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Summary  A popular version of the finite‐strain Maxwell fluid is considered, which is based on the multiplicative decomposition of the deformation gradient tensor. The model combines Newtonian viscosity with hyperelasticity of the Mooney‐Rivlin type; it is a special case of the viscoplasticity model proposed by Simo and Miehe in 1992. A simple, efficient, and robust implicit time‐stepping procedure is suggested. Lagrangian and Eulerian versions of the algorithm are available, with equivalent properties. The numerical scheme is iteration free, unconditionally stable, and first order accurate. It exactly preserves the inelastic incompressibility, symmetry, and positive definiteness of the internal variables and w‐invariance. The accuracy of the stress computations is tested using a series of numerical simulations involving a nonproportional loading and large strain increments. In terms of accuracy, the proposed algorithm is equivalent to the modified Euler backward method with exact inelastic incompressibility; the proposed method is also equivalent to the classical integration method based on exponential mapping. Since the new method is iteration free, it is more robust and computationally efficient. The algorithm is implemented into MSC.MARC, and a series of initial boundary value problems is solved to demonstrate the usability of the numerical procedures.

Efficient time stepping for the multiplicative Maxwell fluid including the Mooney‐Rivlin hyperelasticity