GLY 5705 - Geomorphology Notes Dr. Peter N. Adams Revised: Sept. 2012 10 Flexure of the Lithosphere Associated Readings: Anderson and Anderson (2010), pp. 86-92 Here we ask the question: What is the response of the Earth to various topographic loadings on its surface? The answer is, oversimplified, that the lithosphere behaves as an elastic solid. 10.1 Atomic Springs What supports the load of my body standing on the floor? Atoms have a preferred spacing dictated by the size and charge of the element. Pushing or pulling the atoms away from their spacing results in increased binding energy trying to restore the original configuration. When a load is placed on a rock plate, there is compression of atom spacing on the inner-arc of the flex, and tension of atom spacing at the outer arc of the flex. In strain terms, these motions are shortening and extension. Bigger masses displace more atoms resulting in larger deflections. The weight of a mountain load is offset by a lot of atoms being just a little bit unhappy. In mechanical terms, the sum of the atomic forces are the fibre stresses and the elastic properties within the rock mass determine the degree to which it 1. resists compression (Young s Modulus, E) and 1
2. stretches in one dimension in response to contraction in another dimension (Poisson s Ratio, ν) 10.2 Distribution of Deflection and the Peripheral Bulge 10.2.1 Form of the Deflection Profile For a complete derivation of the general equation governing deflection of a plate, please consult pp. 112-116 of Turcotte and Schubert (2002). The fourth-order differential equation resulting from that derivation is D d4 w dx 4 = q(x) P d2 w dx 2 (1) The solution for the deflection beneath a line load is w = w o e x/α [cos(x/α) + sin(x/α)] (2) where α, the flexural parameter, which sets the horizontal length scale of the response, is given by [ ] 1/4 4D α = (3) (ρ mantle ρ infill )g and D, the flexural rigidity, which relates the material properties (Young s Modulus and Poisson s Ratio), to the plate thickness (Effective Elastic Thickness), is given by D = ET 3 e 12(1 ν) (4) 10.2.2 Important Considerations of the Deflection Profile The sine and cosine terms bounce between -1 and 1. 2
The decaying exponential term monotonically decreases the magnitude of deflection with distance from the center of the load. Question 1: Where is the peripheral bulge located? To obtain an answer to this question, we want to determine where. dw dx = 0 (5) We begin by taking the derivative of the deflection equation: dw { [w dx = o e x/α 1 α sin( x α ) + 1 α cos( x } ] α ) + [ w o e {cos( x/α x α ) + sin( x }] α ) ( ) ( ) ( ) ( ) dw x x x x dx = sin cos + sin + cos α α α α (6) (7) ( ) dw x dx = 2sin = 0 (8) α Where does this occur? It occurs at x = 0, π, 2π,... (9) α Or, in other terms, it occurs at Question 2: x = πα (10) What is the maximum value of the bulge deflection? To obtain an answer to this question, we plug in the position of the bulge, obtained above, directly into the deflection equation: w = w o e π [cosπ + sinπ] (11) w = w o e π ( 1) (12) w = 0.04w o (13) The upward deflection of the bulge is approximately 4% of the maximum downward deflection beneath the load. 3
10.2.3 Maximum Deflection and Degree of Compensation What determines the max flexure, w o? It should increase with the load size and decrease with the plate strength. In addition, the maximum deflection varies with horizontal length scale of the load, λ. w o = ρ m ρ c h o ( ) 4 (14) 1 + D 2π ρ cg λ The degree of compensation, C, which represents the proportion of load weight supported by fluid buoyancy, captures the interplay of these variables: C = w o w = ρ m ρ c ρ m ρ c + D g ( 2π λ ) 4 (15) w represents the deflection if the load were supported entirely by buoyancy, (i.e. the isostatic response). If C = 1.0, the load support is purely isostatic, whereas if C = 0.0, the load is supported entirely by plate flexure. So, for all else being equal, it is the length scale of the load that determines whether the support is flexural or isostatic. For common values of E, ν, and T e, loads narrower than about 400 km are supported by elastic stresses (flexure). Note that Tibet is some 1400 km wide, so we are correct in using an isostatic balance to calculate its support. 4
10.3 Notable Examples of Flexure 10.3.1 Hawaiian Island Line Load and the Cross Seamount 10.3.2 Taconic Orogeny and the Acadian Clastic Wedge 10.3.3 Forebulge of the Amazon Sedimentary Fan References Anderson, R. S., and S. P. Anderson (2010), Geomorphology: The Mechanics and Chemistry of Landscapes, Cambridge University Press. Turcotte, D. L., and G. Schubert (2002), Geodynamics, 2nd ed. ed., Cambridge University Press. 5