Roughening Instability of Broken Extremals

Grabovsky, Yury; Truskinovsky, Lev

doi:10.1007/s00205-010-0377-8

Cited by 21 publications

(17 citation statements)

References 62 publications

(78 reference statements)

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“…We observe that conventional variations, linking both sides of a jump discontinuity and leading to Maxwell condition [8] or roughening instability condition [11], represent combinations of weak and strong variations. This creates unnecessary coupling and obscures the strong character of the minimizer under consideration.…”

Section: Interchange Driving Forcementioning

confidence: 88%

“…In order to prove these algebraic relations only the Weierstrass condition (3.13) will be needed. • Interface roughening condition [11] [[P ]] T a = 0. Next we prove a differentiability lemma that guarantees the existence of rank-1 directional derivatives of quasiconvex and rank-1 convex envelopes at "marginally stable" deformation gradients [12].…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Normality Condition in Elasticity

Grabovsky

Truskinovsky

2014

J Nonlinear Sci

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Strong local minimizers with surfaces of gradient discontinuity appear in variational problems when the energy density function is not rank-one convex. In this paper we show that stability of such surfaces is related to stability outside the surface via a single jump relation that can be regarded as interchange stability condition. Although this relation appears in the setting of equilibrium elasticity theory, it is remarkably similar to the well known normality condition which plays a central role in the classical plasticity theory. arXiv:1309.4708v1 [math.OC]

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Section: Interchange Driving Forcementioning

confidence: 88%

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Normality Condition in Elasticity

Grabovsky

Truskinovsky

2014

J Nonlinear Sci

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“…These conditions are discussed in detail in [54]. One important example is the roughening equilibrium equation [51]…”

Section: Classical Nucleationmentioning

confidence: 99%

“…This system places F on the jump set J (see [51]). The second rank laminate ν 2 is obtained from ν 1 by means of lamination in the sense of Definition 3.12.…”

Section: Microstructure Nucleationmentioning

confidence: 99%

Marginal Material Stability

Grabovsky

Truskinovsky

2013

J Nonlinear Sci

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Marginal stability plays an important role in nonlinear elasticity because the associated minimally stable states usually delineate failure thresholds. In this paper we study the local (material) aspect of marginal stability. The weak notion of marginal stability at a point, associated with the loss of strong ellipticity, is classical. States that are marginally stable in the strong sense are located at the boundary of the quasi-convexity domain and their characterization is the main goal of this paper. We formulate a set of bounds for such states in terms of solvability conditions for an auxiliary nucleation problem formulated in the whole space and present nontrivial examples where the obtained bounds are tight.

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“…The external PTZ-boundaries are the surfaces of the nucleation of new phase plane layers. In the papers [45,46] it was proved that belonging strains to the external PTZ-boundaries is a necessary stability condition.…”

Section: Introductionmentioning

confidence: 99%

Phase transformations surfaces and exact energy lower bounds

Antimonov

Cherkaev

Freidin

2016

International Journal of Engineering Science

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The paper investigates two-phase microstructures of optimal 3D composites that store minimal elastic energy in a given strain field. The composite is made of two linear isotropic materials which differ in elastic moduli and selfstrains. We find optimal microstructures for all values of external strains and volume fractions of components. This study continues research by Gibiansky and Cherkaev [41,42] and Chenchiah and Bhattacharya [15]. In the present paper we demonstrate that the energy is minimized by that laminates of various ranks. Optimal structures are either simple laminates that are codirected with external eigenstrain directions, or inclined laminates, direct and skew second-rank laminates and third-rank laminates. These results are applied for description of direct and reverse transformations limit surfaces in a strain space for elastic solids undergoing phase transformations of martensite type. The surfaces are computed as the values of external strains at which the * Corresponding author optimal volume fraction of one of the phases tends to zero. Finally, we compare the transformation surfaces with the envelopes of the nucleation surfaces constructed earlier for nuclei of various geometries (planar layers, elliptical cylinders, ellipsoids). We show the energy equivalence of the cylinders and direct second-rank-laminates, ellipsoids and third-rank laminates. We note that skew second-rank laminates make the nucleation surface convex function of external strain, and they do not correspond to any of the mentioned nuclei.

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Roughening Instability of Broken Extremals

Cited by 21 publications

References 62 publications

Normality Condition in Elasticity

Normality Condition in Elasticity

Marginal Material Stability

Phase transformations surfaces and exact energy lower bounds

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