On the calculation of crack opening displacement from the stress intensity factor

Engineering Printed

Fracare

in Great

Mechanics

Vol. 27, No. 6, pp. 697-715,

1987

0013-7944/87 @ 1987 Pergamon

Britain.

93.00+ Journals

.oo Ltd.

ON THE CALCULATION OF CRACK OPENING DISPLACEMENT FROM THE STRESS INTENSITY FACTOR Kernforschungszentrum,

Universitlt

T. FETT and C. MATTHECK Karlsruhe West Germany, Institut fur Festkiirperschung IV, F.R.G.

und Materialfor-

D. MUNZ Karlsruhe, Institut fiir Zuverlissigkeit und Schadenskunde im Maschinenbau, F.R.G.

Abstract-For the application of the weight function method the crack opening displacements for a reference case have to be known. An approximate method to derive the crack opening field from the stress intensity factor was proposed by Petroski and Achenbach [Engng Fracture Me&. 10,257 (1978)J. The limited accuracy of their method becomes evident in cases where the stresses differ strongly from the homogeneous loading case (D = const.). By expanding the crack opening displacement field in a power series it is demonstrated here how the approximative solutions can be improved by simple additional conditions.

1. INTRODUCTION THE WEIGHT function

method permits the determination of stress intensity factors, for any chosen stress distribution (T(X) in the untracked component, if the crack opening displacement 4(x, a) and the related stress intensity factor K,(a) of a reference load case, with the stress distribution crJx), are known for the geometry of the component. Then

a(x)

2

(1)

dx

where for plane stress

E H=

(2)

E/j 1- p*‘, for plane strain

(3) is the weight function. The origin crack. Furthermore the displacement

of the x-axis is shown in Fig. 1 for an external field u(x, a), related

to K can be calculated

and an internal from

U, using,

where a1 depends on the load case[l]. Whilst the stress intensity factors are known for many crack geometries and load cases, the displacements are only available for a few special cases. A method published by Petroski and Achenbach[2], which has proved to be very reliable in numerous, also three-dimensional, generalizations[ 1,3,4], enables approximate values for the displacement fields to be calculated from the stress intensity factors. The limited accuracy of this procedure was shown very clearly in a case of loading which differs strongly from c = constant[l]. It has to be noted, however, that in one example (Figs 4 and 5) of ref.[l] the stress intensity factor was calculated incorrectly. This point will be clarified in Section 5.3. Nevertheless the displacement approximation given in[2] leads to false results for inhomogeneous reference stress distribution. It is the intention of this analysis to improve the accuracy of that procedure. 697 EPM 27:6-G

698

T. FETT er al.

2w

Fig. 1. Geometric crack quantities.

2. THE CRACK OPENING DISPLACEMENT

FIELD

Let UNbe the near tip field of the crack opening displacement.

(5) then the crack opening displacement

can be expressed by

(6)

u/UN=f(l-p)

with p = x/a. The function f is one at the crack tip (p = 1) and is developed in a Taylor series with respect to (1- P) U/t.& = 1 +

with Co= 1. The other coefficients

c

"=l

c,(l - p)” =

c

c,(l - p)”

(7)

v=o

C, depend on the relative crack length (Y= a/W:

If the stress intensity factor is written in the form

where a0 is a characteristic is

stress of the untracked

U =

4%z

aF(a)

component,

the crack opening displacement

C C,(l - p)u+l12,

v=o

Crack opening displacement

The first condition in order to determine from eq. (1)

699

from the stress intensity factor

C, is the rule of self-consistency.

For m = a, it follows

(10) Due to

(11) with ujrca=O

and [=x/W

eq. (10) can be rewritten as

(12) Integrating

over a! will show that

(13) Inserting eq. (9) in eq. (13) and using abbreviation

dp)(l - P)‘+~‘~ dp

(14)

leads to

(15) For the special case of constant tension stress (a = uo) 2 2v+3’

(16)

I” = ~

Equation (15) enables only the determination of one of the coefficients, namely C1, upon which the method mentioned in [2] is based. However, further coefficients for crack configurations frequently encountered can be found from simple additional conditions,

3. THE WEIGHT By differentiation

FUNCTION

of eq. (9) it follows for the reference

case of a, = constant

+$$(l-p)‘i”’

(17)

700

T. FSTT et al.

and in a second approximation

the weight function can be determined

4. THE SY~T~CALLY

LOADED

GRIFFITH

as

CRACK

4.1 Additional condition In the case of a symmetrically loaded Griffith crack (Fig. 1) all odd diff erential coefficients of u with respect to p at the point x = p = 0 must disappear, whence (19) Thus for n = 0 the following additional condition i (Zv+l)C,=O v=o is found. This condition enables an additional coefficient Combining eqs (15) and (20) leads to

C2 to be determined.

