Page 302 - Mechanical Behavior of Materials

Section 7.7 Coulomb–Mohr Fracture Criterion 303

These relationships can now be substituted into Eq. 7.42. After doing so, it is useful to make
additional substitutions that arise from trigonometry.

sin φ 2 2
cos 2θ c = sin φ, sin 2θ c = cos φ, μ = tan φ = , sin φ + cos φ = 1 (7.45)
cos φ
After some algebraic manipulation, we obtain three alternative forms of the desired expression:

|σ 1 − σ 3 | + (σ 1 + σ 3 ) sin φ = 2τ i cos φ (a)
√
|σ 1 − σ 3 | + m(σ 1 + σ 3 ) = 2τ i 1 − m 2 (b) (7.46)

|σ 1 − σ 3 | + m(σ 1 + σ 3 ) = σ uc (1 − m) (c)

Equation (b) arises from (a) through the definition of a new constant, m = sin φ, and form (c) will
be derived shortly. It is also useful to note that additional manipulation using the trigonometric
expressions of Eq. 7.45 gives

μ m
, μ = √ (a, b) (7.47)
m = sin φ = 2 2
1 + μ 1 − m
Assume that the failure envelope, as given by Eq. 7.42 or 7.46, is known for a given material.

We can then calculate the strength that is expected in simple compression, σ , where the prime
uc
is included to indicate that the value is calculated from the envelope, as distinguished from the
value σ uc from an actual test. The principal stresses for this situation are σ 3 = σ ,σ 1 = σ 2 = 0.

uc

Substituting these into Eq. 7.46(b) and noting that σ uc has a negative value gives
!
1 + m
2

−σ (1 − m) = 2τ i 1 − m , σ =−2τ i (a, b) (7.48)
uc uc
1 − m
Algebraic manipulation of (a) yields the desired result (b) in explicit form. The corresponding
Mohr’s circle and fracture planes are illustrated in Fig. 7.15(a). Also, substituting Eq. 7.48(a)
into Eq. 7.46(b) gives the envelope equation in the form of Eq. 7.46(c). In the latter, the quantity
|σ |=−σ uc is employed, so that the correct result is obtained regardless of how the sign of σ uc is

uc
entered.
Similarly, the strength expected in simple tension, σ , can be calculated from the failure

ut
envelope by substituting the appropriate principal stresses, σ 1 = σ ,σ 2 = σ 3 = 0, into Eq. 7.46(b).

ut
The result is
!
1 − m

σ = 2τ i (7.49)
ut
1 + m
The corresponding Mohr’s circle and fracture planes for this case are illustrated in Fig. 7.15(b).

Additionally, consider a test in simple torsion, as illustrated in Fig. 7.16, where τ is the fracture
u