Calculating elastic constants Cijkl

[tuomas22]

Homework Statement

Longnitudal and transverse soundwaves in nickel (FCC lattice) moves at velocities 5300m/s 3800m/s. Determine the elastic constants C_ijkl

Homework Equations

$$v =\sqrt{C_{ij}/\rho}$$

The Attempt at a Solution

I guess I can calculate C_ij with that equation...but I don't understand how I get the indices ij, or ijkl...And I don't understand how am I supposed to use the longnitudal AND transverse wave speeds...

 

[lanedance]

been a while, but as I remember it, the elastic stiffness tensor relates stress to strain y
$$\sigma_{ij} = \textbf{C}_{ijkl} \epsilon_{kl}$$

in the anisotropic case, the are upto 21 independentcomponest $$\textbf{C}_{ijkl}$$to see:
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke.cfm

when the material is isotropic, things simplify a fair bit & the stiffness tensor can be represented uniquley by 2 parameters (often E and $\nu$, the young's modulus & poisson's ratio)
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_isotropic.cfm

these should be reasonably easily relatable to the longitudinal & transverse wave speeds...

 

[tuomas22]

Thanks for answer. But I still didnt get it. There was some strange matrices in those links, and we haven't talked about those in the class, so I don't think that's what I'm supposed to use.

I also didnt understand the indices. For example if I have i=2 and j=1 for C₂₁, what does it exactly mean?

 

[tuomas22]

one last cry for help. then i quit :) exam tomorrow :(

 

[lanedance]

do you have any more info?

is the material isotropic & how many dimensions are you working in?

 

[tuomas22]

nope that's all I have :(
but this is an introductory course, so maybe I am supposed to make some assumptions? I don't know...

lets assume it's isotropic

 

[lanedance]

ok i believe Cij is the component of the stiffness matrix as outlined above, there's no reall easy way to go through it... Cij is the ith row, jth component of thw stiffness matrix given in
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_isotropic.cfm
(a condensed matrix form of the full cijkl tensor)

i'm not too sure what C12 means, but in the matrix it relates the stress in the x dir'n to a normal strain in the y dir'n... so 1-3 represent normal stress/strain, 4-6 shear stress/strain..

then i think C11, C22, C33 will represent the longitudinal speeds (all same in isotropic)
while C44, C55, C66 will represnet the shear wave speeds (all same in isotropic)

from a bit of googling on elastic isotropic materials, to remember this stuff...

first shear velocity is relateable to the shear mdoulus
$$v_s = \sqrt{\frac{G}{\rho}}$$
The shear modulus is then relateable to young's modulus & poissons ratio by
$$G = \frac{E}{2(1+\nu)}}$$

now longitudinal velocity is relateable to young's modulus & poissons ratio by
$$v_s = \sqrt{\frac{E(1-\nu)}{\rho(1-2\nu)(1+\nu)}}$$

so you shold be able to solve for E & nu, knowing vp, vs & denisty & assuming linear elastic isotropic

this would then allow you to fill out the stiffness matrix as given... ie the Cij
note i think vp = sqrt(C11/rho) and vs = sqrt(C44/rho) which gives some cofidence that we're on the right track

anyway hope this of some help, if its introductory we may be deving into it a bit much...

 

[lanedance]

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm

this is the only website i found that references the elastic constants as you have...