# The elasticity/stiffness tensor for an isotropic materials

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1. Feb 23, 2016

### Wuberdall

Hi PF,

As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation.
$$C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})$$
Do any of you know a sound argument, mostly relaying on physical intuition, for why this is the most general form of the elasticity tensor for isotropic and homogeneous materials?

Of course one could always impose all the symmetries (for an isotropic material) upon the elasticity tensor and go through all the necessary and tedious computations to derive this result... But this is unfortunately not what I'm looking for.

And furthermore, what are/is the difference between isotropic materials and materials possessing cubic symmetry, where the most general elasticity tensor instead is written
$$C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu\delta_{ik}\delta_{jl} + \nu\delta_{il}\delta_{jk}$$
and is characterized by three (instead of two) independant material parameters.
That is, why is μ = ν for isotopic materials ?

The most general isotropic rank four tensor is of the form with three independent parameters. However, this does not satisfy the symmetries expected from the stiffness tensor. It only does so when $\mu = \nu$.