Consistent tangent operators for constitutive rate equations

A general approach for obtaining the consistent tangent operator for constitutive rate equations is presented. The rate equations can be solved numerically by the user's favourite time integrator. In order to obtain reliable results, the substepping in integration should be based on a control of the local error. The main ingredient of the consistent tangent operator, namely the derivative of the stress with respect to the strain increment must be computed simultaneously with the same integrator, applied to a numerical approximation of the variational equations. This information enables finite-element packages to assemble a consistent tangent operator and thus guarantees quadratic convergence of the equilibrium iterations. Several numerical examples with a hypoplastic constitutive law are given. As numerical integrator we used a second-order extrapolated Euler method. Quadratic convergence of the equilibrium iteration is shown. Copyright (C) 2002 John Wiley Sons, Ltd.