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Predicting crack growth direction in unidirectional composite laminae
Engineering Fracture A4eclranicsVol. 36, No. I, Printed in Great Britain.
pp. 21-37,
PREDICTING CRACK UNIDIRECTIONAL F.
0013-7944/90 $3.00 + 0.00 0 1990Pergamon Press plc.
1990
ELLYIN
GROWTH DIRECTION IN COMPOSITE LAMINAE and
H. El KADI
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8 Abstract-The problem of predicting the direction of crack growth in a centre-notched unidirectional graphite-epoxy is investigated. A homogeneous anisotropic elasticity analysis is presented in conjunction with four crack growth criteria. Comparison between the theoretical prediction of the crack extension direction and experimental results indicate that the proposed criterion based on the critical strain energy density correctly predicts the direction of crack growth for the test cases considered herein.
INTRODUCTION A FUNDAMENTAL problem in predicting the failure of laminated composite materials is the prediction of the direction of crack growth in the individual lamina as well as the laminate. The lamina represents a simpler and perhaps a fundamental problem which should be understood completely before an attempt is made to understand crack growth in laminates. The latter problem is inherently more complex since generally a three-dimensional stress-state exists in a notched laminate. This paper is concerned with predicting the direction of crack growth in unidirectional laminae. The material under investigation will be assumed to be elastic, homogeneous and anisotropic. Three theories that have been used for predicting the direction of the crack extension in the homogeneous, anisotropic materials are: the tensor polynomial criterion[l], the minimum strain energy density criterion[2] and the normal stress ratio criterion[3]. There are certain problems with each of the above mentioned criteria as to be shown later on. A new criterion based on a critical value of the strain energy density will be presented here, and will be shown to accurately predict the direction of crack growth for the test cases considered.
ANISOTROPIC
ELASTICITY
ANALYSIS OF CRACK TIP STRESS FIELDS
The stress analysis of an elliptical hole in an anisotropic plate can be directly related to a crack by reducing the minor axis dimension to zero. This problem has been extensively investigated since the late 1930’s. A summary of these investigations can be found in[4]. Lekhnitskii’s formulation of this problem, which is analogous to that of Muskhelishvili’s counter-part in isotropic elasticity, is adapted in the following discussion. For the plane elastic anisotropic material (shown in Fig. l), the equations of stress equilibrium and strain compatibility can be represented in terms of Airy’s stress function, @, in the familiar form:
where A, are components of compliance tensor for plane stress (or plane strain). Assuming @ = e’+Q, the characteristic equation (1) takes the form: A,, S4 - 2A,$
+ (2A,, + A&S* - 2A2J + A,,= 0. 27
(2)
28
F. ELLYIN
Fig.
1. Infinite
center
and H. El KADI
cracked
plate with far field stresses.
The roots of the characteristic equation, SI and S,, are complex, as are functions of the material properties and the orientation of the crack relative to the principal material direction. Considering the case where S, # S,, evaluation of the complex potential functions near the crack tip yields expressions for the stress and displacement distributions of the form:
(3)
TX
e
I
2ar Re(S, -
[q2q/y2 -
S,)
q, $ ;,‘?I
I
where: $,=coscp+S,sincp
& = cos 4 -i- S, sin d,
p[ =.4,,S:+Arz-A,6S,
~2 = A,, S: + A,2 - AlesS2
41 = (AUS? -1-A22 - AxjSl)/S,
q2 = (A ,?S: + A22- Ax& J/S?
and G X and r 7’ refer to the far field applied stresses. As in the isotropic case, the crack tip stresses exhibit a singularity of Iid’;. However. the magnitude of the stresses is not simply a function of the stress intensity factors. The quantities S, and S, also affect the magnitude of the stresses. This is an important difference between anisotropic and isotropic fracture. In anisotropic fracture, the magnitude of the crack tip stresses is a function of not only the applied load, specimen geometry and crack length, but also the material properties and the orientation of the crack relative to the principal material direction.
Crack growth direction
29
Fig. 2. Infinite center cracked plate under biaxial loading.
The problem under consideration (Fig. 2) has no restrictions as to the orientation of the crack defined by the angle a, or the principal material directions defined by the angle 8. Complete biaxial loading is allowed. The only constraint is that the crack is assumed to have a finite width. The problem defined in Fig. 2, can be solved by transforming the far field stresses to a crack tip coordinate system[5] as shown in Fig. 3, and observing that the far field stress parallel to the crack does not contribute to the singularity (this fact is discussed in[6]). The elasticity solution can then be applied to solve this general problem.
