On Scaling Laws for Multi-Well Nucleation Problems Without Gauge Invariances

Rüland, Angkana; Tribuzio, Antonio

doi:10.1007/s00332-022-09879-6

J Nonlinear Sci

2023

DOI: 10.1007/s00332-022-09879-6

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On Scaling Laws for Multi-Well Nucleation Problems Without Gauge Invariances

Angkana Rüland

Antonio Tribuzio

Abstract: In this article, we study scaling laws for simplified multi-well nucleation problems without gauge invariances which are motivated by models for shape-memory alloys. Seeking to explore the role of the order of lamination on the energy scaling for nucleation processes, we provide scaling laws for various model problems in two and three dimensions. In particular, we discuss (optimal) scaling results in the volume and the singular perturbation parameter for settings in which the surrounding parent phase is in the… Show more

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Cited by 7 publications

References 52 publications

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$$\Gamma $$-Convergence for Plane to Wrinkles Transition Problem

Bella,

Marziani

2024

J Nonlinear Sci

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We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet and identify the $$\Gamma $$ Γ -limit as the sheet thickness goes to 0, thus extending the previous work of the first author (Bella et al. in Arch Ration Mech Anal 218(1):163–217, 2015). The limiting problem is scalar and convex, but constrained and posed for measures. For the $$\Gamma -\liminf $$ Γ - lim inf inequality, we first pass to quadratic variables so that the constraint becomes linear, and then obtain the lower bound using Reshetnyak’s theorem. The construction of the recovery sequence for the $$\Gamma -\limsup $$ Γ - lim sup inequality relies on mollification of quadratic variables and careful choice of multiple construction parameters. Eventually for the limiting problem, we show existence of a minimizer and equipartition of the energy for each frequency.

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$$\Gamma $$-Convergence for Plane to Wrinkles Transition Problem

Bella,

Marziani

2024

J Nonlinear Sci

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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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On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_{3}$ Structure

2023

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In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of $\mathcal{A}$ A -free differential inclusions and for a singularly perturbed $T_{3}$ T 3 structure for the divergence operator. In particular, in the compatible setting of the two-state problem we prove that all homogeneous, first order, linear operators with affine boundary data which enforce oscillations yield the typical $\epsilon ^{\frac{2}{3}}$ ϵ 2 3 -lower scaling bounds. As observed in Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015) for higher order operators this may no longer be the case. Revisiting the example from Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015), we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022); Garroni and Nesi (Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460(2046):1789–1806, 2004, https://doi.org/10.1098/rspa.2003.1249); Palombaro and Ponsiglione (Asymptot. Anal. 40(1):37–49, 2004), we discuss the scaling behavior of a $T_{3}$ T 3 structure for the divergence operator. We prove that as in Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022) this yields a non-algebraic scaling law.

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On Scaling Laws for Multi-Well Nucleation Problems Without Gauge Invariances

Cited by 7 publications

References 52 publications

$$\Gamma $$-Convergence for Plane to Wrinkles Transition Problem

$$\Gamma $$-Convergence for Plane to Wrinkles Transition Problem

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

On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_{3}$ Structure

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