Griffith formulae for elasticity systems with unilateral conditions in domains with cracks

Khludnev, A. M.; Sokołowski, Jan

doi:10.1016/s0997-7538(00)00138-8

Cited by 64 publications

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“…There, the ERR is expressed by the Griffith formula, the J -integral or the Cherepanov-Rice integral, see e.g. [5,[7][8][9][10], or by stress intensity factors, see e.g. [11].…”

Section: Knees and A Mielkementioning

confidence: 99%

Energy release rate for cracks in finite‐strain elasticity

Knees

Mielke

2007

Math Methods in App Sciences

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SUMMARYGriffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: if the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow.In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we rigorously derive the well-known Griffith formula and the J -integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors.

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“…There, the ERR is expressed by the Griffith formula, the J -integral or the Cherepanov-Rice integral, see e.g. [5,[7][8][9][10], or by stress intensity factors, see e.g. [11].…”

Section: Knees and A Mielkementioning

confidence: 99%

Energy release rate for cracks in finite‐strain elasticity

Knees

Mielke

2007

Math Methods in App Sciences

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“…can be interpreted as the derivative of the elastic energy with respect to the crack length, we refer the reader to [15] for the proof, the same result for the Laplacian is given in [5,13,14].…”

Section: Rectilinear Crack In Two Spatial Dimensionsmentioning

confidence: 86%

Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities

Leugering

Sokołowski

Żochowski

2016

Springer Optimization and Its Applications

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A class of nonsmooth shape optimization problems for variational inequalities is considered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith's functional, which is defined in plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, topological optimization can be used for passive control of singularities of weak solutions to variational inequalities.The Hadamard directional differentiability of metric the projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. A domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis.The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. Singular geometrical domain perturbations in an elastic body Ω are approximated by regular perturbations of bilinear forms in variational inequality, without any loss of precision for the purposes of the secondorder shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations → ω ⊂ Ω centred atx ∈ Ω are replaced by regular perturbations of bilinear forms supported on the manifold Γ R = {|x−x| = R} in an elastic body, with R > > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems.

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“…It is possible to prove a solution existence to the problem (2.5)-(2.7) and check that (2.5)-(2.7) is formally equivalent to (1.1)-(1.5) (see [13]). Solution existence to (2.5)-(2.7) can be proved independently of (1.1)-(1.5).…”

Section: Cσ = ε(U)mentioning

confidence: 99%

Shape and topological sensitivity analysis in domains with cracks

2010

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Abstract. Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method [4] can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.Key words. Crack with non-penetration, shape sensitivity, derivative of energy functional, topological derivative AMS subject classifications. Primary 35J85, 74K20 Secondary 35J25, 74M151. Introduction. Shape optimization requires few mathematical results, in the framework of modelling and numerical solution, for any specific class of problems governed by partial differential equations of mathematical physics. Usually, we need to show the well posedness of the specific problem, and also we can propose a numerical method for the effective solution procedure. Hence, in order to solve a shape optimization problem we are obliged to have the results on• the existence and continuous dependence with respect to the shape of solutions to the model, which may result in the existence of optimal shapes, • the differentiability of solutions with respect to the boundary variations, which imply the existence of shape gradients and leads to some necessary conditions for optimality, of the first order and possibly of the second order which leads to the Newton method of shape optimization, • and in addition, perform the asymptotic analysis of the related boundary value problem in singularly perturbed geometrical domains and derive the form of the topological derivative for the shape functional of interest, which allows for the topology changes in the process of numerical optimization, if necessary, • and finally, we may device a numerical method and show its efficiency in numerical examples, and its convergence form the mathematical point of view.

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Griffith formulae for elasticity systems with unilateral conditions in domains with cracks

Cited by 64 publications

References 0 publications

Energy release rate for cracks in finite‐strain elasticity

Energy release rate for cracks in finite‐strain elasticity

Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities

Shape and topological sensitivity analysis in domains with cracks

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