Dependence of stress intensity factor on loading condition

Pergamon

DEPENDENCE

EngineeringFractureMechanicsVol. 55, No. 3, pp. 355-361, 1996 Copyright© 1996ElsevierScienceLtd. Printed in Great Britain.All rightsreserved PII: S0013-7944(96)00031-8 0013-7944/96 $15.00+ 0.00

OF

STRESS

LOADING

INTENSITY

FACTOR

ON

CONDITION

Y. Z. CHEN Division of Engineering Mechanics, Jiangsu University of Science and Technology, Zhenjiang Jiangsu, 212013, People's Republic of China NORIO HASEBE Department of Civil Engineering, Nagoya Institute of Technology, Gokiso-cho, Showaku, Nagoya 466, Japan AbsUaet--Tbe problem of a centrally cracked rectangular plate is taken as a sample problem to evaluate the dependence of the stress intensity factor on the loading condition. The total loading applied on the upper and lower edges is assumed to be 2bp (2b--the width of the plate). With the condition of the same total applied loading 2bp along the upper and lower edges, the following conditions are imposed: (a) uniform tension with intensity p, (b) a pair of concentrated forces with intensity bp, (c) a type of tension-compr.~ssion loading. A novel weight function approach is used to evaluate the stress intensity factor at the crack tip. From the numerical solution it is found that, with the condition of h/b = 3.0 (h/b being the height width ratio of the plate) the maximum deviation of the stress intensity factor from the uniform tension case is 5.8% in the concentrated force loading case, and with the same condition the dependence of the stress intensity factor on the loading condition is negligible in the tension-compression loading case. However,in the case ofh/b < 1.5, the relevant dependence is significant. Copyright (¢.31996 Elsevier Science Ltd.

I. I N T R O D U C T I O N THE SAINT-VENANI principle is a general principle and is widely used in elasticity [1]. However, the principle can not be used in the case that the loading boundary becomes a considerable portion of the total boundary. Generally, the stress intensity factors depend on the loading condition, even though the total loading is the same. The problem of a centrally cracked rectangular plate is taken as a sample problem to evaluate the dependence of the stress intensity factor on the loading condition. The total loading applied on the upper or lower edges is assumed to be 2bp (2b--the width of the plate). Along the upper and lower edges the following loading conditions are imposed: (a) uniform tension with intensity p, (b) a pair of concentrated forces with intensity bp, (c) a type of tension-compression loading. Concerning the solution technique, a complex variable function method was suggested to solve the problem of a centrally cracked plate [2]. If the loading is the concentrated forces on the outer boundary, the relevant solution should be obtained by superposition of three particular solutions. Also, the usual boundary collocation technique is not convenient to solve the boundary value problem with the concentrated forces applied on the boundary [3]. Recently, a novel weight function approach is suggested to evaluate the stress intensity factor at the crack tip [4~. The merit of the weight function is that, once a particular boundary value problem is solved, all the boundary value problems with the same geometry can be solved immediately. In this paper the dependence of the stress intensity factor on the loading condition is evaluated by the suggested weight function approach.

2. ANALYSIS To solve the centrally cracked plate problem, a novel weight function formulation is suggested [4]. The outline of the formulation is introduced below. In the complex variable function method in plane elasticity, the stresses (tr, a,, a , ) , the resultant force functions (X, Y) and the displacements (u, v) can be described by two complex potentials qb(z) and co(z) [4, 5] 355

356

Y. Z. CHEN and N. HASEBE ~Y

Uniform tension IY (U)

P\

,

F!

