Page 322 - Mechanical Behavior of Materials

322 Chapter 7 Yielding and Fracture under Combined Stresses

where μ and τ i are material constants. The Coulomb–Mohr criterion can be considered to be a
shear stress criterion in which the limiting shear stress increases for greater amounts of hydrostatic
compression.
In applying the modified Mohr criterion, values are needed for three material constants. These
can be the ultimate strengths in tension and compression, σ ut and σ uc , and one additional constant,
either μ or the closely related constant m. A value for μ or m can be estimated from the inclination
of the fracture plane in compression tests.
Under high hydrostatic compression, normally brittle materials behave in a ductile manner,
and ductile materials fracture at higher true stresses and strains than otherwise. Such behavior can
be explained by considering yielding and fracture to be independent events with different failure
surfaces. The possibility of either occurring first should generally be considered.
Fracture may be time dependent due to crack growth effects, so caution is needed in applying
failure criteria that use materials constants from short-term tests.

NEW TERMS AND SYMBOLS

(a) Terms
allowable stress design principal normal stress space
anisotropic yield criterion proportional loading
effective stresses: ¯σ NT , ¯σ S , ¯σ H , ¯σ CM safety factor
failure criterion (stress based) ultimate strengths:
failure surface compression, σ uc
fracture criteria: shear, τ u
Coulomb–Mohr tension, σ ut
maximum normal stress yield criteria:
modified Mohr maximum shear stress
load factor design octahedral shear stress

(b) Constants for the Coulomb–Mohr (C–M) and Modified Mohr (M-M) Criteria
μ, τ i Slope and intercept, respectively, of the C–M failure envelope line

m, |σ | Constants for the C–M criterion, expressed in terms of principal normal stresses
uc
φ C–M failure envelope slope angle, tan φ = μ, sin φ = m
◦
θ c Fracture angle, θ c = (90 − φ)/2
σ i For the modified Mohr criterion, stress where the maximum normal and C–M portions
of the failure surface agree

REFERENCES

ASTM. 2010. Annual Book of ASTM Standards, ASTM International, West Conshohocken, PA. See No.
D7012, “Compressive Strength and Elastic Moduli of Intact Rock Core Specimens under Varying States
of Stress and Temperatures,” Vol. 04.09.
BORESI,A.P., andR.J. SCHMIDT. 2003. Advanced Mechanics of Materials, 6th ed., John Wiley, Hoboken,
NJ. (See also the 2d ed. of this book, same title, 1952, by F. B. Seely and J. O. Smith.)