Mode-I crack projection procedure using the strain energy density criterion

"Z ~

.

theoretical and applied fracture mechanics ELSEVIER

Theoretical and Applied Fracture Mechanics 20 (1994) 67-74

Mode-I crack projection procedure using the strain energy density criterion F.D. Fischer Institute of Mechanics, Universityfor Mining and Metallurgy, FrartzJosef Strasse 18, A-8700 Leoben, Austria

F.G. Rammerstorfer Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology, Gusshausstrasse 27-29, A-1040 Wien,Austria

Abstract

A practice used in linear elastic fracture mechanics is the projection of a crack onto a plane normal to the principal tensile stress axes for computing the stress intensity factor K I. The minimum strain-energy criterion is applied for different crack configurations with the introduction of a safety factor S i which is the ratio of the strain energy density factor of the projected crack and that of the original crack. Numerous crack configurations are investigated to illustrate the degree of conservativeness of the crack projection procedure.

1. Introduction The practical application of linear elastic fracture mechanics has considered projecting a crack onto planes normal to the principal tensile stresses. The calculated K I stress intensity factor is considered to be conservative. Such a procedure has been recommended as guidelines in applying fracture mechanics to welded metallic structures [1-3]. In what follows, frictional sliding of the crack surfaces [4] will not be considered since the effect of Coulomb friction for smooth crack surfaces is not significant. Real cracks would have ridges on their surfaces offering substantial resistance against sliding. Examined are cracks with intensity factors KI, K n and K m such that K I > 0, their configurations include a through thickness crack and a plane elliptical crack with plane

strain conditions along the crack periphery. Real crack shapes are simplified to those recommended in the regulations [1-3], the justification of which by experimental methods can be found in [5].

2. Review of mixed-mode fracture criteria The energy release rate criterion in its original form involves calculating the energy released, say G, by the unit extension of a straight crack that grows in a self-similar manner, i.e., normally referred to as Mode I crack extension. If the crack is projected to the plane normal to the jth principal stress axis, Gj can be calculated and compared with G of the original crack, leading to a safety factor for each of these projections, Sj = ( G J G ) 1/2 for j = 1, 2, 3, see [6]. However, this

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68

F.D. Fischer, F.G. Rammerst orfer / Theoretical and Applied Fracture Mechanics 20 (1994) 67-74

procedure has the shortcoming that self-similarity of crack extension must not generally be assumed. This problem was treated in [7] by analysing the energy release rate G6 of a line crack with a small kink. The angle of the kink, ~b, was varied to find the maximum of G with respect to 4~, i.e. G g ax. With G6~,~x denoting th'~ energy release rate for this crack projected to a plane normal to the jth axis an alternative safety factor S = (, (~~ ., mj a x //l~~ m6a x ) l /,2 was introduced in [8]. The method was also applied for 3D crack configurations with the aid of finite element methods [9]. Explored in [8] is also Sj = tr0~ax/tr0m~ with o"0 being the local maximum tangential stress normal to the prospective path of crack growth, a criterion that was first proposed in [10]. The strain energy density criterion [11] is not restricted to self-similar crack growth and applies readily to mixed mode loading. It assumes that crack growth initiates in a direction corresponding to the maximum of the local minima of strain energy density, the application of which, with a safety factor, is outlined in the Appendix. An attempt was made in [12] to combine the local maximum tangential stress criterion with that of the strain energy density. This, however, is not necessary because the precise location of crack initiation can be determined from a unique maximum among many of the minima of the strain energy density function. Besides, the maximum normal stress criterion could contradict itself where the maximum stress is collinear and not perpendicular to the plane of crack propagation [11]. A further modification of the strain energy density criterion was made [13] which was referred to as the T-criterion. It focuses attention of the maximum of the dilatational portion of the strain energy density function. While the T-criterion and strain energy density criterion led to the same order of accuracy when compared with experimental data [14] for cracks inclined more than 45 ° with the applied load, it was argued in [15] that contradicting predictions prevailed in some examples of [13]. Because the strain energy density criterion does not make any a priori requirement on separating the dilatational and distortional component, it applies equally well to mixed mode crack growth

in elastic-plastic materials. Ad hoc parameters involving the combined action of local shear and tensile stresses have been proposed to study mixed mode stable crack growth in an elastic-plastic material [16]. The strain energy density criterion (S-criterion) is further explored in connection with the safety factor concept because of its general acceptance and verification by sufficient number of experiments. Past criticisms on the S-criterion were mainly due to misunderstanding of the concept; they have since been pointed out and explained in the open literature.

