The constant stress term

~

Pergamon

Engineering Fracture Mechanics Vol. 50, No. 5/6, pp. 869 882, 1995 Elsevier Science Ltd 0013-7944(94)E0064-N Printed in Great Britain 0013-7944/95 $9.50 + 0.00

THE CONSTANT STRESS TERM PETER M. HAEFELE and JAMES D. LEE School of Engineering and Applied Science, The George Washington University, Washington, DC 20052, U.S.A. Abstract--The importance of retaining the second, constant stress term of the series expansion for local stress is demonstrated both analyticallyand numerically. It is shown how the standard singular expressions for the stress and displacement field in the vicinity of the crack tip need to be corrected. A load applied parallel to the crack shows up solely and entirely in this constant term. Omitting this term not only denies the physical presence of such a load and hence violates the principle of the uniqueness of the solution of boundary value problems of linear elasticitiy, but also misleads one into thinking that load biaxiality does not influence fracture behavior. It is shown analytically and numerically that the standard expressions are correct only for equal tension-tension loading. The numerical results show that Kxis independent of a horizontal load only when the constant stress term is included.

1.

INTRODUCTION

INVESTIGATIONS[1--1 1], b o t h theoretical a n d experimental, into the question o f how l o a d s a p p l i e d parallel to a c r a c k influence elastic fracture b e h a v i o r have revealed effects which are not p r e d i c t e d by accepted theories[12-14]. These studies have shown that this is because the s t a n d a r d o n e - p a r a m e t e r r e p r e s e n t a t i o n s only in terms o f K~ a n d K . o f the elastic stress a n d d i s p l a c e m e n t field in the vicinity o f the c r a c k tip lack a Constant stress term which only vanishes for special b o u n d a r y conditions. This p a p e r presents the c o r r e c t e d analytical solutions as well as the n u m e r i c a l results for v a r i o u s c r a c k tip p a r a m e t e r s for several plane c r a c k e d geometries, finite and infinite in size. Using a c o m b i n a t i o n o f finite element analysis a n d different analytical crack tip solutions [15], the i m p o r t a n c e o f retaining the second term in the series r e p r e s e n t a t i o n for local stress is shown.

2. ANALYTICALSOLUTIONFOR PLANE CRACKEDG E O M E T R I E S The stresses a n d d i s p l a c e m e n t s for the general p r o b l e m o f elasticity can be given in terms o f two analytic functions q~ a n d [2 as [16] (in u n c o u p l e d form) t,,.,, = Re[q~(z)] + 2y Im[q,'(z)] + Re[f~(z)] t.~x = 3 Re[q)(z)] - 2y Im[q~'(z)] - Re[fl(-~)] tx.,, = - 2y Re[q~ '(z)] + Im[~(z)] - Im[4~ (z)] 2#u,. = x Im[~b(z)] - Re[co(i)] - 2y Im[4~(z)] 2/~y.,, = ~c Im[qS(z)] - Im[co(z)] - 2y Re[q~(z)],

(la-e)

where

co(z)=f P~(z)dz,gp(z)=f 4)(z)dz,

(2)

# is the elastic shear m o d u l u s , v the Poisson ratio, • = (3 - 4v) for plane strain a n d (3 - v)/(l + v) for plane stress conditions. T h e general m e t h o d described by Muskhelishvili in p a r t VI o f ref. [16] to c o n s t r u c t solutions 869

870

P . M . H A E F E L E and J. D. LEE

for the holomorphic functions q~ and f~ can be fairly simplified if one deals with a single straight crack with traction-free edges, giving 2q~(z) -

(2F + F*)z ,/(z ~ -

2f~(z) -

,,~)

(2F + F*)z

P*

(3)

+ F*,

, / ( z ~ - a ~)

where F and F * are constants, generally of a complex nature, and F* denotes the complex conjugate of F. The imaginary part of F can be related to the rotation of an infinitely remote part of the plane and does not affect the stress distribution. Since we are only concerned with the stress distribution, the imaginary part of F can be omitted without losing generality. Using a Cartesian coordinate system where the x-axis extends along the line of the crack (see Fig. 1), F and F* can be expressed either in terms of the stresses t ..... t,., t v at the remote outer boundary of the body or in terms of the principal stresses tll, t22 and the angle ¢p between t~ and the x-axis (see Fig. 1), as

4 Re[F] = t,x(oo) + t.,,>,(oo)= tll (00) +/22(00) 2F* = t,.;(c~) - tx.,-(oo) + 2itv ( ~ ) = - [t~, (oo) - t:2 (oo)] e -2i~,.

(4)

Combining eqs (3) and (4) the introducing the dimensionless parameter

(-~-1 (z-a)=rei 0 a a a (see Fig. 1) yields

2f2 -

~/~

U2

l "3ff

-T-½(t"s- tx~- 2iGs)

= aU,,. t.) 7 w - ~ t + ~ !

/

-T-r*(t ..... t,~. tx,,), (5)

where a(t~,,

tx.,,)-

2F + T'*

,/~

(,,,, - i/xy

,/~

Note that the first term is independent of the load G Fir

2

I

Fig. 1. Cartesian, principal and polar coordinate systems for a single crack.

