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Composites: Part A 38 (2007) 174–185

Three-dimensional stress analysis and Weibull statistics basedstrength prediction in open hole composites

E.V. Iarve a,*, R. Kim a, D. Mollenhauer b

a University of Dayton Research Institute, 300 College Park Avenue, Dayton, OH 45469-0168, United Statesb US Air Force Research Laboratory, AFRL/MLBC, Wright-Patterson AFB, OH 45433-7750, United States

Received 6 January 2005; received in revised form 5 January 2006; accepted 7 January 2006

Abstract

The critical failure volume (CFV) method is proposed. CFV is defined as a finite subvolume in a material with general nonuniformstress distribution, which has the highest probability of failure, i.e. loss of load carrying capacity. The evaluation of the probability offailure of the subvolumes is performed based on the lowest stress and thus provides an estimate of the lower bound of the probability oflocal failure. An algorithm for identifying this region, based on isostress surface parameterization is proposed. It is shown that in the caseof material with strength following Weibull weak link statistics such a volume exists and its location and size are defined both by thestress distribution and the scatter of strength. Moreover the probability of failure predicted by using the CFV method was found tobe close to that predicted by using traditional Weibull integral method and coincide with it in the case of uniform stress fields and inthe limit of zero scatter of strength. Experiments performed on homogeneous epoxy resin plaques with and without holes showed thatthe predictions bound the experimentally measured open hole strength. The Weibull parameters used for prediction were obtained fromtesting only unnotched specimens of different dimensions. The effect of the hole size on tensile strength of heterogeneous materials such asquasi-isotropic carbon–epoxy composite laminates was considered next. Fiber failure was the only failure mechanism taken into accountand a strain-based failure criterion was used in the form of a two parameter Weibull distribution. The stacking sequence was selected tominimize the effect of stress redistribution due to subcritical damage. Not unexpectedly an up to 30% underprediction of the strength ofthe laminates with small (2.54 mm diameter) holes was observed by using classical Weibull integral method as well as Weibull based CFVmethod. It was explained by examining the size of the CFV, which appeared to be below Rosen’s ineffective length estimate. The CFVmethod was modified to account for the presence of a limit scaling size of six ineffective lengths, consistent with recent Monte-Carlosimulations by Landis et al. [Landis CM, Beyerlin IJ, McMeeking RM. Micromechanical simulation of the failure of fiber reinforcedcomposites. Mech Phys Solids 2000;48:621–48] and was able to describe the experimentally observed magnitude of the hole size effecton composite tensile strength in the examined range of 2.54–15.24 mm hole diameters.� 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Laminates; B. Stress concentrations; C. Statistical properties/methods; C. Numerical analysis; Open hole

1. Introduction

Strength of composite materials with stress concentra-tions is a central design issue often dictating the designallowables for the entire structure. In 1970s two empiricalmethods, Waddoups et al. [1] and Nuismer and Whitney

1359-835X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesa.2006.01.004

* Corresponding author. Tel.: +1 937 2559075; fax: +1 937 2588075.E-mail address: endel.iarve@wpafb.af.mil (E.V. Iarve).

[2], emerged to address the pressing needs of notched com-posite design in the aerospace industry. Both approachesare two-parameter models, where the notched strength ispredicted based on the unnotched strength and an addi-tional parameter having a dimension of length. In Waddo-ups et al. [1] the length parameter represents an inherentflaw size, and in the Whitney and Nuismer model, it is thecharacteristic distance from the edge of the hole. It shouldbe mentioned that the inherent flaw concept used in the firstmodel is not a physical model of inherent flaw evolution but

