TWO-PARAMETER CHARACTERIZATION
OF CRACK-TIP FIELDS
DURING THERMAL TRANSIENTS
by
DAGMAR E. HAUF
S.B., Mechanical Engineering
Massachusetts Institute of Technology, 1992
Submitted to the Department of
Mechanical Engineering
in Partial Fulfillment of
the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
April, 1994
(
Massachusetts Institute of Technology 1994
Signature of Author.
--
_-
l
D4epartment of Mechanal Eineering
e,
Certified by
.
) Mach, 1994
David M. Parks
Professor of Mechanical Engineering
-.,.~,...
,Thesis
Supervisor
Accepted
by
-
-
Deartm
Eng.
..
ti
MASSACHt,$jM'NSTf!iU.E
Or c' J.,mrlfv
S
.i
.,
b
LIBRARIE.
_,
Ain A. Sonin, Chairman
Committee on Graduate Studies
Two-Parameter Characterization of Crack-Tip Fields
during Thermal Transients
by
Dagmar E. Hauf
Submitted to the Department of Mechanical Engineering
on April 5, 1994, in partial fulfillment of the requirements for the
Degree of Master of Science in Mechanical Engineering
Abstract
Ample evidence exists that the elastic T-stress, which is the first non-singular term
of the WILLIAMS
eigen-expansion (1957), is the rigorous "second" crack-tip parameter
in well-contained yielding. However, we have no knowledge that the two-parameter
(J-integral and T) characterization has been examined for the case of transient thermal loading. In view of the marked sensitivity of both ductile (void growth) and
brittle (cleavage) fracture mechanisms to crack-tip stress triaxiality, along with the
observed strong dependence of stress triaxiality on T, we investigate the effect of the
T-stress on the plane strain crack-tip fields during a thermal transient. Numerical
techniques are developed to follow the evolution of the T-stress.
To verify the two-parameter characterization under transient thermal loading, we
make use of the plane strain elastic-plastic Modified Boundary Layer (MBL) solutions of WANG(1991). The MBL solutions provide a family of stress states whose
members can be identified by the value of the T-stress. If the two-parameter characterization holds, the elastically calculated T-stress value at any instant in time during
the thermal transient should allow us to uniquely identify a member of the MBL family of crack-tip fields, and that particular MBL field should predict the behaviour of
the corresponding elastic-plastic full-field plane-strain solution.
The ability of the MBL solutions in predicting the stress state of the elastic-plastic
solutions is exceptional considering that the MBL loading is based on the first two
terms of the WILLIAMS
eigen-expansion, which neglects the presence of thermal strains
in its derivation. Simple extraction of the T-stress variation from an elastic analysis of
the problem allows us to predict the triaxiality of the stress state in the elastic-plastic
full-field solution.
Thesis Advisor: David M. Parks
Title: Professor of Mechanical Engineering
2
Acknowledgements
I would like to thank Professor David M. Parks for his incredible support, particularly
during the past six months of this project. I truly enjoyed working for you. Thank
you for being a friend.
I am very grateful to the very dear friends I have made during my time at MIT:
Simona, Richard, Aimee, Stefano, Michele. Thank you for always being there for me.
I would like to thank Attilio for all his help and support during the initial stages of
this project. I wish he could have been here for the rest.
Another, this time professional, thank you to Simona for letting me use her postprocessing program; thanks to Omar, Apostolos and Jian for their system support;
and thanks to Christine for battling the scanner problems with me. Of course thanks
also to the rest of the Mechanics of Materials group. It's been fun working with you!
Final thanks go to my family in Germany for lending me their support from abroad.
This work was supported by the Office of Basic Energy Sciences, Department of
Energy, under Grant #DE - FG02 - 85ER13331. Computations were performed
on SunSparc workstations obtained under ONR Grant #0014 - 89 - J - 3040. The
ABAQUS finite element program was made available under academic license from
Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI.
3
Contents
6
.......................
List of Figures ...........
List of Tables ...................
12
1...............
1 Introduction
13
13
....
1.1 Near Crack-Tip Stress Fields ...................
1.2 Two-Parameter Characterization of Near Crack-Tip Fields ......
1.3
. . . . . . . . . . . . . . . .
Scope of the Work
.
15
.........
18
2 The Elastic T-Stress
2.1
Introduction
2.2
Modified
21
.
. . . . . . . . . . . . . . . .
Boundary
Layer Solutions
. . . . . . . ......
21
. . . . . . . . . . . . . . .....
22
3 Numerical Methods
3.1
Introduction
3.2
Evaluation
3.3
38
. . . . . . . . . . . . . . . .
of the T-stress
.
. . . . . . . . . . . . . . .
3.2.1
Near-Tip Fields and Line-Load Solutions.
3.2.2
The Interaction
Integral.
38
. . . . . . . ......
......
39
.......
. . . . . . . . . . . .
...
..
.
39
41
Finite-Element Formulation for the Domain Integral Method: TwoDimensional
Implementation
. . . . . . . . . . . . . .
4
. . .
..
45
3.4
The Computer Program T-STRESS .
48
4 T-Stress due to Thermal Transients
4.1
Introduction.
58
. . . . . . . . . . . . . . . . . . .
58
4.2 Problem Statement.
. . . . . . . . . . . . . . . . . .
59
4.3
Elastic Analysis ...........
. . . . ..
. . . . .
60
4.3.1
Methods.
. . . . . . . . . . . . . . . . . .
60
4.3.2
Results .
. . . . . . . . . . . . . . . . . .
61
4.4
. ..
..
Elastic-Plastic Analysis .
..
4.4.1
. . . . . . . . . . . . . . . . . .
Methods.
4.4.2 Results .
4.5
..
...........
Loading
..
..
.
...
..
..
..
..
.
.65
65
............. .. 66
Combined Thermal and Mechanical Loading . . . . . . . . . . . . . .
68
4.5.1
68
Methods ...........
4.5.2 Results .
...........
69
5 Conclusions and Future Work
111
5.1
Conclusions.
. 111
5.2
Future
. 113
W ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
121
Appendix
127
A
Derivation of Line-Load Strain and Displacement Fields .......
.
127
B
Temperature Distribution under Convective Cooling .........
.
131
C
Listing of the Program T-STRESS
.
133
5
..................
List of Figures
1.1
Geometry of near crack-tip region .....................
20
1.2
Contour definition of the J-integral.
20
2.1
Plastic Zones, with axes normalized by the characteristic length scale
(KI/Y) 2 (Y = ay: yield strength), of various specimens and BL solution at KI = 0.6 Y a1 /2 (Larsson and Carlsson, 1973).
28
2.2 Plastic Zones at a load level KI = 0.6 Y a1/2 for actual geometries and
the MBL solution (Larsson and Carlsson, 1973) .............
29
2.3
2.4
2.5
2.6
2.7
...................
The variation of stress triaxiality near a crack tip with respect to r =
T/cry in a nonhardening material; ao = uy (Du and Hancock, 1991).
30
Engineering stress vs. plastic strain curve in uniaxial tension, multilinearly modeled from experimental data of ASTM A710 Grade A steel
and used in flow theory continuum finite-element solutions. Also shown
is the power law fit used in Wang's work (1991) for a moderately hardening material with n = 10, and v = 0.3. .................
31
Angular variation of near-tip normalized hydrostatic stress for various
values of r in plane-strain MBL solutions with n = 10 (Wang, 1991).
32
Normalized crack-opening stress profiles in plane-strain MBL solutions
for hardening exponent n = 10, for various values of r. The stresses
marked with circles are HRR-singularity fields (Wang, 1993) ......
33
Comparison of relative accuracy of three-parameter fit to the normalized crack-opening stress a distance 2J/c% ahead of the crack tip in
plane-strain MBL solutions for n = 10 with respect to the finiteelement
solution
(Wang,
1991)
.
. . . . . . . . . . . . . ..
6
. ..
33
2.8
2.9
Variation of maximum plane-strain plastic zone size, normalized by the
plastic zone size at T = 0, at various values of (Wang, 1991) in a
modified boundary layer formulation. ................
34
Angular variation of near-tip equivalent plastic strain for various values
of r in plane-strain MBL solutions with n = 10 (Wang, 1993). ....
35
2.10 Verification of the two-parameter characterization
36
............
2.11 Normalized crack-opening stress profiles in plane-strain MBL solutions
for hardening exponent n = 10 at various values of 633 (KI = constant,
= 0) (Wang, 1993)
37
............................
(a) Line-load applied in the direction of crack advance along the crack
front. (b) Crack tip contour F on the plane locally perpendicular to
the crack front where s represents the location of the crack tip. (c)
Tubular surface St enclosing the crack-front segment. .........
51
3.2
Crack front advance. ...........................
52
3.3
Nodal point numbers of 2-D isoparametric 8-node element
3.4
(a) Schematic of section of body and volume V in the xl - x2 plane
containing a notch of thickness h. (b) Schematic of notch face when
the function Aalj is interpreted as a virtual advance of a notch face
. . . . .. . .
segment in the direction normal to x2 ..................
3.1
and plateau
functions.
. . . . . . . . . . . . . . .
52
.......
. .
.
53
54
3.5
Pyramid
3.6
Flow chart of the program T-STRESS. .................
55
3.7
Domain definition.
56
3.8
Normalized T-variation with respect to the crack depth in SEN specimen under remote tension or bending. .................
57
4.1
Strip with edge crack, subjected to thermal shock along x = 0 .....
71
4.2
Mesh used in elastic analysis of the problem.
4.3
Transient temperature distribution in the strip for
Dt/w
2 )
..
............................
72
.............
= 5 (Fo =
. . . . . . . . . . . . . . . . . . . . . . .
7
73
4.4 Transient temperature distribution in the strip for P/ = 100 (Fo =
Dt/w2 ).
74
Stress intensity factors K1 for a/w = 0.0375 as a function of nondimensional time Dt/w 2 for various values of the Biot number.
75
Stress intensity factors If for a/w = 0.05 as a function of nondimensional time Dt/w 2 for various values of the Biot number .......
75
Stress intensity factors K* for a/w = 0.1 as a function of nondimensional time Dt/w 2 for various values of the Biot number .......
76
Stress intensity factors KI for a/w = 0.3 as a function of nondimen.....
sional time Dt/w 2 for various values of the Biot number .
76
Stress intensity factors K for a/w = 0.4 as a function of nondimensional time Dt/w 2 for various values of the Biot number .......
77
4.10 Stress intensity factors I, as a function of nondimensional time Dt/w 2
for various crack lengths ( = 100)....................
77
4.11 Non-dimensionalized T-stress values for a/w = 0.0375 as a function of
nondimensional time Dt/w 2 for various values of the Biot number. . .
78
4.12 Non-dimensionalized T-stress values for a/w = O.C)5 as a function of
nondimensional time Dt/w 2 for various values of thle Biot number. . .
78
4.13 Non-dimensionalized T-stress values for a/w = 0.1 as a fiunction of
nondimensional time Dt/w 2 for various values of thle Biot number. . .
79
4.14 Non-dimensionalized T-stress values for a/w = 0.3 as a function of
nondimensional time Dt/w 2 for various values of thle Biot number. . .
79
4.15 Non-dimensionalized T-stress values for a/w = 0.4 as a function of
nondimensional time Dt/w 2 for various values of thle Biot number. . .
80
4.16 Non-dimensionalized T-stress values as a function of nondimensional
. .. I. .. . . . .
time Dt/w 2 for various crack lengths ( = 100).
6
4.17 Position of temperature front relative to crack-tip for a/w = 0.0375 at
three values of normalized T ( = 100). Temperature front locations
correspond to (init - E)/(init - Oamb)= 0.01........
81
4.5
4.6
4.7
4.8
4.9
8
,,
4.18 Position of temperature front relative to crack-tip for a/w = 0.05 at
three values of normalized T ( = 100). Temperature front locations
(i,it
-)/(Oiit
-=.....
correspond
to (iit -
)/(O)iit- Oamb)= 0.01. ...........
82
4.19 Position of temperature front relative to crack-tip for a/w = 0.1 at
three values of normalized T ( = 100). Temperature front locations
correspond
to
Oamb)
0.01 .
83
4.20 Position of temperature front relative to crack-tip for a/w = 0.3 and
a/w = 0.4 at maximum value of normalized T ( = 100). Temperature
frontlocationscorrespond
to (iit - )/(init - Oamb)= 0.01. . - -
84
4.21 Non-dimensionalized KI vs. T for a/w = 0.0375, for various values of
the Biot number. Arrows indicate the direction of traverse .......
85
4.22 Non-dimensionalized KI vs. T for a/w = 0.05, for various values of
the Biot number. Arrows indicate the direction of traverse .......
86
4.23 Non-dimensionalized KI vs. T for a/w = 0.1, for various values of the
Biot number. Arrows indicate the direction of traverse .........
87
4.24 Non-dimensionalized KI vs. T for a/w = 0.3, for various values of the
Biot number. Arrows indicate the direction of traverse .........
88
4.25 Non-dimensionalized KI vs. T for a/w = 0.4, for various values of the
Biot number. Arrows indicate the direction of traverse .........
89
4.26 Non-dimensionalized KI vs. T for various relative crack depths (/3 =
100)
90
.....................................
4.27 Non-dimensionalized T-stress values at maximum KI, for various Biot
numbers and relative crack depths. The dashed line indicates the ex-
pected asymptotic T*-values(a/w - 0) .................
4.28 Non-dimensionalized T-stress values as a function of nondimensional
time Dt/w 2 for a/w = 0.0375 ( = 100, E(Oiit - Oamb)/a, = 1.32).
Symbols indicate instances in time for which crack-opening stress profiles are obtained .............................
91
92
4.29 Non-dimensionalizedKI vs. T for a/w = 0.0375 (/3 = 100, Ea(Oiit ,amb)/-y = 1.32). Symbols indicate instances in time for which crackopening stress profiles are obtained ....................
9
93
4.30 Position of temperature front relative to crack-tip for a/w = 0.0375 at
four values of normalized T ( = 100). Temperature front locations
correspond
to (iit -
)/(Oiit- Oamb)
= 0.01............
4.31 Mesh used in elastic-plastic analysis of the problem.
.
.........
4.32 Near-tip region of mesh shown in Fig. 4.31 ...............
94
95
96
4.33 Normalized crack-opening stress profiles at four instances of time and
corresponding MBL solutions (a/w = 0.0375,
= 100, Ea(Oiit -
Oamb)/y = 1.32). KI are stress intensity factors normalized according
to Eq. (4.2) .................................
97
4.34 Enlarged view of Fig. 4.33. Ki are stress intensity factors normalized
according to Eq. (4.2).
..........................
98
4.35 Normalized crack-opening stresses at r/(J/a,)
= 2 vs. T at four
instances of time and corresponding MBL results (a/w - 0.0375,
0/ = 100, Ea(Oi,it - eamb)/Uy = 1.32) ..................
99
4.36 Variation of hydrostatic stress at four instances of time and correspond-
ing MBL solutions(a/w = 0.0375,/ = 100, Ea(Oiit
- O,zmb)/Uy=
1.32). K* are stress intensity factors normalized according to Eq. (4.2). 100
4.37 Variation of equivalent plastic strain at four instances of time and
correspondingMBL solutions(a/w = 0.0375,/ = 100, E'a(Oiit -
= 1.32). K are stress intensity factors normalized according
to Eq. (4.2) .................................
Oamb)/ry
101
4.38 Strain components ahead of the crack at four instances of time during
the thermal transient (a/w = 0.0375,/3= 100, Ea(Oiit - OEmb)/Oy=
1.32).
