Analisis Del Esfuerzo en Compuestos Unidireccionales Laminados

1.20

Strength Analysis ofUnidirectional Compositesand LaminatesC. T. SUN

Purdue University, West Lafayette, IN, USA

1.20.1 INTRODUCTION 2

1.20.2 LAMINA FAILURE ANALYSIS 2

1.20.2.1 Lamina Strength Criteria 21.20.2.2 Comparison Among Lamina Strength Criteria 3

1.20.2.2.1 Bidirectional stress plane 31.20.2.2.2 Off-axis loading 51.20.2.2.3 Pure shear 7

1.20.2.3 Comparison with Experimental Data 81.20.2.3.1 Lamina strength criteria comparison with off-axis tension data 91.20.2.3.2 Lamina strength criteria comparison with tubular specimens 91.20.2.3.3 Discussion 101.20.2.3.4 Concluding remarks 13

1.20.3 LAMINATE STRENGTH ANALYSIS 15

1.20.3.1 Stiffness Reduction 151.20.3.1.1 Parallel spring model 151.20.3.1.2 Incremental stiffness reduction model 16

1.20.3.2 Laminate Failure Analysis Methods 161.20.3.2.1 Ply-by-ply discount method 161.20.3.2.2 Sudden failure method 17

1.20.3.3 Laminate Failure Analysis for Biaxial Loading 171.20.3.3.1 Comparison with data for biaxial loading 171.20.3.3.2 Biaxial failure in the strain plane 181.20.3.3.3 Biaxial testing data for glass woven fabric composite 19

1.20.3.4 Laminate Strength Analysis for Unidirectional Loading 191.20.3.4.1 Selection of laminates and off-axis loading angles 191.20.3.4.2 Laminate coupon specimens and oblique end tabs 201.20.3.4.3 Results and data analysis 201.20.3.4.4 Comparison with test data 21

1.20.3.5 Discussion 23

1.20.4 CONCLUSIONS 25

1.20.5 REFERENCES 26

1

1.20.1 INTRODUCTION

Prediction of the load-carrying capability ofa structure is one of the most important tasks instructural design. In the last four decades, thesubject of strength (failure) criteria for compo-site materials has attracted numerous research-ers who have developed and are continuouslydeveloping criteria for predicting the strengthof composites and their laminates (Nahas,1986; Labossiere and Neale, 1987). Undeni-ably, progress has been made. However, untilnow it does not appear that there is any criter-ion that is universally accepted by compositesresearchers as adequate under general loadingconditions. In fact, there is a growing consensusthat the strength analysis of composite struc-tures is far from being mature. Evidence of thisis the publication of a recent special edition ofComposites Science and Technology, 1998, 58(7)which was entirely dedicated to the subject offailure theories for continuous fiber-reinforcedcomposites.

Although none have been rigorously verifiedexperimentally, many strength criteria for com-posites have been formally introduced in almostall existing textbooks on composite materials.Often, without giving the reader proper warn-ing about the inadequacy of these strengththeories, these textbooks have helped createthe impression that accurate strength analysistools are available. For this reason, there seemsto be a need to evaluate the accuracy of thesestrength criteria that are presented in text-books. Indeed, a recent report by Sun et al.(1996) attempted to perform some comparativeevaluations of the conventionally adopted fail-ure analysis methods.

In this chapter, six representative and mostpopular strength criteria are chosen for study.These criteria are commonly referred to asMaximum Stress, Maximum Strain, Hill±Tsai,Tsai±Wu, Hashin±Rotem, and Hashin criteria.All are based on the assumption that fibercomposites are orthotropic continua, andmacro (averaged) stresses are used in the cri-teria. These criteria are grouped into noninter-active, fully interactive, and partially interactivetheories. These criteria are basically phenom-enological in which detailed failure processesare not described. Moreover, they are all basedon linear elastic analysis.

In this study, only in-plane failure of fibercomposites is considered. Thus, the strengthcriteria are expressed in the state of plane stress.The analysis of laminate strength is performedusing the classical laminated plate theory inconjunction with a stiffness reduction proce-dure. Such a procedure is recognized to beinadequate in laminates where interlaminar

stresses control the failure mechanisms. Freeedge delamination and failure in certain typesof laminate belong to this category.

The purpose of this study is to evaluate andcompare these strength criteria based on avail-able experimental data. Such an exercise maybe useful for revealing the characteristics ofeach criterion considered and providing someuseful guides in using these criteria in practicalapplications.

1.20.2 LAMINA FAILURE ANALYSIS

The purpose of a lamina strength criterion isto determine the strength and mode of failure ofa unidirectional composite or lamina in a stateof combined stress. Many lamina strength cri-teria were developed for three-dimensionalorthotropic materials. In this study, only thereduced two-dimensional criteria are included.In these criteria, the in-plane principal strengthsof the composite are needed.X & X': tensile and compressive strengths infiber direction;Y & Y: tensile and compressive strengths intransverse direction;S: in-plane shear strength.For a strain-based analysis, the correspondingultimate strains are denoted by Xe,Xe',Ye,Ye',and Se, respectively.

The ability of a lamina failure criterion todetermine mode of failure is essential in bring-ing this analysis tool to the laminate level. Ingeneral, the following modes of failure areidentified.Fiber breakage (mode 1): longitudinal stress(s11) or longitudinal strain (e11) dominateslamina failure.Transverse matrix cracking (mode 2): transversestress (s22) or transverse strain (e22) dominateslamina failure.Shear matrix cracking (mode 3): shear stress(t12) or shear strain (g12) dominates lamina fail-ure.It is noted that both mode 2 and mode 3 arematrix failures. The two modes are sometimesseparated because they are caused by differentstress components.

1.20.2.1 Lamina Strength Criteria

Lamina failure criteria can be categorizedinto three groups.

(i) Limit criteria: these criteria predict thefailure load for each mode of failure by

Strength Analysis of Unidirectional Composites and Laminates2

comparing lamina stresses s11, s22 and t12, (orstrains e11, e22, and g12) with correspondingstrengths separately. Interaction among thestresses (or strains) is not considered.(ii) Interactive criteria: these criteria assume

that all the stress components simultaneouslycontribute to the failure of the composite. Theyare usually expressed in the form of a singleequation.(iii) Separate mode criteria: these criteria

separate into a criterion for matrix failuremode and a criterion for fiber failure mode.The criterion for each failure mode may involvestress interactions.

The six lamina strength criteria are given asfollows.

