LOCALIZATION OF SEDIMENTARY ROCKS DURING DUCTILE FOLDING PROCESSES Pablo F. Sanz and Ronaldo I....
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Transcript of LOCALIZATION OF SEDIMENTARY ROCKS DURING DUCTILE FOLDING PROCESSES Pablo F. Sanz and Ronaldo I....
![Page 1: LOCALIZATION OF SEDIMENTARY ROCKS DURING DUCTILE FOLDING PROCESSES Pablo F. Sanz and Ronaldo I. Borja Department of Civil and Environmental Engineering.](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbf5503460f94985760/html5/thumbnails/1.jpg)
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
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Outline of Presentation
• Motivation and objectives
• Kinematics of folding
• Constitutive model
• Stress point integration algorithm
• Finite element implementation
• Numerical simulations
• Ongoing work
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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)
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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
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Field work - Locations
1. Sheep Mt. Anticline, WY
2. Raplee Monocline, UT
1
2
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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
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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
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StratighraphySheep Mountain Anticline, WY Raplee Monocline, UT
Shale Sandstone Limestone
section of units within the exposed anticline
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STRAIN
STRESS
(a) (b)
DAMAGE INITIATION
DAMAGE INITIATION
ONSET OF PLASTICITY
STRESS
STRAIN
Brittle vs. Ductile Behavior
Brittle Ductile-Brittle
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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
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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
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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:
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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
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Stress Point Integration Algorithm
Return mapping algorithm [material subroutine]:
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Local Tangent Operator
Local tangent operator
Local residual
where,
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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
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Parameters:
Numerical SimulationsMesh and geometry:
1,000 elements - 561 nodes
Examples:Boundary conditions and load cases:
I
II
5.0 m
1.0 m
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-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)
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-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):
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Example I : det(a)
Step #100 Step #117
Step #150
Onset of localization:
step #117
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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)
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Step #125 Step #179
Step #185
Example II : det(a)
Onset of localization:
step #179
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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
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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
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No. elements = 250 No. elements = 1,000
Step #170 Step #170
Step #185 Step #185
Step #200 Step #200
Mesh Sensitivity Analysis: det(a)
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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
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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
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• 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
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…???