CRITERIA FOR PREDICTING DIRECTION OF CRACK GROWTH
(1) Tensor polynomial criterion Tsai and Wu[l], Tennyson et a1.[7], and others have presented a tensor polynomial as an anisotropic failure criterion. This criterion is based on the existence of a failure surface in the stress-space in the form of:
f(fli) = F,‘Ji + 4jCriC,+ 4jkBiOjOk+ . . . = 1 (i, j = 1, 2, . . . 6)
(5)
where Fi, Fi, and Fiik are strength tensors of second, fourth and sixth order respectively, and bi is the contracted form of the stress tensor. See the review by Labossiere and Neale[8] for a discussion of various tensor polynomial criteria. The application of the tensor polynomial to fracture problems, utilizes the assumption that the direction of crack extension corresponds to the radial direction of maximumf(aJ The stress components, Giymust be evaluated at a finite distance, r,,, from the crack tip.
--
T --
--
(a) Fig. 3. Transformation
T W
-7
and superposition of far field stresses.
w
30
F. ELLYIN and H. El KADI
(2) minimum strain energy density criterion
The minimum strain energy density criterion is based on variations in the energy stored along the boundary of a core region surrounding a crack. This criterion was first proposed by Sih[9] for isotropic fracture and later modified for the application to anisotropic problems[2]. Sih defined the strain energy factor, S, by the expression: W = (dU/d V) = S/r
(6)
where (dU/dV) is the strain energy density function, and r is the radial distance from the crack tip. For plane stress problems, the strain energy density function can be expressed in terms of the stresses and strains near the crack tip as:
w = tfl,c, + tT,c,. +
~,~Y,X~)P.
(7)
As a result, the strain energy density factor can be defined explicitly as: s = Y(fl,f, + fTLEy + %,Y,)/2.
(8)
The factor S was interpreted as the area under the (dU/d V) vs r curve[lO]. The basic postulates of the minimum strain energy density criterion are: (a) Failure by fracture is assumed to initiate at sites corresponding to the maximum value of the relative minimum of the strain energy density function, i.e. (dU/dV);g
at
# = &
(b) Fracture is assumed to occur when (dU/dV)~~
f9a)
reaches its respective critical value:
(dU/‘dV)E, = (dU/dV),. (c) The failure rate by fracture conditions:
is assumed to occur in accordance
(9b) with the following
(dU/dV), = S, /r, = L&/r, = . . . = S,jr, = 1* . = SC/r,)
(9c)
such that for unstable fracture: r, < r? <. 1. < rj <. . . < r,
and
(3) boreal
stress ratio criterion
Buczek and Herakovich[3] have proposed the normal stress ratio as a crack growth criterion. Using the isotropic results, where the cracks generally grow in the direction of the maximum normal stress, their model assumes that the direction of crack growth is controlled by the ratio of normal stress to tensile strength on a given plane. The normal stress ratio is defined as: R(ro, 4) = noelT+g
(10)
where (r+$= normal stress acting on the radial plane defined by # at a distance r, from the crack tip (Fig. 4), and T,, = tensile strength on the plane fp. Buczek and Herakovich postulated that the crack will grow along the plane for which this ratio is maximum. Since the strength T+,+on an arbitrary plane is very difficult to measure, they defined it as: T,& = XT sin* y + Y, co? y
where y was defined as the angle from the plane of interest to the fibre direction (Fig. 4).
(11)
31
Crack growth direction
Fig. 4. Normal stress ratio parameters (from [SJ).
The definition of T,, satisfies the following necessary conditions: (a) For isotropic materials, T,, does not depend on 4. (b) For crack growth parallel to the fibres in a composite material, T+b is equal to the transverse tensile strength, Y, . (c) For crack growth perpendicular to the fibres of a composite material, T,, is equal to the longitudinal strength, X, . (4) Critical strain energy density method-present
criterion
The critical strain energy density method is based on the assumption that crack initiation takes place when the value of the strain energy density defined by (7) reaches a critical value, i.e. (12)
w = (O,G + a&’ + %yYxy)/2= w,ri, *
The value of W will depend on the material properties and the fibres direction. The direction of crack growth will correspond to the radial direction having the maximum value of the strain energy density. Thus the direction of crack extension 4 = +,, is obtained from:
awla
=o
and
a2w/a42<0.
(13)
It should be stated that although both Sih’s method and the present criterion use strain energy density as a criterion for crack growth direction, each method utilized a unique interpretation, c.f. (9) and (13). This criterion is an extension of Ellyin’s damage function for isotropic solid@ 11. The physical basis of the proposed criterion is as follows: When a material element is loaded by an external agency, part of the supplied energy is dissipated into heat and the remaining part is stored in the material. Damage is caused by the irrecoverable part of the stored energy, i.e. (14)
d Wdamage = d Wsupplid - (dQ + d Wrecovera~e)-
Each material has certain capacity to absorb damage, and failure results as a result of damage accumulation. Because of the difficulty in measuring the heat loss, dQ, it is generally assumed that damage is proportional to the supplied energy, i.e. d Wdamage
0~
d Supplied
(13
*
In front of the crack, the strain energy varies with the distance r and angle 4 (Fig. 4). Its variation at the crack tip, r 4 a, can be calculated from (12). Crack propagation will ensure when the strain energy reaches the material critical value (absorption capacity), Wcrit. The location at which this critical value is first reached is obviously the one where the strain energy is maximum, i.e. condition (13). ANALYSIS OF OFF-AXIS UNIDIRECTIONAL
TENSILE
COUPONS
Numerical results are presented for graphite-epoxy tensile coupons containing central cracks. Lamina properties of the material considered, as well as specimen dimensions, are given in[5, 121.
