Two-Dimensional Bending or Flexure of Plates

The quantity K is the bulk modulus, and its reciprocal is β, the compressibility. The ratio of p to the bulk modulus gives the fractional volume change that occurs under isotropic compression. Because the mass of a solid element with volume V and density ρ must be conserved, any change in volume δV of the element must be accompanied by a change in its density δρ. The fractional change in density can be related to the fractional change in volume, the dilatation, by rearranging the equation of mass conservation δ(ρV ) = 0, (3.51) which gives ρδV + V δρ = 0 (3.52) or −δV V = ∆ = δρ ρ . (3.53)

Equation (3–53) of course assumes ∆ to be small. The combination of Equations (3–50) and (3–53) gives δρ = ρβp. (3.54) This relationship can be used to determine the increase in density with depth in the earth. Using Equations (3–11) to (3–13), we can rewrite the formula for K given in Equation (3–50) as K = 1 β = E 3(1 − 2ν) . (3.55) Thus as ν tends toward 1/2, that is, as a material becomes more and more incompressible, its bulk modulus tends to infinity. 3.9 Two-Dimensional Bending or Flexure of Plates We have already discussed how plate tectonics implies that the near-surface rocks are rigid and therefore behave elastically on geological time scales. The thin elastic surface plates constitute the lithosphere, which floats on the relatively fluid mantle beneath. The plates are subject to a variety of loads – volcanoes, seamounts, for example – that force the lithosphere to bend under their weights.