Anti-plane stress analysis of rectangular plates weakened by multiple cracks and voids

Mechanics

Dissertation to get master's degree

Abstract:

Firstly, the solution of Volterra-type anti-plate displacement in a rectangular plate with limited length and width and different boundary conditions is obtained by the method of separation of variables. Then, the distributed displacement technique is used to obtain the integral equations related to the rectangular plate weakened by various cracks and holes under anti-plane loading. With the help of Buckner's principle, the effect of external loading on the boundaries of the environment on the crack surfaces and cavity boundaries is obtained, with the help of which the terms outside the integral of the integral equations are determined.  The kernel of these equations has a singularity of the Cauchy singularity, which is solved numerically using the numerical method available in the available references. By solving these equations, the dislocation density is obtained and the stress intensity coefficient at the tips of the cracks and the dimensionless environmental stress on the holes are calculated. Several examples are solved to determine the validity and applicability of the applied process.

The presence of defects in the form of cracks and holes in composite materials create areas with high stress changes. These areas are the main places for the emergence of different failure modes in structures, even if the applied load is moderate. Therefore, stress analysis in the vicinity of defects is necessary as the first step in the design process.

In anti-plane elasticity problems, using the anti-plane displacement method to obtain solutions for crack problems in infinite or semi-infinite environments is a common task. This is because the dislocation solution is the same as a Green's function solution for the main crack problems.

Analysis of stress in environments weakened by a set of cracks and holes has been of interest to researchers for a long time. Among the effective techniques in the mentioned analysis is the use of dislocation distribution method. The conducted research has shown that from a mathematical point of view, the crack can be considered as a set of dislocations and by using the principle of the sum of the effects of the relative movement of the crack edges relative to each other, and as a result, the stress intensity factor can be calculated. In fact, the ability to solve dislocations in solving linear fracture mechanics problems is as powerful as Green's solution in solving differential equations. In this thesis, the stress field and location change in different areas due to Volterra anti-plane dislocation are calculated in order to finally use it for the stress analysis of environments containing cracks and holes. The regions that are investigated include a rectangular plate with boundary conditions of two free edges and two clamped edges, a rectangular plate with all four free edges, a rectangular plate with one free edge and three clamped edges, and a rectangular plate with three free edges and one clamped edge. The obtained solutions are used for stress analysis in environments including cracks and cavities. In the case of the cavity, it is shown that the cavity can be considered as a closed crack without singularity, and by applying suitable conditions, the environmental stress can be obtained on it.

The problems related to the rectangular plate are:

Analysis of a straight surrounded crack, analysis of two straight surrounded cracks and an elliptical cavity, analysis of a straight surrounded crack and an edged straight crack along with an elliptical cavity

Each of the above examples once for a rectangular plate The figure with two free edges and two clamped edges, a rectangular plate with one free edge and three clamped edges, as well as a rectangular plate with three free edges and one clamped edge, which have the same loading conditions, have been solved, and once they have been solved for a rectangular plate with four free edges, whose loading conditions are different from the three mentioned cases.

To compare the answers obtained with existing references, the problem of a rectangular plate with all four free edges that is weakened by two straight cracks and an elliptical cavity has been solved. became After setting the length of the rectangular plate with all four free edges to infinity and applying the same loading conditions, the solution obtained for the rectangular plate with all four free edges exactly matched the solution obtained for the beam.

The loading conditions for rectangular plates whose four edges are not free are as points on the upper edge of the rectangular plate, and for the rectangular plate with all four free edges as four points that satisfy the self-equilibrium conditions of the rectangular plate.   

He referred to the analysis of the anti-plane stress of the limited wedge weakened by holes, which was done by Active [1] and colleagues []. Therefore, crack problems in orthotropic rectangular plate as a limited environment can be the subject of a new study. Due to the lack of valuable studies on the problems of cracks in a rectangular plate, the studies conducted on the beam are considered as a semi-confined medium that has the most similar geometry to the rectangular plate.

The stress analysis in a beam including cracks under anti-plane deformation has been the subject of various investigations. Here are some related articles. Zhu[2] et al[] investigated the stress field in the vicinity of two parallel cracks perpendicular to the edges of the isotropic beam. In this problem, the cracks were symmetrical with respect to the center line of the beam and were subjected to anti-plane loading. Lee [3] [] obtained a closed analytical solution for the beam problem mentioned above, except that the beam was orthotropic. Stress analysis in an isotropic beam weakened by two parallel cracks located on the centerline of the beam and under anti-plane shear was performed by Zhou and Ma [4] [].

In the mentioned articles, the application of boundary conditions leads to a set of integral equations that are solved by the Schmidt method [5]. Wu and Dezins[6] [] obtain an analytical solution to calculate

the stress intensity coefficients of the third mode of fracture mechanics related to an edge crack located at the interface of two dissimilar isotropic beams. In another article, Li[] considered a crack located at the interface between two dissimilar orthotropic beams, where the surface of the crack was under anti-plane loading, and the stress intensity coefficients were obtained analytically for the beam under anti-plane loading. In this article, the stress analysis in an orthotropic beam including a Volterra-type dislocation is carried out, and by using the dislocation solution, integral equations for a beam weakened by cracks and holes under anti-plane loading are obtained. By solving these integral equations, the stress intensity coefficients at the tips of the cracks and the dimensionless environmental stress on the boundaries of the holes have been obtained.