(i) Limit criteriaMaximum stress:

s11

X � 1 fiber failure

s22

Y � 1 transverse matrix cracking

t12S � 1 shear matrix cracking

�1�

Maximum strain:

e11Xe� 1 fiber failure

e22Ye� 1 transverse matrix cracking

g12Se� 1 shear matrix cracking

�2�

(ii) Interactive criteria Hill±Tsai (Azzi andTsai, 1965; Tsai, 1965):

s11

X

� �2� s22

Y

� �2ÿ s11

X

� � s22

X

� �� t12

S

� �2 � 1

�3�

Tsai±Wu (Tsai and Wu, 1971):

F1s11 � F2s22 � F11s112 � F22s22

2

� 2F12s11s22 � F66t122 � 1�4�

where

F1 � 1

X� 1

X0;F2 � 1

Y� 1

Y0;

F11 � ÿ1XX0

;F22 � ÿ1YY0

F66 � 1

S2

F12 � experimentally determined

(iii) Separate mode criteria Hashin±Rotem(Hashin and Rotem, 1973):

s11

X � 1 fiber failure

�s22

Y �2 � �t12S �2 � 1 matrix failure

�5�

Hashin (1980):

�s11

X �2 � �t12S �2 � 1 fiber failure (tension)

s11

X0 � 1 fiber failure (compression)

�s22

Y �2 � �t12x �2 � 1 matrix failure

�6�

For Maximum Stress, Maximum Strain,Hill±Tsai, and Hashin±Rotem criteria, thelongitudinal and transverse strengths must bechosen based on the sign of the applied stress.The Tsai±Wu criterion is designed for use in allstress quadrants of the stress plane. The Tsai±Wu criterion requires a biaxial test to experi-mentally determine the interaction term F12. Ithas been suggested to use F12=1/(2XX'). Nar-ayanaswami and Adelman (1977) found thisterm to be insignificant for the most part, andsuggested setting it equal to zero. Cui et al.(1992) also found that F12=0 was within therange satisfying current engineering require-ments. Thus, to avoid ambiguity, F12 is setequal to zero in the present study.

The Hashin criterion listed here is a slightmodification of the two-dimensional criterionpresented in his 1980 paper. In that paper,Hashin suggested using a combination of bothaxial and transverse shear strengths SA and ST

for the compressive matrix failure mode whens225 0. Since it is difficult to find transverseshear strength values in the literature, the ten-sile matrix failure equation given above is usedas the compressive equation by simply repla-cing Y with Y'.

1.20.2.2 Comparison Among Lamina StrengthCriteria

1.20.2.2.1 Bidirectional stress plane

A series of failure envelopes for combinedstresses is presented to graphically show thecharacteristics of the six selected laminastrength criteria. These envelopes are composedof failure stresses normalized by the lamina'srespective tensile strengths X and Y, or shearstrength S. For these graphs, the AS4/3501-6graphite/epoxy system tested by Sun and Zhou(1988) is used (see Table 1 for elastic andstrength constants).

Figure 1 is a plot of the selected criteria in as117s22 stress plane (t12=0). The MaximumStress envelope is a simple rectangle boundedby the failure loads +s11 and +s22. For theanalysis using the Maximum Strain criterion,failure strains are calculated from the strengthsusing the linear relationship:

Lamina Failure Analysis 3

Xe � X

E1;X0e �

X0

E1;Ye � Y

E2

Y0e �Y0

E2;Se � S

G12

�7�

The Maximum Strain envelope is close to thatof the Maximum Stress but is slightly skeweddue to the effect of Poisson's ratio. There isconsiderably more skewing in the vertical (s22)direction because v12> v21 in unidirectionalfiber composites.

Both Hill±Tsai and Tsai±Wu criteria allowquadratic stress interactions; therefore, eachhas a curved failure envelope. Both criteriamatch up with the two limit criteria for allfour unidirectional loading cases (+s11 withs22=0 and +s22 with s11=0) as expected.The Tsai±Wu criterion is a continuous curvethroughout all four quadrants. The Tsai±Wucriterion includes linear stress terms while theHill±Tsai criterion, on the other hand, is apurely quadratic criterion. In order to accountfor differences in tensile and compressivestrengths commonly found in fiber composites,this criterion uses the appropriate strength va-lues in each quadrant (X or X' and Y or Y'accordingly). Though both are interactive,

Tsai±Wu and Hill±Tsai produce different fail-ure envelopes in the stress plane. In the com-pressive s22 quadrants, the Tsai±Wu failureenvelope extends beyond the longitudinalstrengths X and X'.

Finally, both the Hashin and Hashin±Rotemcriteria reduce to the Maximum Stress criterionin the s117s22 plane since t12=0.

A plot of the selected criteria in a s117t12stress plane (s22=0) is shown in Figure 2. TheMaximum Stress envelope in this stress plane isagain a rectangle, bounded by the failure loads+s11 normalized by X and+t12 normalized byS. The Maximum Strain criterion predictsexactly the same loads as the Maximum Stress

Table 1 Moduli and strength values of the AS4/3501-6 graphite±epoxy system.

E1 138.90GPa X 2206.0MPaE2 9.86GPa X' 72013.0MPaG12 5.24GPa Y 56.5MPav12 0.30 Y' 7206.8MPaPly thickness 0.132mm S 110.3MPa

Source: Sun and Zhou (1988).

Figure 1 Comparison of lamina failure criteria under s117s22 biaxial stress.


criterion. The Hashin±Rotem criterion alsoreduces to the Maximum Stress criterion inthis stress plane. Again, Hill±Tsai and Tsai±Wu failure envelopes intersect the other threecriteria for the four unidirectional loading cases(+s11 with t12=0 and +t12 with s11=0). Inthe biaxial loading regions, the two interactivecriteria are nearly identical. The linear stressterm s11 in Tsai±Wu produces a slightly higheror lower failure load than in Hill±Tsai, depend-ing on the quadrant. Interestingly, the Hashincriterion reduces to Maximum Stress with acompressive s11, and reduces to Hill±Tsaiwith tensile s11.

Figure 3 contains a plot of the six criteria inthe s227t12 stress plane (s11=0). MaximumStress and Maximum Strain are identical rec-tangles showing +s22 normalized by Y and+t12 normalized by S. Tsai±Wu and Hill±Tsai produce curved envelopes due tos227t12 interaction. Again, the linear s22

term in Tsai±Wu produces a different shapethan Hill±Tsai, pushing the failure envelopebeyond the lamina shear strength S. Hashinand Hashin±Rotem in this stress plane matchHill±Tsai exactly considering s11=0. As

expected, all six criteria intersect at the fourunidirectional loading cases (+s22 witht12=0 and +t12 with s22=0).

1.20.2.2.2 Off-axis loading

The off-axis tension test is a simple way todetermine the failure of a composite undercombined stresses. Figure 4 shows predictionsof the six criteria for the AS4/3501-6 compositewhose properties are found in Table 1. In Fig-ure 4, the loading angle y is defined as the anglebetween the fiber and loading directions. It isevident that all six criteria predict very similarfailure stress sxx over the entire off-axis range.All six criteria predict a failure load of X aty=08 and Y at y=908. The Maximum Stresscriterion predicts three separate failure regionsrepresenting the three possible modes of failure;i.e., fiber breakage, transverse matrix cracking,and shear matrix cracking. Between y=08 and2.98, fiber breakage is predicted (mode 1). Aty=08, the failure load is simply X. As yincreases in this region, the predicted failure

Figure 2 Comparison of lamina failure criteria under s117 t12 biaxial stress.


Figure 3 Comparison of lamina failure criteria under s227 t12 biaxial stress.

Figure 4 Comparison of lamina failure criteria for off-axis loading.


load actually increases just slightly. This isbecause the s11/X ratio remains dominanteven though the s11 component in the laminagoes down; thus, a larger applied load is neces-sary to satisfy the dominant equation. At thecritical angle y=2.98, the t12/S ratio becomesdominant; therefore, the failure mode switchesto shear matrix cracking (mode 3). This regioncontinues until y=278 where the failure modeswitches again to transverse matrix cracking(mode 2) with the s22/Y ratio becoming domi-nant. This region continues through 908 wherethe failure load is simply Y.