3. Uniaxial loading Let Sj ( j = x , y, z) be defined as a factor of safety .

~

~

1/2

Sj=[So.JS(cb)]

(1)

in which Sp.j is the strain energy density factor computed from the projected crack. The factor 5(4;) is associated with the original crack evaluated at the position ~ for which S is a minimum. A more detailed discussion of Eq. (1) is given in the Appendix. The values of Si will be computed for a number of crack configurations under uniaxial loading.

3.1. Angle crack in infinite body Let an angle crack of length 2a be inclined at angle 0 with the x-axis while a uniform stress tr is applied in the y-direction as shown in Fig. 1. The corresponding stress intensity factors are K I = trvr~a-cos20, K n = trv~-d sin 0 cos 0.

(2)

The crack projected onto the x-axis would experience only Kip,x given by

K,p.x = ~ ¢ ; h - ¢?0s 0 .

(3)

Making use of Eqs. (1)-(3) and (A2), a safety factor S x is obtained: S x = {4(1 - v ) / [ c o s

O(a,lcos20

+a12sin 0 cos 0 +

a22sin20)]} x/z

(4)

69

F.D. Fischer, F.G. Rammerstorfer / Theoretical and Applied Fracture Mechanics 20 (1994) 67-74

llttlltTlo

h

3.0-

AY

I

i

I

,

I

i

I

2.0.

~

w~/w2=l.O _ a/wt=0.8

1.0

x v

4'0

610

810

o [deg] Fig. 3. Variations of safety factor Sr with crack angle 0 in infinite body under uniaxial tension.

Fig. 1. Schematic of angle crack in infinite body. in which an = (3- 4v-cos a12

2 sin ~ ( c o s ~ -- 1 + 2 v ) ,

=

3.2. Finite strip with angle crack

~)(1 +cos ~), (5)

a22 = 4(1 -- v)(1 - - c o s ~ ) + (3 cos ~ -- 1)(1 + cos 4~). The angle q~ is given by Eq. (A3). It can be seen that S x grows monotonically from S x = 1 for 0 = 0 to Sx ~ oo for 0 = 90 °. A numerical study for 0 < v < ~1 shows that S~ is always greater than 1 so that the projected crack solution is always conservative. The same conclusion was reached in

The finite strip of width w contains an offcenter angle crack of length 2 a as shown in Fig. 2. Two dimensionless geometric parameters a / w 1 and W l / W 2 would come into play. Without going into details, a safety factor Sy can be obtained the same way as in the example discussed earlier. The corresponding stress intensity factors for Wl/W2 > 0 and w l / w 2 = 0 are taken respectively from [17] and [18] and those for the projected crack from [19] with w l / w 2 > 0 and [17] with W1//W 2 "~" O.

[8].

Plotted in Fig. 3 are values of Sy against the crack angle 0. A minimum of Sy slightly smaller than one is detected at /9 greater than Og. Refer to Table 1 for the minima of Sy or S~ m and values of 0g. 4

Table 1 Minima of safety factor Sy for an off-center inclined crack in a finite strip a / wa wl / WE Sy in Og 41

(5

G

Fig. 2. Schematic of off-center angle crack in finite strip.

0.8 0.8 0.8 0.5

1.0 1/3 1/9 0

0.931 0.950 0.958 0.948

33° 47° 49° 47°

70

F.D. Fischer, F,G. Rammerstorfer/ Theoreticaland Applied FractureMechanics 20 (1994) 67-74

i~

i

q~

!i

! l .o

b

i

J

o

.

"

Fig. 4. Elliptical crack with major axis tilted.