...................................
102
4.39 Stress intensity factors K for a/w = 0.0375 under combined thermal
and mechanical loading as a function of nondimensional time Dt/w 2
(/ = 100, oa = ay/2, Ea(Oinit - Oamb)/Oy = 1.32) ...........
103
4.40 Non-dimensionalized T-stress values for a/w = 0.0375 under combined
thermal and mechanical loading as a function of nondimensional time
Dt/w2 ( = 100,o = y/2, Ea(O it - )amb)/ay = 1.32) ......
103
4.41 Non-dimensionalized KI vs. T for a/w = 0.0375 under combined
thermal and mechanical loading as a function of nondimensional time
Dt/w2 ( = 100,or = y/2, Ea(Oinit- Oamb)/y = 1.32) ......
10
104
4.42 Normalized crack-opening stress profiles at four instances of time and
corresponding MBL solutions (a/w = 0.0375, = 100, o' = y/2,
Ea(Oinit-Oamb)/y = 1.32). K[ are stress intensity factors normalized
105
. . . . . . . . . . . . . . . . . ......
to Eq. (4.2).
according
4.43 Enlarged view of Fig. 4.42. KIQare stress intensity factors normalized
. . . . . . . . . . . . . . . .
to Eq. (4.2).
according
106
......
.
= 2 vs. r at four
4.44 Normalized crack-opening stresses at r/(J/u)
instances of time and corresponding MBL results (a/w = 0.0375,
= 100,oC = cry/2,Ea(O6jt- eamb)/y
4.45 Variation of hydrostatic
sponding MBL solutions
Ea(Oit - Oamb)/oy =
malized according to Eq.
=
107
1.32). ..........
stress at four instances of time and corre/2,
( a/w = 0.0375, / = 100, ac =
1.32). K are stress intensity factors nor..
(4.2). .....................
108
4.46 Variation of equivalent plastic strain at four instances of time and
= a%/2,
correspondingMBL solutions( a/w = 0.0375,/ = 100, aCO
Ea(Oinit-Oamb)/Oy = 1.32). KI are stress intensity factors normalized
according to Eq. (4.2) ...........................
109
4.47 Strain components ahead of the crack at four instances of time during
the thermal transient ( a/w = 0.0375,P = 100, a' = cy/2, ELa(Ojet-
110
Oamb)/Jy= 1.32). .............................
5.1
Variation of cleavage fracture toughness with r (Beteg6n and Hancock,
1990)
5.2
Variation of fracture toughness KIC with temperature for A533 B Steel
and Corten,
(Sailors
5.3
115
....................................
1972)
.
. . . . . . . . . . . . .
T-K
..
116
Variation of crack-tip temperature during a thermal transient (a/w =
0.0375,/3= 100) .........................
5.4
. . ..
. ....
117
Schematic variation of fracture toughness as a function of r and O and
-
spiral .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
slice of Fig. 5.4 .....................
5.5
Two-dimensional
5.6
Prediction of the behaviour of a 1-in. deep crack during an overcooling
transient for a reactor vessel with (a) severe and (b) light embrittlement
(Stahlkopf,
1982).
.............................
11
118
119
120
List of Tables
3.1
2-D Shape Functions ...........................
3.2
Domain
independence
of J-integral.
12
. . . . . . . . . . . . .
46
...
50
Chapter 1
Introduction
1.1 Near Crack-Tip Stress Fields
Near crack-tip conditions of both linear elastic fracture mechanics (LEFM) and nonlinear elastic fracture mechanics (NLEFM) conventionally are characterized by single
parameters, the stress intensity factor KI in the case of LEFM and the J-integral in
the case of NLEFM.
One of the basic assumptions behind the application of LEFM to elastic-plastic materials is small-scale yielding (SSY). SSY requires the zone of plastic deformation at the
crack-tip to be much smaller than any relevant specimen dimension (cf. ASTME-399).
Then, the stress state outside the plastic zone, but away from the specimen boundary, can be characterized by the first singular term of the WILLIAMSeigen-expansion
(1957)
aij
-fij(),
with KI = cv/r,
(1.1)
where r and 0 are polar coordinates centered at the crack tip as shown .in Fig. 1.1,
f,¥(0) are universal angular variations of the respective stress components, or is the
nominal stress, a is the crack length, and c is a dimensionless function which depends
on the relevant geometrical dimensions. The entire stress field at the crack tip is
13
known when the stress intensity factor in Eq. (1.1) is known. That is, KI completely
defines the crack-tip conditions. K is said to determine the strength of the dominant
elastic singularity.
HIUTCHINSON (1968)
and RICE and ROSENGREN (1968) independently
showed that the
J-integral characterizes crack tip conditions in a nonlinear elastic material. The Jintegral is defined as the energy release rate in a nonlinear elastic body containing
a crack and essentially measures the scale of crack-tip deformation. The line integral expression of J for any contour F encircling the crack tip in a counterclockwise
direction (see Fig. 1.2) is given by
J = F Wdy- Ti ads,
(1.2)
where W is the strain energy density, Ti are components of the traction vector acting
outward on the contour F, ui are the displacement vector components, and ds is a
length increment along the contour F. The J-integral is independent of the path of integration around the crack provided that there is no crack face traction or body forces
and the near crack-tip region undergoes proportional loading. Under SSY conditions
the contour F can be chosen to fall within the region in which the KI-characterized
fields hold, thus allowing a relationship between J and KI to be established as
K2
E"J~~1
E'
~~~~(1.3)
where E' = E for plane stress and E' = E/(1 - v2 ) for plane strain, E is the Young's
modulus, and v is Poisson's ratio.
showed that in order for J to remain path
and RICE and ROSENGREN
HUTCHINSON,
-
independent, the product of stress and strain must vary as 1/r near the crack tip.
In a near-crack tip region, where the plastic strains are much larger than the elastic
strains, the equivalent stress and strain are related by a power law in the form
Cy
c (
14
Yy
) ,
(1.4)
where ay is the effective tensile yield strength, ey = ay/E is the associated tensile
yield strain, n is the strain hardening component, and a is a constant.
Based on
J 2-deformation theory of plasticity and small strain asymptotic analysis, the near
crack-tip fields within the plastic zone are then given as
aij(r,O)
J
_=_
ay( aeyI)n+lij(O,n)-
eij(r,O)
aey(
-
J )
gj+lej(0,n) -
~rHas
R
(1.5)
(1.6)
where In is an integration constant that is a function of n, and &ai and ij are dimensionless functions that depend on 0 and n, and on whether plane strain or plane
stress prevails in the vicinity of the crack tip. Eqs. (1.5) and (1.6) are called the HRR
singularity.
The HRR stress singularity does not apply to points too close to the crack tip; that
is, within a region approximately 2 - 3 crack-tip opening displacements (McMEEKING,
1977). This restriction applies because the asymptotic HRR fields neglect the finite
geometry change at the crack tip. Since the analysis leading to the HRR stress singularity is based on a nonlinear elastic material model and small geometry change, the
HRR fields also do not apply where significant elastic unloading or nonproportional
loading due to the interaction of thermal and mechanical loads, for instance, exists.
1.2
Two-Parameter Characterization of Near
Crack-Tip Fields
Whether a near-crack-tip field is HRR-dominated, that is, whether the asymptotic
HRR fields constitute a sufficiently accurate description of the crack-tip field to a radius which includes the fracture process zone, depends strongly upon geometry, loading condition, and strain hardening. The geometry dependence is especially strong
15
for low-hardening materials in plane strain. The varied ability of attaining HRR dominance at crack tips of different specimens is attributed to the difference in crack-tip
constraint (WANG,1991). A widely used constraint parameter is the stress triaxiality,
which is defined as the ratio of hydrostatic stress, am =
stress,
e,.
Ockk,
to the Mises equivalent
Under plane strain conditions, high levels of crack-tip triaxiality are asso-
ciated with: (a) essentially all states of well-contained yielding; and (b) virtually all
load levels in specimens with sufficiently deep cracks under predominately bending
load. Conversely, low levels of triaxiality occur in large-scale yielding and fully-plastic
flow of single edge-cracked and center-cracked specimens under predominant tension
loading, as well as in shallow edge-cracked specimens under bending (PARKS,1992).
A low level of triaxiality generally manifests itself in high crack-tip ductility and high
macroscopic toughness.
McMEEKINGand PARKS(1979) proposed that crack-tip stress
triaxiality remained sufficiently "high" providing
J <
(1.7)
Ycr
where c,, is a "critical" lower limit, and I is a characteristic length parameter such as
the uncracked ligament in a deeply-cracked specimen. Since J is directly related to
the applied load magnitude, Eq. (1.7) can be interpreted as a limit for applied load
to ensure HRR dominance.
Although the effect of specimen geometry and strain hardening on the attainment of
HRR dominance is a relatively well known subject, there is no established criterion
to define a crack-tip field as "HRR-dominated". Hancock and co-workers (BETEG6N
& HANCOCK, 1991; AL-ANI & HANCOCK, 1991; Du & HANCOCK, 1991) found that the
eigen-expansion of nearelastic T-stress, which is the second term in the WILLIAMS
crack-tip fields (1957), may be valuable in quantifying the deviation from HRR singularity fields. BETEG6N & HANCOCK(1991) showed that the variation of stress tri-
axiality in various plane-strain specimens, as measured locally by the crack-opening
stress profiles, can be adequately predicted by introducing T as the constraint pa16
rameter, even up to large scale yielding. AL-ANI & HANCOCK(1991) demonstrated
that the deviation from SSY of near-tip crack-opening stress-profiles in plane-strain
edge-cracked geometries can be accurately predicted by the two-parameter based solution (J and T) of the respective problems up through (and beyond) moderate-scale
yielding. HANCOCKet al. (1993) showed how
T(-
T/lo,) and associated changes in
crack-tip stress triaxiality change the initial slopes of resistance curves in A710 steel
specimens of varying geometry. WANG (1993) verified the two-parameter characterization of elastic-plastic crack-tip fields (J and T) with a 3-D study of stress fields
along the crack fronts of surface-cracked plates (SCPs). Recently, O'DowD & SHIH
(1991, 1992) proposed a similar approach with Q as the second parameter, measuring
the deviation in crack-tip stress triaxiality from a particular reference value of triaxiality. In small- to moderate-scale yielding Q is isomorphic to the T'-stress (PARKS,
1992). However, Q can strictly be obtained only from detailed near-crack-tip fields
based upon elastic-plastic finite-element solutions. In 3-D applications the required
computational resources and data preparation and reduction make direct implementation of this approach to routine applications all but prohibitive (PARKS, 1992). On
the contrary, the elastic T-stress can be evaluated easily from an elastic solution prior
to a specific, elastic-plastic solution (WANG, 1991).
Although extensive work has been done on the evaluation and correlation of the
elastic T-stress, we have no knowledge that the two-parameter characterization has
been examined for the case of transient thermal loading.
In view of the marked
sensitivity of both ductile (void growth) and brittle (cleavage) fracture mechanisms
to crack-tip stress triaxiality, along with the observed dependence of stress triaxiality
on r, we investigate the effect of the T-stress on the plane strain crack-tip fields during
a thermal transient.
The study has potential applications in the power generation
industry where the design and operation of conventional and nuclear power plants
are founded on rigorous safety requirements. For example, the rules of ASMESection
11 (1974) require the consideration of both mechanical and thermal loads.
17
1.3
Scope of the Work
Chapter 2 introduces the concept of the elastic T-stress and discusses its use as a
second parameter to characterize the near-crack fields. To verify the two-parameter
characterization under transient thermal loading, we make use of the work of WANG
(1991). He performed plane strain elastic-plastic finite element analyses using a Modified Boundary Layer (MBL) formulation under the assumption of small geometry
change at various values of T and constant K or J. In this way he generated a
family of stress states whose members can be identified by the value of the T-stress.
In other words, the same T-stress value, regardless of the specimen geometry and
loading type which produced such value, corresponds to a definite crack-tip stress
field which is a member of the MBL family of fields (WANG, 1991). That is, if the
two-parameter characterization holds for the case of transient thermal loading, the
elastically-calculated T-stress value at any instant in time during the thermal transient should allow us to uniquely identify a member of the MBL family of crack-tip
stress states, and that stress state should predict the behaviour of the corresponding
elastic-plastic full-field plane-strain solution.
Before we can compare the near-crack-tip stress fields with predictions based upon
J and T, the variation of the elastic T-stress during the thermal transient needs to
be evaluated. In Chapter 3 we give a brief overview of numerical methods used to
accurately and reliably evaluate the T-stress as a function of specimen geometry and
loading conditions. Specifically we will use the interaction integral of NAKAMURA&
PARKS
(1992) to track the evolution of the T-stress during a thermal transient. To
evaluate the interaction integral, the so-called domain-integral formulation (LI et al.,
1985; SHIH et al., 1986) is adopted.
Its computational implementation in a post-
processing program for the commercial finite-element-code ABAQUS is discussed.
Given the numerical tools described in Ch. 3, we obtain the variation of the T-stress
in SEN specimens of various crack lengths (Chapter 4). The severity of the thermal
18
shock is varied by varying the Biot number. The Biot number is given by fi = h/wk,
where h is the heat transfer coefficient, w is the width of the specimen, and k is the
material thermal conductivity, and essentially measures the resistance of the body to
heat transfer. A Biot number of infinity corresponds to a step change in temperature
on the surface of the specimen under consideration. We assume that the thermal
stress problem is quasi-static and that inertia effects are negligible. We finally test
the validity of the two-parameter characterization for transient thermal loading using
the steps discussed above.
19
o22
X2
1
X,
Figure 1.1: Geometry of near crack-tip region.
n
Figure 1.2: Contour definition of the J-integral.
20
Chapter 2
The Elastic T-Stress
2.1
Introduction
Under the conditions of SSY, the near-crack-tip stress and deformation fields are
characterized by the stress intensity factor KI, and the crack problem can be solved
by using a boundary layer (BL) approach. This approach considers a semi-infinite
crack in an infinite body and replaces the actual conditions of boundary loading by
the asymptotic boundary conditions that
aij =
K1
fi (0) as r -+ o.
(2.1)
The magnitude of KI is taken from the solution of the elastic boundary value problem
modeling the elastic-plastic specimen. The extent to which the near-crack-tip fields of
the BL solution and those of an actual specimen agree with each other is an indication
of the validity of the KI-based one-parameter characterization of crack-tip fields under
SSY conditions.
LARSSON& CARLSSON
(1973) performed plane-strain elastic-plastic finite-element anal-
yses on four commonly-employed test specimens exhibiting a variety of crack-tip constraint under SSY conditions and compared their computed plastic zones with the
21
appropriately-scaled plane-strain BL solution. They found significant discrepancies
with the BL formulation, even within the range of loads allowed by the ASTMStandard Test Method for Plane-Strain Fracture Toughness of Metallic Materials (E-399).
At the maximum permitted load levels, the computed maximum plastic zone sizes for
the center-cracked specimen and the double edge-cracked specimen, for example, were
greater than that of the BL solution by
25%, respectively (see Fig. 2.1).
50% and
The plastic zones for the different cases would have coincided had the elastic-plastic
crack-tip state been determined by KI alone. LARSSON& CARLSSONshowed that the
observed differences in plastic zone sizes of the specimens, loaded to identical "small"
KI-levels, are due to specimen-to-specimen differences in the T-stress, the second
term of the WILLIAMS(1957) eigen-expansion
The T-stress is not singular as r
-
of near-crack-tip
elastic stress fields.
