Question

According to the distortion energy theory, failure will NOT occur when (symbols have the usual meaning)

• A. ${\left[\frac{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2}{2}\right]}^{\frac{1}{2}}\le {\sigma }_0$
• B. ${[{({\sigma }_1-{\sigma }_2)}^2+4{\tau }^2]}^{\frac{1}{2}}\le {\sigma }_0$
• C. ${\left[\right({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)-\frac{1}{m}({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\left)\right]}^{\frac{1}{2}}\le {\sigma }_0$
• D. ${\left[\frac{({\sigma }_1-{\sigma }_2)+{({\sigma }_1-{\sigma }_2)}^2+4{\tau }^2}{2}\right]}^{\frac{1}{2}}\le {\sigma }_0$

Answer 1

The correct option is A

${\left[\frac{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2}{2}\right]}^{\frac{1}{2}}\le {\sigma }_0$

As per maximum distortion theory, maximum shear strain energy in a body should be less than or equal to maximum shear strain energy under uniaxial loading.
${\left[\frac{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2}{2}\right]}^{\frac{1}{2}}\le {\sigma }_0$