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International Journal of Fracture 27 1985) R8 7-R 91 .
© 1985 Martinus Nijh off Publishers, Dordrecht. Printed in Th e Netherlands
A DUAL-PARAMETER ELASTIC-PLASTIC FRACTURE CRITERION
H.W. Liu and Tao Zhuang
Department of Mechanical Aerospace Engineering, Syracuse University
Syracuse, New York 13210 OSA
tel: (315) 423-3038
In the case of small scale yielding, i.e., when the plastic zone is
small relative to the other specimen dimensions, the stress intensity fac-
tor K is capable of characterizing crack tip stress, strain, and displace-
ment fields even with in the plastic zone. In other words, the same Kv al ue
implies the same crack tip field regardless of the variation of the in-
plane specimen geometry. This concept of characterizing the elas tic~ last ic
crack tip field by a single parameter K as a fracture criterion was first
advance d in 1964 [i]. It was discuss ed in a more detail ed manner and was
extended to cyclic loading in 1972 [2 ]. The fracture crite ria based on
K characterization of crack tip field, sharp notch analysis, and global
energy balance were reviewed by Liu [3]. It is conclud ed that the capa-
bility of K to characterize the crack tip field is the fundamental basis
for the validity of the linear elastic fracture mechanics rather than the
Hutc hins on [4] and Rice and Rosengr en [5] analyzed the crack tip field
in non-linear elastic solids obeying the pure power law stress-strain
relation, E/e = a(o/a )n. In their analysis deform ation theory was used.
Their resu lts°s h0w tha~ the crack tip field in a non-l inear elastic s olid
can be characterized by a single parameter, the J-i nte gra l. Fracture
initiation is caused by crack tip stresses, strains, and displacements.
Therefore J can be used as a fracture criterion in specimens of different
geometry if the Hutchinson, Rice and Rosengren crack tip field is univer-
sally correct.
More recently, McMeeking and Parks [6] and Sih and German [7] have
shown by their plane strain finite element calculations that the J-integral
characterizes the crack tip field only if the crack tip plastic deformation
is not excessive. Wit h extensive plastic deform ation, at the same J value,
the crack tip fields in specimens of different geometry may differ widely.
Furthermore, Hancock and Cowling [8] have shown that the critical values of
crack tip opening displa cement ~_ at fracture in itiatio n in HY80 steel
specimen s of six differe nt geo metry types differ by a factor of ten. These
theoretical and experimental results clearly indicate a need to examine the
capability of a single parameter such as J to characterize the entire tip
field in elastic-plastic solids.
Crack tip stress, strain, and displ acemen t fields in the specimens of
different geometric types used by Hancock and Cowling in their experimental
study are analyzed by using the finite element method both in plane stress
and plane strain cases.
When a single parameter is capable of characterizing the entire crack
tip field, the plots of any comp onent of a.., e.., or u in a given direc-
tion e, versus the distan ce from crack tiv1~ I . •
no~mal mze~ by r , ~, or
(J/Oy) should fall on the same curve. Figure 1 shows the norma lized plots
Fnt Journ of Fracture 27 (1985)
R 8 8
of crack tip stress and strain fields for the plane stress case for e = 0
and 45 deg. The data of four different speci men geometries are shown:
double edge cracked plate, single edge cracked plate, three point bend
specimen, and center cracked panel. All of the curves at various load
levels for all four different specimen g eometries fall on top of each other
including the curves for small scale yielding. The results clearly indi-
cate that in the plane stress case, a single parameter is capable of char-
acterizin g the entire crack tip field. At any given J value, the crack tip
fields in these four different specimen geometries are the same. The cra ck ~
tip field in a small specimen in general yielding corresponds directly to
the crack tip field in a large specimen in small scale yielding, and the
equivalent K or G value of the small specimen in general yielding can be
obtained by direct correspondence. Hence, any one of the following param-
eters, J, ~, r , equivalent K or G can charact erize the entire crack tip
field and can ~e used as a crite rion for fracture initi ation in the plane
stress case.
It has been shown that the plane strain slip line field of Constrained
plastic flow in a double edge cracked plate is significantl y different from
that of non-constra ined plastic flow in a center cracked plate. This dif-
ference in the slip line field induces different stress triaxiality which
is reflected in the difference of the maximum principal tensile stresses
in the crack tip regions of these two specimen types [9] . According to the
slip line field analysis, the maxim um pri ncipal tensile stress in the con-
strained flow is three times the non-con straine d flow. This difference in
the maxim um princi pal stress may explain the variati on of the measure d ~f
by Hancoc k and Cowling [8] . Therefore, it is expected that crack tip
fields in general yielding in different specimen g eometries may differ
widely. Figures 2(a) and 2(b) show the plot of
(a /o ) vs. r/(J/a )
for
center cracked panel in tension and three point be ~ s~ecimens. Beyond
general yi~l~ing, the data at various load levels follows different curves.