Thus for the second approximation

(224 VW and

(234 GW

4.2 Griffith crack in an infinite plate under constant ioad In the case of constant stress a(p) = o. it follows from eq. (15) and (16) because of F(a) = 1 that

Jlo& = =d-b?. C,

Q(ce)

(24)

Crack opening displacement

from the stress intensity factor

701

--- eq.(25al 10 -----

-

eq (25b) and (261

U [pm1

0

1

05

9=x/a Fig. 2. Crack opening displacements

A first

approximation

for a Griffith crack.

follows

The second approximation

leads to

217r

7

32fi

5

cp’,__-+__.

t25b)

Figure 2 shows the crack opening displacement for a crack size u = 10 mm at u = 100 MPa and E = 200 GPa. The first and second approximations are shown using eq. (25a) and eq. (25b) as well as the exact solution

cw Whereas the first approximation still clearly deviates from the exact solution, the deviation of the second approximation as shown in Fig. 2 can hardly be detected. Therefore the relative variation of approximations and exact values are shown in Fig. 3. Whilst the maximum deviation in the first approximation is still as much as +2.0%, in the second approximation it is only -0.27%. 2 0

2 \ k

--” eq 1250)

‘\

5 ,_ f?j

-

\ \

eq (25 bt

I \ = q- \ \ t \ :: 0)

Fig. 3. Deviations

t

between exact and approximative solutions for the Griffith crack Ioaded with constant stress.

102 4.3

T. FETT

et al.

Griffith crack in a plate of finite width

For a central crack in a plate of finite width the geometry function F up to a value of LY= 0.8 can be expressed as F = dsec(ed2).

Integrating

(27)

eq. (15) leads to Q=-&&os(a~/2)

(Mr/2)2”

E, (n + 1)(2n)! n=O

c

(28)

where En stands for the n-th Euler number. With a relative error of only 0.5% up to a = 0.8, eq. (28) can be simplified to Q=~~cos(a?r/2)

2 5T4 l+sa2+1152a4

>

.

(29)

Figure 4 shows the crack opening displacement in second approximation. It can be seen that despite of growing crack opening with increasing (Y the elliptical geometry is more or less retained. To evaluate the weight function one obtains the quantity a aF (Y aF --=-----tan Faa Facu

77

(30)

4

4.4 Stress intensity factor for two symmetrical pairs of forces Considering a Griffith crack under load by two symmetrical pair of forces P according to Fig. 5 the stress distribution is given by u = P[S(x - b) + 6(-x The corresponding

+ b)].

stress intensity factor can then be calculated

Fig. 4. Crack

opening

displacements

for a crack

using eq. (1) as

in a plate with finite width.

Crack opening displacement

from the stress intensity factor

703

b Fig. 5. Stress intensity factor for a Griffith crack loaded with two pairs of forces.

where A = b/a. The function KP = f(b/a) is displayed in Fig. 5 for different CC.In the special case of an infinite plate ((r + 0) an exact solution exists [SJ. In the case of a single pair of forces

-- J

a-i-b a-b'

(33)

Thus for the two pairs of forces (34) This solution is sketched also for different values of b in Fig. 5. The relative differences between eq. (32) and eq. (34) are shown in Fig. 6. They are l/3% at the most. Equation (32) however unlike eq. (34) is not limited to cx= 0.

2

‘.

, ‘\ \ 1. oppmximotion

(%I

i 1

\\

‘1’

‘\

Fig. 6. Deviations between approximative

and exact solutions for the crack loaded with two pairs of forces.

T. FETT et al.

704

4.5 The Griffith crack under strip loading in an infinite plate A Griffith crack under the load case shown in Fig. 7 has the geometry function[l] F(a)=l-arccosa.

