Chapter 3 Principe of Mechanics

1

CHAPTER 3

PRINCIPLES OF MECHANICS

This Chapter is intended as a review on the mechanics principles that relevant to rock

mechanics. Reader who is not familiar with mechanics should start with one of the

many textbook on mechanics, listed at the end of this Chapter.

3.1 Stress, Strain and Deformation Modulus

3.1.1 Normal and Shear Stresses

Stress is commonly defined the internal distribution of force per unit area that balances

and reacts to external loads applied to a body. It is a second-order tensor with nine

components, but can be fully described with six components due to symmetry in the

absence of body moments (Wikipedia).

Stresses can be divided down into normal stresses and shear stresses. Normal stress is

defined as stress perpendicular to the plane where the stress is act on. Shear stress is the

stress parallel to the plane where the stress is act on. They are perhaps best represented

by the diagram shown in Figure 3.1.1a.

Figure 3.1.1a Normal and shear stresses on an infinitesimal cube aligned with the

Cartesian axes. (Hudson 35)

The stresses act on an infinitesimal cube aligned with the Cartesian axes has nine

components: three normal stresses and six shear stresses. The three normal stresses are

σ

xx

, σ

yy

and σ

zz

; and the six shear stresses are τ

xy

, τ

xz

, τ

yx

, τ

yz

, τ

zx

, and τ

zy

.

Often, the nine stress components are often conveniently represented in matrix form,

where the row representing the stresses on a plane and the column representing the

stresses in a direction,

σ

xx

τ

xy

τ

xz

σ

㴠

τ

yx

σ

yy

τ

yz

(3.1.1a)

τ

zx

τ

zy

σ

zz

By considering the state of equilibrium, each pair of shear stresses are equal, hence,

τ

xy

= τ

yx

, τ

xz

= τ

zx

, τ

yz

= τ

zy

(3.1.1b)

The stress components in the matrix can be reduced from nine to six, and the stress matrix

can be written as,

Chapter 3 Principe of Mechanics

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σ

xx

τ

xy

τ

xz

σ

㴠

τ

xy

σ

yy

τ

yz

(3.1.1c)

τ

xz

τ

yz

σ

zz

3.1.2 Principal Stresses

Usually, there are nine stress components acting in three directions on three planes.

However, if the planes where the stresses act on are rotated, there is a position that the

magnitude of shear stresses is zero. Hence, the stresses acting on the three planes are

only normal stresses. In this situation, the three normal stresses are termed as principal

stresses.

σ

1

0 0

σ

㴠

0 σ

2

0 (3.1.2a)

0 0 σ

3

Figure 3.1.2a shows the principle of stress transformation on a 2D element subjected to

normal and shear stresses. Considering a plane a-b, the normal and shear stresses on the

a-b plane is σ

ab

and τ

ab

.

Figure 3.1.2a Stress transformation and Mohr’s circle representation.

Summing the components of forces acting on the element in the direction of a-b plane, it

gives, in the normal direction,

σ

ab

(ab

) = σ

x

sinθ (ab

) sinθ + σ

y

cosθ (ab

) cosθ + 2 τ

xy

(ab

) sinθ cosθ

or

σ

ab

= σ

x

sin

2

θ + σ

y

cos

2

θ + 2 τ

xy

sinθ cosθ

(3.1.2b)

or

σ

y

+ σ

x

σ

y

– σ

x

σ

ab

=

2

+

2

cos2θ + τ

xy

sin2θ

(3.1.2c)

here, σ

y

is assumed to be greater than σ

x

.

Again, in the shear direction along a-b plane, the force equilibrium can be written as,

τ

ab

(ab

) = -σ

x

cosθ (ab

) sinθ + σ

y

sinθ (ab

) cosθ – τ

xy

cosθ (ab

) cosθ + τ

xy

sinθ (ab

) sinθ

Chapter 3 Principe of Mechanics

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or

τ

ab

= σ

y

sinθ cosθ – σ

x

sinθ cosθ – τ

xy

(cos

2

θ – sin

2

θ)

or

σ

y

– σ

x

τ

ab

=

2

sin2θ – τ

xy

cos2θ

To make the a-b plane that only principal stress exist, i.e., τ

ab

= 0, then,

2 τ

xy

tan2θ =

σ

y

– σ

x

The above equation gives two values of θ that are 90° apart. This means that there are in

fact two planes at right angles to each other on which shear stress is zero. These planes

are the principal planes and stresses acting on these planes are principal stresses. By

substituting the above equations, the values of the principal stresses can be found,

