Burgers material

Overview

Maxwell representation

Given that one Maxwell material has an elasticity ${\displaystyle E_1}$ and viscosity ${\displaystyle {\eta }_1}$, and the other Maxwell material has an elasticity ${\displaystyle E_2}$ and viscosity ${\displaystyle {\eta }_2}$, the Burgers model has the constitutive equation

${\displaystyle \sigma +\left(\frac{{\eta }_1}{E_1}+\frac{{\eta }_2}{E_2}\right)\dot{\sigma }+\frac{{\eta }_1{\eta }_2}{E_1{E}_2}\ddot{\sigma }=\left({\eta }_1+{\eta }_2\right)\dot{\epsilon }+\frac{{\eta }_1{\eta }_2\left(E_1+E_2\right)}{E_1{E}_2}\ddot{\epsilon }}$

where ${\displaystyle \sigma }$ is the stress and ${\displaystyle \epsilon }$ is the strain.

Kelvin representation

Given that the Kelvin material has an elasticity ${\displaystyle E_1}$ and viscosity ${\displaystyle {\eta }_1}$, the spring has an elasticity ${\displaystyle E_2}$ and the dashpot has a viscosity ${\displaystyle {\eta }_2}$, the Burgers model has the constitutive equation

${\displaystyle \sigma +\left(\frac{{\eta }_1}{E_1}+\frac{{\eta }_2}{E_1}+\frac{{\eta }_2}{E_2}\right)\dot{\sigma }+\frac{{\eta }_1{\eta }_2}{E_1{E}_2}\ddot{\sigma }={\eta }_2\dot{\epsilon }+\frac{{\eta }_1{\eta }_2}{E_1}\ddot{\epsilon }}$

where ${\displaystyle \sigma }$ is the stress and ${\displaystyle \epsilon }$ is the strain.