The constant stress term

871

Let us now restrict our interset to the region IIc which surrounds the right end of the crack tip in Fig. I. Provided that R < a, the nominator in eq. (5) is analytic at every point z in 1-Ic and can therefore be represented as a power series about the crack tip as 2~(()~ = G(t,.,., txy){(~')-'/2 + 3((),/2 _ 3~(()3/2 + . . . } -T-F*. 2n(~)J "

(6)

In polar coordinates, the series expansion (6) beocmes [8, 10, 11] 2q~ (r, 0) 0)1 = / / i(tyy, txy, a ) f , (O)r -I/2 ~ F*(t ..... ty:., txy)r ° 2~(r, +//,(tyy, txy, a ~ (O)r +1/2 + flz(tyy, txy, a)fz(O)r +3/2 + " "

= fl_l(tyy, tx:., a)f_l(O)r -1/2 -T- T'*(t ..... tyy, txy)r ° + O(rl/2).

(7)

The functions/3, which can be related to G and the crack length a, were introduced to be consistent with standard notations. From eqs (7) and (1) the individual stress components can be expressed as tx.,.= ½{Re[(3fl_, - fl-i)

e

i(0/2)] ..{_ sin

0 Im[fl_ l(tyy, txy, a) e-it3°/z)]}r i/2

+ { --2 Re[F*(txx, t,.y, tx>.)]}r ° + O(r '/2) t,.y = ½{Re[(// i + fl-, ) e i(0/2)]-- sin 0 Im[// , (tyy, t~,., a) e-i(3°/2)]}r- I/2 + O (r i/2) t~y = ½{Im[(fl i - / / - , ) ei(°m]+ sin 0 Re[// i (tyy, txy, a) e-i~3°m]}r -'/2 + O(rl/2),

(8a-c)

where fl_~ denotes the complex conjugate o f / 3 ~. It is seen that the constant stress term, i.e. the coefficient of r °, appears only in the expression for t~. At this point the importance of F* in eq. (3) can be noted. The first term G(tyy, txy) of the stress function (5) is independent of a load txx applied parallel to the crack; it is only the second term F*(tx~, tyy, tx,,) where such a load appears. The functions//~(i = - 1, 1, 2 , . . . ) of the power series representation (7) are related to the function G(ty~., txy) and are therefore independent of txx. The structure of this series as well as one of the stress components (8) starts with a square root singularity r -~/2 for the first term, followed by ascending positive powers of the radial distance r from the crack tip and the F*(t~x, ty, txy) term. For crack tip analysis, i.e. for r ,~ 1, terms of order r ~/z and higher assume values which are small enough to be considered negligible by comparison; all these terms are lumped together in the symbol O(r ~/2) in eqs (7) and (8). The T'* term does not fit in this category for two reasons: (1) unlike the other terms it depends upon the load txx applied parallel to the crack; (2) the F* term is independent of the radial distance to the crack tip, i.e. its contribution to the stress function or the stress is the same, no matter how far or close one is to the crack tip. For crack tip analysis, it is customary to use only the first, singular term of eqs (7) and (8). It has, however, been shown that omitting F* changes the character of the solution since it is only through this term that a load applied parallel to the crack reveals itself. Disregarding the second, constant term of any truncated series representation for the stresses is therefore tantamount to the denial of the physical presence of such a load and misleads one to the assumption that the load txx does not affect fracture characteristics. Many experimentalist, e.g. refs [6, 7, 17-20] to name a few, have, however, shown that this assumption cannot hold. In order to derive the corresponding equations for the local displacements, the integration (2) has to be performed to get o~(z) and th(z). The series representation becomes with eqs (2) and (7) q~(r, 01} F* ~o(r, O) = [3 1 ei~°mr 1/2_~_2 - (ei0r + a) + ~//1 eit3°/2)r3'2 + ' " " + 7 r* = [3 ,fj (O)r 1:2 -~ ~ (r e i° + a) + O (r3"2).

(9)

The complex integration constant 7 in the first line of eq. (9) can be neglected since it represents EFM 50/5 6~R

872

P.M. HAEFELE and J. D. LEE

nothing but inconsequential rigid body motion. Substituting eq. (9) into eqs (ld) and (le) yields the following expressions for the displacements: 1

ux = - {Re[// i (x e i{°/2)- - e i{0/21)]i f _ sin 0 Im[jJ_l ei~°/2)]}r 1/2

+

- R e [ 2 r * ] -g-;-- (r cos 0 + a) + Im[2r*l

r sin 0 + O(r 3/2)

!

uy = ~ {Im[fl_, (• e i~°/2) - e-i(°nl)] - sin 0 Re[? I e"°/2~]}r'/2 +

{

X-3 •+1 } - Re[2F*] ~ r sin 0 - Im[2F*] - - 8 # r cos 0 + a + O ( r 3 n ) .

