rock mechanics and radioactive waste isolation. “one small ... · “one small step for geology,...
TRANSCRIPT
Rock Mechanics and Radioactive Waste Isolation.“One small step for geology, one giant leap for rock mechanics”
Fourth Müller LectureFourth Müller Lecture
presented by
Charles Fairhurst
Professor Emeritus, University of Minnesota, U.S.A.Senior Consultant Itasca Consulting Group IncSenior Consultant, Itasca Consulting Group, Inc.
Minneapolis, MN 55401, U.S.A. [email protected]
Tenth Congress of the International Society for Rock MechanicsTenth Congress of the International Society for Rock MechanicsSandton, South Africa, September 8-12, 2003
International Society for Rock Mechanics
Early Members Journal
1 - Prof. Leopold Müller, Austria2 Mr F Pacher Austria 2 - Mr. F. Pacher, Austria
3 - Prof. L. V. Rabcewicz, Austria 4 - Mr. C. Lorber, Austria 5 - Prof. F. Kahler, Austria6 - Dr. W. Zanoskar, Austria 6 Dr. W. Zanoskar, Austria
7 - Mr. W. Finger, Austria 8 - Dr. A. Fuchs, Austria 9 - Prof. F. K. Müller, Austria10 - Mr. P. Reska, Austria11 - Dr. K. Waschek, Austria12 - Prof. G.B. Fettweis, Austria13 - Dr. Georg Beurle, Austria14 - Neue, Reformbaugesellschaft MbH, Austria
(Supporting Member) (Supporting Member)15 - Dr. Alois Kieser, Austria16 - Prof. H. Seelmeier, Austria17 - Dr. Adolf Bretterklieber, Austria18 - Mr. W. Wessiak, Austria,19 - Prof. A. Watznauer, East Germany20 - Prof. Charles Fairhurst, USA
Ph.D. degrees in Rock Mechanics, University of Minnesota (1960-2003) & Departmental Colleagues & Associates
1960-1969:
Paul GnirkBezalel Haimson
1970-1979:
Francois CornetSteve Crouch
1980-1989:
Gary CallahanChristine Detournay
Honorary Members:
Randall BarnesBarry BradyBezalel Haimson
William HustrulidHassan ImamHerbert KutterWilliam PariseauHil S h f ldt
Steve CrouchJaak DaemenMichael HardyJohn HudsonDarrell PorterJ Cl d R i
Christine DetournayEmmanuel DetournayRoger HartJose LemosErnest LindnerL L i
y yTed BrownNeville CookPeter CundallAndrew DrescherBojan Guzina
1990-present:
Hilmar von SchoenfeldtWolfgang Wawersik
Jean-Claude RoegiersRaymond SterlingEd Van EeckhoutMichael Voegele
Loren LorigPanos PapanastasiouSamuel SharpYongjia Wang
Bojan GuzinaJoe LabuzTom LangYoshi MizutaChuck Nelson1990 present:
Jose AdachiMargaret AsgianNathali BoukpetiIlya Berchenko
Robert SanturbanoAlexei SavitskiSakir SelcukEduard Siebrits
Anthony PearsonFritz RummelMiklos SalamonChris St. JohnTony Starfield
Ali FakimiDimitri GaragashMatt Handley Haiying HuangIlya Berchenko
Bjorn BirgissonMark BoardRoberto CarbonellCarlos Carranza-Torres
Eduard SiebritsKetan ShahKeh-Jian (Albert) ShouYou TianYiming Sun
ff h
Tony StarfieldOtto Strack
Haiying HuangMark LarsonChengho LeeMarc LokenMark Mack
Professorship
Fernanda CarvalhoLiangsheng ChengBranko Damjanac
Jeff WhyattSanchai MaitaimLee PetersenThomas Richard
The problem. Waste Decay
10 7
• Contains the byproducts offi i ti
s
10 6fission reactions.
• Is highly radioactive.
vity
Cur
ves
10 5• Must be isolated fromthe public for many thousands of years.
Rad
ioac
tiv10 4
y
Needed 1 million year
1010
0
3
10 1 10 2 10 3
radionuclide ‘container’
Geological Isolation?
