A shock loading on a bar with a central crack

Linear thermoelasticity is studied of a plane regular truss formed by four families of The paper is concerned with the problem of calculating the time dependence of the stress intensity factor for a plane-strain bar with a stationary central crack caused by the opening mode. A uniformly distributed load has been immediately imposed on the basis of the bar and remained unchanged later. The model of the crack with cohesive forces distributed by Barenblatt’s postulates is used. In this case the stress intensity factor is a result of the calculation of the released energy that is determined by the cohesive forces. The solution is found with a new numerical method that is an adaptation of the method of lines to dynamic fracture mechanics problems. The Crank-Nicolson implicit finite-difference scheme is used for time integration. Boundary problems arising at each step of time integration are solved by the finite element method. The special cohesive finite elements are used, so that the solution of the problem could satisfy Barenblatt’s postulates. Previously these elements were used to solve quasi-static nonlinear fracture mechanics problems. By introducing the additional degrees of freedom of the nodes lying on the crack line, it becomes possible to ensure a smooth closing of the crack edges at its tip; and it is equivalent to the absence of the singularity stress and strain fields at its tip. The cohesive forces are calculated as the constraints. Their field of action (cohesive zone) is localized within the finite element which is adjacent to the tip of the crack. Thus, the smaller the finite element mesh is, the better it satisfies the requirement of Barenblatt’s theory. This requirement concerns the length of the cohesive zone which is small compared to the length of the crack. The stated problem is called Chen’s problem and had earlier been solved by researchers that used different methods. The proximity of the obtained results makes it possible to consider Chen’s problem as a test; and its solution obtained by the developed method agrees well with the data of other researchers.