Page 311 - Mechanical Behavior of Materials

312 Chapter 7 Yielding and Fracture under Combined Stresses

particular, the C–M failure locus from compression-dominated behavior is truncated and replaced
by the maximum normal stress criterion wherever its predictions exceed the latter. This combination
is called the modified Mohr (M-M) fracture criterion.

7.8.1 Details of the Modified Mohr Criterion
For the σ 1 versus σ 2 failure locus for biaxial stresses of Fig. 7.20(a), note that, for simple tension,
and also for biaxial stresses that are both positive, fracture is controlled by σ ut , as measured in a

simple tension test, not by the larger value σ ut expected from the C–M criterion and Eq. 7.49 or
7.56. Looking at the σ versus |τ| failure envelope, as in Fig. 7.20(b), we see that a vertical line at

σ ut truncates the sloping line of the C–M criterion, so that, again, σ ut does not correspond to the
real behavior.
For simple compression, σ uc ≈ σ uc is indicated in Fig. 7.20(a) and (b), with these two quantities

differing only due to statistical scatter in real data, and perhaps due to minor deviations of the
compression-dominated behavior from a linear C–M envelope. (If there are major deviations from
linearity, as when Eq. 7.61 applies, a more general approach is needed.)
Tension-dominated behavior generally extends at least to, and often somewhat beyond, the
σ 1 =−σ 2 line in Fig. 7.20(a) corresponding to simple torsion. (See the data of Figs. 7.11 and 7.13.)

Hence, in torsion, fracture is expected to occur at τ u = σ ut , not at the larger value τ from the C–M
u
criterion and Eq. 7.50.

σ
(a) 2
C-M
max. nor.

M-M σ ut σ' ut
σ 1

τ u τ'
u
torsion
σ i

(b)
τ
max. nor.
σ uc ≈ σ' uc

M-M
C-M
σ
σ uc ≈ σ' uc 0 σ ut σ' ut

Figure 7.20 The modified Mohr (M-M) fracture criterion, formed by the maximum normal
stress criterion truncating the Coulomb–Mohr (C–M) criterion.