Mixed mode crack propagation studied by the Yeh-Stratton criterion

Engineering

Pergamon

0013-7944(94)E0012-6

Fracture

Mechanics

Vol. 48, No. 4, pp. 595-607, 1994 Elsmier Science Ltd. Printed in Great Britain. 0013-7944/94 $7.00 + 0.00

MIXED MODE CRACK PROPAGATION STUDIED BY THE YEH-STRATTON CRITERION HSIEN-YANG

YEHt

Mechanical Engineering Department, California State University at Long Beach, Long Beach, CA 90840, U.S.A. CHANG H. KIM Aerospace Engineering Department, University of Southern California, Los Angeles, CA 90089-l 191, U.S.A. Abstract-The angled crack problem is examined with the Yeh-Stratton criterion in the form of the maximum principal stress theory of isotropic materials. The angle of fracture and the fracture load as a function of a crack angle are computed for simple uniaxial, biaxial and in-plane shear loadings. The predictions are compared with experimental data. The two-dimensional crack tip stresses for a general loading case are obtained by the principle of superposition.

INTRODUCTION IT HASBEENthree decades since Erdogan and Sih [l] pioneered the work later known as the angled crack problem. The problem of predicting the direction of crack propagation of an inclined crack at an angle j? to the loading axis has not been answered satisfactorily yet [2], despite the fact that there are many fracture criteria describing the problem in the literature-to name a few, the maximum tangential stress criterion [l], the maximum energy release rate criterion [l, 3,4], the minimum strain energy density criterion [5], the maximum strain criterion [6], the T-criterion [7l, the maximum tangential principal stress criterion [8], and the determinate of stress tensor criterion [9]. In addition to these, the maximum tangential strain energy density criterion [lo], the distortional strain energy density criterion, called the Y-criterion [l 11, and the mixed-mode fracture criteria for the materials with different yield strengths in tension and compression [12] are currently present in the literature. However, it is not the purpose of this paper to make any comparison between the criteria to determine which criterion is better than the others. Rather, another concept is introduced to examine the 30-year-old angled crack problem. Fracture mechanics of the angled crack problem as a part of the failure analysis can be studied in three steps. First, stress analysis as correct and exact as possible must be carried out. Without reliable stresses acting on a body, it is impossible to continue the analysis further. The main focus of the stress analysis of the fracture mechanics is restricted to calculating the stresses in the immediate vicinity of the crack tip because the region near the tip of the crack is most likely to fail or fracture first due to the nature of singularity. As soon as the stress state at a point of a body is disclosed, the point of interest will be examined next with a failure or fracture criterion to determine whether the point is failed or fractured. A significant error may be present when a failure theory for a ductile material is used for a brittle material. Thus, caution must be exercised in selecting an appropriate theory for a given material. The third step is to calculate the failure or the fracture angle. A good theory of failure or fracture must be able to provide reliable angles of failure or fracture. Such a theory is introduced by the present authors [13], and it is referred to as the Yeh-Stratton criterion, simply called the Y-S criterion. The Y-S criterion is based on the yielding of a homogeneous, isotropic, and linearly elastic material. The failure or yielding surface of the Y-S criterion is shown to be bounded and closed [13]. The Y-S criterion is also designed to work for both ductile and brittle materials, and predicts the zone of failure well which eventually defines the failure angles for all the simple tTo whom correspondence

should be addressed. 595

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YEH and C. H. KIM

tests such as simple tension, compression, and torsion tests. The Y-S criterion is further generalized for the failure analysis of isotropic materials and fibrous composites [13, 141. The objective of this paper is to apply the Y-S criterion to the angled crack problem of simple uniaxial, biaxial, and in-plane shear loading even though the Y-S criterion was developed originally for a untracked body. This may be alarming, but any criterion based on continuum mechanics can be applied to a cracked or untracked body in the valid range of continuum mechanics. The normalized fracture stress and the fracture angle versus the inclined crack angle are shown later, and predictions are compared with the available experimental data. STRESS ANALYSIS The stress analysis for a body with a crack is confined to the immediate vicinity of the crack tip where the local stress has the highest stress concentration, and the stresses increase at the crack tip. Thus, the stresses near the tip of the crack will be of interest here. A thin plate of a homogeneous, isotropic, and linearly elastic material containing a small crack is considered first, and the geometries of the crack and the applied load in tension are shown in Fig. 1. The crack tip stresses are given below by Eftis et al. [15] after some adjustments to the original Westergaard solution.

