Analysis of composite laminates with transverse cracks

Composite Structures 34 (1996) 419-426 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263.8223/96/$15.00 PII:SO263-8223(96)00010-4

ELSEVIER

Analysis of composite laminates with transverse cracks T. E. Tay & E. H. Lim Department of Mechanical and Production Engineering, National University of Singapore, IO Kent Ridge Crescent, Singapore 119260

The analysis of the behaviour of composite laminates with general balanced lay-up sequences and containing distributed transverse cracks is discussed and demonstrated in this paper. It is shown that a constitutive model of damage for composites, which has been successfully used to analyse the effects of uniformly distributed transverse cracks in cross-ply laminates, can be readily extended to the analysis of uniformly distributed transverse cracks in more general symmetric laminates. The further extension of the theory to non-uniformly distributed damage can be achieved by finite element analysis. It is shown that the finite element implementation of the theory is particularly simple since the contribution of the damage to the overall behaviour can be incorporated into the force vectors. Copyright 0 _ 1996 Elsevier Science Ltd.

0” layers. It may be argued that such laminates are of little practical interest, since most laminates used in real structures are quasi-isotopic and generally it is poor design practice to have more than two plies with the same orientation stacked adjacent to each other in a single laminate. Nevertheless, to formulate a truly general failure or damage theory for composite laminates is probably an over-ambitious goal, at least for the present. A more fruitful approach may be to start with a theory or formulation which has been successfully tested and applied to relatively simple cases (such as cross-ply laminates), but has the scope or potential for incremental extension to more complicated cases (such as quasi-isotropic laminates or laminates with holes and cut-outs). The results of using such a theory (or theories) must be carefully compared with as much available experimental or measured evidence as possible. The latter is usually obtained from non-destructive techniques such as ultrasonic scanning or dye-penetration. The monitoring and description of damage in laminated composites by such techniques is by no means a simple task, and is probably a reason for the lack of published data for many practical composite structures. It is nevertheless an essential aspect to damage char-

INTRODUCTION The characterization and analysis of damage progression in practical laminated composites are often complicated due to the multiplicity of failure modes. Characterization is usually possible only in the simplest of cases, where each failure mode can be separated and studied individually by carefully designing and testing the composite specimens. An example of this is in the characterization of delamination toughness of laminated composites.’ As a result of restrictive assumptions made in the loading conditions and idealization of specimen geometry, the laboratory test results are useful only as comparisons between material systems, and have little value for application in analysis of delamination in real composite structures. The same may be said of many studies of transverse matrix cracking in composite laminates.2,” These invariably use cross-ply laminates for their experimental investigations and it is not clear how the results can be extended to other classes of laminates. Most of the developed analytical tools, such as the shear-lag theory, should strictly be used for cross-ply laminates only. Moreover, in order to enhance matrix cracking, many of the cross-ply laminates tested consist of three of more 90” layers sandwiched in between 419

420

7: E. Tay, E. H. Lim

acterization and serves to validate the proposed theory of damage. As more applications of the theory are made in the analysis of damage in practical composite structures, a better understanding of the damage mechanisms, as well as refinements of the theory, will hopefully result. The theory of continuum damage mechanics for composite materials has potential for useful application in the analysis of composite structures because it is sufficiently general. First proposed by Kachanov,4 and later applied to fibre-reinforced composites by Allen et al.” and Talreja,’ it however suffers from the fact that many of the material constants required in the theory are difficult to determine experimentally. Only in special cases of cross-ply composite laminates can the relevant constants be easily determined. However, a variation of the theory incorporates the kinematics of matrix cracks into the formulation5y” and obviates the need to perform extensive experiments to determine the constants. It can be shown instead that the state of damage is adequately described by internal state variables obtainable through a series of finite element parametric studies.6,7 In this paper, a constitutive theory of damage for fibre-reinforced composite laminates, which has been successfully employed to analyse the state and evolution of transverse cracking in cross-ply laminates, is extended for the analysis of laminates with more general ply lay-ups. Furthermore, it is shown that the theory, whose application has mainly been limited to analysis of uniformly distributed damage, can be used analyse problems involving localized to damage through an appropriate finite element formulation.

continuum damage theory to describe a phenomenon that involves sub-laminate local buckling and is highly dependent on structural geometry is entirely appropriate. Mathematically, the general stress-strain relations for a damaged composite material can be written as aii=CijklEkl+I~~lCI;II