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 175

a fracture mechanics-based phenomenological model wherethe size of this ‘‘fictitious flaw’’ is determined based onnotched strength. Once the size of the fictitious flaw is deter-mined for a given laminate for one hole size, the model iscapable of predicting the hole size effect, i.e. a significantincrease of net tensile strength with hole size reduction. Inthe second model a characteristic distance is also deter-mined based on a notched strength for a given hole size,so that the stress at this distance away from the hole edgewould reach the unnotched strength at failure. Besides thisapproach, called the point stress failure criterion, an aver-age stress criterion was also proposed where the stresswas averaged over a certain (characteristic) distance fromthe hole edge. Similar to the fracture mechanics approach,the point and average stress criterion were able to predictthe hole size effect on strength of notched composites. How-ever, none of the additional scaling parameters proved torepresent a material property across a family of compositelaminates with different layups. Nevertheless, both modelsare widely used in the industry for sizing design. A keyaspect of these models is that no notched strength predic-tion can be made based on the data determined by testingunnotched composites only. The knowledge of notchedstrength is required to calculate the value of the additionallength parameter.

A significant body of work has been devoted to strengthprediction in notched composites based on finite elementmethods combined with element property degradationrules. Lee [3] performed the first 3D finite element stressanalysis combined with the property degradation techniquefor damage modeling and strength prediction in open holecomposites. The property degradation was performed byfully degrading (multiplying by D = 10�6) the stiffnesscomponent responsible for the type of damage for whichthe stress component exceeded its ultimate value. Althoughfar preceding its time in its level of analysis fidelity, theapplication of such intuitive property degradation rulesleads to strong mesh sensitivity of the strength predictionfor notched laminates and unrealistic strength predictionsif applied to unnotched laminates. The latter was addressedby developing laminate level property degradation tech-niques based on internal damage variables methods devel-oped by Talreja [4] and Lee et al. [5]. Nguen [6] extendedthis methodology [4] and applied it for 3D strength predic-tion of open hole composite strength. Simplified algorithmsfor notched strength prediction, where the damage vari-ables are determined a priori and fixed, were proposed byTan [7]. A stiffness reduction coefficient D1 (�0.007) wasused to reduce the longitudinal modulus inside the finiteelement, for which the tensile fiber failure criterion wasreached. Camanho and Matthews [8] extended thisapproach to 3D analysis in composite fastener joints. Thecommon shortcoming of these methods in the problemsdealing with stress concentration is the lack of strengthscaling mechanisms. The failure criteria significantly(�30–40%) underpredicts the fiber failure origination loadsin the elements adjacent to the hole, and the subsequent

strength prediction is highly mesh dependent. Shahid andChang [9] offered a practical solution to alleviate the severemesh dependency of the tensile failure prediction by mak-ing the property degradation coefficient dependent uponthe area of the region where tensile fiber failure is detected.Nevertheless it still did not address the underprediction offiber failure initiation.

Weibull’s statistical theory of strength [10] provides anintegral form allowing one to calculate the upper boundof probability of failure in brittle materials in the presenceof stress gradients based on distribution of strengthobtained for uniformly stressed samples. In this case theWeibull modulus, characterizing the scatter of strength,becomes the parameter defining the notch sensitivity. Wu[11] for the first time applied the integral form of the Wei-bull distribution to predict the notched strength in thequasi-isotropic composites. A tensor polynomial function[12] was used in the Weibull integral to account for com-plexity and interaction of the failure modes. It was shownthat the predictions are in agreement with the experimentaldata of Nuismer and Whitney [2], showing the hole sizeeffect on the notched tensile strength. Wetherhold andWhitney [13] applied the integral form of Weibull’s distri-bution to predict the hole size effect in quasi-isotropic lam-inates. A one-dimensional approximation of the Weibullvolume integral in the radial direction was introduced. Asa result of this simplification, the predicted mean valuesof stress appeared significantly lower than the experimentaldata, however the hole size effect trend was clearly demon-strated. In later works Wetherhold [14] extended their workto include interactive failure criterion (see [15]). Gurvichand Pipes [16] developed the theoretical aspects of reliabil-ity of composites in random stress states considering fullanisotropy of random strength properties. A generalizationof this approach was also proposed by Gurvich [17] toaccount for the strength size effect and moderate nonuni-formity of 3D distributions of random stresses. The stressdistribution in the cited works was based on laminationtheory. No estimate of the size of the critical region of mostlikely failure was performed.