1 1 -(I

1 -sinisin:

O,=&COS~

[

fry=&cos:

1 +sinisin; [

_k)o;

rX~=&sin~cos~cos~,

(1)

provided 0 < r/a 4 1. The constant k is the multiplying factor representing the ratio of a,” /a,“, and the stress intensity factor for the opening mode is given as

4 = 0,” ,/W),

(2)

where a is the crack length. Equation (1) can be referred to as a fundamental solution to the opening

Y r

crp=k by” 4

e

/

I

2a

x

I

Fig. 1. The crack geometry under biaxial loads.

Crack propagation studied by YehStratton

Y

criterion

597

r

8 /’ I I

X

2a

4

Fig. 2. The crack geometry under shear load.

The second fundamental solution is given below for the shearing mode, and the corresponding geometries of the crack and the applied load in shear are shown in Fig. 2.

1

2fcos~cos~

OX= -&sin; [ o~=&sin~cos~cos~

~,=$-gjCOS~

[ 1

-sinisinT

1,

where the stress intensity factor for the shear mode is

The 2-D general crack tip stresses of the angled crack of the thin plate can be derived by superimposing the fundamental solutions together since the two solutions in eqs (1) and (3) are linear elastic solutions. The inclined crack at an angle j? with the vertical axis of loading is shown in Fig. 3, and the body is stressed with the normal stresses a;” and a?, and the in-plane shear stress t ;“2at the infInite boundaries. The stresses at infinity can be transformed into the x-y coordinate system by rotating the l-axis 90 - fi degrees counterclockwise. After simplification, a,” = 0 p sin’ B + 02” cos* j3 + r ;” sin 28

~;=(a?-a~)cos/3

sin/I -t;“cos2j?.

(5)

Once the entire boundary stresses are transformed into the x-y coordinate system, we are in a position to apply the principle of superposition to solve the general stress field. The stresses in eq. (5) can be divided into two systems of known solutions, which are demonstrated in Fig. 4. EFM 48,4-l

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YEH and C. H. KIM

Fig. 3. The inclined crack geometry under general loads.

By superimposing two fundamental solutions, the total crack tip stresses of the angled crack problem of Fig. 3 can be written as follows:

ox=&

cos f2

Oy=&

COSB

2

1 -sin 82 sin 22

1

I+sinfsinE2

1

[

[

r,=&sin~cos~cos~+J~~r)cOs~

2

-

+

Jzir)sini

[

2+cos~cos~

1

J~~r)sin~cos~cos~

1, [1-sinisinT

(6)

where K, = [a ;” cos’ /I + 07 sin2 j? - z ;“2sin 2/3]J(rra) K,, = Kc? - a;O)cos /I sin fi - r;o2cos 2j?]J(na).

(7)

For the uniaxial tensile loading, the stress field can be derived from eqs (6) and (7) by letting 01=J,rm ,z = 0. If t;“z= 0, the general solution (6) and (7) will be reduced to the identical solution obtained by solving the complex functions analytically as by Eftis et al. [16]. The stress field for the crack geometry of Fig. 1 can be shown from eqs (6) and (7) for r;“z= 0 and B = 90. The same equations (3) and (4) can also be derived from eqs (6) and (7) if a;” = a? = 0 and /I = 90. It is noted here that the general solution above is based on the two-term approximation of the exact solution. The solutions up to the three-term approximation in the Cartesian and the polar coordinates are illustrated in ref. [17].

