localization of sedimentary rocks during ductile folding processes pablo f. sanz and ronaldo i....

LOCALIZATION OF SEDIMENTARY ROCKS DURING DUCTILE FOLDING PROCESSES

Pablo F. Sanz and Ronaldo I. BorjaDepartment of Civil and Environmental Engineering

Stanford University

8th 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

StratighraphySheep Mountain Anticline, WY Raplee Monocline, UT

Shale Sandstone Limestone

section of units within the exposed anticline

STRAIN

STRESS

(a) (b)

DAMAGE INITIATION

DAMAGE INITIATION

ONSET OF PLASTICITY

STRESS

STRAIN

Brittle vs. Ductile Behavior

Brittle Ductile-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 Wo (for quasi-static loads)

Kirchhoff stress tensor

Consistent tangent operator

Parameters:

Numerical SimulationsMesh and geometry:

1,000 elements - 561 nodes

Examples:Boundary conditions and load cases:

I

II

5.0 m

1.0 m

-30

-25

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25

Step #113

Step #289

0

5

10

15

20

-40 -35 -30 -25 -20 -15 -10 -5 0

Example I: ‘bending/extension’

Meridian plane Deviatoric plane

Stress path

Onset of localization: step #117

(a)(a)

(c)

(b)

(b)

(c)

-1.00E+04

0.00E+00

1.00E+04

2.00E+04

3.00E+04

4.00E+04

5.00E+04

6.00E+04

7.00E+04

8.00E+04

9.00E+04

1.00E+05

1.10E+05

1.20E+05

1.30E+05

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Angle (degrees)

De

t(a

)

Example 1 Step #117

Bifurcation Analysis

Eulerian acoustic tensor:

Onset of localization:

33o 147o

Step #117Element 902

Normal to shear band:

Orientation of shear band:

Expression by Arthur et al.(1977):

Example I : det(a)

Step #100 Step #117

Step #150

Onset of localization:

step #117

0

10

20

30

-120 -100 -80 -60 -40 -20 0

Example II: ‘bending/compression’

Meridian plane

Deviatoric plane

Stress path

Onset of localization: step #179

-80

-60

-40

-20

0

20

40

60

-70 -50 -30 -10 10 30 50 70

Deviatoric plane - Step #169

Deviatoric plane - Step #196

(a)

(a)(c)

(b)

(b)

(c)

Step #125 Step #179

Step #185

Example II : det(a)

Onset of localization:

step #179

Convergence of numerical solution

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1 2 3 4 5 6 7

Iteration

No

rm o

f re

sid

ual

(n

orm

aliz

ed)

Step 1Step 101Step 149Step 212Step 222

Element 2 - Step 173

1.00E-15

1.00E-14

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1 2 3 4 5 6 7

Iteration

No

rm o

f re

sid

ua

l (n

orm

aliz

ed) Iteration 1

Iteration 2

Iteration 3

Iteration 4

Global convergence(finite element)

Local convergence (material subroutine)

• Convergence is asymptotically quadratic

Example IIExample II

Mesh Sensitivity Analysis

No. elements = 250 No. elements = 1,000

Step #0 (undeformed) Step #0 (undeformed)

Step #225

• Plasticity: step #171• Localization: step #184

• Plasticity: step #169• Localization: step #179

Step #225

No. elements = 250 No. elements = 1,000

Step #170 Step #170

Step #185 Step #185

Step #200 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 LayersMesh and geometry: Examples:

I

II

1.0 m

1.0 m

1.0 m

Parameters:

3,000 elements5.0 m

Inner layer Outer layers

• Plasticity: step #125• Localization: step #134 [ = 19o]

• Plasticity: step #114• Localization: step #118 [= 33o]

• Plasticity: step #172• Localization: step #182 [ = 33o]

• Plasticity: step #165• Localization: step #177 [ = 33o]

Numerical Simulations: det(a)Example I Example II

Eouter = 100 MPa

Eouter = 500 MPa

onset of localization

…???