In the pioneering work of Kortshot and Beaumont [18],the strength of laminates with a through-the-thicknesscrack was considered. Two types of damage, namely longi-tudinal splitting and delamination, were parametricallyintroduced into the so-called 21

2D analysis method, where

each delaminated portion as well as the intact laminatewere considered as a separate plain stress plates. Account-ing for these two types of damage was critical to predict thestress relaxation in the fiber direction in the 0� ply. Thelaminate failure load was then predicted by applying theintegral form of Weibull scaling. The authors also reportedthat in calculation of the Weibull’s integral, most of thecontribution came from a small region near the stress con-centration. The method of stress analysis used by theauthors, although not 3D in nature, provided accurate fiberstresses in these regions of the delaminated section of the 0�plies.

176 E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185

Recently de Morais [19] performed 3D/2D global localfinite element analysis in quasi-isotropic composite lami-nates with open holes. His results showed significant differ-ence between the stress concentration factor (in-plane stressin the fiber direction at the edge of the hole) calculated byusing lamination theory and the average stress concentra-tion factors in individual plies computed by the global–local approach. To predict the strength of the notchedcomposite, a simple relationship was introduced for thestrength of the notched composite ruN as

ruN ¼vfruf

Kt1;

where vf is the fiber volume fraction, Kt1 is the stress con-centration in the 0� ply calculated by using the global–localapproach, and ruf is the average tensile strength of the fi-bers on a length scale linked to the hole diameter. Theapparent consequence of this assumption results in predic-tion of the same scatter in their distribution, whereas thefiber strength is characterized by much larger scatter thanthat of the composites for reasons explained by Rosen [20].

The focus of the present paper is on predicting thecatastrophic fiber failure in the presence of stress concen-tration. As shown by Iarve et al. [21], even in the quasi-isotropic composites, significant redistributions of thestress magnitude due to splitting in the 0� ply are possiblefor certain stacking sequences, in particular with the outer0� plies. By selecting the experimental data from the openliterature, we tried to avoid such stacking sequences andassumed that the stress field in the pristine compositedefines its strength. We shall consider the highly stressednear-hole volume, introduce a measure of critical failurevolume and discuss its physical meaning. Two types ofstress analysis will be considered: two-dimensional analy-sis, based on lamination theory, and three-dimensionalply level stress analysis based on displacement splineapproximation developed by Iarve [22].

2. Determination of the critical failure volume

2.1. Definitions and traditional Weibull integral

Consider a population of test samples each having a vol-ume V0 and loaded so that the state of stress in each sampleis uniform. Set r the single stress component or a combina-tion, such as the tensor polynomial function in Wu [11],which controls the strength of that sample. Weibull weaklink statistics is then based on the assumption that theprobability of failure for a sample of similar specimens ofarbitrary volume is