Crack propagation studied by Yeh-Stratton criterion

FRACTURE

599

CRITERION

The hypothesis that the crack propagates from its tip to a point of the body where the tangential stress CT,is maximum is known as the maximum tangential stress criterion [I]. Since the stress state of the point where Q is maximum may not be the principal stress state, another hypothesis known as the maximum tangential principal stress criterion was developed which stated that the crack would propagate from its tip to a point of the body where Z~ = 0 [8]. These two fracture criteria, and other criteria introduced in the previous section, would find themselves in a difficult position to predict the fracture load and angles for an isotropic body which does not have a sizeable crack. It is true that a brittle material without a crack in tension will fracture along the plane perpendicular to the maximum principal stress. This is also argued by Sih and Tzou [18] in a discussion on the maximum tangential principal stress criterion. As far as a failure or a fracture criterion is concerned, there should not be any difficulty or difference in basic procedures to apply the criterion to a body with or without a crack unless the criterion is not based on continuum mechanics. The magnitude of the local stress may be different for the cracked body and the untracked body, but the same criterion must be applicable as long as reliable stresses are provided at a point of the body. The main functions of a strength criterion are to compare the experimentally measured value to the computed value and make a decision as to whether a point of the body in question is fractured or not. Among several theories of failure for isotropic materials, the maximum principal stress theory, the maximum shear stress theory, and the distortional energy theory are well known classical theories. These theories can be utilized to predict the fracture load, but none of these theories predicts the failure or fracture angles for brittle and ductile materials in simple tension, compression and torsion like the Y-S criterion does. In fact, some of the classical theories can be shown to be special cases of the Y-S criterion. Since the crack tip stresses are calculated to be mostly tensile throughout the analyses and the material is assumed to be ideally brittle, only the tensile portion of the Y-S criterion for brittle materials is repeated here from ref. [13]. The principal stress coordinate system is adopted in the Y-S criterion so that 0, > uZ> c3. Let us first define ccc which represents the average of the maximum and the minimum principal stresses 01 0 EC =-.

(8)

2

=Y”

OY"

T_

+a3

%Yoo

t

t =X= -

-_)=4-

=Xm _,+

Fig. 4. Superposition of elastic stresses.

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YEH and C. H. KIM

According to the Y-S criterion, the material will fracture if one of the following is met: Case (1) If

Q
then

2‘

a’= art

1;

(9)

Case (2) If

0
then

aIn3 aI -,+<+-=l; r/

a3

(10)

afi a,, and r/are the fracture strengths from the simple tension and torsion tests. It is noted that the Y-S criterion shown above is not affected by the compressive fracture strength although it is expected when a material is in compression. One main feature of the Y-S criterion is the ability to change its format depending upon what types of stresses are applied to the body and what types of materials are being considered. The crack tip stresses as shown in eq. (6) must be expressed in the principal stress coordinates in order to utilize the Y-S criterion. Thus, transforming the crack tip stresses into the principal stress coordinate system

app,= 0

for plane stress

ap3= p(a, + az) = p(a, + by)

for plane strain,

(11)

where p is the Poisson’s ratio, and a p,, ap2 and a,,, are principal stresses which must be arranged in order. However, for the plane stress problem, a, = ap, (i = 1,2,3) and a, 2 a2 2 ax. Since a thin infinite plate is considered throughout the analysis, the angled crack problem can be treated as a plane stress problem. After substituting a3 = 0 into eqs (9) and (lo), the Y-S criterion reduces to the single equation 2 = 1 for 0 < acr < ofi (12) a/ for the entire tensile part of the stress space. Equation (12) has an identical form to the maximum principal stress theory for isotropic materials, and thus eq. (12) will later be used to calculate the fracture stresses for polymethylmethacrylate (PMMA) and plexiglass. The simple nature of eq. (12) is somewhat fortunate because in order to fully operate the Y-S criterion it is necessary to have three fracture strengths of PMMA and plexiglass in simple tension, compression, and torsion, but strengths other than the tensile strength are not generally provided in the literature or handbooks of materials. Besides, the strengths of these materials are subject to significant change due to variations in humidity and temperature [19]. In order to find the fracture load from eq. (12), the fracture condition where a, = afi must be checked first at a given radius r from the tip of the crack for - 180 < 0 < 180. The point where a, reaches aJ, first on the circular boundary at a given stress is referred to as a fracture point, and the given stress is the fracture stress. It is noted that the angle 8 from the x-axis to the fracture point does not define the direction of crack propagation. In fact, the fracture angle or the direction of crack propagation is dependent upon the principal stress state of the fracture point. Thus, the location of the fracture point may not be important for computing the fracture angle because the behavior of the material is not well defined inside the core region bounded by the radius r where the linear elastic behavior of the material ceases to exist. In other words, linear elastic fracture mechanics (LEFM) allows one to estimate the fracture load and the fracture angle not from the crack tip directly, but from a small distance away from the crack tip.

