  We consider the numerical treatment of one of the most popular finite strain
models of the viscoelastic Maxwell body. This model is based on the
multiplicative decomposition of the deformation gradient, combined with
Neo-Hookean hyperelastic relations between stresses and elastic strains. The
evolution equation is six dimensional. For the corresponding local initial
value problem, a fully implicit integration procedure is considered, and a
simple explicit update formula is derived. Thus, no local iterative procedure
is required, which makes the numerical scheme more robust and efficient. The
resulting integration algorithm is unconditionally stable and first order
accurate. The incompressibility constraint of the inelastic flow is exactly
preserved. A rigorous proof of the symmetry of the consistent tangent operator
is provided. Moreover, some properties of the numerical solution, like
invariance under the change of the reference configuration and positive energy
dissipation within a time step, are discussed. Numerical tests show that, in
terms of accuracy, the proposed integration algorithm is equivalent to the
classical implicit scheme based on the exponential mapping. Finally, in order
to check the stability of the algorithm numerically, a representative initial
boundary value problem involving finite viscoelastic deformations is
considered. An FEM solution of the representative problem using MSC.MARC is
presented.
