First-Order Shape Derivative of the Energy for Elastic Plates with Rigid Inclusions and Interfacial Cracks

Rudoy, E. M.; Shcherbakov, V. V.

doi:10.1007/s00245-020-09729-5

Cited by 20 publications

(10 citation statements)

References 43 publications

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“…Note that to have a upper bound for R, in the WG model, is consistent with the dilute limit theory, on which the WG model is based [40], meaning that the model is valid for small density values. These connection relationships (28)…”

Section: P R E P R I N T a U T H O R C O P Ymentioning

confidence: 99%

“…Recently, some analytical models of hard material interfaces have been also developed [23][24][25][26][27][28][29] and it has been proved that interface models developed for soft adhesives cannot be directly applied in the case of hard adhesives [25].…”

Section: P R E P R I N T a U T H O R C O P Ymentioning

confidence: 99%

“…Equations (28) show that in undamaged conditions D = R = 0 for both damaged-material models. Instead, in fully damaged conditions (D = 1), R → +∞ for the KS model and it is bounded by the value R = 1/2 H nn E 0 for the WG model.…”

Section: P R E P R I N T a U T H O R C O P Ymentioning

confidence: 99%

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A micromechanical model of a hard interface with micro-cracking damage

Raffa

Lebon

Rizzo

2022

International Journal of Mechanical Sciences

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Bonding techniques are increasingly used in many industrial fields. Modelling the under-load damaging behavior of hard structural adhesives is still an open challenge. This work proposes a new hard interface analytical model with evolutive micro-cracking damage. The model is obtained within a rigorous theoretical framework combining asymptotic theory and micromechanical homogenization. Main new features are: (i) the adoption of two dual homogenization approaches; (ii) the formulation of a thermodynamically-based damage evolution law for hard interfaces. The interface model is able to describe both ductile and brittle damage behavior of hard structural adhesives. Provided examples on the structural behavior, under several loads, suggest the suitability of the proposed interface model as a modelling strategy for hard structural adhesives with micro-cracking damage.

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Section: P R E P R I N T a U T H O R C O P Ymentioning

confidence: 99%

Section: P R E P R I N T a U T H O R C O P Ymentioning

confidence: 99%

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A micromechanical model of a hard interface with micro-cracking damage

Raffa

Lebon

Rizzo

2022

International Journal of Mechanical Sciences

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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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“…Our particular methods of non-smooth analysis stem from the variational approach to nonpenetrating cracks in solids developed by Khludnev & Kovtunenko [10] and co-authors (e.g. [11][12][13] and other works related to asymptotic analysis [14][15][16] and numerical techniques [17]).…”

Section: Introductionmentioning

confidence: 99%

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Kovtunenko

2022

Phil. Trans. R. Soc. A.

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The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori , whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton–Hoff and Ramberg–Osgood materials. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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First-Order Shape Derivative of the Energy for Elastic Plates with Rigid Inclusions and Interfacial Cracks

Cited by 20 publications

References 43 publications

A micromechanical model of a hard interface with micro-cracking damage

A micromechanical model of a hard interface with micro-cracking damage

Fully discrete approximation schemes for rate-independent crack evolution

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

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