3.3. Plane elliptical crack in infinite body Consider a flat elliptical crack with semi-major axis a and semi-minor axis b. T h e major axis is tilted at an angle a with the x-axis in Fig. 4. In Fig. 5, the minor axis is tilted by the same angle. Their projections o n t o the xz-plane are shaded. T h e solid is stretched uniaxially in the y-direction. Stress intensity factor solutions for the inclined cracks have b e e n calculated in by application of the quarterpoint concept an extensive finite element study [20,21] and 20 n o d e 3D finite elements. T h e stress intensity factors for the projected elliptical cracks are given by [22]. Variations of Sp, y and S with the angle defined in Fig. 4 are displayed in Fig. 6 for b / a = 0 . 4 and a = 3 0 ° . T h e curve for Sp,y is slightly lower than for S. Those at the polar angles ~o = 0 ° and 180 ° are nearly the same. Figure 7 shows the curves of Sp,y and ~q for the crack in Fig. 5 with b / a = 0.4 and a = 30 °. In this case,

Y

.~o ~---v--'8o

I/O

.

.

.

-~--

.

161)1

1

q~ldeg] Fig. 6. Variations of minimum strain energy density for the actual and projected crack with parametric angle ~ for an elliptical crack in infinite body where b/a = 0.4 and a = 30° (major axis tilted).

Sp,y is always greater than S. T h e projected crack solution is sufficiently accurate for a < 30 °.

3.4. Plane elliptical crack in half-space Suppose that a free surface prevails at a distance h and is parallel to the major axis of the elliptical crack. R e f e r to the schematic in Fig. 8. Use is m a d e o f the projected crack solutions in [23,24]. A comparison of Sp,y with ~q is m a d e in Fig. 8 for values of ~ f r o m 0 ° to 180 ° with b / a = 0.4, h / b = 1.1 and a = 30 °. N o t e that Sp,y

1

~Z

k ~ .

_

I

_.

i

_

I

x

0.5

0

.IlL

I.

121).

16{I,

co Ideg]

Fig. 5. Elliptical crack with minor axis tilted.

Fig. 7. Variations of minimum strain energy density for the actual and projected crack with parametric angle ¢ for an elliptical crack in infinite body where b/a = 0.4 and a = 30° (minor axis tilted).

F.D. Fischer, F.G. Rammerstorfer/ Theoreticaland Applied FractureMechanics 20 (1994)67-74 I

I

4.0-

I b

h

i

I

I

,

71

I

,

x 1.2-

~

2.0

350 o

. . . . 25 o 1.0cp [deg] Fig. 8. V a r i a t i o n s of m i n i m u m s t r a i n e n e r g y density for the a c t u a l a n d p r o j e c t e d c r a c k w i t h p a r a m e t r i c a n g l e ~o for a n elliptical c r a c k in h a l f s p a c e p a r a l l e l to m a j o r axis w h e r e b/a=0.4, h/b=l.1 a n d ~ = 3 0 °.

at ~o = 0 ° is larger than S. If the free surface is parallel to the minor axis of the elliptical crack, Sp,y is always larger than S; Fig. 9 where b / a = 0.4, h / a = 1.1 and a = 30 °.

3.5. Semielliptical cracks Let a semielliptical surface crack be tilted at an angle a with plane normal to the free surface. A uniaxial tensile stress o- is applied to open the crack as shown in Fig. 10. The stress intensity factors for the deepest point at B are given in [25] while for a through crack with b / a -- 0 can ,

i

i

i

L

i

i

i

1.5-

1.0-

0.5-

~0 [deg] Fig. 9. V a r i a t i o n s of m i n i m u m s t r a i n e n e r g y d e n s i t y for the a c t u a l a n d p r o j e c t e d c r a c k w i t h p a r a m e t r i c a n g l e ~0 for an elliptical c r a c k in h a l f s p a c e p a r a l l e l to m i n o r axis w h e r e b/a=0.4, h/a=l.1 a n d a = 3 0 °.

ii

0

•

I 0.4

iJ,

0.8

b/a Fig. 10. Safety factor Sy at B for an i n c l i n e d semi-elliptical surface c r a c k as a f u n c t i o n of b/a for d i f f e r e n t a.

be found in [26]. Values of K I and K n from [25] can be used to obtain the results in Fig. 10 where Sy is plotted against the ratio b/a. As a is varied from 0 ° to 45 °, Sy changes from 1.0 to 1.23. The projected crack solution is again conservative.

