How can i define Hook's Law interm of Tensor

[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
can you all friends help me?
i will be thanksfull to all of you

 

[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 $E_{ijkl}$?

 

[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 a lot

 

[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 $trace(\epsilon)$.

 

[Rousha]

thankx for ur reply
ok i understand this terms and finally i want to know that what is the application of this relation

 

[PerennialII]

You're welcome. If you're interested about Hooke's law in general sense Wikipedia is one possible start:

http://en.wikipedia.org/wiki/Hooke's_Law