Page 313 - Mechanical Behavior of Materials

314 Chapter 7 Yielding and Fracture under Combined Stresses

can be specified by σ uc used with Eq. 7.48; and (3) the ultimate tensile strength σ ut from test data in

simple tension. As already noted, the estimate σ uc ≈ σ uc may be employed if there are insufficient
compression-dominated data for making a C–M envelope fit. But then m or one of its allied constants
must be known, such as an observed fracture angle θ c from simple compression tests. Equation 7.62
presents an additional opportunity to estimate m. In particular, if biaxial data as in Fig. 7.13 are
available and give a reasonably distinct value of σ i , then m may be obtained by solving Eq. 7.62:

|σ |− σ ut + σ i
uc
m = (7.64)

|σ |+ σ ut + σ i
uc
Some caution is obviously needed in employing values of m estimated without the benefit of data
suitable for fitting the C–M failure envelope.
More general, but more complex, methods are available that do not require a linear C–M failure
envelope line. (See books by Nadai (1950), Jaeger (2007), Chen (1988), and Munz (1999) for
discussion and details.) However, the linear assumption is often employed, as in the ASTM test
method for triaxial compression of rock.

7.8.2 Effective Stresses and Safety Factor for the Modified Mohr Criterion

For the M-M criterion, effective stresses for its C–M and maximum normal stress components can be
determined. Each of these gives a safety factor against fracture, the lowest of which is the controlling
one. The effective stress and safety factor for the C–M criterion have already been described by
Eqs. 7.58 and 7.59, which may be employed in the same form here. For the maximum normal stress
component, the effective stress of Eq. 7.13(a) applies to include the three positive faces of the normal
stress cube:

σ ut
¯ σ NP = MAX(σ 1 ,σ 2 ,σ 3 ), X NP = (a)
¯ σ NP (7.65)
¯ σ NP = 0, X NP =∞, if MAX ≤ 0 (b)

Here, the subscripts are changed to NP, as this differs by removal of the restrictions of Eq. 7.13(b).
We will now use the normal stress criterion up to the intersection with the C–M criterion, as at
the stress σ i of Fig. 7.20, which generally exceeds the previous limitation by a small amount. The
situation of Eq. 7.65(b) arises when the combination of stresses is such that a line from the origin
through the point (σ 1 , σ 2 , σ 3 ) never intersects one of the positive faces of the maximum normal
stress cube.
The overall and controlling safety factor for the M-M criterion is then the smallest of the values
from Eqs. 7.59 and 7.65:

X MM = MIN(X CM , X NP ) (a)

1 ¯ σ CM σ NP (7.66)
¯
= MAX , (b)

X MM |σ | σ ut
uc
Form (b) gives the same result and is convenient for numerical calculations, as it avoids generating
infinite values when either or both of ¯σ CM or ¯σ NP are zero.