As-level Deformation of solid

Aslevels Physics Notes

Topic 6: As-level Deformation of Solids

Tension when a stretching force acts on a material, the material is said to be under tension. A stretching for is called a tensile force. Tension is measured in Newton (N).

Compression when a material is being squashed or squeezed, it is under compression and is measured in Newton (N).


Shear is caused by sideways forces acting in opposite directions. If a force acts across the top of an object while the bottom remains fixed, a shearing effect or shear is said to act.

Deformation When the forces are applied to a solid body, its shape changes. The change may be very small, but nevertheless the forces affect the spacing of the atoms in the solid to a tiny extent and its external dimensions change. This change of shape is called deformation. If you think of a metal cylinder, the application of forces along the axis of the cylinder can either stretch it, making it longer, or squash it slightly, making it shorter. We call the deformation produced by these forces a tensile deformation for the stretching case, or a compressive deformation for the squashing case.

Elasticity A material is said to be elastic if it returns to its original size and shape when the load which has been deforming it is removed.

The diagram shows apparatus used to investigate the extension of a spring under a tensile force. The graph shows the results of the experiment. Analyzing the results we see:

• From O to A the extension of the spring is proportional to the applied force. 
• With larger forces, from A to B, the spring extends more easily and the extension is no longer proportional to the load. 
• When the force is reduced, with the spring having been stretched beyond point B, it no longer goes back to its original length.  

From O to A, F is proportional to x: 𝐹 ∝ x 

This can be written as an equality by introducing a constant of proportionality

𝐹 = 𝑘x

Where k is the constant of proportionality, often known as the spring constant. The spring constant is the force per unit extension. It is a measure of the stiffness of the spring. The larger the spring constant, the larger is the force required to stretch the spring through a given extension. The unit of the spring constant is newton per metre (Nm-¹ ). 

Point A, the point at which the spring ceases to show proportionality, is called the limit of proportionality. Very close to this point, there is a point B called the elastic limit. Up to the elastic limit, the deformation of the spring is said to be elastic. This means that the spring will return to its original length when the load is removed. If the spring is stretched beyond the elastic limit it will not return to its original length when the load is removed. Its deformation is said to be plastic.

Hooke’s law sums up the behabiour of many materials that behave in a similar manner to a spring: 

Hooke’s law states that, 

Provided the elastic limit is not exceeded, the extension of a body is proportional to the applied force (load). 

𝑓𝑜𝑟𝑐𝑒 𝐹 ∝ 𝑥 ⟹ 𝐹 = 𝑘x

Note that Hooke’s law also applies to the compression of a body; in this case the quantity x in the equation is the compression rather than the extension.

Stress and strain  

 Stress 𝝈 is the force acting on unit cross – section area for a force F and area A we can write 

The unit of stress is Pascal (Pa)\

Strain 𝜀 is the extension of unit length. If 

Young modulus

Provided the stress is not so high that the limit of proportionality has exceeded, the ratio stress/strain is a constant for a given material and is known as young’s modulus. Thus

Where 𝐸 is young’s modulus (𝑁𝑚−² = 𝑃𝑎).
Young’s modulus is clearly a resistance to change in length.

Extension of wire

The diagram below shows the apparatus that could be used to investigate the stretching of a wire. 

Consider the experimental arrangement shown in the figure. The test wire is loaded (typically up to 100 N in 5N steps), and the resuluting extension is measured as a function of the load. The wires are as long as is convenient (typically 2m) and thin is order to obtain as large an extension as possible. If the test wire is free of kinks at the start and the limit of proportionality is not exceeded, the mearurement can be used to produce a straight line graph. After the elastic limit the graph is no more straight line.  

The graph obtained is similar to that obtained for the spring. This shows the general nature of Hooke’s law. 

For the graph within the limit of proportionality the young modulus can be determined as follows: 

Energy stored in a deformed material

The first graph shows the extension of a body that obeys Hooke’s law. The work done in stretching the body is equal to average force multiplied by distance moved (area under graph). 

The equation of the form 𝑠𝑡𝑟𝑎𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 = 1/2 𝑘𝑥²shows that the energy stored in the spring is proportional to the square of the extension – rather than just the extension itself. This means that if the extension is doubled the energy stored is quadrupled (four times), 

The area uder the graph when loading is larger than when unloading the wire. This means that more energy is stored in the stretched wire than is released when the load is removed. What happens to this energy? It is converted to the internal energy in the wire – the temperature of the wire increases. The energy released is equal to the area in the eclosed lopp made by the two curves. This is called elastic hysteresis