TABLE OF CONTENTS We now begin to outline the procedures and algorithms used in SHEET-S and SHEET-3: static implicit, rigid viscoplastic, finite-element programs for two- and three-dimensional analyses of sheet-forming operations.
For a section analysis like that performed by SHEET-S, a plane-strain line element [5] is the simplest one, as shown in Fig. 11.1. The longitudinal strain increment, ? ? l, during the incremental time step can be written as
| (11.1) |  |
where L and l are respectively element lengths in the current and subsequent configurations, expressed in the x- z plane in terms of nodal coordinates, as defined in Fig. 11.1:
| (11.2) |  |
| (11.3) |  |
The nodal coordinates are related by the incremental displacement, ?u.
| (11.4) |  |
or
| (11.5) |  |
The derivative of the strain increment, ? ? 1, with respect to the incremental displacement ? u, is needed to find the internal force for equilibrium. The required form can be obtained from Eqs. (11.4) and (11.5) as follows:
| (11.6) |  |
In plane-strain analysis, and either rigid plasticity (or ignoring elastic volume change), the strains are simply related: d ? 1 = ?d ? 3, d ? 2 = 0. For any yield function with isotropic hardening, we can write
| (11.7) |  |
where we define F ps
