Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

Davoli, Elisa; Friedrich, Manuel

doi:10.1007/s00526-020-1699-5

Cited by 16 publications

(53 citation statements)

References 62 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The construction fundamentally relies on the fact that the energy of an optimal sequence in (3.5) is concentrated asymptotically arbitrarily close to the interface. (Similar properties can be observed in related phase transition problems, see for example [12,13,15].) As a preliminary step, we need to show that in the definition of ψ we may replace cubes by rectangles.…”

Section: Cell Formula Part I: Relation Of L 1 -Convergence and Boundamentioning

confidence: 79%

“…On the other hand, this calls for carefully refined cut-off constructions since very small modifications in the configurations may induce a lot of energy. However, in contrast to [13,15], a cell problem with converging boundary data turns out to be insufficient in the presence of multiple grain boundaries. Thus, a further step is needed to show that they can be replaced by fixed boundary values.…”

Section: Introductionmentioning

confidence: 97%

“…In such cell problems, it is instrumental to pass from a mere L 1 -convergence to fixed boundary values in order to match thelim inf and -lim sup inequalities. Motivated by [5,25,46] for vectorial problems in liquid-liquid phase transitions and [13,15,36] in solid-solid phase transitions, we use a cut-off construction, the so-called fundamental estimate, to replace an asymptotic realization by the exact attainnment of converging boundary values in a first step. Here, our extremely brittle set-up on the one hand renders geometric rigidity estimates easier as compared to, for example, [13,15].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Friedrich

Kreutz

Schmidt

2021

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $$\Gamma $$ Γ -convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, that is, on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum–solid and solid–solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that, due to the extremely brittle setup, interpolating boundary layers near cracks are energetically not favorable.

show abstract

Section: Cell Formula Part I: Relation Of L 1 -Convergence and Boundamentioning

confidence: 79%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Friedrich

Kreutz

Schmidt

2021

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The effect of higher-order regularizations in material models has been a subject of intense study, and especially, weaker penalizations that do not involve the full Hessian of the deformations (in the second-order case), have come into the focus more recently [5,15,16]. In this spirit, we complement our model with a anisotropic partial regularization, precisely, a uniform bound on the second derivatives in the cross-section variables.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of deformation behavior in high-contrast fiber-reinforced materials: Rigidity and anisotropy

Engl¹,

Kreisbeck²,

Ritorto³

2021

Preprint

View full text Add to dashboard Cite

We identify the restricted class of attainable effective deformations in a model of reinforced composites with parallel, long, and fully rigid fibers embedded in an elastic body. In mathematical terms, we characterize the weak limits of sequences of Sobolev maps whose gradients on the fibers lie in the set of rotations. These limits are determined by an anisotropic constraint in the sense that they locally preserve length in the fiber direction. Our proof of the necessity emerges as a natural generalization and modification of the recently established asymptotic rigidity analysis for composites with layered reinforcements. However, the construction of approximating sequences is more delicate here due to the higher flexibility and connectedness of the soft material component. We overcome these technical challenges by a careful approximation of the identity that is constant on the rigid components, combined with a lifting in fiber bundles for Sobolev functions. The results are illustrated with several examples of attainable effective deformations. If an additional second-order regularization is introduced into the material model, only rigid body motions can occur macroscopically.

show abstract

“…Working with stored energy densities that depend on the Hessian of the deformations has proved successful in overcoming lack of compactness in a variety of applications; see, e.g., [5,21,27,36,38]. Very recently, there has been an effort towards weakening higher-order regularizations: It is shown in [7] that the full norm of the Hessian can be replaced by a control of its minors (gradient polyconvexity) in the context of locking materials; for solid-solid phase transitions, an anisotropic second-order penalization is considered in [23]. Along these lines, we introduce the regularized energies in (1.13) that penalize the variation of deformations only in the layer direction.…”

Section: Introductionmentioning

confidence: 99%

Homogenization in BV of a model for layered composites in finite crystal plasticity

Davoli

Ferreira

Kreisbeck

2019

Advances in Calculus of Variations

Self Cite

View full text Add to dashboard Cite

In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space BV of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.MSC (2010): 49J45 (primary); 74Q05, 74C15, 26B30

show abstract

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

Cited by 16 publications

References 62 publications

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Asymptotic analysis of deformation behavior in high-contrast fiber-reinforced materials: Rigidity and anisotropy

Homogenization in BV of a model for layered composites in finite crystal plasticity

Contact Info

Product

Resources

About