Initiation and growth characterization of corner cracks near circular hole

Theoretical and Applied Fracture Mechanics 13 (1990) 69-80 Elsevier

69

I N I T I A T I O N AND G R O W T H C H A R A C T E R I Z A T I O N OF C O R N E R CRACKS NEAR CIRCULAR H O L E G.C. SIH and C.T. LI Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania 18015, USA

The finite element analysis of crack problems often incorporates the asymptotic character of the local solution into the formulation. Embedment of stress or strain singularities can impose serious restrictions on the outcome and inconsistencies in predicting crack and/or growth. These restrictions are discussed in connection with the problem of two diametrically opposite comer cracks near a circular hole subjected to remote uniform tension. Enforced in the numerical treatment is the 1/r character of the strain energy density function local to the comer crack border where r is the radial distance measured from the crack front. The tendency for the comer crack to become a through crack is predicted by assuming that each point of the crack border extends by an amount proportional to the strain energy density factor. The path would correspond to the loci of minimum strain energy density function. Numerical results are displayed graphicallyand discussed in connection with crack initiation and non-self-similar crack growth.

1. Introduction The infinite subdivision of a continuum is conceptually feasible for simplifying the development of classical continuum mechanics theories. It presumes that all elements in the limit shrink to zero in a uniform manner. N o one element is permitted to vanish in size before the others. A certain smoothness condition is thus invoked on the ways with which the rate change of volume with surface, A V / A A , tends to zero for each element. Irregular and finite subdivision can cause fluctuation in A V / A A beyond the limit assumed in continuum mechanics. Numerical modelling of geometric a n d / o r 10ad discontinuities are known to be problematic because the process of discretization tends to smear the singular behavior that is defined only for systems with infinite degrees of freedom. Three-dimensional crack configurations, when modelled as planes of discontinuities, give rise to orders of stress singularities that depend on whether the crack border is embedded in the solid or intersects with an external boundary. It is necessary to have a prior knowledge of these singularities to guide the numerical analysis. The triaxial crack border stress and energy states were discussed in a recent review [1]. Particular emphases were placed on the three-dimensional elasticity solution near the corner where the 0167-8442/90/$3.50 © 1990, Elsevier Science Publishers B.V.

border of a through crack terminated on a free surface. For certain range of the crack length to plate thickness ratio, the finite element solution attains a certain peculiarity in that the crack border stress intensity factor peaked near the free surface instead at the mid-plane of the plate where the dilatational effect dominates. This unfortunate condition is not unfamiliar to those who have worked on finite elements in fracture mechanics; it should not be disposed simply as the result of numerical inaccuracies. As explained in ref. [1], continuum mechanics theories such as elasticity, plasticity, etc., are not adequate for analyzing situations where the surface and volume interaction effect is not negligible. This depends on the local mechanical constraint and hence the relative dimensions of the crack a n d structural component. Corrective measures can be taken by accounting for the change in the constitutive behavior in the thickness direction. The bulk effects tend to diminish as the surface is approached. Refer to the example of a three-dimensional through crack solved in ref. [2]. Finite element formulation often takes advantage of an a priori knowledge of the local singular behavior. The use of enriched elements [3-5] is a case in point. Embedment of the asymptotic stress or strain field into the finite element procedure, however, makes the treatment case-specific, because the singular character of the

70

G. C. Sih, C. T. Li / Initiation and growth characterization of corner cracks

stresses and strains changes with the angle at which the crack border intersects with the free surface and nonlinear nature of the constitutive relation. Without loss in generality, the finite element formulation used in this work will preserve the 1/r character of the strain energy density function dW/dV, near the crack border with r being the local distance. This can be accomplished by an appropriate shift of the nodal points for all elements adjacent to the crack. The result applies to all crack configurations and constitutive relations as the 1/r behavior of dW/dV remains invariant. Presented are solutions to the three-dimensional problem of two diametrically opposite comer cracks extending from a circular hole in a thick plate subjected to tension. While the findings agreed qualitatively with those in refs. [3,6], they differ from the works reported in refs. [7-11], particularly near the comers where the crack intersects the free surface.

2. Comer cracks The schematic of two diametrically opposite comer cracks near a circular hole of radius b in a finite thickness plate is shown in Fig. 1. The comer crack is in the shape of a quarter of circle with radius a while a uniform stress of magnitude o0 = 1 Pa is applied at y = + h such that the stresses and strains are symmetric about the xzplane. Note that the xy-plane coincides with the backside of the plate. Referring to Fig. 1, the dimensions of the geometric parameters are a = b ---3 mm, h = 1 8 mm, t = l l . 2 mm and w = 7 2

Pl2024- T4

TI

',

I///. P"

AL

re--VERTICAL Y l ] ---J~l PLANE

/ HOLE r"

z

~

/"

.~

,/ i

/ ~ o J

~,

RAOIAC DIRECTION

~"HLOAR~ZgNTAL

Fig. 2. Local coordinatesin the vertical plane.

mm. The plate is made of 2024-T4 aluminum alloy whose elastic properties are given by E = 7 . 2 4 X 1 0 a°Pa,

/ ~ = 2 . 8 X 1 0 ]° Pa,

u = 0.33,

(1)

where E, /~ and ~, are, respectively, the Young's modulus, shear modulus and Poisson's ratio. The surfaces of the crack and circular hole are stress free.

