Applications of finite elements in space and time

Abstract

The paper initiates a major extension of the matrix displacement method concerning the analysis of dynamic phenomena in the presence of material and geometric non-linearities. In particular, elasto-plastic behaviour as well as large displacements are taken into account. An iterative procedure of solution of the nonlinear matrix equations is discussed. The application of the theory is described in detail in two examples. The first considers the simple static problem of a rectangular flat strip in a tensile test. The iterative calculation may be carried out for deformations as large as required and shows clearly the necking effect. More ambitious is the second example which demonstrates the non-linear dynamic theory on a cyclindrial deformable billet under the impact by a heavy rigid body. The momentum of the weight and the property of the billet are such that the latter will undergo large plastic deformations. If so required, it is straightforward to incorporate damping and also allow for friction forces on the contacts. The direct applicability of the technique to forging problems is evident. The solution of the dynamic phenomenon is accomplished by extending the discretisation of space also into time. In particular, the inertia forces are taken to vary over a finite time element as a third order polynomial. Exceptional accuracy is achieved by this method.