A method for calculating the stress–strain state in the general boundary-value problem of metal forming—part 1

Abstract

To simulate metal-forming processes, one has to calculate the stress–strain state of the metal, i.e. to solve the relevant boundary-value problems. Progress in the theory of plasticity in that respect is well known, for example, via the slip-line method, the finite element method, etc.) , yet many unsolved problems remain. It is well known that the slip-line method is scanty. In our opinion the finite element method has an essential drawback. (No one is against the idea of the discretization of the body being deformed and the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechanics equations (these equations are said to be softened, i.e, satisfied approximately) and stress fields can be considered poor for solution of the subsequent fracture problem. We believe that it is preferable to construct an approximate solution by the FEM and soften the constitutive relations (not Newtonian mechanics equations) , especially as, in any event, they describe the rheology of actual deformable materials only approximately. We seem to have succeeded in finding the solution technique. Here we present some new results for solving rather general boundary-value problems which can be characterized by the following: the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical boundary conditions, etc. Solution by the method proposed can be made in two stages: (1) integration in space with fixed time, with an accuracy in respect of some parameters; (2) integration in time of certain ordinary differential equations for these parameters. In the first stage the method is based on the principle of virtual velocities and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution softens (approximately satisfies) the constitutive relations, all the rest of the equations of mechanics being satisfied precisely. The method is illustrated by some test examples.