The Maximum Strain criterion produces asimilar failure curve to that of MaximumStress. From y=08 to 2.98, mode 1 failureoccurs. Maximum Strain predicts a slightlyhigher failure load than Maximum Stress inthis region. In the shear region (mode 3), Max-imum Strain results are identical to the Max-imum Stress results. Maximum Strain's shearregion extends to 298, where failure switches tomode 2. Note that the use of measured ultimatestrains that include nonlinear effects would in-significantly alter the character of this failurecurve.

The two interactive criteria, Hill±Tsai andTsai±Wu, predict the exact three failure regionsas the Maximum Stress criterion, with criticalangles at 2.98 and 278 for mode 1±3 and mode3±2 transitions, respectively. Since these criteriaare completely interactive, their failure curvesremain smooth throughout the entire off-axisloading case. Because X>>S, a criterionwhich couples these stress components (Hill±Tsai and Tsai±Wu) will predict a noticeably

lower value in this area than a limit criterion(Maximum Stress and Strain). For the regionwhere both t12 and s22 have significant contri-butions, the difference between the limit andinteractive criteria diminishes because Y and Sare of similar magnitude. The two interactivecriteria eventually converge with the two limitcriteria at y=908 (only s22 exists).

The two separate mode criteria exhibit char-acteristics of both the limit and interactive cri-teria. They yield the same three failure regionsas the other criteria. Due to its ability to sepa-rate modes, Hashin±Rotem's failure predictionis identical to Maximum Stress in the fiber (s11)dominated region y=0±2.98. After the mode1±3 transition at 2.98, Hashin±Rotem's failureprediction begins to move away from Maxi-mum Stress and towards the prediction ofHill±Tsai.

1.20.2.2.3 Pure shear

The six lamina strength criteria are comparedby using their failure predictions in a pure shearloading situation. The angle between the ap-plied shear load and the lamina fibers is de-noted by y as in the unidirectional off-axisloading previously discussed. The shear loadingis given by +txy(sxx=syy=0). A laminacomposed of the material in Table 1 is usedfor this analysis.

Figure 5 shows a +txy loading case for asingle lamina rotated at an angle y. All sixcriteria predict a failure load of S at y=08

Figure 5 Comparison of lamina failure criteria for positive pure shear.


and 908, as expected. The criteria are all sym-metric about y=458 where a failure load ofapproximately 7Y' (compressive) is predicted.The criteria all predict shear matrix crackingfrom 08 to 30.88 and transverse matrix crackingfrom 30.88 to 458. For the Maximum Stresscriterion, the mode 3±2 transition point drama-tically alters the failure prediction. The s22/Y'term becomes dominant after the transition aty=30.88. Because s22 is smaller at 30.88 thanat 458, it actually requires a larger ultimate txyto satisfy the failure criterion. This explains thedrastic change in the failure curve. MaximumStress and Maximum Strain differ only in thistransverse failure region due to the Poissoneffect, and are plotted as one curve. Theyboth predict a maximum failure load at theshear±transverse failure transition.

The interactive criteria produce smoothcurves for this loading case, even though theydifferentiate between modes of failure. The lin-ear terms in Tsai±Wu gives its failure curve adifferent character than Hill±Tsai, though bothclearly eliminate the cusp formed by the limitcriteria. Both separate mode criteria Hashinand Hashin±Rotem closely match Hill±Tsai.The separate mode criteria deviate slightly dueto the small s11/X contribution found in theHill±Tsai criterion near y=458, where s11 be-comes a maximum. All interactive and separatemode criteria reach a maximum at 458 wheres22 is dominant and at its maximum(txy=7Y').

Figure 6 shows a 7txy loading case for a

single lamina rotated to an angle y. Like the+txy loading case, all criteria are symmetricabout y=458, and switch from a shear matrixcracking failure to transverse matrix cracking aty=13.48. A strength of approximately txy=Y(tensile) is predicted at y=458.

The limit criteria predict an increasing failureload in the shear failure region. This is due to adecrease in the magnitude of the componentt12, even though the t12/S ratio is dominant;higher loads are predicted in order to satisfy thedominant equation. Maximum Stress andMax-imum Strain differ slightly in the transversefailure mode region. Hill±Tsai, Hashin±Rotem, and Hashin are nearly identical withonly a small s11/X contribution in the Hill±Tsaicriterion separating them. Tsai±Wu's failurecurve varies from the other interactive criteriadue to its linear terms, primarily the s22 term.Again, because the shear loading is negative,the lamina is weakest at y=458, in contrastwith the positive shear case in which it is stron-gest at 458.

In contrast to off-axis loading, in the caseof pure shear loading these criteria predict quitedifferent lamina strengths.

1.20.2.3 Comparison with Experimental Data

The accuracy of lamina strength criteria de-pends on reliable material strength data, i.e., X,X', Y, Y', and S, or the corresponding ultimate

Figure 6 Comparison of lamina failure criteria for negative pure shear.


strains. Except for X and Y, good measurementof compressive strength and shear strength arenot easy to obtain, which makes an objectiveassessment of the lamina failure criteria all themore difficult.

1.20.2.3.1 Lamina strength criteria comparisonwith off-axis tension data

Many authors have performed off-axis uni-directional lamina tensile tests. Because all thelamina strength criteria predict very similarfailure loads, correlation with experimentaldata cannot be used as a means of rankingthese criteria. Tests of a boron±epoxy systemby Pipes and Cole (1973) illustrate this point.The strength properties of this material systemare provided in Table 2. Using those values, thetheoretical predictions of the six laminastrength criteria along with the experimentaldata are plotted in Figure 7. For this range ofoff-axis angles, matrix failure dominates thestrength. It is seen that the interactive andseparate mode criteria yielded better predic-tions than the limit criteria. Other sets of off-axis data yield similar conclusions.

1.20.2.3.2 Lamina strength criteria comparisonwith tubular specimens

Use of a tubular specimen allows a biaxialstate of stress to be applied to a compositelaminate. Use of such specimens also eliminatesthe free edge effect found in flat coupon speci-mens (Colvin and Swanson, 1990).

Wu and Scheublein (1974) generated biaxiallamina data using a graphite±epoxy (MorganiteII) system with material constants listed inTable 3. Figure 8 shows the predictions forthe material system vs. experimental data(s117s22 plane). Clearly all criteria match atthe four axis intercepts, considering that thosedata points are used to generate the failureenvelopes. The data in the fourth quadrant isnot sufficient to distinguish any one criterion.

Swanson et al. (1987) obtained strength datafor an AS4/55A unidirectional composite in thes227t12 plane. Table 4 lists the strength prop-erties. Figure 9 shows theoretical predictions ofthe six lamina strength criteria compared withthe experimental data. It is evident that Hill±Tsai, Tsai±Wu, Hashin±Rotem, andHashin (allincluding s227t12 stress interaction) predictthe data very well for tensile s22. However,for the combination of 7s22 and t12, onlyTsai±Wu performs well. In fact, the test dataindicate that lamina shear strength increases asthe s22 component becomes compressive.

In order to verify the aforementioned phe-nomenon, two additional independent sets of

Table 2 Strength values for the boron±epoxymaterial system.

