Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 2 / c / i

For uniaxial extension,

u = (- \frac{ϵ ˙ x}{2}, - \frac{ϵ ˙ y}{2}, \overset{ϵ}{˙} z), a = λ \overset{ϵ}{˙} .

C_{xx} = C_{yy} = \frac{1}{F + a}, C_{zz} = \frac{1}{F - 2 a},

where physical solutions require

F > 2 a

η_{ext} = \frac{τ _{zz} - τ _{xx}}{ϵ ˙} = 3 η_{s} + η_{p} [\frac{2}{F - 2 a} + \frac{1}{F + a}] .

The implicit closure is

T = \frac{2}{F + a} + \frac{1}{F - 2 a}, F = \frac{L ^{2} - 3}{L ^{2} - T} .

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