James A. Craig Omega 2011

Linear Elasticity Stress

Strain

Elastic moduli

Non-Linear Elasticity

Poroelasticity

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).

Normal Stress

F = force exerted

Fn = force exerted normal to surface

Fp = force exerted parallel to surface

A = cross-sectional area

nF

A

Shear StresspF

A

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)

The Stress Tensor

Identifying the stresses related to surfaces oriented in 3 orthogonal directions.

Stress tensor =

Mean normal stress,

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

Stress tensor :

x xy xz

yx y yz

zx zy z

3

x y z

, , ; , ,ij x y z xy yz xz

11 12 13

21 22 23

31 32 33

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.

Choosing θ such that τ = 0

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.

2 2cos sin 2 sin cosx y xy

1

sin 2 cos 22

y x xy

2

tan 2xy

x y

The principal stresses can be ordered so that σ1 > σ2 > σ3.

The principal axes are orthogonal.

2

2

1

1 1

2 4x y xy x y

2

2

2

1 1

2 4x y xy x y

Mohr’s Stress Circle

Radius of the circle:

Center of the circle on σ-axis:

Maximum absolute shear stress:

…occurs when θ = π/4 (= 45o) and θ = 3π/4 (= 135o).

1 2 1 2

1 1cos 2

2 2 1 2

1sin 2

2

1 2

2

1 2

2

1 2

max2

L L L

L L

Normal strain

(elongation)

Elongation is positive

(+) for contraction.

Shear strain

1tan

2

Change of the angle ψ between two initially orthogonal

directions.

The Strain Tensor

Volumetric Strain

Relative decrease in volume

x xy xz

yx y yz

zx zy z

vol x y z

Principal Strains

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.

2

tan 2xy

x y

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.

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).

E

Poisson’s ratio.

A measure of lateral expansion relative to longitudinal contraction.

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.

y

x

General relations between stresses and strains for isotropic materials:

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.

2

2

2

2

2

2

x x y z

y x y z

z x y z

xy xy

xz xz

yz yz

G

G

G

G

G

G

Bulk modulus.

A measure of the sample’s resistance against hydrostatic compression.

The ratio of hydrostatic stress relative to volumetric strain.

If σp = σ1 = σ2 = σ3 while τxy = τxz = τyz = 0:

Reciprocal of K (i.e. 1/K) is called compressibility.

p

vol

K

2

3K G

In a uniaxial test, i.e. σx ≠ 0; σz = σy = τxy = τxz = τyz = 0:

If any 2 of the moduli are known, the rest can be determined.

3 2x

x

GE G

G

2

y

x G

Some relations between elastic moduli:

3 1 2

2 1

9

3

3 2

1 1 2

E K

E G

KGE

K G

GE

G

E

1

3

2 1

3 1 2

2

3

3 2

2 3

2

1 2

K

GK

K G

K Gv

K G

G

H is called Plane wave modulus or uniaxial compaction modulus.

2

1 2

22 1

3 22 1

3 42 2

G

G

G

G

G

G

G

G

G

2

4

3

1

1 1 2

4

3

2

2

H G

H K G

H E

G E GH

E G

H G

H G

The stress-strain relations for isotropic materials can be rewritten in alternative forms:

11

2

11

2

11

2

x x y z

y y x z

z z x y

xy xy xy

xz xz xz

yz yz yz

E

E

E

G E

G E

G E

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:

1 1 1

2 3 3

0 0 0

23 3 21

1

3

1 1

Work = force × distance

Work

1 1Work

2 2

1Work

2

da a d a d a d

E

a a EE

a

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:

W is called the strain energy.

It can also be expressed as:

1 1 2 2 3 3

1

2W

2 2 2

1 2 3 1 2 1 3 2 3

12 2

2W G

Any material not following a linear stress-strain relation.

It is complicated mathematically.

Types of non-linear elasticity: Perfectly elastic

Elastic with hysteresis

Permanent deformation

2 3

1 2 3E E E

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.

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.

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.

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

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.

Zeta (ζ) parameter – increment of fluid, i.e. the relative amount of fluid displaced as a result of stress change.

vol s

Vu

V

p f p f

s f

p f

V V V Pu u

V V K

The simplest linear form of stress-strain relationship is:

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.

volK C

f volP C M

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.

There are no shear forces associated with the fluid.

Shear modulus of the porous system is that of the framework.

0fP

0 vol

vol

C M

C

M

2

volvol vol fr vol

C CK C K K

M M

fr

vol

K

frG G

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.

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:

2p

fr

vol

CK K

M

1

fr

s

KK

C

M

1 1

s f fr

C K

K K K M

Or,

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.:

undrained drained frG G G

1fr f

s s fr s f

K KK

K K K K K K

2

1

1 1

fr

f s

fr

f fr

s s

K

K KK K

K K

K K

1

1 1

fr

f s

f fr

s s

K

K KC

K K

K K

1fr

s

KC M

K

s

s fr

CKM

K K

1

1 1

f

f f

s s

KM

K K

K K

Limit 1 – Stiff frame (e.g. hard rock) Frame is incompressible compared to the fluid:

Finite porosity (porosity not too small):

Then:

, ,fr fr s fK G K K

2

f

s fr

s

KK K

K

frK K 1f fr

s

K KC

K

fKM

Limit 2 – Weak frame For bulk modulus: For porosity:

Then:

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 Mexpressions are used.

, ,fr fr f sK G K K f

s

K

K

f

fr

KK K

fKC M

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 volK f vol

CP C

K

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).

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.

f fr vol

CP K

M fP

1fr

s

KC

M K