Computing Prager's Kinematic Hardening Mixed-Control Equations in a Pseudo-Riemann Manifold

Abstract

Materials' internal spacetime may bear cer- tain similarities with the external spacetime of special relativity theory. Previously, it is shown that material hardening and anisotropy may cause the internal space- time curved. In this paper we announce the third mech- anism of mixed-control to cause the curvedness of inter- nal spacetime. To tackle the mixed-control problem for a Prager kinematic hardening material, we demonstrate two new formulations. By using two-integrating factors idea we can derive two Lie type systems in the product space of M m+1 ⊗M n+1 . The Lie algebra is a direct sum of so(m, 1) ⊕so(n, 1), and correspondingly the symme- try group is a direct product of SOo(m, 1) ⊗SOo(n, 1), which left acts on a twin-cone. Then, by using the one- integrating factor idea we can convert the nonlinear con- stitutive equations into a Lie type system of u Z=CZ with C ∈sl(5, 1,R) a Lie algebra of the special orthochronous pseudo-linear group SL(5, 1,R). The underlying space is a distorted cone in the pseudo-Riemann manifold. Con- sistent numerical methods are then developed according to these Lie symmetries, and numerical examples are used to assess the performance of new algorithms. The measures in terms of the errors by satisfying the consis- tency condition, strain and stress relative errors and ori- entational errors confirm that the new numerical methods are better than radial return method.