2.1. Near fieM behaoior The three-dimensional stresses and strains near the crack border have been studied extensively in refs. [12,13]. For the comer crack configuration in Fig. 2, the y-component of the displacement can be expressed as a function of the local polar coordinates (r, 0) in a vertical plane as

Uy(r, O) = 7rrA(O) + ~ r("+l)/2Bi(O).

(2)

i=1

The function the form

A(O) comes into play and it takes

A(O) = ~

[(7 - 8~,) s i n ( 0 / 2 ) - sin(30/2)],

(3)

q

which corresponds to plane strain, a condition that prevails locally around the crack border from A to B in Fig. 2. The point A is on the plate surface at x = a + b ; z = t and B is at x---b; z = t - a. It follows from eq. (3) that the stress intensity k 1 can be evaluated from

/ . . . . . . . . . . . . . . . . -I

Fig. 1. Schematic of a thick plate with comer crack around a circular hole.

8 A(0) k, = 7~-[(7 - 8u) s i n ( 0 / 2 ) - sin(30/2)] "

(4)

G.C. Sih, C.T. Li / Initiation and growth characterization of corner cracks

71

Only a few terms are needed in the series of eq. (2) for evaluating A (O) and B i(0). Sufficient accuracy can be achieved by taking n equal to 2 or 3. Strictly speaking, the expression of k 1 in eqs. (3) or (4) is not valid at points A and B because Uy is no longer given by eq. (2). It becomes

values of k: or k 3 in addition to that of k 1 for the same material as fracture toughness is a materialspecific quantity and should not be assigned to change with the mode of crack extension.

Uy(r, 8 ) = r~ba* (0) -~ . . . .

An important feature of the finite element calculation for crack problems is to preserve the 1 / r decay in d W / d V as shown by eq. (6). Using the 32 nodes three-dimensional isoparametric elements, an appropriate value of m for the position of the side nodes located at distances ( 1 / 3 ) " and ( 2 / 3 ) " adjacent to the crack border can be chosen. Since details of the finite element formulation can be found elsewhere [17-20], only a brief account of the method will be given for the sake of reference. The use of isoparametric elements allows irregular subdivision of the region in Fig. 1. Physical elements of varying shapes can be mapped to unit cubes with dimensions ~--- + 1, ~ = + 1 and ~ = + 1 referred to a local coordinate system (4, 77, ~). All the nodes are evenly distributed on the edges of the cube except for those near the crack border where the ( 1 / 3 ) " and ( 2 / 3 ) " shift will be applied. An appropriate value of m is 2 for enforcing the character of 1 / r in d W / d V as mentioned earlier. The minimum potential energy principle is commonly used to formulate the problem in terms of the nodal point displacements:

(5)

where ~k= 0.4523 for v = 0.3 [14]. The coefficient associated with A * (8) has no physical meaning in fracture mechanics. For a sufficiently small value of r, the strain energy density function can be written as

dW/dV = S/r,

(6)

where S is known as the strain energy density factor [15,16]. For linear elasticity and r---, 0, S can be expressed in terms of the three stress intensity factors kj ( j = 1, 2, 3) as follows *:

S = al,k 2 + 2a12ktk: + aE2k 2 + a33k2.

(7)

The coefficients aij (i, j = 1, 2, 3) for the isotropic homogeneous elastic solid are given by

1

a u = ~[(3-

1

a12 = ~

az2 = ~

4 v - cos 0 ) ( 1 + cos 0 ) ] ,

sin 0[cos 0 - ( 1 - 2v)], [4(1 - v)(1 - cos 8)

(8)

2.2. Finite element procedure

+ (1 + cos 8)(3 cos 0 - 1)], 1 a33 = 4---~"

u, = E Ni(~, ~1, ~) ue, j=l

The form of eq. (7) would not be used at points such as A and B in Fig. 2 for S can be determined from eq. (6) once d W / d V is known. The critical value Sc can be related to k~c with 0 = 0 ° in eq. (8) for Mode I crack extension:

where N~ (i = 1, 2 . . . . . 32) are the interpolating functions associated with the ith node. The 32 expressions of N~ for the 32 node isoparametric elements can be found in ref. [19]. In matrix form, eq. (10) is given by

(1 + ~)(1 - 2u)k2c Sc = 2E

32

(u)

(9)

The ASTM valid fracture toughness Klc is related to v~'kx~. It would be superfluous to entail critical

-----[ N ] ( u e ) .

