Nuclear Engineering and Design 111 (1989) 109-121
North-Holland, Amsterdam
109
O N T H E P A T H I N D E P E N D E N T T* INTEGRAL IN N O N L I N E A R A N D D Y N A M I C
FRACTURE MECHANICS
T. N I S H I O K A 1, T. F U J I M O T O 1 and S.N. A T L U R I 2
i Department of Ocean Mechanical Engineering~ Kobe University of Mercantile Marine, 5-1-1 Fukae Minamimachi,
Higashinada-ku, Kobe, 658, Japan
2 Center for the Advancement of Computational Mechanics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Received 26 May 1988
Recently the authors have derived new types of path independent integrals in which the theoretical limitations of the
so-called J integral are overcome. For elastodynamic crack problems, a path independent integral J ' (dynamic J integral)
which has the physical meaning of energy release rate was derived. Later the J ' integral was extended to a more general form
of the path independent integral T* (generalized J integral) which is valid for any constitutive relation under quasi-static as
well as dynamic conditions.
This paper presents recent developments in a series of studies on the T* integral in nonlinear and dynamic fracture
mechanics. The first study concerns the behavior of the T* integral in an elastoplastic dynamic fracture initiation test
specimen. The T* integral exhibits excellent path independency despite the impact loadings and large plastic deformations.
The second study reveals the invariance of the T* integral with respect to the shapes and sizes of the fracture process zone.
The last study shows the methodology of a direct measurement of the T* integral using the method of caustics (shadow
pattern method). These results serve to illustrate the validity of the T* integral as a unified crack-tip parameter for various
types of fracture mechanics problems.
1. Introduction
In order to assess the Leak-Before-Break (LBB) condition for a primary piping system of nuclear power
plants, and the integrity of a nuclear pressure vessel
under the Pressurized Thermal Shock (PTS) accident,
the establishment of nonlinear and dynamic fracture
mechanics is essential. The so-called J integral has been
the main focus of elastic-plastic fracture mechanics for
the last fifteen years or so. However it is now well
understood that the J integral [1] is valid theoretically,
only in the context of incipient crack growth in nonlinear elastic materials under quasi-static conditions. Once
crack growth commences, J ceases to be a valid parameter.
Recently the authors have derived various new types
of path independent integrals in which the theoretical
limitations of the so-called J integral are overcome. For
elastodynamic crack problems, Nishioka and Athiri [2]
have derived a path independent integral J ' which has
the physical meaning of the energy release rate. Later,
Atluri, Nishioka and Nakagaki [3] have derived a more
general path independent integral T* which is valid for
any material-constitutive model under quasi-static as
well as dynamic conditions. The T* integral can be
considered to be a natural extension of the so-called
(Rice's static) J integral and the dynamic J integral
(J'). Thus the T* integral is now recognized as a
unified crack-tip parameter governing quasi-static as
well as dynamic propagation of cracks in elastic as well
as elastic-plastic materials.
This paper presents recent studies on the T* integral
in nonlinear and dynamic fracture mechanics. Elasticviscoplastic finite element analyses were carried out for
several types of the problems. The first problem concerns the T* integral in a compact tension (CT) specimen for elastoplastic dynamic fracture testing. The T*
integral exhibits excellent path independency despite
the impact loadings and large plastic deformations.
In this analysis, various experimental techniques to
measure the initiation fracture toughness were also
simulated and were compared with the behavior of the
T* integral. It is found that the area under the load vs.
crack opening displacement curve correlates well with
0 0 2 9 - 5 4 9 3 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.
( N o r t h - H o l l a n d Physics Publishing Division)
T. Nishioka et a L / On the path independent T* integral
110
the T* integral, except for the earlier period of impact
loading. This indicates the possibility of experimental
measurement of the T* integral.
Next, the simulation of elastoplastic dynamic crack
propagation was performed. The invariance of the T*
integral was verified for the shape of the fracture process zone in the vicinity of the propagating crack-tip.
This property is important because the shape of fracture
process zone is not yet fully explored.
This paper also shows the methodology of a direct
experimental measurement of the T* integral. The
method of caustics (shadow pattern method) has proven
to be a powerful optical method to measure stress
intensity factors in static and dynamic fracture mechanics problems. An attempt was made to extend the
method of caustics for nonlinear (elastic-plastic) fracture mechanics problems. The formation of the caustic
pattern for an elastoplastic crack-tip in a compact tension specimen was simulated by the finite element analysis. The relations between the T* integral and the size
of caustic pattern were obtained.
Fig. 2. Global and local coordinate systems.
1. The energy release rate G is given from global energy
considerations as
de
d~
dK"
G= da
da
da
(1.a)
= fst-t,-dda
d u i d S - d fv(W+g)dV,
(1.b)
where ti are the external tractions on S t, u i are the
displacements, and W and K are respectively the strain
and kinetic energy densities.