σ

y

+ σ

x

σ

y

– σ

x

σ

ab

= σ

1

=

2

+ [(

2

)

2

+ τ

xy

2

]

½

as the major principal stress, and,

σ

y

+ σ

x

σ

y

– σ

x

σ

ab

= σ

2

=

2

– [(

2

)

2

+ τ

xy

2

]

½

as the minor principal stress.

The normal stress and shear stress acting on any plane can also be determined by using

the Mohr’s circle, as shown in Figure 3.1.2a. On the Mohr’s circle, the points where the

circle intercept the normal stress axis, where shear stress is zero, are the values of

principal stresses.

The Mohr’s circle can also be used to determine normal and shear stresses acting on a

plane of a given angle to the given principal stresses value and direction. This will be

illustrated later in dealing with the Mohr-Coulomb strength criterion.

The concept of principal stresses is important in rock mechanics. For example, in

excavation, the excavated face is free from shear stress and normal stress, therefore, the

rock at this position is subjected to principal stresses and one (normal to the excavated

face) of the principal stresses is in fact zero.

3.1.3 Deformation and Strain

Chapter 3 Principe of Mechanics

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When a material is subject to stress, the material deforms. The deformation occurs not

only in the direction of the stress, but also in the other two perpendicular directions.

Again, refer the cube in to Figure 3.1.3a, with the normal stress, σ

xx

, the cube deforms

(contracts) in the direction of x-x, and expends in the directions of y-y and z-z.

Figure 3.1.3a Deformations due to stress on an infinitesimal cube.

Strain is defined as the ratio of deformation to the original dimension, i.e.,

ε = δ / l

(3.1.3a)

It therefore has no dimension. Often, it is expressed in term of percentage.

Strain can also expressed in matrix form:

ε

xx

γ

xy

γ

xz

ε

=

γ

xy

ε

yy

γ

yz

(3.1.3b)

γ

xz

γ

yz

ε

zz

ε

xx

, ε

yy

and ε

zz

are normal strain, and γ

xy

, γ

xz

and γ

yz

are shear strain.

Strain is a mechanical property of the material, it only occurs when a stress is applied,

directly or through other means. In another word, when there is a strain, there must be

stress accompanying the strain.

Often, volumetric strain is used in rock mechanics. It is defined as the ratio of volume

change to the original volume, i.e.,

ε

v

= Δ

v

/ v

(3.1.3c)

3.1.4 Poisson’s Ratio

When a sample of material is stretched in one direction, it tends to get thinner in the other

two directions. Poisson's ratio is a measure of this tendency. Poisson's ratio the ratio of the

contraction strain (normal to the applied load) divided by the extension strain (in the

direction of the applied load). Refer to Figure 3.1.3, with a given normal stress in x-x

direction, strains occur in the direction of x-x, as well as in the directions of y-y- and z-z.

The ratio of strain in y-y- or z-z direction to strain in x-x direction is defined as the

Poisson’s ratio,

ν = ε

yy

/ ε

xx

, ν = ε

zz

/ ε

xx

(3.1.4a)

For a perfectly incompressible material, the Poisson's ratio would be exactly 0.5. Most

rock materials have ν between 0.2 and 0.4. The Poisson’s ratio is a mechanical property

Chapter 3 Principe of Mechanics

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of the material. For a homogeneous material, the Poisson’s ratios in y-y and z-z

directions are equal.

3.1.5 Deformation Modulus

Deformation modulus is a mechanical property indicating the rate of change of strain with

change of stress. For an elastic material, it is termed as Modulus of Elasticity, and often

referred as the Young’s Modulus. It is defined as the ratio, for small strains, of the rate

of change of stress with strain. This can be experimentally determined from the gradient

of a stress-strain curve obtained from loading tests conducted on a sample of the material.

E = Δσ / Δε

(3.1.5a)

The Young’s Modulus, E, has the same unit as the stress, σ.

When a material is subject to a stress, it undergoes strain. The stress-strain relationship

can often to be represented by the stress-strain curve, typically shown in Figure 3.1.5a.

