Methods | Part IB, 2003

State the transformation law for an nn th-rank tensor TijkT_{i j \cdots k}.

Show that the fourth-rank tensor

cijkl=αδijδkl+βδikδjl+γδilδjkc_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}

is isotropic for arbitrary scalars α,β\alpha, \beta and γ\gamma.

The stress σij\sigma_{i j} and strain eije_{i j} in a linear elastic medium are related by

σij=cijklekl.\sigma_{i j}=c_{i j k l} e_{k l} .

Given that eije_{i j} is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form

σij=λekkδij+2μeij\sigma_{i j}=\lambda e_{k k} \delta_{i j}+2 \mu e_{i j}

Show that eije_{i j} can be written in the form eij=pδij+dije_{i j}=p \delta_{i j}+d_{i j}, where dijd_{i j} is a traceless tensor and pp is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density E=12σijeijE=\frac{1}{2} \sigma_{i j} e_{i j} to be non-negative for any deformation of the solid are that

μ0 and λ23μ.\mu \geq 0 \quad \text { and } \quad \lambda \geq-\frac{2}{3} \mu .

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