Invariant energy integrals for the non-linear crack problem with possible contact of the crack surfaces

Kovtunenko, Victor A.

doi:10.1016/s0021-8928(03)00021-2

Cited by 40 publications

(45 citation statements)

References 8 publications

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“…Hence, by the boundary conditions (5)- (7), (12) it follows the identity (9). In so doing we change the integration domain γ by since [u] = 0 on γ .…”

Section: Theorem 21 Problem Formulationsmentioning

confidence: 99%

On Timoshenko thin elastic inclusions inside elastic bodies

Khludnev

Leugering

2013

Mathematics and Mechanics of Solids

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The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.

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“…Hence, by the boundary conditions (5)- (7), (12) it follows the identity (9). In so doing we change the integration domain γ by since [u] = 0 on γ .…”

Section: Theorem 21 Problem Formulationsmentioning

confidence: 99%

On Timoshenko thin elastic inclusions inside elastic bodies

Khludnev

Leugering

2013

Mathematics and Mechanics of Solids

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“…Notice that in 3D case the space of infinitesimal rigid inclusions is defined as follows This allows us to differentiate the energy functional in the direction of the field V which implies the formula (5.17) with i, j = 1, 2, 3; see [11], [16].…”

Section: D Problems and Open Questionsmentioning

confidence: 99%

“…ν t = ν. In this case, it is proved that the formula for I can be obtained, namely, (see [11], [16], [19])…”

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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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“…This allows us to differentiate the energy functional in the direction of the field V , which implies the formula (5.58) with i, j = 1, 2, 3; see [69], [79].…”

Section: D Problems and Open Questionsmentioning

confidence: 99%