Years
Dose-response relationships for solid cancer, Alternative hypothetical extrapolations of d th t f hi h t
23E-4
RR
)
ncer
p p ,all types combined, in atomic bomb survivors, 1958-1987 (from UNSCEAR, 1994)
cancer death rate curves from high tolow radiation dose rates
1
2E- 4
lativ
e R
isk (E
R
isk
of fa
tal c
an
LNT
LT
0 1 2 30
0
1E- 4
Exce
ss R
el
Ann
ual r
i
NLH? ?
B0
Weighted intestinal dose, Sv
25 mrem15 mrem4 mrem
0.001 S100 mrem
v 0.005 S500 mrem
v 0.01 S1 rem
vLinear, no thresholdLinear, threshold
LNTLT
Annual dose, Sv
100 mrem 500 mrem 1 remNon-linear, HormesisRange of natural backgroundradiation in U.S. (approximate)
NLHB
Radiation Risk and Waste Isolation Standard. LNTH and Natural Background
R f
Risk of death from radiation
Global Dose rates from Natural Background Radiation
Source of ExposureCosmic Rays
Terrestrial Sources
Radionuclides in Body
0.392.0
0.464.3
0.23
Typical ElevatedDose Rate (mSv/year)
Average Global EffectiveDose Rate (mSv/year)
Reference:ICRP, 1999. "Protection of the Public in Situations of Prolonged Radiation Exposure," Publication 82, Annals of the ICRP, Vol. 29, Nos. 1-2, International Radiation
(except Radon)Radon and its
Decay Products
0.61.3
10.0
Total Global Dose Rate From 2 4 ( d d)
Commission on Radiological Protection, Pergamon Press, New York, NY. Table A.1, page 73, and Table A.2, page 74.
Area
States of Rio de JaneiroEspirito Santo, Brazil
~30 (3.6 average)
Total Global Dose Rate From Natural Background Radiation 2.4 (rounded)
Monazite sand; coastal
Examples of Areas with High Natural
sp o S o,
Volcanic intrusions in 6 km scattered inland areas
~80 (13.3 average)
Kerala and Tamil naduIndia
Central Regionf F
Granitic schistous and sandstone
~6
Mineas Gerais andGoias, Brazil
Monazite sand; coastal area, 200 km long and 0.5 km wide ~30 (9 average)
2
gBackground Dose Rates
of France Granitic, schistous and sandstone
Niue Island(Cook Islands, NZ)
~5
MombasaKenya
Thorium bearing carbonalyte
~30
Ramsar 200
Volcanic soil
Maximum Total Dose Rate (mSv/year) Characteristics of Area
RamsarIran
Areas of radium-226 deposition from spring water
~200
MahallatIran
~20 Areas of radium-226 deposition from spring water
Dose as a function of tume, for SF, for a number of different realisations including those that gave the highest (samplerealisations, including those that gave the highest (sample 164) and the lowest (sample 151) dose maximum out of 1000 realisations.
The Base Case is included to allow an easy comparison. The other curves represent realisations that give the highest dose maximum or the lowest dose maximum in the period between 10 and 10 ears or 10 and 10 ears respecti el The bars4 5 6 710 and 10 years or 10 and 10 years, respectively. The bars beneath the graph indicate the radionuclides that make the highest contribution to dose at any particular time.
4 5 6 7
Predicted Performance of a Repository in Opalinus Clay Spent Fuel —from 02-05, p.229NAGRA Technical Report
“One small step for geology, one giant leap for rock mechanics”
Pre-closure Isolation
A f th E th
• Engineering Problem
Age of the Earth (years)10 10 10 10 100 2 4 6 8
Engineering Problem
Unprecedented Period of Time
Requires Improved Scientific BaseRequires Improved Scientific Base
Periods of Isolation. Geological Isolation Time Scales.
“We don’t know the rock mass strength. That is why we need an International Society” Muller, 24 May 1962
Deformation
Forc
e
Rock mass
De
Intact block
Single joint
Time [10 ~ 10 ye
2
6Forc
e Decreasingdeformationrat e
Continuummechanic s
Localization and disintegration
SIZE
years]
5
Ela sticity
Plasticity
Damagemecha-nics
Ela stic unloading
Energy excess(Seismic
Maximumdesign f
TIME
1 2 3
4
Plasticity (Seismic deformation)
Energy deficit(Aseismicd f ti )
force
Deformation
deforma tion)
Complete Load-Deformation Behavior
“In the field of geomechanics, granular media and block-jointed rockmasses are obvious examples where the concept of the ideal physicalcontinuum −one in which no gaps are formed− cannot be expected to
l h l d l ff l happly... The clastic model offers an alternative approach.
