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)