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

doi:10.3970/cmes.2006.012.161
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 12, No. 3, pp. 161-180, 2006
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Keywords Prager kinematic hardening rule, Integrating factors, Mixed-controls, Pseudo-Riemann manifold, Internal spacetime, Internal symmetry, Consistent numerical schemes
Abstract Materials' internal spacetime may bear certain similarities with the external spacetime of special relativity theory. Previously, it is shown that material hardening and anisotropy may cause the internal spacetime curved. In this paper we announce the third mechanism of mixed-control to cause the curvedness of internal 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${\@mathbb M}^{m+1} \otimes {\@mathbb M}^{n+1}$. The Lie algebra is a direct sum of$so(m,1) \oplus so(n,1)$, and correspondingly the symmetry group is a direct product of$SO_o(m,1) \otimes SO_o(n,1)$, which left acts on a twin-cone. Then, by using the one-integrating factor idea we can convert the nonlinear constitutive equations into a Lie type system of$\mathaccentV {dot}05F{\bf Z}={\bf C}{\bf Z}$ with${\bf C} \in sl (5,1,{\@mathbb R})$ a Lie algebra of the special orthochronous pseudo-linear group$SL(5,1,{\@mathbb R})$. The underlying space is a distorted cone in the pseudo-Riemann manifold. Consistent 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 consistency condition, strain and stress relative errors and orientational errors confirm that the new numerical methods are better than radial return method.
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