The deformation modulus at a specific stress level is the gradient of the stress-strain curve

at point of that specific stress level. For an elastic material, the stress-strain is a straight

line, and the modulus is therefore the gradient of the line.

Figure 3.1.5a Determination of deformation modulus from stress-strain curves.

Stress-strain relationship can be expressed in matrix,

ε

xx

S

11

S

12

S

13

S

14

S

15

S

16

σ

xx

ε

yy

S

21

S

22

S

23

S

24

S

25

S

26

σ

yy

ε

zz

S

31

S

32

S

33

S

34

S

35

S

36

σ

zz

γ

xy

=

S

41

S

42

S

43

S

44

S

45

S

46

σ

xy

(3.1.5b)

γ

yz

S

51

S

52

S

53

S

54

S

55

S

56

σ

yz

γ

zx

S

61

S

62

S

63

S

64

S

65

S

66

σ

zx

ε

=

S

σ

(3.1.5c)

σ

=

D

ε

(3.1.5d)

ε

xx

1

–ν –ν

0 0 0

σ

xx

ε

yy

–ν

1

–ν

0 0 0

σ

yy

ε

zz

1

–ν –ν

1 0 0 0 σ

zz

γ

xy

=

E

0 0 0

2(1+ν)

0 0

σ

xy

(3.1.5e)

γ

yz

0 0 0 0

2(1+ν)

0

σ

yz

γ

zx

0 0 0 0 0

2(1+ν)

σ

zx

Chapter 3 Principe of Mechanics

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3.1.6 Plane Stress and Plane Strain

For a thin wall plate, a 3D problem can be treated as a 2D plane stress problem. The

stresses and strains are expressed as

σ

xx

τ

xy

0

σ

㴠

τ

xy

σ

yy

0 (3.1.6a)

0 0 0

ε

xx

γ

xy

0

ε

=

γ

xy

ε

yy

0 (3.1.6b)

0 0

ε

zz

ε

zz

can often temporarily removed from analysis to make all stress and strain only the in-

plane terms, and effective becomes a 2D analysis.

For a long tunnel, a 3D problem can be approximated by a 2D plane strain problem. The

strains and stresses are expressed as:

ε

xx

γ

xy

0

ε

=

γ

xy

ε

yy

0 (3.1.6c)

0 0 0

σ

xx

τ

xy

0

σ

㴠

τ

xy

σ

yy

0 (3.1.6d)

0 0

σ

zz

σ

zz

is needed to maintain ε

zz

being zero. σ

zz

stress term can be temporarily removed from

the analysis to leave only the in-plane terms, effectively reducing the 3D problem to a

much simpler 2D problem.

3.2 Strength and failure criteria

3.2.1 Basic Definitions

When a material is subjected to a stress, the material deforms. When the stress increases

to a certain state, the material starts to yield and subsequently will either loss the strength

substantially or will undergo large strain without additional stress, as shown typically in

Figure 3.2.1a.

Chapter 3 Principe of Mechanics

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Figure 3.2.1a Stress-strain curve with yield point, peak strength, post-peak ductile and

brittle behaviour.

Strength of a material is the limit states of stress (peak or yield) the material can sustain. It

is often considered in terms of compressive strength, tensile strength, and shear strength,

which are the limit states of compressive stress, tensile stress and shear stress

respectively.

There are various definitions of strength, some represented by yield and some by peak.

In rock mechanics, common definition is by peak strength.

Strain occurred at the peak strength is often defined as strain at failure. This is a useful

measure in rock mechanics and is associated with the brittleness of the material.

Hardness is another commonly used term describing rock material. It is the

characteristic of a solid material expressing its resistance to permanent deformation.

Toughness is the resistance to fracture of a material when stressed. It is defined as the

amount of energy that a material can absorb before rupturing, and can be found by finding

the area (i.e., by taking the integral) underneath the stress-strain curve.

3.2.2 Post-Peak Stress-Strain Behaviour

Post-peak behaviour is the stress-strain behaviour after the peak strength. When a

material has reached the limit of its strength, it usually has the option of either fracture or

deformation, approximately resulting to two typical post-peak behaviours: brittle and

ductile, as shown in Figure 3.2.1a.

