Geodynamics - Plates

Lecture Notes 18 Plates Rock rheology → function of T; T increases with depth. Thin region near surface remains elastic on geologic times. Below this, mantle behaves as a viscous fluid. Plate theory: Assume plate thickness h L, with L a characteristic length scale 3-D equations simplify Applications: Seamount Loading Subduction Zone Fore-arc bulge Trench � Folding Laccolith Magma Figure 18.1 Main Feature y x h z Fibers compressed Neutral surface } Fibers extended } Figure 18.2 “Fiber stresses” large compare to tractions applied to surfaces. Neutral surface – smooth and centered Kirchoff’s assumption – approximate, but useful. σ yy = σ yx = σ yz = 0 h 2 σ xx , σ zz linearly through − ≤ y ≤ h 2 or every straight line originally perpendicular to neutral plane remains straight and perpendicular More accurate formulations lead to nonlinear coupled equations, not solvable analytically. To derive plate equations – consider the following figures: Moments Surface Loads q M Shear Forces End Loads V Figure 18.3 x+dx x q(x) x w+dw w P M P V+dV V Figure 18.4 M+dM V – shear force / unit length P – horizontal force / unit length M – moment / unit length q – force / unit area Force balance – vertical qdx + dV = 0 ⇒ dV = −q dx Torque balance (+ counterclockwise) dM − Pdw −Vdx = 0 dM dw =V + P dx dx Next – relate M to w. y M h/2 o -h/2 Figure 18.5 h/ 2 M= ∫ σ xx ydy −h / 2 For plane stress, σ xx = E exx (1 − ν 2 ) Bending the plate causes fiber strains δl y exx = − = l R � R y �l = -y� = -y l R l � Figure 18.6 Now – let l get small d 2w ε xx = − y 2 dx d� �+d� R l dw � = ­ dw dx dx Figure 18.7 Then, substituting E d 2 w h/ 2 2 M =− ∫ y dy (1 − ν 2 ) dx 2 −h / 2 M =− Eh3 d 2 w 12(1− ν 2 ) dx 2 or d 2w D = dx 2 R Eh3 where D ≡ (flexural rigidity) 12(1− ν 2 ) M = −D Substituting d 4w d2w D 4 = q(x) − P 2 dx dx This equation is called plate equation.