0, but it can alter the elastic-plastic crack-tip
stress state, thus modifying the crack-tip plastic zone. Like KI, the T-stress is a function of geometry and loading conditions, and is proportional to the nominal applied
stress (LARSSON & CARLSSON(1973); LEEVERS & RADON (1982); etc.). For instance, in
shallow-cracked specimens under predominately tensile loading, the proportionality
constant is negative, while deeper-cracked specimens under bending often have a less
negative or even positive T-stress.
2.2
Modified Boundary Layer Solutions
LARSSON & CARLSSONapplied
boundary tractions corresponding to the stress fields of
the first two terms in the WILLIAMS eigen-expansion,
alil(r,0)
a21(r,0)
o 1 2(r,0)
22
(r,0)
_
K
VJ27r
[ f(0)
f 2 L()
f2(0)
1
f 22 (0 )
J
1
T1
0
2.2)
0
on the same plane-strain domain as that in the previous BL solution. The constant
term "Tll" in Eq. (2.2) represents the T-stress. The T-stress was obtained from two
elastic finite-element solutions; the first was a full-field solution with actual specimen
22
and loading, the second a BL solution with applied boundary tractions corresponding
to Eq. (1.1). The T-stress was then calculated by averaging the difference of the
respective x-direction near-tip stresses; that is,
T
a"e(r
)-
(2.3)
BL(r, 7r)
where uallec(r,7r) is the "11"-component stress of the full-field solution, and aBL(r, r)
the stress of the BL solution, respectively. The Modified Boundary Layer (MBL)
Solutions using this two-parameter description of the loading transmitted to the cracktip region were in essentially exact agreement with those of each of the corresponding
specimens for all loads up to those giving KI = 0.6oay/V (see Fig. 2.2).
Recently, extensive work has been done on two-parameter characterizations of nearcrack-tip fields. BETEG6N& HANCOCK(1991) analyzed near-crack-tip
fields of plane-
strain specimens having positive, zero, and negative T-stress. Deep within the plastic
zone, the crack-opening stress profiles (tensile stress distribution on the plane 0 = 0
ahead of the crack) of the specimens closely followed those of the corresponding
MBL prediction up to large-scale yielding. The MBL family of solutions are strongly
affected by the sign and magnitude of the T-stress. A substantial reduction of crackopening stress (relative to SSY) is seen for r < 0; moderate stress elevation above
SSY is observed for r > 0. The effect of the T-stress on the large geometry change
deformation field within two crack-tip opening displacements has been discussed by
BILBY et
al. (1986). Negative T-stresses were shown to reduce the level of maximum
hydrostatic stress ahead of the crack. The MBL solutions predicted this decrease
inside the plastic zone quite accurately. AL-ANI& HANCOCK
(1991) analyzed planestrain crack-opening stress in edge-cracked specimens of various crack depths. Remote
tension or bending loads, ranging from SSY to large scale yielding, were applied to
simulate different levels of crack-tip constraint. The crack-opening stresses were in
excellent agreement with the MBL prediction using the calculated elastic-plastic J
of the specimen and the elastically-scaled T-stress. Du & HANCOCK
(1991) correlated
23
the near-crack-tip hydrostatic stress with the T-stress in a non-hardening material
(see Fig. 2.3). They showed that the limiting Prandtl field, consisting of constant
state regions on the crack flanks connected by centered fans of angular extent 7r/2 to a
constant state region ahead of the tip, was obtained only for sufficiently positive values
of
T.
In contrast, when r < 0, an elastic zone of angular extent > 7r,/4 emerges from
the crack flanks, cutting into the extent of the centered fan within which hydrostatic
stress builds up. The more negative r becomes, the greater is the reduction in fan
extent and crack-tip stress triaxiality (PARKS,1992). Computational results for the
circumferential variation of near-tip stress triaxiality with r are shown in Fig. 2.5 for
the case of strain hardening exponent n = 10 (WANG,1991).
WANG (1993) investigated the influence of the T-stress on the crack-tip opening stress
in a variety of SCPs. He used a plane-strain MBL formulation by applying displacement boundary conditions dictated by KI and T on a semi-circular domain.
His
results are based on a deformation theory plasticity power-law material model ex-
hibiting the tensile stress/strain relation
(
E(
for
)n
<
for >
o
u
;
1<
n < o,(2.4)
with ey - cy/E, and a set of material constants representing a moderately hardening
material, namely, E = 0.0025, n = 10, and v = 0.3. This relation roughly fits
the material data used in the present work (see Fig. 2.4).
Fig. 2.6 shows the
variation of normalized crack-opening stress (22 at 0 = 0) vs. normalized distance
at various values of r. The case r = 0 (thick solid line) is the opening stress profile
at SSY, while the open circles indicate the HRR field. The crack-opening stress
deviates considerably from the SSY stress with decreasing r, while the deviation at
high positive r is less pronounced. At any point outside the crack-tip blunting zone,
the stress profiles for different values of r are roughly parallel to each other, which is
consistent with the observation of BETEG6N& HANCOCK
(1991). This suggests that
the deviation from SSY is essentially independent of normalized distance from the
24
crack tip.
Thus, the variation of the plane-strain crack-opening stress with respect to r, at any
normalized distance in the range 1 < r/(J/ay)
< 6, can be fitted in the following
three-parameter form
4722(r/(J/a8);
) 0S'S(r/(J/ay)) + Antr+ Bnr2 +
B /
))22J)=
o'y
where A,, B, and
Cn,
Cn
a'y
3
(2.5)
are constants dependent upon the strain hardening exponent
n. This three-term polynomial fit was suggested by WANG(1991). Fig. 2.7 compares
WANG'Sresults with the finite-element solution at r = 2J/ay. The fitting parameters
are An = 0.6168, Bn = -0.5646, and Cn = 0.1231 for ey = 0.0025, n = 10, and
v = 0.3.
RICE (1974), SHIHet al. (1993), and WANG(1991) noted a strong variation of plastic
zone size with the T-stress. Fig. 2.8, from the work by WANG,shows the plastic zone
size at various values of T, normalized by the SSY plastic zone size at (r = 0). The
results of the simple shear band yielding model (band shear traction = ry,,a constant)
of RICE(1974) are shown for the purpose of comparison. His results are given in terms
of T and the equivalent tensile strength cry= /ry, with the plastic zone size radius,
rp, as
7rsin2(1+ cos)
rp 64 (1/v + r sin cos )2
(K,2
(2.6)
The angle d in Eq. (2.6) is measured from the crack plane. In practice,
is an implicit function of r,
respect to
X
=
in Eq. (2.6)
(T), which is obtained by maximizing rp with
= 70.6° (RICE, 1974).
at fixed r; at T = 0, this procedure results in
Also shown are the results of SHIHet al. (1993) for a moderately hardening material
(n
=
10). At
= 0, rax
0.15(KI/oy)
2
rSSY,
which differs only slightly from
the generally used nominal plastic zone size, (1/2ir)(Ki/ay)
zone size grows monotonically with decreasing r, reaching
25
2.
The maximum plastic
50rSSY
when r = -1.0.
p
Positive values of r cause the plastic zone size to first decrease, then increase, reaching
lOrSSY when
= 1.0.
The effect of the T-stress on the equivalent plastic strain sp at a distance from the
tip equal to r = 1.22J/oy is shown in Fig. 2.9 (WANG, 1993). The thick solid line
is the SSY solution (r = 0). Negative r is associated with a large increase in sP (at
r = -1.0, the peak sp has increased by
80% compared to the corresponding peak
SP at r = 0) and a shift of the peak to the forward section ( < 90°). A slight decrease
of peak P is observed at a r-value between 0.2 and 0.4. For
increases (for r = -1.0 by
location of the peak at
T
> 0.4, the peak E
25% compared to the peak value at
= 0), while the
= 1.0 shifts back toward the cracked flank.
Based on this noted sensitivity of plastic zone size and orientation to the sign and
magnitude of the T-stress, HAUF et al. (1994) recently formulated a Modified Effective
C(rackLength for plane strain by including effects of T into the definition of the standard effective crack length. They demonstrated that their formulation consistently
extends the load range for which accurate predictions of compliance, J-integral, and
crack-tip constraint are obtained in several plane-strain specimen geometries.
Based on these observations, then, we establish the following criterion to determine
the suitability of the two-parameter characterization of near-crack tip fields during
thermal transients. That is, we considerthe two-parameter (J and T) characterization
to hold if the magnitude and sign of the T-stress given at a particular instant in time
during the thermal transient
allows us to identify a member of the MBL family of crack-tip stress states
and the crack-tip fields of that particular MBL solution suitably describe
the behaviour of the corresponding full-field plane-strain solution in the
range I < 7/(J/oy) < 6.
26
We do not expect the MBL fields to precisely match those of the corresponding
full-field solution, since the WILLIAMS
eigen-expansion, on which the MBL loading
is based, requires the absence of body forces and thermal strains in its derivation.
Nevertheless, we expect that major features of the respective fields will correspond.
Our approach is schematically depicted in Fig. 2.10.
It should be noted that under plane-strain conditions (33 = 0) the presence of thermal
strains results in finite mechanical tension/compression out-of-plane strains tangential
to the crack front which have an effect on the near-crack-tip stress fields. PARKS
(1991) suggested a generalized form of Eq. (2.2) to express the linear-elastic stress
distribution in the vicinity of the crack front; that is,
r11
'12
013
C21
22
023
(731 32
fii(O)
I
033
f2(0)
T11 0 Tll13
f3(0)
f21(0)
f22(0)
f23(0)
f31(0)
f32(0)
f 3 3 (0)
+
0
T31
0
0
0
T33
. (2.7)
Based on Eq. (2.7), WANG(1993) investigated the effects of out-of-plane strains on
the near-crack-tip fields in the context of three-dimensionality of crack fronts for the
special case T13= T31 = 0 and finite T33 by varying the out-of-plane strain
same value of
T.
33
at the
Figure 2.11 from his work shows the effect of out-of-plane strain on
the crack-opening stresses. Clearly, the out-of-plane strain has a much smaller effect
than the T-stress. At r = 2J/a,, the stresses decrease by - 3% at
compared to the value at e33 = 0. The stress profile for
rotated compared to that at
crack-opening stress at
33/Ey
33
=
633/ey
33/Ey
= -0.9
= -0.9 seems slightly
0. That is, for a distance r > 2J/cy the normalized
= 0 decreases more gradually compared to results at
E:33/Ey= -0.9.
27
'IL
2
r
sJ
-2
0
to
r
o
a.
Coordinate /(K:/Y2
Center-cracked specimen
Double-edge-cracked specimen
Bend specimen
~--~-Compact
tension specimen
Boundary layer solution.
Figure 2.1: Plastic Zones, with axes normalized by the characteristic length scale
(KI/Y) 2 (Y = , : yield strength), of various specimens and BL solution at KI =
0.6 Y a 1/2 (Larsson and Carlsson, 1973).
28
U
I
0
C
0
U
0o
ao
-004
Coordinote x/a
-002
0
0 02
0 04
0 06
0 08
Coordinatex/o
(b)
(a)
0
4'O
.
0
C
061
c0
o
o
0
0
U
u
0
*004
·
-002
0
0 02
004
0 06
0 06
08
Coordinate x/a
Coordinate x/a
(d)
(c)
Figure 2.2: Plastic Zones at a load level K = 0.6 Y al/2 for actual geometries and
the MBL solution (Larsson and Carlsson, 1973).
29
2.4
2.0
0
1.6
0
1.2
0.1
0.4
0.;
.1.0
.0.S
0.0
O.S
1.0
Figure 2.3: The variation of stress triaxiality near a crack tip with respect to r = T/ay
in a nonhardening material; a, = ay (Du and Hancock, 1991).
30
800
600
AL
::
b 400
vrn
V)
or
200
n
0.00
0.02
0.04
0.06
Plastic
0.08
strain P
0.10
0.12
Figure 2.4: Engineering stress vs. plastic strain curve in uniaxial tension, multilinearly modeled from experimental data of ASTM A710 Grade A steel and used in
flow theory continuum finite-element solutions. Also shown is the power law fit used
in Wang's work (1991) for a moderately hardening material with n = 10, and v = 0.3.
31
4
3
b
.
2
b
1
-1
0
30
60
90
120
150
180
8 (deg)
Figure 2.5: Angular variation of near-tip normalized hydrostatic stress for various
values of T in plane-strain MBL solutions with n = 10 (Wang, 1991).
32
5
4
b
Cb
3
CQ
b
2
1
1
0
2
3
4
6
5
r/(J/loY)
Figure 2.6: Normalized crack-opening stress profiles in plane-strain MBL solutions
for hardening exponent n = 10, for various values of r. The stresses marked with
circles are HRR-singularity fields (Wang, 1993).
4.0
I
I
I,
I
I
r (.fr -
3.5
I
I
I
I
'
.
.
_
I
I
.
.
= P
b
3.0
C
b
o
o
3-parameter fit
2.5
n
FEM solution
L
-1 .0
I
I
I
I
I
-0.5
I
,
,
I
-
0.0
I
-
I
-
I
-
I
-
I
-
0.5
.
-
-
-
1.0
T
Figure 2.7: Comparison of relative accuracy of three-parameter fit to the normalized
crack-opening stress a distance 2J/ay ahead of the crack tip in plane-strain MBL
solutions for n = 10 with respect to the finite-element solution (Wang, 1991).
33
102
I
I
,
I
I
,
I
I
I
I
I
I
I
I
I
I
I
I
-
101
0
'%'
II
I-
P4
100
-
-
-
x
o data from Wang, n = 10 (1991)
0
curve (Hauf et al., 1994)
-fitted
10 - 1
data from Shih, O'Dowd & Kirk, n = 10, (1993)
yielding model (Rice, 1974)
·-
.shear-band
, ,
1 r-2
-1
.O
LI I ,
-0.5
,
I
]
0.0
I[
0.5
l,
l_
1.0
7 = T/y
Figure 2.8: Variation of maximum plane-strain plastic zone size, normalized by the
plastic zone size at = 0, at various values of r (Wang, 1991) in a modified boundary
layer formulation.
34
90
60
30
n
0
30
60
90
120
150
180
8 (deg)
Figure 2.9: Angular variation of near-tip equivalent plastic strain for various values
of r in plane-strain MBL solutions with n = 10 (Wang, 1993).
35
1
E
V.) Q a
<-0
i
U'
a,u
(I)
0(1)
ul </
.
C,'
=1 0a
2
l-I
-J
z±
w,<
-4-----n
0
._
N
E
t)
..f
cd
N
+
-
I
t4-=
z (,
0
1Ir
k
-0
c
E .°
v-,
C
c
>II
E .
O Y
)
0
b
to
0v
02
=
I
cn
O
'
F')
v
-j
z
2
,v
v
E
LU
\
\
0
a_
0
0
_,,
-J
\\\\\\\\\\\\\
\
'
'
'
'
-0
0
bO
bo
dL
c a)
.o E
a)
:L
_'\\'
-C
* .--
(D'~~~~~~.
Zo
H0
E
0
4-j
yH
36
0
-4
'I
.0 '
\
w. .
Q)
._6
CS
._.V
0
cv
V)
(n
V
0
._
),
4
b
3
b
2
i
0
1
2
3
4
5
6
r/(J/oY)
Figure 2.11: Normalized crack-opening stress profiles in plane-strain MBL solutions
for hardening exponent n = 10 at various values of E33(KI = constant, r = 0) (Wang,
1993).
