Rock failure criteria

FAILURE CRITERIA FOR ROCKS
Jyoti Anischit
MSc in engineering geology
Tribhuvan University

introduction
• The strength of the intact rock is influenced by its origin, structure,
composition, texture, grain size and porosity. It is also affected by the
surrounding pressure, pore pressure, temperature and moisture content.
Laboratory testing methods for determination of strength of intact rock are
generally well established and testing techniques have been recommended
by the International Society of Rock Mechanics (ISRM).
• The most important factors affecting the strength of intact rock are
anisotropy, specimen geometry, confining stress, moisture content and
creep and rate of loading. Rock by nature is anisotropic, consisting of
mineral grains, cracks and pores of random orientation, and hence should
be tested under different orientations and direction of applied load.
• Rocks in the earth’s crust generally exist in a confined state; i.e., surrounded
by other rock, which exerts a stress from all sides on the element under
consideration. Hence, to obtain a more realistic idea of the rock behaviour,
it is tested under various confining stresses. The practical significance of the
presence of water in the rock is the danger that normally stable structure
might become unstable under elevated pore pressures. Hence, it is always
advisable to test the rock under the different moisture and pore pressure
conditions expected to be encountered in the field.

Failure criteria for rocks
• Mohr-coulomb criterion
• Hoek and Brown criterion
• Empirical Rock failure criterion
• Griffith failure criterion
• Bieniawski-Yudhbir criterion
• Ramamurthy’s criterion

Mohr-Coulomb Criterion
• The simplest and best-known failure criterion is given
where, τ is shear strength, c’ is cohesion; σ’is internal friction.
• In terms of effective principal stresses, this leads to a linear
relationship,
where, σc is uniaxial compressive strength, and 'φ is angle of internal
friction. The classical Mohr-Coulomb theory can't be used to predict the
non-linear response of rocks. Keeping in mind this limitation, a
number of investigators suggested empirical strength of
various forms suitable to predict the non-linear response of
jointed rocks.

Effect of water pressure and principle stress ratio
Strength of some rocks get deteriorated in presence of water and reduction of
strength may be upto 15% due to saturation of some friable sandstone. In case of
some rocks like clay shale, it loses strength completely. The pore water and the water
present in cracks and fissures play a crucial role and exerts pressure while loading if
drainage is blocked. Tarzaghi's effective stress law may be applicable in such cases in
rock. As per the law, a pressure of Pw as the pore water in rock will reduce the peak
normal stress by an amount Pw. It is explained as below.

Hoek and Brown criterion
Hoek and Brown (1980b) arrived at an empirical failure criterion capable
of modelling the highly non-linear relationship between the minor and
major principal stresses and also predicting the influence of rock mass
quality on the strength, this criterion is given as
Where, σ’1f and σ’3f are the major and minor effective principal stresses at failure.
σc is the uniaxial compressive strength of the intact rock material, and m and s are
material constants, where s = 1 for intact rock.
Later, Hoek and Brown (2002) have modified the equation to give a generalized
criterion in which the shape of the principal stress plot or the Mohr envelope could be
adjusted by means of a variable coefficient ‘a’ in place of the square root term, which is
as given below.

Comparison of Hoek- Brown and Mohr-Coulomb criteria

Griffith Failure criterion
Griffith failure theory - elliptical cracks

Griffith failure criterion
• Griffith postulated that a crystalline material (like rock) contain a large
number of randomly oriented zones of potential failure in the form of
grain boundaries (microfractures). Griffith hypothesized that stress
concentrations develop at the end of these cracks causing the crack to
propagate and ultimately contribute to the macroscopic failure. The
assumptions are,
• 1. The flaw which is elliptical in shape can be treated as single ellipse in a
semi-infinite elastic medium.
• 2. Adjacent flaws don't interact.
• 3. The material is assumed to be homogeneous.
• 4. Ellipse and stress system are taken to be two dimensional.
• For a thin elastic strip of unit thickness containing a elliptical hole oriented
with its long axis perpendicular to an applied tensile stress σo, the
maximum stress σmax at the apex of the ellipse depends on radius of
curvature of the apex (ρ) and the length of the crack 2c

• If the two dimensional case, randomly distributed and
oriented cracks occur through the body, the criteria for
fracture is as follows.
• If, 𝜎𝜎1> 𝜎𝜎3 and 3𝜎𝜎1+ 𝜎𝜎3 < 0
fracture will occur when,
• (𝜎𝜎1- 𝜎𝜎3)2 = -8To (𝜎𝜎1+ 𝜎𝜎3) [when compressive stress
field is predominant]
• at an angle given by
• If, 𝜎𝜎1> 𝜎𝜎3 and 3𝜎𝜎1+ 𝜎𝜎3 > 0
• fracture will occur when 𝜎𝜎1= To at an angle θ=0o.
[Tension is predominant]
• where, To = Tensile strength of the rock.

• Fracture initiation based on Griffith theory
• Defines the relation between the shear and
normal stresses. τxy and σy at which fracture
initiates on the boundary of an open elliptical
flaw.

Empirical Rock failure criterion
• More precise criterion of failure may be determined for any rock by
fitting an envelope to Mohr's circles representing values of the
principal stresses at peak conditions in laboratory tests. It may be the
best practice, to produce an empirical criterion tailored to given rock
type.

Drucker-Prager yield criterion
Drucker-Prager yield criterion
where α* and k are material constants, e.g., for plane strain
conditions

The end
• THANKING YOU
• JYOTI ANISCHIT

Rock failure criteria

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