Now consider a "core" region, V~, near the crack-tip,
which is enveloped by the path F,. Thus, the region
(V-V,) excludes the crack-tip. Considerations of energy
balance in this region ( V - V,) lead to
fs-r, ti--dUidadS-~a,v_e,d f
(W+K)
dV=0,
(2)
2. Path independent J' integral for dynamic energy
release rate in elastic solids
where ( S - F , ) is the boundary of ( V - I1,). Use of eq. (2)
in eq. (1) results in the following relation for G:
We restrict our attention here to solids that are
linearly or nonlinearly elastic. We consider the dynamic
propagation of a crack in a self-similar fashion, with an
instantaneous crack velocity C(t) =-~, as shown in fig.
G=
- rl i~0
m ~([Jr,
#1"t i - d-dudaiS - ~ - - ~ f v ( W + K
)
dV~.
f
(3)
Using the well-known differential relation for crack
extension d/da = ~/Oa- O/3x 1, and the concept of
"subtracting out singularity" [3,4], the energy release
rate can be rewritten as
G=-r,~0LJv,[lim*f[(W+K)nl-t,~]dS),
(4,
wherein, as seen from fig. 1, x 1 is along the crack axis,
and n I is the direction cosine between the x a axis and a
unit normal to the contour F,.
For a crack propagating in a self-similar fashion at
an angle O 0 measured from the global X 1 shown in fig.
2, the energy release rate G can be expressed as [4]:
G = (CjC)Gk
,,
= GlCOS O 0 + Gzsin O 0
(5)
and
/X)<)<X)<)<
"- Su
Fig. 1. Nomenclature for a cracked body.
where C k and G k are the components of crack velocity
T. Nishiokaet al. / On thepath independentT* integral
and energy release rate with respect to the global coordinates Xk; n, the outward normal direction cosines
with respect to Xk; ( ),k denotes a( )/aX k.
The components of the path independent integral
which are equivalent to the components of energy release rate were derived by Nishioka and Atluri [2]:
J~"=- r,~olim( fr[ ( W + K ) n k - t~U~'k] dS )
= fr+r[(AW+ A K ) n k - (ti+ Ati)AUi,k
--Atiui.k] dS
+f,vr--V,
--AEij(Oij + I AcJij), k
(7a)
+p(n, + an,) au,,~ -
(7b)
where p is the mass density. A superposed dot denotes
a total derivative with respect to time t. F is an arbitrary far-field path which starts at the lower surface of
the crack and ends at the upper surface. The path
independence of the far-field expression in eq. (7b) can
be easily shown using the divergence theorem, equilibrium equations and other basic equations. The body
forces f,. other then inertia forces can be included easily
replacing p/i i by (pigi -fi).
For elastostatic crack problems, the J~ integral reduces to the Jk integral derived by Budiansky and Rice
[5]. However, it should be noted that the definition of J2
in their paper does not involve the crack-face integral,
which accounts for discontinuities of W between the
crack-faces. Thus, the original expression of the J2
integral is not path independent.
3. Path independent T* integral as a crack-tip parameter in nonlinear (elastic-plastic) fracture mechanics
We now consider dynamic crack problems in
elastic-plastic and elastic-viscoplastic materials. Under
an incremental flow theory of plasticity, it is natural to
consider an incremental measure of the strength of the
crack-tip field. Based on this idea, Atluri, Nishioka and
Nakagaki [3] have derived a general form of path independent integral T* which is valid for any materialconstitutive relation under quasi-static as well as dynamic conditions. The T* integral can be written as:
T~ -= f0tT~,* dt = E Ark*,
(8)
ATff =-fr[( aw + AK)n k - ( t i + Ati)Aui, k
-Atiui.,] dS
p ( a, + a , ' , , ) a a , , ,
+ pAi~u~,k - pA~i~.k ] dV,
(f
[(W+K)nk-tiuij, ldS
r,~ot~r+ro
= lim
+ fvr_v[ piiiui,k-- ph,it,,k] dV},
111
(9a)
(9b)
where E denotes the summation along the loading history; AW = (z~eu + ½Aeu)Ac~j is the incremental stress
working density; o~j and c~j are the stress and strain,
respectively. The path independence of the T* integral
can be easily shown without using a constitutive relation. Thus, the T* integral is valid for any constitutive
model.
The T* integral can be also expressed in a total
form as follows:
r: fr[(w+ K ) . , --
,,.,,,1 as
fr+rot(w+K).,-
(10a)
tiui,k] dS
00b)
The near-field path F, in eqs. (9) and (10) will be taken
along the boundary of a fracture process zone. Usually
the size of fracture process zone may be finite for
growing cracks in somewhat ductile (elastic-plastic)
materials. The process zone of brittle (elastic) fracture
may be very small comparing the size of the crack itself.
Thus, an infinitesimally shrinking path to the crack-tip
(limiting path) is used for the J ' integral as given in eq.
(7).
Note that in an elastic-plastic material under arbitrary loading history, W is the total accumulated increments of stress working density. Since ou is not a
single-valued function of cu, in general, we have W k 4:
o,j cu, k. For elastic materials, since W corresponds to
the strain energy density, the Tk* integral reduces to the
J~ integral for elastodynamic cases. Moreover, the Tk*
integral reduces to the Jk integral derived by Budiansky
and Rice [5] for elastostatic cases. Thus, the T* integral
is a natural extension of the so-called (Rice's static) J
integral and the dynamic J integral ( J ' ) .