37
Chapter 3
Numerical Methods
3.1
Introduction
Several methods are available in the literature for evaluating the elastic T-stress in 2-D
specimens under various loading conditions. The most obvious method can be derived
directly from the definition of the T-stress; that is, by using Eq. (2.2). Assuming
the near-crack-tip stress fields can be adequately represented by the first two terms
in the
WILLIAMS
(1957) eigen-expansion, the T-stress can be obtained as
T = aeC,(r0)where aec(r
3.1)
0) is the xl-direction normal stress in the near-crack-tip region of an
actual specimen and loading (WANG,1991). LARSSON
& CARLSSON
(1973) determined
T in this way using two elastic finite-element solutions. The first solution was obtained from an actual specimen and loading analysis. The second was a boundary
layer formulation of a circular domain with a semi-infinite crack, in which the traction
boundary conditions corresponding to the second term on the RHS of Eq. (3.1) were
imposed. The magnitude of K1 applied in the elastic BL solution was determined
from the solution of the elastic boundary value problem modeling the elastic-plastic
specimen. LEEVERS& RADON(1982) calculated the coefficients of the WILLIAMS eigen38
expansion using a variational formulation. WANG& PARKS(1992) obtained approximate estimates of the T-stress distribution in a wide range of surface-cracked plates
under tension and bending using the line-spring method. SHAM(1991) computed the
2-D elastic T-stress using second-order weight functions. Based on a theorem due to
ESHELBY (CARDEWet al., 1984), KFOURI(1986) evaluated T in terms of the difference
in J-integral of two finite-element solutions. The first elastic finite-element solution
was generated from an actual specimen and loading analysis. A second solution was
generated by superposing a point load solution to the first solution. The elastic T was
then obtained from a relation involving the J-integrals of the two solutions. Since
J can be accurately evaluated from a moderately refined elastic finit;e-element analysis, T can be obtained with a mesh much less refined than that of the LARSSON
&
CARLSSON
method. Recently, NAKAMURA
& PARKS(1992) extended this method to
3-D crack fronts using a domain interaction integral. We use the interaction integral of NAKAMURA& PARKS to evaluate the T-stress in 2-D plane-strain
specimens
under transient thermal loading. In the following paragraphs, which closely follow
the derivation presented by NAKAMURA
& PARKS,we describe the near-tip fields and
line-load solutions needed in the evaluation of the interaction integral, present the
resulting expression for the T-stress, and finally use the domain-integral method (LI
et al., 1985; SHIHet al., 1986; MORAN& SHIH,1987, for example) to represent the
interaction integral in a form suited to numerical implementation.
3.2
3.2.1
Evaluation of the T-stress
Near-Tip Fields and Line-Load Solutions
In an isotropic linear-elastic body containing a crack subject to symmetric (mode I)
loading, the leading terms [up to 0(1)] in a series expansion of the stress field very
39
near the crack front are
11a
22
=
cos
V'-F
KI
=
1- sin sin
2
2
0o
cos 2
Ki_
a33 =
KI
~
2/27
O'13 =
sin
30
1 + sin - sin2
2v cos
a12 =
o
0
2
+ T
2
2/
+ T3 3 ,
(3.2)
0
30
2
2'
cos - cos -
23 = 0
where r and 0 are the in-plane coordinates of the plane normal to the crack front,
KI is the local stress intensity factor, and v is the Poisson's ratio. Here x1 is the
direction formed by the intersection of the plane normal to the crack front and the
plane tangential to the crack plane. The associated strain field is given by
==E
cos
i -2-sinsin
- + (T-vT33)
+a(O
-Oinit)
cos-2 1-2v +sin-2 sin--2)
22
EE
27rr
633 =
E33h t3
+
(1 +v)
= a(O -E)
0
K
612
=
623
=
E
E
2
2
(T+ T33)+ a(O- )iit)
(3.3)
) + 33
0
30
/ sin cos cos -
24/=7r
E
2
613 = 0.
where a is the constant linear thermal expansion coefficient, Oiit the reference temperature value at the undeformed state, and 63 and e33 are the mechanical and the
thermal strains in the x3 -direction, respectively. The terms T(= T11 ) and T33 are
the amplitudes of the second order terms in the three-dimensional series expansion of
the crack-front stress-field in the xl- and x3 -directions. We can decompose T3 3 into
T33 = vT + a*, where a* =
mE = [33 - a( - Oinit)]E. This decomposition of
T3 3 is different from the one presented by NAKAMURA & PARKS (1992) as they did not
consider thermal strains in their derivation.
40
rToextract the T-stress, an auxiliary solution corresponding to a plane-strain lineload applied along the crack front in the direction of crack extension is used. The
solution is a special case of a line-load symmetrically applied at the apex of a wedge
of included angle 2r (TIMOSHENKO
& GOODIER,1970). Suppose that a line load with
magnitude f (force per unit length) in the xl-direction is locally applied along the
same crack front segment. Then the stress field in the crack-tip region is given by
aLr
Cos
3
0
11rr
2
r3
C2
13
= -- cos 0 sin2
=sin
-
v
--
=
Cos 0
f2
f cos0
7rr
=
23
(3.4)
sin
=
Here, the superscript "L" denotes the line-load solution. The strain ()
and displace-
ment (uI) fields corresponding to the stress field in Eq. (3.4) are readily calculated
(see Appendix A for a detailed derivation).
The above solutions are valid at any
material point as long as r, its radial distance to the crack front, is sufficiently small
compared to other relevant physical dimensions.
3.2.2 The Interaction Integral
Consider the line-load, fi = fi(s),
to be applied along the crack front as shown
in Fig. 3.1(a). In the figure, s is an arc-length-measuring parameter representing
the location of the crack-tip on the crack front, and
pi(s)
is a unit vector giving the
direction formed by the intersection of the plane normal to the crack front and the
plane tangential to the crack front at s. By superimposing the actual field, Eq. (3.2),
with the field due to the line-load application, Eq. (3.4), a local conservation integral
41
is introduced as
I(s)Ik(s) = lim
[Ljnnkrs -
i
_
CL
]
3U
A path F(s) surrounds the crack front at s and lies in the plane perpendicular to
the crackfront at s. The components ni are those of a unit vector lying in this plane
and normal to the tangent to F, as shown in Fig. 3.1(b). The limit (F -
0) must
be preserved in three-dimensional problems. However, the shape of the path may be
arbitrary as F shrinks onto the tip.
Now suppose that pi(s) is given in the local xl-direction and that the path F is circular
with radius r. Then as the limit is taken (r - 0), the stress fields in Eqs. (3.2) and
(3.4) (and their associated kinematic fields) become applicable in the integrand. After
substituting the fields into the integral, the interaction integral was evaluated with
the help of the program Mathematica
I()
=
TM
as
[T(s) (1- 2) + ECrAO(s)- v*]
= -[T(s) (1- v2)+ EaAO(s)(1 + v)- vEe33(s)]
where AO(s) = O(s)-
(3.6)
Oiit is the temperature difference between the crack-tip
temperature and the reference temperature of the specimen. In the integration, the
terms in the crack-tip fields, Eq. (3.2), containing KI cancel out exactly, and only
the non-singular terms in Eq. (3.2) contribute to I(s). Solving for T(s), we obtain
T(s) =( -v
)f
AO\(s)(+v) +v
33(s)]
(3.7)
Under isothermal conditions , AO = 0, and Eq. (3.7) reduces to the expression given
by NAKAMURA& PARKS.
From a computational point of view, Eq. (3.5) is not suitable for evaluating I(s)
since accurate numerical evaluation of limiting fields along the crack front is difficult.
42
Here the so-called "domain-integral formulation" (LI et al., 1985; SHI:H
et al., 1986) is
adopted. An approximate expression for I(s) may be obtained as follows. The total
interaction energy I(s) released when a finite segment L of the crack front advances
an amount Aa
Ik(S)
(see Fig. 3.2) at the point s in the direction normal to the crack
front is given by
AaI(s) = Aa j I(s)lk(s)iAk(s)ds.
(3.8)
On employing Eq. (3.5) in Eq. (3.8) we obtain
I(s)
=
-=
Lk(S) [lim j
-s
JtL
23
[u
[iji 6Lnk
-
L __
cij3
L Oui ]dl
n - an ] d
ds
Oireinkj n - 3i-nj
a 9Xk ni J k(s)dS,
'7j aXk
(3.9)
where for the case of a sharp crack St is the tubular surface enclosing the crack front
segment as shown in Fig. 3.1(c), and the limiting process consists of shrinking the
"tube" radius to zero. For simplicity, we will model the crack as a notch with notch
thickness h in the following (see Fig. 3.4) and require h
0 in the sharp crack
configuration of interest. The surface of the notch consists of faces SA and SB, with
normals along ±x 2-directions, respectively, and a face with a normal in the x 1 - X3
plane.
Next we identify the arbitrary closed surface S with the surface S1 + S+ + S_ - St
(see Fig. 3.4) and introduce the continuous functions qk defined by
qk
k on St
0 on S1
(3.10)
otherwise arbitrary
Requiring
qk
to be sufficiently smooth in the volume V and invoking the divergence
theorem, the expression for the interaction integral over a domain/volume is,
I
Ji(s)
{
-LaU
43
qk}
-(
23)k
dV.
(3.11)
The stress strain relations for a linear material are
pq =
pqrs
er where Cpqrs = Crspq
are the elastic moduli. Hence,
--= ij klj
5
Cij klE
.
k iij ' '
L
=
=L
L
ij'
m
(3.12)
Using Eq. (3.12), we can express the second term on the RHS of Eq. (3.11) in terms
of the mechanical strains; that is,
L = c~[jj[
O a(O
O'i-E
init)ij
jt6j]
-
=
(3.13)
Using this result and invoking equilibrium
a'ij,j + bi =
L
=
t3 ,3
(3.14)
Oi
(3.15)
i,
where bi is the body force per unit volume, we obtain the desired expression for the
interaction integral
-
I(s) =
)LX _'Y[(sk
()
JV(s)
)oujOk
o
ija qI
ijij
+a(T
9
Lk
-o'ii -b
a-OXk
q
Xk) qdV
(3.16)
We now let h -
0 to obtain the desired expression for the interaction-energy decrease
when a local segment of the crack front advances by Aa
Ik
in its plane. In deriving
Eq. (3.16) we have assumed the crack faces to be traction free. It should be emphasized, that with the presence of thermal strain, the domain of integration for Eq.
(3.16) must include the near-tip region (r -, 0+).
The domain expression, Eq. (3.16) gives the interaction energy per unit of crack
advance over a finite segment of the crack front. In order to calculate the T-stress
with the help of Eq. (3.7), however, we need a local value of the interaction energy.
To a first approximation this value is obtained by assuming that I(s) is constant over
44
some region of the crack front L. This allows us to bring I(s) outside the integral
sign in Eq. (3.8') to yield
I()
- I(s)
Ik(s)#k(s)d.
(3.17)
(s)/ J lk(s)k(s)ds.
(3.18)
or
I(s)
Thus, once I(s) has been calculated from Eq. (3.16) using the computed stress (aij)
and deformation field (ui) of a boundary value problem, and the exact auxiliary
solution of the line load, Eq. (3.4), with unit magnitude (f = 1), the local value of
T-stress at the crack front point s can be determined with Eq. (3.7).
In the case of a plane strain line-crack oriented along the xl-axis (that is, a straight
crack front of length L), AO(s) - AO, T(s) - T, and the interaction energy in
Eq. (3.18) is given by I = I/L.
3.3
Finite-Element Formulation for the Domain
Integral Method: Two-Dimensional Imple-
mentation
The finite-element formulation of the area/volume integral method has been discussed by LI et al. (1985) in the context of the two-dimensional biquadratic (9-node)
Lagrangian element and the three-dimensional triquadratic (27-node) Lagrangian element. We outline their implementation in the context of the two-dimensional isoparametric 8-node element, for which the nodal point numbers are shown in Fig. 3.3.
45
The 2-D expression of the interaction integral in Eq. (3.16) is given by
I(s) =
v
OijA
a,
Ouv
2iJal)
au
i_
..-
(
i
i -ql]
- il)
+(
l
OO
x
.
ui
dA,
12(s)
()
(3.19)
where 2(s) is the contribution to (s) due to thermal strains and body forces.
For isoparametric elements, the coordinates (l,
displacements
in the physical space and the
(u1 , u 2) are written as
8
8
Xi = E NKXiK,
i = 1,2,
ui = E NKUiK,
(3.20)
K=1
K=l1
where N
2)
are the biquadratic shape functions (see Table 3.3), XiK are the nodal
coordinates and UiK are the nodal displacements.
Table 3.1: 2-D Shape Functions
N1 (,7 ,
C)
NV,(,
¢)
N3 (,,
C)
C)
N 4 (,
N 5(y, C)
= (-1/4)(1 - )(
- C)(1
()(1
)(1
)(1
= (-1/4)(1 + r)(1 = (-1/4)(1 + r/)(1+
= (-1/4)(1 - q)(1 +
= (1/2)( -
)(1
+
+
+
+
+
+ ()
)
- )
- ()
/)(1 - C)
= (1/2)(1- ()(1 + 77)(1
N7 (q, ) = (1/2)(1- )(1 +-7)(1
N 8 (q, C) = (1/2)(1- ()(1 + ()(1
+ C)
+
-
-
)
)
t
In 2-D, a suitable choice for the vector qi, i = 1,2 is (ql,q2) = (ql(xl, 2),0), where
1 on St
ql(xl,x2)
=
0 on S1
otherwise arbitrary
(3.21)
Consistent with the isoparametric formulation, we take ql within an element as
8
q = ZNIQ I,
I=1
46
(3.22)
where QI are the nodal values for the Ith node. From the definition of ql, Eq. (3.21),
if the
Ith
node is on St, QIi = 1, whereas if the
Ith
node is on S1 , Q1I = 0. In the area
between St and S1 , QlI will be taken to vary between 1 and 0. It may be noted that a
particular choice of interpolation scheme for QlI is equivalent to selecting a particular
weighting scheme for the field quantities between r and C1. Two possible choices for
ql are a "pyramid" function and a "plateau" function (see Fig. 3.5). For the pyramid
function, q = 1 at the crack tip, ql = 0 on the edge of the domain and ql varies
linearly between the peak and the rectangular edges. In this sense an equal weighting
has been applied to l1(s) in Eq. (3.19) (ql/OXk = piecewise constant), while the
thermal contributions 12(s) have been linearly weighted. The plateau ql function has
a value (or height) of unity everywhere in the domain except in the outermost ring of
elements. Here the value (or height) decreases linearly from unity to zero within one
element width. The "pyramid" and "plateau" ql-functions, together with the virtual
crack extension interpretation of ql as a translation in the xl-direction, is depicted in
Fig. 3.5. It may be noted that in the subdomain where ql is constant (corresponding
to a rigid translation of the subdomain in the context of the virtual crack extension
technique) there is no contribution from Il(s) to the domain integral.
Using Eqs. (3.20) and (3.22) and the chain rule, the spatial gradient of ql within an
element is given by
Oq,
5
2
ONID9k
.a7 IE E
I=1 k=1
a7k a0 Q r ,
(3.23)
where ?0k/0 Xj is the inverse Jacobian matrix of the transformation, Eq. (3.20).
With 3 x 3 Gaussian integration, the discretized form of the domain expression for
the interaction energy for plane-strain problems is
9=
u
LOui
Oql
all elements in A p=lzl
njeL -Oq
323a
l
+
( ao
x,
47
r
II
- b
u
)
q
Jq
dt (Ok
det
-
j
wp (3.24)
3.24)
Here the quantities within { }p are evaluated at the 9 Gauss points, and wp are the
respective weights.
