Victor A. Kovtunenko

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

Generally two-dimensional and three-dimensional formulations of the non-linear crack problem when the crack surfaces do not overlap for a non-uniform anisotropic linearly elastic body are considered. The first derivative of the potential energy function with respect to the perturbation parameter and its representation in the form of an invariant integral over an arbitrary closed contour are obtained for a general form of the differentiable perturbation of a region with a cut, using the method of material derivatives. The sufficient conditions for the existence of an invariant energy integral are derived in general form, and examples of invariant integrals are constructed for different types of perturbations and a different geometry of the cut.

Derivatives of the energy functional for 2D-problems with a Crack under Signorini and friction conditions

We consider the two-dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non-penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non-linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed.

Obstacle Problems with Cohesion: A Hemivariational Inequality Approach and Its Efficient Numerical Solution

Motivated by an obstacle problem for a membrane subject to cohesion forces, constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered. The associated first-order optimality system leads to a hemivariational inequality, which can also be interpreted as a special complementarity problem in function space. Besides an analytical investigation of first-order optimality, a primal-dual active set solver is introduced. It is associated to a limit case of a semismooth Newton method for a regularized version of the underlying problem class. For the numerical algorithms studied in this paper, global as well as local convergence properties are derived and verified numerically.

From shape variation to topological changes in constrained minimization: a velocity method-based concept

The ability of velocity methods to describe changes in topology by creating defects such as holes is investigated. For shape optimization, energy-type objective functions are considered, which depend on the geometry by means of the state variables. The state system is represented by abstract, quadratic and constrained minimization problems stated over domains with defects. The velocity method provides the shape derivative of the objective function due to finite variations of a defect. Sufficient conditions are derived, which allow us to pass the shape derivative to the limit with respect to the diminishing defect and, thus, to obtain the ‘topological derivative’ of the objective function due to a topological change. An illustrative example is presented for a circular hole bored at the tip of a crack.

Sensitivity of Interfacial Cracks to Non-linear Crack Front Perturbations

The SD-elasticity model of an anisotropic, non-homogeneous, bonded solid is considered. The interface is thought of as being a smooth surface comprising the connected part under the transmission condition and the crack under the stress-free boundary condition. We investigate the sensitivity of the model to the non-linear perturbation of the crack front along the interface. Expansions of the energy functionals at least up to the second-order terms are obtained by global derivatives of the solution with respect to the shape of the crack front. These derivatives are constructed over the whole non-smooth domain as iterative solutions of the same elasticity problem with specified fictitious forces. We consider only energetic solutions of the H 1 -class using the week formulation of the elasticity problem. Properties of the constructed derivatives of the energy functionals are discussed.

A hemivariational inequality in crack problems

Interface crack problems arising in quasibrittle fracture due to contact with cohesion or plasticity between the crack faces are considered. These problems are described by a hemivariational inequality. Its solvability is guaranteed by the variational principle, which yields minimization of a nonconvex and nondifferentiable objective functional associated to the total potential energy. To compute solutions of the hemivariational inequality, a primal-dual active-set algorithm is suggested, which obeys global and monotone convergence properties. A numerical example of the quasibrittle fracture is presented.

Sensitivity of cracks in 2D-Lamé problem via material derivatives

Abstract. The Lamé model of a two-dimensional solid with a crack under the stress-free boundary condition of the Neumann type at the crack faces is considered. We investigate the sensitivity of the problem to the crack perturbation. By constructing the material derivatives of the solution as iterative solutions of the same elasticity problem with specified right-hand sides, derivatives of the energy functional and of the stress intensity factors with respect to the crack length of an arbitrary order are obtained providing the corresponding asymptotic expansions. In particular, this implies the local optimality condition for finding of the crack length and the quasi-static model of the local crack propagation by the Griffith rupture criterion.

A Shape-Topological Control Problem for Nonlinear Crack-Defect Interaction: The Antiplane Variational Model

We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in two dimensions. The corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied, resulting in the topological sensitivity of an objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the antiplane strain energy release rate is expressed by means of the mode-III stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions of varying stiffness. The result of shape-topological control is useful either for arrests or rise of crack growth.

Regular Perturbation Methods for a Region with a Crack

The paper considers a model problem for Poissons equation for a region containing a crack or a set of cracks under arbitrary linear perturbation. Variational formulation of the problem using smooth mapping of regions yields a complete asymptotic expansion of the solution in the perturbation parameter, which is a generalized shape derivative. This global asymptotic expansion of the solution was used to derive representations of arbitrary‐order derivatives for the potential energy function, stress intensity factors, and invariant energy integrals in general form and for basis perturbations of the region (shear, tension, and rotation). The problem of the local growth of a branching crack for the Griffith fracture criterion and the linearized problem of optimal location of a rectilinear crack in a body with the energy function as a cost functional were formulated.

High Precision Identification of an Object: Optimality-Conditions-Based Concept of Imaging

A class of inverse problems for the identification of an unknown geometric object from given measurements is considered. A concept for object imaging based on optimality conditions and level sets is introduced which provides high resolution properties of the identification problem and stability to discretization and noise errors. As a specific case, the identification of the center of a test object of arbitrary shape and unknown boundary conditions from