Answer :

Solution :

The Strain Energy of Distortion or Distortion energy per unit volume is given by :

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation)

Total Strain Energy is given by :

$\mathrm{U}=\frac{1}{2\mathrm{E}}\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}$

Energy of dilation :

$U_v=\frac{(1-2\mu )}{6E}\times \frac{({\sigma }_1+{\sigma }_2+{\sigma }_3{)}^2}{3}$

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation)

$\frac{1+\mu }{6E}[({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2]$

In simple tension test:

$\left(\frac{1+\mu }{6E}\right)2{\sigma }_{ym}^2$

Maximum distortion energy theory (Von mises theory)

- According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy per unit volume reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test.

- von misses stress under triaxial condition is given by :

${\sigma }_{vm}=\sqrt{\frac{1}{2}\left\{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right\}}$

Now if we compare von misses stress and distortion energy per unit volume equation then,

U_d = $\frac{1+\mu }{3E}\times {\sigma }_{vm}^2$

U_d ∝ σ²_vm

σvm ∝ $\sqrt{U_d}$

The von Mises stress at a point in a body subjected to forces is proportional to the square root of the distortional strain energy per unit volume.