, we focus on a two‐dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear. We consider the minimization of a weighted sum of volume, perimeter, and compliance (i.e., the work done by the load), focusing on the behavior as the weight ɛ of the perimeter term tends to 0. Our main result concerns the scaling of the optimal value with respect to ɛ. Our analysis combines an upper bound and a lower bound. The upper bound is proved by finding a near‐optimal structure, which resembles a rank‐2 laminate except that the approximate interfaces are replaced by branching constructions. The lower bound, which shows that no other microstructure can be much better, uses arguments based on the Hashin‐Shtrikman variational principle. The regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures. While perimeter has been considered as a selection mechanism in other problems involving microstructure, the example considered here is novel because optimality seems to require the use of two well‐separated length scales.© 2016 Wiley Periodicals, Inc.

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Nucleation Barriers for the Cubic‐to‐Tetragonal Phase Transformation

Cited by 50 publications

References 20 publications

Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

Deformation concentration for martensitic microstructures in the limit of low volume fraction

Optimal Fine‐Scale Structures in Compliance Minimization for a Shear Load

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