Fracture initiation at a sharp grove

J. Mech.Phys.Solids,1969, Vol. 17,pp. 315to 321. Pergamon Press.Printedin GreatBritain.

FRACTURE

INITIATION

AT A SHARP

GROVE

By P. T. HEALD* Department of the Theory of Materials, The University of Sheffield

(Received31st iI&-& 1969)

SUNMAXY

A SINPLE model is presented which enables the plastic relaxation around a sharp groove in the surface of a plate deforming under conditions of anti-plane strain to be discussed. The extent of the relaxed zone and the relative displacement of the material at the groove tip are determined as functions of the applied stress. This enables the initiation of fracture at the tip of the groove to be examined. I.

INTRODUCTION

A MOTELsuitable for discussing the initiation of fracture at the tip of a sharp V-shaped groove situated in the surface of a semi-infinite body has recently been given by SNITI~ (1967). In this model it is assumed that the plastic relaxation is confined to an infinitesimally thin strip ahead of the groove. This assumption is, of course, only a rough representation of the physical situation, SMITI~ (l9G7) has shown, however, that the length of the relaxed zone obtained from this model is in good agreement with the equivalent result of HULT and MCCLINTOCK (19%) who discuss the same problem using a rigid-plastic model. RICE (1966) has extended the analysis of HULT and MCCLINTO~K (1957) to the case of grooves symmetrically situated in the surfaces of a plate of finite thickness. His results are presented in integral form and numerical results are given only for the limiting ease of infinitely sharp notches. KOSKINEN (1963) has treated the same problem using a numerical finite-difference procedure and presents results in graphical form for several groove angles and plate thicknesses. In all these models the body is assumed to deform under conditions of anti-plane strain. In the present paper vve shall discuss both the zones of plastic deformation and the relative displacements at the tips of grooves symmetrically situated in the surfaces of a plate of finite thickness. This enables us to examine the initiation of fracture at the tips. of these grooves. The model used is an extension of that employed by SMITH(1967) but the method of solution is more direct. 2,

V-SHAPED GROOVESIN A PLATE OF FINITE WIDTXX

Consider grooves of included angle 277~~ and depth k symmeterically situatedin the surfaces of a plate of thickness 2d in the xs direction. The body is infinitely extended in the xl and ~3 directions and deforms under an applied stress 321s= o at infinity PresentAddress: Physics Department, The Universityof Surrey, Guildford.

315

1'.T. HEALD

316

(see Fig:. 1). Plastic relaxation is assumed to occur on the plane ;cr = 0 and to extend a distance s from the tips of the grooves. It is simulated in the present model by a planar distribution of infinitesimal screw dislocations.

FIG. 1. The z-plane.

The conformal transformation a-a, sinhs (?-r&Y/24 ax . --. -df; = I sinhs (&$W) - sinh2 p 1

(1)

is used to map the x (= 81 + &a) plane into the I (= [ + in) plane (NEUBER, 1958), t,he surfaces of the pla.te in the x-plane shown in Fig. I becoming the planes 7 = 0 and 2d in the c-plane shown in Fig. 2. The length of the flank of the groove is then

20/a

k set (7~) = 0

t-w sinh2 (7rE/2d) G sinhs /3 - sinhs (7rt/2d) I

(2)

which gives ?A -=cos(nw)sinhp~(X-W’r(f-W’p(~,1-~,~;-sinhs8) ?.$ 2d

(3)

33.7

Fracture initiation at a silarp grove where F (a, b; c; z) is the h~pergeometric

function and F (z) is the gamma function. The extent of the relaxed zone, S, in the x-plane is related to the corresponding length, GI,in the l-plane by the mapping function and is given by a

s=

.r[

0

t-w sin2 (ny/2d) drl* G2 (a~/24 + sinh’J fi 1

On writing u = sin (r~/~d) and 6 = sin (424,

where

M(u)

=

i

equation (4) becomes

u2 zc2+ sinhs fl

t-a,

(6)