(10)

Paying attention to the structure of eq. (10), one sees that the first lines of the expressions for u.~ and u),, i.e. the coefficients of the square root term r ~/2, depend only on the function fl_ t (tyy, tx,., a). A load t.~xapplied parallel to the crack enters the governing displacement equations only through the r * ( t .... t,,, txy) term in the second lines. Terms of order r 3/2 and higher, which are lumped together in the symbol 0(r3/2), show no tx~ dependency since they contain only the functions fl,(t).y, t~y) (i = 1, 2, 3 . . . . ); see eq. (7). Let us now make some simple reflections to show the importance of retaining the second lines in the expressions for the individual stress and displacement components in eqs (8) and (10). It is well known that in the absence of body forces the stress components for a two-dimensional problem of linear elasticity can be expressed as the partial differential coefficient of the second order of the biharmonic Airy stress function U. The stress function U is obtained as a solution of the corresponding boundary value problem. Altering either of the boundary conditions alters U and therefore the stress components. Varying the boundary traction txx influences the stress and displacement expressions (8) and (10) only when the second lines with the F*(t ..... t.,s, tx~,) term are retained. The common practice to omit these terms and retain only the first lines in terms of the functions fl ~(t.,.~.,txy) means that the same stress and displacement field is predicted, independent of the load txx applied parallel to the crack. This obviously violates the principle of the uniqueness of the solution of boundary value problems of linear elasticity. To illustrate this, consider a sheet with a horizontal flat crack subjected only to a uniform load tx.~(c~)= A = c o n s t a n t parallel to the crack. The corresponding stress field in the plate is txx = A, t~.y= tx, = 0. All functions flg(t~.,, tx,.) (i = - l, 1, 2 . . . . ) vanish for this loading case and eq. (4) gives F * = - A . Equation (8a) shows that it is only through retaining the second, constant stress term that the correct stress field t~x = A is obtained. Neglecting this term predicts wrongly a vanishing stress component t~ in the plate, no matter how many terms of the expansion are used, since all the other terms depend on fl~(t,..,., txy) and therefore vanish for this loading case. The stress and displacement expressions (8) and (10) are usually expressed in terms of crack tip parameters. Following the procedure proposed by Paris and Sih [14], the complex elastic stress intensity factor K can be found as K = Kj - iKl, = lim {x/(2rr)r 1/2e~l°/2)2cb(r, 0)},

(1 I)

r~0+

which gives in return with eqs (3), (5) and (7): Ki -- iK, = x/(2~)fl 1(t.,..,., t~,~,a) = x/0za)(2V + r * ) .

(12)

As expected, K~ and Ku are independent of a load t.,..,,applied parallel to the crack. With eq. (12), the first lines in eqs (8) and (10) can be expressed in the well known manner in terms of K~ and K,. For the second lines the parameters A and B are introduced. Let us therefore define the

The constant stress term

873

following crack tip parameters in terms of F and F*, which can be expressed in terms of the applied boundary loading at infinity with eq. (4) as K, = x/(Tra)Re[2F + F*] =

x/(~za)t:,y(OO)

-- x/(xa) [tll (ov)(l -- cos 2~p) + t22(c~)(1 + cos 2q~)] 2 K. - - x/(~a)Im[2F + F*] =

x/(ga )tx:.(oo) x/(na)[(tl~ (vo) - t,2(~))sin 2q~] 2

-

A = -Re[2T'*] = t , - A ~ ) - t . , ( ~ ) -= [tll (O0) - - t22(£Z))]COS 2~0

B - - Im[2F*] = 2t~.(~) (I 3a-d)

= [tlj ( ~ ) - t22(~)]sin 2~0.

If we restrict our region of interest to the crack tip vicinity, terms of order r ~/2 and higher can be neglected in eq. (8) for stress. To have the same order of approximation for the displacements as for the stresses, terms of order and higher have to be omitted in eq. (10). The governing equations for the crack tip region stresses and displacements therefore become with eqs (8), (10), (12) and (13) for 0 < 1:

r3/2

r/a ~

K, ~ cos 02I i _ sin _02sin ~ ] t. . . ~. x/(Z~r----K,

0l-

0 . 30q

t,;._~x/(2xr) c o s -2 I_I + sin 2 s i n 2- -] /

K,, sin O [ 2 + cos ~0 cos ~ ] + A x/(Zlrr~ K, -sin + -x/(2rtr)

K,. sm ~0 cos -~0 cos -~30 + ~ K,, t~,.~-x/(27rr),

K' x/( ~r

cos 5 ½(x -- 1) + sin 25 J +--#-

x+l + A~

x+l (r cos 0 + a) - B ~ - -

u~.~ - -

O[

cos

K, /( ~r ) sm. ~OI,5(~c+

u~,./~ --

o[

0 0 30 2 cos 2 cos ~--

0 ~1 ,

1 - sin -~ sin

0] K,t x/( - r~ ).sm 50[1"~(x+

1) - -

cos

2

1) + cos 2

(14)

O]

r sin 0

0] K,, /( ~r ) COS~OI,5(1 -- K) + sin 2O]

~ J "q"~ -

+l +A ~¢--3 -~ rsinO+B ~ c81~ (rcosO+a).

(15)

Note the significant difference in the standard singular expressions only in terms of K~ and K, 3. H I S T O R I C A L

DEVELOPMENT

OF FRACTURE

In order to gain more insight into how the notion that a load applied parallel to a crack does not influence fracture behavior started, a brief historical review of fracture is presented. A more detailed description is given in ref. [10]. The first work to pay attention to the existence of stress concentrations and stress gradients was that of Inglis in 1913 [2 !]. He determined the stress concentration factor for a plate with an elliptical cavity subjected to equal tension-tension loading. Griffith, in 1921 and 1924 [12, 13], used this solution as a starting point for the development of his global energy rate theory for crack instability. His results do not exhibit any biaxial load dependency. Although derived for the special case of equal tension-tension loading, Griffith made the erroneous conclusion that they are nevertheless generally correct and applicable for any load

874

P.M.