Brittle behaviour indicates a substantial reduction of stress-carrying capacity, reflecting

fracturing of the material under stress. At post-peak region, the material may have

residual strength but that residual strength is generally substantially lower than the peak

strength. A brittle material usually has undergoes small strain before failure, i.e., small

strain at failure. A brittle failure is generally associated with fracture by tension rather

than shear, and there is little or no evidence of plastic deformation before failure.

Ductile behaviour is represented by being capable of sustaining large deformations with

losing stress carrying capability in the post-peak region. Ductility is the physical

property of being capable of sustaining large plastic deformations without fracture.

Post-peak behaviour can be influenced by pressure and temperature. This happens as an

example in the brittle-ductile transition zone at an approximate depth of 10 km in the

Earth's crust, at which rock becomes less likely to fracture, and more likely to deform

ductilely.

Chapter 3 Principe of Mechanics

8

Residual strength is the stress carrying capacity at post-peak region. This strength

indicates that even after failure, rock often has a (much smaller) load carrying capacity,

which can have significant effects on rock engineering.

3.2.3 Yield Strength and Criteria

Yield strength, or the yield point, is defined in engineering and materials science as the

stress at which a material begins to plastically deform. Prior to the yield point the material

will deform elastically and will return to its original shape when the applied stress is

removed. Once the yield point is passed some fraction of the deformation will be

permanent and non-reversible. Knowledge of the yield point is vital when designing a

component since it generally represents an upper limit to the load that can be applied

(Wikipedia).

A yield criterion is a hypothesis concerning the limit of elasticity under any combination

of stresses. There are two interpretations of yield criterion: one is purely mathematical in

taking a statistical approach while other models attempt to provide a justification based on

established physical principles. Since stress and strain are tensor qualities they can be

described on the basis of three principal directions.

The most common yield criteria are briefly outlines below. Details of those criteria are

readily available in textbooks of mechanics and strength.

(a) Tresca-Guest criterion

The Tresca-Guest criterion is the most simple yield surface. In principal stresses it is

expressed as follows:

max(|σ

1

– σ

2

|, |σ

2

– σ

3

|, |σ

3

– σ

1

|) = σ

0

The Tresca-Guest criterion in three dimensional space of principal stresses is a prism of

infinite length and six sides, as illustrated in Figure 3.2.3a. This means that material

remains elastic when all three principal stresses are roughly equivalent (a hydrostatic

pressure), no matter how much compressed or stretched. But when the material is

subject to shearing, one of principal stresses becomes smaller (or bigger), then the yield

surface is crossed and material enters plastic domain. In two dimensional space, it is a

cross diagonally cut section of the prism (Figure 3.2.3b).

Figure 3.2.3a Tresca-Guest criterion in 3D spaces of principal stresses.

Figure 3.2.3b Tresca-Guest and Mises criteria in 2D spaces of principal stresses.

(b) Mises criterion

Chapter 3 Principe of Mechanics

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The von Mises criterion is expressed as follows:

Also it can be expressed in non-principal stresses as below:

The von Mises criterion in three dimensional space of principal stresses is a circular

cylinder of infinite length, with the same angle to all three axes, as seen in Figure 3.2.3c.

In two dimensional space of principal stresses, it is an ellipse that diagonally cut cross the

cylinder (Figure 3.2.3b).

Figure 3.2.3c Mises criterion in 3D spaces of principal stresses.

(c) Mohr-Coulomb criterion

The Mohr-Coulomb criterion is a first two-parametric yield surface, for the maximum

compression and tension. The model is the first one that takes shearing into account. It is

expressed as follows:

or

The parameters are Rc and Rr which are the maximum values for compression and

tension for the given material. it should be noted that the criterion considers the

maximum difference between the major and the minor principal stresses only, and does

not take the intermediate principal stress in the strength criterion.

Chapter 3 Principe of Mechanics

10

The Mohr-Coulomb criterion in three dimensional space of principal stresses is

represented by a conical prism (Figure 3.2.3d). If K = 0 then it becomes Tresca-Guest

criterion, thus K determines the inclination angle of conical surface. In two dimensional

space of principal stresses, it is a cross section of the conical prism (Figure 3.2.3e).

Figure 3.2.3d Mohr-Coulomb criterion in 3D space of principal stresses.