Eq. (3.24) has been implemented in a postprocessing program for the commercial
finite-element code ABAQUS. Some of the main features of the program are discussed
in the next section.
3.4
The Computer Program T-STRESS
The program T-STRESS was developed using the framework of the computer program
DOMAIN (SOCRATE,
1990). Particularly the mesh topology features of DOMAIN were
used.
To evaluate the interaction integral in Eq. (3.24) a domain has to be defined. It should
be noted that the program T-STRESS is developed only for rectilinear meshes and
is thus limited to rectangular domains. A domain is defined by a set of nodes along
the symmetry line of the specimen fixing the base and thus the width of the domain,
and a number of element layers fixing its height (see Fig. 3.7). Once the domain
is defined, the program assigns the chosen perturbation field, plateau or pyramid,
and calculates the interaction energy according to Eq. (3.24) over all elements in
the domain. The T-stress is subsequently calculated using Eq. (3.7). The domain
variables used in the evaluation of Eq. (3.24) are read from the ABAQUS results file.
A listing of the program is given in Appendix C.
In addition to calculating the T-stress for the cases of mechanical and thermal loading,
the program is equipped to calculate the J-integral using an expression similar to Eq.
(3.24). Furthermore, the temperature and stress distributions can be obtained along
the symmetry line of the specimen. The flow chart in Fig. 3.6 shows a rough outline
of the program.
48
We tested our code by calculating calibration factors for T in a SEN specimen subjected to remote tension and bending for various crack depths. Analogously to the
KI-calibration functions k, i.e., KI = k(a, w)
Q, the normalized T-stress can be
expressed as r = ((a, w) Q)/o,, where (a, w) = [iN,tM]1 are T-stress calibration
functions of the specimen under consideration (WANG& PARKS,1992), and w is the
width of the specimen. Q (components = [N, M]) is the vector of generalized load
amplitudes with work conjugate displacements q (RICE, 1972). Using second-order
weight functions, SHAM(1991) has tabulated values for the T-stress calibration functions for various specimens over essentially the entire range of relative crack length
a/w (0.1 < aw < 0.9).
Fig. 3.8 shows the results obtained with T-STRESS compared to SHAM'sdata. The
agreement is exceptional for all relative crack depths.
The two curves practically
coincide. Domain independence obtained with T-STRESS was checked by comparing
our J-integral values to the J-integral values provided by ABAQUS. The results for
six different domains (six different contours in the case of ABAQUS), normalized by
aa/E,
where ae = aE(Oinit - Oamb)/(1
-
2 ),
for a SEN specimen of relative crack
depth a/w = 0.1 are given in Table 3.2. The smallest domain contains two elements
adjoining the crack tip; the second domain, which includes the first domain and the
adjoining layer of elements, contains eight elements. Domains three through six are
also assembled in this fashion. The variation of J over the six domains is less than
2%, an indication of the overall accuracy of the calculation.
1The superscripts N and M denote tension and bending, respectively.
49
M
M
X
0
CD
O
C
cq
C11
cq~
m
CT
CN
X0
r-
t0
CN
c-
.
m
o
o
X
C-1
o10
X
cq
0
0
X0
cl
0
Cl1
o0
c IM
Oe3
o-
cq
MC
Cl
cq
C
C)
cqCl
m
Ci
00
-
o-
0Q)
-
o-
c7
-
©
t
t
0
-
-
O5
o~
03
o
O
00
00
-
.:
Cl1
Cl
E) U
.o
m
50
_
D
C
f
f
(a)
n
(b)
(c)
Figure 3.1: (a) Line-load applied in the direction of crack advance along the crack
front. (b) Crack tip contour F on the plane locally perpendicular to the crack front
where s represents the location of the crack tip. (c) Tubular surface St enclosing the
crack-front segment.
51
Aa
Aa Ik(s)
s
L
k front
Figure 3.2: Crack front advance.
4
A
h1
I PI
3
7h,
h1
1
6
6
i-------
8
I9
h,
1r
PI
I
5
1
l
r1
lI
2
Figure 3.3: Nodal point numbers of 2-D isoparametric 8-node element.
52
(1
SA
SI
(a)
X2
X1
Aa
Segment (
notch face
(b)
Figure 3.4: (a) Schematic of section of body and volume V in the x - X2 plane
containing a notch of thickness h. (b) Schematic of notch face when the function
Aalj is interpreted as a virtual advance of a notch face segment in the direction
normal to x 2.
53
C0
x
4J
x
C-·
.x
-
x
Ir>
C1
o
._
a)
00
co
$4
-..
-I
x
a)
CO
-0
et
cn I
Q)
r-4
C
Ix-.
4-)
u
Or
a)
xN
54
START
Loop over all
the domains
!
Reads ABAQUS mesh data and
assigns local numbering
GEOINP
I
1
!
DOMGEO
Finds the node/element numbers
belonging to the current domain
PERCAL
Applies and stores the chosen
perturbation field (Iplateau or
pyramid) to each r node in the domain
Loop over all
the increments
considered
Loop over all
elements of
the domain
F
Reads and stores nodal and
VARINP
elemental domain variables
Calculates contribution to interaction
DOMINT
and J integral of each element and adds
contributions for all elements in the
domain
l
---
i
,Il
DOMPRINT
Prints results
END
Figure 3.6: Flow chart of the program T-STRESS.
55
element
layers
symmetry line
._/
crack tip
base of domain
Figure 3.7: Domain definition.
56
2
I'
I ,
I
I
'
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I'
solid lines: present work
dotted lines: data from Shar
z
1
tNSN=
T/(N/w)
tSEN=T/(6M/w
z
z]
2)
I
0
I
-1
l
0.0
l
0.2
I
I
jl
0.4
I
I
,I,
0.6
0.8
a/w
Figure 3.8: Normalized T-variation with respect to the crack depth in SEN specimen
under remote tension or bending.
57
Chapter 4
T-Stress due to Thermal
Transients
4.1
Introduction
Chapter 2 showed that the T-stress (especially: negative T-stress) has a strong effect on the near crack-tip fields. Negative T-stresses (
< 0) are associated with a
substantial reduction in crack-tip stress triaxiality (compared to SSY), while positive T-stresses result in only modest elevation of triaxiality above SSY. HANCOCK
&
co-workers (1991) have shown that the variation of stress triaxiality in various planestrain specimens can be adequately predicted by introducing T as the constraint
parameter, even up to large scale yielding. WANG(1991) verified the two-parameter
characterization of elastic-plastic crack-tip fields (J and T) with a 3D study of stress
fields along the crack fronts of SCPs.
Given the numerical tools described in Chapter 3, we will examine/extend the twoparameter characterization for the case of transient thermal loading. Our approach is
as follows: First we evaluate the variation of the T-stress during a thermal transient.
Plane-strain elastic finite-element analyses for single edge-cracked specimens of varying crack depths are carried out and post-processed with the program T-STRESS for
58
this purpose. The severity of the thermal shock is varied by varying the Biot number.
Next, we repeat the analysis for one of the cases assuming elastic-plastic material
behavior. Specifically, we are interested in the crack-opening stress profiles at various
instances of time during the thermal shock. Having obtained values for T and KI as
a function of time from the elastic analysis of the problem, we are now in the position
to make predictions for the stress state using
WANG'S
MBL solutions. If the two-
parameter characterization holds, the MBL solutions will qualitatively predict the
corresponding elastic-plastic results. We finally test the validity of the two-parameter
characterization by analyzing an edge-cracked strip subjected to both thermal and
mechanical loads.
4.2
Problem Statement
The problem of interest is depicted in Fig. 4.1. Consider the edge-cracked strip of
width w and crack length a. Unit thickness in the plane is assumed. The entire strip
is initially at temperature Oinit and is perfectly insulated along the plane x = w. At
time t = 0 the surface is suddenly subjected to Newtonian convective cooling while
the surrounding temperature is kept at (ambient) temperature Oam,. The thermal
conductivity of the material is k, and the heat transfer coefficient of the fluid/solid
interface is h. The strip is assumed to be infinite, thus resulting in one-dimensional
temperature distributions at any instant of time.
We will assume that the resulting transient thermal stress problem is quasi-static;
that is, the inertia effects are negligible. A number of studies on dynamic thermoelasticity have validated this assumption (see, for example STERNBERG
& CHAKRAVORTY,
1959a,b). Thermoelastic coupling effects and temperature-dependence of thermoelastic constants are also neglected.
The material considered is ASTM A710 steel having a Young's modulus of
59
E = 207 GPa (see Fig. 2.4 for tensile stress/strain curve), density p = 7832 kg/m 3 ,
specific heat c = 0.6 kJ/kg°C, thermal conductivity k = 58.8 W/m°C, and Poisson's
ratio v = 0.3. For the elastic-plastic finite-element analyses, the material was modeled as isotropic, obeying J2 flow theory plasticity. Small geometry changes were
assumed. The flow strength was given as a function of the equivalent plastic strain,
with an initial value of ay = 470 MPa and a saturation value of 677 MPa at plastic
strain ep = 0.0538, which essentially corresponds to a strain hardening exponent of
n = 10.
4.3
4.3.1
Elastic Analysis
Methods
Taking advantage of the assumptions discussed in Section 4.2, the problem can be an-
alyzed in two parts. First, the temperature distribution in the material is determined
as a function of time. This temperature distribution is then used as input for the
subsequent stress analysis of the problem to obtain the transient fields at the crack
tip.
The mesh used in the finite-element analysis for a/w = 0.1 is shown in Fig. 4.2.
The problem is symmetric about the y = 0 line; therefore, only half of the strip
needs to be modeled. The crack-tip region is modeled by a rectangular domain for
post-processing with T-STRESS. The inset portion of the mesh contains 32 elements
across the width and 16 elements along the height. 8-node heat transfer elements
were used for the temperature analysis, 8-node plane-strain full integration elements
for the subsequent stress analysis of the problem (ABAQUS element types DC2D8
and CPE8, respectively). Finite-element analyses were performed on the SEC strip at
five different crack depths (alw = 0.0375, 0.05, 0.1, 0.3, and 0.4) and Biot numbers
(
= 1, 5, 10, 20, and 100).
60
For the temperature analysis of the problem, the minimum usable time step was
selected according to the equation
At > PCA12,
-
(4.1)
6k
where At is the time increment, p is the density, c is the specific heat, k is the thermal
conductivity, and Al is a typical element dimension (such as the length of a side of
an element). If time increments smaller than this value are used, spurious oscillations
may appear in the solution (ABAQUS, 1992).
4.3.2
Results
The temperature distributions for the cases
/3=
5 and / = 100 for various values of
nondimensional time Fo, also known as the Fourier number, are shown in Figs. 4.3
and 4.4, respectively. The Fourier number is given as Fo = Dt/w 2 , where
D = k/pc is the thermal diffusivity. Also shown in the Figure are the corresponding
analytical solutions (see Appendix B for derivation). The finite-element results match
the analytical predictions very well, an indication of the adequacy of the mesh design.
Clearly, as the Biot number decreases, the thermal gradient through the strip becomes
less severe. Accordingly, the thermal stresses and thus the stress intensity factors
decrease, as can be seen in Figs. 4.5 through 4.9. The analytical predictions of
NIED
(1983) for KI are shown in Fig. 4.7 for the purpose of comparison. The stress
intensity factors were obtained from the elastic identity, Eq. (1.3), using KI from the
J-integral values calculated with T-STRESS (K = VE-) and normalized as
IK 1
Ea((OEiit-
amb)/ral/(1
-
)(4.2)
For any given crack length ratio a/w, the nondimensional stress intensity factor increases, passes through a maximum, and then decreases as a function of Fourier number. The Biot number strongly controls the maximum stress intensity factor during
61
the thermal shock. Also greatly affected by the Biot number is the nondimensional
time at which this maximum is reached. The lower the Biot number, the later the
peak in KI.
Nondimensional stress intensity factors for various crack lengths at fixed Biot number
(/3 = 100) are shown in Fig. 4.10. As would be expected in a self-equilibrating stress
field, the maximum normalized stress intensity factor is of greatest magnitude for a
very short crack, decreasing with increasing crack length. Thus, decreasing the Biot
number has an effect on K* which is similar to increasing the crack length.
The variation of the nondimensional T-stress values during the thermal shock for the
five different crack lengths is shown in Figs. 4.11 through 4.15. The T-stress values
were calculated with T-STRESS according to Eq. (3.7) and normalized as
T* =
T
T
ECa(Oinit-
Oamb)'
(4.3)
Again, the strong influence of the Biot number in controlling the maximum values
can be observed. More interesting, however, is how differently the T-stress values
evolve for the various crack depths. This difference can be observed in Fig. 4.16,
which shows the nondimensional T-stress values for the five crack depths at fixed
Biot number (/ = 100). For the short crack depths, the T-stress passes through a
positive maximum value early during the transient, before rapidly becoming negative.
Following this rapid "dip", the T-stress reaches a maximum negative value, and finally
returns to zero at the end of the transient. As the crack length increases, the time
at which the transition into the negative regime occurs is more and more delayed.
Along with this trend, the initial maximum positive T-stress value increases, and the
maximum negative values become less negative. For the case of relative crack depth
a/w = 0.1, for example, the T-stress becomes only slightly negative towards the end
of the transient. For even longer cracks, the maximum positive T*-value finally occurs
so late in the transient and is so "positive", that a transition into the negative regime
does not take place at all. The relative decrease in maximum positive T-stress value
62
between the cases of relative crack depth a/w = 0.3 and a/w = 0.4 should be noted.
This decrease seems to be a feature of the self-equilibrating nature of' the stress fields
under consideration.
In order to explore the differences in the evolution of the T-stress for the various
crack depths, we examined the position of the temperature front relative to the crack
tip at the occurrence of maximum T*s for the various crack depths. We define a
nondimensional penetration depth, 6*, which relates the position of the temperature
front to the relative crack depth of the specimen as 6*
(6/w)/(a/w).
The location of
the temperature front itself (6) is determined by ()i.t - O(x = 6))/(Oinit -
amb)=
0.01. Figs. 4.17, 4.18, and 4.19 show the results for a/w = 0.0375, 0.05, and 0.1
for three different values of the Fourier number corresponding to the occurence of
the positive maximum value of T, the zero point, and the maximum negative value,
respectively. Fig. 4.20 shows the position of the temperature front relative to the
crack tip for aw = 0.3 and a/w = 0.4 at the respective maximum values. For the
cases a/w = 0.0375, 0.05, and 0.1, the maximum negative value occurs when the
temperature front has passed the crack tip by a distance approximately ten times its
relative crack depth (6*
10). The maximum negative value of T* for a/w = 0.1
is reached when the temperature front has practically arrived at the backface of the
specimen. The maximum positive value for all cases occurs when the temperature
front has passed the crack tip by a distance approximately half the relative crack
depth of the specimen. Given the fact that the maximum negative T-stress value for
the shallow crack cases occurs when the temperature front has passed the crack tip by
a distance several times its depth, together with the observation that the Biot number
controls the magnitude but not the sign of the T-stress, we identify the relative crack
depth as the controlling parameter of the problem. That is, the position of the crack
tip relative to the surface of the specimen determines whether the specimen sees
negative T-stress values during the thermal transient.
In terms of the.stress fields
under consideration, the relative proximity of the crack-tip stress fields to the surface
63
of the specimen, and thus the interaction of the crack-tip stress fields with the surface
stresses, defines the evolution of the T-stress during the thermal transient.
