Burgers material

Schematic diagram of Burgers material, 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

${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\left(\eta _{1}+\eta _{2}\right){\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}\left(E_{1}+E_{2}\right)}{E_{1}E_{2}}}{\ddot {\varepsilon }}}$

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

Schematic diagram of Burgers material, 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

${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\eta _{2}{\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}}{E_{1}}}{\ddot {\varepsilon }}}$

where ${\displaystyle \sigma }$ is the stress and ${\displaystyle \varepsilon }$ is the strain.[1]