Question

The homogeneous state of stress for a metal part undergoing plastic deformation is

\(T = \left[ {\begin{array}{*{20}{c}} {10}&5&0\\ 5&{20}&0\\ 0&0&{ - 10} \end{array}} \right]\)

Where the stress component values are in MPa. Using von Mises yield criterion, the value of estimated shear yield stress, in MPa is
1. 9.50
2. 16.07
3. 28.52
4. 49.41

Answer 1

Correct Answer - Option 2 : 16.07

Concept:

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 simple tension test.
• yield stress under triaxial condition is given by:

\({\sigma _{y}} = \sqrt {\frac{1}{2}\left\{ {{{\left( {{\sigma _x} - {\rm{\;}}{\sigma _y}} \right)}^2} + {{\left( {{\sigma _y} - {\sigma _z}} \right)}^2} + {{\left( {{\sigma _z} - {\sigma _x}} \right)}^2} + 6\left( {τ _{xy}^2 + {\rm{\;}}τ _{yz}^2 + τ _{zx}^2} \right)} \right\}} \)

Calculation:

Given:

σ_x = 10 MPa, σ_y = 20 MPa, σ_z = -10 MPa, τ_xy = 5 MPa, τ_yz = τzx = 0

putting the value in the equation

\({\sigma _{y}} = \sqrt {\frac{1}{2}\left\{ {{{\left( {{\sigma _x} - {\rm{\;}}{\sigma _y}} \right)}^2} + {{\left( {{\sigma _y} - {\sigma _z}} \right)}^2} + {{\left( {{\sigma _z} - {\sigma _x}} \right)}^2} + 6\left( {τ _{xy}^2 + {\rm{\;}}τ _{yz}^2 + τ _{zx}^2} \right)} \right\}} \)

\({\sigma _{y}} = \sqrt {\frac{1}{2}\left\{ {{{\left( {{10-20}} \right)}^2} + {{\left( {20+10} \right)}^2} + {{\left( {{-10} - {10}} \right)}^2} + 6\left( {5^2 } \right)} \right\}} \)

∴ σ_y = 27.839 MPa 

Shear stress at yield is

 \({τ _y} = \frac{{{\sigma _{y}}}}{{\sqrt 3 }} = 16.07\;MPa\)