(1)

where aii are the applied stresses, C,, are the stiffnesses of the undamaged material, &k[are the strains, I& are elements of the damage matrix, cc;lrare the internal state variables, and &,2,3 ,..., refers to the damage modes. In the above equation, moisture and temperature effects have been excluded. It can be shown that for matrix cracking,5 to a good approximation I$, % - Cijk/

(2)

This assumption considerably simplifies the theory, leaving the complete description of the damage state to the internal state variables in the second term of eqn (1). The constitutive relations can be used in classical lamination theory, in which case, the vector of resultant forces {N} can then be expressed in terms of the vector of mid-plane strains {co>, the inplane stiffness matrix of the laminate [A], and the damage vector {D}, as shown in eqn (3). In this equation, the bending and bending-stretching coupling terms have been excluded. {N}=[A]W)-_(D) The quantities defined as [A]= ;

in

(3) the

above

[@ktk

equation

are

(4)

k=l

STRESS-STRAIN

RELATIONS

The continuum damage mechanics model used in this work is based on that proposed originally by Kachanov4 and later used for composite laminates by Allen et ak5 It employs a set of second-order tensorial quantities called the internal state variables (ISVs), which are strainlike variables containing kinematic features of the distributed transverse cracks or damage. Allen8 has also formulated the ISVs corredelamination in laminated sponding to composites; although delamination can certainly be regarded as a form of damage, the present authors are as yet unconvinced that the use of

t/2

{&*}=

I

{~x+pg.JTd~

--t/2

{D)= ;

[g]k{dktk

(6) (7)

k=l

where the subscripts x and y refer to the global system of coordinates, and the subscripts 1 and 2 to the material longitudinal and transverse directions, respectively. [elk and {a}k are the transformed stiffness matrix and vector of ISVs, respectively, in global coordinates. The usual

421

Composite laminate with transverse cracks

rules of matrix transformation

, Transverse

apply, i.e.

cracks

(8)

Kh=Vlkl [QLP-l/c and G&=[T]k{&

(9)

where [T], is the transformation sponding to each kth ply. The vector of ISVs in material given by

matrix corre-

I

I

I

coordinates

Reprffentative

is

Volume

M {al,=

1

Fig. 1.

Kw&x-

(10)

Transverse cracks in 90” ply.

if=1

where M is the total number of damage types being considered. Since only matrix transverse cracking is considered in this case, it is understood that q=M=l, and for brevity, the superscript q is dropped. By definition,5,6 the ISV for mode I (opening) transverse cracking is

ct22=;

1 ~22~122

(11)

ds

ss

and for mode II (sliding) transverse

cracking is

1 a 12=-

u12n12 0

ds

(12)

ss

In the equations above, u22 is the crack opening displacement, n22 the unit normal to the crack surface, u 12 the crack sliding displacement, 1212 the unit vector in the plane of the crack in the direction of shear, and ZJ the representative cracked volume. Note that all the components of the ISVs so defined are obtainable from geometrical considerations of the crack profile, density and loading conditions, assuming uniform distribution of damage throughout the volume being considered.

DETERMINATION OF THE INTERNAL STATE VARIABLES The ISVs can be very conveniently obtained through simple finite element parametric studies. The idea is to model the repetitive representative cracked volume as shown in Fig. 1. In the figure, the 90” ply is sandwiched between two 0” plies, and all damage in the form of transverse matrix cracking is assumed to be contained within the 90” ply only. Another

assumption is that the damage is evenly distributed throughout the laminate. The size and shape of the representative volume depends on the relative thicknesses of the different plies and the crack density (i.e. number of cracks per unit volume). Although the case illustrated in Fig. 1 is a very special example of cross-ply laminates, more general cases, such as that of alternating 0” and 90” plies, and plies with orientations other than 0” and 90”, can be analysed based on the results of the simplest case, as will be shown later. Thus, what is required is a series of finite element parametric runs only of the simple case of Fig. 1 by varying the thickness ratio and crack density, which can be easily done. Details of such an analysis can be found in a previous paper6 and will not be repeated here. Essentially, the results generated two sets of master curves, one relating the variation of the non-dimensionalized maximum crack opening displacement 6 with the non-dimensionalized crack density p, and the other relating the variation of 6 with 8 the ratio of the thicknesses of the 0” and 90” plies. Here, p is defined as the crack density 5 multiplied by the thickness of the laminate, and 6 is defined as the maximum crack opening displacement divided by the thickness of the cracked ply. Examples of such curves are given in Figs 2 and 3. The value of such curves is that they can be conveniently used in computer codes as part of the material databank. It was noted6 that the form of the curves are dependent on the composite material systems. Furthermore, the crack opening and sliding displacement profiles are needed in the evaluation of the ISVs according to eqns (11) and (12). For mode I, when a fourth-order polynomial fit of the crack opening profile was attempted,6 the resulting expression for the ISV was found to be

422

7: E. Thy, E. H. Lim Y ,

0.5

0

1

1.5

2

2.5

I

3

3.5

P

Fig. 2.