At a given J value, the crack tip fields vary w ide ly from one geometry type to
another, and it varies at different load levels in specimens of different
size but of the same geometry type. Therefore, in the plane strain case,
the crack tip field cannot be characterized by a single parameter. A dual-
parameter elastic-pla stic fracture criterion is proposed. In Fig. 3, crack
tip opening displacem ents at fracture init iation measure d by Hancock a~d
Cowling are plotted against the calculated ratio (~ /5 ) at r=2~. oi s
. m . .
effective stress. The ratio (~ /0) characterlzes ~e maxlmu m tenslle
stress field at a crack tip and ~X re fl ec ts ~he stress triaxiality. The
crack tip openi ng displacement ~_, or the ratio ~f/t, characteri zes crack
tip strain field. t is plate ~hickness.
The maxim um tensile stress in the crack tip region fractures the
brittle particles, brit tle inclusions or embrittled grain boundaries; and
Therefore, the condition for fracture ini tiation should be characterized
by both crack tip tensile stress field and crack tip strain field.
REFERENCES
[i] H.W. Liu, in Fracture Toughness Testing and its Applications, STP 381,
American Society for Testing and Materials, Philadelph ia (1965) 22-
26.
[2] H.W. Liu, An Analysis on Fatigue Crack Propagation, NASA CR-2032
(May 1972).
[3] H.W. Liu, Engineering Fracture Mechanics 17 (1983) 425-438.
Int Journ of Fracture 27 (1985)
R89
[4] J.W. Hutchinson, Journal of the Mechanics and Physics of Solids 16
1968) 13-31.
[5] J.R. Rice and G.V. Rosengren, Journal of the Mechanics and Physics of
Solids 16 1968) 1-12.
[6] R.M. McMeeking and D.M. Parks, in Elastic-Plastic Fracture STP 668,
American Society for Testing and Materials, Philadelphia 1979) 175~94.
[7] C.F. Shih and M.D. German, International Journal of Fracture 17 1981)
27-43.
[8] J.W. Hancock and M.J. Cowling, Metal Science 14 1980) 293-304.
[9] F.A. McClintock and G.R. Irwin, in Fracture Toughness Testing and its
Applications STP 381, American Society for Testing and Materials,
Philadelphia 1965) 84-113.
7 January 1985
3 0 /~ e , o .
-
4 0 - t0 - 2 . 0 - L O 0 0
log r / rp
3 . 0
2 £
1.0
0.~
0 , 4 5
SEC
~ T P B
~ y l C y ~
- 3 D - 2 . 0 - I . 0 0 .0
Figure i. ~ and ~ of the
plane stres~Ycharac~ristic
crack tip field of all four types
of specimen geometries: Single
Edge Cracked, Three Point Bend,
Double Edge Cracked and Center
Cracked Panel. 2HY-80 ~eel.
o = 0.56 kN/mm , E = 200 kN/mm 2,
vY= 0.3, N = 0.ii
a) ~ = 0 deg b) ~ = 45 deg
Int Journ of Fracture 27 1985)
Rg0
~ •
O I I I I
O0 2.0 4.0 6.0 80
i n n ~- THREE POINT BEND
- - I i LA~J/-'-~Y
u ~ - ~ ~ - -
8.0~ - • 27
U x 16
.~
_ _ ~ S M A ~ S C A L E
r / J/o,y )
I
I0.0
|0.01- CENTER CRACKED PANEL
I ~ ~ L
• 184
8D ~ • 62
~ t x 2 0
s o ~ - - - ~ R ~ 71
~ ' ~ SMALL ~ L E
~ 4.0
0 0
~ 0 2 0 4 0 6 0 8 0 ~0 0
, / w / ~ .
Figure 2. Plane strain crack tip stress distribution along the crack line
at various load levels, L/(J/Oy). L is the ligament size.
(a) Three point bend specimen
(b) Center cracked panel in tension
Int Journ of Fracture 27 (1985)
R9]
3.0
ao I. c ~ ~ . . ~
DOUBLEEDGE
C R A ~ S ~ ~
0.0 0.2 0.4 O.S aS ~.O~mm~
Figure 3. The fracture ductility diagram. Based on the dual-parameter
elastic-plastic fracture criterion. HY-80 steel.
Int Jou~n of Fracture 27 (1985)