Paper 3, Section II, A

Vector Calculus | Part IA, 2007

Show that any second rank Cartesian tensor PijP_{i j} in R3\mathbb{R}^{3} can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that PijP_{i j} can be decomposed into the following terms

Pij=Pδij+Sij+ϵijkAk,(†)\tag{†} P_{i j}=P \delta_{i j}+S_{i j}+\epsilon_{i j k} A_{k},

where SijS_{i j} is symmetric and traceless. Give expressions for P,SijP, S_{i j} and AkA_{k} explicitly in terms of PijP_{i j}.

For an isotropic material, the stress PijP_{i j} can be related to the strain TijT_{i j} through the stress-strain relation, Pij=cijklTklP_{i j}=c_{i j k l} T_{k l}, where the elasticity tensor is given by

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}

and α,β\alpha, \beta and γ\gamma are scalars. As in ()(†), the strain TijT_{i j} can be decomposed into its trace TT, a symmetric traceless tensor WijW_{i j} and a vector VkV_{k}. Use the stress-strain relation to express each of T,WijT, W_{i j} and VkV_{k} in terms of P,SijP, S_{i j} and AkA_{k}.

Hence, or otherwise, show that if TijT_{i j} is symmetric then so is PijP_{i j}. Show also that the stress-strain relation can be written in the form

Pij=λδijTkk+μTijP_{i j}=\lambda \delta_{i j} T_{k k}+\mu T_{i j}

where μ\mu and λ\lambda are scalars.

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