Question

b) In general, metals yield (i.e., deform plastically) when the energy of distortion reaches a critical value which is a characteristic of the material. If o and t are the stresses that cause plastic deformation in simple uniaxial tension and simple shear, respectively. What is the relation between o and T. (15 Points)

Answer 1

The freer principal which is mentioned in the equation is also known as the von-mises failure criteria. The distortion energy is given by,

E = (von-mises stress)2/6G

Where G is the modulus of rigidity.

Now during the cases of simple shear and uniaxial tension, plastic deformation starts when the distortion energy reaches a critical value. This critical value is fixed for a material. So, in both the cases here, critical distortion energy should be same, and so, also value of von-mises stress should be same.

Further detail of the calculation are as below-

The simph fled von-mises equation in generalized form (Based on Max. shear strain energy Principle) is- v Hi (16. 5] +6877-) #4033-5m? +6 (72² + 522² +03, 2) For simple uniaxial tension, and 22=832=0,2 = 023 = 3 = 0 Ty = √ 1.2002 so, For simple shear loading, 2-T and ol = 572 533 = 73 031=0. Oy = √ √ 2.6.72 Bt sor Hence, from (1) and (ii) 0 = 37