Energy-consistent time integration for nonlinear viscoelasticity

Abstract

This paper is concerned with the numerical solution of the evolution equations of thermomechanical systems, in such a way that the scheme itself satisfies the laws of thermodynamics. Within this framework, we present a novel integration scheme for the dynamics of viscoelastic continuum bodies in isothermal conditions. This method intrinsically satisfies the laws of thermodynamics arising from the continuum, as well as the possible additional symmetries. The resulting solutions are physically accurate since they preserve the fundamental physical properties of the model. Furthermore, the method gives an excellent performance with respect to robustness and stability. Proof for these claims as well as numerical examples that illustrate the performance of the novel scheme are provided.

Appendix: Time-discrete constitutive laws

The key factor in the proposed consistent method is the use of the discrete derivative operator for the time-approximation of the constitutive laws given by (6). As we have previously mentioned, such operator has a different elaboration depending on the number of internal variables. This fact somehow makes the method slightly different according to this election. For simplicity, we have implemented the consistent method with only one internal variable and, accordingly, we present here the elaborated expressions for the time-discrete approximation of the constitutive laws given by (38) for $\alpha =1$. These expressions should be programmed in order to implement the consistent method in any FE-based software. Taking from [4], the expression for the discrete derivative operator in a partitioned case, we have

$$\begin{aligned}&\tilde{{\mathsf {S}}} = \overline{{\mathsf {S}}}_{n+\frac{1}{2}}^h + \frac{\varDelta \varPsi _{n+1} + \varDelta \varPsi _n - \overline{{\mathsf {S}}}^h_{n+\frac{1}{2}} :\varDelta {\mathsf {C}}^h}{\varDelta {\mathsf {C}}^h:\varDelta {\mathsf {C}}^h}\varDelta {\mathsf {C}}^h,\end{aligned}$$

(57)

$$\begin{aligned}&\tilde{{\mathsf {Q}}} = \overline{{\mathsf {Q}}}^h_{n+\frac{1}{2}} - \frac{\frac{1}{2}\left[ \varDelta \varPsi ^{n+1} + \varDelta \varPsi ^n\right] + \overline{{\mathsf {Q}}}^h_{n+\frac{1}{2}}: \varDelta {\mathsf {\Gamma }}^h}{\varDelta {\mathsf {\Gamma }}^h:\varDelta {\mathsf {\Gamma }}^h}\varDelta {\mathsf {\Gamma }}^h,\nonumber \\ \end{aligned}$$

(58)

where we have introduced the following simplified notation

$$\begin{aligned}&\varDelta \varPsi _i = \varPsi \left( {\mathsf {C}}^h_{n+1},{\mathsf {\Gamma }}^h_i\right) - \varPsi \left( {\mathsf {C}}^h_{n}, {\mathsf {\Gamma }}^h_i\right) \end{aligned}$$

(59)

$$\begin{aligned}&\varDelta \varPsi ^i = \varPsi \left( {\mathsf {C}}^h_i,{\mathsf {\Gamma }}^{\scriptstyle {h}}_{n+1}\right) - \varPsi \left( {\mathsf {C}}^h_i,{\mathsf {\Gamma }}^{\scriptstyle {h}}_n\right) \end{aligned}$$

(60)

$$\begin{aligned}&\overline{{\mathsf {S}}}^h_{n+\frac{1}{2}} =\frac{1}{2}\left[ {\mathsf {S}}_{n+1}^{n+ \frac{1}{2}} + {\mathsf {S}}_n^{n+\frac{1}{2}}\right] \hbox { with } {\mathsf {S}}_i^j \!= \!\left. 2\frac{\partial \!\varPsi ({\mathsf {C}}^h, {\mathsf {\!}}{\Gamma }^h_i)}{\partial {\mathsf {C}}^h}\right| _j\nonumber \\\end{aligned}$$

(61)

$$\begin{aligned}&\overline{{\mathsf {Q}}}^h_{n+\frac{1}{2}} \!=\! \frac{1}{2}\left[ {\mathsf {Q}}^{n+1}_{n+\frac{1}{2}} \!+\! {\mathsf {Q}}^n_{n\!+\!\frac{1}{2}}\right] \hbox { with }{\mathsf {Q}}_i^j\!=\!\left. -\frac{\partial \varPsi ({\mathsf {C}}^h_j, {\mathsf {\Gamma }}^h)}{\partial {\mathsf {\Gamma }}^{h}}\right| _i\nonumber \\ \end{aligned}$$

(62)

As with the case of the Energy-Momentum method for nonlinear elasticity, the inclusion of the above expressions in the FE-implementation leads to an increment of the computational cost (with respect to classical methods) and to non-symmetric tangent matrices.