Loci of KI* vs. '* are shown in Figs. 4.21 through 4.25. Arrows indicate the direction
of traverse during the transient. For any given relative crack depth, the loci are selfsimilar for the various Biot numbers. The K
- T* loci for the five crack depths at
fixed Biot number ( = 100) are shown in Fig. 4.26. The effect of decreasing crack
depth is a rotation of the loci about the origin towards the second quadrant. For the
cases a/w = 0.0375 and aw = 0.05, the maximum value of K* occurs at a negative
value of T*. This result is important in view of the influence of negative T-stress
values on the near-crack-tip stress fields. If the two-parameter characterization holds
for transient thermal loading, the crack-opening stress profiles of the corresponding
elastic-plastic solutions will be lower than predicted by the HRR fields for these cases.
We will further investigate this aspect in the following section.
From these results it is interesting to finally plot the variation of T* corresponding
to the maximum value of K* vs. relative crack depth (see Fig. 4.27). Again, the
strong influence of the Biot number is notable. The stongest relative variation of T*
occurs for / = 100. For this case, the T*-value at maximum K* starts from -0.168
for a/w = 0.0375 and reaches 0.282 for a/w = 0.4. The variation in T* for
/3 =
1
is notably less pronounced, as the T*-value for a/w = 0.0375 starts at -0.036 and
reaches 0.057 for a/w = 0.4. Clearly, the strong effect of both high Biot numbers
and low relative crack depths, resulting in negative T*-values, has not saturated at
a/w
= 0.0375.
Based on the results of HARLIN & WILLIS(1988), we expect the
asymptotic T*-value (a/w - 0) to occur at T*
64
-0.5.
4.4
4.4.1
Elastic-Plastic Analysis
Methods
Given the results from the elastic analysis of the problem, we select the strip of relative
crack depth a/w = 0.0375 at / = 100 to examine the validity of the two-parameter
characterization. This case was chosen because it exhibited the most negative of the
T-stress values. Specifically, we are interested in the crack-opening stress profiles
at four instances in time during the thermal transient.
If the two-parameter char-
acterization holds, the MBL solutions will qualitatively predict the corresponding
elastic-plastic results. The four instances in time and their corresponding T-stress
values are indicated in Fig. 4.28. The T-stress values were normalized as r = T/ay
for comparison with the corresponding MBL results. These values were chosen in
order to examine four distinct variations in crack-opening stress profiles. The value
r = -0.221 corresponds to the maximum value of K* (see Fig. 4.29). We did not
consider any times beyond that of
KImx,
as unloading will occur after this point.
The mesh used in the analysis is shown in Fig. 4.31. In comparison to the mesh
used in the elastic analysis of the problem (Fig. 4.2), the near-tip region is modeled
by the inset shown in Fig. 4.32. There were 32 fans of elements circumferentially,
and 24 rings radially. The ratio of the radius of the outer boundary to the radius
of the first ring of elements was on the order of 103. The first ring of elements
were degenerated, so one side collapsed into a single point at the crack tip. Again,
8-node heat transfer elements were used for the temperature analysis (DC2D8) of
the problem.
Hybrid 8-node plane strain reduced integration elements were used
(ABAQUS element type CPE8HR) in the subsequent stress analysis. Hybrid elements
were used, as we observed oscillations in the crack-opening stress profiles typical of
mesh locking with full integration elements (CPE8)
65
(NAGTEGAAL
et al., 1974).
4.4.2
Results
Fig. 4.33 shows the variation of normalized crack-opening stress
Vs.
(22
at 0 = 0)
normalized distance from the tip at the four values of normalized T-stress. An
enlarged
view of Fig.
4.33 is shown in Fig.
4.34.
Also shown are WANG'S MBL
solutions. WANG'Sresults are given in the range r[ < 1.0 in intervals of 0.1. The
stresses were taken from extrapolated stresses at the nodes on the line 0 = 0, and the
radial distance r from the crack tip was calculated from the nodal coordinate input
files. The thick solid line at r = 0 is the stress profile at SSY. The stresses marked by
the big circles are the HRR singularity fields calculated from Eq. (1.5) using the field
constants given by SHIH(1983). Stresses inside the blunted zone r <- J/oa should be
ignored because the present small-strain analysis does not account for the finite strain
inside the blunting zone. Clearly, the stress profiles for the four r-values behave in the
same way as the corresponding MBL solutions. Substantial stress reduction is seen
for the two negative values of , and moderate stress elevation occurs at
T
= 0.145.
However, the stress profiles of the four r values seem to be slightly rotated compared
to the MBL results. That is, for a distance r > 2J/ay the normalized crack-opening
stress decreases more gradually compared to the MBL results. In this respect they
agree with the HRR fields, which also exhibit this gradual decrease beyond r > 2J/oy.
Neglecting this relative rotation of the four stress profiles, which may be an effect of
the out-of-plane mechanical strain, good agreement with the corresponding MBL
solutions can be noted. The MBL solutions predict the corresponding elastic-plastic
results well. The agreement is remarkable considering that the MBL loading is based
on the first two terms of the WILLIAMS
eigen-expansion, which neglects the presence
of thermal strains in its derivation. It should also be noted that the stress/strain
relationship used in the present work is only an approximation to the one WANG
(1991) used in his analysis.
The variation of the four crack-opening stress profiles with respect to r at a fixed dis-
66
tance r/(J/ay)
= 2 from the crack tip is shown in Fig. 4.35. Also shown are WANG'S
MBL solutions. Again, our results are slighty rotated compared to WANG'Sresults.
Early during the transient, the crack-opening stresses are highest, corresponding to
a value of r = 0.145. By the time the maximum value of the stress intensity factor
is reached, r has decreased to -0.221, and the crack-opening stresses have dropped
considerably below those predicted by the HRR singularity.
Fig. 4.36 shows the circumferential variation of the hydrostatic stress at a fixed
radial distance, r = 1.22J/oa, for the four r-values and the corresponding MBL
results. The thick solid line is the SSY solution (r = 0). Again, the elastic-plastic
results show good agreement with the corresponding MBL results. The hydrostatic
stresses decrease with respect to the SSY solution for the two negative values of r
and increase slightly for r = 0.145.
Fig. 4.37 shows the circumferential variation of normalized equivalent strain (eP) at
r = 1.22J/ty in comparison to the MBL solutions. The thick solid line indicates the
SSY solution. Qualitatively, the strain profiles of the elastic-plastic analysis agree
with the corresponding MBL results. For the two negative r-values, a large increase
in EP and a shift of the peak to the forward section (
< 900) can be observed.
For r = 0.145, the peak P-value slightly shifts backwards (
> 900). However,
the maximum values of the thermal shock problem are approximately 10% higher
than those predicted by the corresponding MBL results. This difference is acceptable
considering the difference in nature of the strain fields under consideration.
We examine the strain components of the transient strain field as a function of normalized distance ahead of the crack in Fig. 4.38 (0 = 0°). Fig. 4.38 (a) shows the
variation of equivalent plastic strain, eP, Fig. 4.38 (b) the "equivalent elastic" strain,
e,/E,
ekk,
where oe is the equivalent Mises stress, Fig. 4.38 (c) the hydrostatic strain,
and Fig. 4.38 (d) the variation of thermal strain, a(O - eit),
respectively.
The strain components are each normalized by the strain at yield, ey. As expected,
67
the variation of equivalent plastic strain is strongly singular at the crack tip, but
the remaining strain components are bounded. Both, the "equivalent elastic" strains
and the hydrostatic strains are finite at the crack tip due to the imposed nonhardening response for large strains (see Fig. 2.4). The hydrostatic strains decrease with
decreasing r, consistent with the behaviour of the hydrostatic stresses. The thermal strains are practically constant ahead of the crack tip, indicating only a slight
temperature variation in the range 0 < r/(J/ro) < 6.
4.5
4.5.1
Combined Thermal and Mechanical Loading
Methods
To further investigate the validity of the two-parameter approach we consider the
case of combined thermal and mechanical loading. A pressure vessel in a nuclear
reactor could see such loading conditions during a small-scale loss of coolant accident
(LOCA), for example (also termed a "pressurized thermal shock"). We apply a fixed
tensile traction equal to half the tensile yield strength to the ends of the edge-cracked
strip during the thermal shock for this purpose. In the analysis we proceed in the
same way as for the case of purely thermal loading. First, we evaluate the variation
of the T-stress during the thermal transient from the elastic finite-element solution of
the problem. Next, we repeat the analysis assuming elastic-plastic material behavior
to obtain the variation of the crack-opening stress profiles during the transient and
compare our results to the corresponing MBL solutions. For the same reasons as
before, we focus our attention on the case of relative crack depth a/uw = 0.0375. For
a pressure vessel of wall thickness 20 cm this corresponds to a crack depth of 0.75
cm, which is both significant and reasonable.
68
4.5.2
Results
The variation of the stress intensity factor and the T-stress during the thermal transient for the combined loading case are shown in Figs. 4.39 and 4.40. The presence
of the mechanical loads cause the two curves to shift upwards and downwards, respectively. These results are not surprising, as we already observed negative T-stress
values for the shallow cracks of the edge crack specimen under pure tension (Fig. 3.8).
In terms of the T - Ki locus, the additional mechanical load results in a translation of
the entire locus into the second quadrant. That is, in the presence of the superposed
mechanical loads, negative T-stress values are seen by the strip for essentially the full
duration of the transient. Thus, assuming the two-parameter approach is valid also
for the case of combined thermal and mechanical loading, we expect the crack-opening
stress profiles of the corresponding elastic-plastic solutions to be considerably lower
than those predicted by the HRR singularity.
Again, we chose four instances in time at which we obtain the crack-opening stress
profiles from the elastic-plastic analysis of the problem. The four instances in time and
their corresponding T-stress values are indicated in Fig. 4.40. The value r = -0.49
corresponds to the maximum value of KI (see Fig. 4.41).
The variation of normalized crack-opening stress (22 at 0 = 0)
s. normalized
distance at the four r values is shown in Fig. 4.42. An enlarged view of Fig. 4.42
is shown in Fig. 4.43. The stress profiles exhibit the expected drop predicted by the
MBL solutions. That is, the two-parameter characterization also holds for the case of
combined thermal and mechanical loading. The slight rotation of the stress profiles
compared to the MBL results was already discussed before.
For completeness, we present the variation of hydrostatic stress, plastic equivalent
strain, and the various strain components also for the combined loading case. Fig. 4.45
shows the circumferential variation of the hydrostatic stress at a fixed radial distance,
69
r = 1.22J/oy, for the four r values and the corresponding MBL solutions. The elasticplastic values agree well with the MBL results. During the decreasing-r portion of
the transient (after
= tmax) the hydrostatic stresses decrease with respect to the
SSY solution as expected.
Fig. 4.37 shows the circumferential variation of normalized equivalent strain (ep ) at
r = 1.22J/ay in comparison to the MBL solution. As before, qualitative agreement of
the elastic-plastic strain profiles with the corresponding MBL solutions and a slight
overestimation (
10%) can be noted.
The normalized radial variation of strain components of the transient strain field
(0 = 0) subjected to combined loading are shown in Fig. 4.47. The strain measures
at the four r-values are similar in magnitude, compared to the results for purely
thermal loading.
70
V
amb
Figure 4.1: Strip with edge crack, subjected to thermal shock along x = 0.
71
11
IIIIIIIIIII .
IIII
IIIIII
Figure 4.2: Mesh used in elastic analysis of the problem.
72
4
I.U
I
C0
0.5
I
0
0.0
_ ._
0.0
0.5
1.0
x/w
Figure 4.3: Transient temperature distribution in the strip for /l = 5 (Fo = Dt/w 2 ).
73
I.U
I0
X
0.5
0I
nn
0.0
0.5
1.0
x/w
Figure 4.4: Transient temperature distribution in the strip for / = 100 (Fo = Dt/w 2 ).
74
0a
0.6
rl
0.5
0.4
cd
tzt
v
0.3
1-
0.2
0.1
0.0
10- 4
10 - 3
10- 2
10-1
100
Dt/w2
Figure 4.5: Stress intensity factors K for a/w = 0.0375 as a function of nondimensional time Dt/w2 for various values of the Biot number.
CQ
0.6
I.
co
0.4
-
I
0.5
I
I
0.3
.5
0.2
(M
0.1
nn
10- 4
10- 2
10-3
100
Dt/w2
Figure 4.6: Stress intensity factors K1 for a/w = 0.05 as a function of nondimensional
time Dt/w 2 for various values of the Biot number.
75
0.5
"I
0.4
1
,--
0.3
I
5
0.2
0.1
An
10- 4
10-
3
10- 2
10-,
10°
Dt/w 2
Figure 4.7: Stress intensity factors K for a/w = 0.1 as a function of nondimensional
time Dt/w 2 for various values of the Biot number.
0.12
t-%_
I
CD
0.08
I
a
CD
%-l
0.04
n n
10- 4
10- 3
10 - 2
1-,
10°
Dt/w 2
Figure 4.8: Stress intensity factors K for a/w = 0.3 as a function of nondimensional
time Dt/w 2 for various values of the Biot number.
76
o
'f
ni
U.UO
0.06
N
V1
a)
I
0.04
4,r-
.1
0.02
n nn
10-4
0- 3
10 - 2
10o-
100
Dt/w 2
Figure 4.9: Stress intensity factors K* for a/w = 0.4 as a function of nondimensional
time Dt/w 2 for vztrious values of the Biot number.
0.6
Cd
0.5
co
0.4
t-
-
0.3
IC
0.2
0.1
0.0
10 -4
10 - 3
10 - 2
10o-
100
Dt/w 2
Figure 4.10: Stress intensity factors KI as a function of nondimensional time Dt/w
for various crack lengths ( = 100).
77
2
u.;J
0.2
0
D0
0.1
0.0
"E-.-0.1
-0.2
10 - 4
10-3
1-2
lo-
100
Dt/w 2
Figure 4.11: Non-dimensionalized T-stress values for a/w = 0.0375 as a function of
nondimensional time Dt/w 2 for various values of the Biot number.
0.3
0.2
l
a)I
0.1
J
I
0.0
E-
-0.1
_n 9
10-4
10-21
10 -
10 - 3
100
Dt/w 2
Figure 4.12: Non-dimensionalized T-stress values for a/w = 0.05 as a function of
nondimensional time Dt/w 2 for various values of the Biot number.
78
I
A
U.,
0.3
0.2
0.1
0.0
-n
1
10 -4
10 - 3
10 - 2
lo-1
100
Dt/w 2
Figure 4.13: Non-dimensionalized T-stress values for a/w = 0.1 as a function of
nondimensional time Dt/w 2 for various values of the Biot number.
0.4
0.3
I
a)
I
9
M
0.2
l
E-
0.1
nn
10 -4
10 - 2
10-3
10-1
100
Dt/w 2
Figure 4.14: Non-dimensionalized T-stress values for aw = 0.3 as a function of
nondimensional time Dt/w 2 for various values of the Biot number.
79
0.4
0.3
It
OD
0.2
0.1
n .10n
-
4
10-4
10 - 2
10-3
10-,
100
Dt/w 2
Figure 4.15: Non-dimensionalized T-stress values for a/w = 0.4 as a function of
nondimensional time Dt/w 2 for various values of the Biot number.
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
10- 4
10- 3
10 - 2
10-1
100
Dt/w 2
Figure 4.16: Non-dimensionalized T-stress values as a function of nondimensional
time Dt/w 2 for various crack lengths ( = 100).
80
---
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oo
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----
I
_
.
I
0
-4
1
0
0
O I_
a
Xiol
40-leiLU
)
0
T
4'
.