HAEFELE

a n d J. D . L E E

t~

TTITIITIT m, "C

ka

k~

C2J •

4

lllllllll a

Fig. 2. Center-crackedinfinitesheet with a single horizontal crack under uniform biaxial load and shear. biaxiality. At this point of the development any effect the horizontal load could possibly have was therefore removed. The next major step in developing a general theory of fracture was done by Irwin between 1957 and 1960 [22-24]. He investigated the elastic stress and strain field near the crack tip and was able to establish a relation between a global parameter, the energy release rate G, and a local crack tip parameter, the stress intensity factor K. Irwin based his solution on the Westergaard equations, presented in 1939 [25]. The first person to point out an error in these equations was Sih in 1966 [26], who showed that a constant term was missing. It was Eftis and Liebowitz in 1972 [1] who traced the error back to a lesser known work of MacGregor in 1935 [27]. They showed that the lacking constant was the result of an oversight by MacGregor, upon which Westergaard and later Irwin based their derivations without noticing the error. When the Westergaard equations are corrected through adding the constant term and the same procedure Irwin used is employed, it is seen that the constant stress term A in eq. (14) emerges exactly from this missing term. Equation (13c) shows furthermore that it is only through A that a load txx parallel to the crack becomes part of the solution. The term A and thus the biaxial load effect vanishes only for the case of equal tension tension loading [see eq. (17) and Fig. 2 for k = 1], i.e. exactly for the loading case upon which Griffith based his derivations. Using the partially incorrect Westergaard equations as a starting point, Irwin inadvertently removed the contribution of loads applied parallel to the plane of a crack to the crack tip elastic stress and displacement field. The series of oversights and errors done by both Griffith and throughout the MacGregor-Westergaard-Irwin chain had the momentous consequence that a load t...... seemed to be immaterial for the fracture process. In this regard both independently derived theories seemed to be consistent, thus supporting the notion about the insignificance of a load applied parallel to the crack. 4. SPECIFIC E X A M P L E S We now derive the expressions for the crack tip parameters (13) and thus the expressions for the stress and displacement field in the crack tip region H~ for some infinite specimen with a single center crack.

4.1. Horizontal crack under biaxial load For the center-cracked infinite specimen with a horizontal crack under both biaxial load and shear, as shown in Fig. 2, we have the following boundary conditions referred to the x - y coordinate system: t.,..,.(~)=k~r,

t,.,.(~)=a,

t,:,.(~)=z.

(16)

The constant stress term

875

A negative value of the biaxial load factor k means tension--compression loading. It is assumed that in this case the (arbitrary) thickness of the sheet is large enough to prevent buckling. The crack tip parameters are found from eqs (13) and (16) as

Ki = ax/(na)

KH = zx/(na)

A = -a(l-k)

B=2~.

(17)

Equation (17) shows again that it is only through retaining the constant stress term A that the boundary load t~x(oo) = ka becomes part of the solution of the boundary value problem. We see furthermore that A vanishes only for the case of equal tension-tension loading, i.e. for k = 1. This means that the standard singular expressions for the crack tip region stresses are only correct for this special loading situation. This will be demonstrated by the numerical results in the next section. 4.2. Inclined crack under biaxial load The crack geometry and the boundary traction for a center-cracked infinite sheet with a single inclined crack under uniform biaxial loading is shown in Fig. 3. In this case it is easier to express the boundary traction applied at infinity in the system of principal axes x'-y'; thus tll(~)=a,

t22(oo)=ka,

~0=0~.

(18)

Combining eqs (18) and (13) yields the crack tip parameters as

K, = ax/(Tra) [(1 + k) - (1 - k)cos 2~] 2

Ku = ax/(na) [(1 - k)sin 2~] 2

A = a(1 - k)cos 2~ B = a(1 - k)sin 2~.

(19)

Equation (19) shows analytically what will be demonstrated numerically in the next section, namely that the standard expressions in terms of K~ and Kn only are correct only for k = 1, since A and B vanish only for this case.

ITTIIITIIo ,

!!I ~ + k ' ~ l ~ )

~s(2~]

Oy=o'~[(l+k')+(l-k') cos(2~)]

Fig. 3. Center-cracked infinite sheet with a single inclined crack suNected to uni~rm biaxial loading.

876

P . M . H A E F E L E and J. D. LEE

,g )

L,I:

-a

91 Fig. 4. Center-cracked infinite sheet with a single inclined crack under pure shear loading.

4.3. Inclined crack under pure shear Figure 4 shows the geometry and the boundary loading conditions for the cracked shear panel. Referred to the x - y coordinate system we have t,.,.(~) = z sin2fl,

t,.>(oo)= - z sin2fl,

t~>.=z cos2fl,

(20)

which yields in return with eq. (13) K~ = - r ~ / ( ~ a ) s i n 2fl, A=2zsin2/L

K. = zx/0za)cos 2fl

B=2zcos2#.

