Stress intensity factors determined by use of westergaard's stress functions

Fracture Mechanics Engkeermg Pnnted m Great Britain.

Vol. 20, No. 1, pp. 65-73,

0013-7944/84 $3.00 + 00 C 1984 Pergamon Press Ltd.

1984

STRESS INTENSITY FACTORS DETERMINED BY USE OF WESTERGAARD’S STRESS FUNCTIONS THEIN WAH Texas A&I University, Kingsville,

TX 78363, U.S.A.

Abstract-A procedure is developed for determining the stress intensity factors in cracked polygonal plates. The method uses eigenfunctions of angular regions and satisfies the boundary conditions exactly. Westergaard’s stress functions are used to satisfy the boundary conditions at the crack. Numerical results are given.

NOTATION a

half length of crack E elastic modulus u, L’ radial and tangential displacements r, 0 polar coordinates lj = v/l + v 1 root of transcendental equation v Poisson’s ratio O. T normal and shear stress Z,(z) stress functions

z=x+iy

INTRODUCTION THE DETERMINATION of stress intensity factors in plates with cracks is a problem of practical importance in fracture mechanics. Although stress intensity factors for plates of infinite extent have been known for some time[l+ very few methods exist for analyzing finite plates with cracks. Kobayashi et al.[5] have proposed a boundary collocation method using complex stress functions. Their method naturally satisfies the conditions on the outer boundaries only approximately. Their method has been extended by Newman[6]. The method proposed here is different in that the boundary conditions are satisfied by every term of the series used. The method applies to any polygonal plate whatever the boundary conditions. The formulation is based essentially on the fundamental work of Westergaard[7]. EIGENFUNCTIONS For any angular region, with the biharmonic equation expressed in polar coordinates, the stress function x is satisfied by an expression of the form[9] ~=~i+~{C,sin(;1+l)+C,cos(~+1)8+C,sin(R-1)8+C,cos(1-1)8)

(1)

where the C, are constants and 1 is a parameter to be determined. The displacements and stresses are given by the expressions ~PU, = ra[ - (A + l)(C, sin (A + l)O + C, cos (A + l)O} + (3 - 4p - R) {C, sin (2 - l)O + C, cos (13.- l)O}]

(2)

2jm, = r”[(A + 1) { - C, cos (A + 1)O + C, sin (A + l)O) +(3-4q+J){-c,

cos (A - l)O + C, sin (A - l)S)]

(3)

bg = r”-‘[A(1 + l)(C, sin (A + l)O + C,cos (A + i)e) + {C, sin (A - i)e + C, cos (A + i)e}] 65

(4)

T. WAH

66 7rfl

=

-it+‘[(l

l){C, cos(i,+l)0-C,sin(i+l)0}

+

+ (J. - l){C, cos (iL - 1)Q - C, sin (1. - 1)0}]

(5)

with E p

=

V

2(1 +v)’

V = 1 +v

The boundary conditions at 0 = 0 at H = M of the angle depend on the specification of the displacements and/or stresses. Assuming that such boundary conditions are homogeneous, substitution of (2)-(5) into them results in 4 homogeneous equations for the constants C,. If one sets to zero the determinant of the coefficients, there results a transcendental equation for the determination of 1. This equation has a denumerably infinite number of roots. For each value of I, real or complex, there is an associated eigenfunction which satisfies the boundary conditions exactly at 0 = 0, c(. In Tables Al and A2 of the Appendix we have listed the transcendental equations and eigenfunction for 10 different sets of boundary conditions. While it is possible to state other types of boundary conditions, they appear to be difficult to realize in practice (Morley[9], in fact, lists only 3 types). In Table 3A of the Appendix we have given a particular integral of the biharmonic equation which satisfies the condition of constant normal stresses (co, 0,) and constant shear stresses (z,, 7,) at the boundaries 0 = 0 and 0 = ~1.We have also given the restrictions on the type of loading that may be applied at a boundary. For example, if u = u = 0 at the boundary the stresses on that boundary must be zero. Westergaard’s stress functions

Following Westergaard[7], we assume that the stress function &, for a plate with a crack of length 2a along the x axis (see Fig. 1) may be written in the form I$,,,,,= Re .?Q,+ y Im zu.

(7)

The expressions of the stresses then follow by differentiation

of (7).

a%$, ” Bx =+=ReZ,-yImZA

av

in which

_dZn

z n

dz ’

Zn=z”+‘G(z), G(z) =

_dZn

z

Z’

dz ’

n

_

’

dZn dz

n=0,1,2...

(z* - a’)-(“‘),

z = x + iy.

(11) (12) (13)

Clearly these equations satisfy the conditions of a stress free crack at - a < x < a and give the correct singularity at x = a. If n is even in eqn (12) the stress function is symmetrical about both the x and y axes. If n is odd it is symmetrical about the x axis only.

