Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Abstract

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth.   Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained.   Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined.   Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations".   Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1.   Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes.   Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.