Stiffness Stress Tensor Question

[ScareCrow271828]

Homework Statement

I am given c11, c12, and c44.

What is poissons ratio ν and the E modulus E [100] for
a single crystal for uniaxial strain in [100] (if Fe is isotropic)?
ii) What is the anisotropy factor A?
(iii) There is: sigma=[100 0 0; 0 100 0; 0 0 0]Mpa
What is the transverse strain in [001] and the change in thickness when the
Sample in [001] is 1mm thick?

Homework Equations

Ansitropy factor= c11-c12-2*c44

The Attempt at a Solution

Firstly I am a little confused why the question statement says it is isotropic and then giives a [1 0 0] diretion. How is that relevant ? Shouldn't E and v be independent of direction since its isotropic? Anyways since I have values for the c's and equations supplied by the matrix for isotropic materials I was able to solve for E and V.

The anistropy factor is -125. How do I interpret a negative value? If it were zero it would be isotrpic right? So the assumption in the beginning was wrong.

I'm not sure how to even start the third part. How does the [0 0 1] direction relate to the sigma matrix? There is no force in that direction right?

Thank you!

 

[Chestermiller]

I'm very puzzled by your analysis of this problem. This is definitely the stress-strain equation for a Hookean isotropic solid. Please show your math for calculating the anisotropy factor. I get zero. You need to look up the elastic modulus and Poisson ratio for iron.

 

[ScareCrow271828]

I'm very puzzled by your analysis of this problem. This is definitely the stress-strain equation for a Hookean isotropic solid. Please show your math for calculating the anisotropy factor. I get zero. You need to look up the elastic modulus and Poisson ratio for iron.

Does the question not ask to calculate it? Why can I not calculate E and V with a system of equations given the matrix and the c values? C11=(E(1+v))/((1+v)(1-2v)) and I can make similliar equations for C12 and C44. Also how did you calculate the anisotropic factor if I did not supply the C values? I used the equation I supplied to calculate it. Thanks a lot Chester!

 

[Chestermiller]

Does the question not ask to calculate it? Why can I not calculate E and V with a system of equations given the matrix and the c values? C11=(E(1+v))/((1+v)(1-2v)) and I can make similliar equations for C12 and C44. Also how did you calculate the anisotropic factor if I did not supply the C values? I used the equation I supplied to calculate it. Thanks a lot Chester!

All I see in your original post is an algebraic (matrix) relationship between the stresses and strains. I thought the problem was asking for actual numbers. Maybe what they want in part (i) is to assume that $\sigma_{11}$ is non-zero, but all the other sigma's are zero; then determine the 6 strains.

Regarding the anisotropy factor, aside from the leading constant in front of the matrix, $$C_{11}-C_{12}-2C_{44}=(1-\nu)-\nu-2\left(\frac{1-2\nu}{2}\right)=0$$