Internal symmetry groups for the Drucker-Prager material model of plasticity and numerical integrating methods

Abstract Internal symmetries for the Drucker–Prager material model and numerical methods based on them are investigated here. We first convert the non-linear constitutive equations to a Lie type system X = A ( X ,t) X , where A ∈sl(5,2, R ) is a Lie algebra of the special orthochronous pseudo-linear group SL(5,2, R ) . The underlying space of Drucker–Prager model in the plastic phase is a pseudo-sphere in the pseudo-Riemann manifold, which is locally a pseudo-Euclidean space E 7 5,2 , and whose metric tensor is indefinite having signature (5,2) and also depends on the temporal component. Then, we split the Drucker–Prager yield condition into two “sub-yield” conditions, which together with two integrating factors idea led us to derive two Lie type systems in the product space M 5+1 ⊗ M 1+1 . The Lie algebra is the direct sum so(5,1)⊕so(1,1), and the symmetry group is thus SOo(5,1)⊗SOo(1,1). These results are essential from computational aspect. Based on these symmetry groups exponential mappings are developed, which update stress points exactly on the yield surface at every time increment without any iteration. As an example, the results calculated by using the two different group preserving schemes are compared to that calculated by the Runge–Kutta scheme together with the Newton–Raphson iterative solution of a non-linear algebraic equation which enforcing the consistency condition.