Having trouble of reaching a specific expression for Maxwell's rheological model with 2 modes from a given formula. I have an expression for a Maxwell viscoelastic rheological model with 2 modes.
I have tried derivation for both sides but I always seem to reach an impasse.
I tried to get second derivation as well but im missing some terms that should have stayed behind.
I know have to get the second derivation of the original expression to get to the first (I CAN SEE THAT) but i cant really get it to match.
Any opinions, inputs on the matter?
Tau is a stress tensor thus irrelevant with time and λ1, λ2 are relevant as they are relaxation times.
original expression
$\boldsymbol{\tau}=\int_{-\infty}^{t}\left(\mathrm{G}_{1} \mathrm{e}^{-\left(t-t^{\prime}\right) / \lambda_{1}}+\mathrm{G}_{2} \mathrm{e}^{-\left(t-t^{\prime}\right) / \lambda_{2}}\right) \dot{\gamma}\left(\mathrm{t}^{\prime}\right) \mathrm{dt}^{\prime}$
final expression $\boldsymbol{\tau}+\left(\lambda_{1}+\lambda_{2}\right) \frac{\partial \boldsymbol{\tau}}{\partial \mathrm{t}}+\lambda_{1} \lambda_{2} \frac{\partial^{2} \boldsymbol{\tau}}{\partial \mathrm{t}^{2}}=\left(\eta_{1}+\eta_{2}\right)\left[\dot{\gamma}+\left(\frac{\lambda_{2} \eta_{1}+\lambda_{1} \eta_{2}}{\eta_{1}+\eta_{2}}\right) \frac{\partial \dot{\boldsymbol{\gamma}}}{\partial t}\right]$
For
$\eta_{1}=\lambda_{1} *{G}_{1}$
$\eta_{2}=\lambda_{2} *{G}_{2}$
 A: Let $\phi_1(t) = \int_{-\infty}^{t}\mathrm{G}_{1} \mathrm{e}^{-\left(t-t^{\prime}\right) / \lambda_{1}} \dot{\gamma}\left(\mathrm{t}^{\prime}\right) \mathrm{dt}^{\prime}$ and  $ \phi_2(t) = \int_{-\infty}^{t}\mathrm{G}_{2} \mathrm{e}^{-\left(t-t^{\prime}\right) / \lambda_{2}} \dot{\gamma}\left(\mathrm{t}^{\prime}\right) \mathrm{dt}^{\prime}$.
Using Leibniz's rule $\frac{\partial }{\partial t} \int_{-\infty}^t f(t,t') \, dt' =f(t,t) + \int_{-\infty}^t \frac{\partial }{\partial t}f(t,t') \, dt'$, we get for $j = 1,2$,
$$\frac{\partial \phi_j}{\partial t} = G_j\dot{\gamma}\left(\mathrm{t}\right)- \frac{1}{\lambda_j}\phi_j(t),\quad  \frac{\partial^2 \phi_j}{\partial t^2} = G_j \frac{\partial \dot{\gamma}}{\partial t} + \frac{1}{\lambda_j^2}\phi_j(t)$$
Thus,
$$\tau = \phi_1 + \phi_2 ,\\\frac{\partial \tau}{\partial t} = \frac{\partial \phi_1}{\partial t}+ \frac{\partial \phi_2}{\partial t} = G_1\dot{\gamma}+ G_2 \dot{\gamma} - \frac{\phi_1}{\lambda_1} - \frac{\phi_2}{\lambda_2},$$
and
$$\frac{\partial^2 \tau}{\partial t^2} = \frac{\partial^2 \phi_1}{\partial t^2}+ \frac{\partial \phi_2^2}{\partial t^2} = G_1\frac{\partial \dot{\gamma}}{\partial t}+ G_2\frac{\partial \dot{\gamma}}{\partial t } - \frac{1}{\lambda_1}\frac{\partial \phi_1}{\partial t} - \frac{1}{\lambda_2}\frac{\partial \phi_2}{\partial t} = \\G_1\frac{\partial \dot{\gamma}}{\partial t}+ G_2\frac{\partial \dot{\gamma}}{\partial t } - \frac{G_1 \dot{\gamma}}{\lambda_1}+  \frac{\phi_1}{\lambda_1^2}- \frac{G_2 \dot{\gamma}}{\lambda_2}+  \frac{\phi_2}{\lambda_2^2}$$
We then have
$$(\lambda_1+ \lambda_2)\frac{\partial \tau}{\partial t}  = -(\phi_1 + \phi_2) - \frac{\lambda_2}{\lambda_1} \phi_1 - \frac{\lambda_1}{\lambda_2} \phi_2 + (\lambda_1+\lambda_2)(G_1+G_2)\dot{\gamma} ,\\\lambda_1 \lambda_2 \frac{\partial^2 \tau}{\partial t^2} = \lambda_1\lambda_2(G_1 + G_2)\frac{\partial \dot{\gamma}}{\partial t } +  \frac{\lambda_2}{\lambda_1} \phi_1 + \frac{\lambda_1}{\lambda_2} \phi_2-  \lambda_2G_1 \dot{\gamma} - \lambda_1G_2 \dot{\gamma}$$
Adding terms, we get
$$ \tau + (\lambda_1+ \lambda_2)\frac{\partial \tau}{\partial t}+ \lambda_1 \lambda_2 \frac{\partial^2 \tau}{\partial t^2} \\=  (\lambda_1 G_1+ \lambda_1G_2 + \lambda_2G_1 + \lambda_2 G_2) \dot{\gamma} - (\lambda_2G_1 + \lambda_1 G_2) \dot{\gamma} + (\lambda_2 \lambda_1 G_1 + \lambda_1 \lambda_2 G_2) \frac{\partial \dot{\gamma}}{\partial t} \\ = (\lambda_1G_1+ \lambda_2 G_2) \dot{\gamma} +  (\lambda_2 \lambda_1 G_1 + \lambda_1 \lambda_2 G_2) \frac{\partial \dot{\gamma}}{\partial t}\\ = (\eta_1 + \eta_2) \dot{\gamma} + (\lambda_2 \eta_1 + \lambda_1 \eta_2)\frac{\partial \dot{\gamma}}{\partial t} = (\eta_1 + \eta_2)  \left[\dot{\gamma} +  \frac{\lambda_2 \eta_1 + \lambda_1 \eta_2}{\eta_1+ \eta_2}\frac{\partial \dot{\gamma}}{\partial t}\right]$$
