On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

Nonsmooth domain optimization for elliptic equations with unilateral conditions

In the paper we consider elliptic boundary problems in domains having cuts (cracks). The non-penetration condition of inequality type is prescribed at the crack faces. A dependence of the derivative of the energy functional with respect to variations of crack shape is investigated. This shape derivative can be associated with the crack propagation criterion in the elasticity theory. We analyze an optimization problem of finding the crack shape which provides a minimum of the energy functional derivative with respect to a perturbation parameter and prove a solution existence to this problem.     

Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

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. Copyright © 2000 John Wiley & Sons, Ltd.     

Shape optimization of rigid inclusions for elastic plates with cracks

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff–Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem.     

Differentiation of energy functionals in the three-dimensional theory of elasticity for bodies with surface cracks

A three-dimensional elastic body with a surface crack is considered. The boundary nonpenetration conditions in the form of inequalities (the Signorini type conditions) are given at the faces of the crack. The convergence is proved of a sequence of equilibrium problems in perturbed domains to the solution of an equilibrium problem in the unperturbed domain in a suitable Sobolev function space. The derivative is calculated of the energy functional with respect to the perturbation parameter of the surface crack.     

Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length

We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.     

Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.     

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

The paper addresses the analysis of a model for a thin shallow linear elastic shell with a smooth vertical through crack that is based on the Kirchhoff–Love shell theory and accounts for cohesive forces acting between the crack faces. We follow the basic idea behind the Barenblatt theory assuming that the density of the total energy spent by the cohesive forces is the sum of longitudinal and transverse contributions, each of which in general is nonconvex. In order to eliminate nonphysical interpenetration of the crack faces, an inequality constraint that involves both the normal component of the longitudinal displacements and the normal derivative of the transverse deflection of the crack faces is imposed. We first prove the existence of minimizers for the total energy and study in detail the Euler–Lagrange system for them. Then we derive the left and right Eulerian shape derivatives of the minimal value of the total energy by developing a fully variational technique. Finally we apply the developed technique coupled with a difference quotient argument to obtain higher differentiability results in Besov and Sobolev spaces for the minimizers.     

On an optimal control problem for the shape of thin inclusions in elastic bodies

An optimal control problem is considered for a two-dimensional elastic body with a straight thin rigid inclusion and a crack adjacent to it. It is assumed that the thin rigid inclusion delaminates and has a kink. On the crack faces the boundary conditions are specified in the form of equalities and inequalities which describe the mutual nonpenetration of the crack faces. The derivative of the energy functional along the crack length is used as the objective functional, and the position of the kink point, as the control function. The existence is proved of the solution to the optimal control problem.     

两个同心旋转圆柱之间的两种流体的交界面几何形状问题

研究两个同心旋转圆柱之间的两种流体的交界面几何形状问题．利用张量分析工具,给出了忽略耗散能量影响下交界面几何形状是一种能量泛函的临界点,其对应的Euler-Lagrange方程是1个非线性椭圆边值问题．对于粘性引起的耗散能量不能忽略的情况下,同样给出了1个带有耗散能量的能量泛函,其临界点是交界面几何形状,相应的Euler-Lagrange方程也是1个二阶的非线性椭圆边值问题．这样,交界面几何形状问题转化为求解非线性椭圆边值问题．     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

Shape sensitivity for the Laplace-Beltrami operator with singularities

This paper considers shape sensitivity analysis for the Laplace-Beltrami operator formulated on a two-dimensional manifold with a fracture. We characterize the shape gradient of a functional as a bounded measure on the manifold and decompose it into a “distributed gradient” supported on the manifold, plus a singular part that we derive as the limit of a “jump” through the crack and Dirac measures at the crack extremities. The important point is that we introduce a technique that is not dimension dependent, and makes no use of classical arguments such as the maximum principle or continuation uniqueness. The technique makes use of a family of envelopes surrounding the fracture which enable us to relax certain terms and to overcome the lack of regularity resulting from the presence of the fracture. We use the min-max differentiation in order to avoid taking the derivative of the state equation and to manage the crack's singularities. Therefore, we write the functional in a min-max formulation on a space which takes into account the hidden boundary regularity established by the tangential extractor method.     

Stability of critical shapes for the drag minimization problem in Stokes flow

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Shape optimization for the Maxwell equations under weaker regularity of the data

We consider a shape optimization problem for Maxwell's equations with a strictly dissipative boundary condition. In order to characterize the shape derivative as a solution to a boundary value problem, sharp regularity of the boundary traces is critical. This Note establishes the Fréchet differentiability of a shape functional.     

Optimal control of the layer size in the problem of equilibrium of elastic bodies with overlapping domains

We consider the problem of equilibrium of a two-layer elastic body. The first of the layers contains a crack,while the second is a circle centered at one of the crack tips. The round layer is glued by its boundary to the first layer. The unique solvability is proved for this problem in the nonlinear formulation. An optimal control problem is also considered. The radius a of the second layer is chosen as a varying parameter under assumption that a takes positive values from a closed interval. It is shown that there are a value of a minimizing the functional that characterizes how potential energy depends on the crack length and a value of a minimizing the functional that characterizes the opening of the crack.