The 7"** integral can be regarded as the Xk components of the vector integral T* emanating from the
112
T. Nishioka et al. / On the path independent T* integral
crack-tip. Thus, the following ordinary coordinate
transformation rule can be used for the T* integral [8]:
T : Io0=0 = ak,.:rd,
(11)
where, for two-dimensional case, aaa = a22 = cos Oo;
alz
=
-a21
= sin 0o.
The features of the T* integral can be summarized
as follows.
(i) It is a path independent integral type crack-tip
parameter even for large amounts of crack growth
and general nonsteady conditions.
(ii) It is a valid crack-tip parameter for arbitrary histories of loading and unloading.
(iii) It is easily defined for nonisothermal, nonhomogeneous material conditions [6].
(iv) It includes the so-called J integral and the dynamic J integral ( J ' ) as special cases. Thus, it can
be considered as the generalized J integral for any
material.
(v) It can be defined for finite deformation [7].
(vi) It can be appfied to mixed mode fracture problems
[81.
It is emphasized that the above features (i)-(iii)
assure the applicability of the T* integral to the Pressurized Thermal Shock (PTS) problems considering dynamic elastic-viscoplastic crack propagation in a thermally strained solid.
! Laj
I
i
J<
8
L=B2.Smm
W-50,Omm
H=30.Omm
B=25.0mm
O=12.5mm
h=19.0mm
h~ -IO.Omm
h2-I ,Omm
a =27.98mm
L
Fig. 3. A compact tension specimen for dynamic fracture
initiation test.
employ a power law type in the form introduced by
Bodner and Symonds [9] which is an appropriate choice
for many metals:
f-f°l"
(13)
*(I)= --ff-o1'
where n is a material constant to be determined by
experiments.
In the case of one-dimension (uni-axial case), the
above constitutive relation expressed by eqs. (12) and
(13) reduces to
~vp = y [ o / o s t ( , p ) - 1]"
(14)
or inversely
4. Behavior of the T* integral in elastoplastic dynamic
fracture testing
In order to investigate the behavior of the T* integral under impact loadings, a compact tension specimen for dynamic fracture initiation testing was analyzed
by a viscoplastic finite element program. The specimen
was assumed to be loaded by pins as shown in fig. 3,
with three different loading velocities of 1 m / s , 4 m / s
and 8 m / s .
The material properties of ASTM A508K steel were
used with Young's modulus E = 206 GPa, Poisson's
ratio p = 0.3 and the initial yield stress o0 ---400 MPa.
In the usual m a n n e r for nonlinear inelastic analyses, it
is assumed that the total strain rate can be separated
into elastic and viscoplastic components. The rate of
viscoplastic strain is defined by
= 3'(q~(f)) : - ~ j ,
(12)
where 3' is the fluidity parameter controlling the
viscoplastic flow rate. For the flow rate function q,, we
o = Ost(,p)[1 q- (~vp//3')l/n],
(15)
where % (c p) is the static stress-strain relation including
the effect of strain-hardening. In the special cases for
3' = 0 and 3' = o0, the above constitutive relation represents, respectively, the elastic and (rate-independent)
elastic-plastic models.
From the results of dynamic uniaxial tensile test [10],
the values of the viscoplastic parameters were chosen as
3' = 194 s -1, n = 1, and the static stress-strain relation
is given by
o,t = o0 + 200 t a n h ( 2 0 ( c - % ) )
[MPa]
(16)
where c is the uniaxial strain and c o = oo/E. Thus,
dynamic yield stress is estimated by eq. (15), for different rates of strain.
Fig. 4 shows the finite element mesh pattern which is
modeled by eight-noded isoparametric elements. Six
contour paths for the calculation of the T* integral are
considered as shown in the figure.
First, the variations of various energies obtained by
elastodynamic analyses are shown in fig. 5, for the
113
T. Nishioka et al. / On the path independent T* integral
IIIIII I I
iMT[I .....l fl
li I-1-1-1-[-[---~-~ - --~l
li-H]] f F-~-ii 1
L
li[ll-III
I! l i - I T l l l
I':1 I I;,1~
!1
ii
1
il
~1 @@
@®
®®
I
a=27,98mm
il 1
Fig. 4. Finite element mesh for the compact tension specimen.
loading velocities of 1 m / s and 4 m / s . In the case of 1
m / s , since the kinetic energy in the specimen is negligible, this case can be considered as a quasi-static one.
Therefore, we can use all the experimental formulas
developed for static fracture toughness evaluation. On
the other hand, in the case of 4 m/s, the kinetic energy
becomes large. Thus fully dynamic analysis is necessary
and the static exi>erimental formulas may not be applicable.
Next, the results for elastic-viscoplastic analysis will
be shown. The computed shapes of plastic zones at
various instants of time are shown in fig. 6 for the case
5 0 -v=4m/s,
•
|¢
/
Total
Work Bone
4 , 0 ~ ~ Kinetlc Energy
|r; 5trainEnergy
eeaee
e
e e
V=lm/s
--o 2 . 0 - ( o Tot.el Nork none
~o'o.