Figure 3.2.3e Mohr-Coulomb and Drucker-Prager criteria in 2D space of principal

stresses.

The Mohr-Coulomb strength criterion can be represented graphically, by Mohr’s circle.

Most of the classical engineering materials, including rock materials, somehow follow

this rule in at least a portion of their shear failure envelope. Application and discussion

of the Mohr-Coulomb strength criterion in rock mechanics is given in Chapter 4.

(d) Drucker-Prager criterion

The Drucker-Prager criterion is expressed as follows:

The Drucker-Prager criterion in three dimensional space of principal stresses is a regular

cone (Figure 3.2.3f). If α = 0 then it becomes the Mises criterion. In two dimensional

space of principal stresses, it is a cross section of this cone, which produces an ellipsioidal

shape (Figure 3.2.3e).

Figure 3.2.3f Drucker-Prager criterion in 3D spaces of principal stresses.

(e) Unified Strength Criterion

Chapter 3 Principe of Mechanics

11

The Unified Strength Criterion by Yu has the following characteristics:

(i)

It is able to reflect the fundamental characteristics of brittle materials (including rock

and concrete), i.e., different tensile and compressive strengths, hydrostatic pressure

effect, the effect of intermediate principal stress and its zonal change and material

dependence.

(ii)

It has a clear physics and mechanics background, a unified mathematical model, and a

simple and explicit criterion, which includes all independent stress components and

simple material parameters.

(iii)

It is also suitable for different types of brittle materials under various stress states, and

is consistent with the triaxial test results.

(iv)

It can be easily applied to analytical and numerical modeling.

The Unified Strength Criterion expressed in terms of three principal stresses is as follows:

σ

1

–

α

(b

σ

2

+

σ

3

)/(1 + b) =

σ

t

, when

σ

2

≤

(

σ

1

+

α

σ

3

)/(1 +

α

)

(

σ

1

+ b

σ

2

)/(1 + b) –

α

σ

3

=

σ

t

, when

σ

2

≥

(

σ

1

+

α

σ

3

)/(1 +

α

)

where

σ

t

is tensile strength,

α

is the strength ratio of tensile to compressive strength

(

σ

t

/

σ

c

), b is the intermediate principal stress parameter.

When b = 0, the Unified Strength Criterion becomes the Mohr-Coulomb (

σ

1

–

α

σ

3

=

σ

t

).

The Unified Strength Criterion can produce a full spectrum of new criteria when the value

of b varies between 0 and 1 (0

≤

b

≤

1), to reflect the characteristics of various different

materials. The Unified Strength Criterion is especially versatile in reflecting the

σ

2

effect to different extents for different materials.

In general, the failure surface of the Unified Strength Criterion is a dodecahedral-shaped

cone about the hydrostatic axis. The failure surface in the plane perpendicular to the

cone axis is shown in Figure 3.2.3g. It is quite obvious that the surface is convex when

0

≤

b

≤

1. When b = 0 or 1, the dodecahedral-shaped cone reduces to hexagonal. When

α

=1, the Unified Strength Criterion is simplified to the Unified Yield Criterion applicable

to these materials with the same yield stress in tension and compression, and the three-

fold symmetric limit surfaces are simplified to six-fold symmetric yield surfaces.

Figure 3.2.3g

Failure surfaces of the Unified Strength Criterion on deviatoric plane.

3.3 Fracture Mechanics

3.3.1 Fracture Initiation and Propagation

Chapter 3 Principe of Mechanics

12

Fracture mechanics is a sub-division of solid mechanics studying the failure of a material

containing a crack, in order to understand the initiation and propagation of cracking. It

uses methods of analytical solid mechanics to calculate the driving force on a crack and

methods of experimental solid mechanics to characterize the material's resistance to

fracture.

Fracturing of a material has to start somewhere within the material. It usually starts at a

location of stress concentration. In most materials, they cannot be perfectly

homogeneous and there are often existing pores and cracks within the material formed at

its creation or through various processes. For example, all rock materials contain micro-

cracks and pores. When loaded, those existing cracks and pores causes stress

concentration at their tips and from there, new cracks will be initiated and further

propagated.

There are three types of crack initiation and propagation (Figure 3.3.1a):

Mode I crack – Opening mode (a tensile stress normal to the plane of the crack), this

mode is most relevant to rock mechanics.

