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There are eight symmetry classes in linear elasticity: triclinic, monoclinic, orthotropic, tetragonal, trigonal, transversely isotropic, cubic, and isotropic. I was wondering if there existed a material in each of those symmetry classes which almost satisfies the Cauchy relations: $C_{ijkl} = C_{ikjl}$ for all $i,j,k,l \in \mathbb{R}^3$ where $[C_{ijkl}]$ is the linear elasticity tensor (basically the elasticity tensor is completely symmetric). Any help would be greatly appreciated.

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It may be easier to analyze this problem using Voigt or reduced notation, in which, e.g., $\mathbb{C}_{1112}=C_{16}$, where $\mathbb{C}$ is the 3×3×3×3 stiffness tensor and $C$ is the reduced 6×6 stiffness matrix. In Physical Properties of Crystals, for example, Nye presents certain implications involving the elasticity tensors using this reduced notation: Constraints on 6×6 elasticity matrices

From physical arguments, we already have $\mathbb{C}_{ijkl}=\mathbb{C}_{jikl}=\mathbb{C}_{ijlk}=\mathbb{C}_{klij}$. You're asking about the additional constraint that $\mathbb{C}_{ijkl}=\mathbb{C}_{ikjl}$. In reduced notation, this can be verified to imply 6 additional relationships:

$$C_{12}=C_{66}$$ $$C_{13}=C_{55}$$ $$C_{14}=C_{56}$$ $$C_{23}=C_{44}$$ $$C_{25}=C_{46}$$ $$C_{36}=C_{45}$$

We can see from the image above that these all hold for any isotropic material (including fully amorphous materials, isotropic crystals, and any crystalline material with sufficiently small and equioriented grains) if $\frac{1}{2}(C_{11}-C_{12})=C_{66}=C_{12}$, or $C_{11}=3C_{12}$. Since

$$C_{11}=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}$$ $$C_{12}=\frac{E\nu}{(1+\nu)(1-2\nu)}$$

where $E$ is the Young's elastic modulus and $\nu$ is the Poisson's ratio, we have $\nu=0.25$, which is allowed. (Stable materials must satisfy $-1<\nu<0.5$.) Ceramics, for example, often exhibit a Poisson's ratio of around 0.25.

Since materials do exist (under this Poisson's ratio constraint) that obey $\mathbb{C}_{ijkl}=\mathbb{C}_{ikjl}$ under this most symmetric and tightly restricted crystal class of isotropy, I'd argue that it wouldn't be physically prohibited for such materials to exist in any of the other crystal classes (although it may require that some of their $\mathbb{C}_{ijkl}$ indices that need not be identically zero are nevertheless approximately zero).

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