# Converting stress-strain curve to shear stress-shear strain

• turpy
In summary, the conversation discusses using data from a monotonic uniaxial tension test to find the corresponding pure shear stress-strain curve for a crystalline metal material. The elastic modulus and Poisson ratio are determined from the test data, and the shear modulus is calculated using an equation. However, it is unclear how to account for the plastic region of the stress-strain curve. Further analysis and consideration of plasticity theories, such as von Mises, may be necessary to determine the shear stress at yield.
turpy

## Homework Statement

For a crystalline metal material
- Elastic modulus E
- Poisson ratio v
- A table with test data of stresses vs. total strains, from a monotonic uniaxial tension test, which generates a stress-strain curve.

How would you use this data to find the corresponding pure shear stress-strain curve?

## Homework Equations

ε_elastic = (σ/E)
γ_elastic = (τ/G)
G=E/[2*(1+v)]

## The Attempt at a Solution

Using the crystal structure of metal, the normal stresses from the table could be converted to shear stresses via the τ_crss equation (http://virtualexplorer.com.au/special/meansvolume/contribs/wilson/Critical.html )

Then, the elastic shear strains can be obtained from τ/G. But what about the plastic shear strains? This is where I'm stuck. Hints/help would be highly appreciated!

Last edited by a moderator:
turpy said:

## Homework Statement

For a crystalline metal material
- Elastic modulus E
- Poisson ratio v
- A table with test data of stresses vs. total strains, from a monotonic uniaxial tension test, which generates a stress-strain curve.

How would you use this data to find the corresponding pure shear stress-strain curve?

## Homework Equations

ε_elastic = (σ/E)
γ_elastic = (τ/G)
G=E/[2*(1+v)]

## The Attempt at a Solution

Using the crystal structure of metal, the normal stresses from the table could be converted to shear stresses via the τ_crss equation (http://virtualexplorer.com.au/special/meansvolume/contribs/wilson/Critical.html )

Then, the elastic shear strains can be obtained from τ/G. But what about the plastic shear strains? This is where I'm stuck. Hints/help would be highly appreciated!
You use the tensile test to determine the Young's modulus and the poisson ratio. Then you use your equation to calculate the shear modulus G from these. Then you plot shear stress vs shear strain with a slope of G.

Chet

Last edited by a moderator:
Hi Chet,
Thanks for the response. That covers the linear elastic region of the shear stress-shear strain curve, but what about the plastic region?

turpy said:
Hi Chet,
Thanks for the response. That covers the linear elastic region of the shear stress-shear strain curve, but what about the plastic region?
I don't have the answer to this immediately up my sleeve. I want to spend a little time playing with the equations.

Chet

Chestermiller said:
I don't have the answer to this immediately up my sleeve. I want to spend a little time playing with the equations.

Chet
If there is a way of doing it in the plastic region, I have not been able to figure out how. It certainly can't be done directly from the experimental measurements because, for all possible plane orientations within the sample, with this kind of uniaxial loading, there is no orientation in which there is a pure shear stress on the plane. There is always a normal component of the stress (except, of course, at 90 degrees to the load, where the shear stress is zero).

Chet

In the same way that 3D elasticity tells you $G$, based on $E$ and $\nu$, plasticity (von Mises) tells you that the material will "yield" in pure shear at a value of $\tau_y$, which is known, based on your known uniaxial yield stress, $\sigma_y$. This value is:
$\tau=\frac{\sigma_y}{\sqrt{3}}$

Again, the assumption there is von Mises plasticity.

Hope that helps

Last edited:

## 1. How is shear stress-shear strain related to stress-strain?

Shear stress-shear strain is a specific type of stress-strain relationship that describes the behavior of a material under shear stress. It is a measure of the resistance of a material to deformation caused by parallel forces acting in opposite directions.

## 2. Why is it important to convert a stress-strain curve to shear stress-shear strain?

Converting a stress-strain curve to shear stress-shear strain allows for a more accurate understanding of a material's behavior under shear stress. This relationship is essential in engineering and materials science, as many structures and materials are subjected to shear stress in various applications.

## 3. How is a shear stress-shear strain curve obtained from a stress-strain curve?

To obtain a shear stress-shear strain curve from a stress-strain curve, the shear stress (τ) is calculated by dividing the shear force (F) by the cross-sectional area (A) of the sample, and the shear strain (γ) is calculated by dividing the displacement (Δx) by the original length (L) of the sample. These values are then plotted against each other to create the shear stress-shear strain curve.

## 4. What can a shear stress-shear strain curve tell us about a material?

A shear stress-shear strain curve can provide valuable information about the mechanical properties of a material, such as its shear modulus, yield strength, and ultimate strength. It can also reveal the material's ductility, toughness, and ability to withstand shear stress without failure.

## 5. Are there any limitations to using a shear stress-shear strain curve?

While a shear stress-shear strain curve is a useful tool for understanding a material's behavior under shear stress, it does have limitations. It only provides information about a material's response to shear stress and does not account for other types of stress, such as tensile or compressive stress. Additionally, it may not accurately represent the behavior of a material under dynamic or extreme loading conditions.

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