H~LJ
O 11
t
C;
o
6
o
o
(e- e)~z/&
84
0'
2 B 9
0.6
I4
v
0.4
-
Ov
I
0.2
I
nn
-0.3
-0.2
-0.1
0.0
T/Ea(
0.1
t,-
0.2
0.3
b)
Figure 4.21: Non-dimensionalized KI vs. T for a/w = 0.0375, for various values of
the Biot number. Arrows indicate the direction of traverse.
85
0.6
Ca
0.4
I
I
v
0.2
0.0
-0.2
-0.1
0.0
T/Ea(9Ot
0.1
0.2
0.3
- eb)
Figure 4.22: Non-dimensionalized KI vs. T for a/w = 0.05, for various values of the
Biot number. Arrows indicate the direction of traverse.
86
U.4
v
0.3
I
0.2
An
-0.1
0.0
0.1
T/Ea(e0t-
0.2
0.3
0.4
eO b)
Figure 4.23: Non-dimensionalized K vs. T for a/w = 0.1, for various values of the
Biot number. Arrows indicate the direction of traverse.
87
0.15
-I-;:
0.10
I
0.05
'-
VD
0.00
0.0
0.1
0.2
0.3
0.4
T/Ea(Oet - eOab)
Figure 4.24: Non-dimensionalized KI vs. T for a/w = 0.3, for various values of the
Biot number. Arrows indicate the direction of traverse.
88
0.08
M
".1
(d
0.06
-
l
0.04
~O
I
V
0.02
0.00
0.0
0.1
0.2
T/Ea(e,.t -
0.3
0.4
eamb)
Figure 4.25: Non-dimensionalized KI vs. T for a/w = 0.4, for various values of the
Blot number. Arrows indicate the direction of traverse.
89
0.6
CQ
I'll
t-
fid
0.4
I
0.2
0.0
-0.4
-0.2
0.0
T/Ea(init -
0.2
0.4
amb)
Figure 4.26: Non-dimensionalized KI vs. T for various relative crack depths (
100).
90
0.6
=
0.3
0.2
*
0
C
I
0.1
.0
0.0
-0.1
P, -0.2
E
-0.3
-0.4
-n
;
0.0
0.1
0.2
0.3
0.4
a/w
Figure 4.27: Non-dimensionalized T-stress values at maximum KI, for various Biot
numbers and relative crack depths. The dashed line indicates the expected asymptotic
T*-values (a/w -, 0).
91
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
n . A-
-v.
10- 4
2
10
100
Dt/w 2
Figure 4.28: Non-dimensionalized T-stress values as a function of nondimensional
time Dt/w 2 for a/w = 0.0375 ( = 100, Ea(ei,,t -
.mb)lay = 1.32). Symbols
indicate instances in time for which crack-opening stress profiles are obtained.
92
A·Y
_ 0.6
0.4
~'0.5
1
0.2
An
0.1
no
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
Figure 4.29: Non-dimensionalizedKI vs. T for a/w = 0.0375 ( = 100, Ea(Oiit Oamb)/cry = 1.32). Symbols indicate instances in time for which crack-opening stress
profiles are obtained.
93
T =0.145
Fo = 8.600 x 10-4
crack tip
T = -0.011
Fo = 1.638 x 10
crack tip
T =-0.135
Fo=2.805x 10
-3
crack tip
T =-0.221
Fo = 4.691 x10
crack tip
Figure 4.30: Position of temperature front relative to crack-tip for a/w = 0.0375 at
four values of normalized T ( =- 100). Temperature front locations correspond to
(init--
o)/(
t - oamb) =
0.01.
94
Figure 4.31: Mesh used in elastic-plastic analysis of the problem.
95
Figure 4.32: Near-tip region of mesh shown in Fig. 4.31 .
96
5
4
b
\
b
3
2
1
1
0
2
3
4
6
5
r/(J/luy)
0.145
***
T =
AA A
T = -0.011
Fo = 8.600 x 10- 4
K I = 0.399
Fo = 1.638 x 10K,*
xx
T
= 0.493
Fo = 2.805 x 10- 3
T = -0.135
Kx*
v v v
3
= 0.543
Fo = 4.691 x 10- 3
K[ = 0.559
= -0.221
Figure 4.33: Normalized crack-opening stress profiles at four instances of time and
corresponding MBL solutions (a/w = 0.0375, 3= 100, Ea(Oi,,it - Oanlb)/yu = 1.32).
IK are stress intensity factors normalized according to Eq. (4.2).
97
4
b
\,C\2 3
b
2
1
2
3
4
5
6
r/(J/uy)
***
T=
A AA
-T= -0.01 1
0.145
Fo = 8.600 x 10- 4
Ki* = 0.399
Fo = 1.638 x 10- 3
K I = 0.493
XX X
T = -0.135
Fo = 2.805 x 10- 3
Ki = 0.543
V
V V
T
=
-0.221
Figure 4.34: Enlarged view of Fig. 4.33. K
according to Eq. (4.2).
98
Fo = 4.691 x 10- 3
KI * = 0.559
are stress intensity factors normalized
4.0
I
r
1
1
1 I
I
I
I
=2
r /(J/)
3.5
I I
b
3.0
b
b\
)
elastic-plastic analysis
2.5
2.0
I
Ii
I
- 1.0
I
I
I
-0.5
i
I
i
0.0
i II
I
0.5
i
1.0
7-
***
T = 0.145
Fo = 8.600 x 10 -4
Ki* = 0.399
AAA
Fo = 1.638 x 10 - 3
T = -0.011
KI = 0.493
Fo = 2.805 x 10 - 3
K = 0.543
x x x
T
v V
- = -0.221
=
-0.135
4.691 x 10- 3
KI = 0.559
Fo
-
Figure 4.35: Normalized crack-opening stresses at r/(J/uy) = 2 vs. T at four
instances of time and corresponding MBL results (a/w = 0.0375, fi = 100,
1((O)init-
amb)/gy =
1.32).
99
4
I
'
I
'
'
I
'
'
I
b
'
I
r I (ilu-)
3'
b
'
2
-
\'t
'
I
= 1.22
"positive
1
0
I-:
-1
II
I
I
. 2, .4, .6, .8,1.
I
I
I
I
I
I
I
I,
I
I
60
30
0
I
I
I
I I
90
I
-';"
I
I
.....
I
120
I
i
'
',"'
I
150
180
8 (deg)
*
* *
7=
A
A
'
A
0.145
=
-0.011
Fo = 8.600 x 10- 4
K = 0.399
Fo = 1.638 x 10- 3
KI = 0.493
x
x
x
V
V
V
'
=
-0.135
=
-0.221
Fo = 2.805 x 10-3
KI = 0.543
Fo = 4.691 x 10- 3
K = 0.559
Figure 4.36: Variation of hydrostatic stress at four instances of time and corresponding MBL solutions (a/w = 0.0375, P = 100, Ea(Oi,it - Oamb)/ay = 1.32). I are
stress intensity factors normalized according to Eq. (4.2).
100
90
60
w
w
30
0
30
0
60
90
120
150
180
8 (deg)
·
·
·
T =
0.145
Fo = 8.600 X 10 - 4
K* = 0.399
A
A
T
=
-0.011
Fo = 1.638 x 10-3
Ki* = 0.493
x x
x
T
=
-0.135
Fo = 2.805 x 10-3
KI*= 0.543
v
v
=
-0.221
A
v
Fo = 4.691 x 10- 3
K* = 0.559
Figure 4.37: Variation of equivalent plastic strain at four instances of time and corresponding MBL solutions (a/w = 0.0375, / = 100, Ea(Oinit - Oamb)/(y = 1.32). K1
are stress intensity factors normalized according to Eq. (4.2).
101
2.0
5
4
w
W
1.5
w
3
1.0
PI
mu
2
b
0.5
1
0
0.0
0
1
2
3
4
5
6
0
1
(a)
(b)
I
.
w
3
,
I
.
I
.El
I
E
VE.-L.
.--
4
5
6
·
-
-
M&AA AA -b
:Wm)(
I
-
I
.
I
N
.
0
A
A
x X
-x
I ivvv v V, v IV 9
0
2
1
v
x
3
4
r/(J/y)
(c)
(d)
0.145
Fo = 8.600 x 10- 4
x----
T =
v v
-0.011
Fo = 1.638 x 10-3
I
M
M.
--vvIV
r/(J/y)
iT =
*-*-A- ·
3
-
-1.0
0
2
.
-0.6
-0.8
1
.
-0.4
A/
1
0
.
6
I
w
2
.
5
r/(J/ay) = 122
-0.2
4!
I
4
r/(J/CY)
0.0
I I
3
r/(J/ry)
5
kU
2
5
6
-r = -0.135
Fo = 2.805 x 10- 3
v
=
-0.221
Fo = 4.691 x 10 - 3
Figure 4.38: Strain components ahead of the crack at four instances of time during
the thermal transient (a/w = 0.0375, d = 100, Ea(O init- eamb)/Cry = 1.32).
102
CQ
-1
1-co
tIz
a-
f%
^
U.V
0.8
v
3
I
CD
0.7
0.6
I
I-W
0.5
I--,
0.4
I--
nR
10- 4
10 - 3
10 - 2
10o-
100
Dt/w2
Figure 4.39: Stress intensity factors KI for a/w = 0.0375 under combined thermal and
mechanical loading as a function of nondimensional time Dt/w 2 ( = 100, am = ay/2,
Eoa(Oit - O.amb)/0y = 1.32).
0.1
0.0
-0.1
I,
-0.2
-0.3
-0.4
-0.5
-n r
10-4
10- 2
10-3
l0-,
100
Dt/w2
Figure 4.40: Non-dimensionalized T-stress values for a/w = 0.0375 under combined
thermal and mechanical loading as a function of nondimensional time Dt/w 2 ( =
100, o, = y/2, E(O)init - Oamb)/Cy = 1.32).
103
1.0
Da
q
N.Cd
I
0.5
I
rzl
a-.
tni
-0.6 -0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
Figure 4.41: Non-dimensionalized K vs. T for a/w = 0.0375 under combined thermal
and mechanical loading as a function of nondimensional time Dt/w 2 ( = 100, oa =
o,/2,E(Oinit- Oamb)/'y= 1.32).
104
5
4
b
C\
3
b
2
1
1
0
2
3
4
5
6
r/(J/o"y)
* * -
T =
A A A
T
=
0.006
Fo = 2.961 X 10 - 4
KI = 0.516
Fo = 1.264
-0.207
X 10 - 3
KI * = 0.752
Xx x
T
=
-0.347
Fo = 2.236
KI*= 0.821
X 10 - 3
V V V
T
=
-0.490
Fo = 4.691
K *I = 0.856
X 10 - 3
Figure 4.42: Normalized crack-opening stress profiles at four instances of time and
= 100, ar = y,/2, E(Oiit
corresponding MBL solutions (a/w = 0.0375,
Oamb)/Cry= 1.32). I are stress intensity factors normalized according to Eq. (4.2).
105
4
b
3
c
C\2
CQ2
b
2
1
2
3
4
5
6
rl(J/gy)
* * *
T = 0.006
Fo = 2.961 x 10-4
KI*= 0.516
A A A
T = -0.207
Fo = 1.264 x 10-3
Ki* = 0.752
X XX
T = -0.347
Fo = 2.236 x 10- 3
KI* = 0.821
V V V
r =
Fo = 4.691 x 10- 3
-0.490
K,*= 0.856
Figure 4.43: Enlarged view of Fig. 4.42. Ki are stress intensity factors normalized
according to Eq. (4.2).
106
4.0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
r/(J/r.) - 2
3.5
b
CQ
3.0
b
)
2.5
n
I
I
-1.0
I
I
II
I I
-0.5
I
III
I
0.0
I
I
I~~~~T
I
I
I
......
.
0.5
II
.
[
1.0
T
* **
= 0.006
Fo = 2.961 X 10 - 4
K, *
A A A
=
= 0.516
Fo = 1.264 x 10 - 3
-0.207
Ki*
= 0.752
x
x x
=
-0.347
Fo = 2.236 x 10 - 3
K I* = 0.821
v
v v
=
-0.490
Fo = 4.691 x 10 -3
Ki*
= 0.856
Figure 4.44: Normalized crack-opening stresses at r/(J/ay) = 2 vs. T. at four instances of time and corresponding MBL results (a/w = 0.0375, = 100,.r =
/2,
E ( nit- Oamb)/Cr,= 1.32).
107
4
3
b
2
b
1
0
-1
30
0
60
90
120
180
150
8 (deg)
·
·
T =
A A A
Fo = 2.961 X 10 - 4
KI = 0.516
0.006
Fo = 1.264
T = -0.207
KI*
x
x
X
T
=
-0.34
7
T = -0.490
10 - 3
= 0.752
Fo = 2.236 x 10- 3
K1 *
V V
x
= 0.821
Fo = 4.691 x 10Ki*
3
= 0.856
Figure 4.45: Variation of hydrostatic stress at four instances of time and corresponding MBL solutions ( a/w = 0.0375,P = 100, r°° = o%/2,Ea(Oinit-Oa,b)/y = 1.32).
K* are stress intensity factors normalized according to Eq. (4.2).
108
90
60
w
w
30
0
30
0
60
90
120
150
180
8 (deg)
**
T = 0.006
Fo = 2.961 X 10 - 4
KI = 0.516
A A A
T = -0.207
Fo = 1.264
KI = 0.752
x
T
x x
= -0.347
X 10-3
Fo = 2.236 X 10-3
Ki* = 0.821
V
V
V
T = -0.490
Fo = 4.691 X 10-3
Ki* = 0.856
Figure 4.46: Variation of equivalent plastic strain at four instances of time and corresponding MBL solutions ( a/w = 0.0375,P = 100, c° = Ory/2,Ea(Oinit-Oamb)/cy =
1.32). KI are stress intensity factors normalized according to Eq. (4.2).
109
2.0
5
4
w
1.5
3
·
I
·
·
.
·
.
·
.
·
= 1.22
r/(J/rI)
-
I
w
b
4
w
·
1.0
2
0.5
1
0.0
0
-
-
0
1
-
-
2
3
r/(a)
0.0
,11
11
-0.2
4
LU
k
I
,-I
3
w.4
1
-0.8
0
- .V
5
6
AA ,A
I
.
0
.I
.
·
1
r/(J/oy)
T =
--
x, ;
0.006
Fo = 2.961 x
I
--
.
A
A
AA
.
I.
2
vvvI
.
I
*.
..
I.
..
3
4
V V
.
li
·
I
.
5
v
,
6
r/(J/o-Y)
(c)
*----
A
A
jvyyy
41 n
4
,
r/(J/o-Y) = 1.22
AAAAA
-0.6
3
I
,1 j
1
2
2
f6I
-0.4
0
w
i
i. I
I
1
5
(b)
5
0
4
r/(J/)
(a)
w
- -
-
Fo = 2.236 x 10 - 3
10-4
-T = -0.207
v VV
Fo = 1.264 x 10 - 3
(d)
T = -0.347
T =
-0.490
Fo = 4.691 x 10-3
Figure 4.47: Strain components ahead of the crack at four instances of time during the
thermal transient ( a/w = 0.0375, 3 = 100,a °° = oy/2, E(Oinit - Oamb)/oy = 1.32).
110
Chapter 5
Conclusions and Future Work
5.1
Conclusions
In summary, we have examined and validated the two-parameter characterization for
the case of transient thermal loading. The ability of the MBL solutions in predicting
the stress state of the elastic-plastic solutions is exceptional considering that the
MBL loading is based on the first two terms of the WILLIAMSeigen-expansion, which
neglects the presence of thermal strains in its derivation. That is, simple extraction
of the T-stress variation from an elastic analysis of the problem allows us to predict
the triaxiality of the stress state in the elastic-plastic full-field solution.