4. Biaxial loading Let the angle crack in Fig. 1 be subjected to uniform stresses Ao" and/xo- in the x- and y-axis, respectively. Recommended in [1-3] is to consider projection of the crack along both the xand y-axis. Selecting for the larger K I factor, it suffices to fix IZ = 1 and vary A in the range -l
-Jr- ,~ sin20),

K n = trx/-~-a [ 1 / 2 ( 1 - A) sinE0].

(6)

F.D. Fischer, F.G. Rammerstorfer / Theoretical and Applied Fracture Mechanics 20 (1994) 67-74

72

The limiting value for 0, 0in for which K I >_ 0 for a negative value of A can be found from the first of Eq. (6) as 0i~=arcsin(l+

I,~1) -~/2,

- I < A < 0_. _

........../0g 611.

(7)

The stress intensity factors for the projected cracks are Kip,x = o ' x / - ~

1.00 -

. ~ 0.90

~minS

~o. ~

Kip,y = Atrx/~--a sv/~ffO .

cv/'~ O,

(8)

0.60

~

00

The safety factors follow from Eq. (1) and can be found numerically for 0 < v < ~ with v being the Poisson's ratio. The denominator of S~ and S r is nonzero for all 0. The numerator vanishes for S x with Klp,x for 0 = 90 ° and Sy with g l p , y for 0 = 0°. Hence the graph for S x starts at unity for 0 = 0° and terminates at zero for 0 = 90 °. This is in contrast to the uniaxial case (A = 0) where S~ is unity at 0 = 0° and becomes unbounded at 0 = 90 ° while Sy vanishes at 0 = 0 ° and becomes unity at 0 = 90 °. 4.1. Positive A (0 < A < 1) Positive A corresponds to a uniform tensile stress applied in the x-direction in addition to that in the y-direction. Numerical values of S x and Sy varying with the crack angle 0 are shown graphically in Fig. 11 for A = 0.4 and /~ = 1. The projected crack solution for Sr is not conservative

i.~

0'.,

.............................

0.6'

0~.,

0

,0

k Fig. 12. Minimum safety factors and angle 0g plotted against positive biaxiality coefficient ,L

for 0 > 78 ° as it becomes less than one. Sy reaches 0.913 as 0 ~ 90 °. Minimum value of S x and Sy can be found for A = 1. This corresponds to K n = 0 in the second of Eq. (6). In view of Eq. (8), it follows from Eqs. (A1) and (1) that Sxl~= 1 = ~

O,

Sy]~=l = ~

0.

(9)

The two curves for Sxtx=l and Syla=l intersect at 0 = 45 ° with a minimum safety factor of 0.841. Figure 12 gives a plot of min S versus A and values of 0g for which the safety factor is less than unity.

I

I_

20

40

I

-~

1.21.2-

0.g 0.8-

Ix=l

d 0.4

0,4-

0-

0 ldeg] Fig. 11. Safety factors Sx and Sy versus crack projection angle 0 for angle crack in biaxial loading with A = 0.4 and ,~ = 1.

00

80

o [deg] Fig. 13. Safety factors S x and Sy versus crack projection angle 0 for angle crack in biaxial loading with A = - 0.5 and ~ = 1.

F.D. Fischer, F.G. Rammerstorfer /Theoretical and Applied Fracture Mechanics 20 (1994) 67-74 i

LeO-

I

I

6. Appendix. Strain energy density criterion (Scriterion)

I

...""

minS 0.95

..............

-so.

