Page 56 - Physical Principles of Sedimentary Basin Analysis

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Linear elasticity and continuum mechanics

3.1 Hooke’s law, Young’s modulus and Poisson’s ratio
Forces that act on a body lead to deformations. A material is said to be linear elastic if
the deformations are proportional to the forces. How forces and deformations are related
is most easily shown in a basic one-dimensional experiment where a rod is stretched a
distance dl by a force F pointing in the same direction as the rod. Figure 3.1 shows such a
stretched rod. The relative elongation of the rod is the longitudinal strain, ε = dl/l 0 , where
l 0 is the initial length of the rod. The stress acting on the rod is the force per cross-section
area, σ = F/A. The stress and the strain for linear elastic materials are related as
stress F l 0
E = = (3.1)
strain A dl
where the constant of proportionality, E,is Young’s modulus. The relation (3.1) is called
Hooke’s law. The radius of a cross-section of the rod becomes reduced by dr = r − r 0
when it is stretched, where r and r 0 are the radius of the stretched and unstretched rod,
respectively. The relative amount of reduction is measured as the transverse strain, −dr/r 0 .
A minus sign has been added because dr is a negative quantity. It turns out that the ratio of
transverse strain and longitudinal strain is a constant for linear elastic materials
transverse strain dr l 0
ν = = (3.2)
longitudinal strain dl r 0
where the constant of proportionality is Poisson’s ratio.
1
2
Exercise 3.1 Show that the work needed to stretch a rod a distance s is W = EV (s/l 0 ) ,
2
1
where V = Al 0 is the volume of the rod. Notice that the energy per volume becomes E 2
2
1 σ , where strain is = s/l 0 and stress is σ = E .
2
3.2 Bulk modulus
A body will have its volume reduced if it is subjected to a normal stress across its whole
surface. This normal stress could, for instance, be produced by immersing the body in a
fluid with pressure p. The volume change caused by the pressure p is denoted dV , and

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