(35)

a

?r

The crack opening displacement is given in [5]. Here we will use eq. (3.5) to calculate the crack opening displacement and compare the results with the exact solution in[5]. We consider the special case with d = 3 mm, (Y= 10 mm. Then Q is Q(a)

=

I

d;u2

i2

a_d F’a’ da’ = 0.083 I.

(36)

From eq. (14) follows 1, =

&

(d/a)“+“‘2.

(37)

Combining eqs (IS) and (20) Cl” = - 1.3406. as a first approximation,

(38)

and C:” = - 1.48924, (39)

Ct2’ = 0 f 69355 2

for the second approximation. Both approximations and the exact solution from [5] are shown in Fig. 7. Although there is still a clear deviation from the exact solution a marked improvement from the method used in [2] is obvious. In all fairness however it has to be pointed out that the procedure used in [2] has not been intended for nonuniformly loaded Griffith cracks. An alternative way of calculation would be to use the weight function derived from the case of h~~mo~eneous loading in Chapter 4.3. Analogous to eq. (4) the crack opening displacement can be calculated using m(c)h(x,

/

,/;

a’)&,(~,

approximation

a’) da’dc;

al = Max(x, c).

\

Fig. 7. Crack opening displacements fnr a strip loaded Griffith crack, calculated with eqs (381, (39) and the exact solution from[S].

(40)

Crack opening displacement

705

from the stress intensity factor +

3

-.-.

1. opproximotlon

0 2. appmxlmatlon [71

exact

r U

lprnl

,.o-. /’

2-

2 ./ /

ir_,_-__.b_._~ l-

*_-‘-

_@._.“’

.*-.--

A.‘.

/

.A./

,”

‘.

\

‘\,

\

‘\

‘\

‘\

‘\ \ i i

I 0 Fig. 8. Crack opening displacement

for a strip loaded Griffith crack, calculated by use of the weight function.

With eqs (3), (28) and (32) the calculation has been carried out for both approximations. For reasons of comparison the calculation was also carried out with the exact weight function given in Appendix IV. The latter was preferred for better comparison of the exact solution in[S], as the numerical inaccuracies were now similar in all solutions. The result for the three weight functions can be seen in Fig. 8. Whilst the first method would only show agreement close to the crack tip, the computations with the weight functions showed very good agreement over the whole crack region. 5. THE EDGE CRACK

In the case of a surface crack of depth a it can be shown (see Appendix If that the curvature of the crack contour at the surface (p = 0) vanishes,

a2u

I

ap2@=*=

0

(41)

“~0(~+1/2)(y-l/2)c”=o.

(42)

it then follows from eq. (9) that

This condition whence

allows the determination

of the coefficient

Cz. Using eqs (14), (15) and (42)

(43)

706

T. FETT et af. -.-

0

_

exact

[71

-

2 approximation

-___

1 app~oxlmotlon

0.5

1

9 =x/a

Fig. 9. Crack opening displacements

for an edge crack.

5.2 Edge crack under constant load The geometry function for an edge crack under tension can be described by[6] t; = 0.265 (1 - a)4 +

0.857 + 0.265tX (1 _ Ly)3,2 .

(44)

The integrals lo, I,, 1, appearing in eq. (43) are given by eq. (16) in the case of u = constant, i.e. 1, = 2/3, 1, = 2/5, 1, = 2/7. In case of the semi infinite plate (a! = 0) it follows from eq. (15) and eq. (44) that Q=&

F(cu = 0) = 0.623114

and thus Cf”=-0.10888,

Cy’=-0.18258,

C:2’=0.103183.

(45)

The crack opening profiles calculated with the coefficients (45) are shown in Fig. 9 together with the exact solution (see Appendix II) given by Wigglesworth [7]. The agreement between the second approximation and the exact solution is very good. As shown in Fig. 10 the maximum deviation is only 0.5%.

.---_

2 ;; : ;

I -I-

I’

’

1. approxlmatlon

2 approximation

I _* _ 1

l

t

-3

i /

c

Fig. 10. Deviations of approximative solutions from the exact solution giveo by[7].

