How can i define Hook's Law interm of Tensor

1. Oct 11, 2005

Rousha

How can i define "Hook's Law interm of Tensor"

i want to define hoos's law interm of tensor
how cn i define it:surprised
can you all friends help me?
i will be thanksfull to all of you

2. Oct 12, 2005

PerennialII

Hi Rousha, welcome to PF!

I'm checking I understood you right .... are you referring to the constitutive tensor of Hooke's law, the elasticity tensor $E_{ijkl}$?

3. Oct 12, 2005

Rousha

need more help

i want to know that how can i define relationship between stress tainsor and strain tensor and also asked that is the transformation law of tensor obey in this relationship
thanx alot

4. Oct 13, 2005

PerennialII

Write the generalized Hooke's law as

$$\sigma^{rs}=E^{rsij}\epsilon_{ji}, r,s,i,j \in (1,2,3),$$

for relating the stress tensor $\sigma$ and the infinitesimal strain tensor $\epsilon$, where E is the elasticity tensor (by postulating the existence of a
strain energy density has 21 independent coefficients).
For homogeneous isotropic material the elasticity tensor, the generalized Hooke's aw can be expressed using the Lame coefficients as

$$\sigma_{ij}=\lambda\delta_{ij}\epsilon_{kk}+2\mu\epsilon_{ji},$$

where the Lame coefficients are given as (by introducing the Young's
modulus E and Poisson's ratio $\nu$ )

$$\lambda= \frac{\nu E}{(1-2\nu)(1+\nu)}$$

$$\mu=E/2(1+\nu) .$$

The generalized Hooke's law in a general coordinate system can be written as (using the Lame constants again)

$$\sigma^{pq}=\lambda I^{\epsilon}_{1} g^{pq}+2 \mu g^{ip} g^{jq} \epsilon_{ij},$$

where $I^{\epsilon}_{1}$ is the first invariant of $trace(\epsilon)$.

Last edited: Oct 13, 2005
5. Oct 20, 2005