.. .
0
...
. ..
.
.. .
50
.. .
. ..
. .
t00
.
.
.
.
.
.
"150
o
5o
t [ ~sec.]
I oo
150
t [ ~sec.]
Fig. 5. Energy variations in the compact tension specimen for the elastic case.
?
I:=56~sec.
t=112~secI
,
?
?
t-84
;7
FI
k._~
asec
I
t=140
~sec.I
,
/)
Fig. 6. Development of plastic zones under the loading velocity of 4 m/s.
f
114
T. Nishioka et al / On the path independent T* integral
v
0.5 V=Imls
2.0
%
•
•
•
+_
7--
o
o t=350 usec.
o
o t=280
o
o t=175 #sec.
•
V=Sm/s
~sec.
o
fl
o
n
--o
n
~ t=85 nsec.
n
e t=80
0.0
PathNo.
e
V=4mls
1.0
o
e
~
t~14O ~sec.
"~
L
m
1.0
c ~
~
o
e
0.5
o
o
,_
o
o
c
c
-
-
~
t=112 ~ec,
~
*~_
o
t=4O usec.
o
t=35 ~sec.
- t=5G usec.
o
t = 3 0 )~sec.
0.0
3 4
5 5
No.
Path No.
Fig. 7. Path independence of T* integral under different loading velocities.
0.0
i
2
Path
V=Im/s
i n t e g r a
120 V=Sm/s
~ 3OI oT'
__
(Iced point, di~olecet~'nt)
4J
t=50 ~sec.
t=70 ~sec.
~
0.5
t = 8 4 ;zsec.
_
usec.
20
J,xp(t)
110' • T' integra]
displacement)
[craCk ~
•
100
(Ioed ~ i ~
J.~lt)
l e r . ~
di~Imcement)
diapl~t)
-~ 10
?
,.#
90
©
0
50
1O0
150
200
250
t [ /lsec.]
--
300
350
-
-
~E
80 ¸
g
70 ¸
%
60
V=4m/s
T' integral
50
•
J,,.Ct){c,,~. ~
di,~l,,=,---,u
/
2
/"
.--
#o~
•
m
~40
30
30
20
20
10
10
O-S
:
4
_50
-~
(load point d i @ l m ~ t l
40.
-4
2
2
Z A
//
..." .....r
///
/
;
;
i-i"
,
+
,
i
,
,
,
,
i
i
I
0
50
t[
0
t [ ~sec. I
Fig. 8. Variations of the T* integral and the experimental J integrals.
50
100
100
~sec. ]
T. Nishioka et al / On the path independent T* integral
of 4 m / s . Relatively large plastic zones around the
crack-tip and the loading point are seen in the figure.
In fig. 7, the values of the T* integral were plotted
against the path number. Excellent path independency
of the T* integral can be seen for all the loading
velocities despite the impact loadings and large plastic
deformations.
Several experimental formulas for the measurement
of initiation fracture toughness were also evaluated by
the finite element analysis. The experimental J integral
[11] which is basically developed for the measurement
of static fracture toughness, is given by
Jcxp(t)
2A l + a
Bb 1 + a 2
(17)
and
a = ¢(2a/b)2+ ( 2 a / b ) + 2 - (2a/b + 1);
b=W-a,
The process zone of brittle (elastic) fracture may be
negligibly small comparing with the size of the crack
itself. Thus, an infinitesimal process zone (V, ~ 0) can
be considered in the evaluations of the J ' integral and
the other integrals given by eq. (18). However, if the J '
integral or the other integrals converge to different
values for different shapes of infinitesimal path (F, --, 0),
those integrals cannot be used as a crack-tip parameter
due to non-uniqueness for the same state of crack-tip
field. For this reason, typical integrals I~ ") (m = - 1 , 0,
and 1) were evaluated for different shapes of infinitesimal process zone using the analytically obtained
asymptotic solutions [2] at an elastodynamically propagating crack-tip.
First, using a circular near-field path F,, it has been
shown [2] for the crack with O 0 = 0 that:
I(1) = j "1_-- ~2~
1 ( A I ( C ) K ? +An(C)K2II
where A is the area under the load vs. displacement
curve. This area is estimated in two different ways using
the load point displacement or the crack mouth displacement.
The behavior of the T* integral was compared in
fig. 8, with those of the experimental J integrals. The
behavior of the experimental J integral obtained by the
crack mouth displacement agrees well with that of the
T* integral. Especially for the time after several periods
of natural frequency of the specimen. This indicates the
experimental measurability of the T* integral.
5. Invariance of the T* integral with respect to the
shape of the fracture process zone
+Am(C)K2I},
Nishioka and Athiri [12] have also shown the existence of many path independent integrals which do
not have the meaning of energy release rate. These
integrals including the J ' integral can be summarized as
,
nk
,
1
11(o)= "~# ( F , ( C ) K ?
