Thin Rigid Inclusion Research Articles

Systems of Inclusions in a Spatial Elastic Wedge

Contact problems are considered for two identical thin rigid elliptic inclusions in a three-dimensional elastic wedge of two-sided angle outer faces of which are subjected to rigid or sliding support. The problems are reduced to integral equations with symmetric kernels. Two dimensionless geometric parameters are introduced to characterize location of the inclusions in the bisecting half-plane of the wedge. Assuming linear connection between the parameters, the regular asymptotic method is used to solve the problems. The asymptotic for two inclusions is compared with corresponding solutions for unit inclusion in the wedge as well as for a periodic chain of inclusions the axis of which is parallel to the wedge edge.

On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition.

The paper investigates a problem concerning the equilibrium of a solid body containing a thin rigid inclusion and a crack. It is assumed that the body is hyperelastic, therefore, it is described within the framework of finite strain theory. One of the peculiarities of this problem is a global injectivity constraint, which prevents the body, the crack faces and the inclusion from both mutual and self penetration. First, the paper deals with the differential formulation of the problem. Next, we consider energy minimization, showing that the latter provides the weak formulation of the former. Finally, the existence of the weak solution is demonstrated through the use of the variational technique.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Elasticity Problem with a Cusp between Thin Inclusion and Boundary

This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. To avoid nonphysical interpenetration of the opposite crack faces, we impose inequality type constraints. Moreover, boundary conditions at the crack faces depend on a positive parameter describing a cohesion. A solution existence of the problem with different conditions on the external boundary is proved. Passages to the limit are analyzed as the damage parameter tends to infinity and to zero. Finally, an optimal control problem with a suitable cost functional is investigated. In this case, a part of the rigid inclusion is located outside of the elastic body, and a control function is a shape of the inclusion.

Stress State Near Cracks Originating from the Edges of a Thin Rigid Inclusion Caused by the Action of Longitudinal Shear Waves

Stress State Near Cracks Originating from the Edges of a Thin Rigid Inclusion Caused by the Action of Longitudinal Shear Waves

On Numerical Solving of Junction Problem for the Thin Rigid and Elastic Inclusions in Elastic Body

On Numerical Solving of Junction Problem for the Thin Rigid and Elastic Inclusions in Elastic Body

Equilibrium problem for an inhomogeneous two-dimensional elastic body with two interacting thin rigid inclusions

A new nonlinear mathematical model is proposed that describes an equilibrium of a two-dimensional elastic body with two interacting thin rigid inclusions. Thin rigid inclusions are defined by straight line segments. In the initial state of the body, both inclusions are in contact at one breaking point. Both inclusions can move, subject to the conditions in the form of systems of inequalities describing a possible contact of the inclusions. These three systems of inequalities for infinitesimal rigid displacements correspond to three possible cases of mutual configurations of inclusions in equilibrium state. Inclusions exfoliate near the breaking point from the elastic matrix, in other words, there are cracks of a given length on both sides of thin inclusions. Nonpenetration conditions of the Signorini type are imposed on the curves defining the cracks. On a part of the outer boundary, clamping conditions are prescribed. The problem is formulated as a minimization of an energy functional over a non-convex set of possible displacements defined in an appropriate Sobolev space. The existence of a solution to the problem is proved. Optimality conditions of the solution have been obtained under the condition of sufficient smoothness of the solution.

The Problem of the Location of an Inclusion in a Two-Dimensional Elastic Body with Two Thin Rigid Inclusions

The Problem of the Location of an Inclusion in a Two-Dimensional Elastic Body with Two Thin Rigid Inclusions

Periodic system of rigid inclusions in a spatial elastic wedge

A periodic contact problem (tangential contact) on a chain of thin rigid inclusions located along the edge of a three-dimensional elastic wedge of a dihedral angle in its middle half-plane (rigid sealing of the wedge faces, the axis of the chain of inclusions is parallel to the edge of the wedge) is considered. To derive the integral equation, the previously obtained fundamental solution of the problem of the action of a concentrated force inside a three-dimensional elastic wedge is used. The nucleus has latent symmetry, then it is reduced to a form with explicit symmetry. To solve the integral equation, a regular asymptotic method is used, which is effective for relatively sparse and far from the edge chains of inclusions in a three-dimensional elastic wedge. Contact characteristics are calculated for different values of dimensionless geometric parameters. Previously, similar problems were considered about one inclusion (stringer) in a strip [1] and a three-dimensional wedge [2], as well as in an unbounded elastic body [1].

Junction problem for thin elastic and volume rigid inclusions in elastic body.

