Stress
Stress (σ) – the force per unit cross sectional area applied to an object.
From this definition we have the following equation:
Where F is the force applied in Newtons (N) and A is the cross sectional area of the object before the force is applied in metres squared (m2).
From the definition we know that the units for stress are Nm-2, in SI units this is the same as Pascals (Pa).
The force can be compressive or tensile and must act perpendicular to the cross sectional area in order to change the length.
Strain
Strain (ε) – the extension (or compression) per unit length resulting from an applied stress.
The strain of an object can be calculated using the following equation:
Where L is the stretched/compressed length of the object and L0 is the original length of the object, both measured in metres (m).
As strain is calculated by dividing a length by a length the units cancel each other out and strain has no units.
NOTE: As long as all your lengths are in the same units, they don’t have to be in metres – as they cancel each other out. For example, they could all be in inches. But if you had L in metres and L0 in inches you would have to convert them to be the same.
Young’s Modulus
When elastic objects are subject to a tensile or compressive force a useful quantity to help describe their properties is Young’s Modulus (also called the modulus of elasticity).
Young’s Modulus – the ratio of stress and strain that a material undergoes.
The value can be calculated using the following equation:
As strain has no units, the units for Young’s Modulus are the same as the units for stress – Newtons per metre squared (Nm-2) or in SI units Pascals (Pa).
IMPORTANT: Young’s Modulus only applies when an object is undergoing elastic deformation NOT plastic deformation.
The value tells us how stiff a material is – the greater the value, the stiffer the material. A material with a greater Young’s Modulus requires a greater force to stretch (or compress it) by the same amount as a material with a smaller Young’s Modulus.
Worked Examples
Example 1
A tensile force of 20N is applied to a wire. Calculate the cross sectional area of the wire if the stress on the wire is 1.5 x107 Pa.
Example 2
Calculate the strain caused by a stress of 1.2 x109 Pa on a steel wire.
Steel has a Young’s Modulus value of 2.0 x1011 Pa.
Example 3
A stress of 4.39 x109 Pa causes a 4m copper wire to stretch by 0.15m. Calculate the Young’s Modulus of copper.
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