The importance of these results becomes even clearer when the T-stress effect on
fracture toughness is taken into consideration.
BETEG6N & HANCOCK,.(1990), for
example, examined the dependence of cleavage fracture toughness on
T
in three-
point-bend specimens of various crack depths. The varying crack depth provided
large variation of crack-tip constraint (thus a large range of r values). Their results
in terms of (Jo, r) at final failure are shown in Fig. 5.1. The experimental data has
a large scatter, as do most cleavage toughness tests. Clearly, though, the cleavage
fracture toughness shows a significant increase at large negative values of r.
111
Thus, for negative r-values, the observed drop in crack-opening stress profiles is associated with a simultaneous increase in fracture toughness. This means that failure,
otherwise predicted using a single-parameter approach, will not occur for these cases.
The temperature dependence of the fracture toughness should not be neglected here.
Namely, toughness generally decreases with decreasing temperature (see Fig. 5.2).
The temperature variation at the crack tip of relative crack depth a/u = 0.375 for an
overcooling transient, for example, is shown in Fig. 5.3. That is, during a pressurized
overcooling transient three interacting effects have to be taken into consideration: (a)
the toughness decreases due to the drop in temperature; (b) the acting dead-loads
shift the KI-r locus into the negative quadrant and thus into a "safer" area with respect to the fracture toughness which is increased due to the influence of the T-stress;
and finally, (c), the K- r - O spiral traversed in this 3-D space, reaches its maximum
K*-value at an even more negative value of r, thus resulting in a further "gain" with
respect to the fracture toughness, counteracting the adverse temperature effect. This
result is schematically depicted in Figs. 5.4 and 5.5.
Fig. 5.6 from a review paper by STAHLKOPF(1982) puts these results into contact with
traditional one-parameter-based
approaches for the assessment of vessel integrity.
Shown in the figure are the cracking response of two hypothetical reactor vessels
exposed to a severe overcooling transient. Vessel (a) represents a case of severe embrittlement; vessel (b) a case of light embrittlement. A one-parameter based approach
predicts grack growth for the highly embrittled reactor vessel, and no crack initiation
for the case of low embrittlement. On the contrary, using a two-parameter approach,
the prediction for the case of severe embrittlement could look like the prediction for
the case of light embrittlement if the relative crack depth were sufficiently small.
In the ongoing efforts aimed at developing two-parameter descriptions of crack-tip
fields, the results of this work confirm the T-stress as the rigorous "second" cracktip parameter in well-contained yielding (PARKS, 1992). Besides this work, ample
evidence exists
(BETEG6N
& HANCOCK(1991), AL-ANI & HANCOCK(1991)) that the
112
T'-stress is valuable in characterizing the stress triaxiality of plane strain and 3-D
elastic-plastic crack-tip fields. This strong dependence of crack-tip stress triaxiality
on r, along with the marked sensitivity of both ductile (void growth) and brittle
(cleavage) fracture mechanisms to stress triaxiality, has profound influences on crack
toughness and growth ductility.
Thus, the parameter r, as elastically calculated
based on the applied loading, plus the parameter J, as calculated based on the actual
elastic-plastic deformation field, rigorously and accurately describe the local crack-tip
stress and deformation (PARKS, 1992).
5.2
Future Work
The most obvious extension of this work is the investigation of the T-effect on the
near-tip stress fields using temperature-dependent material properties. KOKINI(1986),
for example, showed that for the problem of a strip containing an edge.crack using
constant material properties over large temperature ranges can lead to considerable
underestimation of the maximum stress intensity factors.
To realize the full potential the two-parameter characterization of elastic-plastic fields
offers, systematic experimental testing is needed to establish parametric limits. This
needs to be done in conjunction with further numerical investigation of the limits of
the two-parameter approach in predicting the near-crack-tip fields of various specimens (WANG, 1991).
RICE
(1972) showed that the so-called line-spring could be used to calculate KI for
thermal or residual stresses that vary through the thickness of a plate. By applying
the "no-crack" tractions as reverse pressures on the crack faces, the extra elastic compliance and stress intensity factor can be readily computed. PATIL(1993) implemented
these methods into a line-spring program. WANG & PARKS (1992) demonstrated how
this line spring-model can be used to calculate T for mechanical loading of the single-
113
face-cracked specimen subject to combined tension and bending. A natural extension
of their work could explore the case of combined mechanical/transient thermal loading.
The performance of the modified effective crack-length-formulation of HAUF et al.
(1994) for contained yielding in the presence of thermal stresses could also be examined. The nonlinear compliance of mechanical work-conjugate displacements, q,
should be affected by the presence of thermal stresses.
Examination of the T-effect on three-dimensional stress fields and extension to other
specimen geometries constitute further possible areas of future work. Finally, any
future work on this topic aimed at the nulear power industry should examine the
attenuation of fast neutrons through the reactor wall and attendant radiation embrit-
tlement gradients.
114
*CI\A
1
UU
1000
800
'-
600
U
400
200
n
-1.0
-0.8
-0.6
-0.4
-0.2
Figure 5.1: Variation of cleavage fracture toughness with
1(990).
115
T
0.0
0.2
(Beteg6n and Hancock,
CLnn
'UU
160
120
KIc
ksi ,/in.
80
40
0
-200
0
200
TMn1p.°F
Figure 5.2: Variation of fracture toughness KIC with temperature for A533 B Steel
(Sailors and Corten, 1972).
116
m-e- 1.0
0
I I
I
I II I I
I
0.8
I
I I II I
I
II I
I II
a/w = 0.0375
I
I = 100
F_
0.6
0.4
0
0.2
AA
I I iiIii ,
I I,,
..
. Jv.
10- 5
10- 4
,I I ,,
,ll
10-2
10-3
II ,
,
111111
o10-
10°
Dt/w2
Figure 5.3: Variation of crack-tip temperature during a thermal transient (a/w =
0.0375, 3 = 100).
117
KIm
t
Figure 5.4: Schematic variation of fracture toughness as a function of
r- K -
spiral.
118
and
and
2.0
oII
I-
1.5
I-
U
10
1.0
o
0.5
I-
0.0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
7-Figure
5.5:
Two-dimensional
slice of Fig. 5.4.
Figure 5.5: Two-dimensional slice of Fig. 5.4.
119
0.0
0.1
I
Wall
I
~ ~ ~
l
I
thickness (a)
Overcooling transient
A
\Fractur
with pressure
Arrested
crack size
_
e
arrest
severe embrittlement
,N
n.Fracture
initiation
o
Initial
crack size
Time Into Transient
-
Wall
thickness
(b)
Overcooling transient with
pressure light embrittlement
Cn
N
No crack initiation
Fracture
arrest
/
Fracture
/-rinitiation
(C
Initial
crack size
Time Into Transient
Figure 5.6: Prediction of the behaviour of a 1-in. deep crack during an overcooling
transient for a reactor vessel with (a) severe and (b) light embrittlement (Stahlkopf,
1982).
120
References
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J. W., 1991, "J-Dominance of Short Cracks in Tension
and Bending," Journal of the Mechanics and Physics of Solids, Vol. 39, No. 1,
pp. 23-43.
ASME,1974, Rules of in service and inspection of nuclear power plant components,
Boiler and Pressure Vessel Code, Section XI, New York.
ASTM,1983, Standard Test Method for Plane-Strain Fracture Toughness of Metallic
Materials, Annual Book of ASTM Standards, E399-83, American Society for Testing
and Materials, Philadelphia, PA.
BETEG6N,C., and HANCOCK,
J. W., 1990, Proceedings ECF8 Turin, (Edited by D.
Firrao), EMAS, Warley, UK.
BETEG6N,C., and HANCOCK,
J. W., 1991, "Two-Parameter Characterization of ElasticPlastic Crack-Tip Fields," Journal of Applied Mechanics, Vol. 58, pp. 104-110.
BILBY, B. A., CARDEW,G. E., GOLDTHORPE,
M. R., and HOWARD,I. C., 1986, "A Finite
Element Investigation of the Effect of Specimen Geometry on the Fields of Stress and
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CARDEW,G. E., GOLDTHORPE,I. C., HOWARD,I. C., and KFOURI, A. P., 1984, in Fun-
damentals of Deformation and Fracture, Eshelby Memorial Symposium, Cambridge
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CARSLAW,
H. S., and JAEGER,J. C., 1950, Conduction of Heat in Solids, Oxford
University Press.
Du, Z.-Z., and HANCOCK,
J. W., 1991, "The Effect of Non-Singular Stresses on Crack121
Tip Constraint," Journal of the Mechanicsand Physics of Solids, Vol.39, pp. 555-567.
HANCOCK,
J. W.,
REUTER,
W. G., and
PARKS,
D. M., 1993, "Constraint and Toughness
Parameterized by T," Constraint Effects in Fracture, ASTM STP 1171 (Edited by
E. M. Hackett, K.-H. Schwalbe, and R. H. Dodds), American Society for Testing and
Materials, Philadelphia, pp. 21-40.
HARLIN,G., and WILLIS, J. R., 1988, "The Influence of Crack Size on the DuctileBrittle Transition," Proceedings of Royal Society of London A415, pp. 197-226.
HAUF, D. E., PARKS, D. M., and LEE, H., 1994, "A Modified Effective Crack Length
Formulation in Elastic-Plastic Fracture Mechanics," manuscript submitted to Me-
chanics and Materials.
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126
Appendix A
Derivation of Line-Load Strain
and Displacement Fields
Given the stress fields, Eq. (3.2), we derive the corresponding strain and displacement
fields needed in the evaluation of the Interaction Integral of Chapter 2. Using the
strain-displacement relations in polar coordinates and the elastic constitutive equations for plane strain, we have
f cos_
Our _
ar
560=
0
1 (1
=B =
2
r
E r
a
+
2)
c
(1 + )
Uo\
=0.
V
u,
--A +
O
Ou
-r
ar
r
(A.1)
Integrating the first of these equations, we find
=
Ur
- rE
f os(1-v
r
2 )dr
cE
=-
cos(1-v
2 )logr
+ F(O),
(A.2)
where F(O) is a function of 0 only.
Substituting in the second of Eqs. (A.1) and integrating it, we obtain
uo = vff sing(1 + v) + f sin1ogr(Iv
7rE
7r
127
2
)
F(O)d + G(r),
(A.3)
O
in which G(r) is a function of r only.
Partially differentiating Eq. (A.2) by 0 and Eq. (A.3) by r, we have
Our
iO
= ff
sin0(1- 2)+
n'E
Ou
Or
= f sin01(1_v2)
E
r
oF()
+
+
-
(A.4)
()
O0
G(r)
r
(A.5)
Substituting (A.4) and (A.5) in the third of Eqs. (A.1), we conclude that
F(0) =
2
(1-v-v
2'E
)0 s in 0 + Asin0 + Bcos
G(r) = Cr,
(A.6)
(A.7)
where A, B, and C are constants of integration to be determined from the conditions
of constraints. Using Eqs. (A.6) and (A.7) in Eqs. (A.2) and (A.3), the expressions
for the displacements are
fE cos 0 (1 -
=
v2 )
7·rrE
Asin 0 + B cos 0,
uo =
vffrEsin0(1 +v)
2) f
(1-v2V
27rE
logr
--
2)
(1
( - v - 22v 2 ) f sinO+
'~27rE
-
(A.8)
f
+ --EsinOlogr (1-v2) +
[sin 0 -0 cos ] + A cos0 - B sin 0 + Cr.
(A.9)
The constraint is such that the points on the x axis have no lateral displacement.
Then u = 0, for 0 = 0, and we find from Eq. (A.9) that A = 0, C = 0. The constant
B is determined considering a material point lying on the x axis at (listance d, say,
from the origin which does not move out radially. From Eq. (A.8) we then obtain
that B = (1
-
2 ) logd.
By geometric considerations it is possible to derive the displacement field in cartesian
coordinates from the polar components. Namely
[u
1
[u2
=
[
cos0
sin0
-sin
cos
128
1 [
Ur
u
1(A.10)
(A1)
rFaking the derivative of ul and u2 with respect to x2 and xl, respectively, we have
au,
( Our
ar
-0
-
ax2
Ox 2
au2
ax1
cos -
Or sin
vr
auo
9 sin
us cos0 -
00
Ouo
csin9 +
Ox
0 +
ar
09
- usin0}
(A.11)
- usin }
(A.12)
9
cos } +
ax Ir OU
a- sin 0+ Ur COS0 +
Ouo
09
00 cos
Then, after taking the derivatives of u, and uo that appear in Eqs. (A.11) and (A.12),
consideringthat
and
cos 0
=
OXl
0
sin 0
Ox1
r
and substituting everything in Eq. (A.12), we obtain
aul1
sinO
0x2
r
f (1
7rE
+ v)(v -
os2 0 - 1)}
(A.13)
and
aU2
sin 0
Oxl
r
f (I + v)(v - sin2 9)
(A.14)
-xE
The strain components in cartesian coordinates can be obtained directly starting from
the stress field (Eqs. (2)) and using the elastic constitutive equations for plane strain.
Then
Oul
=
X11
= E [(1-
=i[(1 =v)
au2
622
=
Ox
E1~
E33 =
aU3
aX3
2)11
(f
1
E [ ( :1 v 2 ) (-f
- v(1 + v)0 2 2 ]
cos30) +
2
).
22
-
(1 +
(1+ )(
)o1ll]
cos sin2 )+v(1
= 0
129
cos0sin2)]
+v) (
cos 3 )]
(A.15)
l(oul
512
+ OU2
-
(1
V)12
2 \9X2
-
_ (1 + v
a
C
sin
E
513 -= £23= .
130
12
Appendix B
Temperature Distribution under
Convective Cooling
The uncoupled transient temperature distribution for the strip in Fig. 4.1 may be
determined from the solution of the one-dimensional diffusion equation
a 2 o*(x,
1
ao*(x,t)
D
at
t)
0x2
'
where
*
O*(x,t) = O(x,t) - Oamb
(B.2)
O(x, t) is the temperature in the strip at location x and time t, Oambis the ambient
temperature. D in Eq. (B.1) is the thermal diffusivity; that is D = k/pc, where p is
the mass density and c is the specific heat per unit mass of the material.
The initial condition is given by
O*(x, 0) = Oint - Oamb,
(B.3)
where Oinit is the uniform initial temperature throughout the strip. The boundary
condition on the plane x = w is expressed as
a0*(x, t)
ax
0.
~=W
131
(B.4)
The mixed boundary condition
900(0, t)
kO
a ) =
h [Omb
(0, t)].
-
(B.5)
assures continuity of the heat flux on the plane x = 0. That is, the heat flux is
removed from the surface x = 0 by convection to the environment.
Applying the method of separation of variables and making use of the conditions of
Eqs. (B.3), (B.4), and (B.5), Eq. (B.1) can be solved straightforwardely (CARSLAW
&
JAEGER,1950) to give the following non-dimensional temperature distribution along
the strip
O(X*, Fo) - Oamb = 2
Oinit - Oamb
- X*)]] exp(-
[in(An) cos[(1
Fo)
(B.6)
sin(2
n=1
where X* is the dimensionless coordinate, x/w, along the crack, and the non-dimensional
time, Fo = Dt/w 2 , is the Fourier number. The boundary conditions generate eigenvalues, A, that are the roots of the following transcendental equation,
An
tan(An) =
where / is the Biot number defined as /3 = hw/k.
132
,
(B.7)
Appendix C
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