12(o)
where m is an arbitrary number. In the case of m = 1,
lkO) reduces to the J~ integral and has the meaning of
energy release rate. Other integrals I~")1,,,,1 are not
equivalent to the energy release rate except for stationary cracks (C = 0).
Eli ( C ) K ~
+
Fill (C)K?n ),
~ v ( C ) g I KII,
g
11(-1) = ~
(20b)
( F I ( C ) K } -4- FII(C)K?I + Ell I ( C ) K 2 1
},
(21a)
I2~ - ' ) =
F'v(C------))K,K,,,
/*
(21b)
wherein, clearly, the set of functions A M (C), ffM (C),
FM (C); M = I - I V are distinct from each other, and are
given in [2]. We note here only the set A M (C); M =
I-IV:
A I ( C ) = ,81(1 -- ,sff)/D(C),
(22a)
A I I ( C ) = ,82 (1 -- ~82)/D(C),
(22b)
AIIi ( C ) = 1/,82,
(22c)
A,v(C) =
(18b)
+
(19b)
(20a)
=Ffi%{fF+Vc[(W+mK)nk--tiui,k] dS
+ fvr -V, [#iiiui,k--mofiiki.k] d V } ,
(19a)
A I r ( C ) KIK n,
/x
12m = J2
5.1. Elastodynamic crack propagation
hm
115
(,8, _,82)(1_ , 8 2 )
(D(C)} 2
x[ {4,8',82+ (I +,82)}(2+,81 +,82)
[ 2((1 + ,8,)(1+ ,82)
- 2 ( 1 + ,82)[,
]
(22d)
116
T. Nishioka et aL / On the path independent T* integral
&'-Component
~
l{'')
J~- component
Mixed Node (KI=I.O, K~=0.5)
2.5
/
C=O0
-1.0
C=02C~
2.0
C=04C~
~ / - I ~ ' I~'~=
)= J;
= rl
J;=T~
C=0.6C,
-1.5
g[z( 1 ) =J2' =T~
C=06C~
1.5
-2_(
C= 0.2 C~ ~
1.0
~"--
C=O.O
-2.!
L
10"~
i
10"I
i
i
10 0
I0'
~
I 2
,
10 ~
10-2
10-I
100
10'
102
10~
~/~
Fig. 9. The invariance of T* integral or J ' integral with respect to the shape of the infinitesimal process zone.
where D ( C ) = 4fll/~2 - (1 + & 2) 2., p~ 1 - c2/cg; p~
= 1 - C 2 / C 2 and Cd and C~ are the dilatational and
shear wave speeds, respectively.
For rectangular paths, the I2 '') integrals were
numerically evaluated. Fig. 9 shows the variations of
Ik('~) integral with the aspect ratio of infinitesimal rectangular process zone. The integral values for a circular
path were compared with those for a square path as
depicted in figs. 9a and 9b. Only the J [ integrals or the
Tk* integrals show the process-zone-shape invariance
[14].
=
5.2. Nonlinear (elastic-viscoplastic) dynamic crack propagation
ble. Moreover if we take the limit of F, to the crack-tip,
the A T* integral vanishes because of a weaker singularity in the vicinity of a growing crack-tip in elastic-plastic materials [13]. For these reasons, a finite size of
process zone (near field path) should be employed in
the case of elastic-plastic crack growth. However, the
shape of fracture process zone is not yet fully explored.
Therefore, the invariance of the T* integral with respect to the shape of the process zone is very important.
a !i2H200200
Trl T
In the case of elastic-plastic crack growth, the fracture process zone near the crack-tip may not be negligi-
I--]I 2,
I
II ~
W = 400
Ii
/,,,
~Fine Mesh paLterm
(a) Moving Process Zone
(b) Exf,ending Process Zone
Fig. 10. Fracture process zone models.
Fig. 11. An edge cracked strip subject to prescribed displacements and finite element mesh for the strip.
T. Nishioka et aL
Static EIestoplastic Crack Growth
Static Elastoplastic Crack Growth
Moving Process Zone
Extending Process Zone
o;=0.5mm
O~=0.Smm
AE=1.0mm
At=1.0mm
m~=1.5mm
m E=1,Smm
1.0¸
1.0
#
117
/ On the path independent T* integral
Stationary
crac~
Stationary
a |
crack
B
0.5 •
@
0.5
Propagat ing crack
D
m
• Propagatingcrack
@
¢~,B
o.o
m
i~a a
o:s
~///Vo
Lo
0.0
ao
1.0
2.0
o~
1.0
o.o
V/~o
1.o
2.0
&~cmm)
Aacmnu
(a)
(b)
Fig. 12. T* integrals for different sizes of process zone during static crack growth; (a) moving proce~ zone model, (b) extending
process zone model.
scribed displacement o at the edges as shown in fig. 11.
The crack is assumed to propagate with a constant
speed (C = 0.2C s) when the displacement reaches to a
critical value vo = 0.75Oy s H(1 - p ) / E . A relatively fine
mesh pattern as shown in fig. 11, with 232 isoparametric
elements and 771 nodal points was used for the finite
element analysis.