Stress intensity

factors

determined

by use of Westergaard’s

stress functions

67

The preceding equations are suitable for the crack opening mode (Mode I). For Mode II, a set of related functions may be written following Irwin[S] cP%,.~ = -y Re .?!,,

(14)

a,=2ImZ,,+yReZA

(15)

a,=

(16)

-y

ReZ;

z XY= ReZ,-yImZL.

(17)

If n is even these equations satisfy antisymmetry about both axes x and y. Mixed mode stresses may be obtained by superposition. We shall refer to all these functions as Westergaard functions. Crack functions

If we assume a stress free crack extending all along the crack axis, we can derive the following relations +,=yImZ

(18)

a,=2ReZ,-~~ImZ~

(19)

a,,=yImZi

(20)

T I,’

= -(ZmZ,,+yReZA)

zns,

z:=f$

(21) &=z”,

n = 1,2,...

(22)

If n in (22) is even the functions satisfy symmetry about both axes x and y. If n is odd they satisfy symmetry about the x axis only. For antisymmetrical modes, the following functions replace the preceding. qb,=xImZ-(n a,=2(n

+ I)yReZ

+ l)ImZ,-xImZi+(n

(23) + l)yReZL

(24) (25)

z, = n Re Z, - x Re ZA - (M + 1)y Im ZA.

(26)

If n is even in (22) these functions satisfy antisymmetry about both axes x and y. If n is odd they satisfy antisymmetry about the x axis only. We shall refer to these as crack functions. Superposition

Figure 1 shows a quadrilateral plate divided into 5 regions. In region 5, which contains the crack, the stress function may be taken as

(27)

where A, and B, are arbitrary real constants. Here we confine ourselves to Mode I and choose the symmetric Westergaard and crack functions. The modifications required for other modes is obvious. In regions 14 in Fig. 1 the stress field is described by the appropriate eigenfunctions, with

68

T. WAH

arbitrary multiplicative constants. If the boundaries have tractions, we merely add the particular integral (given in the Appendix) to the eigenfunctions. In any event the boundary conditions are satisfied exactly. Collocation It now remains only to establish continuity of the stress functions between regions 5 and the other 4 regions. This requires that the stress function and its 3 derivatives normal to the collocation lines A’B’, B’C’, C’D’ and D’A’ be continuous. It is evident that an exact solution is not possible. We therefore satisfy them in the sense of least squares. The continuity equations are:

a4 -=

-_ a6

dn

an’

(28)

a34 -_a34 -= an3

anf3’

In eqns (28) 4 denotes the stress function in region (5) and 4’ the stress function in any of the other four regions. n and n’ are normal to the collocation line. It will be noted that the first two equations require the functions .?,, and 2, in eqn (7). These are obtained by the integration of Z,, and are given in the Appendix. Stress intensity factor Having obtained the coefficients in eqn (27) the stress intensity factor is given by k = ,/G

Lt (z - a)“‘a,. z-tll

(29)

If we define f(a, b,c)=

5 A,a”

(30)

n=O

wherefis a function of the ratio of the sides of the plate and that of the crack, and A, are the coefficients of the Westergaard functions, then eqn (29) assumes the form k = of(a, 6, c) ,/;;;;.

(31)

We have, of course, essentially limited discussion to the crack opening mode. Extension to other modes is straight forward. Discussion It is important to note that the method will work for any polygonal plate including the triangular plate. It is only necessary that the regions involving the boundaries be triangular so that eigenfunctions may be used. For the case of two collinear edge cracks, the Westergaard functions could be modified by interchanging the roles of z and a in G(z). In the case of a single edge crack, the Westergaard and Crack functions would be replaced by the functions proposed by Williams[lO]. It is also important to add that in applying eqns (28), rigid body rotations and displacements should be given wherever these are possible. Thus in the case considered rigid body rotations are possible in regions 14 (Fig. 1) and rigid body displacements and rotations may be possible in region 5.

Stress intensity

factors

determined

by use of Westergaard’s

stress functions

69

Numerical examples

Table 1 gives the stress intensity factor f(a, b, c) (as defined by eqn (30)) for Mode 1 fracture of a rectangular plate. Two aspect ratios were investigated, namely, c/b (see Fig. 2) of 2 and 1. The results are compared with those of other investigators. The number of eigenroots, Westergaard functions and Crack functions used are noted in the table. The roots of the transcendental equation required for the eigenfunction were obtained from Ref. [ll]. Convergence was quite rapid; that is to say, the first few Westergaard functions gave the stress intensity factor accurately. However, the total number of Westergaard (or other functions) used did affect the final result. Nevertheless the difference was generally within 4% for different values of these functions. The total number of equations used was approx. 65 in all cases. The least squares procedure permitted assembling these equations so that no more than this number was handled at any one time. For comparison, Ref. [6] used on the order of 200 equations. Parts of the program were in double precision and parts in single precision. The results appear to the author to be accurate enough for most engineering applications. If greater precision is desired one must resort to a program entirely in double (or extended) precision.

