How can i define Hook's Law interm of Tensor

Rousha
Messages
3
Reaction score
0
How can i define "Hook's Law interm of Tensor"

i want to define hoos's law interm of tensor:confused:
how cn i define it
can you all friends help me?o:)
i will be thanksfull to all of you
 
Physics news on Phys.org
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}?
 
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
 
Write the generalized Hooke's law as

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

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

<br /> \sigma_{ij}=\lambda\delta_{ij}\epsilon_{kk}+2\mu\epsilon_{ji}, <br />

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

<br /> \lambda= \frac{\nu E}{(1-2\nu)(1+\nu)} <br />

<br /> \mu=E/2(1+\nu) .<br />

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

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

where I^{\epsilon}_{1} is the first invariant of trace(\epsilon).
 
Last edited:
thankx for ur reply:smile:
ok i understand this terms and finally i want to know that what is the application of this relation
 

Similar threads

Replies
4
Views
1K
Replies
2
Views
2K
Replies
8
Views
1K
Replies
18
Views
2K
Replies
3
Views
2K
Replies
7
Views
3K
Replies
14
Views
787
Replies
4
Views
1K
Back
Top