(11)

The superscript e makes reference to quantities at the nodal points. The strain matrix (c} can be related to the displacement matrix ( u } by application of the strain displacement relations ({)=[D](u)

* The specific form of S depends on the theory used in the stress analysis. For a nonlinear material, eq. (6) can still be applied but S is no longer expressible in terms of stress intensity factors.

(10)

=[D][N](ue}.

(12)

If [C] stands for the elastic modulus matrix, then the stress and strain quantities are connected as (o)=[C](,

}.

(13)

72

G.C. Sih, C.T. Li / Initiation and growth characterization of corner cracks

Table 1 Elements in sections with reference to position in a thick plate with a hole and corner cracks Section No.

Element Number A

B

C

1

1

5

9

2 3 4 5 6

2 3 4

6 7 8

10 11 12

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

13 14 15 16

17 18 19 20

21 22 23 24

25 26 27 28 29

30 31 32 33

34 35 36

37 38 39

40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

55 56 57 58 59 60

61 62 63 64 65 66

67 68 69

70 71 72

A p p l i c a t i o n of the v a r i a t i o n a l t h e o r e m leads to a governing e q u a t i o n of the form [K](u}

= {R},

(14)

in which [ K 1 is the stiffness m a t r i x a n d ( R } specifies the l o a d i n g condition. E q u a t i o n (14) gives the d i s p l a c e m e n t s ( u } everywhere in the elastic system from which the stresses, strains, a n d strain energy densities c a n be o b t a i n e d . 2.3. Grid pattern: corner cracks

The system in Fig. 1 is discretized b y a p p l i c a tion of the 32 n o d e i s o p a r a m e t r i c elements. Because of s y m m e t r y in l o a d a n d geometry, o n l y o n e - q u a r t e r of the p r o b l e m needs to b e analyzed. T h e c o r r e s p o n d i n g grid p a t t e r n is shown in Fig. 3 with elements o c c u p y i n g the full thickness t. T h e r e are altogether 72 elements a n d 847 nodes. T h e six sections 1, 2 . . . . . 6 are stacked in layers n e a r the circular hole as shown in Fig. 3. T h e y e x p a n d in

? SECTION

/

/

/'

size a n d r o t a t e in p o s i t i o n at distances a w a y f r o m the hole a n d crack. T h e locations of the 72 elem e n t s are s u m m a r i z e d in T a b l e 1 with reference to the 17 c o l u m n s l a b e l l e d as A, B . . . . . Q. Relative p o s i t i o n s of the elements for sections 1, 2 a n d 3 can be easily i d e n t i f i e d b y the p l a n e view grid p a t t e r n in Fig. 4. Section 4 has n o elements u n d e r c o l u m n s I, J, P a n d Q as the corner crack t e r m i n a t e s with e l e m e n t s 4 a n d 8 facing the surface of the hole. O n l y a relatively few elements are n e e d e d in sections 5 a n d 6 which o c c u p y the r e m a i n i n g region at the b a c k s i d e of the plate. 2.4. Intensification o f stress and energy density

W h i l e eq. (14) is solved b y the c o m p u t e r p r o g r a m in ref. [19] for the d i s p l a c e m e n t s everywhere, o n l y the state of affairs n e a r the c o r n e r crack will be discussed b e c a u s e o f its influence on the condition of fracture initiation. 2.4.1. Local normal stress The stress c o m p o n e n t Oy d i r e c t e d n o r m a l to the p l a n e over which c r a c k extension w o u l d take p l a c e varies a r o u n d the c r a c k b o r d e r . This q u a n t i t y is c a l c u l a t e d as a f u n c t i o n of the angle q~ for small r a n d 0 = 0 ° in the vertical plane, Fig. 2. A c c o r d i n g

6

K

EI

J

H

Q

B J t_~.

Fig. 3. Grid pattern with one-quarter symmetry for corner cracks next to a circular hole.

Fig. 4. Reference letters for locating elements, sections 1, 2 and 3 of the corner crack system.

G.C. Sih, C.T. Li /Initiation and growth characterization of corner cracks

73

Table 2 Angular variations of local normal stress around corner crack Angle (deg)

N o r m a l stress Oy (Pa)

0.0 ° 7.5 o 15.0 o 22.5 o 30.0 ° 37.5 ° 45.0 o 52.5 o 60.0 ° 67.5 o 75.0 ° 82.5 ° 90.0 °

19.902 15.664 14.1342 12.007 11.706 11.227 10.814 10.710 10.413 10.109 10.453 10.548 9.528

E"

x CI o ~J

03x10

10

(m)

CRACK EDGE 8

6

4

2'0

dO

4'0

810

=

(deg) Fig. 6. Variations of crack opening displacement (COD) around the crack border for a distance 0.3 x 10 -3 rn behind the crack edge.

