Example for Higher Order Tensors Could you please give some examples of higher order tensors?
What would be an instance of 2nd, 3rd, 4th or even an higher order tensor in engineering science, especially in mechanics?
An example of a second order tensor is stress tensor.
Thanks!
 A: In (solid) mechanics, some common 2nd order tensors are the deformation gradient $F$, various strain tensors $E$, and various stress measures $T$. There are many more than would be practical to list here.
A useful 3rd order tensor is the Levi-Civita permutation tensor $\epsilon_{ijk}$. This comes up in the cross-product of two vectors:
$$
\mathbf{a} \times \mathbf{b} = a_i\hat{e}_i \times b_j\hat{e}_j = \epsilon_{ijk}a_ib_j \hat{e}_k
$$
Wikipedia has a good explanation of its behavior.
There are a few examples of 4th order tensors in mechanics as well, but the first one that comes to mind is the elasticity tensor. It is the material tangent of the second Piola stress $S$ with respect to the Green strain $E$, and thus is the second derivative of the strain energy density function $\psi$ with respect to strain:
$$
\mathbb{C}= \frac{\partial \mathbf{S}}{\partial \mathbf{E}} = \frac{\partial^2\psi}{\partial \mathbf{E}^2}
$$
In isotropic linear elasticity, the material and spatial descriptions of stress and strain are negligibly different so you get the well-known Hooke's law
$$
\mathbf{\sigma}_{ij} = \mathbb{C}_{ijkl}E_{kl}
$$
where the components of the elasticity tensor are defined as follows:
$$
\mathbb{C}_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu (\delta_{ik}\delta_{jl}+\delta_{ik}\delta_{jk})
$$
where $\lambda$ and $\mu$ are the Lamé constants.
I do not believe that tensors of higher than 4th order are used frequently, if ever, in mechanics.