ANGLE OF FRACTURE As mentioned above, the fracture angle analysis cannot be performed at the crack tip directly because the stresses at r = 0 reach infinity, and thus the direction of crack propagation is

Crack proration

studiedby Yeh-Strattoncriterion

601

analyzed at the fracture point. The angle of crack propagation can be best illustrated through the following examples. For the uniaxial tensile loading case of k = 0, #I = 90 and r/a = 0.05, two fracture points defined at B = 60 and - 60 degrees from the x-axis have reached the maximum value of Q,/afi at the fracture load of 1420 psi. The fracture strength in simple tension is measured to be bfi = 5830 psi for the plexiglass [19]. Figure 5 shows the stress states of these fracture points, and the stresses are found to be principal stresses. Since the material will fracture along the plane which is perpendicular to the maximum principal stress for both cases of 8 = 60 and - 60 degrees, the fracture angle which is measured from the x-axis is zero. As r approaches zero, it is expected that both the fracture planes at 8 = f60 will coalesce. Another case to be examined is k = 0, #I = 70 and r/a = 0.005, and the state of stresses of the fracture point at 8 = -95 degrees is shown in Fig. 6. Once again, the angle from the crack tip to the fracture point will not define the direction of crack propagation. The fracture angle must be determined from the principal stress state of the fracture point. The stress state of the fracture point is calculated to be (TV= 2680, a,, = 4435 and 7, = 2096 psi. The principal stress state of this fracture point can be found by rotating the x-axis 33.68 degrees counterclockwise, and the principal stresses are computed to be Q, = 5830 and a, = 968 psi. The plane which is perpendicular to the maximum principal stress cr, = 5830 psi is fractured. Therefore, the fracture angle is measured to be 0,= -33.68 degrees.

COMPARISON

BETWEEN PREDI~ONS

AND EXPERTS

Kibler and Roberts studied the effect of biaxial loading on fracture behavior of a thin plexiglass plate [19]. The geometry of the crack and the load is shown in Fig. 1, and the angle of the fracture is computed in terms of k = ~,“/a,“. For small values of k, the angle of the initial extension of the crack was observed to be zero from the laboratory. As k increased, the initial crack extension was still zero, but the path began to curve as the crack propagated further. At 5830 psi

I

@90 deg. I

Crack y r

-

524psi

1

Fig. 5. Fracture point and fracture angle (k = 0, /I = 90).

HSIEN-YANG

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YEH and C. H. KIM

Fig. 6. Fracture point and fracture angle (k = 0, /I = 70).

approximately k = 1.8, the path of the crack turned abruptly in the direction of the nearly normal to a,“. From the analysis based on the Y-S criterion, the initial crack extension angle was calculated for -2 < k < 5 by increments of Ak = 0.1 at r/a = 0.05, and the results are plotted in Fig. 7. It 100

80

60

40

\

Jump

20

0.0

-20 -2

2

0

4

6

k Fig. 7. The angle of crack propagation (0,) versus the biaxial loading ratio (/c) for the plexiglass.

Crack propagation studied by Yeh-Stratton

I

criterion

603

+ Experimental data for PMMA [21]

80

60

40

20

0.0 0.0

20

60

40

80

100

P Fig . 8. The crack propagation versus the inclined angle (k = 0).

is significant to note here that there is indeed a sudden jump in fracture angle around 1.9 < k < 2.0. A similar study was done by Eftis et al. [16]. They employed the maximum tangential stress criterion, but a sudden jump was not observed. To demonstrate the fracture angle, two cases of k = 1.9 and k = 2.0 are examined. For k = 1.9, the two fracture points are found at angles of 60 and -90 degrees from the x-axis. Due to the symmetry, only the fracture point at the angle of 60 degrees is considered, and the corresponding stresses of this fracture point are a, = 3220.7, a,, = 5830.1 and r, = 0. They are the principal stresses. Thus, the fracture angle is zero.

10 +ExperimelltaldatafoiPMMA[21] 8

l-

rla=O.OOOS

6

_

rLO.005

+%2

0

20

40

60

’

80

B Fig. 9. Normalized critical load versus the inclined angle (k = 0).