X 1296.2MPaX' 72489.0MPaY 62.1MPaY' 7310.3MPaS 68.5MPa

Source: Pipes and Cole (1973).

Figure 7 Comparison of lamina failure criteria to off-axis data.


s227t12 biaxial data are analyzed. The first setis a T800/3900-2 graphite±epoxy tested bySwanson and Qian (1992). The second set isfrom tests by Voloshin and Arcan (1980) usingglass±epoxy (Scotch-Ply Type 1002). Both setsof material strength constants are given inTable 4. Figures 10 and 11 show the predictionsfrom the six lamina failure criteria vs. experi-mental data. The trend of an increasing shear

strength as the s22 term becomes more com-pressive is seen again. Further discussion onthis phenomenon is given in Section 1.20.2.3.3.

1.20.2.3.3 Discussion

All six lamina strength criteria consideredhere are phenomenological or macromechani-cal in approach. In other words, they are moreor less curve-fitting techniques. Except for theTsai±Wu criterion, these criteria are developedbased on certain assumptions of failure me-chanisms. Thus, the accuracy of each criteriondepends on whether the assumed dominant fail-ure mechanism is included and properly de-scribed by the stresses or strains in the criterion.

Figure 8 Comparison of lamina failure criteria to s117s22 data from Wu and Scheublein (1974).

Table 3 Strength values for the graphite±epoxy

material system.

X 1027.3MPaX' 7710.2MPaY 43.4MPaY' 7125.5MPaS 72.4MPa

Source: Wu and Scheublein (1974).

Table 4 Strength values for material systems used in s227 t biaxial failure comparisons.

AS4/55ASwanson et al. (1987)

Scotch-Ply(Type 1002)

Voloshin and Arcan (1980)T800/3900-2

Swanson and Qian (1992)

X not provided X 1108.0MPa X not providedX' not provided X' ±617.8MPa X' not providedY 26.7MPa Y 19.61MPa Y 65.0MPaY' ±94.7MPa Y' ±137.30MPa Y' ±200.0 MPaS 51.8 MPa S 36.92 MPa S 100.0MPa


Figure 9 Comparison of lamina failure criteria to s227 t12 AS4/55A data from Swanson et al. (1987).

Figure 10 Comparison of lamina failure criteria to s227 t12 T800 data from Swanson and Qian (1992).


(i) On the Tsai±Wu criterion

The Tsai±Wu criterion is the only one amongthe six criteria that strictly follows a curve-fitting approach. Unlike the Hill±Tsai criterion,the polynomial function used in the Tsai±Wucriterion cannot be interpreted as part of defor-mation energy because of the presence of linearterms.

The simplicity in using a single equation topredict failure in a lamina under general load-ing as offered by Tsai±Wu is very attractive.However, in taking such an approach, all thefailure mechanisms must be included simulta-neously for any loading. This may present someconceptual difficulty. It is difficult to arguethat, for example, failure of a compositeunder biaxial tension should depend on itscompressive strength properties and viceversa. Although mathematically more conveni-ent, a practice as adopted by Tsai±Wu maycause unreasonable failure predictions. Asshown in Figure 1, Tsai±Wu suggests that acompressive stress s22 could increase the long-itudinal strength of the composite. There are noknown mechanistic reasons to support this. Infact, this phenomenon is produced by the factthat |Y '|>|Y| which causes translation of thefailure ellipse to the said position.

(ii) On the maximum strain criterion

On the s117t12 and s227t12 planes (seeFigures 3 and 4), the Maximum Stress andMaximum Strain criteria predict identical re-sults. However, for biaxial loading in thes117s22 plane (see Figure 1), these two cri-teria differ significantly. The Maximum Straincriterion predicts that for a tensile longitudinalstress s11, the tensile transverse stress s22

would be greater than Y in order to fail thecomposite. Specifically, for s11 near X, the s22

required to cause failure approaches 2Y. If thetransverse strength of the composite is con-trolled by the fiber/matrix interfacial strength,then this is not realistic.

It is concluded that the Maximum Straincriterion is not adequate for predicting thetransverse matrix cracking failure mode wherea significant s11 is present.

(iii) Dependence of shear strength oncompressive normal stress s22

In all the existing lamina failure criteria, thelamina strengths X, Y, and S are assumed to beconstants. However, from the three s227t12

Figure 11 Comparison of lamina failure criteria to s227 t12 glass±epoxy data from Voloshin and Arcan

(1980).


biaxial plots in Section 1.20.2.3.2 (Figures 9±11), there is strong evidence that when thecomposite is subjected to a combineds227t12 loading, it becomes stronger whens22 is compressive. More specifically, for agiven s22=+s0, the shear stress t12 at failurecorresponding to s22=7s0 is appreciablygreater than the shear stress t12 correspondingto s22=+s0. This behavior indicates that acompressive fiber/matrix interfacial normalstress (which is proportional to s22) could cre-ate a greater fiber/matrix interfacial shearstrength. To reflect this behavior, the matrixfailure criterion of Equation (5) may be mod-ified to

�s22

Y�2 � � t12

Sÿ ms22�2 � 1 �8�

where

m � m0 if s22400 if s2250

��9�

The term m plays a role similar to frictioncoefficients. Equation (8), denoted henceforthas the modified failure matrix criterion, stillyields the expected values of s22=Y att12=0 and t12=S at s22=0.

In the absence of s11, the linear stress termsin Tsai±Wu produce a failure envelope in thes227t12 plane that exhibits a characteristicsimilar to that of Equation (8). However, thiseffect would disappear in the Tsai±Wu criterionif |Y'|5|Y|.

Figure 12 shows AS4/55A data from Figure 9plotted with the Tsai±Wu criterion and themodified matrix failure criterion, Equation(8). It is clear from the comparison that themodified matrix criterion fits the data as well asTsai±Wu if m=0.6 is chosen. Figures 13 and 14are simply replots of Figures 10 and 11, respec-tively, with the modified matrix failure criterionadded. These plots also show that this modifiedmatrix failure criterion can improve thestrength prediction in the s227t12 plane.

1.20.2.3.4 Concluding remarks

From the comparisons with experimentaldata, we conclude that a SeparateMode Failurecriterion in the form

s11

X � 1 for fiber failure

�s22

Y �2 � � t12Sÿms22

�2 � 1 for matrix failure�10�

Figure 12 Comparison of lamina failure criteria and the Modified Matrix Criterion to s227 t12 AS4/55Adata from Swanson et al. (1987).


Figure 13 Comparison of lamina failure criteria and the Modified Matrix Criterion to s227 t12 T800 data

from Swanson and Qian (1992).

Figure 14 Comparison of lamina failure criteria and the Modified Matrix Criterion to s227 t12 glass±epoxy data from Voloshin and Arcan (1980).


is reasonably accurate for lamina failure pre-diction. A modified Tsai±Wu failure criterion,

s11

X� 1 for fiber failure

F2s22 � F22s222 � 2F12s11s22

� F66t122 � 1 for matrix failure

�11�

are also adequate for data fitting for mostadvanced composites (for which |Y'|>|Y|).

