In the previous chapters we have discussed only the applications of flow formulation to the analysis of metal-forming processes. Lately, elastic-plastic (solid) formulations have evolved to produce techniques suitable for metal-forming analysis. This evolution is the result of developments achieved in large-strain formulation, beginning from the infinitesimal approach based on the Prandtl–Reuss equation. A question always arises as to the selection of the approach—“flow” approach or “solid” approach. A significant contribution to the solution of this question was made through a project in 1978, coordinated by Kudo, in which an attempt was made to examine the comparative merits of various numerical methods. The results were compiled for upsetting of circular solid cylinders under specific conditions, and revealed the importance of certain parameters used in computation, such as mesh systems and the size of an increment in displacement. This project also showed that the solid formulation needed improvement, particularly in terms of predicting the phenomenon of folding. For elastic-plastic materials, the constitutive equations relate strain–rate to stress–rates, instead of to stresses. Consequently, it is convenient to write the field equation in the boundary-value problem for elastic-plastic materials in terms of the equilibrium of stress rates. In this chapter, the basic equations for the finite-element discretization involved in solid formulations are outlined both for the infinitesimal approach and for large-strain theory. Further, the solutions obtained by the solid formulation are compared with those obtained by the flow formulation for the problems of plate bending and ring compression. A discussion is also given concerning the selection of the approach for the analysis. In conclusion, significant recent developments in the role of the finite-element method in metal-forming technology are summarized. The field equation for the boundary-value problem associated with the deformation of elastic-plastic materials is the equilibrium equation of stress rates. As stated in Chap. 1 (Section 1.3), the internal distribution of stress, in addition to the current states of the body, is supposed to be known, and the boundary conditions are prescribed in terms of velocity and traction-rate.
Finite element formulations for large strain anisotropic material with inextensible fibers
