Strain Energy of the member is defined as the internal work done in defoming the body by the action of externally applied forces. This energy in elastic bodies is known as elastic strain energy.
STRAIN ENERGY IN UNIAXIAL LOADING:
Let as consider an infinitesimal element of dimensions as shown in Fig .1. Let the element be subjected to normal stress sx.
The forces acting on the face of this element is sx. dy. dz
dydz = Area of the element due to the application of forces, the element deforms to an amount = Îx dx
Îx = strain in the material in x – direction
Assuming the element material to be as linearly elastic the stress is directly proportional to strain as shown in Fig . 2.
\ From Fig .2 the force that acts on the element increases linearly from zero until it attains its full value.
Hence average force on the element is equal to ½ sx . dy. dz.
\ Therefore the workdone by the above force
Force = average force x deformed length
= ½ sx. dydz . Îx . dx
For a perfectly elastic body the above work done is the internal strain energy “du”.
where dv = dxdydz = Volume of the element
By rearranging the above equation we can write
The equation (4) represents the strain energy in elastic body per unit volume of the material its strain energy – density ‘uo' .