(21)

For the case of pure mode II loading, i.e. for/~ = 0, both K~ and A vanish. 5. N U M E R I C A L RESULTS To demonstrate numerically the importance of retaining the constant stress term A, a finite element program was developed. The program combines the analytical crack tip solution in the circular core region Hc surrounding the crack tip (see Fig. 1), with a conventional finite element analysis in the outer region Hr. Various crack tip parameters are computed as part of the solution. A detailed description is given in ref. [15]. The following three analytical solutions to describe the local stress and displacement in the core region II e have been used and the numerical values for K~ and KH have been compared:

Option 1. The standard singular expressions for stress and displacement only in terms of Kt and K. are used, i.e. eqs (14) and (l 5) with A = B = 0. The two consequences of neglecting A and B are (i) a load t,,. applied parallel to the crack does not become part of the solution of the boundary value problem and (ii) the crack tip (r = 0) is treated as a fixed point, i.e. ux(tip) = u.,.(tip) = 0. The first consequence violates the principle of the uniqueness, as mentioned above. Treating the crack tip as a fixed point is not in accordance with the physical behavior. For a center-cracked specimen with a horizontal fiat crack under tensile load, i.e. pure mode I conditions, for example, the crack tip obviously has to move towards the center due to the Poisson effect. Option 2. Both A and B are again omitted. However, the need to account for a displacement

The constant stress term

877

of the crack tip is recognized. Therefore (integration) constants are added to the displacement equations (15) (with A = B = 0 ) as additional parameters. Similar methods have already been employed by several investigators, who introduced either only a horizontal crack tip displacement 6 or a displacement vector to at the crack tip as an additional parameter to be determined as part of the solution [28-30]. A shortcoming of this option can be seen when a specimen with a horizontal flat crack under only uniform load t.~x(~)=,4 applied parallel to the crack is considered. In this case the crack does not influence the stress field in the plate; both Kj and K , vanish. The resulting stress distribution is t,.,. = A = constant, t,. = t,.y = 0 in the whole plate. The expression for stress employed in this option [eq. (14) with `4 = 0] predicts, however, tx,. = 0 in the crack tip region, which is obviously incorrect. The predicted horizontal displacements along the x-axis are in this case u,. = constant, i.e. every point moves by the same amount (rigid body motion). Option 3. The corrected eqs (14) and (15), which give the best approximation for the stress and displacement field in the crack tip region, are used. In ref. [31] a constant stress term sxx was introduced as an unknown parameter for the stress field description to determine stress intensity factors (SIFs). Thus, one expects to get bad numerical results for the SIFs for Option 1, better ones for Option 2 and the best if the analytical solution to Option 3 is employed. The numerical results are presented in a dimensionless, normalized form. Therefore the dimensionless SIFs K/Numerical K * - - K/Analytical

i = I , II

(22)

are defined. When K/Analytica I ~--- 0, a " - " is used in the tables.

5.1. Horizontal crack under biaxial load and shear This loading case, as shown in Fig. 2, has already been discussed in ref. [15]. In Table I and Fig. 5, we therefore present only the case where a and z are kept constant, whereas the load ktr parallel to the crack is varied. Varying the biaxial load factor k should not alter K~, since K, depends only on the load applied perpendicular to the crack, eq. (17). From Table 1 one sees that Option 1, which does not allow crack tip displacements, results in poor results for both Kj and K . . Figure 5a shows that KI depends strongly on the biaxial load factor k. It can be shown analytically [15] that the crack tip only moves horizontally for equal tensiontension loading. This is also seen from eq. (17), since A vanishes only for k = 1. The arrow in Fig. 5a indicates that Option 1 gives a reasonable value for K* only for k = 1. This demonstrates numerically that the standard singular expressions used in Option 1 are only correct for this loading case. Both Options 2 and 3 yield good results. The two lines in Fig. 5a for K* for those two options are hardly distinguishable. However, a closer look (Fig. 5b) reveals that Kl still depends on the load applied parallel to the crack for Option 2. This is due to the fact that `4 is the only parameter which accounts for the presence of the load ka applied parallel to the crack, which depends upon k. Failure to include `4 in the analytical crack tip solutions as done in Options 1 and 2 therefore results in a dependency of Kj on k which is physically incorrect. It is only in Option 3 where K~ is independent of k, depending solely on the load applied perpendicular to the crack, as one would expect (see Fig. 5b). Table 1. Results for center-cracked infinite sheet with a horizontal crack under biaxial load and shear, with variable biaxial load factor k Option 1 Option 2 Option 3 k -4 -3 -2 -- 1 0 +1 +2 +3 +4

K( -2.078 - 1.446 -0.814 -0.181 0.451 1.084 1.717 2.349 2.982

K~ 0.018 0.018 0.018 0,018 0,018 0,018 0,018 0.018 0,018

K~ 0.958 0.968 0.979 0.989 0.999 1.010 1.020 1.030 1.041

K~ 1.013 1.013 1.013 1.013 1.013 1.013 1,013 1.013 1.013

K~ 0.999 0.999 0.999 0.999 0.999 0.999 1.000 1.000 1.000

K~ 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013

P . M . H A E F E L E and J. D. LEE

878

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3

4

B i a x i a l l oa d fa c t or k Fig. 5. Normalized stress intensity factors versus biaxial load factor k for an infinite sheet with a horizontal crack under biaxial loading and shear. (a) K* for Options 1-3. (b) K* and K* for Options 2 and 3.

