Page 82 - Physical Principles of Sedimentary Basin Analysis

64 Linear elasticity and continuum mechanics

porous matrix is thus compressed by the normal forces when the pore fluid pressure is
less than the average normal stress. Finally, the pore fluid pressure and the average normal
stress balance each other when they are equally large.

Exercise 3.20 The fluid pressure acting on any (infinitesimal) surface with area dS and
orientation n (given by the outward unit vector n)is p =− f gzn. (The minus sign is
needed because the force from the hydrostatic pressure acts in the direction opposite to the
outward unit normal vector.) Use the divergence theorem

a · n dS = ∇· a dV (3.134)
S V
to show that the total hydrostatic force acting on the surface of a body with volume V and
arbitrary shape is F =− f gV e z .
Solution:

∂
F i =− f gzδ ij n j dS =− ( f gzδ ij )dV =− f gV δ zi . (3.135)
S V ∂x j
The divergence theorem is applied to each vector component of the force separately. We

used that f gzn i can be written as a scalar product of a j = f gzσ ij and n j , that ∂z ∂x j =
δ zj and that δ zj δ ij = δ zi .

3.16 Euler and Lagrange coordinates
A moving or deforming continuum like the stretching of the crust or the subsidence and
compaction of the sediments in a basin can be modeled in two alternative reference frames
(coordinate systems). One option is to relate the motion of the “particles” constituting the
continuum to a fixed coordinate system. Such a fixed reference frame is called an Eulerian
coordinate system. The alternative coordinate system is one that follows the motion of the
“particles.” In other words, the particles are at rest in this reference frame, which is called
a Lagrangian coordinate system. The Euler and the Lagrange coordinates are related. The
Lagrange coordinate a of a particle is now taken to be its position at time t = 0. The Euler
position of this particle is

x = x(a, t) (3.136)
where the a-vector is its unique label, which therefore remains constant through time. (See
Figure 3.16.) We will assume that the inverse of equation (3.136) exists,
a = a(x, t), (3.137)

which tells where the particle in position x was at time t = 0. The velocity of a particle
following the path x = x(a, t) is
∂x(a, t)
V(a, t) = (3.138)
∂t