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We consider the numerical treatment of one of the most popular finite strainmodels of the viscoelastic Maxwell body. This model is based on themultiplicative decomposition of the deformation gradient, combined withNeo-Hookean hyperelastic relations between stresses and elastic strains. Theevolution equation is six dimensional. For the corresponding local initialvalue problem, a fully implicit integration procedure is considered, and asimple explicit update formula is derived. Thus, no local iterative procedureis required, which makes the numerical scheme more robust and efficient. Theresulting integration algorithm is unconditionally stable and first orderaccurate. The incompressibility constraint of the inelastic flow is exactlypreserved. A rigorous proof of the symmetry of the consistent tangent operatoris provided. Moreover, some properties of the numerical solution, likeinvariance under the change of the reference configuration and positive energydissipation within a time step, are discussed. Numerical tests show that, interms of accuracy, the proposed integration algorithm is equivalent to theclassical implicit scheme based on the exponential mapping. Finally, in orderto check the stability of the algorithm numerically, a representative initialboundary value problem involving finite viscoelastic deformations isconsidered. An FEM solution of the representative problem using MSC.MARC ispresented.

An explicit solution for implicit time stepping in finite strain viscoelasticity