Elasticity tensor

Remarks

Example: the isotropic tensor $C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk})$ with $\mu > 0$ and $3\lambda + 2\mu > 0$. Clauses 1 and 2 hold by inspection. For clause 3, $\varepsilon : C : \varepsilon = \lambda (\tr \varepsilon)^2 + 2\mu \, \varepsilon : \varepsilon$; splitting $\varepsilon$ into its trace part and its trace-free part $\varepsilon'$ gives $(\lambda + \tfrac23 \mu)(\tr \varepsilon)^2 + 2\mu \, \varepsilon' : \varepsilon'$, which is positive for $\varepsilon \ne 0$.

Non-example: the same formula with $\mu = 0$ and $\lambda > 0$ fails clause 3, since any nonzero trace-free $\varepsilon$ gives $0$. Such a "material" offers no resistance to shear.

Clause 1 says that $C$ maps $\mathrm{Sym}$ to $\mathrm{Sym}$ and ignores the skew part of the strain. Clause 2 holds when the material has a stored energy $W(\varepsilon) = \tfrac12 \varepsilon : C : \varepsilon$; then $C$ is a symmetric positive definite operator on the $6$-dimensional space $\mathrm{Sym}$, which has $21$ independent entries. In Voigt notation $C$ is written as a symmetric $6 \times 6$ matrix. Some authors include only clause 1 in the definition and state clauses 2 and 3 as extra assumptions.