LOCALIZATION OF SEDIMENTARY ROCKS DURING DUCTILE FOLDING PROCESSES Pablo F. Sanz and Ronaldo I. Borja Department of Civil and Environmental Engineering Stanford University 8 th US National Congress on Computational Mechanics

Outline of Presentation Motivation and objectives Kinematics of folding Constitutive model Stress point integration algorithm Finite element implementation Numerical simulations Ongoing work

Motivation In the geoscience community the study of folding processes is carried out with kinematic models or simple mechanical models Better representation of rock behavior can be achieved using more realistic mechanical models Kinematic models by Johnson et al. (2002)

Objectives Formulate and implement a finite deformation FE model using a three-invariant plasticity theory to capture ductile folding of rocks Demonstrate occurrence of localized deformation in different numerical examples

Field work - Locations 1. Sheep Mt. Anticline, WY 2. Raplee Monocline, UT 1 2

Sheep Mountain Anticline, WY Sediments are 100 million years old Folding occurred approx. 65 Ma 12 km long 1 – 2 km wide 300 m structural relief (height) Upward fold Anticline

Raplee Monocline, UT Single upward fold Sediments are 300 million years old Folding occurred approx. 65 Ma 14 km long 3 km wide 500 m of structural relief (height) Monocline

Stratighraphy Sheep Mountain Anticline, WY Raplee Monocline, UT ShaleSandstoneLimestone section of units within the exposed anticline

Brittle vs. Ductile Behavior BrittleDuctile-Brittle

Assumptions – Kinematics of Folding Thermal and rheological effects not considered Folding is driven by imposing displacements at the bottom and at the ends Vertical load (dead load) remains constant throughout the deformation

Constitutive model Features to capture: Elastic and plastic deformations Yielding is pressure-dependent and non-symmetric in deviatoric stress plane  Three-invariant model Shear-induced dilatancy  Non-associative plastic flow Onset of localized deformations

Matsuoka-Nakai yield criterion : Hardening law: Plastic potential: Translated principal stresses and invariants: Flow rule : Elastoplastic model Material parameters: Yield surface in principal stress space:

Stress Point Integration Algorithm Return mapping algorithm: Integration scheme is fully implicit and formulated in principal stress axes Based on spectral representation of stresses and strains Finite deformation formulation is based on multiplicative plasticity using the left Cauchy-Green tensor and Kirchhoff stress tensor Isotropic hardening three-invariant plasticity model

Stress Point Integration Algorithm Return mapping algorithm [material subroutine]:

Local Tangent Operator Local tangent operator Local residual where,

Finite Element Implementation Variational form of linear momentum balance Linearization of W respect to the state W o (for quasi-static loads) Kirchhoff stress tensor Consistent tangent operator

Parameters: Numerical Simulations Mesh and geometry: 1,000 elements nodes Examples:Boundary conditions and load cases: I II 5.0 m 1.0 m

Example I : ‘bending/extension’ Meridian planeDeviatoric plane Stress path Onset of localization: step #117 (a) (c) (b) (c)

Bifurcation Analysis Eulerian acoustic tensor: Onset of localization: 33 o 147 o Step #117 Element 902 Normal to shear band: Orientation of shear band: Expression by Arthur et al.(1977):

Example I : det(a) Step #100Step #117 Step #150 Onset of localization: step #117

Example II : ‘bending/compression’ Meridian plane Deviatoric plane Stress path Onset of localization: step #179 (a) (c) (b) (c)

Step #125Step #179 Step #185 Example II : det(a) Onset of localization: step #179

Convergence of numerical solution Global convergence (finite element) Local convergence (material subroutine) Convergence is asymptotically quadratic Example II

Mesh Sensitivity Analysis No. elements = 250No. elements = 1,000 Step #0 (undeformed) Step #225 Plasticity: step #171 Localization: step #184 Plasticity: step #169 Localization: step #179 Step #225

No. elements = 250No. elements = 1,000 Step #170 Step #185 Step #200 Mesh Sensitivity Analysis: det(a)

Ongoing Work Formulation and numerical implementation of a coupled elastoplastic damage constitutive model Modeling of several rock layers with distinct constitutive properties (elastic, ductile, brittle) Numerical simulations in 3-D

Numerical Simulations: 3 Layers Mesh and geometry: Examples: I II 1.0 m Parameters: 3,000 elements 5.0 m Inner layerOuter layers

Plasticity: step #125 Localization: step #134 [  = 19 o ] Plasticity: step #114 Localization: step #118 [  = 33 o ] Plasticity: step #172 Localization: step #182 [  = 33 o ] Plasticity: step #165 Localization: step #177 [  = 33 o ] Numerical Simulations: det(a) Example I Example II E outer = 100 MPa E outer = 500 MPa onset of localization

…???