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Stress and Strain

Siemens Experimenter Siemens Experimenter
Siemens Experimenter

What are stress and strain? 

 

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

 

Picture 1: Stress/strain test of metal coupon sample.  Left: Original sample.  Middle: Sample with necking.  Right: Sample with failurePicture 1: Stress/strain test of metal coupon sample. Left: Original sample. Middle: Sample with necking. Right: Sample with failure

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:

 

Picture 2: Force being applied to metal cylinder or couponPicture 2: Force being applied to metal cylinder or coupon

 

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

 

stress_formula.png

 

 

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:

 

Picture 3: By changing from area A to area A1, the stress on the part is reduced.Picture 3: By changing from area A to area A1, the stress on the part is 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.

 

Picture 4: Geometry change reduces the effective cross-sectional area and creates a stress concentrationPicture 4: Geometry change reduces the effective cross-sectional area and creates a stress concentration

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.

 

Picture 5: Strain is defined as a change in length over the original lengthPicture 5: 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.

 

strain_formula.png

 

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.

 

Picture 6: Relationship between stress and strain in the elastic region of a materialPicture 6: Relationship between stress and strain in the elastic region of a material

 

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.

 

youngs_modulus_equation.png

 

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.

Picture 7: Stress vs strain relationship in both plastic and elastic regions of materialPicture 7: Stress vs strain relationship in both plastic and elastic regions of material

 

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.

 

 

Picture 8: Differences in cross-sectional area as metal coupon undergoes increasing stress level (from left to right)Picture 8: Differences in cross-sectional area as metal coupon undergoes increasing stress level (from left to right)

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.

 

 

Picture 9: The True Engineering stress curve (Blue) versus the Engineering Stress curve (Red) for a coupon undergoing increasing load levelsPicture 9: The True Engineering stress curve (Blue) versus the Engineering Stress curve (Red) for a coupon undergoing increasing load levels

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.

 

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