Normalized

Fig. 4.

maximum COD, crack density, p.

6 vs normalized

0.25,

1

005-

0

1

2

3

15

6

7

8

-

Gl/E,

t-

G,lEp

9

to

11

8

Fig. 3.

Normalized

y-$&p-J \

maximum COD, 6 vs thickness 0.

a22 =pt

displace-

presence of transverse cracks in the 4” ply, where the local l-2 material directions are oriented at an angle 4” to the global x-y directions. If the laminate is loaded in tension in the x-direction, the resultant displacement vector of a point on the crack surface in the direction of load {u,} can be resolved into two components, {u12} and {Use}, parallel and perpendicular to the fibre direction, respectively. The opening mode ISV can then be defined as

ratio, ct -

8

Transverse crack opening and sliding ments in an off-axis ply.

84 l

(14)

22- 5 sin d,

(13)

where r is the crack density and U, is the maximum crack opening displacement. Assuming the material properties remain elastic throughout the loading history, the maximum crack opening displacement is directly proportional to the applied load. Similar parametric finite element studies can be performed for [90;/0;],-type laminates instead of [0;/90”,],-type laminates. One would expect some difference in the results, since the cracks in the 90” ply in the former class of laminates is less constrained from opening than in the latter case. This is indeed the case, with the maximum crack opening displacement for the [90;/00,], laminate about 1.107 times greater than that for the [0;/90”,], laminate. For constrained angle-ply laminates of the types LOi@iIs and [+i/Oi]s, the sliding mode must be taken into account in addition to the opening mode. Here, as in the cross-ply laminate case, damage is assumed to occur entirely in the constrained ply. In general, this is a more complicated problem; however, in the case of uniaxial tensile loading it turns out that the opening and sliding displacements are related through specimen geometry. Figure 4 shows the

where now u,, is the maximum crack opening displacement in the 2-direction. From the figure, we see that u,2=u22

cot

(b

so that

84 t a12=

5 tan 4 sin 4

(16)

As in the cross-ply case, if the constraining 0” plies are the inner plies, the resulting value of u, is assumed to be greater by a factor of 1.107. It is noted that eqns (14)-(16) are applicable only for the case of uniaxial tension, since u, is zero for compression (i.e. the crack does not open). However, the sliding displacement of the crack faces is still non-zero. For the compression case, we let U, assume a negative value, but clearly it is no longer interpreted as the maximum crack opening displacement. Hence eqn (16) still applies, but SI22--0

(17)

The foregoing may be extended to biaxial loading conditions, assuming that crack face displacements are proportional to the applied

423

Composite laminate with transverse cracks

loads, and that the material remains elastic throughout the loading history, the only nonlinear effects being due to the accumulation and propagation of damage. If the constraining plies are not oriented at O”, we may compensate for the reduced constraining effects. The curve for graphite-epoxy shown in Figs 2 and 3 can be fitted by equations of the form 6= -4.51 x 10-2~1.4870_0.1652~-6% x lO~‘O 0.3029

+

0.96’

0

* 02

’

0.1

0.6

0.8 Crack

Fig. 5.

1

12

Densely

11

1.6

\l/mm)

loss E/E,, vs crack density laminate.

Stiffness

I 2

16

for [O/90],Y

(18)

a= -8.&j ,/ 10-2e-‘.95~+0.21e-1).74p +2.2x

1o-2

.

(19)

-

E/E0 IExpl-Ad

91

E/E, lPnrent1

Instead of 0 in eqn (18), one may use Qg,which can be defined as

%&,=+(%, +0,)

(20)

L

090’

0

I

0.2

0.1

where

0.6

0.6

Crack

Fig. 6.

Here, the damaged ply being considered has a thickness t, and is oriented at an angle tic from the global x-direction. It is sandwiched in between two plies, of orientations $, and &,, and thicknesses t, and t,, respectively.

Stiffness

for [O/90],T

J 0

Fig. 7.