Crack opening displacement from the stress intensity factor

701

5.3 Edge crack with inhomogeneous stress distribution An even better assessment of the quality of the approximation-as the example of strip stress loading with the Gri~th crack has shown -should be possible with a non-homogeneous distribution. Therefore the load cases already described in [l] will be regarded: Case I: stress decreases linearly from a value cro at the plate surface to a value (+ = 0 at the crack tip CT =

ob(l - p).

(46)

Case 11: stress increases linearly from a value cf = 0 at the surface of the plate to the crack tip (+ =

uop.

(47)

The agreement of the superposing of cases I and II with a separately calculated case III shall be a criterion for the accuracy of the approximation. Computation of the geometry function for all three load cases will be carried out with the weight function by Biickner [8] (Appendix III). Setting a = 10 mm, ff = 0.5. Case I: Inserting the weight function in eq. (1) and combining with eq. (3) leads to the geometry function

(48) eq. (14) produces 1”

=2

(49)

2v+5’

Thus Q = 0.55422,

C’,” = 0.53977,

Ci”’ = 0.57780,

C’,” = -0.04889.

The crack opening profiles are shown in Fig. 11 for both approximations.

+ 1. approximation a 2,approximarion -

weightfunction

--- FE-calculation

I

I81

[ll

Fig. 11. Crack opening displacements for an edge crack: Case I.

(501

For the reason of

T. FETT et al.

708

comparison the results of the FE-calculations [l] are shown also, as we11as the curves resulting directly from the weight function by Biickner using eq. (4). The crack opening profile described is equivalent for all procedures. Case II: The analogous computation

results in

1 v+5/2’

I,=-.-!--_-v-?-3/2

O(O.5) = 0.25434

and the coefficients C:” = 0.09926.

C:” = -0.1430,

C’,” = -0.10786,

(53)

The results are shown in Fig. 12. A distinct deviation of the first approximation from the other curves can be seen close to the plate surface. It has to be noted that in [l] Figs 4 and 5 are not correct. The curves for the approximate method for the two loading cases (here the first approximation) have to be interchanged. Case 111: Here the geometry

function is

F is identical to the geometry function in [6]-see deviation of 1.5%. using

eq. (44)-up

to a value of a = 0.5 within a

2 2v+3

I, = -

(55)

and Q(0.5) = 0.76588 leads to Cl” = 0.24803, The corresponding

C”:’ = 0.2338,

profiles and the superposition

C:” = 0.0199 1.

(56)

of cases I and II are shown in Fig. 13.

d,

1 + 1. a~oximatiofl 70

~

0 2. a~oximation

~

-

weight

----

FE-

?=+a+ --p‘;**

I 0

“t.

%*

function

[81 [ll

calculation

*NW k

I

O5

9

Fig. 12. Crack opening displacements

\ 1

case

for an edge crack: Case Ii.

Ii.

Crack opening displacement

from the stress intensity factor

709

20 ” [pm1

Fig. 13. Crack opening ~splacements

for an edge crack, loaded with e = const.

5.4 The edge-crack under single force loading The weight function for the edge crack can be calculated differentiation of eq. (9). In the case of (Y= 0 this leads to

by use of eq. (3), found by

(57) A pair of forces f P in distance b from the plate surface having the following stress distribution CT= PS(x - 6)

(58)

will cause the stress intensity factor

K,=P

(59)

Figure 14 shows in suitable way this function for the exact solution by Wigglesworth[7] as well as for the first and second approximations with coefficients C, according to eq. (45). Additionally the stress intensity factors found by using the weight function method by Biickner[g] were included. The second approximation showed the best agreement with the exact solution, The weight function by Biickner displayed somewhat larger deviations. Clear differences however can be seen for the first approximation, especially for singular forces close to the ptate surface (b = 0). For the special case of singular forces which can be evaluated quite easily analytically the crack opening displacement was calculated to be h(x, ~‘)&,(a’)

da’

(60)

(61) Integration

leads to C, (l-p)“+“2+In

(62)

T. FETT et al.

S-J a

P

I “3

L&

f ‘-“A*

--I

-.-

b-I

exact

I71

*

1. a~~~irnat~~n

0

2. apprcximatlan

b la1

b/a

Fig. 14. Stress intensity factor for an edge crack loaded by a single farce pair. Comparison different solutions.

between

12 HlT

2p” 10 -.-- exact [71 + 1. opproximotion

8

0 2. ~~ra~irnatian b

[81

6

L

2

Fig. 15. Crack opening displacement

for an edge crack loaded by a force pair at the surface.

