A STUDY OF THE DYNAMIC RESPONSE OF A FRACTURED TUNNEL

TO PLANE WAVES

Pei-cheng Xu

SwRI, SanAntonio,TexasSept. 7, 2000

x

z

y

vacuum

host medium

tunnel

fracture

Objectives of Study

• Develop a computational tool to predict dynamic response of a fractured tunnel to plane waves

• Investigate the effects of fracture properties on the displacement and stress field, using the slip interface model

Method of Approach:The Boundary Integral Equation Method

• Discretize the boundary only

• Suitable for an unbounded host medium

• Additional efficiency when large numbers of sources and detectors are involved

)()()(),()()(2

1 0 dSnuTuu iik

ijkk

The Boundary Integral Equation

}{}{][ 0u = u M

Solve BIE for boundary displacements

Traction free

x

The Boundary Integral

)()(),()()( 0 dSnuxTxuxu iik

ijkk

)(2,,);(3,1,,, SHkjiSVPkjix

* Displacement off boundary: direct integration

* Stresses: Hooke’s law / differentiation

jkisikjs

ijkspkijkpk

ij

rkHrkH

rkHrkHrkHiTs

,0,0

,002

,02

)]([)]([

)]()([)]()[1(4 22

Green’s Functions and Associated Stresses

iksikspkik rkHrkHrkHGiS

)()]()([4 0,0012

Uniform, isotropic

Layered (fractured)

dkxikxkFI )](exp[|]|exp[)( 1133

)Re(||22 kkkk

)Re(||22 kkkki

x1

x3 3

1

r

x

wh b

a

1

2

3 4 5

6

7

8

1

2

3

4 5

6 7 8

9

10

1112

Boundary Mesh

Basic type of elements

n1 columns

n 2 ro

ws

detector

fracture

tunnel

Detector Mesh

The Slip Model of Fracture

n

s

s

sss

n

nnn

sss

nnn

Kuu

Kuu

)2(

Stiffnesscoefficients

The Stiffness Calculation in theE-Model of Fracture

Kn(

Virtual thicknessin BIE

Slip line stiffness

Combined stiffness across the slip line =

1Kn

I-model

ks

kn

E-model

ks

kn

Fractured host medium

I-model

ks kn

Uniform host medium

E-model

ks kn

Comparison of I-Model and E-Model of Fracture

• Zero thickness• Infinite length• Longer computer time• Numerically unstable

when elements are small

• No mesh on the fracture

• Finite thickness• Finite length• More efficient• Numerically unstable

when virtual thickness is too small

• Mesh on the fracture

I-Model E-Model

Features of the Current Program

• Predicts displacements and stresses on the boundary and in the surrounding area

• Handles multiple sources and a large number of detectors efficiently.

• Two alternative slip interface models for the fracture: Implicit and Explicit.

• A user friendly interface for input data.

• A variety of display means of output data, including deformed meshes, hoop stresses on the boundary, and quiver and contour plots of field stresses.

Limitations of the Current Program

• The Implicit slip model encounters numerical instability when boundary elements are small.

• The Explicit slip models works but the input value of virtual thickness may depend on the frequency and geometry.

• Needs a user friendly output interface.

= +

1 1/2 1/21/2 -1/2

Decomposition based on Symmetry

nn n/2n/2 n/2n/2

Frequency loop

Symmetry loop

Incidence loop

Boundary mesh

Decomposed boundary Matrix

Input

Decomposed boundary response w.o. tunnel

Decomposed boundary response w. tunnel

Detector mesh

Total boundary response

PROGRAM STRUCTURE

Detector response Output files

Solving equation

SUBROUTINE LAYOUT

IN P U T M E S H TM E S H N E T

G R E E NG R E E N S

IN C D N TIN C S Y M

COEFF

CGARCFAR

M A TD O M G R E E N 0

CSUM

W A V E

O M E G A

S O U R C E

B A N D

F U L L

G L O B A L

L O C A TEL O C A TE 3

S O R T

G R E E N Z D IV ID E IN S E R T M C C F S C

S TM S C 2

S F S C

S TM 2

IN TX R

M C C IN T

C H E B S HC H E B S H 3

TC N P P O S ITIO N

F ITA IL R O TA TR O TA T3

E X P F C N C O N F C T

R E C IN TR E C IN T3

F ITIN T

G R E E N SG R S N 2

IN TG F 0IN TG F 0 2

G F IK XG F IK X 2

G A U S S 4

IN TL IN

W E IG H T

M A TR IXM A TS Y M

C E Q S L V

N O R M P L O T R A D IP L O T

P R B IE B

COEFF

CGPARRCFPAR

IN TG F SIN TG F S Q

G F IK SG F IK S Q

IN TL IN

W E IG H T

M F IE L D SM F IE L D S Q

CVF

G R E E N P

CSUM

G R E E N S

R E S U L TS D E F O R M M O H R H O O K E

M A IN

INPUT FILESparalst2.h ------------ basic numeric settingstunnel.in ------------ general input data

