Page 69 - Physical Principles of Sedimentary Basin Analysis

3.8 Mohr’s circles 51

J = σ σ = (R ik σ kl R −1 )(R ip σ pq R −1 )
ij ij lj qj
−1 −1
= (R R ik )(R R jq )σ kl σ pq
pi lj
= δ pk δ lq σ kl σ pq
= σ pq σ pq
= J. (3.60)

(3) The third invariant of σ is

I =|σ |=|RσR −1 |=|R| |σ| |R −1 |=|σ|= I 3 (3.61)
3
because |R|=|R −1 |= 1.

3.8 Mohr’s circles
The principal stresses can be used to express the normal stress and the shear stress on any
given plane. The normal and shear stresses are obtained by adding the forces that act normal
to and tangential to a plane. Figure 3.10 shows a plane in 2D with a unit normal vector that
makes an angle θ with the direction of the largest principal stress σ 1 . The largest principal
stress is horizontal and the least principal stress is vertical, and could for example be the
lithostatic stress. The horizontal force acting on the vertical side is F 1 = σ 1 A cos θ and the
vertical force on the base is F 3 = σ 3 A sin θ, where A is the area of the plane. These forces
are decomposed normal to the plane and tangential to the plane, and the normal force (F n )
and the tangential force (F t )are
F n = F 1 cos θ + F 3 sin θ (3.62)
F t =−F 1 sin θ + F 3 cos θ (3.63)

which lead to the normal stress σ = F n /A and the shear stress τ = F t /A
2
2
σ = σ 1 cos θ + σ 3 sin θ (3.64)
τ =−(σ 1 − σ 3 ) sin θ cos θ. (3.65)

σ 3

θ n θ σ t
σ 1 σ 1 σ 1
σ n
θ θ

σ 3 σ 3
θ
(a) (b)
Figure 3.10. (a) A plane has a unit normal vector that makes an angle θ with the largest principal
stress. (b) The normal and the tangential stresses on the slanted surface are obtained by adding
forces.