Micromechanical Modeling of Composites Fracture
Abstract The paper considers the new micromechanical models of fracture of fiber-reinforced composites. The main emphasis is made on the determination of the exact analytical solutions of appearing fracture problems, which allow derivation of the closed functional formulas for limit fracture stresses. These expressions are suited for subsequent use in the formulations of optimal design problems. The model presented here describes the deformation and tearing of cloth. The material under study is made up of two families of orthogonal fibers. The object of investigation is the distribution of stresses in the neighborhood of the end of a semi-infinite tear. The description of the stress-strain state in the neighborhood of the end of the tear reduces to the solution of a mixed boundary-value problem for an infinite system of difference equations. When cast in terms of generating functions for an infinite vector of unknowns, the problem reduces to a RIEMANN-HELBERT problem. The analytical solution to the problem shows that the character of tearing of the material depends critically on the stresses in the fibers parallel to the direction of the tear: if the fibers parallel to the tear are stretched, then a finite rip in the material has an elliptical shape and the asymptotic behavior of the stresses around the end of the tear is similar to behavior of a solid elastic body with a crack. Conversely, if there is no stretching of the fibers parallel to the tear, then the sides of the tear intersect at a right angle, and the first unbroken fiber bears a considerably larger load than when stresses in parallel fibers are absent.