chapter 2: failure mechanics

Chapter 2: Failure Mechanics

2.1. Basic Concepts

This chapter discusses the most

elementary and well known models for rock failure.

2.1.1. Strength and related concepts

The stress level at which a rock typically

fails is commonly called the strength of the rock.


a uniaxial or triaxial test.

Principle sketch of stress versus deformation in a

uniaxial compression test.


Several important concepts are defined in the figure:

• Elastic region

• Yield point

• Uniaxial compressive strength

• Ductile region

• Brittle region


Triaxial testing


uniaxial and triaxial tests


2.1.2. The failure surface

1 2

3 ≥ ≥

The failure surface

may be described by

the equation:

1 2 3, , 0f

（2.1）

2.2. Tensile Failure

Tensile failure occurs when the effective tensile stress

across some plane in the sample exceeds a critical limit.

This limit is called the tensile strength, it is given the

symbol T0, and has the same unit as stress.


The failure criterion, which specifies the stress condition

for which tensile failure will occur, and identifies the

location of the failure surface in principal stress space, is

given as:

0T (2.2)


For isotropic rocks, the conditions for tensile failure

will always be fulfilled first for the lowest principal

stress, so that the tensile failure criterion becomes

3 0T (2.3)

2.3. Shear Failure

max f

This assumption is called

Mohr’s hypothesis.

（2.4）

Fig. 2.7: Failure line, as specified by Eq(2.4)in the shear

stress–normal stress diagram. Also shown are the Mohr

circles connecting the principal stresses 1

2 3

2.3. Shear Failure

Pure shear failure, as defined by Mohr’s hypothesis,

depends only on the minimum and maximum principal

stresses and not on the intermediate stress.

2.3. Shear Failure

2.3. Shear Failure

By choosing specific forms of the function f(σ′), various

criteria for shear failure are obtained. The simplest

possible choice is a constant. The resulting criterion is

called the Tresca criterion. The criterion simply states that

the material will yield when a critical level of shear stress

is reached:

S0 is the inherent shear strength (also called

cohesion) of the material.

max 1 3 0

1

2S (2.5)

2.3.1. The Mohr–Coulomb criterion

0S

Assumption : f(σ′) is a linear function of σ′

μ is the coefficient of internal friction.

(2.6)

The latter term is clearly chosen by analogy with

sliding of a body on a surface, which to the first

approximation is described by Amontons’ law:


(2.7)

In the Figure, Mohr’s circle touches the failure

line. The angle is called the angle of internal

friction and is related to the coefficient of internal

friction by

The Tresca criterion can be considered as a

special case of the Mohr–Coulomb criterion, with

υ = 0.

tan

（2.8）


The attraction

（2.9）

is defined as the distance from the intersection

point to the origin .


0 cotS

The angle 2β gives the position of the point where

the Mohr’s circle touches the failure line. The shear

stress at this point of contact is

1 3

1sin 2

2

while the normal stress is

1 3 1 3

1 1cos2

2 2


（2.11）

（2.10）


22

4 2

The allowable range for υ is from 0º to 90º

（2.13）

（2.12）

Introducing the expressions (2.10)and (2.11) for σ′ and τ

into the failure criterion Eq. (2.6), we find

1 3 0 1 3 1 3

1 1 1sin 2 cos2

2 2 2S

（2.14）


1 3 0 1 3 1 3

1 1 1cos tan tan sin

2 2 2S

Replacing β and μ by υ, we obtain:

Multiplying with 2 cos υ and rearranging, we find

2 2

1 3 0 1 3cos sin 2 cos sinS

1 0 31 sin 2 cos 1 sinS

1 0 3

1 sin2 cos

1 sinS


（2.15）

（2.16）

（2.17）

（2.18）

The angle γ in the ( , ) plane is related to by 1

3

1 sintan

1 sin

（2.19）

or

tan 1sin

tan 1

（2.20）


An expression for the uniaxial compressive strength

C0 is obtained by putting in Eq. (2.18), giving 3 0

0 0 0

cos2 2 tan

1 sinC S S

（2.21）


Making use of Eqs. (2.21)and (2.12), we note that

Eq. (2.18) may be writtenin the following simple way:

2

1 0 3 tanC （2.22）


2.3.2. The Griffith criterion

1 3 0 1 38T 1 33 if 0

3 0T if 1 33 0

The theory is scaled in terms of the uniaxial tensile

strength T0, and the resulting failure criterion can be

written

(2.23)

(2.24)

The uniaxial compressive strength C0 is given by

Eq. (2.23) as

In τ–σ′-coordinates, the Griffith criterion is given by

only one equation:

0 08C T (2.25)

2

0 04T T (2.26)


A result from this modified theory is that the ratio

between uniaxial compressive strength and the tensile

strength is given by

0

20

4

1

C

T

(2.27)


2.4. Compaction Failure

In principal stress space, this type of failure is

represented by an ―end cap‖ that closes the

failure surface at high stresses, as seen on Fig.

