 Solution

ID: past-exam-of-the-mathematical-mechanical-biology-course-of-the-university-of-oxford/2023/3/a/iii/solution

The kinematic relation is unchanged:

α = 1

before contact and

α = λ / (1 + c t)

afterwards. Therefore the nonlinear constitutive equation gives

n (t) = ⎩ ⎨ ⎧ 0, E_{s} [\frac{λ ^{2}}{( 1 + c t ) ^{2}} - \frac{1 + c t}{λ}], 0 \leq t \leq t_{c}, t > t_{c} .

The first term in the post-contact expression is

O (t^{- 2})

n (t) = - \frac{E _{s} c}{λ} t - \frac{E _{s}}{λ} + O (t^{- 2}), n (t) \sim - \frac{E _{s} c L _{0}}{L _{1}} t (t \to \infty) .

In this model the compressive axial stress diverges in magnitude, because the

- α^{- 1}

term penalizes vanishing elastic stretch ratio without bound.

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