A survey on structured deformations

Baía, Margarida; Matías, José; Santos, Pedro Miguel

doi:10.11606/issn.2316-9028.v5i2p185-201

Cited by 5 publications

(6 citation statements)

References 23 publications

(37 reference statements)

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“…We describe here a few essential elements of the treatment of structured deformations by Choksi and Fonseca [4]. The articles [3], [1], [2], and [15] also provide summaries of that treatment, and [1], [2], and [15] provide alternative settings for structured deformations. The summary in [3] is intended for those interested in immediate applications in continuum mechanics, while [1] sets the stage for applications of structured deformations to thin bodies [11].…”

Section: Structured Deformations and Disarrangement Densities In The mentioning

confidence: 99%

Explicit formulas for relaxed disarrangement densities arising from structured deformations

Barroso

Matías

Morandotti

et al. 2017

Math. Mech. Compl. Sys.

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In this paper we derive explicit formulas for disarrangement densities of submacroscopic separations, switches, and interpenetrations in the context of firstorder structured deformations. Our derivation employs relaxation within one mathematical setting for structured deformations of a specific, purely interfacial density, and the formula we obtain agrees with one obtained earlier in a different setting for structured deformations. Coincidentally, our derivation provides an alternative method for obtaining the earlier result, and we establish new explicit formulas for other measures of disarrangements that are significant in applications.

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Section: Structured Deformations and Disarrangement Densities In The mentioning

confidence: 99%

Explicit formulas for relaxed disarrangement densities arising from structured deformations

Barroso

Matías

Morandotti

et al. 2017

Math. Mech. Compl. Sys.

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“…Using a Lusin-type result due to Alberti Here, ∇ f n denotes the absolutely continuous part of the distributional derivative of f n . (See also [4,15] for alternative settings for the Approximation Theorem, and [14] for the approximation of second order structured deformations). For each "simple" deformation f n it is natural to consider a total energy which is sum of a bulk term and of an interfacial term.…”

Section: Relaxed Bulk and Interfacial Energy Densities For Structuredmentioning

confidence: 99%

Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations

Owen

Paroni

2015

Arch Rational Mech Anal

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We establish the unexpected equality of the optimal volume density of total flux of a linear vector field (Formula presented.) and the least volume fraction that can be swept out by submacroscopic switches, separations, and interpenetrations associated with the purely submacroscopic structured deformation (i, IÂ +Â M). This equality is established first by identifying a dense set (Formula presented.) of (Formula presented.) matrices M for which the optimal total flux density equals |trM|, the absolute value of the trace of M. We then use known representation formulae for relaxed energies for structured deformations to show that the desired least volume fraction associated with (i, IÂ +Â M) also equals |trM|. We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of (Formula presented.) and the volume fraction swept out by submacroscopic separations alone, with common value (trM)+. Similarly, the optimal volume density of the negative part of the flux of (Formula presented.) and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)â\u88\u92

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“…Let x 0 ∈ S g be such that dµ dH N −1 S g (x 0 ) exists and is finite, denote by ν := ν g (x 0 ) and assume the point x 0 also satisfies 6) and…”

Section: Second-order Structured Deformations: Relaxation Integral Rmentioning

confidence: 99%

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

Barroso

Matías²,

Morandotti

et al. 2017

Arch Rational Mech Anal

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ABSTRACT. Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.

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A survey on structured deformations

Abstract: In this work we briefly describe the theory of (first-order) structured deformations of continua as well as the variational problems arising from this theory.2000 Mathematics Subject Classification. 4E15, 49J45, 28A33, 46T30.

Cited by 5 publications

References 23 publications

Explicit formulas for relaxed disarrangement densities arising from structured deformations

Explicit formulas for relaxed disarrangement densities arising from structured deformations

Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

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