J

tt

ttt

2h ' ; B[,,_~Oa_J A

_._~ i ¢ __i_L_,

E

r

P X

I

Fig. I. An internal crack problem. C

{,

2b ............. D

Fig. 2. A centrally cracked rectangular plate in uniform tension p.

a , + a,, = 4 R e [ ~ ( z ) ]

a.,.- a., + 2ia,,. f = -

= 2( -

4~(z) -

(z -

,O~'(z)

(l)

+ fl(z))

(2)

Y + iX = 4~(z) + (z - zT)~'(z) + co(z)

2G(u + iv)

(3)

= x~b(z) - (z - ~?)~b'(z) - og(z),

where q~(z) = qV(z), fl(z) = co'(z)

(4)

a n d G is the s h e a r m o d u l u s o f elasticity, x = 3 - 4 v for the p l a n e x = (3 - v)/(1 + v) for the p l a n e stress p r o b l e m a n d v is P o i s s o n ' s ratio. In the a n a l y s i s we d e n o t e

strain

problem,

q~(z) = S,(x,y) + iS2(x,y). After some manipulations, f o l l o w i n g e q u a l i t y [4]

for the b o u n d a r y

(5)

v a l u e p r o b l e m s h o w n in Fig. I, we o b t a i n the

-frS~,)aonjdS=£Re[(K,A-iKza)(~o+g,a)+(K,B-iK2a)(~o-~,a)

1,

(6)

w h e r e a,,nj is the b o u n d a r y t r a c t i o n in the p h y s i c a l p r o b l e m (Fig. 1), K,A -- iK2^ a n d g m iK2a are the stress i n t e n s i t y factors at the tips A a n d B, respectively, a is the h a l f c r a c k length. In eq. (6), the f u n c t i o n Se> (i = 1,2) is the i n v e s t i g a t e d w e i g h t f u n c t i o n a n d it is c o m p o s e d o f S ~ = S,~, + S~¢~ (i = 1,2).

(7)

A / ~ j - f i e l d is d e r i v e d f r o m the f o l l o w i n g c o m p l e x p o t e n t i a l [4] ~b~,(z) -

eo +

etz

X(z)

Table 1. The calculated

h/b 0.5 1.0 1.5 2.0 3.0

'

o~,~,,(z)-

do + ~tz X(z)

(8)

'

E;A(h/b,a/b) values [see Fig. 2 and

eq. (12)]

a/b 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.000 t 1.000 1.000 tl.000 1.000 1.000 1.000

1.045 1.046 1.014 1.014 1.007 1.006 1.006

1.174 I. 175 1.055 1.055 1.029 1.024 !.023

1.371 I. 371 1.123 1.123 1.066 1.057 1.054

1.630 1.629 1.216 1.216 1.121 1.107 1.101

1.966 1.967 1.332 1.334 1.199 1.180 1.171

2.422 2.424 1.476 1.481 !.311 i.287 1.276

3.035 3.04 1.661 1.68 1.480 1.454 1.444

3.737

4.281

1.941

2.595

1.773 1.752 1.752

2.493 2.497 2.509

t In handbook [71.

Dependence of stress intensity factor on loading condition 2bp F

Loadig Case

[

I

3 c,

~2p ~

v~

357

2.2p I.,o~g Case

, E rc, 2.4p

l

Ca

•

2.6p F

Ibp

bPl

E z

(73

I

i

W__b ~

2.8p

, q •

1.5b

F

z ~

E TC, 3.0p

"~

,L_ (a)

Co)

Fig. 3. Various loading cases on the edge, (a) the concentratedloading case, (b) the tension--compression loading case.

where e0 and et take a complex value in general and - a 2 (taking the branch Lim X ( z ) / z = 1).

X(z) = ~

(9)

z~ao

The fit-field is a siagular field in the sense that the displacements have an order O(r-t/2) at the vicinity of the crack tip. From the notation shown by eq. (5) and the function Stpt~(z) in eq. (8) we can get S~at~ immediately. A particular physical fl2-field is introduced. In the fl:-field the tractions applied along the outer boundary F have the same magnitude but opposite direction as those derived from the fl~-field. The complex potentials for the fl2-field can be assumed in the form [4] M

ok(z) = ~., (a., + i b . . ) X ( z ) z " - '

M

+ ~., (c,. + id,.)z"

¢n=l

M

to(z) = ~ (a,~ - i b , , ) X ( z ) z m - ' m=l

m=l

M

~ (c,~ - idm)z '~ .

