Lattice of infinite bending-resistant fibers

Abstract

This article present the double-periodical lattice made of infinite elastic fibers that withstand bending and tension. The model describes the elastic properties of flat periodic structure. With this model the behavior of a two-dimensional array of infinite fibers is simulated. The material that contains a row of broken fibers is considered. These broken fibers form the failure in the material that shapes like a long straight crack. The lattice is tensioned in the direction, which is orthogonal to the direction of straight crack. The conditions of fracture of this lattice are investigated. The closed form expression for the stress in the first unbroken fiber and the expression for fracture toughness are given. These values are the functions of mechanical parameters of lattice and tensions in both families of fibers. The closed form solution demonstrates a notable behavior of the material. Namely, the fracture behavior of two-dimensional lattice is cardinally depends upon the pre-stress in the material in the direction, parallel to crack direction. If the tension in fibers that parallel to the crack direction exists, it stabilizes the crack growth and makes the load distribution in the unbroken fibers more even. The two-dimensional lattice behaves in the presence of tension in both directions similarly to the plane elastic media. The finite length crack assumes the shape of the elongated elliptic split. Another behavior of lattice occurs if the fibers, parallel to crack direction, are unstressed. The character of stress concentration near the crack differs. The load distribution at the crack tip varies considerably. The first unbroken fiber carries higher load. The crack is lens-shaped and the crack borders form at the tip the finite angle.