A computer code based on the classical lamination theory and the preceding equations was written to compute the stiffness loss of various composite laminates, and where possible, comparison with experimental data from the work of other researchers were made. In Figs 5-7, experimental results from Groves et d9 are shown together with predictions of the stiffness loss using the present theory. The results are for [0”/90”],, [0”/900,], and [0”/90”,], AS4-3502 laminates, respectively. Although Groves et al. reported that some curved transverse cracks were observed, the prediction using the present theory where only straight transverse cracks are

1.2 (l/mm)

loss E/E, vs crack density laminate.

0.1

0.2

0.3 Crack

COMPARISON WITH EXPERIMENTAL DATA

1

Oensity

Stiffness

0.1 Density

0.5

0.6

0.7

0.6

[l/mm)

loss E/E,, vs crack density laminate.

for [O/90],T

assumed seems to be fairly accurate, as shown in the figures. Leong and King’” reported some stiffness loss data for [0”/90”],, and [90”/0”],, graphite-epoxy laminates, which are presented in Figs 8 and 9. The predictions are quite good, although at higher loads, some delamination was observed. Two examples of stiffness loss in laminates other than cross-plies are extracted from Refs 11 and 12 and are shown in Figs 10 and 11, respectively. In the case of a quasiisotropic laminate” of lay-up [0”/90”/ + 45”],, the model over-predicted the stiffness loss initially (N < 3000 cycles) but under-predicted the results later (for N>7000 cycles). The next

424

7: E. Tay, E. H. Lim

.

0.96

-

0.95 0.911 5

E/E, IExpt-Ret 10) E/E, IPresent

6.5

6

Applied

Fig. 8.

I 8.5

1

55

7

7.5

8

0.97

A

EIE,IExpt-Ret E/E,iPresentl

1000

2000

-

0975 0

111

3000

toad (kN)

Stiffness loss E/E,, vs applied load for [O/9O],Tlaminate.

1.02

Fig. 10.

LOO0

5000

Fatique

Cycles

6000

.

- - - - G,,IG,,, (Present1 -.vxr /v”y(llPrrsentl . I,/E,,(FE-Rel 121

0.96 0.91 0 90

3

. 35

E/E,lExpt-Ret 101 E/E, (Prrsenll 1

1.5

c . 5 Applied

Fig. 9.

9000

Stiffness loss E/E, vs number of fatigue cycles for [O/90/&45], laminate.

090 -

0.92 -

woo

1

1 -

5

7000

55 toad

6

h5

7

75

8

“, 2

. p OfJ05

*

0

1

Stiffness loss E/E,, vs applied load for [90/O],vlaminate.

comparison is with the finite element results for [55”/ - 55”15, graphite-epoxy laminates obtained by Zang and Gudmundson.‘2 It should be noted that there is no independent experimental confirmation of the finite element results in this case. The finite element method enables the effective longitudinal, transverse and shear moduli as well as the effective Poisson’s ratio to be computed. These are presented in Fig. 11 where the corresponding predictions of the current model show that agreement is generally good for low crack densities. The results for very high crack densities are usually not of much interest since laminates rarely sustain crack densities greater than 3.

DAMAGE IN

The foregoing discussion and development of the model have been limited so far to analysis of uniformly distributed transverse cracks in composite test coupons. In order for the model to be really useful in analysis of composite structures, the theory must be extended to cases

3

Fig. 11.

Stiffness

1

5

Density Wmm)

Crack

loss E/E, vs crack [55/-55],,, laminate.

density

for

of non-uniformly distributed or localized transverse cracking. The most convenient way of achieving this is through the use of the finite element method, which is now an established method of analysing local stress concentrations in structures. The present theory lends itself readily to the formulation of the standard plate and shell finite elements. We begin by stating the plate equilibrium equations’” aN, -+-=

ax

aN, -+

ax

ANALYSIS OF LOCALIZED COMPOSITE STRUCTURES

2

IkN)

a24, a2

-+2-

aN,

0

(23)

ay

aN,

(24)

-=()

ay a2bfq

a%f, + -= axay ay2

0

(25)

where N,, NY, Nxy are the force resultants, and M,, My, Mxy are the moment resultants. Body forces have been neglected in eqns (23)-(25). If eqns (3) and (7) are substituted into the above, we obtain

Composite laminate with transverse cracks

ae,o

agy

a&;

A,, -+A12

ax

a&;

-f&j

ax

-+A,,

-

a&;

w?y

ax

aY +

(26)

