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