F. ELLYIN and H. El KADI
32
In comparing the predictions of aforementioned criteria for the crack growth direction, the tensor polynomial criterion will be excluded. This is mainly due to the dramatic dependence of the result of this method on the distance from the crack tip, rO. Herakovich er aZ.151have shown that a small change in the value of r, can affect the predicted crack growth direction by as much as 40P. The independence of $J, on r, allows the prediction of the crack extension direction in unidirectional composites without prior knowledge of a proper r0 value. The remaining three methods are compared with the experimental results of different off-axis tests in Table 1. It is noted that the present model (critical strain energy density) predictions are very consistent with the experimental results. This is in contrast to the minimum strain energy criterion of Sih which was shown in[S] as well as in Table 1, to be inconsistent in predicting the direction of crack growth. The variation of the three crack growth criteria as a function of 4, for the various tests are shown in Figs 5-7. UNIDIRECTIONAL
LAMINAE
SUBJECTED
TO MIXED-MODE
LOADING
In an attempt to verify the consistency of the proposed critical strain energy density criterion, and to study crack growth of fibrous composites under more general loading conditions, unidirectional laminae were analysed for various mixed-mode loading. The predicted values of 4,. together with the experimental results[12, 131 are presented in Table 2. From the comparison, it is clear that the present model predicts fairly well the crack growth direction for all of the five cases studied. The largest difference between the experimental and the predicted direction of crack growth is 2”. The variation of the crack growth criteria as a function of Cp,for some of the cases presented in Table 2, are shown in Figs 8-l 1. Any criterion for the anisotropic materials has to degenerate to an isotropic one at the limit. In the case of the maximum normal stress ratio (eq. lo), we meover the maximum principle stress criterion which is more appropriate for brittle materials. The critical strain energy criterion (12) reduces to W=WD+W*
(16)
where the first term on the right-hand side ( WD) is the distortion energy and the second term is the energy involved with the volume change of the material element. Both Tsai-Hill[l4] and the Table
I. Comparison
Test data
between different criteria and experimental unidirectional tensile coupons Normal stress ratio
Minimum strain energy density
results for off-axis
Present model
Experimental results[5]
Case (1): T300/5208 8= 120” ? =O” 2a = 0.2032“ r, = 0.002” LT,.=i.Oksi
301
350.’
297
300,
Case (2): AS4/3.501-6 f3= 105” c( = o2a = 0.2032” r. = 0.002” q= l.Oksi
286
60-
284
285
Case (3): AS4/3501-6 B = 105” a = 15O 2a = 0.2032” r0 = 0.002” 0, = l.Oksi
271
320’
268”
270
15’
88”
90
Case (4): AS4/3501-6 e = 90” u. = 0’ 2a = 0.102” r0 = 0.042%” CT,= 1.0 ksi
88’
Crack growth direction
OLLV~SSI8JS?VX80N
33
(a)
(Degrees)
0
90 9
no
(c)
(Degrees)
1eo
0.000
-
0
360
Wf
120 270 up (Degrees)
SO
Fig. 7. Variation of crack growth criteria as a function of 4 for 15” .off-axis graphite/epoxy coupons (case 3). (a) Normal stress ratio (from [5, 121. (b) Mnumum strain energy density. (c) Critical strain energy density (present model).
Q
Rz E
.
2a = r, = o, = r,, =
0.2032” O.OOI” 9.38 ksi 0.378 ksi
0=75” ct= -15’
Case (9): AS4/3501- 6
Case (8): AS4/3501L6 0=:75”a=-t5” 2a = 0.2032” To= 0.001” u,,= ll.Oksi rXy= 1.18 ksi
Case (7): AS4/3501-- 6 0 = 15” a = 0 2a = 0.2032” r0 = 0.001” ov= ll.Oksi t,). = 1.18 ksi
Case (6): AS4/3501-6 $=7.5” a = -15” 2a = 0.2032” r, = 0.001” g,*= 15.9 ksi rzF = 3.53 ksi
Case (5): AS4/3501L6 6=75” a=O” 2a = 0.2032” ro = 0.001” (TV = 15.9 ksi try = 3.53 ksi
Test data
Table 2. Comparison
88
88’
13”
88”
72
Normal stress ratio
320
330”
315’
20’
335”
Minimum strain energy density
91
90
74’
89
73”
Present model
90
90
75’
90”
15”
Experimental resuitslr)]
between different criteria and experimental results for unidirectional laminae subjected to mixed-model loading
Crack growth direction
35
tIO.LDV6 rUISN3a A383N3 NIVXLS
kLISN3a AL383N3 NIVZLLS
oLLV2?ss3xzs
” H
“!
a3ZI7VHlIoN
9
?