100

HSIEN-YANGYEH and C. H. KIM

604 100

+EqrimentaldataforPMMAl21] 80

0.0

20

0.0

40

60

So

100

Fig. 10. The crack propagation versus the inclined angle (k = 0.5).

For k = 2.0, the fracture point is located only at 0 = 0,and the stresses are calculated to be o, = 5830.1, a,, = 4429.4 and r,,, = 0. In this case, a, has the maximum principal value. Therefore, the fracture occurs in the plane perpendicular to the direction of c,, which implies the fracture angles are f 90 degrees. Since the geometry and the stresses are symmetrical, the sign of the fracture angle will be insignificant. Figures 8, 10 and 12 show the fracture angle as a function of the inclined crack angle for k = 0, 0.5and 1, respectively. Figures 9, 11 and 13 show the fracture load versus the inclined crack angle for k = 0,0.5and 1, respectively. The results of these figures are compared with experiments for 3.0 +ExperimentaldaraforPMMA[21]

* 2.5

0.0

J

++

t

J 0.0

20

40

60

80

100

B Fig. 11. Normalized critical load versus the inclined angle (k = 0.5).

Crack propagation studied by YehStratton

605

criterion

W

t l3quimentaldatafor FMMA [21] 40

-

20

-

0.0

I

r/a=0.05.0.005.0.ooO5 :

.,

l

+

*

+ +

/ *

-Bf

-20

-

-40

_

-60

II 0.0

20

+-z

60

40

80

100

Fig. 12. The crack propagation versus the inclined angle (k = 1).

PMMA [20], and correlations are shown to be in reasonable agreement. All fracture loads are normalized with the fracture load at /I = 90. The results of the circular tubes under torsion are shown in Figs 14 and 15, which show the curves of the fracture angle and the fracture strength as a function of the inclined crack angle. The computed results are shown to be in reasonable agreement with experiments [21]. In Fig. 15, both the predicted and experimental values of the fracture load are normalized with the fracture load at /I = 45 and the value of 1.12 MN/m3j2, respectively. This was necessary since the failure stresses in ref. [21] were plotted as r(wa)‘l* versus 8.

1.5

rla=o.05.0.005.0.0005 ..

wJlcl2

+

1.0 *-

.

0.5

+

+

J-:

+

$

+

::

+

+

+

+

+

+

-

0

20

60

40

80

100

B Fig. 13. Normalized critical load versus the inclined angle (k = 1).

606

HSIEN-YANG

YEH and C. H. KIM

+ Experimentaldatafor PMMA [22]

0

20

40

60

80

100

B Fig. 14. The crack propagation versus the inclined angle for shear loading.

The analyses in this paper were carried out at the values of r/a = 0.05, 0.005 and OAK@5for comparison purposes. The theoretical curves with the different values of r/a can be approximated because the general trends of the curves should remain similar. For Figs 8-13, the curves with the smaller values of r/a fit the experimental data well, except the curves shown in Fig. 10. It can be seen from Figs 14 and 15 that the theoretical curves become more symmetrical as r/a becomes smaller.

+ Experimental data for PMMA [22]I

r/a=o.O005

r/a=O.O5

0

20

40

60

80

100

B Fig. 15. Normalized critical load versus the inclined angle for shear loading.

