Rock mechanics

1. James A. Craig
Omega 2011

2.  Linear Elasticity
 Stress
 Strain
 Elastic moduli
 Non-Linear Elasticity
 Poroelasticity

3.  Ability of materials to resist and recover from deformations
produced by forces.
 Applied stress leads to a strain, which is reversible when the
stress is removed.
 The relationship between stress and strain is linear; only
when changes in the forces are sufficiently small.
 Most rock mechanics applications are considered linear.
 Linear elasticity is simple
 Parameters needed can be estimated from log data & lab tests.
 Most sedimentary rocks exhibit non-linear behaviour,
plasticity, and even time-dependent deformation (creep).

4. F = force exerted
Fn = force exerted normal to surface
Fp = force exerted parallel to surface
A = cross-sectional area
Normal Stress
Shear Stress
Fn

A

Fp
A

5.  Sign convention:
 Compressive stress = positive (+) sign
 Tensile stress = negative (-) sign
 Stress is frequently measured in:
 Pascal, Pa (1 Pa = 1 N/m2)
 Bar
 Atmosphere
 Pounds per squared inch, psi (lb/in2)

6. The Stress Tensor
 Identifying the stresses related to surfaces oriented in
3 orthogonal directions.

7.  Stress tensor =
  x  xy  xz 


  yx  y  yz 

 zy  z 
zx


 Mean normal stress,


x
 y  z 

3
 For theoretical calculations, both normal & shear
stresses can be denoted by σij:
 “i” identifies the axis normal to the actual surface
 “j” identifies the direction of the force
 ij   x ,  y ,  z ; xy , yz , xz
  11  12  13 
 Stress tensor :  
    21  22  23 



 31  32  33 

8.  Principal Stresses
 Normal & shear stresses at a surface oriented normal to a
general direction θ in the xy-plane.
The triangle is at rest.
No net forces act on it.

9.    x cos2    y sin 2   2 xy sin  cos
1
   y   x  sin 2   xy cos 2
2
 Choosing θ such that τ = 0
tan 2 

2 xy
x
 y 
 θ has 2 solutions (θ1 & θ2), corresponding to 2 directions
for which shear stress vanishes (τ = 0).
 The 2 directions are called the principal axes of stress.
 The corresponding normal stresses (σ1 & σ2)are called
the principal stresses.

10. 2
1
1
2
 1   x   y    xy   x   y 
2
4
2
1
1
2
 2   x   y    xy   x   y 
2
4
 The principal stresses can be ordered so that σ1 > σ2 > σ3.
 The principal axes are orthogonal.

11.  Mohr’s Stress Circle

12. 1
1
   1   2    1   2  cos 2
2
2
 Radius of the circle:
1
    1   2  sin 2
2
 1   2 
2
 Center of the circle on σ-axis:
 1   2 
 Maximum absolute shear stress:
2
 max
 1   2 

2
 …occurs when θ = π/4 (= 45o) and θ = 3π/4 (= 135o).

13. Normal strain
(elongation)
L  L L


L
L
Elongation is positive
(+) for contraction.

14. Shear strain
1
  tan 
2
Change of the angle ψ between two initially orthogonal
directions.

15.  The Strain Tensor
 x

  yx

 zx
 xy
y
 zy
 xz 

 yz 
z 

 Volumetric Strain
 vol   x   y   z
 Relative decrease in volume

16.  Principal Strains
tan 2 

2 xy
x
y 
 In 2-D, there are 2 orthogonal directions for which the
shear strain vanishes (Γ = 0).
 The directions are called the principal axes of strain.
 The elongations in the directions of the principal axes of
strain are called the principal strains.

17.  A group of coefficients.
 They have the same units as stress (Pa, bar, atm or psi).
 For small changes in stress, most rocks may normally be
described by linear relations between applied stresses
and resulting strains.
 
  E
 Hooke’s law.
 E is called Young’s modulus or the E-modulus.
 A measure of the sample’s stiffness (resistance against
compression by uniaxial stress).

