 # Energy in stress-strain graphs

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## Energy in stress-strain graphs

#### Evaluation of strain energy from stress - strain graph

We know that when a material behaves elastically, the work done on straining it is stored as energy in it. We call this (elastic) strain energy. We can derive the strain energy density (ρe) in a material by calculating the area under its stress - strain graph. The definition of the density of energy is analogous to the definition of the density of mass. It is the energy stored per unit volume (how many joules are stored in 1m3 of the material).  Jm-3

Where:

F is the applied force,

e is extension obtained at force F,

A is the area of the cross section of the object and

l is the length of the object

With the knowledge of ρε we can calculate the total energy stored in an object (i.e. that given by the area under the force - extension graph) if we know the volume of the object.

We can demonstrate this by calculating the work done per unit volume from the total work (W) done on the object derived from the force - extension graph. Work done per unit volume = total work done/ total volume, V = But V = Al

Work done per unit volume = ( )( )

#### Worked Example

A mass of 200N is hung from the lower end of a steel wire hanging from the ceiling of the laboratory. The length of the wire 5m, its diameter is 1mm, Young's modulus is: 2 x 1011 Nm-2. Calculate the strain energy density of the wire and the total energy stored in it.

Apply the formulae for strain density and total work done F = 200N

d = 1mm = 1 x 10-3m

A = ¼ (πd2) = 0.785 x 10-6m2

l = 5m Strain energy density =1.62 x105Jm-3

Total strain energy stored = 0.637 J

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