$$\sigma(t) =\int_{0}^{t}E(t-\tau) \frac{d\epsilon(\tau)} {d\tau} d\tau$$
I am trying to integrate the above equation and impliment the same in FORTRAN, to solve for the material behavior of a visco elastic material. $\epsilon(t)$ is the strain on the material and $\sigma$ is the stress developed.
Here, $E(t-\tau)=E_0*exp(t-\tau)$.
I tried to solve it using integration by parts with no success. Could someone please point me right direction in how to numerically integrate this equation so that I can impliment it in FORTRAN. Thanks in advance.
Since $\frac{\partial}{\partial\tau}E(t-\tau)=-E(t-\tau)$, integrating by parts gives $$ \sigma(t)=E(0)\epsilon(t)-E(t)\epsilon(0)+\int_0^t E(t-\tau)\epsilon(\tau)\,\mathrm{d}\tau $$ Note that with equally-spaced nodes, you can easily reuse the previous evaluations since $E(t+h-\tau)=e^h E(t-\tau)$.