Crack propagation studied by YehStratton

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607

CONCLUSION The application of the Y-S criterion to the angled crack problem is satisfactory in the form of the principal stresses. Unlike most other fracture criteria which are applicable only to a cracked body, the Y-S failure criterion developed for an untracked body is applied to a cracked body. Although the approach is somewhat different from the conventional methods used in fracture mechanics, the analysis is done within the valid range of LEFM. The nature of the fracture of a cracked body is different from that of an untracked body, but as long as the linear elasticity is applicable, there should not be any difficulty or difference in applying the criteria to a cracked or untracked body. However, some of the fracture criteria cannot be applied to an untracked body. Definitely, no angle of fracture is perceived for there is no crack tip. This is what makes fracture criteria in fracture mechanics so special. In the analysis shown in this paper, the behavior near the crack tip inside the core region is simply ignored for two reasons. Firstly, it is reasonably believed that the crack path is measured experimentally well beyond the core region, and the behavior of the material within the core region cannot be detected with the naked eye unless a microscope is used. This may be apparent as the core region gets smaller and smaller. Secondly, the material behavior is not well defined mathematically inside the core region. Thus, defining a fracture angle inside the core region may not be appropriate according to LEFM. All analyses must be performed outside the core region. For example, the fracture stresses are calculated in the valid range of LEFM, not at the crack tip. It is shown in this paper that for an isotropic body with or without a crack, the Y-S criterion can be used for estimating the fracture load and angle. Due to the nature of the singularity at the crack edge, the fracture load and angle are approximated at a small distance away from the crack tip, although the crack tip is one possible fracture point, but the direct analysis cannot be performed at r =O. REFERENCES [1] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. Trans. ASME. J. has. Engng SD, 519-527 (1963). [2] G. C. Sih, Mechanics of FractureInitiation and Propagation. Kluwer Academic, Dordrecht (1991). [3] K. Palaniswamy and W. G. Knauss, Propagations of a crack under general in-plane tension. Inr. J. Fracture Mech. 8, 114117 (1972). [S] M. A. Hussain, S. L. Pu and J. Underwood, Strain energy release rate for a crack under combined mode-1 and -II. ASTM STP 560, 2-28 (1974). [5] G. C. Sih, Some basic problems in fracture mechanics and new concepts. Engng Fracture Mech. 5, 365-377 (1973). [6] K. J. Chang, On the maximum strain criterion-a new approach to the angled crack problem. Engng Fracture Mech. 14, 107-124 (1981). [7] P. S. Theocaris and N. P. Andrianopoulos, The T-criterion applied to ductile fracture. Int. J. Fracture Mech. 20,

Rl25-R130 (1982). [S] S. K. Maiti and R. A. Smith, Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field. Part I: slit and elliptical cracks under uniaxial tensile loading. Inf. J. Fracfure Mech. 23, 281-295 (1983). [9] G. A. Papadopoulos, New concepts on the Det.triterion. Engng Fracture Mech. 32, 351-360 (1989). [IO] J. M. Koo and Y. S. Choy, A new mixed mode fracture criterion: maximum tangential strain energy density criterion. Engng Fracture Mech. 39, 44349 (1991). [11] N. A. B. Yehia, Distortional strain energy density criterion: the Y-criterion. Engng Fracture Mech. 39, 477-485 (1991).

[12] X. Yan, Z. Zhang and S. Du, Mixed-mode fracture criteria for the materials with different yield strengths in tension and compression. Engng Fracture Mech. 42, 109-I 16 (1992). [13] C. H. Kim and Hsien-Yang Yeh, Development of a new yielding criterion: the YehStratton criterion. Engng Fracture Mech. 47, 569-582 (1994). [14] H.-Y. Yeh and C. H. Kim, The Yeh-Stratton criterion for composite materials. Submitted to J. compos. Mater. [15] J. Eftis, N. Subramonian and H. Liebowitz, Crack border stress and displacement equations revisited. Engng Fracture Mech. 9, 189-210 (1977). [16] J. Eftis and N. Subramonian,

The inclined crack under biaxial load. Engng Fracture Mech. 10, 43367 (1978). [17] D. Maugis, Stresses and displacements around cracks and elliptical cavities: exact solutions. Engng Fracture Mech. 43,

217-255 (1992). [18] G. C. Sih and D. Y. Tzou, Discussion on “criteria for brittle fracture in biaxial tension” by S. K. Maiti and R. A. Smith. Engng Fracture Mech. 21, 977-981 (1985). [19] J. J. Kibler and R. Roberts, The effect of biaxial stresses on fatigue and fracture. J. Engng Ind., pp. 727-734 (1970).

[20] Y. Ueda, K. Ikeda, T. Yao, M. Aoki, T. Yoshie and T. Shirakura, Brittle fracture initiation characteristics under biaxial loading, in Fracfure 1977, Volume 2, ICF4 (pp. 173-182), Waterloo, Canada (1977). (211 P. D. Ewing and J. G. Williams, The fracture of spherical shells under pressure and circular tubes with angled cracks in torsion. Znr. J. Fracture Mech. 10, 537-544 (1974). (Received 30 April 1993)