(10)

m=l

The involved undetermined coefficients am, b,,, c, and d~, can be evaluated by some numerical approach [4]. The following four particular cases are of interest to evaluate the individual stress intensity factor KLA [taking e0 = 1/2, et = l / 2 a in eqs (6) and (8)] K2A [taking e0 -- - i/2, et = - i/2a in eqs (6) and (8)] . Kja [taking e0 = - 1/2, et = l / 2 a in eqs (6) and (8)] K2B [taking e0 = - i/2, et = i/2a in eqs (6) and (8)]

(ll)

After the function Saat is obtained, the stress factors at the crack tips can be evaluated by simply performing integration. 3. N U M E R I C A L RESULTS The dependence of stress intensity factors on the loading condition is illustrated by the calculated numerical results. As mentioned before, the problem of a centrally cracked rectangular plate is taken as a sample problem (Fig. 2). Various loading conditions at the upper and lower

358

Y . Z . CHEN and N. HASEBE Table 2. The calculated

F,^(h/b, a/b) values [see Fig.3(a) and eq. (13), C,, C:, C~, C4, C~ being the loading condition in Fig.3 (a)].

a/b

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3.514 3.675 2.259 0.394 - 0.057 - 0.512

3.584 3.641 2.474 0.655 - 0.037 - 0.525

3.662 3.630 2.768 1.064 0.037 - 0.555

3.755 3.699 3.098 1.578 0.208 - 0.607

3.938 3.905 3.474 2.157 0.518 - 0.662

4.311 4.304 3.954 2.799 0.996 - 0.647

4.912 4.915 4.576 3.515 1.658 - 0.402

5.550

5.656

5.199 4.220 2.467 0.320

5.373 4.629 3.334 1.750

1.922 1.927 1.654 1.038 0.418 - 0.127

1.933 1.933 1.680 1.090 0.476 - 0.078

1.951 1.944 1.719 1.171 0.574 0.010

1.975 1.962 1.769 1.275 0.711 0.144

2.008 1.989 1.827 1.395 0.888 0.331

2.055 2.033 1.896 1.531 1.108 0.582

2.130 2. I 18 1.992 1.698 1.382 0.913

2.287

2.788

2.176 1.958 1.752 1.384

2.741 2.617 2.463 2.301

1.303 1.294 1.215 1.015 0.818 0.615

1.312 1.301 1.226 1.034 0.850 0.652

1.328 1.315 1.246 1.069 0.904 0.714

1.355 1.341 1.279 1.120 0.982 0.802

1.401 1.387 1.331 1.193 1.087 0.921

1.476 1.466 1.414 1.300 1.228 1.080

1.606 1.608 1.555 1.466 1.424 1.303

1.855

2.521

1.822 1.763 1.735 1.662

2.524 2.505 2.153 2.480

1.083 1.070 1.052 0.994 0.964 0.928

1.098 1.084 1.068 1.011 0.986 0.952

I. 125 1.111 1.096 1.042 1.023 0.993

1.168 1.154 1.140 1.091 1.078 1.053

1.233 1.221 1.206 1.162 1.157 1.139

1.330 1.326 1.306 1.269 1.271 1.260

1.485 1.506 1.466 1.439 1.443 1.440

1.766

2.48 I

1.759 1.746 1.746 1.753

2.496 2.513 2.491 2.511

1.029 1.011 0.981 0.992 1.064

1.047 1.028 0.998 1.009 1.081

1.077 1.059 1.029 1.040 1.111

1.123 1.106 1.077 1.087 1.157

1.191 1.176 1.150 1.158 1.222

1.291 1.279 1.259 1.266 1.318

