The Bresler–Pister yield criterion (Ref. 4,
Ref. 19) was originally devised to predict the strength of concrete under multiaxial stresses. This failure criterion is an extension of
The Drucker–Prager Criterion to brittle materials, and it can be expressed in terms of the stress invariants as
here, k1,
k2, and
k3 are parameters obtained from the uniaxial compressive strength
σcs, the uniaxial tensile strength
σts, and the biaxial compressive strength
σbc
The Willam–Warnke criterion (Ref. 12) is used to predict failure in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. Just as
The Bresler–Pister Yield Criterion, it depends only on three parameters. It was developed to describe initial concrete failure under triaxial conditions. The failure surface is convex, continuously differentiable, and it is fitted to test data in the low compression range.
where σcs is the uniaxial compressive strength,
σts is uniaxial tensile strength, and
σbc is the biaxial compressive strength (all parameters are positive). The octahedral normal and shear stresses are defined as usual; see
Other Stress Invariants
so the criterion in Equation 3-67 can be written in term of stress invariants as
The dimensionless function r(
θ) describes the segment of an ellipse on the octahedral plane as a function of the Lode angle
θ
Here, the tensile and compressive meridian rt and
rc are defined in terms of the strengths
σcs,
σbc, and
σts:
The function r(
θ) can be interpreted as the friction angle which depends on the Lode angle
θ (
Ref. 12).
Here, σcs is the uniaxial compressive strength of concrete, and
a > 0 and
b > 0 are dimensionless parameters. The dimensionless function
λ(θ) depends on the Lode angle
θ and two dimensionless parameters
k1 > 0 and
k2 > 0.
The parameter k1 is called the
size factor. The parameter
k2 (also called
shape factor) is positive and bounded to 0
≤ k2 ≤ 1(
Ref. 19,
Ref. 20).
Typical values for these parameters are obtained by curve-fitting the uniaxial compressive strength σcs, uniaxial tensile strength
σts, and from the biaxial and triaxial data (for instance, a typical biaxial compressive strength of concrete is 16% higher than the uniaxial compressive strength). The parameters
σcs,
σbc, and
σts are positive.
The ratio rt/
rc =
λc/
λt normally lies between 0.54~0.58 for concrete.
where σ1 ≥ σ2 ≥ σ3 ≥ 0 are the principal stresses at failure (as defined in geotechnical engineering; that is, an
absolute value),
σcs is the uniaxial compressive strength of the intact rock, and
m and
s are positive material parameters.
with σcs,
m, and
s positive material parameters. (In this case, note that
σ1 < sσcs/m).
As developed originally, there is no relation between the parameters m and
s and the physical characteristics of a rock mass measured in laboratory tests. However, for intact rock,
s = 1 and
m = mi, which is measured in a triaxial test.
For jointed rock masses, 0 ≤ s < 1 and
m < mi. The parameter
m usually lies in the range
5 < m < 30 (
Ref. 9).
The generalized Hoek–Brown criterion was developed to fit the Geological Strength Index (GSI) classification of isotropic rock masses (Ref. 8). A new relationship between GSI,
m,
s, and the newly introduced parameter
a was developed to give a smoother transition between very poor quality rock masses (GSI
<
25) and stronger rocks
where σcs is the compressive strength of the intact rock. In terms of the invariants
J2 and the Lode angle
0 ≤ θ ≤ π/3 this equals
where σ1 ≥ σ2 ≥ σ3 are the principal stresses (using the Structural Mechanics Module conventions) of the
effective stress tensor (that is, the stress tensor minus the fluid pore pressure).
The positive parameter mb is a reduced value of the material constant
mi:
s and
a are positive parameters for the rock mass given by the following relationships:
The disturbance factor D was introduced to account for the effects of stress relaxation and blast damage, and it varies from
0 for undisturbed in situ rock masses to
1 for very damaged rock masses.