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?

Many thanks for your help.

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?

  • $\begingroup$ Can you write mathematically what you are trying to do? $\endgroup$
    – nicoguaro
    Apr 3, 2020 at 15:37
  • $\begingroup$ @nicoguaro edited. $\endgroup$
    – Shayan
    Apr 4, 2020 at 17:04
  • $\begingroup$ I don't see any tensor that might be called "elasticity tensor" in your expression. $\endgroup$
    – nicoguaro
    Apr 4, 2020 at 17:06
  • $\begingroup$ @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 $\endgroup$
    – Shayan
    Apr 4, 2020 at 17:08

1 Answer 1


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).


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

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