1. No category

Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
www.elsevier.com/locate/tafmec
Fatigue crack opening stress based on the strip-yield model
J.H. Kim, S.B. Lee *
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology ± KAIST, 373-1 Kusong-Dong, Yusong-Gu,
Taejon 305-701, South Korea
Abstract
The modi®ed strip-yield model based on the Dugdale model and two-dimensional approximate weight function
method were utilized to evaluate the e€ect of in-plane constraint, transverse stress, on the fatigue crack closure. The
plastic zone sizes and the crack opening stresses considering transverse stress were calculated for four specimens: single
edge-notched tension (SENT) specimen, single edge-notched bend (SENB) specimen, center-cracked tension (CCT)
specimen, double edge-notched tension (DENT) specimen under uniaxial loading. And the crack opening behavior of
the center-cracked specimen under biaxial loading was also evaluated. Normalized crack opening stresses rop =rmax for
four specimens were successfully described by the normalized plastic zone parameter Dx0rev =x0 considering transverse
stress, where Dx0rev and x0 are the size of the reversed plastic zone at the moment of ®rst crack tip closure and the size of
the forward plastic zone for maximum stress, respectively. The normalized plastic zone parameter with transverse stress
also was satisfactorily correlated with the behavior of crack closure for CCT specimen under biaxial loading. Ó 2000
Elsevier Science Ltd. All rights reserved.
1. Introduction
The fatigue crack closure phenomenon is an
intrinsic aspect of the mechanics of growing fatigue cracks, and, in many applications, closure
provides a powerful ®rst-order correction to the
crack driving force that facilitates more accurate
prediction of crack growth rates. Although closure
can be induced by several di€erent physical
mechanisms, including roughness or oxides on the
crack surface, extensive experimental and computational studies have shown that crack wake
plasticity is often the dominant contribution to
crack closure, particularly at high stress intensi®-
*
Corresponding author. Tel.: +42-869-3029; fax: +42-8693210.
E-mail address: sblee@kaist.ac.kr (S.B. Lee).
cation. Hence, it is important to determine the size
and shape of the crack tip plastic zones as for
characterizing crack tip plastic deformation.
Transverse stress has been found [1,2] to have a
signi®cant in¯uence on the plastic zone size and
the behavior of the crack opening stress. Such an
e€ect was evaluated using center-cracked tension
(CCT) and single edge-notched bend (SENB)
specimen [3]. The biaxial stress a€ects the crack
growth rate signi®cantly [4±7].
In [8,9], the normalized stress intensity
factor
p
parameter Kmax =K0 , where K0 ˆ r0 pa and r0 is
the ¯ow stress, correlates well with the normalized
crack opening stresses rop =rmax under small-scale
yielding condition (SSY). However, it is dicult to
correlate the normalized crack opening stress in
terms of the stress intensity factor KI , where SSY
condition does not apply. Normalized stress intensity factor is also not e€ective for biaxial
0167-8442/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 2 5 - 2
74
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
Fig. 1. Four di€erent specimen geometries: (a) SENT specimen; (b) SENB specimen; (c) CCT specimen and (d) DENT specimen.
loading because KI is not a€ected by the transverse
stress.
In this paper, the crack opening stresses for
single edge-notched tension (SENT), SENB, CCT
and double edge-notched tension (DENT) specimens in Figs. 1(a)±(d) are obtained by the modi®ed
strip-yield model considering the e€ect of transverse stress. The relation between the normalized
plastic zone size, Dx0rev =x0 , and the normalized
crack opening stress, rop =rmax , was also obtained.
Moreover, the opening stress for the CCT specimen subjected to biaxial loading is evaluated.
2. Modi®ed strip-yield model
The strip model is expedient for ®nding the
plastic zone size and crack surface displacement
using the superposition of two elastic problems.
They correspond to a crack subjected to remote
applied load and a crack subjected to a uniform
stress applied over a segment of the crack surface.