Crack opening displacement

711

from the stress intensity factor

This relationship is plotted in Fig. 15 for the four weight functions shown in Fig. 14. Again the first approximation shows a clear deviation from the exact solution whilst the second approximation is almost identical.

5.5

The edge crack in the semi-infinite plate under strip-loading

Let us consider a crack under load 0 o.

w=

forx
(63)

The crack opening displacement observed shall be computed using the weight function as well as by direct evaluation of eqs (14), (15) and (42). Furthermore FEM-calculations are carried out to verify the quality of the approximations numerically. (a) Direct evaluation of eq. (15). By inserting the weight function eq. (57) into eq. (1) the stress intensity factor for the load case in eq. (63) K,,, is found and thus the corresponding geometry function F “eW=$zo

Cv[(d/a)“1’2-~(d/a)v13’2]

(64)

If one uses the weight function calculated from the exact solution, then one must set Cv=A,(v=O ,..., 12). In the case d/a = I-i.e. homogeneous loading of the crack-eq. (64) will reproduce the value of F = 1.122, introduced by eq. (44) for (Y= 0, as F,,,,(d/a

= 1) = 1.1218

+

1. approxlmatlon

0

2

.- -

U

opproximahon

FE - calculation

[pm1

2

4/O -a+

_----

---_------

0

0

+

0 o

+

+‘,

a,

0

“t

+

\

\ \

I

0

0

.\ 0

:

o + 0

,,*-, +

9

\i

1

+

0 +

0

-2

-0

+

0

!

+ +

-4

Fig. 16. Crack opening displacements

for a strip loaded edge crack; direct evaluation.

T. FETT et al.

712

-.--exact

)

[71

+ 1

approximation

0

opproxlmation

2

_

\

0

I

/

05

1

g=x/a

Fig. 17. Crack opening displacements for a strip loaded edge crack using the weight function method.

exact in[7].

to within

0.02%.

This agreement

From eq. (15) it follows

supports

the high accuracy

of the power

series given

for d/a = 0.3 0 = 0.07683

and CI”=-1.6591,

Cy’=-1.7483,

C:2’=0.4163.

(65)

The crack opening profiles determined with these data are shown in Fig. (16). Again-as in the case of the first approximation of the strip loaded Griffith crack-negative crack opening displacements are observed close to the plate surface. Here also the second approximation shows this effect, although in a limited way. (b) Determination using the weightfunction eq. (57). Using the weight function eq. (57) with the stress intensity factor K, for the unit force pair loading according to eq. (59) the crack opening displacement can be determined. The results for the first and second approximations with the coefficients eq. (65) are shown in Fig. 17. Additionally-making use of the solution given by[7l_the exact weight functions and the exact crack opening profiles have been determined and included as a reference in Fig. 17. The agreement of all so-calculated solutions is very good.

6. DISCUSSION

OF THE

RESULTS

Because of the fact that especially in the case of strip loading the approximation of Petroski and Achenbach[2] may fail, the methods used here will be discussed in a more detailed way for this special loading case. (i) Direct method: Only non-singular elements in (1 - p) appear in the power series eq. (9). But the leading term of the a-shaped stress distribution (single force) becomes I/K-p. Such singularities as well as unsteady changing stress distributions can generally only be described reasonably accurately by considering a very large number (in theory an infinite number) of terms of the type as in eq. (9). The ‘direct method’ based on eqs (I 5) and (20) respectively (41) will therefore fail here. The

Crack opening displacement

differential originally securing the l/q the derivation of eq. (13).

from the stress intensity factor

713

term in eq. (10) was lost through integration

(ii) Weight function method: By differentiating the power series according to eq. (3) the term l/G weight function. Then a good accuracy will be reached. The general procedure should be:

during

will appear in the

providing the geometry function for the case CT= const. computation of the coefficients Cl”‘, Cy) for CT= const. using eqs (15) and (20) respectively (42). determination of the weight function using eq. (18). computation of the stress intensity factors using eq. (1) computation of the crack opening displacement using eq. (40) if required. As a correction

of some wrong statements

in [l] it has to be concluded

here:

In the case of linear varying load along the complete crack area the errors of the method[2] are negligible for practical use. In the case of only partially loaded crack surfaces and in the case of very steep reference stress profiles the theory[2] may fail as mentioned in[l]. In this sense the limitations of [2] are not so striking as outlined in [l]. The difficulty is to quantify the range of validity of [2]. Therefore the best way is to use a constant reference load which is also the first considered loading case for the main practical crack geometries.

