Page 62 - Physical Principles of Sedimentary Basin Analysis

44 Linear elasticity and continuum mechanics

z z
F
dz n n
−F x v 1
v n z
−F y
A y
dy
dx

x −F z
(a) (b)
Figure 3.7. (a) The force acting on an arbitrary oriented triangle is balanced by the forces on the
orthogonal sides of the tetrahedron. (b) A cutting plane through the tetrahedron in which both the
z-axis and the normal vector lie. Notice that n is first translated to the point on the triangle where it
1
passes through the origin. It is then seen that dx dy is equal to A n z .
2
The components of the stress vectors form the matrix

⎛ ⎞
σ xx σ xy σ xz
σ = ⎝ σ yx σ yy σ yz ⎠ . (3.19)
σ zx σ zy σ zz

We will refer to this matrix as the stress tensor, although tensors are not formally intro-
duced. The knowledge of the stress state in terms of these components allow us to compute
the force and the stress state on any (infinitesimal) plane with any orientation. The stress
vector S on the triangle in Figure 3.7, oriented by an arbitrary unit normal vector n, can
be computed by first finding the force F on the orthogonal sides. We have from Newton’s
second law that forces on all sides of the tetrahedron have to cancel each other for the
tetrahedron to be at rest. The force F on the triangle is therefore
1
F = S A = σ x dy dz + σ y dx dz + σ z dx dy (3.20)
2
where S is the stress vector on the triangle, and where dx, dy and dz are the corner positions
of the triangle relative to the origin. The areas of the orthogonal sides are related to the area
1
1
1
A as dx dy = An z , dx dz = An y and dy dz = An x . Using these areas, the stress
2 2 2
vector S is written in a compact way using the summation convention as
S i = σ xi n x + σ yi n y + σ zi n z = n j σ ji (3.21)
Premultiplication of the stress tensor by the unit normal vector (of a plane) gives the stress
vector on the plane. The stress in a particular direction with unit vector m is the scalar
product of the stress vector and m,

S m = S · m = n j σ ji m i . (3.22)