The T* integral was numerically evaluated by using
the far-field path expression given in eq. (9b). In the
We consider two types of process zone models as
shown in fig. 10. In the moving process zone model (fig.
10a), the process zone moves with the crack-tip, while in
the extending process zone model (fig. 10b), the process
zone remains as a wake of the advancing process zone.
A viscoplastic analysis of dynamic crack propagation
was carried out to investigate the invariance of the T*
integral for different models and for different sizes of
finite process zone. The specimen is subject to a pre-
(Viscoplastic)
(Viscoplastic)
C=0.2Cs
C=0.2Cs
Moving Process Zone
Extending Process Zone
o~=0.5mm
O~=0.5mm
AE=1.0mm
AE=1.0mm
e E=1.5mm
e E=1.5mm
Stationary
crack
,*
@
e ~ ~ r ~ c I ; ~ W ~ @
%
~
~
~
~
;
~
0.5
_a ~
B
B
Stat i onary
• crack
1.0
~acmm]
(a)
2.0
~
@ °~°~=~D~
^.~__
_ ~
.
0.5i
@
"
G
oJ V//Vo
i~@
~.o
oD
a
=
m
d.s
v//~o
8
Propagati ng crack
=
0.0
w
1.0
1.0
0.5
Propagating crack
~C)
i
0
1.0
20
z~acmm)
(b)
Fig. 13. T* integrals for different sizes of process zone during dynamic crack propagation; (a) moving process zone model, (b)
extending process zone model.
T. Nishioka et aL / On the path independent T* integral
118
present analysis, F is taken as the external boundary of
the specimen. Fig. 12 shows the behavior of the T*
integral for static crack growth (C = 0.0). The values of
the T* integral are nearly the same for both the models
and for different sizes of the process zone (~ = 0.5, 1.0
and 1.5 mm). Fig. 13 shows the results for dynamic
crack propagation ( C = 0.2Cs). The values of the T*
integral in this case are also nearly the same for all the
types of process zone. It is noted that the size of the
active plastic zone rp around the crack-tip was about 14
mm.
These results indicate the invariance of the T* integral with respect to the shape of the fracture process
zone. As shown here, the incremental form of the T*
integral can be used for both the models. However, it
should be pointed out without showing the results here
that the total form of the T* integral given by eq. (10)
in conjunction with the moving process zone model
cannot be used due to numerical difficulties [15].
i
Path
No.
The method of caustics (shadow pattern method) has
proven to be a powerful optical method to measure
stress intensity factors in static and dynamic fracture
mechanics problems. An attempt was made to extend
the method of caustics to measure directly the T*
integral in elastic-plastic materials. Fig. 14 illustrates
the formation of caustic pattern. The surface of the
specimen around the crack-tip deforms due to the contraction in the thickness direction• Incident parallel
light beams are reflected by the polished surface of the
2
3,4
Fig. 15. Finite element mesh for a CT specimen.
specimen and the fight beams no longer will be parallel
and will gather on the so-called caustic surface as
depicted in fig. 14. This bright light concentration on a
screen, which surrounds a shadow area, is called caustic
curve.
The positions of the beams on a screen can be
calculated by the following relation [16]:
W = r + w,
6. Methodology of a direct experimental measurement of
the T* integral
1
(23)
where W and r are the corresponding position vectors
of a beam on the screen and the specimen, respectively.
The deviation of a beam w can be evaluated by
w = - 2 z 0 grad f ( x l, x2),
(24)
where z 0 is the distance between the screen and the
specimen and f ( x 1, x2) is the total displacement of the
specimen surface in the thickness direction x 3. The
surface deformation f ( x 1, x2) is calculated by using the
generalized plane stress condition as follows:
f ( x a , x2) = y ' A f ( x l ,
x2)
(25)
and
A f ( x a , x2) = foB/ZA¢33(x,, x2) dx3
~-~ #.SCREEN
"
~
= B ( ( A°aa + A ° z z ) 1 -
P
PARALLEL
LIGHT
-- ( A { l l
+
A[22)} ,
(26)
•
IC SURFACE .~ POLISHED
~-'-- S PEC IMEN
SURFACE
Fig. 14. Schematic of the optical setup for caustics by reflection [16]•
where ~ will be taken over the loading steps or time
steps; B is the specimen thickness.
Fig. 15 shows the finite element mesh pattern of a
C T specimen used for the numerical simulation. Due to
the symmetry of the problem, only the upper half of the
specimen was modeled. Fig. 15 also shows the contour
paths for the calculation of the T* integral.
Fig. 16a shows the points of incident light beam on
the specimen, which are actually the Gaussian integra-
119
T. Nishioka et a L / On the path independent T* integral
".'.'.1:::.'.'."
•
. . . . . . .
: ::
:
::
: ::
:
::
:: : : : : : : : : : : : : : : :
::'.::..
: :: :
:
::
: ::
:
::
:::!!
: ::'.:":::'~,i;:.::":.'::
:
:::.#.~-':!~:.~!:.~::
..............
:: ~!,..~';:
...
,...
.... . . . . . . . . . . . . . . . . . . . . . . . .