Indeed, it is this writer’s view that only with clastic models or some further development thereof can the problem of predicting the complete load-deformation behaviour of solids be tackled optimistically.”
D.H. Trollope (1968)p ( )
Choose a geologically simple, stable site!(low tectonic strain rate)
Geological Isolation
Overlying and underlying strata:Flat terrain
Borehole
Overlying and underlying strata:- Infrequent joints/faults.- Far-field stresses close to isotropic.- Groundwater discharge to large body of non-potable water (ocean, sea).
σ σ
σ
h v
v
= k (k 1)~
>200 m Repository stratum, simple host medium::- Low permeability.- Low hydraulic gradient.- High ion-exchange potential.
Establish and close the repository
Backfill EDZ (excavation disturbed zone)surrounds seal and backfillDetail of
seal and
SealSeal1 mm gap
seal andbackfill
0 5Exothermic
Seal1 mm gap
90
KK
1010
Hydraulic Conductivity
no gap
1mm gap
~~
-9
-3m/sm/s
o
0.5 m
Drift andcanister
Backfill 2.5 mg p
Compaction (6%) due to particlefriction reduction after t~10,000 years
Rock Charac eteris ticsRock Type
P bi lit I E h Therm al St th D tilit
Clay Low
Salt Low
Very Low
Im perme able
Permeabi lity
Very High
None
Ion Exchange
Low
High
Therm alConducti vity
Strength
High
High
Ductility
++ − − +
Crystalline High
Volc anic Tuff Low
p
Matrix LowFractures High
High
UsuallyNone
High
g
Variable
Low
g
Low
Moderate
+
ο
−
+
−
−
ο
−+ +
−ο +
+
Current Research (and Operational) Sites
Clay Indurated Bure, Franc eMont Terri, Switzerland
ο + − − ο
,Benken, Switzerl andNon-Indurated
Mol, BelgiumSalt DomalGorleben, Germany
Bedded WIPP, Carlsbad, NM, USA(intermediate level ope rational)
Be low water table:Re ducing environment.
( p )Crystalline Unfractured Granite Pinawa, Manitoba, Canada
Fractured Granit e Aspö, Sweden, F inland (cons truction approved May 2002)
Volcanic Tuff Yucca Mountain NV USA Above water table:Volcanic Tuff Yucca Mountain, NV, USA (license applicati on construct Dec 2004)
Preferred Respository Host Rocks and Current Research Sites
Oxidizing environment.
S ifi i
• Rock testing (and prediction)
Some specific issues
• Rock testing (and prediction)
- samples → coring, slotting- full scale → underground excavations
• Excavation technology
full scale → underground excavations
• Excavation damage• Time dependent deformation /rock mass strength• Transparent earth• Transparent earth
σ1 Limit surface
12(a)
Unloading path effects on coring,slotting and underground excavation
3(b)
σ
σ2
31Path 3 Coring (surface or underground)
Sl i (bl k )(a)(b)σ
σ
σ σ−
3
v
v H
031 2 3
Slotting (block center)(b)
σσ σ
σ σ−
h h
H vH
003
1Path 2 Underground excavation
D/R = 0.1 D/R = 0.2
σ σt o/ σ σt o/Maximum Maximum
Potential for core damage during coring operation
0 4
0.6
σ σσ σt
t t
o
o o
// /0.00
0.25
0.00
0.25
Maximum Maximum= 0.05 = 0.1
σσ
to
/or
e ,
coring operation
0.2
0.40.50 0.50
tens
ile st
ress
in c
oFa
r-fie
ld st
ress
DNumerical resultsBest fit curve
00
2 4
Core Depth / Core Radius, D/R
Max
. t F
R
D/R = 1D/R = 2
σ σσ σt to o/ /Maximum Maximum
= 0.5 = 1.0
Let n σ
σ
σ= =t
i
c
tensile strength
induced tension in core
compressive strength[n ~ 0.1]
σ σt o/0.0
0.5
σ σt o/0.0
0.5
Let m σσ= =t
o
induced tension in corein-situ horizontal stress
[m ~ 0.5]
Then core damage occurs if
1.0 1.0σ
σ
σ
σ
c
c
o
o
>
>
n/m
0.2i.e.,
Discing of a 1.3 m diameter core from highly stressed rock (Courtesy of URL/AECL, Canada)
N N
5545o
o
σ
σσ
σ
σ
σ 22
2112
11
2
1
= 45 MPa
= 60 MPa
σσ
σ σ
σσ2
22
21
2112
12
σσ 111
SE SENW NW
Directionof advance