2.4.3. Stress intensity factor to the data in Table 2, Oy is the largest at ~ = 0 ° that corresponds to the point B on the hole. Its value decreases gradually with ~ as indicated in Fig. 5 and becomes nearly constant for ~ >/70 °. It would suggest a strong tendency for crack growth to start at ~, = 0 ° if oy were taken as a criterion for crack initiation.

2.4.2. Crack opening displacement Displayed in Fig. 6 is the crack opening displacement (COD) at a radius distance 0.3 x 10-3 m behind the crack border. It follows the same trend as oy. The C O D obtains a maximum at B or = 0 ° and decreases to a minimum at A or ,/, = 90 o. The corresponding numerical data can be found in Table 3.

The crack border stress intensity factor defined in eq. (4) is also computed for different values of the angle ~. The m a x i m u m value of k 1 occurs near g, = 0 ° but not at ~ = 0 o where the order of the stress singularity differs from 1 / ¢ ~ and hence k I ceases to exist. The same applies to the point at = 90 °. Excluding these corner points, Fig. 7 shows that k 1 decreases as the point P in Fig. 2 travels from B to A. This is expected because of the increase in mechanical constraint as the plate surface is approached. Table 4 gives the numerical values of k 1. A slight increase in k 1 is detected near the c o m e r A before it vanishes at ~ = 90 o. This feature has been discussed in great detail [21] and will not be elaborated any further. Because k 1 cannot be applied consistently to all points of the Table 3 Corner crack opening displacement next to border space

0

2'o

4'o

do

do

'~

(deg) Fig. 5. Variations of normal stress around the crack border with angle ~.

Angle (deg)

Displacement C O D x 10-15 (m)

0.0 o 7.5 o 15.0 o 22.5 o 30.0 ° 37.5 o 45.0 o 52.5 ° 60.0 o 67.5 o 75.0 o 87.5 o 90.0 o

105.800 91.370 82.054 75.402 71.370 68.054 65.356 63.342 62.290 62.252 61.604 63.130 67.638

G.C. Sih, C.T. Li / Initiation and growth characterization of corner cracks

74

10

4t

!8 a.

2 x >

o "

61

2

,i

1

20

40

60

80

o

4) (deg)

crack border, the stress intensity factor criterion would not yield a complete account of crack initiation.

2.4.4. Strain energy density function Once the stresses a n d / o r strains near the crack border are known, the local strain energy density function follows immediately from the expression

dV

t

(15)

= ~°ijEij"

Since the singular character of d W / d V in eq. (6) applies to every point on the comer crack including A and B, it can be used as a failure criterion [15,16] to predict crack growth in three dimensions without any conceptual difficulties. The smooth decay of d W / d V with ~ is shown in Fig. 8 where ( d W / d V ) ~max n occurs at B ( ~ = 0 °) and min (dW/dV)r~n at A (d? = 90 o ). The relative minimum of d W / d V denoted by (dW/dV)min is taken with reference to the angle 0 in Fig. 2 and it corresponds to 0 = 0 °. Their numerical values for from 0 o to 90 o are summarized in Table 5.

Table 4 Variations of stress intensity factor around the comer crack Angle (deg) 22.5 o 45.0 o 67.5 ° 90.0 °

Intensity factor k 1 x 10 -2 (Pa f ~ ) 5.09612 4.41727 4.13853 4.32813

4o

~o

8~ -~'-

¢~ (deg)

Fig. 7. Angular variations of stress intensity factors around the crack border.

dW

2'0

Fig. 8. Angular variations of the strain energy function around the crack border.

2.5. Non -self similar crack growth Crack growth is said to be non-self-similar when the profile of the crack border changes from each segment of growth depending on the variations of the stored energy in elements prior to failure. The shape of the new crack border must be determined; it cannot be assumed arbitrarily. The release of local energy must conform to the nonuniform distribution found from the analysis; the strain energy density criterion [15,16] can be most easily applied to this. Elements local to the crack border are assumed to fail when the volume energy density reaches a critical value (dW/dV)c

Table 5 Relative m i n i m u m of strain energy density functions around the comer crack Angle ep (deg) 0.0 ° 7.5 ° 15.0 ° 22.5 ° 30.0 o 37.5 ° 45.0 ° 52.5 ° 60.0 o 67.5 ° 75.0 o 87.5 ° 90.0 °

Energy density

. ( d W / d V ) m i n × 10 -9 (Pa) 3.3276 2.1284 1.5868 1.1915 1.1024 1.0175 0.96717 0.91863 0.87805 0.86737 0.86310 0.87755 0.83502

G.C. Sih, C.T. Li / Initiation and growth characterization of corner cracks

BY• ~ERTICAL ANE

2¢ o

'o

75

/ - - ~ =0"

15

>

I

~

LOCIOF (dWldV)mi n

10 i , ~22.5"

5

67"5~ o°

PLANE Fig. 9. C o m e r crack growing along the hole wall from B to B'.

