Lusin's condition and the distributional determinant for deformations with finite energy

Henao, Duvan; Mora-Corral, Carlos

doi:10.1515/acv.2011.016

Cited by 39 publications

(48 citation statements)

References 31 publications

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“…In the mathematical theory that has developed, hole-creating deformations are obtained as singular minimizers of the stored energy of the material, whenever sufficiently large tensile loads are applied to the body (see, e.g., the seminal work by Ball [4], the review paper by Horgan and Polignone [20], or the more recent works by Müller and Spector [28], Sivaloganathan and Spector [36], Conti and De Lellis [10], and Henao and Mora-Corral [16,17,18]).…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Method for Cavitation in Nonlinear Elasticity

Henao

2011

Math. Models Methods Appl. Sci.

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This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

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Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Method for Cavitation in Nonlinear Elasticity

Henao

2011

Math. Models Methods Appl. Sci.

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“…for example, is also a problem with lack of compactness. Indeed, for a deformation u ∈ W 1,2 ∩ L ∞ (Ω, R 3 ), E(u) = 0 implies that Det Du = det Du (see Proposition 7.1 in [21] [21]) such that u n ⇀ u in W 1,2 and Det Du = det Du + π 6 δ p − π 6 δ O for two points P, O in the domain they considered, hence E(u) = 0. This example corresponds to what is called a dipole phenomenon, in harmonic maps theory for example.…”

Section: Introductionmentioning

confidence: 99%

On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting

Henao¹,

Rodiac²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Let Ω be a smooth bounded axisymmetric set in R 3 . In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis in [12, Section 6], a phenomenon of concentration of energy can occur preventing the strong convergence in W 1,2 (Ω, R 3 ) of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where Ω is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are W 1,2 -quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for 3D neo-Hookean materials.

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“…Using tools from geometric measure theory, they defined a concept of new surface created by the deformation, and showed that the corresponding energy due to cavitation or fracture was proportional to the area of the created surface. Moreover, using the work by Conti and De Lellis [14] to cover the critical exponent, they showed in [38] that, when restricted to Sobolev deformations (so that cavitation was permitted but fracture was not), the model turned out to be essentially equivalent to that of [51].…”

Section: Introductionmentioning

confidence: 99%

Quasistatic Evolution of Cavities in Nonlinear Elasticity

Mora-Corral¹

2014

SIAM J. Math. Anal.

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We show the well-posedness of a free-discontinuity model in nonlinear elasticity allowing for cavitation. The model, based on global minimization, takes into account the noninterpenetration of matter and the inhomogeneties of the material. Then we prove the existence of a quasi-static evolution corresponding to this variational model that takes into account the irreversibility of the process of the cavity formation.

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Lusin's condition and the distributional determinant for deformations with finite energy

Cited by 39 publications

References 31 publications

An Efficient Numerical Method for Cavitation in Nonlinear Elasticity

An Efficient Numerical Method for Cavitation in Nonlinear Elasticity

On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting

Quasistatic Evolution of Cavities in Nonlinear Elasticity

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