symmetry - Is the coefficient of thermal expansion a symmetric tensor?

Is the coefficient of thermal expansion (CTE) a symmetric tensor? For thermal stresses, CTE has to be symmetric otherwise the stress tensor would not be symmetric. So does this mean that CTE tensor is always symmetric?

Answer

The stress tensor is commonly taken as a symmetric tensor, and it is a consequence of the conservation of angular momentum. In that line, the thermal expansion tensor should be symmetric. From Reference 1:

If the temperature of a crystal is changed, the resulting deformation may be specified by the strain tensor $[\epsilon_{ij}]$. When a small temperature change $\Delta T$ takes place uniformly throughout the crystal the deformation is homogeneous, and it is found that all the components of $[\epsilon_{ij}]$ are proportional to $\Delta T$; thus

$$\epsilon_{ij} = \alpha_{ij} \Delta T,$$

where the $\alpha_{ij}$ are constants, the coefficients of thermal expansion. Since $[\epsilon_{ij}]$ is a tensor, so also is $[\alpha_{ij}]$, and, moreover, since $[\epsilon_{ij}]$ is symmetrical so also is $[\alpha_{ij}]$.

There are some theories that allow for non-symmetric stress tensors, though. But they are much less common.

References

1. Nye, John Frederick. Physical properties of crystals: their representation by tensors and matrices. Oxford university press, 1985.