1.20.3 LAMINATE STRENGTHANALYSIS

Classical laminate strength analysis is basedon the two-dimensional plane stress field in thelaminate. Laminate failure is the eventual resultof progressive failure processes taking place inthe constituent laminae under loading. Concep-tually, a ply-by-ply failure analysis should yieldthe desired failure load for the laminate. Inreality, however, the failure mechanisms in la-minates are a great deal more complicated thanthose in a unidirectional composite under in-plane loading. New failure mechanisms areadded to the three intralamina failure modesoccurring at the lamina level. The most notablethree-dimensional failure modes include dela-mination and failure induced by free edge sin-gular stresses. Classical laminate strengthanalysis is restricted to those laminates whosefailure is not dominated by three-dimensionalfailure modes.

Another complication encountered in classi-cal laminate failure analysis is that the laminastrengths in the laminate may be quite differentfrom those obtained from the unidirectionalcomposite panel. Such variations are attributedto the constraining effect from the adjacentlaminae in the laminate. For better resultsfrom laminate failure analysis, the ªin situºlamina strengths should be used.

The effects of free edge stresses are usuallytreated separately from classical laminate fail-ure analysis. Thus, it is generally assumed thatthe laminate is either free of free edge stresses,or laminate failure does not initiate from thefree edge. Some authors have utilized tubularspecimens to avoid the effect of free edge stres-ses. The use of layers of film adhesive at theinterlayers can also toughen the interface, for-cing failure to occur in in-plane modes. Thislatter approach is taken in this study to enablethe use of laminate coupon specimens for test-ing laminate strength.

As lamina failure is progressive in nature, theprogressive loss of lamina stiffness must also

be accounted for in the laminate analysis.However, the local stress concentration effectdue to matrix cracks is usually neglected exceptfor laminates with thick laminae such as a[0/908/0] laminate. This local stress concentra-tion effect on laminate strength was discussedby Sun and Jen (1987).

1.20.3.1 Stiffness Reduction

Some of the laminate failure analysis meth-ods consider a laminate capable of load bearingafter an individual ply within the laminate hasfailed. These methods require a procedure forªdiscountingº the failed ply and reducing thelaminate stiffness. Two methods for achievingthis are as the parallel spring model and theincremental stiffness reduction model.

1.20.3.1.1 Parallel spring model

Each lamina is modeled with a pair of springsrepresenting fiber (longitudinal) and matrix(shear and transverse) deformation modes.The entire laminate is modeled by groupingtogether a number of parallel lamina springsets as shown in Figure 15. When fiber break-age occurs, the longitudinal modulus isreduced. When matrix cracking occurs, theshear and transverse moduli are reduced. Var-ious approaches and models have been pro-posed to investigate stiffness reduction causedby matrix cracking in a laminate (for references,see Tao and Sun, 1996). In fiber dominatedlaminates, such laminate stiffness reductionsare usually small, and, for the sake of simpli-city, the matrix dominated moduli, E2 and G12,are usually set equal to zero.

This model is also capable of differentiatingbetween types of matrix failure if desired; i.e.,the transverse and shear moduli can be reducedseparately depending on the specific type ofmatrix failure mode. The model which reducesE1 for fiber failure and E2 and G12 for eithertransverse or shear matrix failure is denoted thePSM. The model which reduces E1 for fiberfailure, E2 for transverse matrix failure, andE2 and G12 for shear matrix failure is denotedthe PSMs. The idea behind the PSMs is that atransverse matrix failure does not necessarilyinhibit the ability of the lamina to carry sig-nificant shear loads. Creating these two differ-ent reduction models has little micromechanicalbasis, and is done mainly for curve fittingpurposes.

Laminate Strength Analysis 15

1.20.3.1.2 Incremental stiffness reductionmodel

To avoid the sudden jump in strain at plyfailure seen in the parallel spring model, amodel resembling the bilinear hardening inclassical plasticity can be formulated. Laminatestiffness reduction is achieved similar to that inthe parallel spring model. However, it is as-sumed that the reduced laminate stiffness gov-erns only the incremental load±deformationrelations beyond immediate ply failure.

Both of these stiffness reduction models haveflexibility. Instead of reducing the appropriatemoduli suddenly after a ply failure, a nonlinearfunction such as an exponential function maybe used to gradually reduce these values. Thisprogressive softening approach may model cer-tain laminates better than others, i.e., thoselaminates whose failure is dominated by matrixcracking.

For most fiber-dominated composites, set-ting the stiffness constants directly to zeroafter the corresponding mode of failure occursis simple and unambiguous. The use of suchreduction can be justified by regarding the la-minate analysis to be at the location wherematrix cracking occurs. Consider a 908 lamina(within a laminate) containing a number oftransverse matrix cracks, as shown in Figure16. The 908 ply still retains some stiffness in theloading direction (E2 direction locally). How-ever, the assumption is made that ensuing 08fiber failure will occur at the weakest point.This point is where matrix cracking hasoccurred in the 908 plies, or where locallyE2=0. Thus, it is acceptable in the ultimate

strength analysis to reduce E2 directly to zeroafter transverse matrix cracking.

Since matrix cracks are discrete, between twocracks a failed lamina would still contributesubstantially to laminate stiffness. It is obviousthat such drastic lamina stiffness reduction, ifassumed to be true over the entire laminate,would overestimate the ultimate strains of thelaminate. In fiber-dominated laminates, the ef-fect of matrix cracks on overall laminate stiff-nesses is usually very small. It is reasonable toestimate the laminate ultimate strains by usingthe virgin laminate stress±strain relations andthe laminate failure stresses obtained from thelaminate failure analysis.

1.20.3.2 Laminate Failure Analysis Methods

As with lamina failure analysis, a variety oflaminate failure analysis methods have beenproposed. The following is a description ofeach methodology.

1.20.3.2.1 Ply-by-ply discount method

This is a very common method for laminatefailure analysis. Laminate is treated as a homo-geneous material and is analyzed with a laminastrength criterion. The laminated plate theory isused to initially calculate stresses and strains ineach lamina. A lamina strength criterion is thenused to determine the particular ply which willfail first and the mode of that failure. A stiffnessreduction model is used to reduce the stiffness

Figure 15 Schematic of the parallel stiffness model.


of the laminate due to that individual ply fail-ure. The laminate with reduced stiffnesses isagain analyzed for stresses and strains. Thelamina failure criterion predicts the next plyfailure, and laminate stiffness is accordinglyreduced again. This cycle continues until ulti-mate laminate failure is reached.

A number of definitions have been proposedon how to determine ultimate laminate failure.One common way is to assume ultimate lami-nate failure when fiber breakage occurs in anylamina. Another way is to check if excessivestrains occur (i.e., yielding of the laminate stiff-ness matrix). Matrix-dominated laminates suchas [+45]s may fail without fiber breakage.Others have suggested a ªlast plyº definitionin which the laminate is considered failed ifevery ply has been damaged. For this study,laminate failure is defined as fiber breakage inany ply or the reduced stiffness matrix becomessingular.

1.20.3.2.2 Sudden failure method

In highly fiber-dominated composite lami-nates, laminate stiffness reduction due to pro-gressive matrix failures insignificantly affectslaminate ultimate strength. This suggests thatin such laminates the progressive stiffness re-duction seen in the previous method may beunnecessary, and laminate failure may be takento coincide with fiber failure of the load-carry-ing ply. To perform this analysis, a laminastrength criterion is chosen, and the failureload is determined by calculating the load re-quired for fiber failure in the dominant lamina.No stiffness reductions are included in the pro-cess. The laminate strength predicted by thesudden failure method is usually higher thanthe laminate strength predicted by the ply-by-ply discount method.