Figs. 2(a) and (b) show the crack con®guration at
maximum and minimum applied stress. The model
consists of three regions: (1) a linear-elastic region
containing an imaginary crack of half-length
a ‡ x, (2) a plastic region of length x ahead of the
physical crack length a, and (3) a residual plastic
deformation region along the crack surface. The
body is treated as an elastic continuum. The
shaded regions in Fig. 2(a) indicate material that is
in a plastic state. The compressive plastic zone in
Fig. 2(b) is Dxrev . The plastic zone and residual
plastic deformation regions are composed of rigid
perfectly plastic (constant stress) bar elements. At
any applied stress level, the bar elements are either
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
75
Fig. 2. Schematic of the modi®ed strip-yield model.
intact or broken. The broken elements carry only
compressive loads, and then only if they are in
contact.
To approximate the e€ects of strain hardening,
the ¯ow stress r0 is taken to be the average of the
yield stress and ultimate tensile strength. The elements are in contact yield in compression when the
contact stress reaches ÿb0 r0 (b0 is the constraint
factor to represent Bauschinger's e€ect). Those
elements that are not in contact do not a€ect the
calculation of crack surface displacements. In this
model, the e€ects of the state of stress on the
plastic zone size and displacements are approximately accounted for by using a constraint factor
a0 . The constraint factor was used to elevate the
¯ow stress for the intact elements in the plastic
zone to account for three-dimensional stress states.
The plastic zone x ahead of the physical crack
tip under the maximum applied loading is modeled
by using 10 elements 1±10 as shown in Fig. 2(a).
The plastic zone size is computed by assuming ®nite stresses at the imaginary crack tip [10]. Let
d ˆ a ‡ x, under the maximum applied loading,
the crack surfaces may be assumed fully open and
the extent of the plastic zone x may be computed
from
Z d
Z d
rmax …x†m…x; d†dx ÿ
a0 r0 m…x; d†dx ˆ 0; …1†
0
a
where rmax …x† is the remote maximum cyclic stress
distribution along the crack line direction.
rmax …x† ˆ rmax for SENT, CCT, and DENT specimen, and rmax …x† ˆ 2rmax …0:5 ÿ x=W † for SENB
specimen. Here, rmax is de®ned as the maximum
outer ®ber cyclic stress for SENB specimen, and it
represents the constant maximum cyclic stress for
three specimen geometries. m…x; d† is the weight
function for cracked geometries.
The aspect ratios of the element width to forward plastic zone size …2wj =x; j ˆ 1; 2; . . . ; 10† are
0.01, 0.01, 0.02, 0.04, 0.06, 0.09, 0.12, 0.15, 0.20
and 0.30, respectively [11]. Near the crack tip, elements with smaller widths are used. Under the
maximum applied loading, with the crack surface
fully open, the element lengths are equal to the
crack surface displacement calculated as follows.
76
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
If the crack surface displacement solution
rS f …x† due to remote stress rS and the crack surface displacement solution rd g…x0 ; x† due to a
uniform stress rd acting on a segment of the crack
surface located at x0 are obtained, the displacement of crack surface V …x† at any point x can be
determined by superposition as
V …x† ˆ rS f …x† ÿ rd g…x0 ; x†:
…2†
When all the constant stress elements with the
constraint factor a0 along the crack surface are
considered, the displacement Vi at element i can be
written as
Vi ˆ rS f …xi † ÿ
n
X
ÿ
a0 rj g xj ; xi for i ˆ 1; . . . ; n;
jˆ1
…3†
where n is the total number of elements, rj is the
stress on element j; f …x† is the crack surface displacement due to unit remote applied stress, and
g…xj ; xj † is the displacement at element i due to unit
stress acting on element j. The element length for
remote maximum applied stress can be written as
Li ˆ rmax f …xi † ÿ
10
X
ÿ
a 0 r0 g x i ; x j :
…4†
jˆ1
The closure model provides no information about
the amount of crack growth per cycle. The crack
increment Da considering the plastic zone size can
be estimated [12] as
2
Da ˆ 0:2…x=4†…1 ÿ R† ;
…5†
where R is the stress ratio. For stress ratios less
than zero, R is set to be zero. Each cycle, a new
plastic zone x is computed at maximum stress. A
new element k is thus created with 2wk ˆ Da and
Lk ˆ L10 . When the applied load is reduced to a
minimum, the element length remains the same as
those computed from Eq. (5) because element
yielding does not occur. Any yielding requires the
computation of a new element length. In this
manner, residual deformations …L11 ; L12 ; L13 ; etc:†
are left in the wake of the growing crack to preserve a history of prior loading.