Directionof advance
65
60 60
65 65
70 70
Tunnel parallel to Tunnel at ~10 toσ
σ σσ σ− −
σ2
1 12 2
2o
contours contours
Asymmetric breakout produced by shear stresses acting parallel to excavation (R.S., Read, 1995)
σ σσ
1 3o
−Contours of Maximum Shear Stress developed during slotting of a rock block
σ1
horizontalslot a/b=20
horizontalslot a/b=10
σ3
2.0
horizontalslot a/b=40
verticalslot a/b=10
3
1.0
0.0
verticalslot a/b=40
verticalslot a/b=20
σ σ σ1 3 oNote: Results are for = =
Wire saw slotting tool (Courtesy of URL/AECL, Canada)
W
W
WN
N
R< 1
= Stiffness of excavated slot (N)
W
N
R
( )
= Rock stiffness
[slot can be left open or backfilled]
WW WW
N
N R
R
< <
11
Slotting technique proposed by Kvapil (1962) to increase the stability of ‘critical’ underground excavations (e.g., intersections)in highly stressed rock.
Tunnel Near-field. Disintegration of EDZ
σc = 13 MPa σc = 30 MPawater circulated in inner borehole dry inner borehole
c 13 MPa
σ σ
c
σ σc c
Effect of water on the EDZ in the Opalinus Clay, Mont Terri, Switzerland (Courtesy of )
Bulkhead Open Drift8.0 1.5 5.0
1 81.8
7.01.53.0
5.79.0 1.0
Permeability [m/s]
10-21 10-20 10-19 10-18 10-17 10-16 10-15
ALOHA2: Permeability Distribution Around the Bulkhead Drift (Courtesy of )
FF
TBM Disc CMM Disc(undercutting)
Excavation face
Excavation face
(undercutting)
Plastic region Plastic region
Probablechip path
Excavationface
Note: Tensile crack can not propagate
Unstablecrack propagation
Stress distribution beneath a TBM Disc Indenterafter the stable vertical crack has developed
Crack path produced byCMM Disc Indenter
Disc tipradius
CMM Indenter Force (N/mm of contact length)
S = 25 mm S = 50 mm S = 100 mm S = ∞
until plastic region reaches critical size
6 mm
8 mm
12 mm
1,700 (74%) 2,310 (100%)
2,500 (100%)
3,150 (100%)
2,700 (117%)
3,150 (126%)
3,600 (114%)
3,000 (130%)
(3,500?) (140%?)
4,200 (133%)
N t I d t F I i ll f l i i tti d th SNote: Indenter Force Increase is small for large increase in cutting depth S.
Comparison of chip formation in classical TBM and in CMM (undercutting)
CMM Indentation F
L
λ
plasticrack
) len
gth,
λ
Stable crack propagation( i i )λ
L/λ increases
λ
plasticzone
flawcrack
Flaw
(c (F increases as increases)λ
Stable crack propagation(F decreases as increases)λ
2
TBMIndentation F
Force F
λ'πK
= i2
2θ
T
Infinite
= fracture toughness= pre-existing crack
= tensile strength of rock (q/K )= unconfined compressive strength Mohr Coulomb
Force, F
λ'K T
qi pθ
λ length in rock = (1+sin )/(1 sin )− φφ
φ = friction angle
Mohr-CoulombmaterialKp
Relationship between applied force on an indenter and the length of edge crack ( ) beneath the plastic crushed zone
λ
WIRTH-HDRK Continuous Mining Machine (CMM)
Rock cuttings produced by CMM
α
σ
αq q
F
F q
l
σ
2 a q qo
0.6
Wing crack growth under compression, Fairhurst and Cook (1966) model
KK FII oo = = = = 2π σ− π/ q ; a (1)0.64n
m F l lσ 0 ; crack closed
Crack growtharrested
0.4
K KI IC c
=
σ σCrack growth if
Rewriting (1)
> >[ ]
(after Germanovich and Dyskin, 2000)
n
mq
/ σ
00
0.2
5 10 15 20K
KK
I
I ICI
o o
oo=
= =
=
= σ − π
πa 2
2 0.64
nn
m
where n m
m
l / a q /
0
K
if ~
0.9
1σ
>KK
II IC o=
= = πn nn
ml / a
0K
if
Supression of crack growth by small confining pressure
( )>
aa
1
2σ
σσmax
minmax
3 0
= 0.6=‘harm onic’ hole:
a2a ξ2
σ i
2
= 2.0
1.0
3 .0
q u
σmean
a1
2 σ min
1
Inelastic region
aa
1
2= 1.5 ~ 1/0.6
a ξ 1
σ max
region
ξ 2
= 2.0
1 0
3.0 q u
σmean
a
a
1
2
a ξ 2
1
2 1 .0 σ min
Inelastic
a ξ 1region
“Harmonic Hole” Optimum shape of hole to minimize the extent of excavation damage?
