Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks

Shcherbakov, V. V.

doi:10.1016/j.nonrwa.2021.103505

Cited by 11 publications

(4 citation statements)

References 43 publications

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“…In the literature, a rigorous mathematical derivation of the energy release rate for solid mechanics models predominantly relies on the technique of shape sensitivity analysis. When it comes to curvilinear cracks whose faces are subjected to non-penetration constraints, this poses certain technical difficulties that were examined with the help of variational and primal–dual approaches in [11,26–28] for linear elastic materials with quadratic energy densities, in [29] for a Timoshenko elastic plate model, in [30,31] for Kirchhoff–Love elastic plate models, and in [7] for an elastic material with strictly convex energy density satisfying the standard (two-sided) p-growth condition.…”

Section: Setting Of the Problemmentioning

confidence: 99%

Fully discrete approximation schemes for rate-independent crack evolution

Knees

Schröder

Shcherbakov

2022

Phil. Trans. R. Soc. A.

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Over the past two decades, several distinct solution concepts for rate-independent evolutionary systems driven by non-convex energies have been suggested in an attempt to model properly jump discontinuities in time. Much attention has been paid in this context to the modelling of crack propagation. This paper studies two fully discrete (in time and space) approximation schemes for the rate-independent evolution of a single crack in a two-dimensional linear elastic material. The crack path is assumed to be known in advance, and the evolution of the crack tip along it relies on the Griffith theory. On the time-discrete level, the first scheme is based on local minimization, whereas the second scheme is a regularized version of the first one. The crucial feature of the schemes is their adaptive time-stepping nature, with finer time steps at those points where the evolution of the crack tip might develop a discontinuity. The set of discretization parameters includes the mesh size, crack increment, locality parameter and regularization parameter. In both cases, we explore the interplay between the discretization parameters and derive sufficient conditions on them ensuring the convergence of discrete interpolants to parametrized balanced viscosity solutions of the continuous model. To illustrate the performance of the approximation schemes, we support our theoretical analysis with numerical simulations. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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Section: Setting Of the Problemmentioning

confidence: 99%

Fully discrete approximation schemes for rate-independent crack evolution

Knees

Schröder

Shcherbakov

2022

Phil. Trans. R. Soc. A.

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“…Starting from the well-known work by G. Fichera [1], a huge number of results were obtained related to such problems, see [2] and the references therein. The last two decades are characterized by activity in the crack theory with nonlinear conditions imposed at the crack faces [3][4][5][6][7][8]. Moreover, this also concerns elastic bodies with inclusions of different nature in the cases with delaminations, i.e.…”

Section: Introductionmentioning

confidence: 99%

Thin inclusion at the junction of two elastic bodies: non-coercive case

Khludnev

2024

Phil. Trans. R. Soc. A.

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This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity. This article is part of the theme issue ‘Non-smooth variational problems with applications in mechanics’.

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“…The variational formulation (1.1) was employed earlier [28,30] for the description of nonpenetrating cracks (implying that Σ t has singularity at the crack tip), and supported by appropriate numerical methods [23,29]. The surface energies were specified taking into account for adhesion [15] and cohesion [32,41], where the latter results in non-smooth and non-convex functionals E. For non-differentiable energies, see respective hemi-VI approaches in [19,42]. We cite [43] for the concept of the conical differential of a solution to the Signorini VI.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian approach and shape gradient for inverse problem of breaking line identification in solid: contact with adhesion

Kovtunenko

2023

Inverse Problems

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A class of inverse identification problems constrained by variational inequalities is studied with respect to its shape differentiability. The specific problem appearing in failure analysis describes elastic bodies with a breaking line subject to simplified adhesive contact conditions between its faces. Based on the Lagrange multiplier approach and smooth Lavrentiev penalization, a semi-analytic formula for the shape gradient of the Lagrangian linearized on the solution is proved, which contains both primal and adjoint states. It is used for the descent direction in a gradient algorithm for identification of an optimal shape of the breaking line from boundary measurements. The theoretical result is supported by numerical simulation tests of destructive testing in 2D configuration with and without contact.

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Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks

Cited by 11 publications

References 43 publications

Fully discrete approximation schemes for rate-independent crack evolution

Fully discrete approximation schemes for rate-independent crack evolution

Thin inclusion at the junction of two elastic bodies: non-coercive case

Lagrangian approach and shape gradient for inverse problem of breaking line identification in solid: contact with adhesion

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