To keep the number of elements to a reasonable
size (45±50), an element-lumping procedure was
also implemented [11]. The lumping procedure
combined adjacent elements …i±i ‡ 1† to form a
single element if
2…wi ‡ wi‡1 † 6 a ÿ xi‡1 ‡ Da:
…6†
Therefore, the width of the newly created element
is taken to be the sum of the widths of the two
adjacent elements, and the new element length is
taken as the following weighted average,
Lˆ
wi Li ‡ wi‡1 Li‡1
:
wi ‡ wi‡1
…7†
When the applied loading is reduced to its minimum value, some elements in the plastic zone may
yield in compression. In addition, some elements
along the crack surface may come into contact and
carry compressive stress. Some of these elements
may also yield in compression. For all elements
that yield, new element lengths must be computed.
According to the compatibility equation …Vi ˆ Li †,
which is expressed as
n
X
ÿ
rj g xi ; xj ˆ rmin f …xi † ÿ Li
for i ˆ 1; . . . ; n;
jˆ1
…8†
the contact stresses along the crack surfaces and
the stresses in the plastic zone rj can be obtained
numerically using the Gauss±Siedel iteration
method under the following constraints. For elements in the plastic zone …xj > a†
ÿb0 r0 6 rj 6 a0 r0 ;
…9†
and, for elements along the crack surface …xj 6 a†
ÿb0 r0 6 rj 6 0:
…10†
For those elements that yielded, the compatibility
requirement Li ˆ Vi must be satis®ed with rj from
above. The new element lengths are found again
using the following equation:
Li ˆ rmin f …xi † ÿ
n
X
ÿ
rj g xi ; xj :
…11†
jˆ1
For the unyielded elements, their element lengths
remain unchanged.
On the other hand, the applied stress level at
which the crack surfaces are fully open is denoted
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
as the crack opening stress rop . This stress can be
calculated by using the crack surface displacements [12]. To ®nd the applied stress level needed
to open the crack surface at any point, the displacement due to an applied stress increment
…rop ÿ rmin † is set to be equal to the displacement
at that point due to the contact stresses at rmin .
Thus,
ÿ
rop
i
ˆ rmin ÿ
n
X
ÿ
r j g x i ; x j = f …xi †:
…12†
The maximum value of …rop †i gives the crack
opening stress rop .
3. Crack surface opening displacements using weight
function
The modi®ed strip model is applied to predict
fatigue crack closure. This depends on whether or
not the basic relations in Eq. (3) can be found for
the corresponding geometries. This corresponds to
the crack surface displacement f …x† under remote
applied load and the crack surface displacement
rd g…x0 ; x† due to a uniform stress over a segment of
the crack surface. However, very limited information about the crack surface displacement is
available in the literature. This problem can be
alleviated by the development of the weight function method [13]. The weight function m…x; a† is
de®ned as
m…x; a† ˆ
E0 ou
;
K oa
1
E0
x
0
a
77
rS …x†m…x; a†dx m…x; a† da:
…14†
The stress intensity factor Kr …x0 ; a† for the uniform
stress r acting on a segment of the crack surface
with the center at x0 can be solved by integration
Z x0 ‡w
rm…x; a† dx;
…15†
Kr … x 0 ; a † ˆ
x0 ÿw
jˆ11
for i ˆ 11; . . . ; n:
VS …x; a† ˆ
a Z
Z
…13†
where u is the crack surface displacement, K the
known stress intensity factor and a is the crack
length, E0 ˆ E for plane stress and E0 ˆ E=…1 ÿ m2 †
for plane strain (E is Young's modulus and m is
Possion's ratio).
