= Solution
Let $y(x,t)$ be downward deflection, $0\le x\le\ell$, with $y(0,t)=y_x(0,t)=0$. Take a uniform circular <elastic beam>, $I=\pi r^4/4$, small slopes, and linear material response. Write $E$ for its axial elastic modulus and $\eta_s$ for its axial viscous coefficient; these are material quantities, not the solvent viscosity. The extension of the printed parallel model has moment law
$$
M=EIy_{xx}+\eta_sIy_{xxt},\qquad \tau=\eta_s/E.
$$
With <mass> per unit length $\rho A$, transverse momentum balance in the interior is
$$
\boxed{\rho Ay_{tt}+EIy_{xxxx}+\eta_sIy_{xxxxt}=0.}
$$
The free-tip conditions are $M(\ell,t)=0$ and $M_x(\ell,t)=-F(t)$ with this downward-positive convention; the clamp supplies the remaining reactions.
Neglecting <inertia> gives $M=F(t)(\ell-x)$. Twice integrating the moment law with the clamp conditions gives
$$
y+\tau y_t=\frac{F(t)x^2(3\ell-x)}{6EI},\qquad \delta+\tau\dot\delta=\frac{F(t)\ell^3}{3EI},\quad\delta=y(\ell,t).
$$
For a step $F_0$ and an initially relaxed <elastic beam>, <Kelvin-Voigt cantilever creep> is
$$
\boxed{\delta(t)=\frac{F_0\ell^3}{3EI}(1-e^{-t/\tau}),\qquad t\ge0.}
$$
Thus the equivalent tip <spring> and <dashpot> are $k_{\rm eff}=3EI/\ell^3$ and $\mu_{\rm eff}=3\eta_sI/\ell^3$.
If the intended material is instead a true series <Maxwell fluid>, its moment satisfies $\dot M+M/\tau=EIy_{xxt}$. Together with momentum balance $\rho Ay_{tt}+M_{xx}=0$, this gives the full interior equation $\rho A(y_{ttt}+y_{tt}/\tau)+EIy_{xxxxt}=0$, with the moment law retained for boundary and initial conditions. In the quasistatic limit,
$$
\dot\delta=\frac{\ell^3}{3EI}\left(\dot F+\frac F\tau\right),\qquad \boxed{\delta(t)=\frac{F_0\ell^3}{3EI}(1+t/\tau)\quad(t>0).}
$$
This <Maxwell cantilever creep> has an initial elastic jump followed by linear creep, not a finite plateau. The jump is the ideal massless step-load limit; <inertia> resolves the transient. A two-element series model has no nonzero relaxed modulus, so “Maxwell solid” is not an additional specification of a solid with finite long-time stiffness. Such a solid would need an additional <spring>. The small-deflection approximation requires $\delta\ll\ell$ and eventually fails for unbounded creep.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-71-mechanical-response.png]
{title=Thermal displacement crossover, parallel and series <elastic beam> creep, and diffusion-limited versus reaction-limited ratchet slowdown}
Back to article page