Figure 5b and Table 1 show furthermore that the results for K* are the same for Option 2, where rigid body motion is possible, and for Option 3, since A vanishes for pure mode II loading, eqn (17). The poor results for Option 1 are due to the fact that the crack tip is fixed. 5.2. Inclined crack under biaxial loading The results for this loading case, as shown in Fig. 3, are presented in Table 2 and Fig. 6 for an angle ct = 30 ° and k from - 4 to +4. Figure 6a shows K* versus k for Options 1-3; the results for K* and K* versus k for Options 2 and 3 are shown in Fig. 6b. Option 1 again yields very poor results with the exception of K* for k = 1 (see Fig. 6a). It is only for this loading case that the constant stress term A and the crack tip displacement vanish [see eqs (14), (15) and (19)]. Thus, the corrected analytical expressions (14) and (15) reduce to the standard expressions only in terms of K~ and K~ which are used in Option 1.

Table 2. Results for center-cracked finite sheet with an inclined crack under uniform biaxial load, with variable biaxial load factor k Option I

30

Option 2

Option 3

k

K?

K~

K~

K~

K?

K~

--4 --3 --2 -- 1 0 + 1 +2 +3 +4

0.513 0.455 0.325 --0.181 2.350 1.084 0.903 0.835 0.796

0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018

1.000 0.999 0.997 0.989 1.030 1.010 1.007 1.006 1.005

1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013

1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000

1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013 1.013

T h e c o n s t a n t stress t e r m

879

(a) 2.5 - °° • ° ;

2.0 o

1.5

;

_

• . ..... •

I

I

I

-3

-2

-1

@......

•

-- • - Option I -- A. --•~

,"

"3 -4

,I 0

-

Option 2 Option 3

I

I

I

I

1

2

3

4

Biaxial load factor k

(b) 1.03

"-- . . . . . .

.** "- * = .

--

-o.s

o

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,.

0.5, . . . . .

Z

k=l

".

"

1.o

o

°.

--

,,

1.02 -----m----m-----m--7.m---'Am.---

t----m--..m it....

I .oi

.$ F

I .o0

•

/

~A

0.99 _

•

--$ ......

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•

-- A, - KI*, Option 2 m ~ - - KI*, Option 3

I ~' ~.~

- - n . - KII*, Option 2 and 3 0.98 Z

] -3

--4

I -2

I

I

I

I

I

I

-1

0

1

2

3

4

Biaxial load factor k Fig. 6. N o r m a l i z e d stress intensity f a c t o r s versus b i a x i a l l o a d f a c t o r k for a n infinite sheet w i t h a n inclined c r a c k u n d e r b i a x i a l l o a d i n g (ct = 30°). (a) K * f o r O p t i o n s 1-3. (b) K ? a n d K~ for O p t i o n s 2 a n d 3.

With the coarse scale used in Fig. 6a, both Options 2 and 3 seem to give the same good results for K*. However, if we zoom in and use the fine scale of Fig. 6b, we see that the apparent value K* is influenced by the biaxial load factor k for Option 2. Again, it is only when the corrected equations (14) and (15) are used to describe the stress and displacement field near the ends of the crack that K* remains constant. KS is independent of k for both Options 2 and 3, since A assumes the value zero for mode II loading (see Fig. 6b). 5.3. Inclined crack under pure shear Varying the angle ~ from 0 ° to - 9 0 ° for the cracked shear panel (Fig. 4) yields the results shown in Table 3. The error in calculating the SIFs is very high for Option 1 and is reduced to about 1% with Option 2. Describing the stresses and displacements in the vicinity of the crack tip through the corrected equations (14) and (15) as done in Option 3 reduces the error for K~ to less than 1%. T a b l e 3. R e s u l t s f o r c e n t e r - c r a c k e d infinite sheet w i t h a n inclined c r a c k u n d e r p u r e s h e a r f o r v a r i o u s angles p Option 1

0.0 - 15.0 -22.5 --30.0 - 45.0 --60.0 --75.0 --90.0

Option 2

Option 3

K~

K~

K~

X~

K~

K~

---0.177 -0.177 --0.177 - 0.177 --0.177 -0.177 --0.177

0.018 0.018 0.018 0.018 -0.018 0.018 0.018

-0.989 0.989 0.989 0.989 0.989 0.989 0.989

1.013 1,013 1.013 1.013 -1.013 1.013 1.013

-0.999 0.999 0.999 0.999 0.999 0.999 0.999

1.013 1.013 1.013 1.013 -1.013 1.013 1.013

880

P . M . H A E F E L E and J. D. LEE Table 4. Dimensionless values K* for a quadratic finite plate subjected to a load perpendicular to the crack for various ratios W/a for Options 1-3 W

Option l

Option 2

Option 3

0.419 0.439 0.437 0.448

0.952 0.989 0.982 0.991

0.953 0.990 0.983 0.992

2H

a

a

6 12 18 36

Again, since A assumes the value zero for pure mode II loading, eqn (17), the value for KH is the same no matter whether A is included or not, as long as rigid body rotation is not prevented (Options 2 and 3). 5.4. Finite plate In reality one has to deal with cracked specimens having finite dimensions. Determining the SIFs for these specimens is of great importance in fracture mechanics. For a center-cracked finite sheet with a horizontal crack under pure mode I loading, K~ can be calculated as [32] Kl = a ~/(~a)F, (~, fl) 2a = -W'

fl

2H W'