+A26- fA66 -=d, aY aY aq ac: &j--+&j---+A~~---++A,~-ax ax

a%; D,, -+DIz ax2

(~lhC(rx+~26~jy+~hhClx)i)k

ag

a?; ax

a6; +A22 +A aY aSo

(27)

a%0 + 2066 2+D,, ax aY

a%; ax aY a2ti;

-

(32)

@~={d.,d,dJ

(33)

which can be regarded as a body force-like vector, but not to be confused with the damage vector of eqn (7). The form of these equations is particularly suited to the displacement-based finite element method. The explicit formulation of the finite elements, taking into account the internal state variables and the demonstration of its use in analysis will form the subject of a forthcoming paper.

a2

a%; -+2D,,axaY

(31)

Clearly, the effects of the damage on the constitutive relations are embedded in the vector

a%;

-

ax2

+24,

{dk={&,oc,,x,>k

aY

a?; -=dy aY

l+D,,

425

aY’

CONCLUSION a%; +D22

-

+D26

aY’

a%; -=d, 3Y’

assuming the [B] matrix is zero, and where

dy=;i

tk
16’%x+~26’;(ly+~66%_y)k

k-l

N

+A1 aY

k=l

l,(e,,~~+e22~+Q26~~)k

(28)

Stiffness loss prediction of cross-ply and angleply composite laminates with transverse cracks using a simple damage model has been presented. The model utilizes a set of strain-like internal state variables to characterize the damage state. The internal state variables are related to crack geometry, loading conditions and crack face displacements. The information can be obtained easily from a series of finite element parametric runs. The model developed has been shown to work well with both cross-ply and quasi-isotropic laminates, when compared with experimental evidence from other researchers. Furthermore, it is shown that the theory can be incorporated into the finite element method, since the contribution of the damage to the structural behaviour can be embedded in an equivalent damage force-like vector. This is significant as it enables the use of this theory in analysis of localized damage in composite structures. REFERENCES 1. Tay, T. E., Williams, J. F. & Jones, R., Characterization of pure and mixed mode fracture in composite laminates. Theor: & Appl. Fruct. Mech., 7 (1987) 115-23. 2. Nairn, J. A., The strain energy release rate of composite microcracking: a variatibnal approach. J. Co&p. Mat., 23 (1989) 1106-29.

426

7: E. Tay, E. H. Lim

3. Talreja, R., A continuum mechanics characterization of damage in composite materials. Proc. R. Sot. Land., A 399 (1985) 195-216. 4. Kachanov, M., Continuum theory of media with cracks. Mekhanika Tverdogo Tela, 7 (1972) 54-9 (in Russian). 5. Allen, D. H., Harris, C. E. & Groves, S. E., A thermomechanical constitutive theory for elastic composites with distributed damage theoretical development. ht. J. Solids Strut., 23 (1987) 1301-18. 6. Tay, T. E. & Lim, E. H., Analysis of stiffness loss in cross-ply composite laminates. Comp. Strut., 25 (I 993) 419-25. 7. Tay, T. E. & Lim, E. H., Analysis of stresses in crossply composite laminates containing distributed transverse cracks. Finite Elements in Analysis and Design, 18 (1994) 301-8. 8. Allen, D. H., Damage evolution in laminates. In Damage Mechanics of Composite Materials. ed. R. Talreja, Composite Materials Series Vol. 9. Elsevier

Science, The Netherlands, 1994, pp. 79-116. 9. Groves, S. E., Harris, C. E., Highsmith, A. L., Allen, D. H. & Norvell, R. G., An experimental and analytical treatment of matrix cracking in cross-ply laminates. Experimental Mech., 27 (1988) 73-9. 10. Leong, K. H. & King, J. E., An investigation of damage accumulation in cross-ply glass-epoxy laminates. In Proc. FEFGJICF Int. Con& on Frac. Engng. MateK Struct., Elsevier Science, 1991, pp. 251-6. 11. Jamaison, R. D., Schulte, K., Reifsnider, K. L. & Stinchcomb, W. W., Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite-epoxy laminates. Effects of Defects in Composite Materials, ASTM STP 836, (1984) 21-55. 12. Zang, W. & Gudmundson, P., Damage evolution and thermoelastic properties of composite laminates. Int. J. Damage Mech., 2 (1993) 290-308. 13. Ochoa, 0. 0. & Reddy, J. N., Finite Element Analysis of Composite Laminates. Kluwer Academic, The Netherlands, 1992.