Numerical modeling of rock deformation: 0 Analytical methods - Folding Stefan Schmalholz schmalholz@erdw.ethz.ch NO E 6 AS 2009, Thursday 0-2, NO D
Overview Application of linear stability analysis Dominant wavelength concept
Application to lithosphere The deflection of the elastic oceanic lithosphere at an ocean trench can be described with a simple bending equation. Note that the so-called forebulge is a typical feature of elastic beam bending D 4 x w 0 4 m w gw Turcotte & Schubert, Geodynamics
Various bending equations Bending of elastic crust due to topography under gravity Buckling of elastic crust under gravity 4 w x D 4 m cgw cghcsin 2 x w w D P gw x x 4 2 4 2 m c 0 Buckling of elastic layer in viscous media 4 2 w w D P 4 w 4 2 2k 0 x x t Folding of viscous layer in viscous media 5 2 H w w w 4 H 2 2k 4 4 0 xt x t
Folding: Mechanism Single-layer folds are presumably the best studied fold types. Natural single-layer folds are characterized by a more or less constant layer thickness and a more or less regular and periodic shape. Questions: Is there a mechanical explanation for this observation? Does the geometry provide information on the rheology? How are these folds generated anyway?
Folding: Linear stability analysis Basic state - no perturbation (flat layer) Matrix Viscosity Pure shear shortening causes homogeneous thickening. Layer Matrix 50 x viscosity Viscosity Homogeneous thickening in pure shear shortening is a flow field that fulfills the continuum mechanics conservation equations for mass, linear momentum and angular momentum. However, to occur in nature a flow field MUST also be stable. Is homogeneous thickening a stable flow field? Stable means here that small perturbations of the flow field must, once they appear, decay with time. We have to test the stability of homogeneous thickening.
Folding: Linear stability analysis Pure shear shortening causes homogeneous thickening. Basic state - no perturbation Matrix Layer Matrix Viscosity 50 x viscosity Viscosity Add small geometrical perturbation (sinus) Perturbed state w x w cos kx 0 Testing for stability means adding a perturbation on the basic flow field and determine whether this perturbation decays with time (or deformation).
Folding: Linear stability analysis Pure shear shortening causes homogeneous thickening. Basic state - no perturbation Matrix Layer Matrix Viscosity 50 x viscosity Viscosity Add small geometrical perturbation (sinus) Perturbed state w x w cos kx 0 Pure shear shortening causes buckle folding.
Folding: Linear stability analysis Basic state - no perturbation Matrix Layer Matrix Add small geometrical perturbation (sinus) Perturbed state Viscosity 50 x viscosity Viscosity w x w cos kx 0 Pure shear shortening causes homogeneous thickening. Homogeneous thickening is NOT a stable flow field. Pure shear shortening causes buckle folding. Small perturbations, always present in nature, cause folding of a stiff layer.
Equation for viscous folding w w D P q xt x 5 2 0 4 2 5 2 H w w w 4 H 2 2k 4 4 0 xt x t xx w xx 2 2y 2 t x t H D P 4 w 2 H P 4 H q 4 k 2 w t 2 k
Folding: Linear stability analysis Thin-plate equation for viscous folding. 5 2 H w w 2 w 4 H 2 2 xt Partial differential equation. 4 4 0 x t x H w 2 2 Is a layer with small sinusoidal perturbations stable under compression? Do small perturbations grow very fast?, exp cos2 / w x t w t x 0
Folding: Linear stability analysis Thin-plate equation for viscous folding. 5 2 H w w 2 w 4 H 2 2 xt Partial differential equation. 4 4 0 x t x H w 2 2 Is a layer with small sinusoidal perturbations stable under compression? Do small perturbations grow very fast?, exp cos2 / w x t w t x 0 Assuming exponential growth with time. Homogeneous pure shear thickening of the layer is: stable if < 0 shortening/thickening neutrally stable if = 0 shortening/thickening and unstable if > 0 folding/buckling
Original partial differential equation 5 2 H w w 2 w 4 H 2 2 xt w x, t w exptcoskx 4 4 0 x t After substituting the solution Ansatz into the governing equation we get H 4 2 k 4 Hk 42 k 0 The equation includes both the growth rate and the wave number. Solving for the growth rate provides 2 Hk Hk 2 2 which is a dispersion relation, because it relates the growth rate of a perturbation to its wave number, or alternatively, wavelength. The maximum of is found by setting the derivative of with respect to k to zero and solving for k. Substituting the wavelength corresponding to the maximum back into the dispersion relation yields the maximal value of. 2 4 2 2 H 6 H k 2 4 0, 2 2 H, 2 2 Hk 22 6 2 2 k H k k Solution Ansatz 0 Dispersion curve
Numerical verification ~ ~ Dispersion curve 4 6 2 2 Biot, 957 Ramberg, 96 2H 62
Necking: Linear stability analysis Pure shear extension causes homogeneous thinning. Rheology must be power-law. No necking in Newtonian viscous materials. Pure shear extension causes necking.
Analytical solutions and numerical algorithms Analytical solutions are important to: test the numerical algorithm to optimize the numerical resolution to optimize the time step to optimize iterations to get physical insight Test for nonlinear power-law folding.