0

5

110 15 r x10-'*(m)

20

215

Fig. 10. Decay of the near field strain energy density function with radial distance for different angular positions.

being characteristic of the material that can be obtained from a uniaxial tensile test, i.e., dW (-~--~-)c = f0%o de,

(16)

developing a new profile. The corresponding strain energy density factors are

si(k', s(2k) ..... S) k' ..... S~k). in which o and c are, respectively, the uniaxial stress and strain. In eq. (16), cc stands for the critical strain at fracture. The location of the critical dements are assumed to coincide with the loci of dW/dV, i.e., (dW/dV)mi,. These minima of d W / d V are taken with reference to the angular position of the vector r originating from the point P in Fig. 9. If the load is applied symmetrically across the crack plane, the loci of (dW/dV)min will lie in the horizontal plane and hence r is directed in the plane 0 = 0 °. The new profile along which (dW/dV)mm is constant will be different from the original shape APB in Fig. 9. Each segment of growth will be different. The growth condition for the k th crack profile can be written as

dW ...

r(k)

'

r)k)

Examples on the use of eq. (17) can be found in refs. [22,23] for the cases of a through crack [22,23] and semi-elliptical surface crack [24]. Application of the growth condition in eq. (17) requires a knowledge of d W / d V as a function of r. This has been done for five of the angles ¢ = 0 o, 22.5 o, 45 o, 67.5 o and 90 o where ¢ = 0 o corresponds to B and q 5 = 9 0 ° to A in Fig. 9. The curves in Fig. 10 show how (dW/dV)mi, decreases with r for different ¢. For a constant d W / d V value, more growth would occur at B, say from B to B' because the curve for ¢ = 0 ° gives the largest r for a fixed dW/dV. The amount of growth decreases as the point A is approached. Figure 9 shows the case of no growth at qb= 90 o. This trend can also be seen from the numerical data in Table 6.

"..

(17)

where k = 1, 2 . . . . . m are the number of crack growth steps taken to obtain the final crack configuration just prior to the onset of rapid fracture. The radial distances rl(k), rE(k). . . . , r) k) . . . . . r~(k)

(19)

(18)

correspond to the number of segments used in

Table 6 M i n i m u m strain energy density function versus distance for different q~ around a corner crack Distance

(dW/dV)i,~, × 10 -11 (Pa)

r × 1 0 - 3 (m)

0o

22.5 °

45 °

67.5 °

90 °

3.0333 3.1333 3.3000 4.0000 4.7000 5.4000

194.100 47.969 13.352 10.918 9.555 8.914

66.556 14.572 3.137 1.655 1.152 0.985

56.567 12.423 2.832 1.471 0.994 0.752

49.150 11.001 2.463 1.421 1.144 1.079

40.503 7.693 1.955 1.290 1.079 1.023

G.C. Sih, C. "1~ Li / Initiation and growth characterization of corner cracks

76

f 2024-T4AL

/~

THROUGH Ylj--~/~HOLE

P T

U

/

!

I-

w

Fig. 11. Schematic of the thick plate with through cracks around the circular hole.

3. Through cracks

AJ La Fig. 13. Reference letters for locating elements in all four layers of the through crack system.

to a load of o0 = 1 Pa applied on the plate boundaries at y = + h .

Because of local imperfections and higher stress or energy concentration at the hole boundary, cracks tend to nucleate and initiate at these locations. Such a situation has been assumed in Fig. 1 where two diametrically opposite corner cracks occurred. Further increase in load will cause the corner crack to grow. As the point B in Fig. 9 will be subjected to more extension than that at A, a through thickness crack will eventually be developed as indicated in Fig. 11. The dimensions of the geometric parameters are the same as those in Fig. 1 for the corner crack problem. That is, a=b=3 ram, h = 1 8 mm, t = 11.2 m m and w = 72 ram. Material properties for the 2024-T4 aluminum are given in eq. (1). The same finite element procedure will be used to solve the problem of two through cracks around a hole subjected

3.1. Grid pattern." through cracks A schematic of the grid pattern with one-eighth symmetry is shown in Fig. 12 for the system in Fig. 11. The four layers 1, 2 . . . . . 4 elements occupy only half of the plate thickness, t/2. Note that the x- and y-axis are in the mid-plane of the plate. Each layer contains 21 elements making a total of 84 elements. They are connected by 936 nodes. Figure 13 locates the position of the 21 columns of elements labelled as A, B , . . . , U. Each column has four elements following the numbering system in Table 7. Columns A and B next to the crack border would contain the elements 1 to 4 and 5 to 8, respectively. In the same way, 9 to 12 and 13 to 16 are those in the respective columns C and D.

/

L~YER /

Fig. 12. Grid pattern with one-eighth symmetry for through edge cracks next to the circular hole.