1.450 1.444 1.435 1.439 1.469

1.740 1.747 1.758 1.758 1.746

2.460 2.492 2.547 2.539 2.439

h/b = O.5 Ci

3.484 ~3.692 2.180 0.305 - 0.060 - 0.509

C2 C~ C, Cs

h/b = 1.0 CI

1.919 "1"1.925 1.645 1.021 0.399 - 0.143

C2 C~ C4 Cs

h/b = 1.5 C,

1.301 "1"1.292 1.211 1.008 0.808 0.603

C2 C~ C4 (.'~

h/b = 2.0 C,

1.079 "11.065 1.047 0.988 0.957 0.921

C: C~ (;4 C,

h/b = 3.0 C~ C2 C~ C~ C~

1.024 1.005 0.975 0.986 1.058

1" In handbook [7].

edges are assumed. As mentioned before, the fl~-field has a definite form which was shown by eq. (8). For the fl2-field, 47 terms are truncated in eq. (10) and 47 unknowns [am, b,,, c~, (m = l, 2 .... 12), dm (m = 2, 3 .... 12)] are evaluated by using the weight residue approach in an integration form [4]. In addition, the Simpson rule with 100 divisions is used to evaluate the integral in eq. (l 1) [6].

2.6

FIA F,A 3 2.5

h/b-l.0

1.8

2 1.5

1.4 I

0.5 0 -0.5 0

h/b=1.5

2.2

0.2

0.4

0.6

0.8 0.9 a/b Fig. 4. The calculated F,^(h/b, a/b) values in the case ofh/b = 1.0. [see 9 . (13), the loading conditions U, C,, C2, C3, C, and Cs referred to Fig. 2 and Fig. 3(a)].

0.61 0

CI C3 C, . . 0.2

.

0.4

.

.

. . 0.6

0.8 0.9 a/b

Fig. 5. The calculated F,^(h/b, a/b) values in the case ofh/b = 1.5 [see eq. (13), the loading conditions U, C,, C2, Cj, C, and C~ referred to Fig. 2 and Fig. 3(a)].

D e p e n d e n c e o f stress intensity factor on l o a d i n g c o n d i t i o n 2.6

Fj^

359

2.6

h/b=2.0

2.2

Fl^ 2.21.41.8 h/b=3"O 1.8

1.4

q 0.2

0.6

0.4

0.8

c~

c~.~/..~"

0.9

a/b

0.2

0.4

0.6

0.8

0.9

a/b Fig. 7. The calculated F~^(h/b, a/b) values in the case ofh/b = 3.0 [see eq. (13), the loading c o n d i t i o n s U, C,, C,, C~, C~ a n d Cs referred to Fig. 2 a n d Fig. 3(a)].

Fig. 6. The calculated F~^(h/b, a/b) values in the case ofh/b = 2.0 [see eq. (13), the l o a d i n g c o n d i t i o n s U, C , C:, C~, C4 a n d C5 referred to Fig. 2 a n d Fig. 3(a)].

Loading case l--the uniform tension case (U) In the first case (Fig. 2), the uniform tension with intensity p is assumed at the upper and lower edges. The h/b ratio (height width ratio) is changed from 0.5, 1.0, 2.0, 3.0 and the ratio a/b is c h a n g e d from 0.1, 0.2 ..... 0.9. T h e calculated results are expressed by

K,, = E,,(h/b,a/b)pw/-~.

(12)

The calculated E~A values are listed in Table 1.

T a b l e 3. The calculated

G~^(h/b, a/b) values [see Fig. 3(b) and eq. (15), TC, TC:, TC~, TC4, TCs being the l o a d i n g c o n d i t i o n in Fig. 3(b)].

a/b

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.069 0.500 - 0.068 -0.637 - 1.205

