Fatigue failure is a failure mechanism that results from the formation and growth of cracks under repeated cyclic stress loading, leading to fracture. It can result in failure at stress levels well below the material yield or ultimate strengths.
The video below discusses the mechanisms behind fracture, and how S-N curves can be used to predict the fatigue life of a component. It also discuss the effect of having a tensile mean stress, and how we can use techniques like the Rainflow counting method and Miner’s rule to consider complex cyclic stresses.
Strength, ductility and toughness are three very important and distinct material properties, but understanding the differences between them can sometimes be confusing. This page and the video below should help clear things up!
Strength is a measure of the stressa material can withstand. Two different measurements are used to define the strength of a material:
The ultimate strength, which is the maximum stress the material can withstand before fracturing.
The yield strength, which is the stress at which the material begins to deform plastically (meaning that permanent deformation remains after the applied load is removed).
Both the ultimate strength and the yield strength express how much force the material can withstand per unit area, so they have the same units as stress, which is force/area. In the SI system strength and stress have units of pascals ($1 \mathrm{Pa} = 1 \mathrm{N} / \mathrm{m^2}$), but are often expressed in megapascals (MPa).
Yield and ultimate strengths can be determined from the stress-strain curve of a material, that is obtained by performing a tensile test.
The yield and ultimate strengths on a stress-strain curve
Many structures and components are design to ensure that they only deform elastically (i.e. there is no permanent deformation after the applied load is removed). This makes the yield strength a commonly used criterion for defining failure in engineering design codes. The failure theories page goes into more detail about the different ways failure is defined.
Here are some typical strength values for a few different materials.
Material
Yield Strength [MPa]
Ultimate Tensile Strength [MPa]
Copper
33
210
Aluminium Alloy 3033
126
200
Steel, stainless AISI 302
275
620
Titanium 6Al-4V
880
950
Determining the Yield Strength from a Stress-Strain Curve
The transition from elastic to plastic deformation is not always easy to identify on a stress-strain curve. For this reason standard methods have been developed to determine the yield strength from tensile test data. One of the more common methods that is often used for metals is the 0.2% offset method. Here’s how it works:
Step 1 – draw a line with a slope equal to the Young’s modulus of the material
Step 2 – shift this line 0.2% to the right on the stress-strain curve
Step 3 – the intersection between this line and the stress-strain curve gives you the material’s 0.2% offset yield strength (also called 0.2% proof stress)
This process is illustrated in the image below. This material has a 0.2% offset yield strength of 330 MPa.
How to determine the yield strength of a material using the 0.2% offset method
The Efficient Engineer Summary Sheets
The Efficient Engineer summary sheets are designed to present all of the key information you need to know about a particular topic on a single page. It doesn’t get more efficient than that!
Ductility is a measure of the ability of a material to deform plastically before fracturing.
A material is ductile if it undergoes a large amount of plastic deformation before it breaks. These materials fracture at very large strains. Examples of ductile materials include mild steel, aluminium, and gold.
A material is brittle if it fractures at low strains with little or no plastic deformation. Examples of brittle materials include glass and ceramics.
The differences between ductile and brittle materials can be seen in the stress-strain curves shown below.
Comparison between stress-strain curves for brittle and for ductile materials
Ductile and brittle materials have very different stress-strain curves, with brittle materials exhibiting little or no plastic deformation before fracturing. Because they don’t deform plastically, the concept of yield strength is irrelevant for brittle materials.
Typically a material that has a strain at fracture of less than 5% is considered to be brittle.
Compressive vs Tensile Loading
Another interesting difference between ductile and brittle materials is how the behave under compressive versus tensile loading.
For ductile materials like mild steel the yield and ultimate strengths are very similar in the tensile and compressive directions. Compression tests are more difficult to carry out than tensile tests because buckling can be an issue, so compression tests aren’t often performed for ductile materials.
For brittle materials like concrete and ceramics the material strength is much larger in compression than in tension. Separate tensile and compression tests usually need to be carried out to properly characterised these materials.
DUCTILE MATERIALS
compressive strength $\simeq$ tensile strength
BRITTLE MATERIALS
compressive strength > tensile strength
Ductility vs Temperature
The ductility of a material can vary with temperature. A lot of different types of steel for example are ductile are room temperature but become brittle when the temperature drops to below the ductile-to-brittle transition temperature. This is an important design consideration because ductile failure is normally preferred to brittle failure.
Material Toughness
Toughness is the ability of a material to absorb energy up to fracture. Materials that can absorb a lot of energy before fracturing have high toughness.
Toughness can be thought of as the area under the stress-strain curve. If the area is large, the material will have high toughness and will be able to absorb a large amount of energy before fracturing.