(23)

where 2a is the crack length, W the width and 2H the height of the plate. The correction factor F~ was found from ref. [32] using linear interpolation. Table 4 lists the dimensionless values for K~' for a quadratic finite plate, i.e. fl = 1, subjected to a uniform load a perpendicular to the crack only, for various ratios 1/~ = W/a = 2H/a. The stress intensity factor K~ is obtained with almost the same good accuracy for both Options 2 and 3, but not for Option 1. In order to investigate the influence of a load ka applied parallel to the crack in addition to the load a applied perpendicular to it, a finite plate with W/a = 12 was subjected to biaxial loading. Table 5 lists the results for values of k from - 4 to + 4. Figure 7a shows K* versus k for Options 1-3, Fig. 7b only for Options 2 and 3. As expected, the results are similar to the ones for the infinite sheet under biaxial load in Table 1' poor results for Option 1 with the exception of k = 1 (see Fig. 7a), a value of K~ which depends on the load ka parallel to the crack for Option 2 and finally, for Option 3, a result of K~ which depends only on the load applied perpendicular to the crack, which is not altered by k (see Fig. 7b). The results show that the developed finite element program computes the SIFs for finite specimens with good accuracy, provided that A is included in the analytical crack tip solutions. 6. CONCLUSIONS It was shown analytically and numerically that the standard singular expressions only in terms of Kt and K. for the stress and displacement field in the vicinity of the crack tip cannot be regarded as valid approximations in a complete general sense. These equations, in which a load applied Table 5. Dimensionless values K~ for a quadratic finite plate with W/a = 12 subjected to biaxial loading for Options 1 3, with variable biaxial load factor k k

Option 1 K*

Option 2 K*

Option 3 K*

-4 - 3 - 2 - 1 0 + 1 +2 +3 +4

- 2.029 - 1.430 - 0.800 -0.170 0.439 1.068 1.706 2.335 2.967

0.936 0.949 0.963 0.975 0.989 1.000 1.013 1.026 1.039

0.987 0.988 0.989 0.990 0.990 0.991 0.991 0.992 0.994

The constant stress term

8

881

(a) 3 -k=l 2 --

o

• o.

~..•.o

l I *=,=-'*i==*-,i ,

•

•

.-'°'-'*

•o

.

..o

"

.•"

*~O'd"'~O

o°

•

oO

0

-

-I

o.o"

--

.g

0""

"" • "

Option 1

o

- - A. - O v t i o n 2

~ O - - Option 3

-3 8

-4

Z

I

I

I

I

I

I

I

I

-3

-2

-1

0

I

2

3

4

(b)

Biaxial

load factor

k

1.04

,s

t.o2

__

j,

jA ~

s

1.00 e~ "~" "Y"

s

0.98 s

0.96

j,

- - •" " O p t i o n 2

.•s

,s

0.94 Z

~@--

.s

0.92 -4

Option 3

I

I

I

I

I

I

I

I

-3

-2

-I

0

I

2

3

4

Biaxial

load factor k

Fig. 7. Normalized stress intensity factor K~ versus biaxial load factor k for a finite sheet with a horizontal crack under biaxial loading. (a) K~' for Options 1-3. (b) K~' for Options 2 and 3.

parallel to the crack does not reveal itself, need to be corrected through retaining the second, constant stress term of the series expansion for local stress in eq. (8). The biaxial load effect enters only through this term. Omitting this term, which is still the general practice, therefore has the particular effect of nullifying the physical presence of this load. The numerical results for a sheet with a horizontal crack under biaxial load show that K, is independent of horizontal load only when the constant stress term A is included in the analytical crack tip solutions. If A is not included, one obtains mistaken values for KI which increase as k is increased (see Figs 5 and 7). As predicted by the analytical results, eqs (17) and (19), the leading terms of the series expansion can adequately describe the state of stress and displacement near the ends of the crack only for the special case of equal tension-tension loading, i.e. for k = 1 (see Figs 5a, 6a and 7a). It is only for this case that the constant stress vanishes. Bearing this in mind and considering that Griffith based the development of his theory upon results for exactly this loading case, it makes sense that he did not encounter a biaxial load sensitivity. His incorrect conclusion was, however, to generalize his results and declare them applicable for any load biaxiality. Irwin and Westergaard based their work on the results of MacGregor, who mistakenly neglected (or forgot) a real valued constant in this derivations. This had the momentous consequence that the constant stress term A could never appear in the series expansion. The presence of a load applied parallel to the plane of the crack therefore seemed to be immaterial, thus giving the impression of being fully consistent with Griffith's theory. Provided that the crack tip is not treated as a fixed point (Options 2 and 3), the presence or non-presence of A in the finite element analysis does not affect the values for Kl~ for mode II loading (Tables 1-3 and Figs 5b, 6b), since A assumes the value zero for this case, eq. (17). However. the accuracy for calculating Kl is always improved by including the constant stress term A. Acknowledgement--The authors and comments.

