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The Kelvin (or Kelvin-Voigt) material model consists of a spring having modulus $E$ and a dash pot with viscosity $\eta $, coupled in parallel, as shown in Figure below.
Figure: The Kelvin-Voigt Model
It is assumed that the strain experienced by the spring is the same as that experienced by the dash pot,
\begin{equation} \label{eq:1}
\[{{\varepsilon }_{spring}}={{\varepsilon }_{dashpot}}=\varepsilon
\end{equation}
and that the applied external stress equals the sum of the stress within the spring and dashpot elements,
\begin{equation} \label{eq:2}
\[\sigma ={{\sigma }_{spring}}+{{\sigma }_{dashpot}}=E{{\varepsilon }_{spring}}+\eta {{\dot{\varepsilon }}_{dashpot}}
\end{equation}
Hence, from Eqs. (\ref{eq:1}) and (\ref{eq:2}) the following governing differential equation for the Kelvin Voigt model can be constructed:
\begin{equation} \label{eq:3}
\[\sigma =E\varepsilon +\eta \dot{\varepsilon }
\end{equation}
Equation (\ref{eq:3}) may now be used for predicting the creep and relaxation behavior associated with the Kelvin-Voigt model by inputting the appropriate initial condition.
Creep Behavior in Kelvin-Voigt Model
If a sudden constant stress ${{\sigma }_{0}}$ is applied, then Eq. (\ref{eq:3}) leads to the following first order differential equation,
\begin{equation} \label{eq:4}
\[{{\sigma }_{0}}=E\varepsilon +\eta \dot{\varepsilon }
\end{equation}
The homogeneous solution and particular solution of (\ref{eq:4}) is,
\begin{equation} \label{eq:5}
\[0=E{{\varepsilon }_{h}}+\eta {{\dot{\varepsilon }}_{h}}\text{ }\Rightarrow \text{ }{{\dot{\varepsilon }}_{h}}=-\frac{E}{\eta }{{\varepsilon }_{h}}\text{ }\Rightarrow \text{ }{{\varepsilon }_{h}}(t)={{A}_{1}}{{e}^{-\frac{E}{\eta }t}}
\end{equation}
where, ${{A}_{1}}$ and ${{A}_{2}}$ are constants. The solution for strain is,
\begin{equation} \label{eq:6}
\[\varepsilon (t)={{\varepsilon }_{h}}(t)+{{\varepsilon }_{p}}(t)={{A}_{1}}{{e}^{-\frac{E}{\eta }t}}+{{A}_{2}}
\end{equation}
Substituting Eq. (\ref{eq:6}) into Eq. (\ref{eq:4}) gives,
\begin{equation} \label{eq:7}
\[{{\sigma }_{0}}=E\left( {{A}_{1}}{{e}^{-\frac{E}{\eta }t}}+{{A}_{2}} \right)+\eta \left( -\frac{E}{\eta }{{A}_{1}}{{e}^{-\frac{E}{\eta }t}} \right)\text{ }\Rightarrow {{A}_{2}}=\frac{{{\sigma }_{0}}}{E}
\end{equation}
Now by substituting ${{A}_{2}}$ from Eq. (\ref{eq:7}) and initial strain $\varepsilon (0)=0$ into Eq. (\ref{eq:6}), one obtains,
\begin{equation} \label{eq:8}
\[\varepsilon (t)={{A}_{1}}{{e}^{-\frac{E}{\eta }t}}+\frac{{{\sigma }_{0}}}{E}\text{ }\Rightarrow \varepsilon (0)=0={{A}_{1}}+\frac{{{\sigma }_{0}}}{E}\Rightarrow {{A}_{1}}=-\frac{{{\sigma }_{0}}}{E}
\end{equation}
Hence from Eqs. (\ref{eq:6}), (\ref{eq:7}) and (\ref{eq:8}), the solution for strain can be obtained as,
\begin{equation} \label{eq:9}
\[\varepsilon (t)=\frac{{{\sigma }_{0}}}{E}\left( 1-{{e}^{-\frac{E}{\eta }t}} \right)={{\sigma }_{0}}{{J}_{creep}}(t)
\end{equation}
where, \[{{J}_{creep}}(t)=\frac{1}{E}\left( 1-{{e}^{-\frac{E}{\eta }t}} \right)\] is the relaxation modulus for the Kelvin-Voigt Model.
If the stress is completely removed, ${{\sigma }_{0}}=0$ at time $t=\tau$, as shown in Fig. *, the strain can be computed numerically or by Laplace transform. While assuming the zeros initial conditions, the strain can be computed as,
\begin{equation} \label{eq:10}
\[\varepsilon (t)=\left\{ \begin{array}{*{35}{l}}
\frac{{{\sigma }_{o}}}{E}\left( 1-{{e}^{-(E/\eta )t}} \right), & 0<t<\tau \\ \frac{{{\sigma }_{o}}}{E}{{e}^{-(E/\eta )t}}\left( {{e}^{(E/\eta )\tau }}-1 \right), & \quad \ t>\tau \\
\end{array} \right
\end{equation}
Fig. 8. Creep Behavior of the Kelvin-Voigt Model
Relaxation Behavior in Kelvin-Voigt Model
This model is not used for predicting relaxation behavior because an instantaneous change in strain $d\varepsilon ={{\varepsilon }_{0}}$ with $dt\to 0$ gives infinite strain $\sigma (0)\to \infty $at time t=0 implying that the dashpot is rigid and an infinite stress is required to obtain a finite value of strain. For $t>0$ the strain is constant implying $\varepsilon ={{\varepsilon }_{0}}$ and $\dot{\varepsilon }=0$. Hence, from Eq. (\ref{eq:3}),
\begin{equation} \label{eq:11}
\[\begin{matrix}
\sigma =E{{\varepsilon }_{0}}, & t>0 \\
\end{matrix}\
\end{equation}
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