?
9
0
0
0
0
---------_ [ z d
2lOCX’d ALISNBa b383N3 NIVKLS
52
E
kLISN3a XQl3N3
‘_
a
G
9
0
0
0
6
4
OIJW ss3zLs EFM 36,I-c
9 n
9
(Y
0
s
a3ZrIVmoN
0
d
NIVEJS
36
F. ELLYIN and H. El KADI
ZIOKWil LLISN3a A383N3 NIVKLS
OILVZI SSB?uS a3ZIIvHl?ON
2
: d
d
2 d
2
d
:: d
2
d
?IOL3Vd LLISN3a A3X3N3 NIWIJS
OILVll ss3lus
a3ZI?vmION
8 d
Crack growth direction
37
term of the tensor polynomial (5) will reduce to WD for an isotropic material. It is seen therefore that the critical strain energy density criterion is more general than that of the maximum normal stress ratio. The extension of the critical strain energy density criterion to laminates would require further work since adjacent layers can affect the crack growth direction.
first
CONCLUSIONS A criterion based on the critical strain energy density has been presented to predict the crack growth direction in a Iamina. The comparison of the experimental results with the different criteria for the crack growth direction indicates that the proposed critical strain energy density method can be used to analyse the problem of a through-the-thickness central notch in a unidirectional composite. The present approach leads to more consistent results than the minimum strain energy density of Sib. The advantage of the energy approach is in its universality by being a frame indeferent quantity. It is more general than the normal stress ratio criterion. The latter is based on the maximization of the normal stress ratio (eq. lo), which gives results comparable to our criterion. However, the validity of eq. (11) for anisotropic materials has to be demonstrated. Acknowlegemenrs-The work presented here is part of a general investigation into materials behaviour under complex loading and adverse environment. The research is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC Grant No. A-3808).
REFERENCES S. W. Tasi and E. M. Wu, A general theory of strength for anisotropic materials. J.
Compos. Muter. 5, 58-80 (1971).
::; G. C. Sih, E. P. Chen, S. L. Huang and E. J. McQuillen, Material characterization on the fracture of filament-reinforced composites. J. Compos. Mater. 6, 167-186 (1975). [31 M. B. Buczek and C. T. Herakovich. Direction of crack arowth in fibrous comnosites. Mech. Comoos. Mater. 58.75-82 (1983). [41 S. G. Lckhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, translated by P. Fern (Edited by J. J. Brandstatter). Holden-Day, Inc., San Francisco (1963). PI M. A. Gregory and C. T. Herakovich, Predicting crack growth direction in unidir~tional composites. J. Compos. Mater. 20, 67-85 (1986).
PI E. M. Wu, Fracture mechanics of anisotropic plates, in Composite Material Workshop (Edited by S. W. Tsai, J. C. Halpin and N. J. Pagano), pp. 20-43, Technomic publishing Co., fnc, Lancaster, Pennsylvania (1968). t71 R. C. Tennyson, D. MacDonald and A. P. Nanyaro, Evaluation of the tensor polynomial failure criterion for composite materials. J. Compos. Mater. 12, 63-75 (1978). PI P. Labossiere and K. W. Neale, Macroscopic failure criteria for fibre-reinforced composite materials. SM Arch. 12, 439-450 (1987). [91 G. C. Sib; A special theory of crack propagation, in Method of Analysis and Solutions to Crack Problems (Edited by G. C. Sib). DV. XXI-XLV. Wolters-Noordhoff. The Netherlands (19721. PO1 G. C. Sib, Dynamics of cor&osites with cracks, in ~an~ook of Co~~s~~es, Vof. 3, Failure Mechanisms ofcomposifes (Edited by G. C. Sib and A. M. Skudra), pp. 127-176, Nosh-Holland, The Netherl~ds (1985). [I 11 F. Ellyin and K. Gotos, Multiaxial fatigue damage. J. Engng Mater. Technol. 110, 63-68 (1988). 1121M. A. Gregory, J. L. Bcuth, Jr, A. Barbe and C. T. Herakovich, Application of the normal stress ratio theory for predicting crack growth direction in unidirectional composites. Fracture Fibrous Compos. 74, 33-42 (1985). [I31 J. L. Beuth, Jr, M. A. Gregory and C. T. Herakovich, Crack growth in unidirectional graphite-epoxy under biaxial loading. Exper. Mech. 26, 245-253 (1986). [‘41 R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). I.
