# Is there any deference between the directional derivatives of isotropic and anisotropic tensors?

I need to calculate an elasticity tensor which can be expressed by the directional derivate of a tensor with respect the other. Is the anisotropy of the tensor affects this derivative and how?

Edit: for clarification:

Assume $$S$$ and $$E$$ are the symmetric spacial stress and strain tensors. If for the variation of the displacement $$\delta u$$ we assume: $$u_{\epsilon} = u + \epsilon\delta u$$ for any small amount of $$\epsilon$$, then the elasticity tensor can be used by the chain rule:

$$\frac{\mathrm d S}{\mathrm d E} = \frac{\partial S}{\partial E}\frac{\mathrm d E}{\mathrm d \epsilon}$$

I suspect that the above equation is also applicable for the anisotropic models because the variation of displacement for these materials can be defined only in certain directions. Is that right?

• Can you write mathematically what you are trying to do? – nicoguaro Apr 3 at 15:37
• @nicoguaro edited. – Shayan Apr 4 at 17:04
• I don't see any tensor that might be called "elasticity tensor" in your expression. – nicoguaro Apr 4 at 17:06
• @nicoguaro $\frac{\partial S}{\partial E}$ here is called the elasticity tensor in many publications, for example: link.springer.com/10.1007/s00033-016-0623-5 – Shayan Apr 4 at 17:08

If we assume that there is an energy density function $$U$$, then we have

$$S_{ij} = \frac{\partial U}{\partial E_{ij}}\, ,$$

where $$S$$ and $$E$$ are thermodynamically conjugated, i.e., their product represents work done to deform the body.

For small strains, we can use a Taylor expansion and conclude that we can write the stiffness tensor as

$$c_{ijkl} = \frac{\partial^2 U}{\partial E_{ij} E_{kl}} = \frac{\partial \sigma_{ij}}{\partial E_{kl}}\, .$$

This is described in the Wikipedia's article for Hooke's law.

Answering your question, this expression is valid independently of the symmetries of the material (anisotropy).

## References

1. James Doyle and C.T. Sun (2007). Theory of Elasticity, Purdue University.