f ðr; V Þ ¼ 1� e� V

V 0BðrÞ

. ð1Þ

Eq. (1) can also be generalized [10] for nonuniform stressfields as

F ¼ 1� e� 1

V 0

RV

BðrðxÞÞ dv; ð2Þ

where dv = det(J)dxdydz in Cartesian coordinates and J isthe Jacobian matrix. Integral (2) is derived by subdividingthe nonuniformly loaded specimen into an infinite numberof infinitesimal cells, each being loaded by stress r(x),which can be assumed constant within the cell. The expo-nent in Eq. (2) expresses the probability of simultaneoussurvival of all mesh cells (each under a different stress),assuming that the probability of failure of each mesh cellis given by (1). Physically F can be interpreted as the prob-ability of damage initiation anywhere in the volume of thespecimen and therefore interpreted as an upper bound ofthe probability of failure. Eqs. (1) and (2) relate eventson the same scale. Indeed if the specimens tested to definethe function B(r) were unnotched quasi-isotropic lami-nates, then Eq. (2) can be used to estimate the upper boundof the probability of failure of the same notched quasi-iso-tropic laminate. In this case the volume integration is re-duced to 2D in-plane integral since the thicknessdirection in both cases is the same. The integrand in Eq.(2) is generally a very nonuniform function of coordinatesand often dominated by the contribution of only a smallvolume near the stress concentration. We shall discussthe physical meaning of such a critical volume in the nextsection. It is conceivable that this volume is small as com-pared to the characteristic size of the scale at which theproblem is considered. In the example at hand, i.e. notchedstrength of the quasi-isotropic laminates, the characteristicscale size is the thickness of the laminate. Indeed, if most ofthe contribution to integral (2) is coming from a small re-gion near the hole edge with a characteristic dimension ofless than a laminate thickness, the accuracy of the predic-tion is questionable, because of the 3D stress distributionin this region, which is neglected in the 2D (lamination the-ory) analysis.

In the case of the shape function B(r) in traditionalpower form

BðrÞ ¼ rb

� �a

; ð3Þ

one can rewrite Eq. (2) in the form

F ¼ 1� e�rwbð Þ

a

; ð4Þ

where

rw ¼1

V 0

ZV

rðxÞð Þa dv� �1=a

ð5Þ

is called the Weibull stress [29]. A direct observation whichcan be made from (5) is that in linear elasticity, Weibullstress is proportional to loading, and thus the distribution(4) and consequently (3) will have exactly the same shapeparameter a as (1). This was shown earlier by Wetherholdand Whitney [13], where instead of introducing the Weibullstress, a similar manipulation was applied to recalculate b.In the case of multiaxial stress state failure in composites isdescribed by a failure criteria, i.e. Tsai and Wu [12]. Suchformal extensions of Eqs. (1) and (2) are discussed by

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 177

Wu [11]. However, the conceptual difficulty is that differentfailure modes in composite materials do not exhibit similarstatistical/scaling properties as reviewed by Wisnom [30]and thus no physical basis was established to applicationof Eqs. (1) and (2) in general multiaxial loading cases.

2.2. Critical failure volume in Weibull media

In this section we shall discuss the physical meaning ofthe critical or most likely local failure region in the presenceof stress concentration. As mentioned previously, integral(2) provides the probability of failure initiation, i.e. failureof an infinitesimal volume. We shall determine below theprobability of failure of a finite volume and its size.

The failure or loss of load carrying capacity was definedonly for uniformly loaded specimens as their apparentstrength, described by distribution function (1). Transition-ing to a nonuniform stress state is based on the assumptionthat the probability of failure P of a nonuniformly stressedspecimen with stress distribution r(x) and the probabilityof failure (1) of the same specimen under uniform stateof stress ru are related

P P f ðru; V Þ; ð6Þ

if

ru ¼ minx2VðrðxÞÞ. ð7Þ

The estimate given by Eq. (6) is not very useful when ap-plied to the entire volume of the specimen. On the otherhand, one can select a finite region in the nonuniformlyloaded specimen, which has a volume Vi and minimumstress of ri, and calculate the probability of failure for thissubvolume f(ri,Vi). Suppose that we have found a subre-gion with volume Vc and minimum stress rc, for which thisprobability is the highest, i.e.

f ðrc; V cÞ ¼ maxi

f ðri; V iÞ; ð8Þ

where index i scans all subregions of the specimen. Thenthe subregion Vc will have the highest probability of localfailure, and we will call it critical failure volume (CFV).

Identification of the CFV and calculation of its failureprobability is easily performed in the case of finite maxi-mum stress. Although two-parameter function (3) is usedin this study, other more complex distributions may beconsidered as well. Such distributions, e.g. Gurvich et al.[28], may be helpful to capture more sophisticated nonlin-ear size effects, i.e. where relationships between logarithmsof average strength and size parameter are nonlinearfunctions.