.
.
(Received 10 January 1989)
pp. 21-37,
PREDICTING CRACK UNIDIRECTIONAL F.
0013-7944/90 $3.00 + 0.00 0 1990Pergamon Press plc.
1990
ELLYIN
GROWTH DIRECTION IN COMPOSITE LAMINAE and
H. El KADI
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8 Abstract-The problem of predicting the direction of crack growth in a centre-notched unidirectional graphite-epoxy is investigated. A homogeneous anisotropic elasticity analysis is presented in conjunction with four crack growth criteria. Comparison between the theoretical prediction of the crack extension direction and experimental results indicate that the proposed criterion based on the critical strain energy density correctly predicts the direction of crack growth for the test cases considered herein.
INTRODUCTION A FUNDAMENTAL problem in predicting the failure of laminated composite materials is the prediction of the direction of crack growth in the individual lamina as well as the laminate. The lamina represents a simpler and perhaps a fundamental problem which should be understood completely before an attempt is made to understand crack growth in laminates. The latter problem is inherently more complex since generally a three-dimensional stress-state exists in a notched laminate. This paper is concerned with predicting the direction of crack growth in unidirectional laminae. The material under investigation will be assumed to be elastic, homogeneous and anisotropic. Three theories that have been used for predicting the direction of the crack extension in the homogeneous, anisotropic materials are: the tensor polynomial criterion[l], the minimum strain energy density criterion[2] and the normal stress ratio criterion[3]. There are certain problems with each of the above mentioned criteria as to be shown later on. A new criterion based on a critical value of the strain energy density will be presented here, and will be shown to accurately predict the direction of crack growth for the test cases considered.
ANISOTROPIC
ELASTICITY
ANALYSIS OF CRACK TIP STRESS FIELDS
The stress analysis of an elliptical hole in an anisotropic plate can be directly related to a crack by reducing the minor axis dimension to zero. This problem has been extensively investigated since the late 1930’s. A summary of these investigations can be found in[4]. Lekhnitskii’s formulation of this problem, which is analogous to that of Muskhelishvili’s counter-part in isotropic elasticity, is adapted in the following discussion. For the plane elastic anisotropic material (shown in Fig. l), the equations of stress equilibrium and strain compatibility can be represented in terms of Airy’s stress function, @, in the familiar form:
where A, are components of compliance tensor for plane stress (or plane strain). Assuming @ = e’+Q, the characteristic equation (1) takes the form: A,, S4 - 2A,$
+ (2A,, + A&S* - 2A2J + A,,= 0. 27
(2)
28
F. ELLYIN
Fig.
1. Infinite
center
and H. El KADI
cracked
plate with far field stresses.
The roots of the characteristic equation, SI and S,, are complex, as are functions of the material properties and the orientation of the crack relative to the principal material direction. Considering the case where S, # S,, evaluation of the complex potential functions near the crack tip yields expressions for the stress and displacement distributions of the form:
(3)
TX
e
I
2ar Re(S, -
[q2q/y2 -
S,)
q, $ ;,‘?I
I
where: $,=coscp+S,sincp
& = cos 4 -i- S, sin d,
p[ =.4,,S:+Arz-A,6S,
~2 = A,, S: + A,2 - AlesS2
41 = (AUS? -1-A22 - AxjSl)/S,
q2 = (A ,?S: + A22- Ax& J/S?
and G X and r 7’ refer to the far field applied stresses. As in the isotropic case, the crack tip stresses exhibit a singularity of Iid’;. However. the magnitude of the stresses is not simply a function of the stress intensity factors. The quantities S, and S, also affect the magnitude of the stresses. This is an important difference between anisotropic and isotropic fracture. In anisotropic fracture, the magnitude of the crack tip stresses is a function of not only the applied load, specimen geometry and crack length, but also the material properties and the orientation of the crack relative to the principal material direction.
Crack growth direction
29
Fig. 2. Infinite center cracked plate under biaxial loading.
The problem under consideration (Fig. 2) has no restrictions as to the orientation of the crack defined by the angle a, or the principal material directions defined by the angle 8. Complete biaxial loading is allowed. The only constraint is that the crack is assumed to have a finite width. The problem defined in Fig. 2, can be solved by transforming the far field stresses to a crack tip coordinate system[5] as shown in Fig. 3, and observing that the far field stress parallel to the crack does not contribute to the singularity (this fact is discussed in[6]). The elasticity solution can then be applied to solve this general problem.
