Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 13C / b / i

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- Solution i

Solution

For the displacement gradient tensor specified in the question,

\nabla u : \nabla u^{T} = tr (\nabla u \nabla u^{T}) = u_{x}^{2} + u_{y}^{2} + v_{x}^{2} + v_{y}^{2},

while the divergence is

\nabla \cdot u = u_{x} + v_{y}

. Hence the isotropic linear-elastic energy density is

\frac{μ}{2} (u_{x}^{2} + u_{y}^{2} + v_{x}^{2} + v_{y}^{2}) + \frac{λ + μ}{2} (u_{x} + v_{y})^{2} .

Expanding the square and collecting terms yields

J = \iint_{Ω} [(\frac{λ}{2} + μ) (u_{x}^{2} + v_{y}^{2}) + \frac{μ}{2} (u_{y}^{2} + v_{x}^{2}) + (λ + μ) u_{x} v_{y}] d x d y .

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