Cauchy tensor tensor components from Energy function I hope someone could help me.
Let $W\left( \lambda_1,\lambda_2,\lambda_3  \right) = \sum_{p=1}^N \frac{\mu_p}{\alpha_p}\left( \lambda_1^{\alpha_p} + \lambda_2^{\alpha_p} + \lambda_3^{\alpha_p} -3 \right)$ be a strain energy density function. How can I find the non-diagonal elements of the Cauchy tensor?
Edit 1: Basically I want to derive the equations motion for hyperelastic material. In order to achieve that I first need the Cauchy Tensor.
Edit 2: As far as I know, the $ \lambda_k \frac{\partial W(\lambda_1,\lambda_2,\lambda_3)}{\partial \lambda_k} + p = \sigma_{kk} = \sigma_{k}$ (no summing convention) for incompressible and isotropic hyperelastic materials[Ogden 2001]. But what abount the $\sigma_{kj}$ | $k \neq j$? How can I calculate them?
Edit 3: I think I have found the answer. When calculating $\sigma_k$, using the description above, I am implicitly using a frame of reference which the tensor $\sigma$ is diagonal. This frame of reference concedes with the principal directions of the deformation.
[Ogden 2001] Nonlinear Elasticity: Theory and Applications (Ogden)
Thanks
 A: Dvorkin and Goldschmit ([1], p.120) define a hyperelastic material as one whose stress power is an exact differential and for which thus the strain $ \boldsymbol{\sigma} $ is given by
$$ \boldsymbol{\sigma} = \frac{\partial U}{\partial \boldsymbol{E}} $$
with $\boldsymbol{E}$ the strain and $U$ the elastic energy function per volume.
In general, this translates into
$$\sigma_{ij} = \frac{\partial U}{\partial E^{ij}} $$
Thus, your concern about off-diagonal components is quite right in general.
Usually, the variables $\lambda_1,\lambda_2,\lambda_3$ are used for the principal strains and there is a coordinate system in which $\boldsymbol{E}=\sum_j \lambda_j \phantom{a} \vec{e}_j \otimes \vec{e}_j$. In this coordinate system, there should be no off-diagonal components of $\boldsymbol{\sigma}$ either because of its definition via the partial derivatives.
So yeah, I think your edits are correct.  
[1] Eduardo N. Dvorkin, Marcela B. Goldschmit: Nonlinear Continua; Springer-Verlag Berlin Heidelberg 2005
