A criterion for plane strain cumulative fracture

Scripta

METALLURGICA

Vol. 13, pp. 1 5 3 - 1 5 5 , P r i n t e d in the U.S.A.

1979

P e r g a m o n Press, Ltd. All r i g h t s r e s e r v e d .

A CRITERION FOR PLANE STRAIN CUMULATIVE FRACTURE

E. Smith Department of Metallurgy, University of Manchester, U.K. (Received

December

20,

1978)

Introduction Cottrell (i) has highlighted the two extreme ways in which a crack can propagate in a solid subject to Mode I plane strain deformation conditions. At one extreme, crack extension can be represented by the injection of dislocations into the material from the crack tip along sllp lines that are inclined at 45 ° to the tensile axis, when each dislocation of Burgers vector b contributes an increment b//2 to both the opening and extension of the crack. A non-cumulative situation exists in that each crack growth increment requires the injection of extra dislGcations, and more importantly, these dislocations push the existing ones further away from the crack tip, whereupon the plastic zones spread much more rapidly across the section than does the crack. In this situation the material at the crack tip slides off along specific planes rather than actually fracturing, and it is this sliding-off which is responsible for the crack extension, which is stable and obviously limited in magnitude. At the other extreme~ crack extension can be represented by the climbing of edge dislocations along the crack plane ahead of the crack tip, a process that does not require the injection of extra dislocations into the material, the same dislocation group describing the fracture processes at all stages of crack propagation. A"cumulative" situation exists, and it is not neeessar 9" to increase the crack tip stress intensification during propagation, whereupon the fracture is unstable for most practical loading systems ; the classic example of cumulative fracture is crack propagation in a perfectly brittle solid. Cottrell has emphasised that cumulative fracture can also occur in a plastically deforming material, provided cracks or holes form ahead of the main crack thereby allowing the plastic deformation to be confined to the rupturing of the ligaments between the cracks or holes within a fracture process zone. If q ~s the fractional area of material failing by a rupture process, the average cohesive stress within the fracture process zone is mqY where Y is the tensile yield stress, and m is the constraint factor for the plastic material. For the operation of a cumulative fracture mode, Cottrell argues that this average stress mqY should be less than Y, or otherwise the non-cumulatlve fracture mode will operate. Implicit in this cumulative fracture condition is the requirement that plastic deformation be confined to the fracture process zone. However, experimental evidence (2) for various types of fracture: stress corrosion, liquid metal e=brittlement, hydrogen embrittlement, etc., shows that cracks can propagate without an increase in crack tip stress intensification, even when there is plastic deformation, albeit to a limited extent, away from the crack plane. It is against this background that the present paper develops a criterion for the operation of this crack extension mode, thereby broadening Cottrell's description of plane strain cumulative fracture. Development of Cumulative Fracture Criterion Figure 1 provides a simple physical picture of a propagating crack; a fracture process zone, within which fracture processes operate, ~s surrounded by an adjacent plastic region, which forms to accommodate the local strains associated with the fracture processes. To facilitate a very simple quantitative discussion of this model, the crack tip is assumed to move forwards in discrete jumps (Figure 2), the crack extension increment Aa being the fracture process zone Size.

153 0036-9748/79/020153-03502.00/0 Copyright (c) 1979 P e r g a m o n Press,

Ltd.

154

PLANE

STRAIN

CUMULATIVE

FRACTURE

Vol.

13,

No.

2

FIG. 1

PLQS'TIC

RE&ION

I C~ ~CK

I= R t:::tCTO R E P ROC..E E~S

ZONE

The physical picture of a propagating crack, w~th its fracture process zone and plastic region. FIG. 2

The simple model of a crack moving forward in discrete jumps of magnitude Aa; the plastic deformation occurring at the end of each jump is slmulatedby the emission of edge dislocations alon E planes that are incllned at q5 to the crack plane. •

•

U

At the end of each jump, the material in the vicinity of the crack tip plastically deforms, and this ~lasticlty is simulated by the emission of edge dislocations alon E planes that are inclined at 45 to the crack plane• In this type of situation a cumulative state is defined as one for which the crack tip stress intensification needed for propagation does not increase with crack length. It is argued that this will be the case if dislocations move away from the crack tip without pushing the dislocations associated with previous relaxations further away from the crack plane, i.e. the relaxed zone size is some small multiple, say unity, of the jump distance. If the dislocations associated with each crack jump produce a displacement A@, the plastically relaxed zone size is approximately (E/Y) A@ where E is Young's modulus, and the cumulative fracture condition becomes (E/Y) A~/Aa < i; otherwise the fracture mode becomes non-cumulatlve. It is clearly difficult to determine the precise value of the number on the right hand side of this inequality; however, the main thrust of this simple analysis, and indeed the paper, is to highlight the importance of the parameter (E/Y) (A#/Aa) with respect to the operation of a cu~ulative fracture mode. FIG. 3

CR~CK,

Y

FR BCTL)R~" PROC~S~

\

V

i~a,.

