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[ Progress in the theory of plasticity in that respect is well known\ for example\ via the slip!line method\ the _nite element method\ etc[#\ yet many unsolved problems remain[ It is well known that the slip!line method is scanty[ In our opinion the _nite 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 _elds 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\ satis_ed approximately# and stress _elds 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 _nding 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^ _nite deformations^ non!isothermal ~ow^ rapid ~ow\ 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] "0# integration in space with _xed time\ with an accuracy in respect of some parameters^ "1# integration in time of certain ordinary di}erential equations for these parameters[ In the _rst 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 satis_es# the constitutive relations\ all the rest of the equations of mechanics being satis_ed precisely[ The method is illustrated by some test examples