Biomaterials: An Introduction to Materials Science – Stress vs. Strain

Two of the biggest concepts that one must intimately understand to understand materials science are stress and strain.

Stress is the normalized load applied to a material.  Stress is expressed as σ = F_n / A.  Stress is the force/area applied.

σ = F_n / A         (1)

where

σ = normal stress ((Pa) N/m², psi)

F_n = normal component force (N, lb_f)

A = area (m², in²)

Strain, similarly, is the normalized deformation that results due to a certain stress.

ε = dl / l_o = σ / E         (3)

where

dl = change of length (m, in)

l_o = initial length (m, in)

ε = unitless measure of engineering strain

E = Young's modulus (Modulus of Elasticity) (Pa, psi)

There are 3 types of principal loads: tension, compression, and shear.

 Tension/Compression involve situations where the area of the load is perpendicular to the loading direction and the change in length of the material is parallel to the original length.  Or put more simply, when the tensile stress is perpendicular to the length and the tensile strain is parallel we are dealing with either tension or compression.

Tensile stress will try to stretch the material whereas compressive stress will try to compress the material.

Shear Stress is a special type of stress that involves the applied load being parallel to the load area.  The strain will be in the same direction as the applied load.

The formula to calculate a shear stress is:

where

τ = the shear stress

F = the force applied

A = the cross sectional area

An example of shear stress is seen below:

 

Deformation

I’ve been using the term deformation to describe strain, but have yet to properly explain it.  Deformation is the change that results in the material after the application of a stress.  There are two types of deformation: plastic and elastic.  The difference is that elastic deformation is recoverable, while plastic deformation is permanent.  Think a rubber band stretching (elastic) vs. a glass breaking (plastic).

Stress Strain Curve

The most important diagram to understand the macroscopic properties of a material is probably the stress strain curve.  This is literally a graph of the stress on a material vs. the strain that results in the material.

So we see in the figure that stress is charted on the y-axis and strain is charted on the x-axis.  In my opinion, it’s rather odd because intuitively I would think the stress is the independent variable and the strain is the dependent variable.  But whatever.  Scientists way smarter than I set this up, so I will go with it.  (BTW, if you happen to know why the axes are as they are, do let me know in the comments!)

In this graph, 2 is the point where the material breaks.  And we can see that this will occur at a stress of 1.  1 is referred to as the tensile strength.  2 is referred to as the yield.

Stress vs. Strain Vocabulary

• Tensile Strength: the stress at which a material breaks or permanently deforms.  There are 3 types of tensile strength:
  • Yield strength: The stress at which a material changes from elastic to plastic deformation
  • Ultimate strength: The maximum stress a material can withstand. (max Y on the stress-strain curve)
  • Breaking strength: The stress coordinate on the strain-strain curve at the point of rupture (breaking)

In the figure above, point 4 is the tensile strength, 1 = ultimate strength, and either 2 0r 3 can be considered the yield strength.  The graph above is the stress strain curve for aluminum.  We see 2 general regions: a straight line region, and a curved line region.  The straight line region is the area under which the material is undergoing elastic (reversible) deformation.  The curved line region is representative of plastic deformation.

To be more “rigorous” (as a Professor of mine would love to say), point 2 is the offset yield strength.  This is due to a offset strain that is seen as the dotted line from point 5 to point 2.  Point 3 is then referred to as the Proportionality limit yield strength. All this bullocks is because yield strength can be defined in several ways.

• 1 = True elastic limit – the lowest stress at which dislocations in the material move.  This value is rarely used in engineering applications, but does provide some sort of chemical information
• 2 = Proportionality limit – up to this point, stress and strain are related by Hooke’s Law (defined below)
• 3 = Elastic limit – up to this point, the material only undergoes elastic deformation, past this is all plastic
• 4 = Offset Yield Strength – usually a value of 0.2% offset from the zero point.  Sometimes materials don’t even have a linear region, and the offset becomes very handy there.  It is helpful in making the stress-strain curves of different materials comparable

Hooke’s Law

Hooke’s Law is used to relate stress and strain through the Young’s Modulus.  If you didn’t notice, the “Hooke’s Law” title is gigantic. Yup, I did it on purpose.  Everyone note that Hooke’s Law is simultaneously important and fundamental and simple.

Most physics students first encounter Hooke’s Law through the equation F = – kx.  In words: the force equals the spring constant times the distance the spring is stretched or compressed.  The negative sign simply means that it is a “restorative” force.  So when you pull the spring and make the change in x positive, then the force is negative to pull it on the opposite direction.

So if we can describe elastic deformation with a force law used to describe springs then how should we visualize elastic deformation?  Like a spring, duh.  Silly.

There are two analogs to Hooke’s law for springs.  One for tension/compression and one for shear.

σ = Eε

The equation for Hooke’s Law for tension/compression.  Sigma equals the tensile stress.  E is Young’s Modulus (the analog to the spring constant) and epsilon is the tensile strain.

τ = G γ

The equation for Hooke’s Law for shear stress.  In words: Shear stress equals the shear strain times the shear modulus.

Young’s and Shear Modulus

The basic idea behind these two moduli is that the higher the modulus, the more resistive the material will be to the stress.  The higher the modulus, the higher the slope, thus there will be less of a strain for a given stress.

The Young’s and Shear Moduli are inherent macroscopic properties of a material.  Thus, they are often used to compare one material to another.

Poisson’s Ratio

Poisson’s Ratio is another thing that you just have to get familiar with before we dive into biomaterials.  Sorry y’all.

Poisson’s Ratio is the ratio of the lateral to the axial strain.

In the figure above, is the transverse strain, which is the same as the lateral strain.

Poisson’s Ratio is important because it relates the deformation in one plane to the deformation in another plan.  When you take a rubber band and stretch it, you should notice that the more you stretch it length wise, the thinner it becomes width wise.  It’s a very intuitive relationship.

Most materials will have a positive Poisson’s Ratio, meaning that an expansion in one direction will result in a compression in another.  The reason is that most materials will resist a change in volume more than they resist a change in shape. ( The Bulk Modulus K is generally higher than the Shear Modulus G)

And yes, in the rare case that the Poisson Ratio is negative, that implies that the material will actually get larger when stretched.  Some wine corks operate in this manner.  Therefore, when the cork needs to be reinserted into the bottle, when it is compressed (pushed in) the cork will become thinner allowing it to slide into the opening.  Once the compressive force is gone, then the cork should expand to complete close the opening.