:.:.!!!
:::~
!
-"~::;~.: A... :.
Screen
Spec i men
:
•
,
...............
. :'-~! :'. ! ::.: ::.: : "::.'.~-~C.:'::. '.'', -.
. . . . . . . . . . . . . . . . . . . . . . .
: ::
........'..:.
?
,
.
.
...
. . . .
: . . . . . . . . . . . .
. : . . .
. . . . . . . . . . . . . . . . . . . . . . .
(a)
lb)
Fig. 16. Numerical simulation for the formation of caustic and shadow patterns; (a) parallel light beams on the specimen, (b)
reflected light beams on the screen.
tion points. First, an elastic analysis was made and the
beam positions on the screen were calculated as shown
in fig. 16b. The theoretical caustic curve was also indicated by a solid line. The simulated shadow pattern
agrees very well with the theoretical one.
A n elastic-plastic analysis was made, using the same
material properties used in Section 4. The development
of plastic zone around the crack-tip is shown in fig. 17.
In fig. 18, the shadow patterns obtained by elastic-plastic analysis are compared with those by elastic analysis.
In the elastic-plastic case, the shadow patterns become
flatter due to the plastic deformation around the cracktip.
Fig. 19 shows the relations between the T* integral
and the maximum size of shadow pattern in the x 2
direction, D. For an elastic case, this relation is theoretically given as
wherein M is a constant determined by the optical
setup and the material properties; F ( C ) is a correction
factor for non-zero crack velocities [17]. For lower speed
crack propagation ( C < 0.2C~), this correction factor
can be neglected ( F = 1.0). The theoretical curves for
elastic material calculated by eq. (27) were also indicated in fig. 19.
elastic
elsaEic-pleaEic
(a)
case
~:~
(b)
.
... ,j.:,
.
r* =F(C)M
case
.:
.... ,~;~..,
,,.
.,.
.
:.
•.
.
:
:.! k
:'
- -,
(27)
D5
ZO = 0 . 2 5
LOAD
(e)
m
ZO = 0 . 2 5
= 28.0
KN
RO = 2 . 5 2
mm
LOAD
..
•
•
•
= 28,0
:...
::.::~::....
- . ~;%'. ~l..'..
".' 4~.. ": :". ,,~Y.'.
..:,<~. . . . .
~.,.,:.
.
,.
,.
•
' -:.,
,.....,.
•
.
• : :.~ ~t,.a'~:.
ZO = 0 . 2 5
LOAD
e
///If \\
12
16
20
24
~
= 32.0
RO = 2 . 6 5
32
(KN)
Fig. 17. Development of plastic zone around the crack-tip.
KN
(d)
.
.
.•
..
~ :i.: ~i~..",:.
...
'= f -¢:L t , ~
~ p l .I
• .. : ,;,y...
".
m
" •
....
""
.
..
.?,?:
.:.
:
m
ZO = 0 . 2 5
m
KN
mm
LOAD
= 32.0
KH
Fig. 18. Comparison of caustic (shadow) patterns for the
elastic and elastic-plastic cases.
120
T. Nishioka et al. / On the path independent T * integral
EiasLoplasLic Analysis
24
2O
I ~ r l ~ l reeul~ of
~ G e l u t a p l u t l c enelysle
-- El~.lc
b~
c
r'ellLIone
/
,~
~,JZo ~ 0.25 m
ir
e /
,
16
©Jo
12
/
,
Ceuetlc curve
/ /
'/,
//
8
/~
0
2
~
j
4
B
/~
/'*-~
///
/
~Zo
/
f(xl, x2)
= 0.5m
,
/'/
8
10
12
D
E
/
; //
,/
,' /
/ Za = [3.75 m
, /
/
///
e~,,'
4
0
/
, ,
t
c~
c~
,
p/
jI
14
D [mm]
Fig. 19. T* integral vs. the maximum size of caustic pattern.
f,
FI( C ), etc.
ffi (C),
H
I,II,III
J~xp
J;
K
K 1, etc.
M
The generalization of T* vs. D formula for elasticplastic materials is now under way. Once the similar
formula with eq. (27) is established, the T* integral can
be measured directly in laboratory fracture specimens.
nk
P
ti
T~
S
s,
7. Closure
Ui
V
Recent developments in the path independent T*
integral which is applicable to nonlinear and dynamic
fracture mechanics problems were presented here. The
results presented in this paper serve to illustrate the
validity of the T* integral as a unified crack-tip parameter. To validate the T* integral as a powerful fracture
parameter, further experimental-numerical hybrid type
studies are needed.
K
Uo
W
W
Xk
x~
Zo
Olkm
Acknowledgments
Y
F
A series of studies presented here was supported by
the Science and Technology Grant from Toray Science
Foundation. The first author acknowledges this support. He would also like to thank graduate students Mr.
M. Kobashi and Mr. H. Yagami for their numerical
calculations.
rc
r,
¢ij
¢0
Cp
oo
Nomenclature
a
A
A i (C), etc.