Toughness can be thought of as the area under the stress-strain curve
For a material to have high toughness it should have a good balance of both high strength and high ductility. Low strength and brittle materials tend to have low toughness.
Strength vs Ductility vs Toughness
Strength, ductility and toughness are separate but linked material properties. The five boxes below summarise the definitions of these properties, and a fifth parameter, resilience, is introduced.
Ultimate strength
the maximum stress which is reached during the tensile test
Yield strength
the stress at which a material begins to deform plastically
Toughness
a measure of the ability of a material to absorb energy up until fracture
Ductility
a measure of the ability of a material to deform plastically
Resilience
a measure of the ability of a material to absorb energy while deforming elastically
These five properties can be identified on the stress-strain curve, as shown below.
Strength, ductility, toughness and resilience represented on a stress-strain curve
Related Topics
Young's Modulus
Young's modulus is a measure of the stiffness of a material – it describes the relationship between stress and strain.
Young’s modulus is an important material property in engineering:
It is a measure of the stiffness of a material (i.e. a measure of how much a material will deform when acted on by a force).
Metals and ceramic materials tend to have high Young’s modulus values, whereas polymers have much lower values (they deform more for the same applied load)
In the elastic region of the stress-strain curve the slope of the curve is equal to Young’s modulus. Young’s modulus can be calculated as an applied stress divided by the resulting strain – this equation is called Hooke’s Law.
Young’s modulus is a really important material property that appears all the time in engineering and physics. It’s used for everything from calculating when a column will buckle to determining how much a beam will deflect when forces are applied to it. So let’s learn more about it!
What is Young’s Modulus?
Young’s modulus, also called the elastic modulus, is a material property that describes how much a material will deform when a load is applied to it. It is essentially a measure of how stiff a material is.
The easiest way to visualise Young’s modulus is on a stress-strain curve, shown in the image below, that can be obtained by performing a tensile test on a sample. The curve can be divided into two regions, an elastic region and a plastic region. If the applied stress is low and we remain in the elastic region – the original dimensions of the material will be completely recovered when the applied load is removed. For larger stresses that take us into the plastic region, permanent plastic deformation will remain after the applied load is removed.
The stress-strain curve is a straight line in the elastic region for most materials. The slope of this curve is equal to Young’s modulus, which is denoted using the letter $E$. Young’s modulus describes the relationship between stress and strain in the elastic region. The higher the Young’s modulus of a material, the smaller the elastic deformations (strain) will be for a given applied load (stress).
A material’s Young’s modulus is equal to the slope of its stress-strain curve in the elastic region
Hooke’s law gives us the relationship between stress and strain in the linear elastic region of the stress-strain curve.
Hooke’s Law
$$\sigma = E \varepsilon$$
Since strain is dimensionless, Young’s modulus has the same units as stress. Steel has a high Young’s modulus of around 200 GPa, whereas Polystyrene has a low Young’s modulus of around 3 GPa.
Check out the video below if you want to watch an animated summary of all of this information about Young’s modulus. If not, just keep reading!
Young’s modulus is very important when it comes to engineering, because so often engineering design and analysis involves determining how much something will deform. A high Young’s modulus value is good if you want deformations to be very small – you wouldn’t want a bridge to deform much when a car passes over it, for example, so it would make sense to construct the bridge from materials with a high Young’s modulus.
Because it defines the relationship between stress and strain, Young’s modulus appears in all kinds of different equations that are commonly used in mechancal and civil engineering. Here are a few examples:
Buckling Formula
This equation describes the critical load at which a column will buckle
$$P_{cr} = \frac{\pi^2 EI}{{L_e}^2}$$
Beam Deflection
This equation describes how much a beam will deflect due to applied loads
$$\frac{d^2y}{dx^2} = \frac{M}{EI}$$
Axial Deformation
This equation describes how much a bar will deform when an axial load is applied to it
$$\delta = \frac{FL}{AE}$$
How to Determine Young’s Modulus
If you have data from tensile tests performed for a particular material, it’s easy to figure out what the Young’s modulus for that material is. All you have to do is take two points in the linear elastic part of the stress-strain curve, and use those points to measure the slope of the curve. This is illustrated in the figure below.
How to calculate Young’s modulus from test data
Young’s Modulus Values for Different Materials
The image below shows stress-strain curves for a few different materials. Material 1 has the steepest slope in the elastic region, which means that it has the largest Young’s modulus – you will need to apply a much larger force to deform Material 1 than you would to deform Materials 2 or 3 by the same amount. The stress-strain curve for Material 1 is typical of diamond, for example, whereas Material 2 is more like steel, and Material 3 is typical of a material like rubber.