1.20.3.3 Laminate Failure Analysis for BiaxialLoading

The six lamina strength criteria are used inconjunction with the above laminate failureanalysis methods to predict laminate strengthfor biaxial loading. In view of the limitations ofthe present method, laminates whose failure iscaused by free edge stresses or delamination areavoided. The ply-by-ply discount method isused in conjunction with the parallel springmodel (PSM) for laminate stiffness reduction.In this analysis, the appropriate lamina modulireduce to zero at the individual ply failures. Thelaminate is assumed to reach ultimate failurewhen any ply within the laminate fails by fiberbreakage.

1.20.3.3.1 Comparison with data for biaxialloading

Recently, Swanson and Trask (1989) per-formed biaxial testing on an AS4/3501-6graphite±epoxy [0/+45/90]s laminate usingtubular specimens. The ply properties givenby Swanson and Trask (1989) are listed inTable 5. Note that they are slightly differentfrom those given in Table 1.

Figure 17 shows the failure envelopes in thesxx7syy plane. The predicted envelopes are

Figure 16 Schematic of laminate with matrix cracks.

Table 5 Moduli and strength values of

AS4/3501-6 graphite±epoxy system.

E1 134.60GPa X 1986.0MPaE2 11.03GPa X' ±1193.0MPaG12 5.52GPa Y 47.9MPav12 0.28 Y' ±168.0MPaPly thickness 0.13mm S 95.7MPa

Source: Swanson and Trask (1988).


generated with the six lamina failure criteria inconjunction with the ply-by-ply discountmethod. Comparing with the data, it is evidentthat the predictions of all six lamina failurecriteria agree with the data very well. However,the fully interactive criteria such as Hill±Tsaiand Tsai±Wu are very sensitive to the sequenceof lamina matrix failures and can be signifi-cantly affected by the sudden reduction ofmatrix stiffness resulting in jumps in strength.These jumps are evident in Figure 17.

Swanson and Qian (1992) also testedlaminate tubes in the sxx7syy plane. The[0/+45/90]s, [03/+45/90]s, and [0/(+45)2/90]sstrength data (I and IV quadrants only) werefound to match well with the Maximum Strainanalysis. Considering that all lamina failurecriteria are in close agreement in these quad-rants for the p/4 laminate, it is clear that theyshould all be adequate for these laminates.

1.20.3.3.2 Biaxial failure in the strain plane

The failure strains corresponding to the fail-ure stress envelopes of Figure 17 are plotted in

Figure 18. It is interesting to note that the fail-ure strain envelopes predicted by MaximumStress, Maximum Strain, Hashin, and Hashin±Rotem criteria essentially coincide with the lim-its set by the ultimate tensile strain (Xe) andcompressive strain (Xe') of the unidirectionalcomposite. However, these limit strains shouldnot be automatically taken as the ultimatestrains of the laminate at failure. For the p/4quasi-isotropic laminate under biaxial loadingin the third stress quadrant (compressive sxx

and syy), there are no ply matrix failures beforeultimate laminate failure. Thus, there is nostiffness reduction before laminate failure, andthe calculated failure strain is the laminate fail-ure strain.

Under tensile biaxial loading (tensile sxx andsyy), all plies in the laminate have sufferedmatrix failure before laminate final failure.The drastic reduction (to zero) of ply trans-verse and shear stiffnesses tends to slightlyoverestimate the ultimate strains. Therefore,the strain failure envelope as shown in Figure18 should not be used directly in laminatestrength design without accounting for thestiffness reductions.

Figure 17 Comparison of ultimate stress envelopes with experimental data, Swanson and Trask (1989) for a[0/+45/90]s laminate under biaxial loads.


1.20.3.3.3 Biaxial testing data for glass wovenfabric composite

Recently, Wang and Socie (1992, 1994) per-formed biaxial testing on NEMA G-10 E-glassplain woven fabric composite laminates. Theyused both tubular specimens and flat squarespecimens. Strength data were obtained for allfour biaxial loading quadrants. Their test re-sults clearly indicate that the laminate failurestrain envelope is bounded by the uniaxial ulti-mate strains of the laminate. Specifically, theyfound that failure in one direction was notaffected by loading in the transverse direction.Hence, the failure strain envelope appears to berectangular as predicted by the MaximumStrain and Maximum Stress criteria.

1.20.3.4 Laminate Strength Analysis forUnidirectional Loading

Laminate strength data are available for cou-pon specimens under uniaxial loading. In mostcases though, free edge stresses control theinitiation and final failure of these laminates.Data as such are not suitable for evaluating thelaminate failure analysis methods as attemptedhere. However, by placing film adhesive at the

interfaces of laminate coupon specimens, it ispossible to suppress these three-dimensionaleffects. This type of coupon specimen is usedto generate additional laminate strength datafor the purpose of evaluating the strengthcriteria.

1.20.3.4.1 Selection of laminates and off-axisloading angles

A variety of laminates was theoretically ex-amined in advance of actual strength testing.Only the ply-by-ply discount method was usedto determine ultimate laminate strength. Lami-nate layups and off-axis loading angles werechosen to provide a comparison among thesix lamina strength criteria. The focus of theexperiment was to examine whether thesestrength criteria could predict the correcttrend of laminate strength vs. loading angle.

Table 6 shows the laminates and off-axisloading angles selected. Included in these testswere unidirectional laminate specimens forprincipal material properties. Five or moretests were performed at each off-axis loadingangle to provide accurate mean results. In thetable, ªAº indicates the location of an adhesivefilm in the laminate.

Figure 18 Comparison of ultimate strain envelopes with experimental data, Swanson and Trask (1989) for

a [0/+45/90]s laminate under biaxial loads.


1.20.3.4.2 Laminate coupon specimens andoblique end tabs

The material used for testing was AS4/3501-6graphite±epoxy from Hercules. A 0.13mmthick ply of film adhesive was added at eachlamina interface except the middle interfacebecause of symmetry. The adhesive film wasFM 1000 marketed by American Cyanamid.The elastic and strength properties of this ma-terial (FM 1000) were determined by Sun andZhou (1988) and are listed in Table 7.

For those laminate specimens with anisotro-pic stiffness, it was important not to overlookthe effect of shear±extension coupling. Highlyanisotropic laminate coupon specimens can failprematurely due to stress concentrations in thetab region if rectangular tabs are used. In orderto accommodate the deformation induced bythe extension±shear coupling, oblique end tabssuggested by Sun and Chung (1993) were used.The oblique angle f (see Figure 19) was derivedusing extensional laminate stiffnesses A11 andA16 as

cotf � ÿA16=A11 �12�

It was demonstrated by Sun and Chung(1993) that, using oblique end tabs, an almostuniform state of stress corresponding to uniax-ial loading could be generated in off-axislaminate coupon specimens rigidly grippedduring loading. In quasi-isotropic laminates,extension±shear coupling is absent, and stan-dard rectangular tabs were used.

