Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

On bending an elastic plate with a delaminated thin rigid inclusion

Under study is the problem of bending an elastic plate with a thin rigid inclusion which may delaminate and form a crack. We find a system of boundary conditions valid on the faces of the crack and prove the existence of a solution. The problem of bending a plate with a volume rigid inclusion is also considered. We establish the convergence of solutions of this problem to a solution to the original problem as the size of the volume rigid inclusion tends to zero.     

Numerical simulation of the non‐linear crack problem with non‐penetration

Here the numerical simulation of some plane Lamé problem with a rectilinear crack under non‐penetration condition is presented. The corresponding solids are assumed to be isotropic and homogeneous as well as bonded. The non‐linear crack problem is formulated as a variational inequality. We use penalty iteration and the finite‐element method to calculate numerically its approximate solution. Applying analytic formulas obtained from shape sensitivity analysis, we calculate then energetic and stress characteristics of the solution, and describe the quasistatic propagation of the crack under linear loading. The results are presented in comparison with the classical, linear crack problem, when interpenetration between the crack faces may occur. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimal rigid inclusion shapes in elastic bodies with cracks

The paper concerns the control of rigid inclusion shapes in elastic bodies with cracks. Cracks are located on the boundary of rigid inclusions and in the bulk. Inequality type boundary conditions are imposed at the crack faces to guarantee mutual non-penetration. Inclusion shapes are considered as control functions. First we provide the problem formulation and analyze the shape sensitivity with respect to geometrical perturbations of the inclusion. Then, based on Griffith criterion, we introduce the cost functional, which measures the shape sensitivity of the problem with respect to the geometry of the inclusion, provided by the energy release rate. We prove existence of optimal shapes for the problem considered.     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

An Optimal Size of a Rigid Thin Stiffener Reinforcing an Elastic Two-Dimensional Body on the Outer Edge

The equilibrium problem for a two-dimensional body with a crack is studied. We suppose that the body consists of two parts: an elastic part and a rigid thin stiffener on the outer edge of the body. Inequality-type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. For a family of variational problems, dependence of their solutions on the length of the thin rigid stiffener is investigated. It is shown that there exists a solution of an optimal control problem. For this problem, the cost functional is defined by a continuous functional on a solution space, while the length parameter serves as a control parameter.     

圆形界面刚性线夹杂的平面问题

研究了圆弧形界面刚性线夹杂的平面弹性问题．集中力作用于夹杂或基体中的任意点,并且无穷远处受均匀载荷作用．利用复变函数方法,得到了该问题的一般解答．当只含一条界面刚性线夹杂时,获得了分区复势函数和应力场的封闭形式解答,并给出刚性线端部奇异应力场的解析表达式．结果表明,在平面荷载下界面圆弧形刚性线夹杂尖端应力场和裂纹尖端相似具有奇异应力振荡性．对无穷远加载的情况,讨论了刚性线几何条件、加载条件和材料失配对端部场的影响．     

Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion

Under consideration is a 2D-problem of elasticity theory for a body with a thin rigid inclusion. It is assumed that there is a delamination crack between the rigid inclusion and the elastic matrix. At the crack faces, the boundary conditions are set in the form of inequalities providing mutual nonpenetration of the crack faces. Some numerical method is proposed for solving the problem, based on domain decomposition and the Uzawa algorithm for solving variational inequalities.We give an example of numerical calculation by the finite element method.     

Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.     

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.     

The conjunction problem for thin elastic and rigid inclusions in an elastic body

Under consideration is the conjunction problem for a thin elastic and a thin rigid inclusions that are in contact at one point and placed in an elastic body. Depending on what kind of conjunction conditions are set at the contact point of inclusions, we consider the two cases: the case of no fracture, where, as the conjunction conditions, we take the matching of displacements at the contact point and preservation of the angle between the inclusions, and the case with a fracture in which only the matching of displacements is assumed. At the point of conjunction, we obtain the boundary conditions for the differential formulation of the problem. On the positive face of the rigid inclusion, there is delamination. On the crack faces, some nonlinear boundary conditions of the type of inequalities are set, that prevent mutual penetration of the faces. The existence and uniqueness theorems for the solution of the equilibrium problem are proved in both cases.     

圆形界面刚性线夹杂的反平面问题

研究了在反平面集中力和无穷远纵向剪切作用下,不同弹性材料圆形界面上有多条刚性线夹杂的问题．运用Riemann-Schwarz解析延拓技术与复势函数奇性主部分析方法,首次获得了该问题的一般解答,求出了几种典型情况的封闭解,并给出了刚性线夹杂尖端的应力场分布A·D2结果表明,在反平面加载的情况下圆形界面刚性线夹杂尖端应力具有平方根奇异性,无奇异性应力振荡;应力场与刚性线夹杂的形状,加载方式和材料性质有关．退化结果与已有的解答完全吻合．     

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.     

裂纹与线夹杂垂直接触问题的奇性分析

以短纤维复合等作为工程背景,采用现有文献中单裂纹和单夹杂的基本解,对于限平项（基体）上,裂纹和线夹杂的垂直接触问题从断理解力学的角度作了研究,得到了问题的积分方程,推出了接触点的性线指数,奇性应力及以此表示的接触点附近三个区域内的应力强度因子表达式,并给出一些数值结果,可供工程实际参考。     

刚性线夹杂与弹性圆夹杂的相互作用

本文将刚性线夹杂与弹性圆夹杂的相互作用，归为解一个标准的柯西型奇异积分方程，获得了刚性夹杂端点的应力强度因子及夹杂的界面应力．     

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.     

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.