*

7. O N E

~ON~

<

P u~

>

I]BCS type model of a crack with its associated fra~.ture process and yield zones.

Vol.

13,

No.

2

PLANE

STRAIN

CUMULATIVE

FRACTURE

155

Proceeding beyond this very simple discussion, consider the DBCS type model shown in Figure 3 (3,4). There is a long crack and a fracture process zone of length Aa, ahead of which is a simulated yield zone of length p. There is a close similarity between this model and that in Figure 2, the main difference being that all the non-elastic processes are confined to the crack plane with the new model, and the coplanar plastic zone simulates the relaxatien at the end of each crack extension increment. The cohesive stress within the fracture process zone is assumed to be Y (this assumption does not alter the general conclusions), and if #s is the displacement at the start of the fracture process zone, where it is adjacent to the yield zone, and ~f is the displacement at the end of the zone, i.e. at the crack tip, standard results for the DBCS model are: = i/~/--u- u log ~l 1 ~+ Ji/~--u~ ..........

WE~s {(l-vZ)Yw

(i)

Y

~E#f

= i ..........

(2)

8(l-v2)yw where E is Young's modulus, v is Poisson's ratio, w = Aa + p, with Aa and p being the sizes of the fracture process and plastic zones, and u = Aa/w. Furthermore, the critical stress intensification K required for crack propagation is given by: K

- - -

. .........

(3)

Following the same argument as that developed in the preceding paragraph, and supposing that cumulative crack extension occurs if Aa > p (i,e, u > 0.5), relation 1 gives the cumulative crack extension condition as: ~E$ s < 0.27 .......... 8(l-vZ)YAa

(4)

or with the earlier notation, Ss ~ A¢, E A¢ Y Aa

< i ..........

(5)

in accord with the condition obtained in the preceding paragraph. Discussion This paper has developed a criterion (relation 5) for the operation of a plane strain cumulative fracture mode, when the crack tip stress intensification does not have to increase for propagation to proceed. The key factors which promote cumulative fracture are a high yield stress Y and a low value of A~/Aa, i.e. a small number of dislocations generated per crack extensier. increment, which is equivalent to a small crack tip opening angle. The magnitude of A~/Aa will be controlled primarily by the ease with which holes or cracks form and grow within the fracture process zone. As indicated in the Introduction, implicit in Cottrell's criterion is the requirement that plastic deformation be confined to the fracture process zone, which essentially implies that A~/Aa is zero; in a general sense, therefore, his criterion is encompassed w~thin that developed in this paper. It should be emphasised that the criterion is a necessary condition for cumulative fracture, in that a crack is assumed to propagate in a cumulative mode, and the condition for continued cumulative propagation, rather than non-cumulative propagation, has been derived. The criterion may not be sufficient, however, since the condition for the onset of cumulative fracture at a stress concentration could be different, primarily because of the effects of the plastic region which forms to accommodate the local strains needed for the onset of crack extension. Despite this limitation of the implications of the present paper's conclusions, there are many situations where plane strain cumulative crack propagation does occur, e.g. stress corrosion fracture, liquid metal embrittlement, and crack propagation in thick steel sections. In the context of these situations, the main implication of this paper's consideration is the emphasis given to the material's yield stress, for it has been demonstrated that the susceptibility of a material to cumulative, and therefore unstable fracture increases with its yield stress, a prediction that is in agreement with general experience of material behaviour. References i. 2. 3. 4.

A.H. S.P. D.S. B.A.

Cottrell, Fracure, Ed. C.J. Osborn, Butterworths, London, p. 1 (1965). Lynch, Fracture 1977, University of Waterloo, Volume 2, p. 859 (1977). Dugdale, Jnl. Mechanics and Physics of Solids, 8, i00 (1960). Bilby, A.H. Cottrell and K.H. Swinden, Proc. Roy. Soc. A 272, 304, (1965).