Glass±epoxy and graphite±epoxy quasi-iso-tropic laminates were used for oblique end tabs.For the rectangular tabs, material stiffness isnot as critical, and chopped fiberglass circuitboard was used. All specimens were 19mm

wide with a 171mm gage section. This yielded a9:1 aspect ratio.

1.20.3.4.3 Results and data analysis

Tests were performed in the Composite Ma-terials Laboratory (CML) at Purdue Univer-sity. The first set of coupons tested were 08 and908 unidirectional laminates. All specimenswere strain gauged to determine the appropri-ate elastic constants. Table 8 lists the results.Only the moduli E1, E2, n12, and the strengthvalues X and Y were determined in the presenttest. The strengths S, X', and Y' were takenfrom a similar AS4/3501-6 material systemgiven by Sun and Zhou (1988). The shear mod-ulus G12 in Sun and Zhou (1988) was obtainedfrom testing [+45]s laminate and is lower than6.9GPa given by Daniel and Ishai (1994). Thelatter value is listed in Table 8.

For laminates containing layers of film ad-hesive, the calculation of ultimate stress mustaccount for the thickness and stiffness of the

Table 6 Laminates and off-axis loading angles tested.

Laminate Off-axis angles tested

[0]8 0 8 and 90 8[0/A/+45/A/745/A/90]s 0 8 ± 22.5 8 every 7.5 8[90/A/0/A/90/A/0]s 0 8 ± 7.5 8 every 1.5 8, 15 8, 22.5 8[0/A/+45/A/745]s 0 8 ± 30 8 every 7.5 8, 26 8, 458[90/A/+30/A/730]s 0 8 ± 22.5 8 every 7.5 8

Table 7 Material properties for FM 1000 filmadhesive.

sult 38MPaE 1.724GPaG 0.648GPav 0.33Ply thickness 0.127mm

Figure 19 Example of oblique end tabs.


film adhesive in determining the true graphite±epoxy laminate strength. First, it is assumedthat the total measured ultimate load is thesummation of the load carried by the compo-site, PC, and the load carried by the adhesive,PA, i.e.,

Pexp � PC � PA �13�

For such laminates under uniaxial loads, it isreasonable to assume that the composite andadhesive loads can be separated by the rule ofmixtures:

PA � EAhA

A11 � EAhA

� �Pexp �14�

PC � A11

A11 � EAhA

� �Pexp �15�

where EA is the adhesive modulus, hA is totalthickness of all adhesive layers in the laminate,and A11 is the total extensional stiffness calcu-lated just for the graphite±epoxy plies.

Equation (15) is used to obtain the truecomposite ultimate load from the experimentalload. The laminate strength is determined bydividing PC by the total cross-sectional area ofthe composite plies. Table 9 lists all the aver-aged ultimate stress data for the tested lami-nates.

Though the majority of the laminates failedby fiber breakage, a few laminates in whichfiber failure did not occur were matrix domi-nated. In the matrix dominated coupon tests,the specimen never fractured into two (or more)pieces. Instead, the deformation continued until

reaching a strain of approximately 3% at whichthe test was stopped. As seen in Figure 20, a[90/A/0/A/90/A/0]s laminate loaded at 22.58exemplifies this behavior. For comparison, thestress7strain curve for a [0/A/+45/A/±45]slaminate loaded at 458 is also plotted, whichdemonstrates the typical behavior involvingfiber failure.

1.20.3.4.4 Comparison with test data

Theoretical predictions using the six laminafailure criteria in conjunction with the ply-by-ply discount method for laminates under off-axis loading are compared with the experimen-tal data. Only the results obtained with thePSM stiffness reduction method are reportedhere. Results with the PSMS method can befound in the report by Sun et al. (1996).

Figure 21 shows theoretical predictions forthe [0/A/+45/A/745/A/90]s laminate obtainedusing the PSM stiffness reduction procedure.The experimental data (CML data) for the[0/A/+45/A/745/A/90]s laminate shows anincrease in strength as the off-axis loadingangle rotates from 08 to 22.58. MaximumStress, Maximum Strain, and Hashin±Rotemall predict this increase, supporting the idea ofseparating fiber failure (governed by fiber stresss11) from matrix failure (governed by matrixstresses s22 and t12).

Figure 22 compares CML data with the the-oretical predictions for the [90/A/0/A/90/A/0]slaminate. The fully interactive criteria Hill±Tsaiand Tsai±Wu underestimate the strength of thelaminate. The Hashin criterion is closer to the

Table 8 Moduli and strength values for the tested AS4/3501-6 graphite±epoxysystem.

E1 153.7GPa X 2171.0MPaE2 11.0GPa X' ±2013.0MPaG12 6.9GPa Y 67.0MPav12 0.32 Y' ±206.8MPaPly thickness 0.13mm S 110.3MPa

Table 9 Ultimate laminate stresses (MPa).

Off-axis loading angle 0 8 7.5 8 15 8 22.5 8 26 8 30 8 45 8

[0/A/+45/A/±45/A/90]s 765 752 774 832[0/A/+45/A/±45]s 883 843 929 1028 1129 1074 818[90/A/+30/A/±30]s 966 908 837 807

Off-axis loading angle 0 8 1.5 8 3 8 4.5 8 6 8 7.5 8 15 8 22.5 8

[90/A/+30/A/±30]s 1126 1140 1074 1018 861 713 394 288


data than the fully interactive criteria. Again,the criteria (Maximum Stress, MaximumStrain, and Hashin±Rotem) which separatefiber failure from matrix failure best matchthe data.

Data from the [0/A/+45/A/745]s laminateare displayed in Figure 23. Overall, MaximumStress, Maximum Strain, and Hashin±Rotemare clearly the best fit for the data. In the off-

axis region from 08 to 208, the Tsai±Wu criter-ion closely fits the data. However, the overalltrend of Tsai±Wu's prediction is quite differentfrom the experimental data.

Data from the [90/A/+30/A/730]s laminateare displayed in Figure 24. Once again, Max-imum Stress, Maximum Strain, and Hashin±Rotemmatch both the magnitude and the trendof the data.

Figure 20 Stress±strain curves characterizing fiber and matrix failures.

Figure 21 Comparison of ultimate strengths for a [0/A/+45/A/745/A/90]s laminate under unidirectional

loading using different lamina failure criteria.


1.20.3.5 Discussion

From the four different sets of laminatestrength data presented in the previous section,it appears that the interactive criteria (Hill±Tsaiand Tsai±Wu) and Hashin criterion (whichcouples s11 and t12) not only underestimateultimate laminate failure, but cannot correctly

predict the trend of the data. On the other hand,Maximum Stress, Maximum Strain, andHashin±Rotem criteria all perform quite well.

Simply put, those criteria which separatefiber failure completely from matrix failureare relatively insensitive to inaccurate laminastrengths Y and S. There is some matrixstrength sensitivity in these criteria from the

Figure 22 Comparison of ultimate strengths for a [90/A/0/A/90/A/0]s laminate under unidirectional loading

using different lamina failure criteria.

Figure 23 Comparison of ultimate strengths for a [0/A/+45/A/745]s laminate under unidirectional loadingusing different lamina failure criteria.


effects of intermediate ply failures (failure pre-ceding ultimate fiber failure). However, thefully interactive criteria and the Hashin criter-ion are considerably more sensitive to inaccu-rate matrix strengths. This can be illustrated byvarying the lamina matrix strengths used in thetheoretical analysis of the laminates.

