The Maxwell material model consists of a spring of modulus E and a dashpot with viscosity 
Figure 3: The Maxwell Model
Under an applied external stress 
Taking the time derivative above gives,
From the constitutive relationship of spring and dashpot, we have
and,
with
for a constant applied stress.
Substituting (\ref{eq:3}) and (\ref{eq:4}) into (\ref{eq:2}) gives the following governing first order differential equation,
\begin{equation} \label{eq:5}
\dot{\varepsilon }=\frac{{\dot{\sigma }}}{E}+\frac{1}{\eta }\sigma
\end{equation}
which may now be used to predict the creep and relaxation behavior of a material by introducing suitable input conditions.
Creep Behavior in the Maxwell Model
When the material is subjected to a constant stress condition, $\sigma ={{\sigma }_{0}}$, then $\dot{\sigma }=0$ between time 0 to t s. The output (strain versus time) corresponding to this input condition can be obtained by integrating (\ref{eq:5}) as
\begin{equation} \label{eq:6}
\int\limits_{\varepsilon (0)}^{\varepsilon (t)}{d\varepsilon }=\frac{1}{E}\int\limits_{\sigma (0)}^{\sigma (t)}{d\sigma }+\frac{\sigma }{\eta }\int\limits_{0}^{t}{dt}
\end{equation}
where, $\varepsilon (0),\ \sigma \left( 0 \right)$ and $\varepsilon (t),\ \sigma \left( t \right)$ represents the initial stress and strain and its subsequent value at time t respectively. By substituting $\dot{\sigma }=0$ in (\ref{eq:6}) we can obtain,
\begin{equation} \label{eq:7}
\varepsilon (t)-\varepsilon (0)=\frac{{{\sigma }_{0}}}{\eta }t
\end{equation}
When a constant stress is applied the arrangement, it is assumed that the dashpot takes finite time to respond, whereas the stretching of the spring is instantaneous. Therefore, the initial strain results from the spring alone,
\begin{equation} \label{eq:8}
\varepsilon (0)=\frac{{{\sigma }_{0}}}{E}
\end{equation}
By substituting (\ref{eq:8}) into (\ref{eq:7}), the strain can be computed as,
\begin{equation} \label{eq:9}
\varepsilon (t)=\frac{{{\sigma }_{0}}}{\eta }t+\frac{{{\sigma }_{0}}}{E}\text{ }\Rightarrow \text{ }\varepsilon (t)={{\sigma }_{0}}\left( \frac{1}{\eta }t+\frac{1}{E} \right)\text{ }\Rightarrow \text{ }\varepsilon (t)={{\sigma }_{0}}{{J}_{creep}}(t)
\end{equation}
where, ${{J}_{creep}}(t)=\left( t/\eta +1/E \right)$ is the creep compliance for Maxwell Model.
Relaxation Behavior in Maxwell Model
When the material is subjected to a constant strain, $\varepsilon ={{\varepsilon }_{0}}$, between time 0 to t s., then substituting $\dot{\varepsilon }=0$ into (\ref{eq:5}) gives
\begin{equation} \label{eq:10}
0=\frac{{\dot{\sigma }}}{E}+\frac{1}{\eta }\sigma \text{ }\Rightarrow \text{ }\frac{1}{E}\frac{d\sigma }{dt}=-\frac{\sigma }{\eta }\Rightarrow \text{ }\frac{d\sigma }{\sigma }=-\frac{E}{\eta }dt
\end{equation}
Integration of (\ref{eq:10}) gives,
\begin{equation} \label{eq:11}
\int\limits_{\sigma \left( 0 \right)}^{\sigma \left( t \right)}{\frac{d\sigma }{\sigma }}=-\frac{E}{\eta }\int\limits_{0}^{t}{dt}\text{ }\Rightarrow \text{ }\ln \sigma \left( t \right)-\ln \sigma \left( 0 \right)=-\frac{E}{\eta }t
\end{equation}
When a constant strain is applied to spring, the stretch of spring is instantaneous, whereas the dashpot remains undeformed. Therefore, the initial stress is
\begin{equation} \label{eq:12}
\sigma \left( 0 \right)=E{{\varepsilon }_{0}}
\end{equation}
By substituting (\ref{eq:12}) into (\ref{eq:11}), the stress can be computed as,
\begin{equation} \label{eq:13}
\ln \sigma \left( t \right)=\ln \left( E{{\varepsilon }_{0}} \right)-\frac{E}{\eta }t\text{ }\Rightarrow \text{ }\sigma \left( t \right)=E{{\varepsilon }_{0}}{{e}^{-\left( \frac{E}{\eta } \right)t}}\text{ }\Rightarrow \text{ }\sigma \left( t \right)={{\varepsilon }_{0}}{{E}_{relax}}(t)
\end{equation}
where, ${{E}_{relax}}(t)=E{{e}^{-\left( \frac{E}{\eta } \right)t}}$ is the relaxation modulus for the Maxwell Model.