G.C. Sih, C.T. Li / Initiation and growth characterization of corner cracks

77

Table 7 Elements in layers with reference to position in the thick plate with through cracks around the hole Layer

Element number

no.

A

B

C

1

1

5

9

2 3 4

2 3 4

6 7 8

10 11 12

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

13 14 15 16

17 18 19 20

21 22 23 24

25 26 27 28

29 30 31 32

33 34 35 36

37 38 39 40

41 42 43 44

45 46 47 48

49 50 51 52

53 54 55 56

57 58 59 60

61 62 63 64

65 66 67 68

69 70 71 72

73 74 75 76

77 78 79 80

81 82 83 84

Table 8 Variations of normal stress component with thickness coordinate for a through crack

2.0

1.a

~

~ v 1.6

1.4

, _a_ 11.2

,

221.4

331.6

441.8

511.0 " -

z ~ 1 o-' (,.) Fig. 14. Variations of normal stress with thickness coordinate for the through crack.

T h e r e m a i n i n g e l e m e n t s can b e l o c a t e d in the s a m e way.

z × 10 -3 (m)

oy (Pa)

5.6000 5.1333 4.6667 4.2000 3.7333 3.2667 2.8000 2.3333 1.8667 1.4000 0.9333 0.4667 0.0000

14.866 18.582 19.253 19.038 19.005 19.318 19.620 19.577 19.513 19.466 19.490 19.505 19.534

8.5

3.2. Local normal stress T h e local stress c o m p o n e n t ay near the crack edge at x = a + b + r w h e r e r is small is p l o t t e d against the thickness c o o r d i n a t e z as in Fig. 14. It decreases with increasing z w h e r e z = 0 coincides w i t h the m i d - p l a n e of the plate. A sharp d r o p in Oy occurs as z a p p r o a c h e s the plate free surface at z = 5.6 )< 10 -3 m. T h e r e is a slight waviness in the c ur ve f r o m z = 1.8667 to 2.8000 × 10 -3 m w h i c h c a n be m o r e readily seen f r o m the d a t a in T a b l e 8. T h e effect, however, is insignificant.

8.0 o el.

7 o

7.5

7.0

6.5

o

d.2

2~.4

3~.e

4;..8

5~.o"

z x l O - ' (m)

Fig. 15. Variations of stress intensity factor with thickness coordinate for the through crack.

3.3. Stress intensity factor S h o wn in Fig. 15 is a plot of k I versus the z-coordinate. M a x i m u m stress intensity o c c u r r e d near the m i d - p l a n e w h e r e z - 0. T h e decay of kl b e c o m e s m o r e a p p r e c i a b l e as the p l a t e free surface is a p p ro ach ed . This q u a li ta ti v e feature is to be e x p e c t e d as the free surface p r o v i d e s less constraint. N u m e r i c a l values of k 1 for six distinct locations of z are given in T a b l e 9.

Table 9 Variations of stress intensity factor along the crack edge X 10 - 3 ( m )

k1 X 10 -2 (Pa v ~ )

5.6 4.2 2.8 1.4 0.0

6.980002 8.001565 8.200173 8.183315 8.194400

z

78

G.C. Sib, C. T. Li / Initiation and growth characterization

of corner cracks

t

141.

t (3

Fig. 16. Schematic

of medium with edge cracks around in two dimensions.

12L____I-,.. 11.2 0

the hole

22.4

33.6

zx10-'on)

A comparison with the two-dimensional solution in ref. [25] can be made where k, is given in the form

(20) where u is the applied Referring to Fig. 16 and function F( A, c/b) can mm and c = 6 mm. This eq. (20) leads to

stress in the y-direction. Table 10, the normalized be found for X = 0, b = 3 gives F(0, 2) = 0.967 and

k, = 0.0749 Pa 6,

(21)

for u = 1 Pa. This is approximately 8.55% lower than k, = 8.1944 X lo-’ Pa 6 at z = 0 in Table

Table 10 Normalized stress intensity around a circular hole [25] c/b

F(h = -1,

1.01 1.02 1.04 1.06 1.08 1.10 1.15 1.20 1.25 1.30 1.40 1.50 1.60 1.80 2.00 2.20 2.50 3.00 4.00

0.4325 0.5971 0.7981 0.9250 1.0135 1.0775 1.1746 1.2208 1.2405 1.2457 1.2350 1.2134 1.1899 1.1476 1.1149 1.0904 1.0649 1.0395 1.0178

c/b)

factor

F(X =l, 0.3256 0.4514 0.6082 0.7104 0.7843 0.8400 0.9322 0.9851 1.0168 1.0358 1.0536 1.0582 1.0571 1.0495 1.0409 1.0336 1.0252 1.0161 1.0077

solution

c/b)

for

edge

cracks

Fig. 17. Variations thickness

of the strain energy density function coordinate for the through crack.

