The Method of Smooth Domains in the Equilibrium Problem for a Plate with a Crack (original) (raw)
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Japan Journal of Industrial and Applied Mathematics
The equilibrium problems for homogeneous and inhomogeneous plates with a crack is considered. We impose the nonpenetration condition, which has the form of inequality (such as Signorini type condition), on the crack faces. In this paper, we deal with the cases that the crack intersects the external boundary at zero angle (on the mid-plane). Using the fictitious domain method, we establish the unique solvability of four equilibrium problems for different cases of non-Lipschitz domains. In these cases a precise connection between equilibrium problems for a plate contacting with a rigid obstacle and for a plate with a crack is identified.
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.
2020
The paper focuses on nonlinear problems describing the equilibrium of Kirchhoff–Love plates with cracks. We assume that under an appropriate load, plates have special deformations with previously known configurations of edges near a crack. Owing to this particular case, we propose two types of new nonpenetration conditions that allow us to more precisely describe the possibility of contact interaction of crack faces. These conditions correspond to two special cases of configurations of plate edges. In each case, the nonpenetration conditions are given in the form of a system of equalities and inequalities. For initial variational statements, we prove the existence and uniqueness of solutions in an appropriate Sobolev space. Assuming that the solutions are sufficiently smooth, we have found differential statements that are equivalent to the corresponding variational formulations. The relations of the obtained differential statements are compared with the well-known setting of an equi...
Equilibrium of elastic media with internal nonFiat cracks
2008
A class of mixed boundary problems on equilibrium of three-dimensional bodies weakened by nonflat cracks on parts of the second-degree surfaces is examined. The general approach to these problems is developed. The solution of Lame vector equation of equilibrium is presented in the form of eigenfunction expansions. The unknown coefficients are found from boundary conditions transferred to a crack surface according to the superposition principle. The principle of displacements and stresses fields continuity is used out of the crack. The result is a coupled system of dual series equations or integral equations. Data obtained are applicable to the study of material damage.
Crack on the boundary of two overlapping domains
Zeitschrift für angewandte Mathematik und Physik, 2010
In this paper, we consider an overlapping domain problem for two elastic bodies. A glue condition of an equality-type is imposed at a given line. Simultaneously, a part of this line is considered to be a crack face with an inequality-type boundary condition describing mutual non-penetration between crack faces. Variational and differential formulations of the problem are considered. We prove a differentiability of the energy functional in the case of rectilinear cracks and find a formula for invariant integrals. Passage to the limit is justified provided that the rigidity of the body goes to infinity. . 49J40 · 49K10 · 74R10.
Journal of Siberian Federal University. Mathematics & Physics, 2021
The paper considers equilibrium models of Kirchhoff-Love plates with rigid inclusions of two types. The first type of inclusion is described by three-dimensional sets, the second one corresponds to a cylindrical rigid inclusion, which is perpendicular to the plate’s median plane in the initial state. For both models, we suppose that there is a through crack along a fixed part of the inclusion’s boundary. On the crack non-penetration conditions are prescribed which correspond to a certain known configuration bending near the crack. The uniqueness solvability of a new problems for a Kirchhoff-Love plate with a flat rigid inclusion is proved. It is proved that when a thickness parameter tends to zero, the problem for a flat rigid inclusion can be represented as a limiting task for a family of variational problems concerning the inclusions of the first type. A solvability of an optimal control problem with a control given by the size of inclusions is proved
Ukrainian Journal of Mechanical Engineering and Materials Science, 2020
Purpose. A two-dimensional mathematical model for the problem of elasticity theory in a three-component plate containing rectilinear crack due to the action of mechanical efforts is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks, we determined stress intensity factors (SIFs) at the tops of the crack, which are subsequently used to determine critical values of the tension. Therefore, the aim of present work is to determine the two-dimensional elastic state in plate containing an elastic two-component circular inclusion and crack under conditions of power load in the case of unidirectional tension of the plate perpendicular for the crack line. This makes it possible to determine th...
An integral-equation solution for a bounded elastic body containing a crack: Mode I deformation
International Journal of Fracture, 1978
An integral-equation solution for the two dimensional problem of a stress-free crack in a bounded, linearly elastic, isotropic medium subjected to in-plane forces is presented. A set of coupled equations, involving integrals over the outer boundary and the line of the crack, is obtained by the simultaneous solution of a crack problem in an unbounded medium-the perturbed problem-and a problem in an unttawed region having the same outer boundary as the medium containing the crack-the equable problem. The solution for the perturbed problem is given in quadrature form in terms of the derivative of the normal displacement of the crack surface. The solution for the equable problem is given by a set of boundary integral equations. The character of the stress-field singularity at the crack tips is provided by the solution for the perturbed problem. For a numerical evaluation the set of coupled integral equations is approximated by a set of simultaneous, linear algebraic equations. Since the stress intensity factors and the crack surface displacements are incorporated into the integral equations, the values of these quantities are obtained directly from the numerical solution. Several sample problems are presented in order to determine the versatility and accuracy of this approach.
Fracture and Structural Integrity, 2023
This is the first part of a short three-paper series, aiming to revisit some classical concepts of Linear Elastic Fracture Mechanics. The motive of this first paper is to highlight some controversial issues, related to the unnatural overlapping of the lips of a 'mathematical' crack in an infinite plate loaded by specific combinations of principal stresses at infinity (predicted by the classical solution of the respective first fundamental problem), and the closely associated issue of negative mode-I Stress Intensity Factor. The problem is confronted by superimposing to the first fundamental problem of Linear Elastic Fracture Mechanics for an infinite cracked plate (with stress-free crack lips) an 'inverse' mixed fundamental problem. This superposition provides naturally acceptable stress and displacement fields, prohibiting overlapping of the lips (by means of contact stresses generated along the crack lips, which force the overlapped lips back to naturally accepted position) and, also, non-negative mode-I Stress Intensity Factors. The solutions of this first paper form the basis for the next two papers of the series, dealing with the respective problems in finite domains (recall, for example, the cracked Brazilian disc configuration) weakened by artificial notches (rather than 'mathematical' cracks), by far more interesting for practical engineering applications.
An Integral-Equation Solution for a Bounded Elastic Body with an Edge Crack: Mode I Deformations
1978
An integral-equation solution for the two dimensional problem of a stress-free crack in a bounded, linearly elastic, isotropic medium subjected to in-plane forces is presented. A set of coupled equations, involving integrals over the outer boundary and the line of the crack, is obtained by the simultaneous solution of a crack problem in an unbounded medium-the perturbed problem-and a problem in an unttawed region having the same outer boundary as the medium containing the crack-the equable problem. The solution for the perturbed problem is given in quadrature form in terms of the derivative of the normal displacement of the crack surface. The solution for the equable problem is given by a set of boundary integral equations. The character of the stress-field singularity at the crack tips is provided by the solution for the perturbed problem. For a numerical evaluation the set of coupled integral equations is approximated by a set of simultaneous, linear algebraic equations. Since the stress intensity factors and the crack surface displacements are incorporated into the integral equations, the values of these quantities are obtained directly from the numerical solution. Several sample problems are presented in order to determine the versatility and accuracy of this approach.