The weight function is related only to the geometry, independent of loading. For the mode I
problem, once the weight function and the crack
surface distribution load rS …x† are obtained, the
corresponding crack surface displacement can be
derived from Eq. (13) as
where 2w is the width of the crack surface on
which a uniform stress r is applied. The corresponding crack surface displacement can also be
solved from Eq. (13) by the integration
Z a Z x0 ‡w
1
rm…x; a†dx m…x; a† da:
…16†
Vr ˆ 0
E x
x0 ÿw
Thus, the displacement functions f …x† and g…x0 ; x†
in Eq. (4) are obtained as
f …x† ˆ VS …x; a†=rS ;
…17†
g…x0 ; x† ˆ Vr …x; x0 ; a†=r:
For the cracked specimens considered in this paper, the approximate weight function m…x; a† is
given in Appendix A.
4. E€ect of transverse stress
The transverse stress is the second non-singular
term in Williams' expansion of the elastic stress
®eld at the crack tip [17]. For a crack in an isotropic elastic material subjected to plane stress
mode I loading, the stress ®eld near the crack tip is
given by
KI
rij ˆ p fij …h† ‡ rxx0 dxi dxj
2pr
…i; j ˆ x; y†;
…18†
where KI is the mode I stress intensity factor, fij …h†
the non-dimensional angular function, and dij is
the Kronecker delta. The transverse stress rxx0 that
is independent of r acts in a direction parallel to
the plane of crack and approximately constant
over the crack tip region. The crack tip singularity
is not a€ected by the transverse stress. However,
since the transverse stress can alter the plastic
zone of crack tip, the ¯ow stress r0 in the modi®ed
78
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
Fig. 4. Stress biaxiality ratio b for four cracked specimens as a
function of a=W [19].
Fig. 3. Change of the forward plastic zone size x and the reversed plastic zone size Dxrev for modifying ¯ow yield stress r00 :
(a) stress distribution under rmax ; (b) stress distribution under
compressive load ÿDr and (c) stress distribution after unloading from rmax ÿ Dr by adding (b) to (a).
strip-yield model was modi®ed to account for
transverse stress.
For a cracked geometry subjected to uniaxial
stress, the modi®ed ¯ow stress r00 can be expressed
by the application of the von-Mises yield criterion
for plane stress condition [5] as
q
1
rxx0 ‡ 4r20 ÿ 3r2xx0 :
…19†
r00 ˆ
2
In case of the CCT specimen subjected to biaxial
loading, the modi®ed ¯ow stress also can be obtained by above equation with transverse stress
rxx .
The superposition procedure for the present
modi®ed model is shown in from Figs. 3(a)±(c).
They explain schematically the change of forward
plastic zone size x to x0 and reversed plastic zone
size Dxrev to Dx0rev in crack tip ®elds. The forward
plastic zone size refers to the region of material
experiencing plastic deformation when the cracked
geometry is subjected to the maximum stress of the
cycle. The reversed plastic zone size is de®ned as
the smaller region of material within the forward
plastic zone that undergoes reversed plasticity as
unloading corresponds to the minimum stress.
In a cracked specimen subjected to mode I
loading, the transverse stress can be obtained by
scaling with the applied load. The biaxiality ratio b
relates transverse stress to stress intensity factor as
[18]
p
rxx0 pa
:
…20†
bˆ
KI
Fig. 4 shows the values of b for four specimen
geometries [19] with H =W ˆ 6, while KI remains
the same for all specimens. Transverse stresses are
di€erent from each other. As a result, those different values result in the di€erence of plastic zone
size for each specimen.
5. Results and discussion
All cracked specimens were assumed to be made
of 2024-T351 aluminum alloy. The mechanical
properties of the material are ultimate tensile stress
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
ru ˆ 480 MPa, yield stress ry ˆ 379 MPa, and
Young's modulus E ˆ 70 GPa. The strain hardening e€ect was simply considered by means of the
constant ¯ow stress r0 , which is taken to be the
average of the yield and ultimate tensile stress. For
simplicity, the plane stress condition is assumed
for four specimens, and both b0 and a0 are set to
equal to unity. Therefore, the material was assumed to be elastic perfectly plastic.