Denote the finite maximum stress magnitude as rm.Introduce a set of isostress surfaces qirm, q0 = 1 > q1 >q2 > q3 � � � > 0. Consider a continuous function v(q),0 6 q 6 1:

vðqÞ ¼ volðV qÞ; x 2 V q () rðxÞP qrm

� �. ð9Þ

This function is equal to the volume of the specimen withstress higher or equal to qrm. In this case

V c ¼ vðqcÞ; rc ¼ qcrm; ð10Þ

where

f ðqcrm; vðqcÞÞ ¼ maxq

f ðqrm; vðqÞÞ. ð11Þ

The probability value (11) will be denoted fc for brevity

fc ¼ f ðqcrm; vðqcÞÞ. ð12Þ

Without limitation of generality one can assume that thestress distribution is continuous, which means that thefunction f(qrm,v(q)) is as well. Depending upon the stressdistribution which defines the volume function v(q), thisfunction can have complex shape. For a typical open holeproblem and shape function B(r) in the form (3), one ob-tains f = 0 for q = 0 and q = 1, meaning that the functionf(qrm,v(q)) will have at least one local maximum (f P 0)for 0 < q < 1. The fact that f = 0 for q = 1 follows fromthe premise that the maximum stress is attained at a pointassociated with zero volume, i.e. v(1) = 0.

For an arbitrary volume Vq with an isostress qrm

boundary, one can writeZV q

BðrðxÞÞdv P BðqrmÞV q;

where B(r) is a monotonic function, and qrm is the mini-mum value of stress in the integration volume. Thus theprobability F calculated by integral (2) always satisfiesthe inequality

F P fc. ð13Þ

Direct calculation shows that for homogeneous state ofstress, F and fc reduce to (1), where v(q) = V (total volume)for any q.

As will be shown in examples, the probabilities F and fc

will be close and approach each other for a!1. Besidesthe probability of failure, we will be interested in the aver-age values of strength resulting from criteria (2) and (12).As mentioned before, both criteria will result in Weibull-type stress distributions with the same shape parameter a.For Weibull distribution (1) with shape function (3), theaverage value of strength ra and the coefficient of variationx (standard variation divided by average value) are givenby well known equations:

ra ¼ bV 0

V

� �1=a

C 1þ 1

a

� �; ð14aÞ

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCð1þ 2=aÞC2ð1þ 1=aÞ

� 1

s; ð14bÞ

where C-denotes the gamma function. By using Eqs. (1), (3)and (14a), one can find the average strength value for aknown a if the probability of failure is known for justone value of r, i.e. f is equal to f1 for r = r1. In this case

178 E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185

ra ¼ r1 � lnð1� f1Þð Þ�1=aC 1þ 1

a

� �. ð15Þ

Eq. (15) will be used to calculate the average strength forboth the upper F and lower fc estimates of the probabilityof failure, which will inversely result in lower and upperaverage strength estimates. Finally, it is worth mentioningthat Eq. (14a) provides a formal way to calculate the Wei-bull modulus a for a set of experimental data with knowncoefficient of variation by solving a nonlinear equation.

3. Stress analysis and v(q) function calculation

Consider a rectangular orthotropic plate containing acircular hole having a diameter D, as shown in Fig. 1.The plate consists of N plies of total thickness H in the z-direction and has a length L in the x-direction and widthA in the y-direction. The following displacement boundaryconditions were applied to the specimen on lateral sides

� uxð0; y; zÞ ¼ uxðL; y; zÞ ¼ e0L=2;

uyð0; 0; 0Þ ¼ uzðx; y; 0Þ ¼ 0.ð16Þ

Traction-free boundary conditions are present on all othersurfaces. The dimensionless loading parameter e0 corre-sponds to relative elongation of the specimen. The z-direc-tion displacement component on the bottom surface isconstrained due to symmetric lay-up of the laminates con-sidered, which allows one to model only half of the speci-men. The constitutive relations of each ply are as follows:

rij ¼ Cpijklðekl � ap

klDT Þ; i ¼ 1; . . . ;N ;

where Cpijkl and ap

kl are elastic moduli and thermal expan-sion coefficients of the pth orthotropic ply, and DT is thetemperature change. The average applied traction was thencalculated as

r0 ¼Z

y;zrxxð0; y; zÞdy dz. ð17Þ

A cylindrical coordinate system is defined originating fromthe center of the hole:

x ¼ r cos hþ xc; y ¼ r sin hþ yc; z ¼ z; ð18Þwhere xc, yc are the coordinates of the center of the hole.

