Quasistatic Evolution of Cavities in Nonlinear Elasticity

Mora-Corral, Carlos

doi:10.1137/120872498

Cited by 7 publications

(7 citation statements)

References 65 publications

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“…The second possibility is more realistic, due to the dynamic and irreversible nature of the fracture processes and due to local vs. global minimization considerations. What prevents an arbitrarily large number of cavities from being created are the energies required for fracture (see [MC14]) and the tension associated to the presence of an ever increasing inner surface (which is especially large since what matters is its state in the deformed configuration, as pointed out by Müller & Spector [MS95]).…”

Section: Observe Thatmentioning

confidence: 99%

“…Lopez-Pamies, Idiart & Nakamura [LPIN11] and Negrón-Marrero & Sivaloganathan [NMS12] discussed the onset of cavitation under non-symmetric loadings. Mora-Corral [MC14] studied the quasistatic evolution of cavitation. We refer to [FGLP,KFLP18,PLPRC,KRCLP], the Introduction in [HS13], and the references therein for a more complete guide through the literature on this fracture mechanism.…”

Section: Introduction 1cavitation and Spherical Symmetrymentioning

confidence: 99%

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A lower bound for the void coalescence load in nonlinearly elastic solids

Cañulef-Aguilar

Henao

2019

Interfaces Free Bound.

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The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having n microvoids of radius ε. The constraint is added that the cavities should reach at least certain minimum areas υ 1 , . . . , υ n after the deformation takes place. They can be thought of as the current areas of the cavities during a quasistatic loading, the variational problem being the way to determine the state to be attained by the elastic body in a subsequent time step. It is proved that if each υ i is smaller than the area of a disk having a certain well defined radius, which is comparable to the distance, in the reference configuration, to either the boundary of the domain or the nearest cavity (whichever is closer), then there exists a range of external loads for which the cavities opened in the body are circular in the ε → 0 limit.

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Section: Observe Thatmentioning

confidence: 99%

Section: Introduction 1cavitation and Spherical Symmetrymentioning

confidence: 99%

A lower bound for the void coalescence load in nonlinearly elastic solids

Cañulef-Aguilar

Henao

2019

Interfaces Free Bound.

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“…We refer to [13,14,16,17,20,21] and references therein for developments concerning equilibrium or quasistatic cavitating solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Ball [2] in a seminal paper proposed to use continuum mechanics for modeling cavitation and used methods of the calculus of variations and bifurcation theory to construct cavitating solutions for the equilibrium version of (1.2): There is a critical stretching λ cr such that for λ < λ cr the homogeneous deformation is the only minimizer of the elastic stored energy; by contrast, for λ > λ cr there exist nontrivial equilibria corresponding to a (stress-free) cavity in the material with energy less than the energy of the homogenous deformation [2]. We refer to [17,20,21,13,16,14] and references therein for developments concerning equilibrium or quasistatic cavitating solutions.…”

Section: Introductionmentioning

confidence: 99%

On the Construction and Properties of Weak Solutions Describing Dynamic Cavitation

Miroshnikov

Tzavaras

2014

J Elast

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We consider the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions d = 2, 3 we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity. Our analysis treats both stress-free cavities and cavities with contents.

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“…In [31,32] the two approaches were shown to be conceptually equivalent in the Sobolev setting 2 . A variant of this strategy, analyzed by Sivaloganathan & Spector [53] in the static problem and by Mora-Corral [45] for quasistatic evolutions, is to assign a fixed quantized energy at any new cavitation point.…”

mentioning

confidence: 99%

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

Barchiesi¹,

Henao²,

Mora-Corral³

et al. 2021

Preprint

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In a previous work, Henao & Rodiac (2018) proved an existence result for the neo-Hookean energy in 3D for the essentially 2D problem of axisymmetric domains away from the symmetry axis. In the relaxation approach for the neo-Hookean energy in 3D, minimizers are found in the space of maps that can be obtained as the weak H 1 -limit of a sequence of diffeomorphisms; see Malý (1993). Among the singular maps contained in that H 1 -weak closure, a particularly upsetting pathology, constructed upon the dipoles found in harmonic map theory, was presented by Conti & De Lellis (2003). Their example involves a cavitation-type discontinuity around which the orientation is reversed, the cavity created there being furthermore filled with material coming from other part of the body. Since physical minimizers should not exhibit such behaviour, this raises the regularity question of proving that minimizers have Sobolev inverses (since the inverses of the harmonic dipoles have jumps across the created surface). The first natural step is to address the problem in the axisymmetric setting, without the assumption that the domain is hollow and at a distance apart of the symmetry axis. Attacking that problem will presumably demand a thorough analysis of the fine properties of singular maps in the H 1 sequential weak closure, as well as an explicit characterization of the relaxed energy functional. In this paper we introduce an explicit energy functional (which coincides with the neo-Hookean energy for regular maps and expresses the cost of a singularity in terms of the jump and Cantor parts of the inverse) and an explicit admissible space (which contains all weak H 1 limits of regular maps) for which we can prove the existence of minimizers. Chances to succeed in establishing that minimizers do not have dipoles are higher in this explicit alternative variational problem than when working with the abstract relaxation approach in the abstract space of all weak limits.Our candidate for relaxed energy has many similarities with the relaxed energy introduced by Bethuel-Brézis-Coron for a problem with lack of compactness in harmonic maps theory. The proof we present in this paper for the lower semicontinuity of the augmented energy functional, partly inspired by the prominent role played by conformality in that context, further develops the connection between the minimization of the 3D neo-Hookean energy with the problem of finding a minimizing smooth harmonic map from B 3 into S 2 with zero degree boundary data.

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Quasistatic Evolution of Cavities in Nonlinear Elasticity

Cited by 7 publications

References 65 publications

A lower bound for the void coalescence load in nonlinearly elastic solids

A lower bound for the void coalescence load in nonlinearly elastic solids

On the Construction and Properties of Weak Solutions Describing Dynamic Cavitation

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

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