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Stress and Strain

Siemens Valued Contributor

07-11-2016
08:59 AM

Stress and strain are two measurable engineering quantities that are important in understanding the durability or fatigue life of a product.

**Stress**

Stress, represented by the Greek sigma symbol, can be simply thought as a force distributed over an area. Stress has units of MPa.

For example, the force could be applied to metal cylinder (ie, a metal coupon) like so:

To calculate the stress, one would divide the force by the cross-sectional area of the cylinder:

How can the stress on the metal cylinder be reduced? There are only two options:

- Reduce the force
- Increase the area

By increasing the diameter, the stress on the metal coupon would be reduced:

This illustrates a classic issue with when designing for increased fatigue life: added weight and cost.

**Weight**

By increasing the area, the weight and material cost of the part has been increased. Not only is the part more expensive, it is also heavier. The added weight will effect the energy efficiency of operating the final product – it will take more fuel and/or power to move the heavier design.

It is of great importance to really understand the loading environment extremely well. If the actual loads are smaller than anticipated, than the part would not require as large of an cross-sectional area to survive for the intended life.

Many manufacturers today are undergoing "lightweighting" initiatives with their products. In order to increase energy efficiency, the weight of the product must be reduced. A key step in this process is to verify the assumed loads expected during the product lifetime to ensure that they are appropriate and not causing an over-design situation.

**Geometry**

Another engineering challenge may be that another department, perhaps the design department, may decide that a “coke bottle” glass shape would be better than a plain cylinder. They may decide to “neck down” in the middle of the cylinder for a more “pleasing” appearance.

By doing this “necking” and reducing the effective cross-sectional area, the stress has increased, despite the fact the area had previously been made larger to decrease the stress.

The neck itself also creates a stress concentration area. Because of the sudden change in geometry, the chances of a failure have been increased in that area.

**Strain**

Imagine that the same metal cylinder, after having a force applied, becomes a little bit longer (ie, elongated). Strain is defined as a change in length over the original length.

The strain is the change in length of the cylinder divided by the original length of the cylinder.

Because strain has a unit of length in both the numerator and denominator, it can be thought of as dimensionless.

However, because the change in length of parts, like the metal cylinder, are typically so small, it is common to use “units” of microstrain (sometimes abbreviated “muE”) to describe the change in length.

Microstrain changes the decimal place by a million, or 6 digits. For example, a strain “value” of 0.000050 becomes 50 muE or microstrain.

**Linear Relationship between Stress and Strain**

When applying a load to a part, initially the relationship between stress and strain is linear. While the relationship remains linear, it is considered the *elastic region* of the material.

In the elastic region of the material, when the stress is removed, the part returns to its original shape.

This linear stress-strain relationship yields “E”, which is the Young’s Modulus (or spring rate) of the part or material. The Young’s modulus is the change in stress over the change in strain.

This linear relationship is described by Hooke’s Law, which was proposed by Sir Robert Hooke in 1660.

**Non-Linear Relationship between Stress and Strain**

With a high enough load, the relationship between stress and strain becomes non-linear. Instead of linear elastic behavior, the relationship becomes non-linear plastic behavior as shown in Picture 7.

The point beyond which the relationship between stress and strain becomes non-linear is called the *yield strength*. Applying loads beyond the yield strength results in “plastic” deformation of the material.

While the yield strength is thought of as a single number or point on the curve, in reality, there is a small transition zone between the elastic and plastic region; it is not an instantaneous transition. Therefore the yield strength is defined by using a line offset 0.2% from the elastic line and plotting it's intersection on the stress/strain curve as shown in Picture 7.

In the *plastic region* of the material, the part deforms permanently, and will not return to the original shape when the stress is removed.

At another point of increasing load/stress, there is a point where the part starts to fail, or “neck”. This is the *ultimate strength* of the material.

With a high enough load/stress applied, the part will eventually pull apart, fracture or fail.

** **

**Engineering Stress vs True Stress**

As the part under load deforms, the original area (Ao) decreases as the load increases. When stress vs strain is plotted against the original area (Ao), it is called the *Engineering Stress* curve. Engineering Stress does not take into account that the area is changing.

If the stress and strain relationship are plotted using the actual cross-sectional area of the part, the plot result is called the *True Stress* curve , rather than the *Engineering Stress *curve.

Looking at True Stress versus strain, one can observe that the stress in fact increases in the part with increasing load. However, features like the ultimate strength are difficult to observe in the True Stress curve, and are easier to see in the Engineering Stress curve.

Questions? Email john.hiatt@siemens.com.

Be sure to check out the free on-demand webinar Fundamentals of Fatigue.

**More Fatigue and Durability links: **

- History of Fatigue
- Stress and Strain
- Calculating Damage with Miner's Rule
- What is a SN-Curve?
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- Difference between 'Range-Mean' and 'From-To' Counting
- Power Spectral Density
- Some Thoughts on Accelerated Durability Testing
- Goodman-Haigh Diagram for Infinite Life
- Measuring strain gauges in Simcenter Testlab
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