GEOL5700 Class notes: FlexureFlexure of the lithosphere brings together two end notions of how the Earth behaves: therigid behavior of the Earth (after all, we don't worry about buildings or landfills causing the crustto sink down isostatically (we might worry about them sinking into peat or clays, though)) withthe observation of isostatic support of the lithosphere at very long wavelengths. By bridging thisgap, we find ourselves in possession of a tool capable of explaining many phenomena of interestin tectonics, from the geometry of trenches to foredeeps in front of mountain belts.General Equations for Flexure of Plates (Turcotte & Schubert, sec 3-9):Consider a thin plate of elastic material. If the plate is thin enough, we may approximate theforces acting on and within the plate as follows:ww+dwxx+dxq(x)VV+dVPPMM+dMThe displacement of the plate from some horizontal axis is w(x). The forces and momentsacting upon a small segment of the plate are as illustrated above, where q is a force per unitlength acting on the plate, V is the shear force acting on the edges of a piece of the plate(integrated shear stresses on the edge), P is the pressure on the edges (integrated normal stresses;must balance because there is no motion of the plate to the right or left), M is the moment actingon the edges of the piece of the plate (moment of the normal stresses). If we balance forces inthe vertical direction, we findq(x)dx + dV = 0dVdx=−q(1)Balancing the forces in the horizontal (x) direction shows that P is constant. Summing themoments acting on the segment (and noting that we can drop the second order term dV dx) wefindGEOL5700 Flexure Notes p. 2dM − Pdw = VdxdMdx= V + Pdwdx(2)We drop V by combining (1) and (2) to getd2Mdx2=−q + Pd2wdx2(3)If we examine M more closely, we find it is related to w:σxxMMhz=0zFirst, consider the stresses acting on the edge of the plate (these are deviatoric stresses; wehave subtracted out the mean stress P/h). The total moment isM =σxxzdz−h /2h /2∫(4)Now if we consider the plate in its original unbent state and compare it to the diagram above,we can see that there has been shortening towards the top and lengthening below. The dashedline is the neutral surface, which is unstrained by the bending. From Hooke's law relating stressto strain in a linear, isotropic elastic medium we haveεxx=1Eσxx−νσyy()εyy=1Eσyy−νσxx()(5)where E is Young's modulus (of order 70 GPa for the crust) and ν is Poisson's ratio (of order0.25 for the crust) and y is perpendicular to x in the horizontal plane. For our purposes now, weare going to say that all variations are in our cross section, that is, εyy = 0. We may then combinethese equations to getσxx=E1 −ν2()εxx(6)this can be placed into (4) to getM =E1−ν2()εxxzdz−h /2h/2∫(7)Now the strains are geometrically related to the radius of curvature of this piece of plate, R,assuming the radius of curvature remains large:GEOL5700 Flexure Notes p. 3εxx=−∆ll=zR(8)With a little geometry we find that the angle φ subtended by the length l (about dx) isφ= dα=dαdxdx =ddx−dwdx⎛ ⎝ ⎞ ⎠ dx =−d2wdx2dx(9)Since φ = l/R we can find that1R=φl≈φdx=−d2wdx2(10)Substitution back into (8) gets usεxx=−zd2wdx2(11)This then can go into (7) to get the moment:M =−E1−ν2()d2wdx2z2dz−h /2h /2∫=−E1−ν2()d2wdx2z33⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −h /2h/2=−Eh312 1 −ν2()d2wdx2(12)This last gives us the flexural rigidity of the plate, D:D ≡Eh312 1−ν2()(13)If we now use this in equation (3) we getDd4wdx4= q(x) − Pd2wdx2(14)This is now a differential equation that can be solved for the deformation of a plate in twodimensions. It is the general equation governing deflection of a thin plate without largedeflections. We shall now consider how it can be applied to deformation of the lithosphere, theproblem we have at hand. This is the basis for deflection of sheets of elastic materials inengineering (which may well be one of the most common applications of this piece of physics)Basic application to the Earth's lithosphere (Turcotte and Schubert sec. 3-13)The very first aspect that we encounter for our purposes that is different that those typicalengineering applications is that our plates are not in air. We treat the upper part of the Earth asan elastic plate overlying an inviscid substrate in the asthenosphere. That is to say that ourelastic plate is over a fluid. Thus if we pushed down on part of the plate, we would not only beGEOL5700 Flexure Notes p. 4resisted by the forces that keep the plate from bending, as above, but also the pressure of thefluid asthenosphere on the base of the plate.Consider lithosphere of thickness h and density ρm over asthenosphere of density ρa inisostatic equilibrium under a column of water thickness hw. At a depth z under the lithosphere,the pressure will be the integral of the weight above, or (per unit area) gρwhw + gρmh + gρa (z - hw- h). Let us assume that this defines the pressure in the asthenosphere. Now consider part of thelithosphere that has been depressed a distance w under a load. If a material of density ρf fills inabove the plate, we have the weight above the base of the lithosphere as being ρw ghw+ gρfw +gρmh. The force per unit area acting on the base of the lithosphere is the asthenopheric pressureat hw + h + w minus the weight of the column above, org [ρwhw + ρmh + ρa (hw + h + w - hw - h) - {ρwhw +ρfw + ρmh }] = (ρa - ρf)gw (15)This force is directed upwards. If we are defining down as positive for deflection and force,then the net force per unit area acting on the lithosphere isq(x) = qa(x) - (ρa - ρf)gw (16)where qa(x) is the load applied at the top of the lithosphere. Our equation (14) now becomesDd4wdx4+ Pd2wdx2+ρa−ρf()gw = qax()(17)If the material filling in the top has a density of crust, say ρc, then ρc can be substituted for ρf .Periodic topography (Turcotte and Schubert, sec. 3-14)The simplest place to start is with a solution that requires little effort and yet gives us our firstreal insights into the scales where isostasy takes over from rigidity. Consider some region wheretopography is sinusoidal with x such that the elevation e(x) = e0sin (2πx/λ) and does not varywith y. The load on the lithosphere, qa, is then the variation of weight that accompanies thisdeflection:qa(x) =ρcge0sin2πxλ(18)where ρc is the density of the crust associated with the height variation.