relaxation-Malkin¶
Module: solid
Category: materialprop
Type string: "relaxation-Malkin"
Parameters¶
| Name | Description | Default | Units |
|---|---|---|---|
tau1 |
min. relaxation time | 0 | [t] |
tau2 |
max. relaxation time | 0 | [t] |
beta |
power exponent | 0 | [] |
Description¶
This relaxation function is derived from the continuous relaxation spectrum given in 1. The material type for this relaxation function is relaxation-Malkin.
These parameters must satisfy \(\tau_{2}>\tau_{1}>0\). When \(\beta>1\), the reduced relaxation function for this material type is given by
\[
g\left(t\right)=\frac{\left(\beta-1\right)t^{1-\beta}}{\tau_{1}^{1-\beta}-\tau_{2}^{1-\beta}}\left[\Gamma\left(\beta-1,\frac{t}{\tau_{2}}\right)-\Gamma\left(\beta-1,\frac{t}{\tau_{1}}\right)\right]
\]
where \(\Gamma\left(a,z\right)\) is the incomplete gamma function. In the limit as \beta\to1 this function reduces to the form given by 2,
\[
g\left(t\right)=\frac{\Gamma\left(0,\frac{t}{\tau_{2}}\right)-\Gamma\left(0,\frac{t}{\tau_{1}}\right)}{\ln\frac{\tau_{2}}{\tau_{1}}}=\frac{\text{Ei}\left(-\frac{t}{\tau_{2}}\right)-\text{Ei}\left(-\frac{t}{\tau_{1}}\right)}{\ln\frac{\tau_{1}}{\tau_{2}}}
\]
where \(\text{Ei}\left(z\right)\) is the exponential integral function.