with

9. Larger differences are expected if the ratio c/b is decreased which corresponds to shorter crack length in comparison with the hole radius. In such a case, the interaction between the crack border and hole would be more pronounced in addition to the three-dimensional thickness effect. 3.4. Strain energy density function Also computed are the strain energy density functions dW/dV along the crack border in the z-direction. The trend is the same as that for uY where dW/dV decreased as z is increased except for the small fluctuation occurring at z = 1.8667 and 2.8000 X 10e3 m. Refer to Fig. 17. This can be seen from the data in Table 11. The strain energy density criterion [15,16] would predict crack

F(A = 0, c/b) 0.2188 0.3058 0.4183 0.4958 0.5551 0.6025 0.6898 0.7494 0.7929 0.8259 0.8723 0.9029 0.9242 0.9513 0.9670 0.9768 0.9855 0.9927 0.9976

Table 11 Distribution of the strain through crack front z X

1O-3 (m)

5.6000 5.1333 4.6667 4.2000 3.7333 3.2667 2.8000 2.3333 1.8667 1.4000 0.9333 0.4666 0.0000

energy

density

dW,‘dV X 1O-9 (Pa) 1.2262 1.7680 1.8835 1.8363 1.8330 1.8919 1.9529 1.9436 1.9304 1.9216 1.9266 1.9293 1.9347

function

along

the

G.C. Sih, C.T. Li / Initmtion and growth characterization of corner cracks

g r o w t h to initiate f r o m the m i d - p l a n e of the crack. A t h u m b - n a i l - s h a p e d c r a c k profile w o u l d b e dev e l o p e d b e c a u s e the energy d e n s i t y level in the m a t e r i a l n e a r the p l a t e surface is lower a n d hence less crack g r o w t h w o u l d take place.

79

t i a t i o n a n d / o r g r o w t h criteria c a n n o t b e chosen a r b i t r a r i l y . T h e y s h o u l d b e free f r o m inconsistencies a n d a p p l y e v e r y w h e r e o n the c r a c k front a n d to each stage of the c r a c k g r o w t h process.

References

4. Concluding

remarks

T h e solutions for two d i a m e t r i c a l l y o p p o s i t e c o m e r cracks i n i t i a t e d f r o m a circular hole are o b t a i n e d b y the t h r e e - d i m e n s i o n a l finite e l e m e n t c o m p u t e r p r o g r a m [19]. A p p r o p r i a t e shifts of the n o d a l p o i n t s a d j a c e n t to the crack b o r d e r are m a d e to enforce the 1 / r a s y m p t o t i c c h a r a c t e r of the strain energy d e n s i t y function such that every p o i n t on the surface c o m e r crack can b e a n a l y z e d b y a single criterion for p r e d i c t i n g fracture init i a t i o n a n d crack growth. P a r a m e t e r s such as the stress intensity factor a n d energy release rate cann o t b e a p p l i e d in a consistent fashion for analyzing the initiation a n d growth of t h r e e - d i m e n s i o n a l surface cracks for at least two f u n d a m e n t a l reasons: - the stress i n t e n s i t y factor a n d energy release rate d o n o t exist at the c o m e r s where the crack b o r d e r intersects with the free surfaces. - N o n - s e l f - s i m i l a r crack g r o w t h c o u l d n o t b e m a d e b y the classical fracture mechanics criteria that lack the predictive c a p a b i l i t y for d e t e r m i n ing the d i r e c t i o n of crack initiation. T h e a d v a n t a g e of using the strain energy density criterion [15,16] is that there is n o longer the n e e d to d e t e r m i n e the i n d i v i d u a l stress i n t e n s i t y factors k 1, k 2 a n d k 3 that m a y all occur simultaneously on the c r a c k b o r d e r . It suffices to c o m p u t e for the p a t h of m i n i m u m strain energy density function which is a s s u m e d to c o i n c i d e with the profile of the new c r a c k b o r d e r . O b t a i n e d in this w o r k is the p a t h of ( d W / d V ) m i n a h e a d of the c o m e r cracks a n d have shown that m o r e g r o w t h w o u l d take place at the interior c o m e r along the hole wall. Less g r o w t h w o u l d occur at the exterior c o m e r that t e r m i n a t e s at the p l a t e surface b e c a u s e the ( d W / d V ) m i n value is m u c h lower. A t h r o u g h c r a c k with straight front w o u l d eventually b e developed. T h e results for two t h r o u g h cracks a r o u n d a circular hole are also o b t a i n e d a n d the n e a r field stress a n d energy states are discussed in c o n n e c tion with p o s s i b l e a d d i t i o n a l c r a c k growth. E m p h a s i z e d t h r o u g h o u t this w o r k is that c r a c k ini-