Fig. 1. Definition

Fig. 2. Illustrative

sketch.

Table

1. (Ref. 2)

.6

1.2767

1.3043

15, 16, 16

.7

1.4535

1.4891

15, 16, 16

.a

1.6777

1.8161

15, 17, 17

2.4428

~ 2.5482

.9

*R = Eigen, **Approximate EFM Vol 20.No I E

W = Westergaard, (Figures

from

15, 17, II7

C = Crack graph).

2.5293

-

problem.

T. WAH

REFERENCES G. R. Irwin, Analysis

of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 9, 361-364 (1957). G. Sih, P. C. Paris and F. Erdogan, Crack tip stress intensity factors for plane extension and plate bending problems. J. Appl. Mech. 29, Trans. ASME 84, Ser. E. 306312 (1962). F. Erdogan, The stress distribution in an infinite plate with two colinear cracks subjected to arbitrary loads in its plane. Proc. 4th U.S. National Cong. Appl. Mech. 547-553 (June 1962). Paul C. Paris, Application of Muskhelishvili’s Methods to the Analysis of Crack Tip Stress Intensity Fuctors for P/me Problems-IIZ. A Handbook of Crack Tip Stress Intensity Factors. Institute of Research, Lehigh University, June, 1960. A. S. Kobayashi, R. D. Cherepy and W. C. Kinsel, A numerical procedure for estimating the stress intensity for a crack in a finite plate. Trans. ASME, J. Bus. Engng 86, 681-684, 1964. J. C. Newman, Jr., An Improved Method of CoNocation,for the Stress of Cracked Plates with Various Shaped Boundaries. NASA TN D-6376 (1971). H. M. Westergaard, Bearing pressures and cracks. J. Appl. Mech. 6, 49-53 (1939). G. R. Irwin, Fracture mechanics. Proc. 1s~ Symp. on Naval Structural Mechanics, pp. 557-594. Pergamon Press, Oxford (1960). L. S. D. Morley, Skew Plates and Structures. Pergamon Press, Oxford (1963). M. L. Williams, On the stress distribution at the base of a stationary crack. J. AppL Mech. 24, 109-l 14 (1957). Thein Wah, Roots of Transcendental Equations. National Technical Information Service, Accession No. PB272929/AS (1977) U.S. Dept. of Commerce, Springfield, VA 22161, U.S.A. H. Tada, P. Paris and G. Irwin, The Stress Analysis qf Cracks Handbook. Del Research Corporation (1973).

(Received

26 May

1983)

APPENDIX Westergaard

functions

2 and _? (see eqn 7). zni 1 ___

Z(n) =

@2

5 r(n)

=

--

(z2

-

_

dz = z”(z2 - a2)1,’ - nr(n - 1)

az)li:

a>)312 z”-I+

(n + 2)

i

ha”C,(n)

+

Cl* c C,(n,k)z’“~zk-

uazk

k=l

1

z(z2 - a2)“2 - a2 in {z + fl

CAlI

a’)

-1

L n*=(n-I)/2

n=odd

(A3

= (n - 2)/2 (n - l)(n - 3)

c,(n,k)=------------

_ . (n~~ - 2k + 1)

n(n-2).....(n-2k+2)

t.43) (n - l)(n - 3).

.

1

C”(n)=(n;2)n(n-2)...4’ For n = 0,

t=

:[7(z2-a2)li2 - a2 In {z f z=

JXT}]

(A4)

pdz=F(n)-n!(n-1) s

(z2 - a2)3/2zm- 3r(n + 1)

f(n)=’

n

(n + 2)

‘* C,(n, k) ___ a2k{(zZ - a2)3~2z”-x - 3f(n - 2k + l)] +C k=l(n -2k) I

+: C,(n)a”

;

(=2 _

a2)32 _ a22 In (2 + &?Z)

+ a2(z2 -

a2)li2 1

C,(O) = I C,(n) = 0, n odd. For n = 0, only the last term survives

in eqn (A5)

Stress intensity

factors

determined

Table Al.