-

In the t-plane we have a plate of thickness 2d deforming under an applied Relaxed zones extend a distance u, on the plane stress pp, = cs(&+-I[)[,,,. f = 0, from the sides of the plate. If we simulate these relaxed zones by a distribution of screw dislocations D (7) the stress in these regions is pc3 = u1 (dz]d{)\,=,, where al is the (constant) stress opposing the motion of dislocations in the z-plane. The equation which expresses the requirement that the resultant shear stress on any dislocation in the distribution is zero when they are in equilibrium is (L~SIBFRIED,

A

L951)

s

OL I) ($)

--a

cos (~~~lzd) dT’ sin (~~/2d) - sin (wq’,/2d)

+ u2

dx

d5 e=oo = *I 2

t=*

0 -=I 171 -=c a,

(7)

where D (7) dy is the number of dislocations of Rurgers’ vector b between q and 7 + d7 and A = Gb/Zn where G is the shear modulus. On letting u = sin (nT/gd) and 6 = sin (m/2d) in (7) we obtain

s

'D (u’) du’ ‘11_ u,

A

--6

+

(3 =

Q_x (4,

0 <

lu[ <

6.

The condition for a solution to (8) is (~~S~I~~LISI~VI~I, 1953) s

-_= 5% ?rc7

s

M (u) du

*

(S2-

u2p

1-m

s

(1 -

ZS(4

w) 7r+ -

W)

F (+ -

w, 1. -

w; $- -

w; -

P/sinl$ 6).

(9)

For the special case of a slit crack, w = O, equations (3), (5) and (9) may be combined, with the aid of the relation F&l;*;

-22)

2

z tan-1 x,

to give cos (424

= sin (~~~2d)isi~ [a (k + s)/2d].

(10)

P. T,

318

FIG. 5. The relation

HEALD

between the applied stress and the extent of the relaxed zone for a notch with w = 2.

This result was first given by BILBY et.al. (1964). For the general case the relaxed zone size must be determined numerically. Typical plots of s/d against u[al are shown in Fig. 3 for w = g. The solution to (8) may be written (MUSKHELISHVIIJ, 2953) 6

The relative ~splacement dislocations is

produced at the tip of the groore by this distribution of G(

8

(12) Substituting (11) into (12) gives rr@G -=crl=-rr

4

’ ’ (62 - &‘)I udu . M (v) dv oss o (1 - 3&y (62 - zy (ZP - z&2) *

(13)

Integration over u reduces (13) to $.E==Glnp
--IIr:“_‘~)~~n/:I:I::1:Ii:,_:~i:jd~-

When the relaxed zone has spread across the plate 6 = 1 and (14) becomes

(14)

Fracture

319

initiation at a sharp grove 1

&G

2

Values

against

M (v)

In

so (l-q*

zd’&=Y-

of @ must be determined numerically. 42q are shown in Fig. 4 for a groove

Fig. 4 is a plot of (15); it represents spread completely across the plate.

0.2

0.4

0.8

I.0

(15)

*

I

Typical plots of (774/2d) . (G/al) with w = &. The broken line in

the displacement

0.6

&

?I2

Ii-z?

1.2

when the relaxation

has

1.4

FIG. 4. The relation between the applied stress and the relative displacement at the tip of a notch with w = t.

DISCUSSION

3.

The relative

displacement

since the attainment tion

at the groove

assumption

af the tip of the groove

of a critical

is of special

Gc may be used as a condition

tip (see, for example,

is likely to be particularly

COTTRELL, 1961;

applicable

importance,

for fracture initia-

WELLS, 1961).

This

when fracture initiates in a ductile

manner. For a given size d of structure we may select a critical displacement at the tip of the groove as being sufficient for fracture. Then a vertical line

may be drawn on Fig. 4, which gives the critical stress for a given notch If w = 0 then (15) reduces to n@G -

2a*;=-rr

a/2

2

In s

II,

I

depth.