CRITERIA FOR PREDICTING DIRECTION OF CRACK GROWTH
(1) Tensor polynomial criterion Tsai and Wu[l], Tennyson et a1.[7], and others have presented a tensor polynomial as an anisotropic failure criterion. This criterion is based on the existence of a failure surface in the stress-space in the form of:
f(fli) = F,‘Ji + 4jCriC,+ 4jkBiOjOk+ . . . = 1 (i, j = 1, 2, . . . 6)
(5)
where Fi, Fi, and Fiik are strength tensors of second, fourth and sixth order respectively, and bi is the contracted form of the stress tensor. See the review by Labossiere and Neale[8] for a discussion of various tensor polynomial criteria. The application of the tensor polynomial to fracture problems, utilizes the assumption that the direction of crack extension corresponds to the radial direction of maximumf(aJ The stress components, Giymust be evaluated at a finite distance, r,,, from the crack tip.
--
T --
--
(a) Fig. 3. Transformation
T W
-7
and superposition of far field stresses.
w
30
F. ELLYIN and H. El KADI
(2) minimum strain energy density criterion
The minimum strain energy density criterion is based on variations in the energy stored along the boundary of a core region surrounding a crack. This criterion was first proposed by Sih[9] for isotropic fracture and later modified for the application to anisotropic problems[2]. Sih defined the strain energy factor, S, by the expression: W = (dU/d V) = S/r
(6)
where (dU/dV) is the strain energy density function, and r is the radial distance from the crack tip. For plane stress problems, the strain energy density function can be expressed in terms of the stresses and strains near the crack tip as:
w = tfl,c, + tT,c,. +
~,~Y,X~)P.
(7)
As a result, the strain energy density factor can be defined explicitly as: s = Y(fl,f, + fTLEy + %,Y,)/2.
(8)
The factor S was interpreted as the area under the (dU/d V) vs r curve[lO]. The basic postulates of the minimum strain energy density criterion are: (a) Failure by fracture is assumed to initiate at sites corresponding to the maximum value of the relative minimum of the strain energy density function, i.e. (dU/dV);g
at
# = &
(b) Fracture is assumed to occur when (dU/dV)~~
f9a)
reaches its respective critical value:
(dU/‘dV)E, = (dU/dV),. (c) The failure rate by fracture conditions:
is assumed to occur in accordance
(9b) with the following
(dU/dV), = S, /r, = L&/r, = . . . = S,jr, = 1* . = SC/r,)
(9c)
such that for unstable fracture: r, < r? <. 1. < rj <. . . < r,
and
(3) boreal
stress ratio criterion
Buczek and Herakovich[3] have proposed the normal stress ratio as a crack growth criterion. Using the isotropic results, where the cracks generally grow in the direction of the maximum normal stress, their model assumes that the direction of crack growth is controlled by the ratio of normal stress to tensile strength on a given plane. The normal stress ratio is defined as: R(ro, 4) = noelT+g
(10)
where (r+$= normal stress acting on the radial plane defined by # at a distance r, from the crack tip (Fig. 4), and T,, = tensile strength on the plane fp. Buczek and Herakovich postulated that the crack will grow along the plane for which this ratio is maximum. Since the strength T+,+on an arbitrary plane is very difficult to measure, they defined it as: T,& = XT sin* y + Y, co? y
where y was defined as the angle from the plane of interest to the fibre direction (Fig. 4).
(11)
31
Crack growth direction
Fig. 4. Normal stress ratio parameters (from [SJ).
The definition of T,, satisfies the following necessary conditions: (a) For isotropic materials, T,, does not depend on 4. (b) For crack growth parallel to the fibres in a composite material, T+b is equal to the transverse tensile strength, Y, . (c) For crack growth perpendicular to the fibres of a composite material, T,, is equal to the longitudinal strength, X, . (4) Critical strain energy density method-present
criterion
The critical strain energy density method is based on the assumption that crack initiation takes place when the value of the strain energy density defined by (7) reaches a critical value, i.e. (12)
w = (O,G + a&’ + %yYxy)/2= w,ri, *
The value of W will depend on the material properties and the fibres direction. The direction of crack growth will correspond to the radial direction having the maximum value of the strain energy density. Thus the direction of crack extension 4 = +,, is obtained from:
awla
=o
and
a2w/a42<0.