1.862 1.318 0.775 0.231 - 0.313

2.888 2.481 2.075 1.668 1.261

h/b = 0.5 TCI TC2 TC~ TC~ TCs

- 0.518 - 0.952 - 1.386 - 1.820 - 2.254

-

0.509 0.954 1.399 1.844 2.289

-

0.474 0.946 1.417 1.889 2.361

-

0.383 0.886 1.389 1.892 2.395

-

0.208 0.737 1.265 i.793 2.321

-

0.072 0.473 1.018 1.563 2.109

-

0.484 0.075 0.634 1.193 1.752

h/b = 1.0 TCI TC~ TC3 TC, TC~

0.187 - 0.047 - 0.281 - 0.515 - 0.750

0.207 - 0.025 - 0.258 - 0.490 0.722

0.268 0.042 - 0.185 - 0.412 - 0.639

0.371 0.154 - 0.063 - 0.280 - 0.497

0.517 0.315 0.113 - 0.088 - 0.290

0.708 0.527 0.346 0.165 - 0.016

0.946 0.792 0.638 0.484 0.330

1.245 1.123 1.001 0.879 0.757

i.651 1.564 1.478 1.391 !.304

2.427 2.374 2.321 2.268 2.215

0.750 0.674 0.599 0.523 0.448

0.784 0.713 0.641 0.569 0.498

0.843 0.777 0.711 0.646 0.580

0.926 0.869 0.811 0.753 0.696

1.039 0.990 0.942 0.894 0.846

I. 188 1.150 !. 113 1.075 1.037

1.395 1.368 1.341 1.313 1.286

1.723 1.705 1.688 1.670 1.652

2.472 2.461 2.450 2.439 2.428

0.954 0.938 0.921 0.904 0.888

0.977 0.961 0.945 0.930 0.914

1.015 1.001 0.987 0.974 0.960

1.073 1.061 1.049 1.038 1.026

1.155 1.146 1.136 I. 127 I.I18

1.272 1.265 1.258 1.252 1.245

1.448 1.444 !.440 1.435 1.431

1.757 1.755 1.752 1.750 1.748

2.513 2.513 2.513 2.513 2.513

1.008 1.007 1.005 1.004 1.003

1.025 1.024 1.023 1.021 1.020

1.056 1.055 1.054 1.052 1.051

1.104 1.102 1.101 1.100 i.099

1.175 1.174 1.172 1.171 1.170

1.282 1.281 1.280 1.279 1.279

1.453 1.453 1.453 1.453 1.454

1.769 1.771 1.773 1.775 1.776

2.545 2.551 2.556 2.562 2.567

h/b = 1.5 TCI TC2 TC3 TC, TCs

0.738 0.661 0.585 0.508 0.431

h/b = 2.0 TC~ TC2 TC3 TC, TCs

0.947 0.930 0.913 0.896 0.879

h/b = 3.0 TG TC2 TC3 TC, TC5

1.002 1.001 1.000 0.998 0.997

360

Y. Z. CHEN and N. HASEBE GIA3 2.5

2.5 Gl^ 2

h/b=1.0

1.5

j

15

1 0.5 (3 -0.5 -1

h/b=l 5

I 0.5 0.2

0.4

0.6

0.8 0.9 a/b Fig. 8. The calculated GLA(h/b,a/b) values in the case ofh/b = 1.0 [see eq. (15), the loading conditions U, TC, TC2, TC~, TC, and TC~ referred to Fig. 2 and Fig. 3(b)].

0

0.2

0.4

0.6

0.8 0.9 afo Fig. 9. The calculated G~^(h/b, a/b) values in the case ofh/b = 1.5 [see eq. (15), the loading conditions U, TC, TC,, TC3, TC, and TC~ referred to Fig. 2 and Fig. 3(b)].

Loading case 2--the concentrated force case (C~ C2 C3 C4 C5)

In the second case, a pair of concentrated forces with intensity bp is applied at the upper edge and the lower edge of the cracked plate [Fig. 3(a)]. The five particular loading cases are denoted by C,, C2, C~, C4 and C5, respectively [Fig. 3(a)]. Similarly, the calculated results can be expressed by (13)

K,A = F . ^ ( h / b , a / b ) p x / ~ .