Material 1 has the largest Young’s modulus (it has the highest stiffness)
The table below shows typical Young’s modulus values for a few different materials. Ceramics and metals generally tend to have quite high Young’s modulus values, meaning that they are quite stiff. Polymers on the other hand have much lower values.
Material
Young’s Modulus [GPa]
Rubber
0.1
Polystyrene
3
Aluminium Alloy
69
Titanium
120
Copper
130
Carbon Steel
207
Tungsten Carbide
600
Diamond
1200
Understanding Young’s Modulus Using the Ball-and-Spring Model
The Young’s modulus of a material is closely related to the strength of the bonds between its atoms. We can think of these interatomic bonds as small springs connecting adjacent atoms, represented by balls.
In the ball-and-spring model the atoms are represented by balls and the interatomic forces are represented by springs.
When a force is applied to the material, deformation of the material is resisted by the stiffness of the interatomic bonds. Young’s modulus is a measure of how strong the interatomic bonds are, or in other words it is a measure of how stiff the springs between the atoms are. When the force is removed the atoms return to their initial position – this is elastic deformation.
The mechanism behind plastic deformation is very different, because it involves the breaking of bonds and the re-arrangement of the position of the atoms.
Elastic Deformation
caused by the stretching of the bonds between atoms
deformation is reversed when the load is removed
Plastic Deformation
caused by interatomic bonds breaking at dislocations
deformation is permanent and remains after the load is removed.
As well as helping us understanding the differences between elastic and plastic deformation, the ball-and-spring model can help us answer a few important questions about Young’s modulus.
Why is Young’s Modulus Different for Different Materials?
The strength of the bonds between the consituent atoms or molecules of a material (i.e. the stiffness of the springs in the ball-and-spring model) plays a big role in determining the Young’s modulus of a material.
The strength of these bonds is affected by atomic/molecular structure and chemical composition, which is why Young’s modulus values can be so different for different materials.
The way the atoms and molecules are bonded also plays a big role. Polymers for example have lower Young’s modulus values compared to metals and ceramics because the material stiffness is determined by the much the weaker intermolecular bonds, instead of the strong interatomic bonds.
Polymers have low Young’s modulus values because the stiffness of the material is governed by weak intermolecular forces instead of strong interatomic forces
Does Young’s Modulus Change for Different Metal Alloys?
The ball-and-spring model also explains why Young’s modulus values tend to be very similar for different alloys of the same metals. Carbon steel for example has many different alloys that have very different yield and tensile strengths, but they all have very similar Young’s modulus values. This is because these alloys usually contain only very small concentrations of the alloying elements, that have little effect on the overall stiffness of the atomic bonds. Carbon steel is an interstitial alloy, but the same is true for substitional alloys, like the one shown below.
This substitional alloy contains only very small concentrations of the blue alloying elements, so the alloying elements don’t significantly affect the overall stiffness of the material
Test Your Understanding
Which of the following significantly affect the Young's modulus of steel?
None of these
Alloying
Heat treatment
Work hardening
Explanation
The Young's modulus of a material is related to the strength of its atomic bonds. This is not significantly affected by alloying in steels due to the small proportion of alloying elements, and it is also not affected by heat treatment or work hardening. This is why all steels have a Young's modulus of around 207 GPa.
For most materials, Young’s modulus decreases as temperature increases. This is because an increase in temperature causes an increase in the vibrations of atoms, which increases the spacing between them, reducing the strength of the interatomic bonds.
Carbon steel for example has a Young’s modulus of 200 GPa at room temperature, but this reduces to around 150 GPa at 600°C.
Young’s Modulus vs Tangent Modulus
The tangent modulus is a material property that is closely related to Young’s modulus. Whereas Young’s modulus is the slope of the stress-strain curve in the elastic region, the tangent modulus is defined as the slope of the stress-strain curve at any point along the curve, whether that’s in the elastic or the plastic region. The value of the tangent modulus changes as you move along the stress-strain curve.
In the linear elastic region of the stress-strain curve the tangent modulus is equal to Young’s modulus. Outside of this region the tangent modulus is smaller than Young’s modulus, because materials tend to soften and become less stiff as they deform plastically.
The tangent modulus is defined as the slope of the stress-strain curve. Unlike Young’s modulus, which is a constant value, the tangent modulus varies along the stress-strain curve.
The tangent modulus can be calculated from stress-strain curves obtained from testing, or can be calculated analytically using methods like the Ramberg-Osgood equation.
Related Topics
Strength, Ductility & Toughness
Strength, ductility and toughness are three important material properties that can be determined from a stress-strain curve.