All analyses discussed at present are carried out
under the plane stress condition. The ratio of initial crack to specimen width …a0 =W ˆ 0:15† was
used. The crack closure behavior for four specimens subjected to uniaxial stress and CCT specimen subjected to biaxial loading was evaluated at
two stress ratios…R ˆ ÿ1; 0†, and the e€ect of
transverse stress was considered. Calculated crack
opening stresses showed that the present model
requires some ®nite amount of crack growth to
develop a plastic wake, but once a sucient deformation history has been established, the crack
opening behavior is characterized by a stable
opening level which does not change according to
the crack growth. Therefore, in the following discussion, the stable crack opening stresses were
de®ned as crack opening stresses rop .
The normalized crack opening stresses rop =rmax
at R ˆ 0 and R ˆ ÿ1 for all specimen geometries
are given as a function of the normalized maximum applied stress in Figs. 5(a) and (b). These
®gures, like other researcher's results [7,8], also
showed that it is dicult to describe the behavior
of the crack closure between di€erent specimen
geometries with rmax =r0 parameter. To consider
the e€ect of specimen geometry, another parameter is expressed [8] as
p
Kmax F rmax pa F rmax
p ˆ
ˆ
;
…21†
K0
r0
r0 pa
where Kmax is the stress intensity factor including
the geometry correction factor F, and K0 is the
stress intensity factor for ¯ow stress.
The crack opening results for both stress ratios
are shown in Fig. 6 in terms of the correlating
parameter Kmax =K0 . The opening stresses for the
four specimen geometries converge very closely to
a common value. This is especially true for small
79
Fig. 5. Normalized crack opening stresses as a function of
normalized maximum stress: (a) R ˆ 0 and (b) R ˆ ÿ1.
values of Kmax =K0 . For large Kmax =K0 values, the
data tend to spread, especially at R ˆ ÿ1. This
suggests that Kmax =K0 applies only to SSY conditions. Better correlation can be obtained by
0
=K00 , where
replacing large Kmax =K0 values by Kmax
0
a is the modi®ed crack length.
1
a ˆa‡
2p
0
Kmax
r00
2
:
…22†
A new value of F 0 was also computed based on
a0 =W . This parameter gave slightly improved correlation of crack opening stresses at R ˆ 0 but no
improvement at R ˆ ÿ1.
On the other hand, the calculated crack opening
stresses at both stress ratios for CCT specimen
subjected to biaxial loading were obtained from
80
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
Fig. 6. Normalized crack opening stresses as a function of
normalized maximum stress intensity factor: (a) R ˆ 0 and (b)
R ˆ ÿ1.
Fig. 7. Normalized crack opening stresses as a function of
normalized maximum stress for biaxial loading: (a) R ˆ 0 and
(b) R ˆ ÿ1.
the modi®ed ¯ow stress. Fig. 7 shows that the
crack opening levels are the smallest for outof-phase stress ®eld (k ˆ ÿ1, where k ˆ rxx =ryy ) The
largest occurs for in-phase stress ®eld with k ˆ 1.
It is also found that the crack opening stresses for
R ˆ ÿ1 are smaller than that for R ˆ 0, but the
scatter of normalized opening stress rop =rmax for
R ˆ ÿ1 is larger than that of R ˆ 0. These results
are consistent with the tendency, their faster crack
growth prevails when the out-of-phase stress ®eld
…k ˆ ÿ1† was applied and slower crack growth for
in-phase stress ®eld with k ˆ 1 [4,6]. The crack
opening results for both stress ratios were replotted in Fig. 8 in terms of the Kmax =K0 parameter.
Since this parameter is not related to the transverse
stress, the crack opening behavior under biaxial
Fig. 8. Normalized crack opening stresses as a function of
normalized maximum stress intensity factor for biaxial loading
of R ˆ ÿ1 and 0.
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
loading could not be successfully described by the
stress intensity factor parameter.