The sensitivity of the interactive strengthcriteria to matrix strength (Y or S) is demon-strated by the ultimate strength curves for the[0/A/+45/A/745/A/90]s laminate in Figure 25.The Hill±Tsai strength criterion is employed forthe laminate strength analysis using three dif-ferent lamina shear strength values. The curve

Figure 24 Comparison of ultimate strengths for a [90/A/+30/A/730]s laminate under unidirectionalloading using different lamina failure criteria.

Figure 25 Comparison of ultimate strengths for [0/A/+45/A/745/A/90]s with different lamina shearstrengths using Hill±Tsai Criterion.


ªHill±Tsai with S*1.0º is the same as the Hill±Tsai curve from Figure 21. The other twocurves in Figure 25 are for the shear strengths26S and 2.56S, respectively, with all othermaterial constants kept the same. It is clear thatby simply increasing the shear strength of thelamina, the Hill±Tsai criterion takes on a com-pletely different trend, predicting an increase inlaminate strength instead of a decrease as theoff-axis loading angle increases from 08 to22.58. With these high shear strengths, theHill±Tsai criterion now correctly predicts thetrend and magnitude of the experimental data.

Another issue often raised is whether matrixcracking actually occurs in composite laminateswith well-dispersed laminae. To answer thisquestion, we tested [0/90/0] and [0/90/0]s lami-nates of AS4/3501-6 graphite±epoxy compo-site. Coupon specimens were tested undertension. Figure 26 clearly shows the presenceof matrix cracks in the 908 plies at the loadabout 95% of the laminate strength. Thus, weconclude that matrix cracking does occur andply-by-ply discount in laminate failure analysisis justified.

1.20.4 CONCLUSIONS

Six commonly used composite strengthcriteria were investigated and compared forapplications in unidirectional fiber compositesand their laminates. These strength criteriawere evaluated using lamina and laminatestrength data which do not involve any out-of-plane deformation failure modes such asdelamination. Moreover, all the experimentaldata used in the evaluation were obtained fromspecimens under a state of uniform stress; i.e.,no stress gradient effects were included. Basedon the foregoing environment, the followingconclusions have been obtained.

(i) At the lamina level, those criteria (such asthe Hashin±Rotem criterion) which separatethe fiber failure mode from the matrix failuremode are the most reasonable and accurate.

(ii) For fiber-dominated laminates, Maxi-mum Stress, Maximum Strain, and Hashin±Rotem failure criteria outperform other cri-teria. These criteria are insensitive to variationsin matrix strengths (Y and S) which are verydifficult to obtain in situ.

Figure 26 Evidence of 90 8 ply matrix cracking in [0/90/0] and [0/90/0]s laminates of AS4/3501-6 graphite±

epoxy composite.

Conclusions 25

(iii) The interactive failure criteria (Hill±Tsai, Tsai±Wu, and Hashin) are sensitive tovariations of the matrix-dominated laminastrengths (i.e., Y and S). Accurate in situ com-posite strengths are critical to the use of thesecriteria. Because of the interaction among allstress components, sudden switching of failuremodes makes the failure envelope (in stress orstrain) very jumpy.

(iv) To predict lamina matrix failure in alaminate, the in situ transverse strength (Y)and shear strength (S) should be used. In addi-tion, thermal residual stresses must be includedin the laminate analysis.

(v) Experimental results indicate that matrixcracking does take place even in laminates withwell-dispersed laminae. Thus, the ply-by-plydiscount of stiffnesses in failed laminae is justi-fied.

(vi) The Parallel Spring model for stiffnessreduction is adequate for analysis of laminatestrength. The drastic ply stiffness reduction (theconcerned stiffness is set equal to zero after plyfailure) does not cause appreciable errors in thepredicted laminate strength for fiber-domi-nated laminates.

1.20.5 REFERENCES

V. D. Azzi and S. W. Tsai, Experimental Mechanics, 1965,5, 283±288.

G. E. Colvin and S. R. Swanson, J. Eng. Mat. Technol.,1990, 112, 61±67.

W. C. Cui, M. R. Wisnom and M. Jones, Composites,1992, 23, 158±166.

I. M. Daniel and O. Ishai, `Engineering Mechanics ofComposite Materials', Oxford University Press, NewYork, 1994.

Z. Hashin, J. Appl. Mechanics, 1980, 47, 329±334.

Z. Hashin and A. Rotem, J. Composite Materials, 1973, 7,448±464.

P. Labossiere and K. W. Neale, Solid Mechanics Archives,1987, 12(2), 65±95.

M. N. Nahas, J. Composites Technology & Research,1986, 8(4), 138±153.

R. Narayanaswami and H. M. Adelman, J. CompositeMaterials, 1977, 11, 366±377.

R. B. Pipes and B. W. Cole, J. Composite Materials, 1973,7, 246±256.

C. T. Sun and I. Chung, Composites, 1993, 24(8), 619±623.

C. T. Sun and K. C. Jen, J. Reinforced Plastics andComposites, 1987, 6, 208±222.

C. T. Sun, B. J. Quinn, J. Tao and D. W. Oplinger, USDepartment of Transportation Federal Aviation Ad-ministration Report No. DOT/FAA/AR-90/1-9, 1996.

C. T. Sun and S. G. Zhou, J. Reinforced Plastics andComposites, 1988, 7, 515±557.

S. R. Swanson, M. J. Messick and Z. Tian, J. CompositeMaterials, 1987, 21, 619±630.

S. R. Swanson and B. C. Trask, Composites, 1988, 19,400±406.

S. R. Swanson and B. C. Trask, Composites Science andTechnology, 1989, 34, 19±34.

S. R. Swanson and Y. Qian, Composites Science andTechnology, 1992, 43, 197±203.

J. X. Tao and C. T. Sun, Mechanics of CompositeMaterials and Structures, 1996, 3, 225±239.

S. W. Tsai, `Strength Characteristics of Composite Mate-rials', NASA CR-224, National Aeronautics and SpaceAdministration, 1965 April.

S. W. Tsai and E. M. Wu, J. Composite Materials, 1971,5, 58±80.

A. Voloshin and M. Arcan, Experimental Mechanics,1980, 20, 280±284.

J. Z. Wang and D. F. Socie, in `Composite Materials:Testing and Design', ASTM STP 1206, vol. 1, Amer-ican Society for Testing and Materials, Philadelphia, PA,1992, pp. 136±149.

J. Z. Wang and D. F. Socie, J. Composites Technology &Research, 1994, 16(4), 336±342.

E. M. Wu and J. K. Scheublein, in `Composite Materials:Testing and Design 3rd Conference', ASTM STP 546,ASTM, Philadelphia, PA, 1974, pp. 188±206.


Comprehensive Composite MaterialsISBN (set): 0-08 0429939

Volume 1; (ISBN: 0-080437192); pp. 641±666

Copyright # 2000 Elsevier Science Ltd.All rights reserved. No part of this publicationmay be reproduced, stored in anyretrieval system or transmitted in any form or by any means: electronic,electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise,without permission in writing from the publishers.

Documents

Analisis Del Esfuerzo en Compuestos Unidireccionales Laminados