Page 303 - Mechanical Behavior of Materials

304 Chapter 7 Yielding and Fracture under Combined Stresses

(a) σ = σ'
3 uc

σ = 0
1
μ τ
θ c

(a) τ i
uniaxial (b) (b) σ = σ' ut
compression uniaxial 1
tension
2θ c 2θ c
σ = 0
3
σ' 0 σ' σ
uc ut
θ c

Figure 7.15 Fracture planes predicted by the Coulomb–Mohr criterion for uniaxial tests in
tension and compression.

τ = τ' u
τ

τ
i
τ' u σ 1

45 o θ
2θ c μ c
σ
σ 3 0 σ 1 σ 3

Figure 7.16 Pure torsion and the fracture planes predicted by the Coulomb–Mohr criterion.

strength in shear expected from the failure envelope. Substituting the appropriate principal stresses,

σ 1 =−σ 3 = τ ,σ 2 = 0, into Eq. 7.46(b) gives
u

τ = τ i 1 − m 2 (7.50)

u
If experimental data from several triaxial compression tests at various stress levels are available,
then a linear least squares fit can be employed to obtain constants for the failure envelope line.
Two constants are needed: (1) the slope, as specified by any one of μ, φ, θ c ,or m, and (2) the