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.

Under harmonic stress, strain amplitude is ${\sigma }_0/\sqrt{E^2+{\omega }^2{\eta }^2}$ and its lag is $\arctan (\omega \eta /E)$. Low frequencies are nearly elastic; high-frequency stress-controlled deformation is small and nearly a quarter cycle behind stress.

For $E\epsilon +\eta \dot{\epsilon }={\sigma }_0\Theta (t)$ with initially zero strain, $\epsilon =({\sigma }_0/E)(1-e^{-t/\tau })$ and $\tau =\eta /E$. Strain is continuous at load application, approaches a bounded value and recovers exponentially after unloading.

For a rod with unit mass density and elastic modulus, the Kelvin-Voigt model gives ${\varphi }_{tt}-{\varphi }_{xx}-\gamma {\varphi }_{txx}=0$, $\gamma >0$. For the operator with the opposite overall sign, $-{\partial }_t^2+{\partial }_x^2+\gamma {\partial }_t{\partial }_x^2$, the inverse under the $e^{-i\omega t}$ Fourier transform convention is the displayed expression. Its poles are ${\omega }_{\pm }=-i\gamma k^2/2\pm \sqrt{k^2-{\gamma }^2{k}^4/4}$. With ${\Omega }_k=\sqrt{k^2-{\gamma }^2{k}^4/4}$ real, the retarded fundamental solution in spatial Fourier variables is ${\tilde{G}}_R(k,t)=-\Theta (t)e^{-\gamma k^{2}t/2}\sin ({\Omega }_{k}t)/{\Omega }_k$. Its initial derivative is $-1$, fixed by the leading coefficient $-{\partial }_t^2$. At a repeated root take the continuous limit; for imaginary ${\Omega }_k$ the sine becomes a hyperbolic sine.

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.