Generally speaking, fatigue crack growth and
crack closure are related to plastic strains at the
crack tip. The present investigation attempts to
®nd a new parameter related to plastic zone for
correlating the crack opening behavior for specimen geometries subjected to arbitrary remote
stress ®eld. A simple model for estimating the reversed plastic zone size for any stress ratio was
proposed [20]. The model can be modi®ed with the
¯ow stress r00 of Eq. (19) to consider the e€ect of
the transverse stress as
Dx0rev
ˆ
x0
(
81
)2
ÿ
1 ÿ rop =rmax
ÿÿ
:
2 ÿ rop =rmax ÿ R …rmax =r00 †
…23†
For four specimen geometries subjected to uniaxial stress and CCT specimen subjected to biaxial
loading, Figs. 9(a) and (b) show the crack opening
behavior rop =rmax for Dx0rev =x0 parameter at stress
ratios R ˆ 0 and R ˆ ÿ1, respectively. It is also
found that the normalized plastic zone size parameter describes successfully the behavior of
crack opening for four cracked specimen geometries at each stress ratio. At R ˆ ÿ1 and R ˆ 0,
the normalized crack opening stresses for CCT
specimen subjected to biaxial loading could also
be correlated satisfactorily by the parameter
Dx0rev =x0 regardless of biaxial loading ratios
…k ˆ ÿ1; 0; 1†.
On the other hand, the equation of the crack
opening stress for CCT specimen subjected to
uniaxial constant amplitude loading is written [21]
as
rop
ˆ A0 ‡ A1 R ‡ A2 R2 ‡ A3 R3 ; R P 0;
rmax
…24†
rop
ˆ A0 ‡ A1 R; ÿ1 6 R < 0;
rmax
where R is a function of stress ratio, rmax maximum stress, and r0 three-dimensional constraint
Fig. 9. Normalized crack opening stresses as a function of
normalized reversed plastic zone size for four specimens, and
CCT specimen subjected to biaxial loading: (a) R ˆ 0 and (b)
R ˆ ÿ1.
Fig. 10. The comparison of normalized crack opening stresses
for four specimens, and CCT specimen subjected to biaxial
loading with Eq. (24) in [21] for CCT specimen at R ˆ ÿ1; 0.
82
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
factor. The coecients Ai are given in Appendix A.
The equation is also modi®ed by the replacement
¯ow stress r0 with the new ¯ow stress, Eq. (19).
Fig. 10 shows the behavior of crack opening stress
for four specimens, and CCT specimen subjected
to biaxial loading at two stress ratios R ˆ ÿ1, 0
using Eqs. (23) and (24). This shows that Eq. (24)
for CCT specimen can be applied to predict the
normalized crack opening stress in terms of the
normalized plastic zone size parameter Dx0rev =x0 .
This is an especially useful result, because the behavior of crack opening stresses can be expressed
as the function of variable Dx0rev =x0 for a wide
range of stress ratios, rmax =r0 values, remote stress
®elds regardless of biaxial loading ratio k, and
specimen types.
6. Conclusions
· For the SENT, SENB, CCT, and DENT specimens and CCT specimen (biaxiality R), the
stress ratio R a€ects the crack opening behavior.
The e€ect is more pronounced for lower ratio of
R, especially at R ˆ ÿ1.
· The normalized stress intensity parameter
Kmax =K0 correlates well with the normalized
crack opening stresses for only four specimens
at R ˆ 0. However, the normalized plastic zone
size parameter Dx0rev =x0 provides good description for the crack opening stresses at two stress
ratios for four cracked specimens and CCT
specimen (biaxial loading).
· The Dx0rev =x0 can be used to predict crack opening stresses in di€erent crack geometries and
loads.