Mode II crack - Sliding mode (a shear stress acting parallel to the plane of the crack and

perpendicular to the crack front);

Mode III crack - Tearing mode (a shear stress acting parallel to the plane of the crack and

parallel to the crack front).

Figure 3.3.1a Three modes of crack initiation and propagation.

3.3.2 Griffith Crack Theory

The Griffith crack theory is a theory of brittle fracture, using elastic strain energy

concepts. It describes the behaviour of crack propagation of an elliptical nature by

considering the energy involved. The equation basically states that when a crack is able to

propagate enough to fracture a material, that the gain in the surface energy is equal to the

loss of strain energy, and is considered to be the primary equation to describe brittle

fracture. Because the strain energy released is directly proportional to the square of the

crack length, it is only when the crack is relatively short that its energy requirement for

propagation exceeds the strain energy available to it. Beyond the critical Griffith crack

length, the crack becomes dangerous.

For the simple case of a thin rectangular plate with a crack perpendicular to the load

Griffith’s theory becomes:

G = π σ

2

a / E

G

c

= π σ

f

2

a / E

where G is the strain energy release rate, σ is the applied stress, a is half the crack length,

and E is the Young’s modulus. The strain energy release rate can otherwise be understood

Chapter 3 Principe of Mechanics

13

as the rate at which energy is absorbed by growth of the crack. When G is greater than

the critical value G

c

, crack will begin to propagate.

This concept is often referred as Griffith’s energy instability concept. This concept was

applied in rock mechanics to develop the Griffith strength criterion, as outlined in te next

chapter.

3.3.3 Stress Intensity and Fracture Toughness

A modification of Griffith’s theory was made by Irwin. In the modified criterion, strain

energy release rate is replaced by stress intensity (K

I

)and surface energy is replaced by

fracture toughness (K

c

). Both of these terms are simply related to the energy terms in the

original Griffith theory:

K

I

= σ (π a)

½

K

c

= (E G

c

)

½

for plane stress

K

c

= [E G

c

/(1 – ν)]

½

for plane strain

where ν is the Poisson’s ratio. It is important to note that K

c

has different values when

measured under plane stress and plane strain conditions

Fracture occurs when K

I

≥ K

c

. For the special case of plane strain deformation, K

c

becomes K

Ic

and is considered a material property. The subscript I refers to mode I

cracking.

3.4 Mechanics of Discontinuity

3.4.1 Discontinuity and Global Continuum Law

Discontinuity in a continuous material is defined a plane feature where continuity of the

materials is no longer hold. They are usually significant mechanical breaks in a material.

Continuity may continue to exist after the discontinuity. Typical discontinuities are

fractures, interfaces and joints.

In rock, discontinuities are fractures, joints, faults and bedding plane. While the rock

materials are often considered as continuous materials, rock fractures and joints are

commonly treated as discontinuities.

While the mechanics at discontinuity is discontinuous, certain laws of continuum are still

hold.

3.4.2 Stress and Strain at Discontinuity

Chapter 3 Principe of Mechanics

14

Stresses are often disturbed by a discontinuity. For example, at non-fully-contacted

fracture, opening exists. Normal stress on the opening is zero and there are stress

concentrations on the contact points of the fracture surface. The stress field is no longer

the same as in the continuous material.

The only exception is a fully-contact plane perpendicular to a uniaxial load. In this case,

the stress field is continuous although strain may not.

Similarly, displacement at discontinuity is not continuous. For example, at a fracture

plane, sliding or shear displacement may occur. There may be much greater normal

displacement at fracture than those of the material.

Discontinuities can range from a fully-contacted interface to an opening containing

different material. The mechanics of are vary different.

For a fully-welded interface between two different materials, the interface actually has the

continuities both is stress and displacement. The discontinuity is presented in the change

of mechanics of the two materials on each side of the interface.

For a fully-contacted smooth interface, the interface is subjected to shear displacement.

For a locally-contacted fracture, i.e., there are void between the two sides, both stress and

displacement discontinuous are expected.

3.4.3 Mechanics of Discontinuities for Normal Stress and Deformation

Normal stress and displacement of fully-contact discontinuity is continuous and therefore

can be dealt with continuum approach.