@,

320

I’ * T . Hk‘ul, _.

where J(,= n%c/2a3. The integral may be evaluated in terms of the Lobatschefsky function L (27) = -

il

(cos t> dt

(I$] < $77)

s

0

to give -x@ G 2d * &

--

z. ?r

[L (n/2) -

2L (nk/Zd) - L (742 -

n?qd)].

This result differs from the corresponding result given by BILBY et al. (196h) because those authors omit the minus sign in the devotion of the Lobatschefsky function and give incorrect arguments in the last two terms. If we identify the yield stress appearing in the van Mises yield criterion used by KOSKINEN (1063) with 01 and identify the distance from the plastic zone extremity to the groove tip in the rigid plastic model with the distance s of the present model, we may compare the results of the two treatments. A convenient measure of the applied stress, for this comparison, is X = a/al . d/(d - k) and in Table 1 values of s/k are given for various values of S, lejd and w. The agreement is

T.~IJLE1. Vakes

\ s li\ Y\ w = TO 3

w=z

1

qf s/k for variow

0.36 0.38 057

1 4,

values of S, k/d and w

0.61 O%S

OS3

0.91 OW

1.00 Model

0.11

0.31

o*sn

2.30 3.00 Koskinen(K)

0.11

0.32

o*ss

1%?1 3.00 IXslocation (D)

1

0.03

0.11

0.28

0.51

1.00

I(:

T

0.03

0.12

0.30

0.51

1.00

11

3

0.015

0.05

0.12

0.25 0.33

I<

4,

O*OlS

0.05

0.12

0.21 0.33

D

1 z

0.065

0.28

o+E?

2.29 3.00

K

O.OG5

0.30

0.82

1.92 3.00

u

1

0.02

0.09

0.26

?

0.02

O-09

0.26

3

0.01

0.04

0.115

z

0.01

0.04

0.105

0.53

1.00

I(:

0.49

1.00

n

0.25 0.33

K

0.21 0.33

I)

satisfactory, the values calculated from the present model tend to be a little higher for small stresses and low for large stresses. The gaps in the Table arise because the numerical results from the rigid plastic treatment are not avaitable for those values OfS.

321

Fracture initiation at a sharp grove

Finally we note that as d + to equations (2), (4) and (9) respectively reduce to 2d & k set (nw) - sinh /3 = r (1 - w) r (4 + w) = p’ T

These results are equivalent to those given by SMITH (1967). If, in addition, o/al, and hence the relaxed zone, is small we may combine these equations to give rr* r (1 -

rrcl -= 201

W) 2 cos (rTTw) r (2 -

2r(+-- w) -t

w)

r (* + w)

7rt

. s (1-2~)'(2-2~) '-ii >

SMITH(1967) has shown that this result is in good agreement with the equivalent expression given by HULT and MCCLINTOCK (1957).

ACKNOWLEDGMENT My thanks are due to Professor for financial assistance.

B. A. BILBY for many useful discussions

and to the S.R.C

REFERENCES 1964 BILBY, B. A., COT~RELL, A. H., SMITH, E. and SWINDEN, K. H.

Proc. R. Sot. A. 279,l.

COTTRELL, A. H.

1961

HULT, J. A. H. and

1957

Iron Steel Inst. Spec. Rept. no. 69, p. 281. 9th ht. Congr. AppZ. Mech. Vol. 8, p. 51. Brussels Free University of Brussels, Brussels. Tram Am. Sot. me&. Engrs 85 D, 585. 2. Phys. 130, 214. Singular Integral Equations (Translation by -OK, J. M., 1953), P. Noordhoff, N.V., Groningen, Holland. Kerbspannungslehre Springer, Berlin. Int. J. Fract. Mech. 2, 426. Int. J. Engng Sci. 5,791. Proc. Crack Propagation Symposium College of Aeronautics, Cranfield.

MCCLINTOCK, F. A. KOSKINEN,

M. F.

LEIBFRIED,

G.

MUSKHELISHVILI, NEUBER,

H.

1963 1951 N. I.

1946 1958

RICE, J. R.

1966

SMITE, E.

1967

WELLS, A. A.

1961