2.13. An elliptical form is often used for such an

end:

2 2

2 2

1 11

1

q

p p

(2.28)


Boutéca et al. (2000) argued that the following simple

form of Eq. (2.28) (obtained by choosing γ = 0 and δ = 1)

may be an acceptable approximation for many rocks:

2 2 2q p (2.29)


It is to be expected that the critical effective

pressure p* decreases with increasing porosity. From

theoretical considerations, Zhang et al. (1990) derived

the relation

3

2p

(2.30)


The two cases > and < can be represented

by two lines that are symmetric around the line

as shown in Fig. 2.14.

1 3

1 3

1 3

2.5.1 Failure Criteria in Three Dimensions

Fig. 2.14

Fig. 2.15: The Mohr–Coulomb failure surface in principal

stress space. Fig. 2.15


Fig. 2.16: Cross-section of the Mohr–Coulomb criterion in a

π-plane. The arrows represent projections of the principal

stress axes onto the plane. The friction angle φ = 30º.


Typically, it is found that rocks are stronger

when > > than for the situations

where = or = .

One simple solution is to implement

rotational symmetry for the π-plane cross-

section. This approach has no physical

foundation, however it is mathematically

attractive, and is the basis for some of the most

commonly used criteria shown below.

2.5.2. Criteria depending on the intermediate

principal stress

2 1

3

2 1

3 2


principal stress

One of these criteria is the von Mises criterion, which

can be written

C is a material parameter related to cohesion.

2 2 2 2

1 2 1 3 2 3 C

(2.31)


principal stress

2 2 2 2

1 2 1 3 2 3 1 1 2 3 2C C

The corresponding generalization of the Mohr–

Coulomb criterion is the Drucker–Prager criterion. It

can be formulated as

where C1 and C2 are material parameters, related to

internal friction and cohesion.

(2.32)


principal stress

2 2 2

1 2 1 3 2 3 0 1 2 324T

ending on the planes

1 0 2 0 3 0, ,T T T

The extended Griffith criterion predicts that the relation

between uniaxial compressive strength and tensile strength

is given by

0 012C T

(2.33)

(2.34)

(2.35)


principal stress

A modified version of the empirical failure criterion

was presented by Ewy (1999):

33

3

3 L

I

I

1 1 2 3L L LI S S S

3 1 2 3L L LI S S S (2.38)

(2.37)

(2.36)


principal stress

SL is a material parameter related to the cohesion and

the friction angle of the rock:

while ηL is related only to the internal friction:

0

tanL

SS

2 9 7sin4 tan

1 sinL

(2.40)

(2.39)


principal stress


principal stress

π-plane representation

1 2cos3LJ f f

The radial distance to a point on the failure surface is

given by Eq. (1.50). Defining two general functions f1

and f2 we may thus write a general failure criterion as

(2.41)


principal stress

A representation of the modified Lade criterion can

be obtained by expressing

and in terms of σ′, q and . and defining a

new variable x = q/(SL + σ′)

1I 3I

(2.42)


principal stress

2

3 3 3

3 cos 3 cos 3 31 ...

3 9 3 27 3

L L L

L L L

2

1 3

3 1 2 cos 31 2cos arccos 1

2cos3 3 3 3

L

L

f

2

1

3Lf S (2.44)

(2.43)


principal stress

A general, simple form for f1 may be convenient for

practical purposes. We may for instance choose

1 1 cos3f k k (2.45)

1 3 sin1

2 3 sink

(2.46)


principal stress

2 0

1 sin sin2

3 sin 3 sinf C

(2.47)

may be used as a rough approximation for the Mohr–

Coulomb criterion.

By choosing f1 appropriately, it is possible to formulate

the Mohr–Coulomb criterion exactly in terms of the

invariants. The expression

1

1

1 sin3 cos 3sin

3 sinf

(2.48)


principal stress

Physical explanations


principal stress

2.6. Fluid Effects 2.6.1. Pore pressure

2.6.2. Partial saturation

moist sand is partially water saturated

cp nw wep p p

The effects of partial saturation occur whenever the

pore space is filled with at least two immiscible fluids,

like for instance oil and water.

where pwe is the pressure in the wetting fluid, pnw is

the pressure in the non-wetting fluid, and pcp is called

the capillary suction.

(2.50)


The capillary suction has some effect on the effective

stresses in the rock, and we may define a generalized

effective stress

nw we cpp S p (2.51)


2.6.3. Chemical effects

6yield

w

MPaa

If the pore fluid is changed, for instance during drilling

or production, the chemical equilibrium may be disturbed,

and a net dissolution or deposition of minerals may occur.

