GeoMechanics Course Notes

MNE/GEN 427/527 Formula Sheet No. 2

Kemeny

Formula Sheet on Hooke’s Law
9. Hooke’s Law (isotropic, homogeneous)
Hooke’s law can be written in matrix or equation form. Below I give it in the form of 6 equations (and six unknowns). It is fine in this form since as a matrix it contains mostly zeros. 1⎛
⎞
⎜ σ − νσ y − νσ z ⎟
E⎝ x
⎠
1
= ⎛ − νσ + σ − νσ ⎞
⎜
x y z⎟
E⎝
⎠
1
= ⎛ − νσ − νσ − σ ⎞
⎜
x y z⎟
E⎝
⎠

ε = x ε ε y z τ

xy ε = xy 2G

τ

ε = xz xz 2G

ε

yz

=

τ

yz
2G

Where E = Young’s modulus, ν = Poisson’s ratio, and G = shear modulus.
A typical E for rock is between 10 and 60 GPa. A typical ν for rock is between 0.15 and 0.25.
Note: isotropic means the properties do not depend on direction, and homogeneous means the properties are the same at every point in the material. In general neither of these two properties are true for rocks.
10.

Some Relationships Between the Elastic Constants

G=

E
2(1 + ν )

λ=

Eν
(1 + ν )(1 − 2ν )

K=

E
3(1 − 2ν )

Where λ = Lame’s constant and K = bulk modulus.
Note: The five elastic constants listed above (E,ν,G,λ,K) are the most common for reasons that will be shown. However, there are only two independent elastic constants for isotropic conditions and Hooke’s law can be written with any two of the 5 constants (or others not shown).
11.

Reduced Versions of Hooke’s Law

Many special cases of Hooke’s law are used often in rock mechanics. The most famous is probably the special case of uniaxial compression. For this case the boundary conditions are: Hooke’s Law

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MNE/GEN 427/527 Formula Sheet No. 2

Kemeny

σ y = σ and σ x = σ z = τ xy = τ xz = τ yz = 0 and Hooke’s law reduces to:

εx = −

νσ
E

εy =

σ
E

εz = −

νσ
E

ε xy = ε xz = ε yz = 0

Note: This is the basis for determining E and ν with uniaxial tests in the lab.
However, the boundary condition assumption that all shear