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

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. Pr ogress in the theory of plasticity in that respect is well known, for examp le, via the slip-line method, the finite element method, etc.), yet many un solved problems remain. It is well known that the slip-line method is scant y. 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 an d the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechani cs equations (these equations are said to be "softened", i.e, satisfied app roximately) 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 desc ribe the rheology of actual deformable materials only approximately. We see m 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; f inite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical b oundary 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 velociti es and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution "softens" (approximately satis fies) the constitutive relations, all the rest of the equations of mechanic s being satisfied precisely. The method is illustrated by some test example s. (C) 1998 Elsevier Science Ltd. All rights reserved.