%

-

E

73

'.40, gt~ 0.90 -

For a local state of plane strain where o-~= v(~rx + (ry), the relative strain energy density S is given by [27]: = (3 - 4v - cos 4))(1 + cos 4))KIz

F 0.85

-

-1.o

.ko

-~.0

-~.,

J,.~

L

+ 2 sin 4)(cos 4) - 1 + 2v)KIKII O

+ [4(1 - u)(1 - cos 4))

o.o

+ ( 3 cos

Fig. 14. Minimum safety factor and angle O~ plotted against negative biaxialitycoefficientA.

4.2. Negative h ( - 1 < h < O)

- 1)(1 +cos

+ 4K2I .

(A1)

The angle ~ is found by enforcing dS/d~b = 0 and d2S/d4) 2 > 0. The former yields a transcendental equation 2(1 - 2 u ) [ ( K ~ - K~) sin q~ - KIKixcos q~]

If a compressive stress is applied in the x-direction, then A takes negative values. The results in Fig. 13 pertain to A = - 0 . 5 and ~ = 1 with 0t. = 54.7 °. For 0 < 0 < 0g, the projected crack solution is slightly nonconservative with a minimum S of 0.962 at dg = 37.5 °. The discontinuities in S x and Sy are artificial as they arise from taking minimum of S outside the range of - w / 2 to ~r/2 for the angle 4;. Figure 14 again illustrates the variations of min S and 0g with A where the projected crack solution is not conservative.

5. Conclusions

The Mode I crack projection method has been used for assessing the safety factors of cracks with a variety of configurations. The strain energy density criterion [11] is applied in this work to demonstrate that many of the cracked structural components can be analyzed by using simplified approaches and to obtain conservative estimates. Slightly nonconservative estimates should not discriminate the Mode I crack projection method for its simplicity as long as additional compensation is made to correct for the difference.

+ ( K ~ - 3K 2) sin 2~ + 2KxKncos 2~ = 0

(A2) solving for 4~ while the latter gives for 4, = 4~: dzS

d4)Z = 2(1 - 2v)[(KI] - K~) cos 4~ +K[Klxsin 4~] + 2 ( K 2 - 3KI2I) c0s23 - 4KiKi, sin 2(~.

(m) For nonzero values of K I and Kn, the solution for Eq. (A2) with d2S/d4)2> 0 must be found numerically in the interval - 4 ) / 2 < 4; < ,rr/2. For K n = 0, Eq. (A2) gives ~1 = 0 and 4~z = arc cos(1 - 2u). Inserting 4)1 into Eq. (A3) shows immediately that 4~ corresponds to the minimum. If a crack is projected onto the i-axis (i = x , y, z), there results the stress intensity factor Kip,i with = 0. A safety factor S i can be defined as ~

.

~

Sj= [Sp,.i/S(qb)]

i12

(A4)

which is Eq. (1). The quantity Sp,j takes the form Sp,j = 4[(1 - u)K~,,i +

gi21i]

(A5)

while S(~) corresponds to the minimum of S.

74

F.D. Fischer, F. G. Rammerstorfer / Theoretical and Applied Fracture Mechanics 20 (1994) 67-74

7. Acknowledgements

Both authors are deeply indebted to Professor K.L. Maurer, University of Mining and Metallurgy, Leoben; they dedicated this work to his 65th anniversary. The authors gratefully acknowledge the contributions by Chr. Nagl who performed all the numerical work. Discussions with Dr. K. Mayrhofer, former research assistant at the Institute of Mechanics, and now with VOEST-ALPINE Industrieanlagenbau, Hot Rolling Mills, Linz, are also appreciated.