The calculated F~A(h/b, a/b) values are listed in Table 2. Meantime, the normalized stress intensity factors are also plotted in Figs 4 - 7, for h/b = 1.0, 1.5, 2.0 and 3.0, respectively. From Table 2 we see that the maximum deviation of the stress intensity factor from the uniform tension case is 5.8% in the case of h/b = 3.0. However, it is significant in the case of h/b <_ 1.5. Loading case 3--the tension-compression loading case (TC~ TC2 TC~ TC, TCs)

In the third case, tension-compression loading is applied at the upper edge and lower edge of the cracked plate [Fig. 3(b)]. A compression loading with intensity 2c~p is applied at the center part of the edges and a tension loading with intensity 2(I + ~)p is applied at the side part of the edges [Fig. 3(b)]. The resultant force will be R = 2(1 + a)bp - 2otbp = 2bp,

(14)

which is equal to the total force in the uniform tension case (Fig. 2). If choosing • = 0.2, 0.4, 0.6, 0.8 and 1.0, the corresponding loading case is denoted by T C , TC2, TC3, TC, and TCs, respectively. Similarly, the calculated results can be expressed by (15)

K,^ = G,^(h/b,a/b)pq/'~ . 2.6 Gt^. IVb-2.0

2.6

2.2:

Gl^ 2.2

h/b=3u'

1. 4 FC~T 1.81 0

0

T

C

~

1.8 C

~

1.4

1 0.2

0.4

016

0.8 0.9 a/b Fig. 10. The calculated GtA(h/b, a/b) values in the case of h/b = 2.0 [see eq. 05), Ihe loading conditions U, TC, TC2, TC;, TC, and TC~ referred to Fig. 2 and Fig. 3(b)].

0

012

0~4

0~6

018 0

thecase

Fig. I I. The calculated G~^(h/b, a/b) values in of h/b = 3.0 [see eq. (15), the loading conditions U, TC. TC. TC, TC, and TCs referred to Fig. 2 and Fig. 3(b)].

Dependence of stress intensity factor on loading condition

361

The calculated G~^(h/b, a/b) values are listed in Table 3. Meantime, the normalized stress intensity factors are also plotted in Figs 8 - 11, for h/b = 1.0, 1.5, 2.0 and 3.0, respectively. As before, from the figures we see that the dependence of the stress intensity factor on the loading condition is negligible in the case of h/b = 3.0. However, it is significant in the case of h/b < 1.5. 4. C O N C L U S I O N S From the above mentioned analysis and numerical examples, the following conclusions can be reached: In the case of h/b > 3.0 (h/b being the height width ratio of the plate), the dependence of the stress intensity factor on the loading condition is rather small, even though the loading distribution is seriously deviated from the uniform tension. In the concentrated loading case [Fig. 3(a)], the maximum deviation of the stress intensity factor from the.uniform tension case is 5.8% (see h/b = 3.0 case in Table 2). Meanwhile, in the tension-compression loading case [Fig. 3(b)], the relevant deviation is negligible (see Fig. 11). However, the relevant dependence is significant in the case of h/b < 1.5. In addition, the presented numerical examples show how the Saint-Venant principle plays a role in the concrete crack problem. REFERENCES [I] Sokolnikoff, I. S., Mathematical Theory of Elasticity. McGraw-Hill, New York, 1956. [2] lsida, M., Arbitrary symmetric loading problems of centrally cracked rectangular plates. Trans. Jpn Soc. Mech. Engrs 1976, 42, 3019-3030. [3] Chen, Y. Z., A mixed boundary problem of a finite cracked plate. Engng Fracture Mech. 1981, 14, 741-751. [4] Chen, Y. Z. and H~sebe, N., Novel weight function approach for plane elasticity crack problem. Int. J Engng Science, 1996, to appear. [5] Muskhelishvili, N. I., Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen, 1953. [6] Hildebrand, F. B., Introduction to Numerical Analysis. McGraw-Hill, New York, 1974. [7] Stress Intensity Factors Handbook, ed. Y. Murakami. Pergamon Press, New York, 1989. (Recei~'ed 31 March 1995)