An explicit solution for implicit time stepping in multiplicative finite strain viscoelasticity

Abstract

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.

Dissipative heating of elastomers: a new modelling approach based on finite and coupled thermomechanics

“…Therefore, the presented material model is implemented in the open-source FE-code PANDAS [12]. Therefore, the local balance equations of internal energy (45) and of the momentum are transferred to their weak form by multiplying them with a testing function and integrating them over the calculated body's volume.…”

“…1.2), this self-heating is dependent on both the frequency and the amplitude of the deformation. Equation (45) contains both of these influences. The reason for the self-heating is the non-equilibrium stress, the inelastic deformation and its rate.…”

“…This evolution equation is solved in the context of a finite element calculation by the method of [45]. For the inclusion of the temperature-dependent mechanical behaviour, the standard WLF-equation is introduced with the standard parameter C 1 = 17.5, C 2 = 52 K, cf.…”

An explicit update formula for implicit time integration within finite strain viscoelasticity

“…The numerical integration of the inelastic flow rule is accomplished using the implicit integration scheme of Euler-backward method. To this end, Shutov et al (2013) recently proposed a new closed form solution for the corresponding time stepping algorithm. The algorithm retains all favourable characteristics of the Euler-backward method.…”

“…The evaluation of the free energy function leads to eq. (1) 2 for the 2 nd Piola-Kirchhoff stress tensor T as well as a remaining dissipation (1) 3 which has to be non-negative [3].Here, (·) = d(·)/dt is the Lagrangian (material) time derivative. Unimodular and deviatoric parts of a tensor X are denoted by X and X , respectively.…”