Page 282 - Mechanical Behavior of Materials

Section 7.4 Maximum Shear Stress Yield Criterion 283

The yield stress in shear, τ o , for a given material could be obtained directly from a test in simple
shear, such as a thin-walled tube in torsion. However, only uniaxial yield strengths σ o from tension
tests are commonly available, so that it is more convenient to calculate τ o from σ o . In a uniaxial
tension test, at the stress defined as the yield strength, we have

σ 1 = σ o , σ 2 = σ 3 = 0 (7.17)

Substitution of these values into the yield criterion of Eq. 7.16 gives
σ o
τ o = (7.18)
2
In the uniaxial test, note that the maximum shear stress occurs on planes oriented at 45 with respect
◦
to the applied stress axis. This fact and Eq. 7.18 are easily verified with Mohr’s circle, as shown in
Fig. 7.4.
Equation 7.16 can thus be written in terms of σ o as

σ o |σ 1 − σ 2 | |σ 2 − σ 3 | |σ 3 − σ 1 |
= MAX , , (at yielding) (7.19)
2 2 2 2
or

σ o = MAX(|σ 1 − σ 2 | , |σ 2 − σ 3 | , |σ 3 − σ 1 |) (at yielding) (7.20)

The effective stress is most conveniently defined as in Eq. 7.3, so that it equals the uniaxial strength
σ o at the point of yielding. That is,

¯ σ S = MAX(|σ 1 − σ 2 | , |σ 2 − σ 3 | , |σ 3 − σ 1 |) (7.21)
σ
1

σ 1 τ (σ', τ')

o
90
σ
0 (σ , 0)
σ 1
1

45 o

σ
σ σ' = 1
τ' = 1 2
2

Figure 7.4 The plane of maximum shear in a uniaxial tension test.