Appendix A. Weight function and coecients Ai
A.1. Weight function
The approximate weight function m…x; a† is
expressed as [14]
m…x; a† ˆ
r 4
iÿ3=2
8 X ÿ
bi a ÿ x0
;
paL iˆ1
…A:1†
where
1
b1 ˆ a1=2 ;
2
1 a dfr 3
‡ C1 aÿ1=2 ;
‡
b2 ˆ
2 fr da 2
1 a dfr
dC1 5
‡ C2 aÿ3=2 ;
‡a
b3 ˆ C1 ÿ ‡
da 2
2 fr da
3 a dfr
dC2 ÿ5=2
‡a
a
;
b4 ˆ C2 ÿ ‡
da
2 fr da
…A:2†
where a and x0 are dimensionless parameters for
the crack length and the x coordinate, a ˆ a=L and
x0 ˆ x=L, where L is a scale factor. The stress intensity factor fr …a† for the reference crack surface
load rr …x† with the characteristic load rS is expressed as
p
…A:3†
Kr ˆ rS fr …a† pa;
where the dimensionless stress intensity factor are
given polynomially as
fr …a† ˆ
n
X
Ai a i :
…A:4†
iˆ0
The coecients C1 and C2 are given as
5… Q ÿ I 0 † ‡ I 2
;
5I1 ÿ 3I2
3… Q ÿ I 0 † ‡ I 1
C2 ˆ ÿ
5I1 ÿ 3I2
C1 ˆ
…A:5†
for the center crack and
15…Q ÿ I0 † ÿ I2
;
15I1 ÿ 3I2
3… Q ÿ I 0 † ÿ I 1
C2 ˆ ÿ
15I1 ÿ 3I2
C1 ˆ
…A:6†
for the edge crack.
The Q in Eqs. (A.5) and (A.6) is given as
n
X
p
ai‡j
;
Ai Aj
Q…a† ˆ p
i‡j‡2
8fr …a† i;jˆ0
…A:7†
where Ai are the polynomial coecients of the
dimensionless stress intensity factor in Eq. (A.4),
and Ii is expressed as
J.H. Kim, S.B. Lee / Theoretical and Applied Fracture Mechanics 34 (2000) 73±84
Ii ˆ
n
X
Sj 2j‡1 j!aj
j
Y
kˆ0
jˆ0
1
:
2…i ‡ j† ‡ 3
…A:8†
Sj in Eq. (A.8) are the polynomial coecients for
the reference crack surface load rr …x† with the
characteristic load rS , which is expressed as
n
x i
rr …x† X
ˆ
Si
:
…A:9†
rS
L
iˆ0
For derivatives in Eq. (A.2), a simple relation can
be written for dQ=da as
dQ
pfr …a†
2
1 ofr …a†
ˆ p ÿ Q
‡
:
…A:10†
da
a fr …a† oa
8a
For SENT and SENB specimen, the dimensionless
reference stress intensity factor can be written as
[15]
fr …a† ˆ 1:12 ÿ 0:231a ‡ 10:55a2 ÿ 21:72a3 ‡ 30:39a4 ;
…A:11†
and the corresponding reference crack surface load
is
rr …x†
ˆ 1:
rS
…A:12†
For CCT specimen, the dimensionless reference
stress intensity factor is written as [16]
fr …a† ˆ 1 ‡ 0:128a ÿ 0:288a2 ‡ 1:525a3 ;
…A:13†
and is related to uniform crack surface load as
given by Eq. (A.12).
For DENT specimen, the dimensionless reference stress intensity factor is written as [16]
fr …a† ˆ 1:12 ÿ 0:0251a ‡ 1:849a2 ÿ 13:318a3
‡ 40:97a4 ÿ 53:42a5 ‡ 26:12a6 ;
…A:14†
and is related to uniform crack surface load as
given by Eq. (A.12).
A.2. Coecients Ai
The coecients are
1=a0
ÿ
p rmax
A0 ˆ 0:825 ÿ 0:34a0 ‡ 0:05a20 cos
;
2 r0
…A:15†
A1 ˆ …0:415 ÿ 0:071a0 †
rmax
;
r0
83
…A:16†
A 2 ˆ 1 ÿ A0 ÿ A 1 ÿ A 3 ;
…A:17†
A3 ˆ 2A0 ‡ A1 ÿ 1;
…A:18†
where ¯ow stress r0 is taken to be the average
between the uniaxial yield stress and uniaxial ultimate tensile strength of the material.
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