Thin rigid inclusions with delaminations in elastic plates (original) (raw)
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Delaminated thin elastic inclusions inside elastic bodies
Mathematics and Mechanics of Complex Systems, 2014
We propose a model for a two-dimensional elastic body with a thin elastic inclusion modeled by a beam equation. Moreover, we assume that a delamination of the inclusion may take place resulting in a crack. Nonlinear boundary conditions are imposed at the crack faces to prevent mutual penetration between the faces. Both variational and differential problem formulations are considered, and existence of solutions is established. Furthermore, we study the dependence of the solution on the rigidity of the embedded beam. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain a rigid beam inclusion and cracks with nonpenetration conditions, respectively. Anisotropic behavior of the beam is also analyzed.
On bending an elastic plate with a delaminated thin rigid inclusion
Journal of Applied and Industrial Mathematics, 2011
ABSTRACT 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.
On elastic bodies with thin rigid inclusions and cracks
Mathematical Methods in the Applied Sciences, 2010
This paper is concerned with the analysis of equilibrium problems for 2D elastic bodies with thin rigid inclusions and cracks. Inequality type boundary conditions are imposed at the crack faces providing a mutual non-penetration between 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.
Crack on the boundary of a thin elastic inclusion inside an elastic body
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2012
We propose a model for a 2D elastic body with a thin elastic inclusion in which delamination of the inclusion may take place, thus forming a crack. Non-linear boundary conditions at the crack faces are imposed to prevent mutual penetration. We prove existence and uniqueness of the equilibrium configuration, considering both the variational and the differential formulations. Moreover, we study the dependence of solutions on the rigidity of the beam and we prove that in the limit corresponding to infinite and zero rigidity, we recover the case of a semi-rigid inclusion and the case of a crack with non-penetration conditions, respectively. The convergence of solutions is proved both using variational inequalities and Γ-convergence.
Junction problem for elastic and rigid inclusions in elastic bodies
Mathematical Methods in the Applied Sciences, 2015
Communicated by W. L. Wendland 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.
2013
An equilibrium problem for an elastic Timoshenko type plate containing a rigid inclusion is considered. On the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that at both crack faces, boundary conditions of inequality type are considered describing a mutual nonpenetration of the faces. A solvability of the problem is proved, and a complete system of boundary conditions is found. It is also shown that the problem is the limit one for a family of other problems posed for a wider domain and describing an equilibrium of elastic plates with a vertical crack as the rigidity parameter goes to infinity.
Interference Analysis between Crack and General Inclusion in an Infinite Plate by Body Force Method
Key Engineering Materials, 2013
A Continously Embedded Force Doublet over the Particular Region can be Regardedas the Distributing Eigen Strain. this Fact Implies that many Sorts of Inelastic Strain can Bereplaced by the Force Doublet. in the Present Paper, the Force Doublet is Used to Alter the Localconstitutive Relationship. as a Result, a Method for Analyzing the General Inclusion Problem Inwhich the Material Properties of the Inclusion are Not only Different from those of the Matrixmaterial but also can be even a Function of Spacial Coordinate Variables is Proposed. Thetheoretical Background of the Present Analysis is Explained Followed by some Representativenumerical Results.
Contact problems for elastic bodies with rigid inclusions
2012
This paper is concerned with a new type of free boundary problems for elastic bodies with a rigid inclusion being in contact with another rigid inclusion or with a non-deformable punch. We propose correct problem formulations with inequality type boundary conditions of a non-local type describing a mutual non-penetration between surfaces. Solution existence is proved for different types of inclusions and different geometries. Qualitative properties of solutions are analyzed provided that rigidity parameters are changed.
Singular invariant integrals for elastic body with delaminated thin elastic inclusion
Quarterly of Applied Mathematics, 2014
We consider an equilibrium problem for a 2D elastic body with a thin elastic inclusion. It is assumed that the inclusion is partially delaminated, therefore providing the presence of a crack. Inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration of the faces. Differentiability properties of the energy functional with respect to the crack length are analyzed. We prove an existence of the derivative and find a formula for this derivative. It is shown that the formula for the derivative can be written in the form of a singular invariant integral.
On the extension of a crack due to rigid inclusions
1979
Dislocation layers have been utilized to study the effect of rigid inclusion on the crack extension in an infinite body and thereby to derive the crack extension condition from Irwin's criterion. Some particular cases have also been considered.