REFERENCES

[2] [3] [4] [5] [6] [7] [8]

F. Gijrner, C. Mattheck, P. Morawietz and D. Munz, Limitations of the Petroski-Achenbach crack opening displacement approximation for the calculation of weight function. Engng Fracture Mech. 22, 269-277 (1985). H. J. Petroski and J. D. Achenbach, Computation of the weight function from a stress intensity factor. Engng Fracture Me& 10,257-266 (1978). C. Mattheck, P. Morawietz and D. Munz, Stress intensity factor at the surface and at the deepest point of a semi~lliptic~ surface crack in plates under stress gradients. fnt. J. Fracture 23, 201-212 (t983). C. Mattheck, P. Morawietz and D. Munz, Calculation of the stress intensity factor of a circumferential crack in a tube originating from a hole under axial tensile and bending loads. Engng Fracture Mech. 22, 645-6.50 (1985). H. G. Hahn, Bruchmechanik. Teubner, Stuttgart (1976). H. Tada, The Stress Analysis of Cracks Handbook. Del. Research Corporation (1973). L. A. Wigglesworth, Stress distribution in a notched plate. Mathematika 4, 76-96 (1957). H. F. Bueckner, Field singularities and related integral representations, in Mechanics of Fracture I-Methods of Analysis and Solution of Crack Problems (Edited by G. C. Sih). Noordhoff, Leyden (1973).

APPENDIX

I

Let us consider displacements u and u in the vicinity of the point (0,O). Because the x-axis is an axis of symmetry no shear stresses will be acting there 7(x, 0) = 0.

(Al)

40, y) = UX(O,y) = 0.

(A21

In the free surface of the plate it holds

For stresses a, and 7 developabie

by power series as (‘43)

it follows due to conditions (Al) and (A2) B,, = I$,, = AOY= 0.

(A41

714

T. FETT et al.

The relationship between deflections u, u and shear distortion y is

au au -+-=y. ax ay Therefore it follows, using the strain component

645)

cX= au/ax

a2u ay a20 ay k _=---=_-_ ax2 ax axay ax ay.

646)

Considering Hooke’s law

m=

-

l/E (l-

“Z),Ei

n=

-

(A7)

VIE v(l

+

for plane stress for plane strain

v)/E

and the equilibrium condition au Y+!T,o ay

(A81

ax

it follows from (A6)

a2u

s=

1

( 1!!_msay E+n

(A9)

and at point (0,O)

NO) i.e. directly at the surface of the plate the curvature of the crack contour disappears.

APPENDIX

II

Wigglesworth[7] determined the crack opening displacement solutions are given as a power series

of the surface crack for tx = 0 analytically. The

m

u(p) = c A,(1 -p)“+r’=. “=”

(All)

The coefficients up to the 12th order are A0 = 1.OOOOO

A, = 0.006254

A, = -0.143719

A, = 0.002993

A; = 0.019965

A; = 0.001256

A, = 0.019665

A, = 0.000390

A; = 0.011856

A; = -0.00001

APPENDIX

AI, = -0.000172 A,, = -0.000213 A;; = -0.000212

III

The weight function according to Biickner is

6412) with m, = 0.6147 + 17.1844~~~+ 8.7822a6 m, = 0.2502 + 3.2889~~~+ 70.0444~1~

(A13)

Crack opening displacement

from the stress intensity factor

APPENDIX The weight function for the G~th

715

IV

crack in an i~fiRite~y wide plate is given by 1 hix,n)=G ~~

(

a+x a_x

“2

for x>O

> for x
(Receipt

29 October 1986)

fAl4)