Stress and Strain in Solids

Think of continents as blocks of wood floating on a sea of mantle rock, as illustrated in Figure 2–2. The mean density of the continent, say ρc = 2750 kg m−3, is less than the mean upper mantle density, say ρm = 3300 kg m−3, so that the continent “floats.” Archimedes’ principle applies to continents; they are buoyed up by a force equal to the weight of mantle rock displaced. At the base of the continent σyy = ρcgh, where ρc is the density of the continent and h is its thickness. At this depth in the mantle, σyy is ρmgb, where ρm is the mantle density and b is the depth in the mantle to which the continent “sinks.” Another statement of Archimedes’ principle, also known as hydrostatic equilibrium, is that these stresses are equal. Therefore we find ρch = ρmb. (2.2) The height of the continent above the surrounding mantle is

h− b = h− ρc ρm

h = h

(

1 − ρc ρm

)

Using the values given earlier for the densities and the thickness of the continental crust h = 35 km, we find from Equation (2–3) that h− b = 5.8 km. This analysis is only approximately valid for determining the depth of the oceans relative to the continents, since we have neglected the contribution of the seawater and the oceanic crust. The application of hydrostatic equilibrium to the continental crust is known as sostasy; it is discussed in more detail in Chapter 5. An average thickness of the oceanic crust is 6 km. Its density is 2900 kg m−3. This is overlain by 5 km of water (ρw = 1000 kg m−3) in a typical ocean basin. Determine the normal force per unit area on a horizontal plane at the base of the oceanic crust due to the weight of the crust and the overlying water. Problem 2.2 A mountain range has an elevation of 5 km. Assuming that ρm = 3300 kg m−3, ρc = 2800 kg m−3, and that the reference or normal continental crust has a thickness of 35 km, determine the thickness of the continental crust beneath the mountain range. Assume that hydrostatic equilibrium is applicable.