A metal piece under the stress state of three principal stresses 30, 10 and 5 kg/mm² is undergoing plastic deformation. The principal strain rates will be in the proportions of

Concept:

Levi Von Mises flow rule: Relates principal strain rates during plastic loading to principal stress

\(\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{2}}}}\,}{{{\sigma }_{1}}-{{\sigma }_{2}}}=\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{{{\sigma }_{1}}-{{\sigma }_{3}}}=\frac{\overset{}{\mathop{{{\epsilon }_{2}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{{{\sigma }_{2}}-{{\sigma }_{3}}}\)

Calculation:

Given, three principal stresses, σ₁ = 30 kg/mm², σ₂ = 10 kg/mm² and σ₃ = 5 kg/mm²

Then

\(\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{2}}}}\,}{30-10}=\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{30-5}=\frac{\overset{}{\mathop{{{\epsilon }_{2}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{10-5}\)

\(\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{2}}}}\,}{20}=~\frac{\overset{}{\mathop{{{\epsilon }_{1}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{25}=~\frac{\overset{}{\mathop{{{\epsilon }_{2}}}}\,-\overset{}{\mathop{{{\epsilon }_{3}}}}\,}{5}~\)

The option can be put and checked to satisfy the condition obtained in the above equation.