Question 6.3: For plane stress (σ3 = 0), what is the largest possible rati......

Mechanical Behavior of Materials [1002814]

For plane stress ( $\sigma_{3}=0$ ), what is the largest possible ratio of $\sigma_{1}/Y$ at yielding and at what stress ratio, $\alpha$ , does this occur?

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Inspecting Equation (6.12),

$\sigma_{1} = Y/(1-\alpha +\alpha^{2} )^{1/2}.$ (6.12)

it is clear that the maximum ratio of $\sigma_{1}/Y$ corresponds to the minimum value $1-\alpha +\alpha^{2}.$ Differentiating and setting to zero, $d(1-\alpha +\alpha^{2} )/ d \alpha=-1+ 2\alpha ; \alpha =-1/2$ . Substituting into equation 6.12, $\sigma_{1}/Y = [1 − 1/2 + (1/2)^{2} ]^{-1/2} =√(4/3) = 1.155.$

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