# Energy in stress-strain graphs

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## Energy in stress-strain graphs

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 1m^{3} 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 ρ_{ε} 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** = ( )( )

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 10^{11} 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^{-3}m

A = ¼ (πd^{2}) = 0.785 x 10^{-6}m^{2}

*l* = 5m

**Strain energy density =1.62 x10 ^{5}Jm^{-3}**

**Total strain energy stored = 0.637 J**