Thin Elastic Inclusion Research Articles

Thin inclusion at the junction of two elastic bodies: non-coercive case.

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

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

Method of Direct Cutting-Out in Modeling Orthotropic Solids with Thin Elastic Inclusions Under Longitudinal Shear

By the method of direct cutting-out, the problems of longitudinal shear of an orthotropic half space, a layer, and a wedge with thin elastic orthotropic inclusions are reduced to the basic problem of interaction of thin inhomogeneities in the orthotropic space. We establish the conditions of interaction of loaded elastic anisotropic inclusions with the matrix of the body. We study the influence of elastic moduli both of the inclusion and of the body, as well as of the geometric parameters of the problems on the generalized stress intensity factors. The level lines of stresses are plotted in the vicinity of the inclusion.

Noncoercive problems for elastic bodies with thin elastic inclusions

The paper concerns boundary value problems for an elastic body with a thin elastic inclusion for a noncoercive case. The inclusion is assumed to be delaminated thus forming a crack between the inclusion and the surrounding elastic body. To provide a mutual nonpenetration between crack faces, we consider inequality type boundary conditions with unknown set of a contact. In the paper, a solution existence of the equilibrium problems is proved for the cases when one or two given points of the inclusion are fixed as well as and for the case without these restrictions.

Asymptotic Approach in the Dynamic Problems of the Theory of Elasticity for Bodies with Thin Elastic Inclusions

Asymptotic Approach in the Dynamic Problems of the Theory of Elasticity for Bodies with Thin Elastic Inclusions

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'.

Optimal control of thin elastic inclusion in an elastic body

Optimal control of thin elastic inclusion in an elastic body

Elastic equilibrium of anisotropic bimaterial bodies with thin elastic anisotropic inclusions under longitudinal shear

Elastic equilibrium of anisotropic bimaterial bodies with thin elastic anisotropic inclusions under longitudinal shear

T-shape inclusion in elastic body with a damage parameter

We consider an equilibrium problem for a 2D elastic body with a thin elastic T-shape inclusion. A part of the inclusion is delaminated from the elastic body forming a crack between the inclusion and the surrounding elastic body. Inequality type boundary conditions are imposed at the crack faces preventing interpenetration between the crack faces. The model is characterized by a damage parameter. This parameter is responsible for connection at the junction points between different parts of the considered structure. Dependence of solutions on the damage parameter is investigated, in particular, a passage to infinity and to zero is analyzed. Inverse problems are considered provided that the damage parameter and Lamé parameters of the elastic body are unknown. In this case, a displacement of the tip point of the inclusion is assumed to be known. A solution existence of the inverse problems is proved.

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.

Direct cutting-out method in modelling of orthotropic solids with thin elastic inclusions under longitudinal shear

Direct cutting-out method in modelling of orthotropic solids with thin elastic inclusions under longitudinal shear

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.

Asymptotic approach in dynamic problems of the elasticity theory for bodies with thin elastic inclusions

Asymptotic approach in dynamic problems of the elasticity theory for bodies with thin elastic inclusions

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact The problem of the interaction of harmonic waves with a thin elastic circular inclusion, which is located in an elastic isotropic body (matrix), is solved. On both sides of the inclusion between it and the body (matrix), the conditions of smooth contact are realized. The solution method is based on representing the displacements in the matrix through discontinuous solutions of the Lamé equations for harmonic vibrations. This made it possible to reduce the problem to Fredholm integral equations of the second kind with respect to functions associated with jumps in normal stress and radial displacement to included ones. After the realization of the boundary conditions on the sides of the inclusion, a system of singular integral equations is obtained to determine these jumps. Keywords: elastic inclusions, cylindrical waves, matrix, stress intensity factor.

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.

Inverse problem for elastic body with thin elastic inclusion

Abstract An inverse problem for an elastic body with a thin elastic inclusion is investigated. It is assumed that the inclusion crosses the external boundary of the elastic body. A connection between the inclusion and the elastic body is characterized by the damage parameter. We study a dependence of the solutions on the damage parameter. In particular, passages to infinity and to zero of the damage parameter are investigated. Limit models are analyzed. Assuming that the damage and rigidity parameters of the model are unknown, inverse problems are formulated. Sufficient conditions for the inverse problems to have solutions are found. Estimates concerning solutions of the inverse problem are established.

On Equilibrium of the Elastic Bodies with Cracks Crossing Thin Inclusions

Under study is the equilibrium problem of a two-dimensional elastic body with a crack crossing a thin rigid inclusion at some point. Nonpenetration conditions in the form of inequalities are put on the crack faces and at the intersection point of the crack with the rigid inclusion. The equilibrium problem of an elastic body with a crack crossing a thin elastic inclusion is also considered. The theorems of unique solvability of these problems are proved, and some complete systems of boundary conditions are obtained. The equivalence of the two formulations, variational and differential, is examined. We establish that the limit transition with respect to the rigidity parameter in the problems on the equilibrium of an elastic body with an elastic inclusion leads to the equilibrium problem of an elastic body with a rigid inclusion.

On Contact Between a Thin Obstacle and a Plate Containing a Thin Inclusion

We consider problems governing a contact between an elastic plate with a thin elastic inclusion and a thin elastic obstacle and study the equilibrium of the plate with or without cuts. We discuss various statements and establish the existence of a solution. We analyze the limit problem as the rigidity parameter of the elastic inclusion tends to infinity.

Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method

A model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal is discussed. The atomic arrangements of the matrix and the inclusion are both periodic along the $$x_{1}$$ -direction and quasiperiodic along the $$x_{2}$$ -direction in the $$ox_{1}x_{2}$$ -coordinate system. Using the hypersingular integral equation method, the inclusion problem is reduced to solving a set of hypersingular integral equations. Based on the exact analytical solution of the singular phonon and phason stresses near the inclusion front, a numerical method of the hypersingular integral equation is proposed using the finite-part integral method. Finally, the numerical solutions for the phonon and phason stress intensity factors of some examples are given.

Antiplane Deformation of a Bimaterial with Thin Interfacial Nonlinear Elastic Inclusion

Abstract The problem of longitudinal shear of bimaterial with thin nonlinear elastic inclusion at the interface of matrix materials is considered. Solution of the problem is constructed using the boundary value problem of combining analytical functions and jump functions method. The model of the thin inclusion with nonlinear resilient parameters is built. Solution of the problem is reduced to a system of singular integral equations with variable coefficients. The convergent iterative method for solving such a system is offered for various nonlinear strain models, including Ramberg-Osgood law. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analysed for the tensely-deformed matrix under loading a uniformly distributed shear stresses and for a balanced system of the concentrated forces.