18.  Poisson’s ratio.
 A measure of lateral expansion relative to longitudinal
contraction.
y
 
x
 σx ≠ 0, σy = σz = 0.
 Isotropic materials
 Response is independent of the orientation of the
applied stress.
 Principal axes of stress and the principal axes of strain
always coincide.

19.  General relations between stresses and strains for
isotropic materials:
 x     2G   x   y   z
 y   x     2G   y   z
 z   x   y     2G   z
 xy  2G xy
 xz  2G xz
 yz  2G yz
 λ and G are called Lamé’s parameters.
 G is called shear modulus or modulus of rigidity.
 G is a measure of the sample’s resistance against shear
deformation.

20.  Bulk modulus.
 A measure of the sample’s resistance against hydrostatic
compression.
 The ratio of hydrostatic stress relative to volumetric
strain.
p
K
 vol

2
If σp = σ1 = σ2 = σ3 while τxy = τxz = τyz = 0: K    G
3
 Reciprocal of K (i.e. 1/K) is called compressibility.

21.  In a uniaxial test, i.e. σx ≠ 0; σz = σy = τxy = τxz = τyz = 0:
x
3  2G
E
G
x
 G
y

  
 x 2  G
 If any 2 of the moduli are known, the rest can be
determined.

22.  Some relations between elastic moduli:
E  3K 1  2 
E  2G 1  
9 KG
E
3K  G
3  2G
E
 G

E  1  1  2 

 1   
K
3
2G 1   
K
3 1  2 
2
K  G
3
3K  2G
v
2  3K  G 

2

G 1  2

23. 2 

 G
G
1  2 
 G
  2G
2 1   
 G
3  2G
2 1   
 G
3  4G
2  2   
 G
H    2G
4
H K G
3
1  
H E
1  1  2 
G  E  4G 
H
 E  3G 
H  2G

2H  G
 H is called Plane wave modulus or uniaxial
compaction modulus.

24.  The stress-strain relations for isotropic materials can be
rewritten in alternative forms:
E x   x     y   z 
E y   y     x   z 
E z   z     x   y 
1   
1
 xy 
 xy 
xy
2G
E
1   
1
 xz 
 xz 
xz
2G
E
1   
1
 yz 
 yz 
yz
2G
E

25.  Strain Energy
 Potential energy may be released during unloading by a
strained body.
 For a cube with sides a, the work done by increasing the
stress from 0 to σ1 is:
Work = force × distance
1
1
1
d
 a 2    a  d     a 3   d   a 3  
Work   
d

E
0
0
0
1 3  12 1 3 2
Work  a
 a E 1
2
E 2
1 3
Work  a  11
2

26.  When the other 2 principal stresses are non-zero,
corresponding terms will add to the expression for the
work.
 Work (= potential energy) per unit volume is:
1
W   11   2 2   3 3 
2
 W is called the strain energy.
 It can also be expressed as:
1
2
2
W     2G  12   2   3   2 1 2  1 3   2 3 
2

27.  Any material not following a linear stress-strain
relation.
 It is complicated mathematically.
  E1  E2 2  E3 3 
 Types of non-linear elasticity:
 Perfectly elastic
 Elastic with hysteresis
 Permanent deformation

28.  Perfectly Elastic
 Ratio of stress to strain is not the same for all stresses.
 The relation is identical for both the loading and
unloading processes.

29.  Elastic with Hysteresis
 Unloading path is different from the loading path.
 Work done during loading is not entirely released
during unloading, i.e. part of the strain energy dissipates
in the material.
 It is commonly observed in rocks.

30.  Permanent Deformation
 It occurs in many rocks for sufficiently large stresses.
 The material is still able to resist loading (slope of the
stress-strain curve is still positive), i.e. ductile.
 Transition from elastic to ductile is called the yield point.

31.  Sedimentary rocks are porous & permeable.
 The elastic response of rocks depend largely on the
non-solid part of the materials.
 The elastic behaviour of porous media is described by
poroelastic theory.
 Maurice A. Biot was the prime developer of the theory.
 We account for the 2 material phases (solid & fluid).
 There are 2 stresses involved:
 External (or total) stress, σij
 Internal stress (pore pressure), Pf

32.  There are 2 strains involved:
 Bulk strain – associated with the solid “framework” of
the rock. The framework is the “construction” of grains
cemented together with a certain texture.
 vol
  V
  s 
u
V
 Zeta (ζ) parameter – increment of fluid, i.e. the relative
amount of fluid displaced as a result of stress change.
  