For a locally-contacted fracture shown in Figure 3.4.3a, i.e., there are void between the

two sides, stress-displacement function is discontinuous. For an idealised pillar-like

locally-contacted fracture shown below, assume the ratio of contact area to intact area (A)

is k (k=0~1), i.e., contact area at fracture is kA.

Figure 3.4.3a Idealised fracture with local contact of constant area.

Assume the normal stress in the intact portion is N and the total load is NA. The normal

stress in the contact area is NA/mA = N/m. Since 0<m≤1, stress in the contact area is

always higher than that in the intact portion. Assume the material modulus (E) is the

same for the intact portion and the contact area, then strain of the contact area is N/kE.

If there is no damage of the contact element, the contact area will remain the same with

change of load. Therefore, the load-deformation of the contact zone is linear. Stiffness

(k) of the contact zone is a constant, is given as,

Chapter 3 Principe of Mechanics

15

Load N A m A E

k =

Deformation

=

(N / m E) d

=

d

(3.4.3a)

For a prism shape locally-contacted fracture shown in Figure 3.4.3b, the contact area has

an initial contact area of A

0

, and a thickness of d. The contact area in this case is no

longer constant; it in fact increases with increasing displacement (closure) of the contact

zone. i.e., ΔA∝Δh

2

.

Figure 3.4.3b Idealised fracture with local contact of changing area.

With increase of loading, and discontinuity closes, the contact area increases. Therefore,

at beginning of loading, the contact area is small, rate of deformation is large. At higher

load, the contact area increases, the rate of deformation becomes smaller. Clearly the

load-displacement of the contact zone is no longer linear. This non-linear load-

displacement characteristics is typical for rock fractures, as shown in Figure 3.4.3c.

Figure 3.4.3c Typical non-linear load-displacement characteristics of rock fractures.

3.4.4 Mechanics of Discontinuities for Shear Stress and Deformation

The most common known shear phenomenon of a discontinuity is the sliding between

two contact surfaces (Figure 3.4.4a), i.e., the friction theory. It gives the relationship

between the friction angle φ, the normal force (N) and shear force (S), as S = N tanφ.

Figure 3.4.4a Sliding between two horizontal contact surface.

When slipping at the surface of contact is about to occur, the maximum static frictional

force is proportional to the normal force. When slipping is occurring, the kinetic

frictional force is proportional to the normal force, as shown in Figure 3.4.4b.

Figure 3.4.4b Mobilisation of shear stress with applied shear load.

If the contact surface is at an inclined up angle (i), shown in Figure 3.4.4c, from the force

diagram, along the sliding direction, the normal force is N cos(i) + S sin(i), the shear force

is S cos(i) – N sin(i).

Figure 3.4.4c Sliding between two contact surface at an inclination.

Chapter 3 Principe of Mechanics

16

By friction theory,

S cos(i) – N sin(i) = [N cos(i) + S sin(i)] tanφ,

S – N tan(i) = N tanφ + S tanφ tan(i),

S = N [tanφ + tan(i)] / [1 + tanφ tan(i)],

S = N tan(φ+i)

3.5 Flow Mechanics of Porous Material and Parallel Plates

3.5.1 Flow in Porous Materials

Consider a cylindrical sample of porous material (e.g., soil or rock) under the different

water pressure heads h

1

and h

2

, shown in Figure 3.5.1a. In one dimension, steady water

flows through the fully saturated sample without affecting the structure of the soil or rock,

in accordance with Darcy's flow law,

Q = A k i (3.5.1a)

where Q = volume of water flowing per unit time, A = cross sectional area of sample

corresponding to the flow, k = coefficient of permeability, i = hydraulic gradient = (h

1

–

h

2

)/L, and L = length of sample.

Figure 3.5.1a Darcy's flow experiment on porous material.

Hence

Q Q L

k =

A i

=

A (h

1

– h

2

)

(3.5.1b)

The coefficient of permeability k is a constant. Experiments have shown, however, that

its value depends not only upon the character of the material, but also upon the properties

of the fluid percolating through it. The value of k is inversely proportional to fluid

kinematic viscosity, ν, which can be expressed as,

K g

k =

ν

(3.5.1c)

where g = gravitational acceleration, K = the intrinsic permeability, and is a property of

the material only, with dimension of L

2

.