(13)
It should be stated that although both Sih’s method and the present criterion use strain energy density as a criterion for crack growth direction, each method utilized a unique interpretation, c.f. (9) and (13). This criterion is an extension of Ellyin’s damage function for isotropic solid@ 11. The physical basis of the proposed criterion is as follows: When a material element is loaded by an external agency, part of the supplied energy is dissipated into heat and the remaining part is stored in the material. Damage is caused by the irrecoverable part of the stored energy, i.e. (14)
d Wdamage = d Wsupplid - (dQ + d Wrecovera~e)-
Each material has certain capacity to absorb damage, and failure results as a result of damage accumulation. Because of the difficulty in measuring the heat loss, dQ, it is generally assumed that damage is proportional to the supplied energy, i.e. d Wdamage
0~
d Supplied
(13
*
In front of the crack, the strain energy varies with the distance r and angle 4 (Fig. 4). Its variation at the crack tip, r 4 a, can be calculated from (12). Crack propagation will ensure when the strain energy reaches the material critical value (absorption capacity), Wcrit. The location at which this critical value is first reached is obviously the one where the strain energy is maximum, i.e. condition (13). ANALYSIS OF OFF-AXIS UNIDIRECTIONAL
TENSILE
COUPONS
Numerical results are presented for graphite-epoxy tensile coupons containing central cracks. Lamina properties of the material considered, as well as specimen dimensions, are given in[5, 121.
F. ELLYIN and H. El KADI
32
In comparing the predictions of aforementioned criteria for the crack growth direction, the tensor polynomial criterion will be excluded. This is mainly due to the dramatic dependence of the result of this method on the distance from the crack tip, rO. Herakovich er aZ.151have shown that a small change in the value of r, can affect the predicted crack growth direction by as much as 40P. The independence of $J, on r, allows the prediction of the crack extension direction in unidirectional composites without prior knowledge of a proper r0 value. The remaining three methods are compared with the experimental results of different off-axis tests in Table 1. It is noted that the present model (critical strain energy density) predictions are very consistent with the experimental results. This is in contrast to the minimum strain energy criterion of Sih which was shown in[S] as well as in Table 1, to be inconsistent in predicting the direction of crack growth. The variation of the three crack growth criteria as a function of 4, for the various tests are shown in Figs 5-7. UNIDIRECTIONAL
LAMINAE
SUBJECTED
TO MIXED-MODE
LOADING
In an attempt to verify the consistency of the proposed critical strain energy density criterion, and to study crack growth of fibrous composites under more general loading conditions, unidirectional laminae were analysed for various mixed-mode loading. The predicted values of 4,. together with the experimental results[12, 131 are presented in Table 2. From the comparison, it is clear that the present model predicts fairly well the crack growth direction for all of the five cases studied. The largest difference between the experimental and the predicted direction of crack growth is 2”. The variation of the crack growth criteria as a function of Cp,for some of the cases presented in Table 2, are shown in Figs 8-l 1. Any criterion for the anisotropic materials has to degenerate to an isotropic one at the limit. In the case of the maximum normal stress ratio (eq. lo), we meover the maximum principle stress criterion which is more appropriate for brittle materials. The critical strain energy criterion (12) reduces to W=WD+W*
(16)
where the first term on the right-hand side ( WD) is the distortion energy and the second term is the energy involved with the volume change of the material element. Both Tsai-Hill[l4] and the Table
I. Comparison
Test data
between different criteria and experimental unidirectional tensile coupons Normal stress ratio
Minimum strain energy density
results for off-axis
Present model
Experimental results[5]
Case (1): T300/5208 8= 120” ? =O” 2a = 0.2032“ r, = 0.002” LT,.=i.Oksi
301
350.’
297
300,
Case (2): AS4/3.501-6 f3= 105” c( = o2a = 0.2032” r. = 0.002” q= l.Oksi
286
60-
284
285
Case (3): AS4/3501-6 B = 105” a = 15O 2a = 0.2032” r0 = 0.002” 0, = l.Oksi
271
320’
268”
270
15’
88”
90
Case (4): AS4/3501-6 e = 90” u. = 0’ 2a = 0.102” r0 = 0.042%” CT,= 1.0 ksi
88’
Crack growth direction
OLLV~SSI8JS?VX80N
33
(a)
(Degrees)
0
90 9
no
(c)
(Degrees)
1eo
0.000
-
0
360
Wf
120 270 up (Degrees)
SO
Fig. 7. Variation of crack growth criteria as a function of 4 for 15” .off-axis graphite/epoxy coupons (case 3). (a) Normal stress ratio (from [5, 121. (b) Mnumum strain energy density. (c) Critical strain energy density (present model).
Q
Rz E
.
2a = r, = o, = r,, =
0.2032” O.OOI” 9.38 ksi 0.378 ksi
0=75” ct= -15’
Case (9): AS4/3501- 6
Case (8): AS4/3501L6 0=:75”a=-t5” 2a = 0.2032” To= 0.001” u,,= ll.Oksi rXy= 1.18 ksi
Case (7): AS4/3501-- 6 0 = 15” a = 0 2a = 0.2032” r0 = 0.001” ov= ll.Oksi t,). = 1.18 ksi
Case (6): AS4/3501-6 $=7.5” a = -15” 2a = 0.2032” r, = 0.001” g,*= 15.9 ksi rzF = 3.53 ksi
Case (5): AS4/3501L6 6=75” a=O” 2a = 0.2032” ro = 0.001” (TV = 15.9 ksi try = 3.53 ksi
Test data
Table 2. Comparison
88
88’
13”
88”
72
Normal stress ratio
320
330”
315’
20’
335”
Minimum strain energy density
91
90
74’
89
73”
Present model
90
90
75’
90”
15”
Experimental resuitslr)]
between different criteria and experimental results for unidirectional laminae subjected to mixed-model loading
Crack growth direction
35
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” H
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a3ZI7VHlIoN
9
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?