aa
1
2
σσ
minmax
= 0.6=Mechanical propertiesThermal properties
φ = 30κ = 3 W/m C ooo q = 1 MPac = 844 83 J/kg Cσ
σminmax
Wall temperature T=100 C (fixed)
o
aa
1
2
= 1.2 m= 2.0 m
= 1.19 MPa= 0.71 MPa
o
o 3-6up
t
q = 1 MPac = 844.83 J/kg Cρ = 2650 kg/mα = 40 x10 1/ C
σσ
maxmax
100 o
a1 0o
σ σmin
min
Contours of t t
a2
1 0o
Extent of failure region (after 15 days)
temperature(after 15 days of heating)
“Harmonic Hole” Effect of thermal load on excavation damage
(a) (b)
(c) (d)
“Harmonic hole” - Kolar Gold Fied (Caw, 1956)
a)
b)
(a) Empirical stand-up time relations for tunnels as a function of span and(a) Empirical stand-up time relations for tunnels as a function of span and rock quality defined by the Rock Mass Rating System; (b) Theoretical stand-up time based on joint strength deterioration with time (after Fakhimi 1992).
Fracture mechanics models, a) single rock bridge under far fieldnormal and shear stresses; b) multiple rock bridge under far fieldnormal and shear stresses; b) multiple rock bridge under far fieldnormal and shear stresses.
Cohesion and safety factor as a function of time.From Kemeny, 2003
Constitutive behavior at contact between two particles
Linear contact lawContact law
Kn UnFn .=F
F
F
n
ncbond
breaks
(tension)Slip condition
nKs Us
F
F
F
F n
s
s
s
.
=
=Δ Δ
Fnμ
1
Kn
U
contact bondDeformation is assumed to occurat contact point only.
Un
Un
slip model(overlap)
contact
Contact bond Parallel bond2r
Fscontact bond
Fsmax
1
Ks n
Us
slip modelwhen U > 0
Models adhesion over vanishinglysmall area of contact point (does not resists bending moment); breaks ifnormal or shear force exceeds bond strength.
Models additional material deposited after particles are in contact (resists bending moment); breaks if normal or shear stress exceeds bond strength.
s
Particulate mechanics (PFC -Cundall and Potyondy)
Parallel-bonded stress corrosion modelParallel bonded stress corrosion model
2 r(t)Stress-dependent corrosion reaction occurs at micro-tension sites.
vCorrosion reaction occurs at the periphery of parallel bonds and removes bond material
vReaction rate is determined by local driving force and local energy barrier.
1) Driving force: bond stress (sigma)2) Energy barrier: micro-activation stress (sigma_a)
Express corrosive-front velocity as:
v =v =
β 1β 2 ( c)/σσe
0 if sigma < sigma_a
PFC simulation of time dependent weakening of rock
Cluster of particles
Compression testpon assembly ofparticles
PFC model of a particulate assembly (rock) and compression loading
y = -24.386x + 23.491R2 = 0.8779y = -20x + 2014
y = -31.056x + 31 0142
R2 = 1ds
]
12
14
y = -13.541x + 13.503R2 = 0.738
R2 = 0.7583
g( t
) f[s
econ
d
8
10
log
6
LdB (Pc=0)
2
4
Tuff (Pc=0) Assumed
Tuff (no fail)Tuff (Pc=5 MPa)LdB (Pc=5 MPa)
σ σ/
0.00
0.2 0.4 0.6 0.8 1.0
σ σ/ c
Static Fatigure Curves used as input to the UDEC analysis of collapse over time at Yucca Mountain
MagnitudeMagnitudeof displacements[meters]
0.0
After 1 year After 100 years
0.25
> 0.5
Collapse predicted around
After 1,000 years After 10,000 years
drifts over 10,000 years at Yucca Mountain assuming Category 2 Tuff Static Fatigue Curve.Note: thermal and seismic effects have not been considered.
