Past exam of the mathematical mechanical biology course of the University of Oxford / 2023 / 3 / a

Before contact with both walls, the elastic rod can grow freely, so its elastic stretch ratio is $\alpha =1$ and its length is its stress-free grown length,The first contact occurs when $\ell (t_c)=L_1$. For the intended growth case $c>0$,The condition $L_1>L_0$ ensures $t_c>0$. If $c=0$, contact never occurs; negative $c$ describes shrinkage rather than the stated continuing growth.

Write $\lambda =L_1/L_0>1$. The one-dimensional multiplicative decomposition of the deformation gradient reads $\ell /L_0=\alpha \gamma$. Before contact, $\alpha =1$; afterwards, the imposed fixed length givesSubstituting in the linear constitutive equation yieldsHere negative axial stress means compression. The two branches agree at $t_c$ because $1+c{t}_c=\lambda$. At late times,Thus the law permits arbitrarily severe elastic shortening at a finite compressive stress. It is a small-strain approximation, appropriate near $\alpha =1$, and is not a credible material law as $\alpha \to 0$. The one-dimensional constraint rules out buckling in this particular model; the finite limiting stress is a defect of extrapolating the linear constitutive equation, even with that constraint retained.

The kinematic relation is unchanged: $\alpha =1$ before contact and $\alpha =\lambda /(1+ct)$ afterwards. Therefore the nonlinear constitutive equation givesThe first term in the post-contact expression is $O(t^{-2})$. Hence its asymptotic expansion isIn this model the compressive axial stress diverges in magnitude, because the $-{\alpha }^{-1}$ term penalizes vanishing elastic stretch ratio without bound.