Question

Explain strain energy per unit volume in terms of stress and strain, stress and young's modulus

Answer 1

Dear Student,

The energy stored in a body, when a work is done on it by an external force for stretching (deforming) the body, is known as strain energy and strain energy per unit volume is strain energy density.

Strain energy is given by , U = 0.5 F l (1), where 'F' is the external force applied to produce an extension 'l'.

The strain energy density, u = U/V = (0.5 F l)/V

or $u=\frac{1}{2}\frac{F}{A}\frac{l}{L}$ (where A and L are the area and length of the body before applying the external force and V = A L)
i)
or $u=\frac{1}{2}\times stress\times strain$ (2) ( because F/A = stress and l/L = strain)
Thus equation (2) gives the energy density of a strained wire or potential energy per unit volume of a stretched wire.

ii)

Young's modulus (Y) of a material is expressed as , Y = stress/strain , using this in equation (2)

$u=\frac{1}{2}\frac{stres{s}^2}{Y}\left(3\right)$ (strain = stress/Y)

or we can say $u\alpha stres{s}^2$ (4) (Since for a given material its Young's modulus of elasticity will be constant)




Regards