Chapter 3 Principe of Mechanics

17

3.5.2 Flow between Parallel Plates

For flow of a viscous fluid through a narrow interspace between two closely spaced

parallel plates (e.g., rock fractures), shown in Figure 3.5.2a, the Darcy’s flow law is

applicable when the flow is laminar.

Figure 3.5.2a Darcy's flow experiment on parallel plates.

The intrinsic permeability for laminar flow between parallel plates is (Todd 1959; Verruijt

1970),

d

2

K =

12

(3.5.2a)

where d is the thickness of the fluid lamina, i.e., the aperture of the two parallel plates.

Hence for laminar flow through smooth parallel plates, flow equation can be written as

g d

2

k =

12 ν

(3.5.2b)

This is often called the "parallel plate theory" in the flow mechanics of rock joints. A

laminar flow through smooth parallel plates is a potential flow with its Reynolds number

≤2300.

By combining the above equation with Darcy’s equation, it gives

A i g d

2

Q =

12 ν

(3.5.2c)

Since the area A is the flow passage which is equal to the width w times to the aperture of

the parallel plates, d, the above equation may be written

w i g d

3

Q =

12 ν

(3.5.2c)

The above equation is identified as the "cubic flow law" which is widely used to describe

the flow of fluid through parallel plates. It is also used to describe flow in rock joints.

For a planar array of parallel smooth openings, the equivalent permeability parallel to this

array is given as,

Chapter 3 Principe of Mechanics

18

d

3

k =

12 b

(3.5.2d)

where b is the spacing between openings.

The above equations show that the flow rate and permeability are extremely sensitive to

the aperture of the opening.

3.6 Empirical Approaches

3.6.1 Use of Empirical Equations

Empirical approaches are often used when theoretical approaches are limited due to many

reasones, for example, the complexity of the material which leads to non-conformable to

theory. Empirical equations then are obtained usually based on extensive experiment

results, by regression. New regression techniques, e.g., neural network, have also been

applied to analysis large amount variables and data.

Empirical criteria commonly used in rock mechanics are the Hoek-Brown criterion for

both rock material and rock mass, JRC-JCS shear strength equation for rock fractures

(Figure 3.6.1a) and several others.

Figure 3.6.1a Empirical JRC-JCS shear strength criterion.

3.6.2 Linear and Multiple Regression

3.6.3 Neural Network and other New Methods

Further Readings

Gere JM, Timoshenko SP, Mechanics of Materials, 2nd Edition. PWS-Kent, Boston,

1984.

Jaeger JC, Cook NGW, Fundamentals of Rock Mechanics, 3rd Edition. Chapman and

Hall, London, 1979.

1

σ

x

τ

yx

τ

xx

σ

z

σ

y

τ

zy

τ

xz

τ

xy

τ

yz

x

y

z

3.1.1a

σ

x

, -τ

xy

σ

y

, τ

xy

σ

2

σ

1

O

C

2

θ

½ (σ

x

+ σ

y

)

√

[½(σ

x

+σ

y

)]

2

+ τ

xy

2

S

M

θ

σ

n

τ

a

b

σ

x

τ

xy

τ

xy

σ

y

τ

ab

σ

ab

c

3.1.2a

2

σ

x

3.1.3a

δ

x

δ

y

δ

z

ε

1

σ

1

E

1

3.1.5a

3

σ

ε

3.2.1a

Yield Point

Peak Strength

Ductile

Brittle

3.2.3a

4

3.2.3b

3.2.3c

5

3.2.3d

3.2.3e

6

3.2.3f

3.2.3g

7

3.3.1a

Mode I

Mode II Mode III

3.4.3a

Stress N, Area A, Modulus E

Contact area mA, thickness d

8

3.4.3b

Stress N, Area A, Modulus E

Initial contact area A

o

,

thickness d

3.4.3c

9

3.4.4a

N

S

3.4.4b

F

P

N

P

F

F

s

F

k

F=P

N

10

3.4.4c

S

i

N

L

A

h

1

h

2

Q

3.5.1a

11

h

1

h

2

L

Q

w

d

3.5.2a

3.6.1a

τ = σ

n

tan[JRC JMC log(JCS/σ

n

+φ

r

)]

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