This may have a strong effect on the rock properties,

typically a reduction in strength of 30–100% is seen in

many rocks due to deterioration of the cement (Broch,

1974). (2.52)

The q–p′ plot essentially displays the maximum shear

stress versus the mean effective stress. It is based on the

parameter q, usually called the generalized shear stress

and defined as

and the parameter p′ which is identical to the mean

effective stress σ′

2.7. Presentation and Interpretation of Data from

Failure Tests

2 2 2

1 2 2 3 1 3

1

2q

1 2 3

1

3p

(2.54)

(2.53)

2.7. Presentation and Interpretation of Data from

Failure Tests

The von Mises criterion

2 21

2q C

the Drucker–Prager criterion

22 2

1 2

13

2q C p C

(2.55)

(2.56)

2.7. Presentation and Interpretation of Data

from Failure Tests

the modified Lade criterion (Eq. (2.36)—with f1( , ηL)

given by Eq. (2.42); note the similarity with Eq. (2.56):

1 , L Lq f S p

the extended Griffith criterion

2

036q T p

and the compactive yield criterion

2 2 2p q p

(2.57)

(2.59)

(2.58)

2.7. Presentation and Interpretation of Data

from Failure Tests

2.8. Beyond the Yield Point 2.8.1. Plasticity

The theory of plasticity is based on four major

concepts:

1.Plastic strain

2.A yield criterion.

3.A flow rule.

4. A hardening rule.

e p

ij ij ijd d d

1 2 3, , , 0f

(2.60)

(2.61)

2.8.1. Plasticity

p

ij ij kld d h

where λ is a scalar not specified by the flow rule.

Plastic flow

(2.62)

The function of the flow rule is to describe the

development of the plastic strain increments.

The basic assumption concerning plastic flow

dates back to Saint-Venant in the nineteenth

century. It states

Plastic flow

• The assumption that the plastic strains are

independent of stress increments, is

intuitively understandable in the following

simple example.

0p

kl ij

ij

d (2.63)

p

ij

ij

gd d

(2.64)

Plastic flow

0p

ij ij

ij

d d

p

ij

ij

fd d

One solution to the problem is Drucker’s (1950)

definition of a stable, work hardening material. Such a

material is defined by a more strict version of Eq. (2.63):

(2.65)

(2.66)

Associated flow

Associated flow

1 cospd d

From the figure we see that

3 csinpd d

3 1 tan 0p pd d

(2.67)

(2.68)

(2.69)

Associated flow

1 1 tanp p

vold d

0p

vold

(2.70)

(2.71)

Eq. (2.16) may be written as:

1 30 1 3

1tan

2cos 4S

(2.72)

Associated flow

Associated flow

• It can thus be concluded that the inclination of

the arrow in Fig. 2.26 determines the dilatancy

of the material as follows:

• If the arrow is tilted to the left (υ ≥ 0), the

material is positively dilatant.

• If the arrow is vertical (υ = 0), the material does

not change volume.

• If the arrow is tilted to the right (υ ≤ 0), the

material is negatively dilatant , or contractant .

Non-associated flow

1 3 1 0 3 1 0 3

1 sin, tan

1 sinf C C

(2.73)

1 3 1 0 3 1 0 3

1 sin, tan

1 sing C C

(2.74)

Hardening

Hardening

• There are two common ways to relate κ to the plastic

strain. One way is to assume that κ is a function of the

total plastic strain. This is called strain hardening, and

can be expressed as

p

ijSd (2.75)

where ∫S symbolizes integration over the stress path.

Another way is to relate κ to the total plastic work:

p

ij ijS

d (2.76)

This is called work hardening.

Hardening

2.8.2. Soil mechanics

• Voids ratio e is the volume of voids Vvoid relative to

the volume of the solid grains V solid in the material:

• Specific volume υ is the total volume (grains + voids)

divided by the volume of the solid grains, that is

void

solid

Ve

V (2.77)

1solid void

solid

V Ve

V

(2.78)


• The specific volume and the voids ratio are

related to the porosity by

1

1

e

e

(2.79)


Normally consolidated clay:

• No peak stress in the stress–strain diagram, i.e.

the shear stress does not fall below previous

value as the strain increases.

• The sample contracts throughout the loading.

• The loading path in a q–p′-plot curves to the

left, i.e. the pore pressure increases.


Strongly overconsolidated clay:

• A peak is normally observable in the stress–strain

diagram, i.e. at some point the shear stress falls

below previous values as the strain increases.

• The sample contracts initially, then dilates as

failure is approaching.

• The loading path in a q–p′-plot curves to the right

near failure, i.e. the pore pressure decreases

Normally consolidated clays

Overconsolidated clays

2.9.2. The plane of weakness model

2.9.2. The plane of weakness model

• The corresponding failure angle is given as

• Isotropic failure criterion

• Weak plane failure criterion

4 2

ww

(2.80)

0 31 3

cos sin2

1 sin

S

(2.81)

0 3

1 3

cos sin2

sin 2 cos cos2 1 sin

w w w

w w

S

(2.82)

2.9.3. Fractured rock


• Hoek and Brown (1980) derived an empirical failure

criterion for fractured rocks:

• The uniaxial compressive strength of the fractured

rock is given by

2

1 3 0 3 0bm C sC (2.83)

2

0 0fC sC (2.84)


0.321

b im m s (2.85)

31 3 0

0

a

bC m sC

where s → 0 and a → 0.65. Note that Eq. (2.86) is

identical to Eq. (2.83) when a = 0.5.

(2.86)

2.10. Stress History Effects

• 2.10.1. Rate effects and delayed failure

2.10.2. Fatigue

• Thank you

chapter 2: failure mechanics

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