wish to express their appreciation to Professor John Eftis for his many useful suggestions

882

P . M . HAEFELE and J. D. LEE

REFERENCES Ill J. Efiis and H. Liebowitz, On the modified Westergaard equations for certain plane problems, int. J. Fracture Mech. 8, 383-392 (1972). [2] J. Eftis, N. Subramonian and H. Liebowitz, Crack border stress and displacement equations revisited. Engng Fracture Mech. 9, 189-210 (1977). [3] J. Eftis, N. Subramonian and H. Liebowitz, Biaxial load effects on the crack border elastic strain energy and strain energy rate. Engng Fracture Mech. 9, 753-764 (1977). [4] J. Eftis and N. Subramonian, The inclined crack under biaxial load. Engng Fracture Mech. 10, 4 3 ~ 7 (1978). [5] J. Eftis and N. Subramonian, The cracked shear panel. AIAA J. 18, 324-332 (1980). [6] D. L. Jones and J. Eftis, Fracture and fatigue characteristics of aircraft structural materials under biaxial loading. AFOSR-TR-81-0856. ADA 10954, AFOSR, Washington, DC (1981). [7] J. Eftis and D. L. Jones, Influence of load biaxiality on the fracture load of center cracked sheets. Int. J. Fracture Mech. 20, 267-289 (1982). [8] J. Eftis, Influence of load biaxiality on the fracture characteristics of two collinear cracks. Int. J. Fracture Mech. 24, 59-80 (1984). [9] J. Eftis, On the fracture stress for the inclined crack under biaxial load. Engng Fracture Mech. 26, 105-125 (1987). [10] J. Eftis, Load biaxiality and fracture: a two sided history of complementing errors. Engng Fracture Mech. 26, 567-592 (1987). [I I] J. Eftis, D. L. Jones and H. Liebowitz, Load biaxiality and fracture: synthesis and summary. Engng Fracture Mech. 36, 537-574 (1990). [12] A. A. Griffith, The phenomena of rupture and flaw in solids. Phil. Trans. R. Soc. A221, 163-198 (1921). [13] A. A. Griffith, The theory of rupture. Proc. First Congress of Applied Mechanics, pp. 55~3, Delft (1924). [14] P. C. Paris and G. C. Sih, Stress analysis of cracks. A S T M STP 381, 30-81 (1965). [15] P. M. Haefele and J. D. Lee, Combination of finite element analysis and analytical crack tip solution for mixed mode fracture. Engng Fracture Mech. 50, 849-868 (1995). [16] N. I. Muskelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, 3rd edn. P. Noordhoff, Groningen, Holland (1953). [17] J. M. Etheridge and J. W. Dally, A critical review of methods for determining stress intensity factors from isochromatic fringes. Exp. Strain Anal. 254-284 (July 1977). [18] J. C. Radon, P. S. Leevers and L. E. Culver, Fracture toughness of PMMA under biaxial stress, in Fracture 77, Vol. 3, pp. 1113-1118. Univ. of Waterloo Press, Waterloo, Ontario, Canada (1977). [19] J. J. Kibler and R. Roberts, The effects of biaxial stresses on fatigue and fracture. J. Engng Ind. 92, 727-734 (1970). [20] C. D. Hopper and K. J. Miller, Fatigue crack propagation in biaxial stress field. J. Strain Anal. 12, 23-28 (1977). [21] C. E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners. Trans. Inst. Nay. Archit. 60, 219-230 (1913). [22] G. R. Irwin, Analysis of stress and strains near the end of a crack traversing a plate. J. appl. Mech. 79, 361 (1957). [23] G. R. Irwin, Fracture, in Encyclopedia of Physics, Band VI, Vol. VI (Edited by S. Flfigge). Springer, Berlin (1958). [24] G. R. Irwin, Fracture mechanics, in Structural Mechanics (Edited by J. N. Goodier and N. HolT). Pergamon Press, New York (1960). [25] H. M. Westergaard, Bearing pressure and cracks. J. appl. Mech. 6, A49-A53 (1939). [26] G. C. Sih, On the Westergaard method of crack analysis. Int. J. Fracture Mech. 2, 628 (1966). [27] C. W. MacGregor, The potential function method for the solution of two dimensional stress problems. Trans. Am. math. Soc. 8, 177-186 (1935). [28] J. J. Oglesby and O. Lomacky, An evaluation of finite element methods for the computation of elastic stress intensity factors. J. Engng Ind. 177-185 (Feb. 1973). [29] S. G. Papaioannou, P. D. Hilton and R. A. Lucas, A finite element method for calculating stress intensity factors and its application to composites. Engng Fracture Mech. 6, 807-823 (1974). [30] P. D. Hilton, L. N. Gifford and O. Lomacky, Finite element fracture mechanics analysis of two dimensional and axisymmetric elastic and elasti~plastic cracked structures. NSRDC Report 4493, Bethesda, MD (Nov. 1974). [31] Y. Yamamoto and N. Tokuda, Determination of stress intensity factors in cracked plates by the finite element method. Int. J. numer. Meth. Engng 6, 427-439 (1973). [32] Y. Murakami (Ed.), Stress Intensity Factors Handbook. Pergamon Press, Oxford (1987).