The article concerns a junction problem for two-dimensional elastic body with a thin elastic inclusion and a volume rigid inclusion. It is assumed that the inclusions have a common point. A delamination of the thin inclusion from the surrounding elastic body is assumed thus forming an interfacial crack. Constraint-type boundary conditions are imposed at the crack faces to prevent interpenetration between the faces. Moreover, a connection between the crack faces is characterized by a positive damage parameter. Limit transitions are justified as the damage parameter tends to infinity and to zero. In addition to this, a transition to limit is analysed as a rigidity parameter of the thin inclusion tends to infinity. Limit models are investigated. In particular, junction conditions at the common point are found for all cases considered. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Non-coercive problems for Kirchhoff–Love plates with thin rigid inclusion

Non-coercive problems for Kirchhoff–Love plates with thin rigid inclusion

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

A new type of non-classical 2D contact problem formulated over non-convex admissible sets is proposed. Specifically, we suppose that a composite body in its undeformed state touches a wedge-shaped rigid obstacle at a single contact point. Composite bodies under investigation consist of an elastic matrix and a rigid inclusion. In this case, the displacements on the set, corresponding to a rigid inclusion, have a predetermined structure that describes possible parallel shifts and rotations of the inclusion. The rigid inclusion is located on the external boundary and has the form of a wedge. The presence of the rigid inclusion imposes a new type of non-penetration condition for certain geometrical configurations of the obstacle and the body near the contact point. The sharp-shaped edges of the obstacle effect such sets of admissible displacements that may be non-convex. For the case of a thin rigid inclusion, which is described by a curve and a volume (bulk) rigid inclusion specified in a subdomain, the energy minimization problems are formulated. The solvability of the corresponding boundary value problems is proved, based on analysis of auxiliary minimization problems formulated over convex sets. Qualitative properties of the auxiliary variational problems are revealed; in particular, we have found their equivalent differential formulations. As the most important result of this study, we provide justification for a new type of mathematical model for 2D contact problems for reinforced composite bodies.

Stress state near the cracks coming out from the ends of a thin rigid inclusion that is caused by the action of longitudinal shear waves

Stress state near the cracks coming out from the ends of a thin rigid inclusion that is caused by the action of longitudinal shear waves

An Iterative Method for Solving the Problem of Diffraction of Longitudinal Shear Waves on A System of Thin Rigid Inclusions

An Iterative Method for Solving the Problem of Diffraction of Longitudinal Shear Waves on A System of Thin Rigid Inclusions

Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary

Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary

A contact of two elastic plates connected along a thin rigid inclusion

A contact of two elastic plates connected along a thin rigid inclusion

The conjugation thin inclusions problem in elastic bodies with crack

A problem of pairing thin elastic and rigid inclusions with possible delamination in elastic bodies is considered. On the crack and at the intersection of the crack with a hard inclusion, boundary conditions are imposed in the form of inequalities, excluding mutual penetration of the crack faces and thin inclusions. The existence and uniqueness of the solution of the problem is established. It is shown that a corresponding variational statement is equivalent to a differential setting. A passage to the limit as stiffness parameter of thin inclusions tends to infinity is investigated.

Plastic Exfoliation of a Periodic System of Thin Near-Boundary Inclusions

We obtain a numerical-analytic solution of an antiplane problem of the stress-strain state of elastoplastic half space containing a periodic system of thin rigid tunnel inclusions parallel to the boundary of the half space. Prior to loading, these inclusions were in unilateral mechanical contact with the medium. We determine the stress-strain state and study the phenomenon of plastic exfoliation of the inclusions. The diagrams of the critical load are plotted on the coordinates of the geometric parameters of the system of inclusions under the condition of constancy of the critical length of interface plastic strips.

On junction problem with damage parameter for Timoshenko and rigid inclusions inside elastic body

AbstractThe paper is concerned with an equilibrium problem for 2D elastic body with a thin elastic Timoshenko inclusion and a thin rigid inclusion. The elastic inclusion is assumed to be delaminated from the elastic body thus forming an interfacial crack with the matrix. Inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration. A connection between two inclusions at a given point is characterized by a positive damage parameter. Existence of solutions of the problem considered is proved, and different equivalent formulations of the problem are analyzed; junction conditions at the connection point are found. A convergence of solutions as the damage parameter tends to zero and to infinity as well as the rigidity parameter of the elastic inclusion tends to infinity is analyzed. An analysis of the limit models is performed. A solution existence of an inverse problem for finding the damage and rigidity parameters is proved provided that an additional information concerning the derivative of the energy functional with respect to the delamination length is given.

Optimal location of a thin rigid inclusion for a problem describing equilibrium of a composite Timoshenko plate with a crack

We consider equilibrium problems for a cracked composite plate with a thin cylindrical rigid inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that does not touch the rigid inclusion. In order to describe mutual nonpenetration of the crack faces we impose a boundary condition in the form of inequality on the crack curve. For a family of appropriate variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the location parameter of inclusion is chosen as the control parameter. The existence of a solution to the optimal control problem and a continuous dependence of the solutions in a suitable Sobolev space with respect to the location parameter are proved.

Optimal Control of Parameters for Elastic Body with Thin Inclusions

In this paper, an equilibrium problem for 2D non-homogeneous anisotropic elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. A connection between the inclusions at a given point is characterized by a junction stiffness parameter. The elastic inclusion is delaminated, thus forming an interfacial crack with the matrix. Inequality-type boundary conditions are imposed at the crack faces to prevent interpenetration. Existence of solutions is proved; different equivalent formulations of the problem are discussed; junction conditions at the connection point are found. A convergence of solutions as the junction stiffness parameter tends to zero and to infinity as well as the rigidity parameter of the elastic inclusion tends to infinity is investigated. An analysis of limit models is provided. An optimal control problem is analyzed with the cost functional equal to the derivative of the energy functional with respect to the crack length. A solution existence of an inverse problem for finding the junction stiffness and rigidity parameters is proved.