[1] G.C. Sih and Y.D. Lee, "Review of triaxial crack border stress and energy behavior", J. Theor. Appl. Fract. Mech. 12 (1) (1989) 1-17. [2] G.C. Sih and C. Chen, "Non-self-similar crack growth in elastic-plastic finite thickness plate", J. Theor. Appl. Fract. Mech. 3 (2) (1985) 125-139. [3] P.D. Hilton, B.V. Kiefer and G.C. Sih, "Specialized finite element procedures for three-dimensional crack problems", Numerical Methods in Fracture Mechanics, eds., A.R. Luxmoore and D.R.J. Owens (University of Swansea Publication, 1978) pp. 411-421. [4] P.D. Hilton and D.C. Perice, "The enriched element formulation for 3-D combined mode elastic crack problem", Nonlinear and Dynamic Fracture Mechanics, eds., N. Perrone and S.N. Atluri, Vol. 35 (ASME, AMD, New York, 1979) pp. 65-76. [5] P.D. Hilton and B.V. Kiefer, "The enriched element for finite element analysis of three-dimensional elastic crack problems", Pressure Vessels and Piping Division Conference ASME, San Francisco, June 1979. [6] D.M. Tracey, "Finite elements for three-dimensional elastic crack analysis," Nucl. Engrg. Des. 26 (1973) 282-290. [7] T.A. Cruse, "An improved boundary-integral equation method for three-dimensional elastic stress analysis", J. Comput. Structures 4 (1974) 741-754. [8] T.A. Cruse, "Boundary-integral equation method for three-dimensional elastic fracture mechanics analysis", AFOSR-TR-75-0813, Accession No. ADA011660, May 1975. [9] I.S. Raju and J.C. Newman, Jr, "Improved stress intensity factors for semi-elliptical surface cracks in finite thickness plates", NASA Technical Memorandum 72825, 1977. [10] I.S. Raju and J.C. Newman, Jr, "Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite thickness plates", J. Engrg. Fracture Mech. 11 (1979) 817-829. [11] S.N. Atluri and E. Kathiresan, "3-D analysis of surface flaws in thick walled reactor vessels using displacement hybrid finite element method", NucL Engrg. Des. 51 (1979) 163-176. [12] R.J. Hartranft and G.C. Sih, "The use of eigenfunction expansions in the general solution of three-dimensional crack problems", J. Math. Mech. 19 (1969) 123-138. [13] G.C. Sih, "A review of the three-dimensional stress problems for a cracked plate", lnt. J. Fract. Mech. 7 (1971) 39-61. [14] J.P. Benthem, "State of stress at the vertex of a quarter-infinite crack in a half-space", Int. J. SoL Struct. 13 (1977) 479-492. [15] G.C. Sih, Introductory Chapters of Mechanics of Fracture, Vols. I to VII, ed., G.C. Sih (Martinus Nijhoff Publishers, The Netherlands, 1972-1983).

80

G.C. Sih, C.T. Li /Initiation and growth characterization of corner cracks

[16] G.C. Sih, "Fracture mechanics of engineering structural components", Fracture Mechanics Methodology, Eds., G.C. Sih and L.O. Faria (Martinus Nijhoff Publishers, The Netherlands, 1984) pp. 35-101. [17] O.C. Zienkiewicz, The Finite Element Method (McGrawHill, London, 1979). [18] E. Hinton and J.S. Campbell, "Local and global smoothing of discontinuous finite element functions using a least squares method", Int. J. Numer. Meths. in Engrg. 8 (1974) 461-480. [19] Anisotropic Three-Dimensional Elasticity Analysis (ATDEA) Computer Program, Institute of Fracture and Solid Mechanics, IFSM-88-155, 1988. [20] Stress and Energy Design Analysis (SEDA) Computer Program, Institute of Fracture and Solid Mechanics, Library of Congress No. 88-081940, 1987. [21] R.J. Hartranft and G.C. Sih, "Alternating method applied to edge and surface crack problems", Methods of Analysis and Solutions to Crack Problems, Vol. I, ed., G.C. Sih

[22]

[23]

[24]

[25]

(Noordhoff International Publishing (Now Martinus Nijhoff Publishers), The Netherlands, 1973) pp. 177-238. G.C. Sih, "Mechanics of crack growth: geometrical size effect in fracture", Fracture Mechanics in Engineering Application, eds., G.C. Sih and S.R. Valluri (Sijhoff and Noordhoff International Publishers, The Netherlands, 1979) 3-29. G.C. Sih and B.V. Kiefer, "Nonlinear response of solids due to crack growth and plastic deformation", Nonlinear and Dynamic Fracture Problems, eds., N. Perrone and S. Atluri, Vol. 35 (American Society of Mechanical Engineers, AMD, 1979) pp. 135-156. G.C. Sih and B.V. Kiefer, "Stable growth of surface cracks", Engrg. Mech. Division, ASCE 106 (EM2) (1980) 245-253. O.L. Bowie, "Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular hole", J. Math. Phys. 35 (1956) 60-71.