Let L be length of body, A the area of cross-section and

l

the increase in length.

S t r e s s equals fraction numerator F over denominator A to the power of ´ end exponent end fraction s t r a i n equals fraction numerator l over denominator L end fraction

Force necessary to deform the body is

F equals fraction numerator Y A over denominator L end fraction l

If body is deformed by a distance, then

W o r k blank d o n e equals F cross times d l equals fraction numerator Y A over denominator L end fraction l d l

W equals not stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator Y A over denominator L end fraction l d l equals fraction numerator Y A over denominator L end fraction open square brackets fraction numerator l to the power of 2 end exponent over denominator 2 end fraction close square brackets subscript 0 end subscript superscript l end superscript equals fraction numerator 1 over denominator 2 end fraction Y A fraction numerator l to the power of 2 end exponent over denominator L end fraction

equals fraction numerator 1 over denominator 2 end fraction open parentheses Y fraction numerator l over denominator L end fraction close parentheses open parentheses fraction numerator l over denominator L end fraction close parentheses open parentheses A L close parentheses

equals fraction numerator 1 over denominator 2 end fraction open parentheses s t r e s s cross times s t r a i n close parentheses cross times v o l u m e

Hence, work done for unit volume is

W equals fraction numerator 1 over denominator 2 end fraction s t r e s s cross times s r a i n.