 Vp Pf 
Vp  V f
   us  u f  



 V
V
Kf 
p



33.  The simplest linear form of stress-strain relationship is:
  K  vol  C
Pf  C vol  M 
 This is Biot-Hooke’s law for isotropic stress conditions.
 C and M are poroelastic coefficients. They are moduli.
 C → couples the solid and fluid deformation.
 M → characterizes the elastic properties of the pore fluid.

34.  Drained Loading (Jacketed Test)
 A porous medium is confined
within an impermeable “jacket.”
 It is subjected to an external
hydrostatic pressure σp.
 Pore fluid allowed to escape during
loading → pore pressure is kept
constant.
 Stress is entirely carried by the
framework.


35. Pf  0
0  C vol  M 
C vol
 
M
  K  vol
 C vol
C
 M

C2 
 
  vol  K fr  vol   K fr
 K 
M
 
vol
 There are no shear forces associated with the fluid.
 Shear modulus of the porous system is that of the
framework.
G  G fr

36.  Drained Loading (Unjacketed Test)
 A porous medium is embedded
in a fluid.
 Pore fluid is kept within the
sample with no possibility to
escape.
 Hydrostatic pressure on sample
is balanced by the pore pressure.

37.  The following equations are combined to give the
elastic constants K, C and M in terms of the elastic
moduli of the constituents of rock (Ks & Kf) plus
porosity φ and Kfr:
p
C2
K
 K fr
 vol
M
K fr
Ks 
C
1
M
 1
1  CK
 

K K  K M

f 
fr
 s

38. K fr
K
1 Kf


K s  K K s  K fr  K s  K f
 Or,
2
 K fr 
1 

Kf
Ks 

K  K fr 
Kf 
K fr 

1
1   

 Ks 
Ks 
 This is known as Biot-Gassmann equation. Biot
hypothesized that the shear modulus is not influenced
by the presence of the pore fluid, i.e.:
Gundrained  Gdrained  G fr

39.  K fr 
C  1 
M
Ks 

CK s
M
K s  K fr
C
M
Kf

1
K fr
Ks
Kf 
K fr 
1
1   

 Ks 
Ks 
Kf
1

Kf 
Kf 
1 
1   

Ks 
Ks 

40.  Limit 1 – Stiff frame (e.g. hard rock)
 Frame is incompressible compared to the fluid:
K fr , G fr , K s  K f
 Finite porosity (porosity not too small):
 Kf 
   2   K s  K fr 
 Ks 
 Then:
K  K fr
K f  K fr 
C
1 

  Ks 
M 
Kf


41.  Limit 2 – Weak frame
 For bulk modulus:
For porosity:
K fr , G fr , K f  K s
 Then:
K  K fr 

Kf

Kf
Ks
CM 
Kf

 K is influenced by both rock stiffness and Kf.
 In a limiting case when Kfr → 0 (e.g. suspension): K = C=
M (≈ Kf /φ) are all given mainly by fluid properties.
 For practical calculations, complete K, C and M
expressions are used.

42.  Undrained Test (Effective Stress Principle)
 Jacketed test with the pore fluid shut in.
 No fluid flow in or out of the rock sample.
 Increase in external hydrostatic load (compression) will
cause an increase in the pore pressure.
 No relative displacement between pore fluid and solid
during the test.
 0
  K  vol
Pf  C vol
C
 
K

43. C
  Pf  K fr  vol
M
   Pf   
 σp = total stress
σ’p = effective stress
 The solid framework carries the part σ’p of σp, while the
fluid carries the remaining part αPf. This is called the
Effective stress concept (Terzaghi, 1923).
K fr
C

 1
M
Ks
 α is called Biot constant.
 φ < α ≤ 1.
 In unconsolidated or weak rocks, α is close to 1.
 Upper limit for Kfr is (1- φ)Ks. The lower limit is zer0.