19.2.2 Double Power Law

Another example of a creep function J(t,$\tau$) is the Double Power Law which reads

$$\tau {\frac{{1}}{{E(\tau)}}} \left(\vphantom{ 1 + \alpha \: \tau^{-d} ( t - \tau )^{p} }\right. \alpha \tau^{{-d}}_{} \tau \left.\vphantom{ 1 + \alpha \: \tau^{-d} ( t - \tau )^{p} }\right)$$ (19.23)

This relation is restricted to p > 0 , therefore the momentary compliance at time t is J(t, t) = ${\frac{{1}}{{E(t)}}}$ . To circumvent storage of the entire history, the part f (t) = (t - $\tau$)^p is expanded into a Taylor series around t - $\tau$ = t_d . Truncating the Taylor series to powers of 5 and collecting equal powers of $\tau$ results in

$$\tau \sum_{{r=0}}^{{5}} \tau^{{r}}_{}$$ (19.24)

With h_r a function of t - t_d , depending on the power p . The Taylor series converges to the required creep function at the interval 0 < t < 2t_d . The development point t_d should therefore be taken halfway the maximum analysis time. Substitution of (19.23) and (19.24) into (19.13) and (19.14) yields

$$\begin{split}\frac{1}{\tilde{E}(t^{*})} &= \frac{1}{E(t^{*})}... ...+1} h_{r} ( t + \Delta t - t_{\mathrm{d}} ) \right) \end{split}$$ (19.25)

and

$$\tilde{{\boldsymbol{\sigma}}} \tilde{{E}} \alpha \sum_{{r=0}}^{{5}} \left(\vphantom{ h_{r} (t+\Delta t - t_{\mathrm{d}}) - h_{r} (t-t_{\mathrm{d}})}\right. \Delta \left.\vphantom{ h_{r} (t+\Delta t - t_{\mathrm{d}}) - h_{r} (t-t_{\mathrm{d}})}\right) \tilde{{\boldsymbol{\varepsilon}}}_{{r}}^{}$$ (19.26)

where

$$\tilde{{\boldsymbol{\varepsilon}}}_{{r}}^{} \int_{{0}}^{{t}} {\frac{{1}}{{E(\tau)}}} \tau^{{r-d}}_{} \dot{{\boldsymbol{\sigma}}} \tau$$ (19.27)

The item $\tilde{{\boldsymbol{\varepsilon}}}$ can be calculated by summation during the analysis as

$$\begin{split}\tilde{\boldsymbol{\varepsilon}}_{r} (t+\Delta t... ...t } \frac{ (t+\Delta t)^{r-d+1} - t^{r-d+1}}{r-d+1} \end{split}$$ (19.28)