9
0
0
0
0
---------_ [ z d
2lOCX’d ALISNBa b383N3 NIVKLS
52
E
kLISN3a XQl3N3
‘_
a
G
9
0
0
0
6
4
OIJW ss3zLs EFM 36,I-c
9 n
9
(Y
0
s
a3ZrIVmoN
0
d
NIVEJS
36
F. ELLYIN and H. El KADI
ZIOKWil LLISN3a A383N3 NIVKLS
OILVZI SSB?uS a3ZIIvHl?ON
2
: d
d
2 d
2
d
:: d
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Crack growth direction
37
term of the tensor polynomial (5) will reduce to WD for an isotropic material. It is seen therefore that the critical strain energy density criterion is more general than that of the maximum normal stress ratio. The extension of the critical strain energy density criterion to laminates would require further work since adjacent layers can affect the crack growth direction.
first
CONCLUSIONS A criterion based on the critical strain energy density has been presented to predict the crack growth direction in a Iamina. The comparison of the experimental results with the different criteria for the crack growth direction indicates that the proposed critical strain energy density method can be used to analyse the problem of a through-the-thickness central notch in a unidirectional composite. The present approach leads to more consistent results than the minimum strain energy density of Sib. The advantage of the energy approach is in its universality by being a frame indeferent quantity. It is more general than the normal stress ratio criterion. The latter is based on the maximization of the normal stress ratio (eq. lo), which gives results comparable to our criterion. However, the validity of eq. (11) for anisotropic materials has to be demonstrated. Acknowlegemenrs-The work presented here is part of a general investigation into materials behaviour under complex loading and adverse environment. The research is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC Grant No. A-3808).
REFERENCES S. W. Tasi and E. M. Wu, A general theory of strength for anisotropic materials. J.
Compos. Muter. 5, 58-80 (1971).
::; G. C. Sih, E. P. Chen, S. L. Huang and E. J. McQuillen, Material characterization on the fracture of filament-reinforced composites. J. Compos. Mater. 6, 167-186 (1975). [31 M. B. Buczek and C. T. Herakovich. Direction of crack arowth in fibrous comnosites. Mech. Comoos. Mater. 58.75-82 (1983). [41 S. G. Lckhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, translated by P. Fern (Edited by J. J. Brandstatter). Holden-Day, Inc., San Francisco (1963). PI M. A. Gregory and C. T. Herakovich, Predicting crack growth direction in unidir~tional composites. J. Compos. Mater. 20, 67-85 (1986).
PI E. M. Wu, Fracture mechanics of anisotropic plates, in Composite Material Workshop (Edited by S. W. Tsai, J. C. Halpin and N. J. Pagano), pp. 20-43, Technomic publishing Co., fnc, Lancaster, Pennsylvania (1968). t71 R. C. Tennyson, D. MacDonald and A. P. Nanyaro, Evaluation of the tensor polynomial failure criterion for composite materials. J. Compos. Mater. 12, 63-75 (1978). PI P. Labossiere and K. W. Neale, Macroscopic failure criteria for fibre-reinforced composite materials. SM Arch. 12, 439-450 (1987). [91 G. C. Sib; A special theory of crack propagation, in Method of Analysis and Solutions to Crack Problems (Edited by G. C. Sib). DV. XXI-XLV. Wolters-Noordhoff. The Netherlands (19721. PO1 G. C. Sib, Dynamics of cor&osites with cracks, in ~an~ook of Co~~s~~es, Vof. 3, Failure Mechanisms ofcomposifes (Edited by G. C. Sib and A. M. Skudra), pp. 127-176, Nosh-Holland, The Netherl~ds (1985). [I 11 F. Ellyin and K. Gotos, Multiaxial fatigue damage. J. Engng Mater. Technol. 110, 63-68 (1988). 1121M. A. Gregory, J. L. Bcuth, Jr, A. Barbe and C. T. Herakovich, Application of the normal stress ratio theory for predicting crack growth direction in unidirectional composites. Fracture Fibrous Compos. 74, 33-42 (1985). [I31 J. L. Beuth, Jr, M. A. Gregory and C. T. Herakovich, Crack growth in unidirectional graphite-epoxy under biaxial loading. Exper. Mech. 26, 245-253 (1986). [‘41 R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). I.
.
.
(Received 10 January 1989)