The criteria of Stress and Strain

Hooke's Law

Hooke's Law states that, for certain elastic materials, force is proportional to extension, when a sample is stretched. This means that the extension of the sample increases linearly with the amount of force applied. Materials that obey Hooke's law are called Hookean Materials. Springs behave like Hookean Materials.

Hooke's law can be written as F = kx, where F is Force, x is extension, andk is the Stiffness Constant of the sample. The stiffness constant describes the stiffness of a material, and is measured in Nm^-1 (or Kgs^-2).

F = kx

Hooke's law can be demonstrated with the use of Force-Extension graphs.

However, no sample follows Hooke's law indefinitely, and there comes a point, called the Limit of Proportionality, where there is no longer a linear relationship between force and extension. After yet more force is applied, the Elastic Limit will be reached. This means that the sample will no longer return to its original shape when the force ceases to be present. Eventually, the force will become so great that the material snaps. This is called the Yeild Point.

Hooke's Law

Forces can cause objects to deform (i.e. change their shape). The way in which an object deforms depends on its dimensions, the material it is made of, the size of the force and direction of the force.

If you measure how a spring stretches (extends its length) as you apply increasing force and plot extension (e) against force (F);

the graph will be a straight line.

Note: Because the force acting on the spring (or any object), causes stretching; it is sometimes called tension or tensile force.

This shows that Force is proportional to extension. This is Hooke's law. It can be written as:

F = ke

Where:.

F = tension acting on the spring.

e is extension = (l-l_o); l is the stretched length and l_o is original length, and.

k is the gradient of the graph above. It is known as the spring constant.

The above equation can be rearranged as

Spring constant = Applied force/extension

The spring constant k is measured in Nm^-1 because it is the force per unit extension.

The value of k does not change unless you change the shape of the spring or the material that the spring is made of.

A stiffer spring has a greater value for the spring constant

We can apply the concept of spring constant to any object obeying Hooke's law. Such an object is called (linearly) elastic.

An elastic object will return to its original form if the force acting on it is removed.

Deformation in an elastic object increases linearly with the force.

In fact, a vast majority of materials obey Hooke's law for at least a part of the range of their deformation behaviour. (e.g. glass rods, metal wires).

In the diagram above, if you extend the spring beyond point P, and then unload it completely; it won't return to its original shape. It has been permanently deformed. We call this point the elastic limit - the limit of elastic behaviour.

If a material returns to its original size and shape when you remove the forces stretching or deforming it (reversible deformation), we say that the material is demonstrating elastic behaviour.

If deformation remains (irreversible deformation) after the forces are removed then it is a sign of plastic behaviour.

Before the elastic limit is reached, the sample is experiencing Elastic Deformation, where it will return to its original shape when the load(force) has been removed. However, once the material passes that point, it experienced Plastic Deformation, where its shape is permanently changed.

If two springs are used in series, the effective stiffness constant of both of them is less than either of them. In fact, it can be worked out by the formula: 
If two springs are in parallel, their effective stiffness constant isgreater: 

Stress[]

Stress is a measure of the internal force an object is experiencing per unit cross sectional area. Hence, the formula for calculating stress is the same as the formula for calculating pressure: but

σ=FA{\displaystyle \sigma ={\frac {F}{A}}}

where σ is stress (in Newtons per square metre or, equivalently, Pascals). F is force (in Newtons, commonly abbreviated N), and A is the cross sectional area of the sample.

Tensile Strength[]

The (ultimate) tensile strength is the level of stress at which a material will fracture. Tensile strength is also known as fracture stress. If a material fractures by 'crack propagation' (i.e., it shatters), the material is brittle.

Yield Stress[]

On a stress strain graph beyond the yield point (or elastic limit) the material will no longer return to its original length. This means it has become permanently deformed. Therefore the yield stress is the level of stress at which a material will deform permanently. This is also known as yield strength.

Strain[]

Stresses lead to strain (or deformation). Putting pressure on an object causes it to stretch. Strain is a measure of how much an object is being stretched. The formula for strain is:

 ϵ=Δll0=l−l0l0=ll0−1{\displaystyle \ \epsilon ={\frac {\Delta l}{l_{0}}}={\frac {l-l_{0}}{l_{0}}}={\frac {l}{l_{0}}}-1},

where l0{\displaystyle l_{0}} is the original length of a bar being stretched, and l is its length after it has been stretched. Δl is the extension of the bar, the difference between these two lengths.

Young's Modulus[]

Young's Modulus is a measure of the stiffness of a material. It states how much a material will stretch (i.e., how much strain it will undergo) as a result of a given amount of stress. The formula for calculating it is:

E=σϵ{\displaystyle E={\frac {\sigma }{\epsilon }}}

The values for stress and strain must be taken at as low a stress level as possible, provided a difference in the length of the sample can be measured. Strain is unitless so Young's Modulus has the same units as stress, i.e. N/m² or Pa.

Stress-Strain Graphs[]

Stress–strain curve for low-carbon steel.

Stress (σ) can be graphed against strain (ε). The toughness of a material (i.e., how much it resists stress, in J m^-3) is equal to the area under the curve, between the y-axis and the fracture point. Graphs such as the one on the right show how stress affects a material. This image shows the stress-strain graph for low-carbon steel. It has three main features:

Elastic Region[]

In this region (between the origin and point 2), the ratio of stress to strain (Young's modulus) is constant, meaning that the material is obeying Hooke's law, which states that a material is elastic (it will return to its original shape) if force is directly proportional to extension of the material by BRYAN ESSIEN

Hooke's Law[]

Hooke's law of elasticity is an approximation that states that the Force (load) is in direct proportion with the extension of a material as long as this load does not exceed the proportional limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic

F=−kx{\displaystyle F=-kx\,}

The relation is often denoted

F∝x{\displaystyle F\propto x}

The work done to stretch a wire or the Elastic Potential Energy is equal to the area of the triangle on a Tension/Extension graph, but can also be expressed as

12kx2{\displaystyle {\tfrac {1}{2}}kx^{2}}

Plastic Region[]

In this region (between points 2 and 3), the rate at which extension is increasing is going up, and the material has passed the elastic limit. It will no longer return to its original shape. After point 1, the amount of stress decreases due to necking at one point in the specimen. If the stress was recorded where the necking occurs we would observe an upward curve and an increase in stress due to this reduction in area(stress = Force / area, thus stress increases during necking). The material will now 'give' and extend more under less force.

Fracture Point[]

At point 3, the material has been fractured.

Other Typical Graphs[]

In a brittle material, such as glass or ceramics, the stress-strain graph will have an extremely short elastic region, and then will fracture. There is no plastic region on the stress-strain graph of a brittle material.

Questions

1.     100N of force are exerted on a wire with cross-sectional area 0.50mm². How much stress is being exerted on the wire?

2.     Another wire has a tensile strength of 70MPa, and breaks under 100N of force. What is the cross-sectional area of the wire just before breaking?

3.     What is the strain on a Twix bar (original length 10cm) if it is now 12cm long?

4.     What is this strain, expressed as a percentage?

5.     50N are applied to a wire with a radius of 1mm. The wire was 0.7m long, but is now 0.75m long. What is the Young's Modulus for the material the wire is made of?

6.     Glass, a brittle material, fractures at a strain of 0.004 and a stress of 240 MPa. Sketch the stress-strain graph for glass.

7.     (Extra nasty question which you won't ever get in an exam) What is the toughness of glass?

8.     Wire has a tensile strength of 0.95Mpa, and breaks under 25N of force. what is the cross-sectional area of the wire before and after breaking?