Page 310 - Mechanical Behavior of Materials

Section 7.8 Modified Mohr Fracture Criterion 311

The plane of fracture in compression is often observed to be an acute angle relative to the
◦
◦
loading axis on the order of θ c = 20 to 40 . (See the fractured compression specimens of cast iron
and concrete in Figs. 4.23 and 4.24, and compare with Fig. 7.15.) From Eq. 7.43, this corresponds
approximately to μ values in the range 1.2 to 0.2.
However, the fracture planes predicted for a tension test are incorrect. Brittle materials generally
fail in tension on planes near the plane normal to the maximum tension stress—that is, normal to the
specimen axis—not on planes as shown in Fig. 7.15. Failures of brittle materials in torsion generally
also occur on planes normal to the maximum tension stress, not on the planes predicted by the C–M
theory, as in Fig. 7.16. (See the broken tension and torsion specimens of cast iron in Figs. 4.13 and
4.42.) Moreover, the fracture strengths in tension, compression, and shear are not typically related
to one another as predicted by a single value of m used with the previous equations.
The situation of the maximum tension stress controlling the behavior in tension and torsion, in
disagreement with the C–M criterion, can be handled by using the C–M criterion in combination
with the maximum normal stress fracture criterion. This combination, called the modified Mohr
fracture criterion, will be discussed in Section 7.8.
An alternative form of Eq. 7.46 is sometimes employed. Returning to the σ 1 ≥ σ 2 ≥ σ 3
assumption, so that only Eq. 7.46(c) is needed, some algebraic manipulation yields
1 + m

σ 3 = hσ 1 −|σ |, where h = (7.60)
uc
1 − m
In some cases, a linear relationship does not fit the data very well, so Eq. 7.60 is generalized to a
power equation:
a
σ 3 =−k(−σ 1 ) −|σ uc | (σ 3 ≤ σ 2 ≤ σ 1 ) (7.61)
The quantities k and a are fitting constants, and σ uc is the strength in simple compression from test
data. Where this nonlinear relationship is needed, the value of a is typically less than unity and in
the range 0.7 to 0.9. A nonlinear relationship between σ 3 and σ 1 implies a curved failure envelope
line, rather than a straight line as in Eq. 7.42 and Fig. 7.14. A curved failure envelope is indeed
sometimes observed, especially for tests under rather large confining pressures, where failure of the
normally brittle material is controlled by ductile yielding rather than by fracture.
In most tests for obtaining C–M envelope fits, the σ 3 at failure is a larger compressive value
than σ 1 = σ 2 from lateral pressure; these are called Type I tests, as in Ex. 7.7. Another option is to
increase σ 3 = σ 2 to fracture while σ 1 is held at a smaller compressive value; this is called a Type II
test. The C–M envelopes for Type II tests in general appear to be above those for Type I tests. So
the intermediate principal stress σ 2 does have an effect, contrary to the assumption implicit in the
C–M criterion that it does not. Although a more general approach would be desirable, it appears to
be reasonable to use the envelope from Type I tests as a conservative approximation.

7.8 MODIFIED MOHR FRACTURE CRITERION

As already noted, the Coulomb–Mohr fracture criterion does not agree with behavior of brittle
materials in tension and torsion. This difficulty can be handled by using the C–M criterion in
combination with the maximum normal stress fracture criterion, as illustrated in Fig. 7.20. In