MagnitudeMagnitudeof displacements[meters]
0.0
After 1 year After 100 years
0.25
> 0.5
Collapse predicted around
After 1,000 years After 10,000 years
drifts over 10,000 years at Yucca Mountain assuming Category 2 Lac du Bonnet Static Fatigue Curve.Note: thermal and seismic effects have not been considered.
v
Upper block
Bonds breaking in shearingBonds breaking
Upper block
Bonds breaking in tension
Rough joint
u
0.6
al
cem
ent,
u [m
m]
0.2
0.4
Dilation response of the joint
Horizontal displacement u [mm]
Verti
cadi
splac
0.0
0.0
0.5 1.0 1.5 2.0 2.5 3 .0
sample as measured duringthe PFC shearing experiment(no rel ative rotation of upper andlower blocks allowed)
Use of PFC to examine the shear behavior of a rough joint.
Horizontal displacement, u [mm]
220
230
Pink altered zone
230
240
250
-1 MPa
6 MPa
240 Level
250
260
270
Dep
th (m
) Major fracture zone
28 MPa
21 MPa24 MPa
RL
shaf
t
N l t ti F t
270
280
290
-1 MPa Cataclastic zonewith clay gouge
30 MPa19 MPa
? ? UR
Normal stress acting on FractureZone as determined from triaxialovercore measurements
290
300
30 MPa
Observed variability of normal stress across a thrust fault at the URL, Pinawa,Canada.
Subsidence?
Collapsed P
Monitoringsystem
prock
oad
capa
city
,
Pillar Pillar
Pillar degradationrate
Stable roomarch
Pilla
r l
Ti t ( )rate
P di ti f l t (50 100 ) b h i f Pill d L E ti i b d d i
Time, t (years)
Prediction of long-term (50~100 yr) behaviour of Pillars and Large Excavations in abandoned mines
Log10 seismicmoment
7.3
Time
55 days
Log10 seismicmomentTime
Mar 2 10:51 1992Feb 27 00:32Feb 22 14:14Feb 18 03:56
3 90 hours
4.80
4.353.90
3.453 00
Feb 18 03:56Feb 13 17:38Feb 9 07:20Feb 4 21:01Jan 31 10:43Jan 27 00:25Jan 22 14:07Jan 18 03:49
3.93.00Jan 13 17:31
Breakout PFC analysis
Keyed highly compactedclay-block bulkhead
3.5 m
12 MS7
3.5 m high by4.4 m wideelliptical tunnel
2.6 m
12 m Concretebulkhead key
MS4
MS13MS9
MS5
MS8
Mine-byTunnel
MS2
MS11MS7
Keyed concretebulkhead
MS3MS6
MS10
MS9 MS8
MS12
MS14MS1
AE13-16
AE1-4
AE5-8Claybulkhead key
σσ
13
Steel supportSand filler Highly compacted
backfill
Pressure supplyand withdrawalheaders
Room425
MS15
MS16
σ
σ
1
2
Transparent Earth. URL TSX AE (Courtesy of Professor Paul Young, University of Toronto)
C l iConclusions
• Remarkable and continuing computational and• Remarkable and continuing computational and observational advances.
• Predictive models should build from empirical rules; p ;rationalize and extend them.
• Advances will benefit all of rock mechanics.
• Waste isolation offers unprecedented opportunity to “know” the rock mass —Professor Müller’s prime concern.concern.
Slides left out
D/R = 0.1 D/R = 0.2
σ σt o/ σ σt o/Maximum Maximumσ σσ σ
σo
t to o/ /0.00
0.25
0.00
0.25
Maximum Maximum= 0.05 = 0.1
Dσt
σo0.50 0.50
R
/D/R = 0.4 D/R = 1
D/R = 2
σ σ σσ σ σ
t t to o o
/ / /Maximum Maximum Maximum
= 0.25 = 0.5 = 1.0
σ σt o/0.0
0.5
σ σt o/0.0
0.5
σ σt o/0.0
0.5
1.0 1.0 1.0