OUTPUT FILES tunnel.out ----------- general output

mesh.dat ------------ coordinates of boundary nodesmeshr.dat ----------- coordinates of Gaussian points on

boundarymeshnet.dat -------- coordinates of network nodes

bform.dat ------------ deformation of boundary meshform.dat ------ deformation of network nodes

ub.dat ---------------- nodal displacements on boundarybplot.dat ------------- nodal stress on boundary for plottingps.dat ---------------- principal stresses at network nodescross.dat ------------ principal stresses at network nodes for

plotting

I/O Files

10 2

4

6 7 8

53

Geometry ofCavities orInclusions

9

12 13 14

15 16 17

1011

Geometry ofCavities orInclusions

Select type of medium structureCavityFree solidHost & inclusion

Select type of geometry

Select type of dynamic sourceConcentrated line force in the inclusion Concentrated line force in the host domainIncident plane P-wave in the host domainIncident plane SV-wave in the host domainIncident plane SH-wave in the host domain Pressure center in the host domain

Torsion center in the host domain

Determine boundary conditionFree surfaceFixed surfaceFree surface & internal slip interfaceFixed surface & internal slip interface

Input Parameters (1)

Enter maximum number of nodes

Is the geometry symmetric?

Select type of medium

Number of sources 2

Are the detectors aligned in a network? NY

If no, then enter number of individuallyspecified detectors 10

Number of frequencies 1

Choose one of the following calculation options:BIE solution onlyMesh without solution

Choose the component of response to plot along the boundaryHorizontal or radial displacementVertical or tangential displacementRadial normal stressTangential shear stressTangential normal (hoop) stress

Input Parameters (2)

Is the geometry symmetric? Y/N

Are you using an Implicit (I)or Explicit (E) model of fracture? I/E

Parameters of fracture for the Explicit Model (used when nfrct=0 and id3=3)Virtual normal thickness oft fracture [m](suggested range=0.1-0.5) 0.2

Virtual shear thickness oft fracture [m](suggested range=0.1-0.5) 0.2

Define dividing value of weak and strong slip coefficient(suggested range = 0.8-2.0 ) 1.0

Input Parameters (3)

Parameters controlling element and mesh sizesEnter number of elements per wavelength(suggested range=2-5) 2.0

Define far field in terms of number of wavelengths(suggested range=2-5) 2.0

Parameters associated with physical properties of the mediumSource domainDensity P-wave vel. S-wave vel. Q Q[g/cm**3] [km/s] [km/s] (P-wave) (S-wave)1.00 1.754 1.200 100.0 50.0

The other domain (used only when med.=2 but must be present for all cases)Density P-wave vel. S-wave vel. Q Q[g/cm**3] [km/s] [km/s] (P-wave) (S-wave)0.80 1.500 1.000 100.0 50.0

Input Parameters (4)

Parameters associated with dynamic sourcesEnter x-coordinate of sources or reference point in the order of their labels [m] 0.00 0.00Enter z-coordinate of sources or reference point in the order of their labels [m] 0.00 1.00

Components of dynamic sourcesEnter x-component of force [Newton/m] or plane wave in the order of labels of dynamicsources 0.00 1.00Enter z-component of force [Newton/m] or plane wave in the order of labels of dynamicsources 1.00 0.00Enter y-component of force [Newton/m] or plane wave in the order of labels of dynamicsources 0.00 0.00

Input Parameters (5)

Parameters associated with the of detectors aligned in a networkEnter number of divisions in direction parallel to the fracture in the network of detectors 20

Enter number of divisions in direction perpendicular to the fracture in the network of detectors 20

Enter minimum value of rotated x-coordinate in the network of detectors 4.0

Enter maximum value of rotated x-coordinate in the network of detectors 4.0

Enter minimum value of rotated z-coordinate in the network of detectors 2.0

Enter maximum value of rotated z-coordinate in the network of detectors 6.0

Parameters associated with frequencyEnter minimum value of frequency [kHz] 0.25

Enter maximum value of frequency [kHz] 0.50

Enter increment value of frequency [kHz] 0.25

Input Parameters (6)

Welded Half Spaces

x

y

2

x

y

Circular

Round-head Semi-infinite Layer

x

y

h

Infinite Layer

x

z

12

Pinch-out Layer

x

z

h

h2

h1

2

1

Split Layer

xz

h

w

d

Wash-out Layer

xz

2

1

12h2

h1

Faulted Layer 1

z

xd

h2

h1

Faulted Layer 2

x

zd

w

h 0 0

Rounded Block/Slot

x

y

w

h

Dome

x

z

Loaf Shaped

2 2

1

3

Half Space with Semi-Circular Notch

x

z

Cracked Circular Hole

w w

2 x

z

Half Spaces Divided by Slip Line

x

z

s n

Circular Hole Crossed by Slip Line

h x

z

ns

w

h

h1

Dome Crossed by Horizontal Slip Line

x

z

s n

w

h

h1

x

z

Dome Crossed by Oblique Slip Line

sn

Lahey FORTRAN 95