B
C
E
crack length,
area under load-displacement curve,
functions of crack velocity,
specimen thickness,
crack velocity,
ff
p
ff
Oij
Oo, %s
~,(%)
etc.
dilatational wave velocity,
shear wave velocity,
maximum size of caustic pattern,
Young's modulus,
deformation of specimen surface in the
thickness direction,
components of body force,
functions of crack velocity,
functions of crack velocity,
half height of specimen,
fracture modes,
experimental J integral,
components of J ' integral,
kinetic energy density,
total kinetic energy,
stress intensity factors,
index for fracture mode,
normal direction cosines,
input energy (work done),
components of traction force,
components of T* integral,
surface of cracked solid,
mechanical boundary,
components of displacement,
domain of cracked solid,
domain of fracture process zone,
prescribed critical displacement,
strain energy density or stress working
density,
total strain energy,
local coordinate system attached at
crack-tip,
global coordinate system,
distance between specimen and screen,
coordinate transformation tensor,
viscoplastic fluidity parameter,
far field path,
path on crack surface,
near field path or boundary of process
zone,
half size of fracture process zone,
components of strain,
yield strain,
plastic strain,
viscoplastic strain,
angle between x 1 axis and X 1 axis,
shear modulus,
mass density,
equivalent stress,
components of stress,
yield stress,
static stress-strain relation,
T. Nishioka et al. / On the path independent T* integral
grad( )
( )
( ),k
f( ) dS
f( ) dV
~.
W
A(
)
gradient operator,
time derivative,
partial derivative with respect to X k,
surface integral,
volume integral,
s u m m a t i o n over loading steps or time
steps,
vector,
incremental quantity.
References
[1] J.R. Rice, A path independent integral and approximate
analysis of strain concentration by notches and cracks, J.
Appl. Mech. 35 (1968) 379-386.
[2] T. Nishioka and S.N. Atluri, Path-independent integrals,
energy release rates, and general solutions of near-tip
fields in mixed mode dynamic fracture mechanics, Engrg.
Fracture Mech. 18 (1983) 1-22.
[3] S.N. Atluri, T. Nishioka and M. Nakagaki, Incremental
path-independent integrals in inelastic and dynamic fracture mechanics, Engrg. Fracture Mech. 20 (1984) 209-244.
[4] S.N. Atluri, Path-independent integrals in finite elasticity
and inelasticity, with body forces, inertia, and arbitrary
crack-face conditions, Engrg. Fracture Mech. 16 (1982)
341-364.
[5] B. Budiansky and J.R. Rice, Conservation laws and energy release rates, J. Appl. Mech. 40 (1973) 201-203.
[6] S.N. Atluri, M. Nakagaki, T. Nishioka and Z.B. Kuang,
Crack-tip parameters and temperature rise in dynamic
crack propagation, Engrg. Fracture Mech. 23 (1986)
167-182.
[7] F.W. Brust, T. Nishioka, S.N. Atluri and M. Nakagaki,
Further studies on elastic-plastic stable fracture utilizing
the T* integral, Engrg. Fracture Mech. 22 (1985)
1079-1103.
121
[8] T. Nishioka and S.N. Atluri, On the computation of
mixed-mode K-factors for a dynamically propagating
crack, using path-independent integrals J~, Engrg. Fracture Mech. 20 (1984) 193-208.
[9] S.R. Bodner and P.S. Symonds, Experimental and theoretical investigation of the plastic deformation of cantilever
beams subjected to impulse loading, J. Appl. Mech. 29
(1962) 719-728.
[10] Y. Ando (editor), A study of evaluation of integrity of
primary system of light water reactor using fracture mechanics (Part II), Report of Atomic Energy Research Committee, JWES-AE-8603, Japan Welding Engineering
Society (1986).
[11] ASTM standard E813 (1983), ASTM.
[12] T. Nishioka and S.N. Atluri, A numerical study of the use
of path-independent integrals in elasto-dynamic crack
propagation, Engrg. Fracture Mech. 18 (1983) 23-33.
[13] J.D. Achenbach, M.F. Kanninen and C.H. Popelar,
Crack-tip fields for fast fracture of an elasto-plastic
material, J. Mech. Phys. Solids 29 (1981) 211-225.
[14] T. Nishioka, Invariance of the elastodynamic J integral
( J ' ) , with respect to the shape of an infinitesimal process
zone, Engrg. Fracture Mech. (in press).
[15] T. Nishioka and H. Yagami, Invariance of the path independent T* integral in nonlinear dynamic fracture mechanics, with respect to the shape of a finite process zone,
Engrg. Fracture Mech. 31 (1988) 481-491.
[16] A.J. Rosakis and L.B. Freund, Optical measurement of
the plastic strain concentration at a crack tip in a ductile
steel plate, J. Engrg. Mater. and Technol. 104 (1982)
115-120.
[17] J. Beinert and J.F. Kalthoff, Experimental determination
of dynamic stress intensity factors by shadow patterns, in:
Experimental Evaluation of Stress Concentration and Intensity Factors, G.C. Sih, Ed. (Martinus Nijhoff,
Dordrecht, 1981) 281-330.