Kelvin-Voigt model ($F=kx+\mu \dot{x}$)

The Kelvin-Voigt model puts a spring and dashpot in parallel. Their common displacement makes their forces add. With the convention $e^{i\omega t}$, the complex stiffness is $k+i\omega \mu$, giving storage stiffness $k$ and loss stiffness $\omega \mu$. Under a force step, $x=(F_0/k)(1-e^{-t/\tau })$ with $\tau =\mu /k$. The series spring–dashpot is a Maxwell fluid analogue and has different relaxation and creep behavior.

For a quasistatic cantilever of Kelvin-Voigt material, the bending moment is $M=EI{y}_{xx}+{\eta }_{s}I{y}_{xxt}$. A tip load gives $M=F(t)(\ell -x)$, so the free-tip displacement obeys the displayed equation with $\tau ={\eta }_s/E$. A step load gives a finite asymptotic deflection. If inertia is retained, the interior equation is $\rho A{y}_{tt}+EI{y}_{xxxx}+{\eta }_{s}I{y}_{xxxxt}=0$, with the corresponding moment and shear conditions at the tip.