Computation in finite-strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations Original Research Article

In this article the interpretation of current non-linear finite element calculations applied to constitutive equations of evolutionary type as a system of differential–algebraic equations is continued [P. Ellsiepen, S. Hartmann, Int. J. Numer. Methods Engrg. 51 (2001) 679]. This procedure is applied to a model of finite-strain viscoelasticity which incorporates a particular model of non-linear rate dependence. Three specific topics are studied: firstly, we investigate a possible treatment of the interpretation in view of mixed finite elements. Secondly, the application of stiffly accurate diagonally implicit Runge–Kutta methods, which yield on element level the same structure as a Backward-Euler-based integration step, is analysed. It is shown that the resulting stress algorithm is reducible to the solution of only three non-linear equations at each Gauss point (one Maxwell element). Lastly, the effect with respect to expense and achievable accuracy of a time-adaptive procedure is focused, which is necessary in the case of different time scales such as relaxation or creep dominated processes.