Question 6.4: Find the ratio of the principal strains that result from yie......
Find the ratio of the principal strains that result from yielding if the principal stresses are \sigma_{y} =\sigma_{x}/4 , \sigma_{z} =0 . Assume the von Mises criterion.
According to Equation (6.15),
d\varepsilon _{1} =d\lambda [2( \sigma_{1}- \sigma_{2})-2( \sigma_{3}- \sigma_{1})]/4=d\lambda \cdot \left[\sigma_{1}-(\sigma_{2}+\sigma_{3})/2\right]
d\varepsilon _{2} =d\lambda [ \sigma_{2}-( \sigma_{3}+ \sigma_{1})/2]
d\varepsilon _{3} =d\lambda [ \sigma_{3}-( \sigma_{1}+ \sigma_{2})/2].^† (6.15)
d\varepsilon _{1} : d\varepsilon _{2} : d\varepsilon _{3} =\left[\sigma_{1} -(\sigma_{2}+\sigma _{3})/2 \right] : \left[\sigma_{2} -(\sigma_{3}+\sigma _{1})/2 \right] : \left[\sigma_{3} -(\sigma_{1}+\sigma _{2})/2 \right] =(7/8)\sigma_{1} : (-1/4)\sigma_{1}: (-5/8)\sigma_{1} =7:-2:-5.