8. References [1] DVS Merkblatt 2401 Teil 1, 2: Bruchmechanische Bewertung von Fehlern in Schweil3verbindungen (Deutscher Verband fiir SchweiBtechnik e.V.: Diisseldorf, 1989). [2] PD 6493:1991 Guidance on methods for assessing the acceptability of flaws in fusion welded structures (British Standards Institution: 1991). [3] Empfehlungen zur bruchmechanischen Bewertung von Fehlern in Konstruktionen aus metallischen Werkstoffen, 2. Ausgabe (Osterreichischer Stahlbauverband: 1992). [4] S. Melin, Which is the most unfavourable crack orientation?, Int. J. Fracture 51,255-263 (1991). [5] D. De Vadder, M.G. Silk and D. Bouami, Review of knowledge about diffraction of US by crack tips, J. Mech. Behaviour Mater. 3, 1-34 (1990). [6l H.P. Rossmanith, Uber Sicherheitsabschiitzungen bei der bruchmechanischen Fehlerbewertung, Materialpriif 22, 337-340 (1980). [7] M.A. Hussain, S.L. Pu and J. Underwood, Strain energy release rate for a crack under combined Mode I and Mode II, in: Fracture Analysis, ASTM STP 560 (1974) 2-28. [8] H.P. Rossmanith and M. Tiefenb6ck, fiber Sicherheitsabschiitzungen bei bruchmechanischer Fehlerbewertung ffir biaxiale Beanspruchung, Materialpriif 25, 181186 (1983). [9] P.W. Claydon, Maximum energy release rate distribution from a generalized 3D virtual crack extension method, Eng. Fracture Mech. 42, 961-969 (1992). [10] F. Erdogan and G.C. Sih, On the crack extension of plates under plane loading and transverse shear, J. Basic Eng. 85, 519-527 (1963).

[11] G.C. Sih, Mechanics of Fracture Initiation and Propagation (Kluwer Academic Publishers: Dordrecht, 1991). [12] X. Yan, Z. Zhang and S. Du, Mixed-mode fracture criteria for the materials with different yield strengths in tension and compression, Eng. Fracture Mech. 42, 109116 (1992). [13] P.S. Theocaris and P.D. Panagiotopoulos, On a mathematical comparison of the S- and T-criteria in fracture mechanics, ZAMM 72, 341-345 (1992). [14] F. Taheri and A.A. Mufti, Numerical/graphical implementation of maximum strain energy density criterion, Trans. CSME 16, 33-46 (1992). [15] C.P. Spyropoulos, Some remarks on the Det.-criterion and caustics, Eng. Fracture Mech. 43, 1053-1061 (1992). [16] K. Bose and P. Ponte Castaneda, Stable crack growth under mixed-mode conditions, J. Mech. Phys. Solids 40, 1053-1103 (1992). [17] Y. Murakami, Stress Intensity Factors Handbook (Pergamon Press: Oxford, 1987) 169-170 and 225-227. [18] F. Erdogan and O. Aksogan, Bonded halfplanes containing an arbitrarily orientated crack, Int. J. Solids Structures 11,569-585 (1974). [19] M. Isida, Stress-intensity-factors for the tension of eccentrically cracked strip, J. Appl. Mech. 33, 674-675 (1966). [20] C. Manu, Three-dimensional finite element analysis of cyclic fatigue crack growth of multiple surface flaws, Ph.D. Thesis, Cornell University, 1960. [21] C. Manu, Quarter-point elements for curved crack fronts, Computer&Structures 17, 227-231 (1983). [22] H.G. Hahn, Bruchmechanik (B.C. Teubner: Stuttgart, 1976) 57 ft. [23] K. Mayrhofer, Die Variation des Spannungsintensit~itsfaktor entlang der Randkontur von ebenen zur Oberfl~iche senkrecht liegenden Rissen, Ph.D. Thesis, University for Mining and Metallurgy, Leoben, Austria, 1989. [24] K. Mayrhofer and F.D. Fischer, Derivation of a new analytical solution for a general two-dimensional finitepart integral applicable in fracture mechanics, Int. J. Num. Meth. Eng. 33, 1027-1047 (1992)~ [25] Y. Murakami, Analysis of stress intensity factors of Modes I, II and III for inclined surface cracks of arbitrary shape, Eng. Fracture Mech. 22, 101-114 (1985). [26] N.I. Ioakimides and P.S. Theocaris, A system of curvilinear cracks in an isotropic elastic half plane, Int. J. Fracture 15, 299-309 (1979), [27] G.C. Sih, Fracture mechanics of engineering structural components, in: Fracture Mechanics Methodology, G.C. Sih and L. Faria, eds. (Martinus Nijhoff: The Hague, 1984) 35-101.