# Stress and strain in crystal structures

1. Dec 11, 2008

### darkelf

Are there stress and strain graphs for different crystal structures?

2. Dec 11, 2008

### Mapes

Can you give some more information about what you're looking for? The elastic properties of metals aren't strongly dependent upon crystal structure. There's much more of a correlation between, say, stiffness and melting temperature because both are correlated with bond strength. There's some correlation between crystal structure and yield behavior in metals, because the number of slip systems differ between crystal structures. But again, the dependence is weak compared to the influence of other variables such as impurity concentration and dislocation density.

3. Dec 11, 2008

### darkelf

It should be tensile stress - strain curves for different crystal structures. I'm not sure but I think it has something to do with Taylor's work hardening theory.

4. Dec 11, 2008

### Mapes

I'm still not getting it. You mean a universal stress-strain curve for fcc metals, one for bcc metals, one for hcp metals, etc.?

5. Dec 11, 2008

### darkelf

Not sure what you mean by universal, Just a plastic stress-strain curve for crystal structures derived from Taylor's theory.

6. Dec 11, 2008

### Mapes

OK, I think I know what you mean. See here for the stress-strain relationship in an fcc single crystal. Taylor modeled the strength increase due to dislocation interactions, also known as work hardening or forest hardening, which is shown in stage II of the graph.

7. Dec 11, 2008

### darkelf

Thanks Mapes,

He had an equation as well didnt he?

8. Dec 11, 2008

### Mapes

$$\Delta \tau\propto Gb\rho^{1/2}$$?

9. Dec 11, 2008

### darkelf

More like T = To + K(G)1/2

Sorry about the way the equation is
T= Shear stress
To = critical resolved stress
K = constant
G = shear strain

Can a graph for f.c.c be ploted from this equation?

10. Dec 11, 2008

### Mapes

I think you're trying to write

$$\tau=\tau_0+KGb\rho^{1/2}$$

which is equivalent to what I have above; the increase in shear strength $\tau$ is proportional to the square root of the dislocation density $\rho$. Here's how you would plot the strength vs. dislocation density for a specific fcc sample:

1) As a reference point, experimentally measure the shear strength $\tau$ and the dislocation density $\rho$ of the sample.
2) Look up the shear modulus G.
3) Calculate the Burgers vector b from the atomic spacing.
4) From (1), (2), and (3), calculate the constants K and $\tau_0$.
5) Plot $\tau=\tau_0+KGb\rho^{1/2}$.

11. Dec 11, 2008

### darkelf

OK think I might be getting a bit confused here. I think you're right about that being Taylor's theory, almost sure of it. But I know I'm looking for a shear strain, shrear stress relationship in connection to that in the form of hall-petch and then a tensile stress-strain graph from that equation. Am I sounding crazy?

12. Dec 11, 2008

### Mapes

We're not getting anywhere. You mentioned "work hardening" and then "hall-petch"; these are two different strengthening mechanisms. To reduce the back-and-forth, can you state your ENTIRE homework problem / project assignment / train of thought / personal goal in as much detail as possible? This will help people determine whether to provide a

- Description of a concept
- Formula
- Graph of an idealized relationship
- Photograph
- Chart of actual data
- Literature reference

13. Dec 11, 2008

### darkelf

sorry about that Mapes, just sent you a PM with the problem.

Thanks