Case

by use of Westergaard’s

Transcendental

equations

Conditions B”“~d~ry 0=Cl

/

Clamped u=o,v=o

1

71

stress functions

Equation Symmetrical: (3-411) sin ha - A sin a = 0

Clamped

i ~;_'~_~~f~~t"~'i

qj,

Free

Symmetrical: sin AT + k sin a = 0 Antisymmetrical: sin Aa - h sin a = 0

Free

(3-4~1) sin* ia + h2 sin* a-4(1-n)*

a

=

0

I

Clamped

3

Iss

6

= 0

Simply Supported u=oo=o

(3-4ri) sin 2Xa + h sin

Free

,I sin

ss

sin

2

Zu-sin

2u = 0

2ia = 0

ha - sin*a

= 0

+ / Guided

i cos

2

ha-sin2u

I

= 0

t Free

sin Zhu-i

sin

I

Guided

I (3-4~) sin

Z&z-h

I/

u

=

Poisson's

ratio

n = v/(l+v)

Table A2. Eigenfunctions General

1.

(b)

rh+l

x=

F(0)

=

(3-4n-X)

cos(h-1)

+

(h+l) cce(h+l)

; cos(h+l) (o-q)

; cos

(A-1)(0 - 4)

Antisymmetrical F(o)

2.

Form:

clamped-Clamped (a) symmetrical: F(O)

=

(3-4n-A)

+

iktl)sin(A+l):

sin(x-1);

sin(X+l) (O- t)

sin(X-1) (a- ;)

Free-Free (a) Symmetrical F(Q)

= cos(A-1);

COS(h+l) (+

- cos(hc1);

ccl5 (X-1) (0- g)

(b) Anti-symmetrical F(O)

=

20 = 0

I

sin(X-1);

sin(I+l) (0-G)

- sin(A+l)$

sin(A-1) (0-t)

sin2a

= 0

/

T. WAH

72

Table A2. (Cord) 3.

Clamped-Free -R1(3-4n+X)

F(O) =

sin(h+l)Q

R2(3-4n-X)

t

cosCX+l)O

+

Rl(h+L) sin(h-1)Q

+ R2(li+f) cos(h-l)ff = (X-l) sin(h-1)o + (3-411-h) sin(Xtl)a

*1

R2 = (X-1) cos(a-l)a 4.

- (3-4n+h) cos(htl)o

clamped - ss a f n/2 P(O) = (3-417+X) cos(A-l)a - (x+1) cos(h-l)a

sin(A+l)o+

sin(h-1)0+

(3-4u-h) sin (h-1),1 cos(;c+l);

(hi-17 sin(h-l)a cos(i-l)O

il= n/2 x

= +

Cl

rzm [ll+m-2r1) sin

c2

L 2m+1

m

5.

=

1(3-4q-2m)

2mO

-

m

sin

cos(2mtl)~

2(m-l)o

+

(2m+l)

j cos(Z!m-1103

1,2,3....

5s - Free u # n/2 F(o) = sin[,I-l)o sin(x+l)e a

=

- sin(A+l)rr

sin(X-1)B

n/2

x = Cl r2m [(m-l) sin 2mQ t m sin 2(m-1)QI 2m+l

+C

2'

[sin(2m+l)Q + sin (2~l)Qj

m = 1,2,3.... 6.

SS-6s nn/a [Cl + C2f 2 I sin !I.?.? 0

x=r

Symmetrical, n = odd Anti-symmetrical, n = even 7.

Guided - Guided nr/a cc, + c2r*1 cos y

x=r

Symmetrical, n = even Antisymmetrical, n = odd 8.

SS - Guided x = r(2n-l)n/2a

[Cl + C2r2"J sin (2n-'J)'e

n = 1,2,3.... 9.

Guided - Free u # n/2 F(0) = cos(L-1)~

cos(X+l)~

- cos(x+l)u

costa-l)Q

a = n/t x = Cl rZm [cos2mf3 + co8 2fm-l)el + c2 r *m+l

~0s

(2m+l)o + (2m+l) cos(2m-1)Ql

1,2,3...

m = 10.'

[(2m-1)

CBamped - Guided 0 # n/2 F(O) = (3-4q+A) aj.n(x-l)a sin(X+l)o - (3-4n-X) -

a

=

(h+lf

sin(X-1)o

sin(X-1)0-(A+11

CostA-l)a

COS(i-117

COS(A-lb0

n/2 2m

x=c1r +

C2f2m+1

m

=

[(2-2n-m)

1,2,3...

cos2mB

+

m

00s

[(3-4"+2m) sin(2m+l)O

-

Z(m-l)ol (Zm+l)

sin(2m-l)Ol

COS(i+l)q

Stress intensity

factors

determined

by use of Westergaard’s

Table A3. Particular

F

P

=r ' [A1 cos

A2 =

+ A2

(sina cosu

A1 =\(T~-T~)

- (aa +

20

2~. TV - oo)

(~~-7~)

Restrictions

sin a +

Condition

integrals

sin 20 + A3 0 + A4;

-a

)

sin a "5

/ S1 sin a

(aa + 2ar

on particular

Boundary

stress functions

o-

ao) co*

a / s1

integral:

Restriction

Clamped

o=r=o

Free

NOIll?

SS

r=o

Guided

o=o

13