Observations of plate tectonics imply that the thin near-surface rocks, that constitute the lithosphere, are rigid, and therefore behave elastically on geological time scales. From the observed bending, or flexure, of the lithosphere, due to known surface loads, we can deduce the elastic properties and thickness of the plates. Linear elasticity can be used to explore the response of thin plates, assuming that h << L, h is thickness, and L is length scale, to load and horizontal forces. The displacements have to be small, w<<l, in order for the linear elasticity theory to work. The resulting equations give inside into buckling due to horizontal stresses, deflection due to loading, and intrusions. A Plate bending The general equation for plate deflection in 2-D is a fourth-order differential equation, D d4 w dx 4 = q(x) P d2 w dx 2, (1) Eh 3 D 12(1 ν 2 ), (2) is flexural rigidity [D] = Nm, M = D d2 w dx 2, (3) is bending moment, q(x) is downward force per unit area (load), and P is a horizontal force. Figure 1 shows a small section of a thin curved elastic plate. There is a net shear force V per unit length in the z direction, which is related to the load, dv dx = q. (4) Buckling Consider the case there is no load q = 0, but the plate is subjected to horizontal force P. Using Equation (1) and the boundary conditions, w =0 d 2 w dx 2 =0 atx =0,L, atx =0,L, Throstur Thorsteinsson (th@turdus.net) 1
M P V h q(x) P M+dM s xx V+dV Figure 1: Forces, torques, and normal stresses on a small section of a thin curved elastic plate. we find that ( ) 1 P 2 w = c 1 sin x, (5) D P = n2 π 2 D, (6) L2 since w =0atx = L. The plate will not deflect unless there is a critical (minimum) horizontal force, P = P c = π2 L 2 D. The amplitude of deflection cannot be determined by linear analysis. Intrusions Here we will only consider a two-dimensional case of intrusions. Consider the case there is an intrusion that is able to overcome the overburden to form a dome like structure. In that case the load is q = p + ρgh. We set the coordinate system such that the center of the dome is at x = 0. The boundary conditions are w = dw/dx =0atx = ±L/2, and the geometry is shown in Figure 2. w/w 0 1 0.5 0 L/2 0 L/2 x/l Figure 2: The geometry of a laccolith. Using Equation (1) we find, after some algebra, w = ρgh p 24D (x 4 L2 x 2 2 ) + L4. (7) 16 Throstur Thorsteinsson (th@turdus.net) 2
The maximum deflection occurs at x = 0, and is (ρgh p)l4 w 0 = 384D. (8) These features are called laccolith, that is, a laccolith is a sill-like igneous intrusion in the form of a round lens-shaped body much wider than it is thick. B Application to the Earth s lithosphere When applying Equation (1) to determine the downward deflection of the lithosphere, we must include in q the hydrostatic restoring force due to the effective replacement of mantle rocks by material of smaller density. The net force is thus q = q a (ρ m ρ i )gw, (9) q a is the applied load, and ρ i is the density of the overlying material (ρ m is the mantle density, g gravity, and w the deflection, as before). Periodic loading Consider now topography, with elevation h, that has a characteristic wavelength λ, and is periodic, ( ) 2πx h = h 0 sin. (10) λ Since the amplitude of the topography is small compared to the thickness of the elastic lithosphere, the influence of the topography on this thickness can be neglected. The applied load is then q a = ρ c gh, and the net force per unit area is q = q a (ρ m ρ c )gw. (11) Using Equation (1) with P = 0, we find that ( ) 2πx w = w 0 sin, (12) λ h 0 w 0 = ρ m ρc 1+ D ( 2π ) 4. (13) ρ cg λ ( ) 1/4 If λ<<2π D ρcg then w0 << h 0, that is there is virtually no deformation of the lithosphere. In the limit of λ we find w = w 0 = ρ ch 0, ρ m ρ c which is the isostatic result. In the long wavelength limit the lithosphere thus has no rigidity and the topography is fully compensated. The degree of compensation, C = w 0 /w 0, (14) as a function of wavelength for a typical lithosphere is plotted in Figure 3, using E = 70 GPa, h = 25 km, ν =0.25, ρ m = 3300 kg m 3, and ρ c = 2800 kg m 3. Throstur Thorsteinsson (th@turdus.net) 3
1 0.8 0.6 C 0.4 0.2 0 0 200 400 600 800 1000 λ (km) Figure 3: Degree of compensation C as a function of wavelength λ for given lithospheric conditions (see text). Point and end loads An infinitely long lithospheric plate (L ) will become unstable and deflect into sinusoidal shape if the horizontal force P exceeds a critical value, Eh P c = 3 (ρ m ρ c )g 3(1 ν 2 = σ c h. (15) ) The wavelength corresponding to the critical force P c is, 2D λ c =2π. (16) P c Now consider a line load at x = 0 on an infinitely long lithosphere. The deflection is given by, ( w = w 0 exp x )( cos x α α + sin x ), (17) α w 0 = V 0α 3 8D, V 0 is the line load applied at x = 0, and α is known as the flexural parameter, ( ) 4D 1/4 α =. (18) (ρ m ρ c )g Throstur Thorsteinsson (th@turdus.net) 4
w/w 0 0 0.2 0.4 0.6 0.8 1 0 2 4 6 x/α Figure 4: Theoretical deflection profile (Eq. 17) for a floating elastic plate supporting a line load at x =0. Here α = 100 km, and we plot the ratio w/w 0 versus x/α. READING ASSIGNMENT Plates Good sections to read are: T+S 3.9 to 3.18 M+W on plates Remember that reading one of these is in some cases sufficient, but it can be very useful to see different approaches. PROBLEMS 1 What is the displacement of a plate pinned at both ends (w =0atx =0,L) with equal and opposite bending moments applied at the ends? The problem is illustrated in Figure 5. M L x=0 x=l M Figure 5: Bending of a plate pinned at both ends. 2 Calculate V and M by carrying out force and torque balances on the section of the uniformly loaded plate shown in Figure 6. 3 Find the deflection of a uniformly loaded beam pinned at the ends, x =0,L. Where is the maximum bending moment? What is the maximum bending stress? Throstur Thorsteinsson (th@turdus.net) 5
V q -M x=0 x=l Figure 6: Section of a uniformly loaded plate. 4 Calculate the magma pressure of a basaltic intrusion if the height w 0 = 200 m, the plate thickness h = 20 km, the length L = 200 km, E =0.6 10 11 Pa, ν =0.25, ρ = 2950 kg m 3. 5 Consider periodic loading of a lithosphere with the following properties E = 50 GPa, ν = 0.25, ρ m = 3300 kg m 3, ρ c = 2950 kg m 3, and h 0 = 10 km. a) Find the wavelength λ the compensation C =0.5 b) How much of Vatnajokull (λ = 100 km) is compensated (very rough calculation) c) What is the amplitude of deflection (w 0 ) 6 Explain the assumptions, boundary conditions, and results obtained in T+S for: a) Horizontal load (buckling, stability of lithosphere) b) Line load and bending (island chains, ocean trench) c) Periodic loading and intrusions Throstur Thorsteinsson (th@turdus.net) 6