Three-dimensional ply level stress analysis in realisticcomposite laminates containing holes represents a formida-

Fig. 1. Schematics of the open hole tension specimen.

ble problem if using standard finite element programs. A B-spline displacement approximation approach developed byIarve [22] was shown to provide highly accurate stress solu-tions in the immediate vicinities of the ply interface andhole edge intersections, where there is singular stressbehavior.

Three-dimensional approximation is built by using thetensor product of one-dimensional approximations. Con-sider an elementary cube [0, 1]3 in local x1, x2, x3 coordinatesystem, then the 3D displacement approximation can bewritten as

uðx1x2x3Þ ¼X

i

Xj

Xk

X iðx1ÞY jðx2ÞZkðx3ÞUijk; ð19Þ

where u is the displacement vector and Uijk are vectors ofdisplacement approximation coefficients not necessarilyassociated with nodal displacements, and indexes i, j andk in Eq. (19) change from 1 to the total number of approx-imation functions in each direction. Depending upon theapplication and geometry, different orders of splines (from1 to 8) can be used in each direction. Besides changing theorder of splines, one can also change their defect (maxi-mum number of discontinuous derivatives) in the node,thus being able to apply standard linear or a higher orderp-type finite element approximation if desired. Curvilinearcoordinate transformation x = x(x1x2x3), xT = (x,y,z)with Jacobian matrix J(x1x2x3) is used to map the unit vol-ume into the global x, y, z, coordinate system. The Gauss-ian integration procedure is used to calculate thecomponents of the stiffness matrix. For the details of vari-ational formulation and detailed verification of stress pre-diction accuracy, the reader is referred to Iarve [22]. Forpurposes of the present study, we shall describe the proce-dure of calculating the overstressed volume function v(q).After the solution is completed and all vectors Uijk aredetermined, a post-processing step is performed when eachintegration point of the structure is examined twice. Firstthe stress and strain components are computed, and themaximum value rm of the component of interest is foundby searching through all integration points. A large num-ber M (in our analysis M = 101 and 201) is then prescribed,and a sequence

qi ¼ 1� i=M ; i ¼ 0; . . . ;M

defined. The overstressed volume function v(q) is then cal-culated in M points as

vðqiÞ ¼X

g1

Xg2

Xg3

wg1wg2

wg3det Jðxg1

1 ; xg22 ; x

g33 Þgðr� qirmÞ

ð20Þby using the Heaviside step function

gðrÞ ¼1; r > 0;

0; r 6 0.

�ð21Þ

In Eq. (20) indexes gi, i = 1, 2, 3 denote Gauss integrationpoints in x1, x2 and x3 directions, respectively, and wgi

arerespective Gaussian weights. Step function (21) cuts off the

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 179

contribution from all integration points where the stress islower than the threshold qirm. For low values of the thresh-old value, v(q) will include almost all integration points in(20) and become close to the entire volume. The probabilityfc is then calculated according to Eqs. (11) and (12) byscanning through all f(qi) values. The integral in Eq. (2)can be calculated directly by using a formula similar to(20), where the step function is replaced byBðrðxg1

1 ; xg22 ; x

g33 ÞÞ.

4. Homogeneous plate with a hole

4.1. Variation of CFV location and size and comparison with

Weibull integral method

The size and location of the CFV as functions of scatterof strength will be investigated in homogeneous isotropicplates with an open hole. The specific size and elastic mod-ulus of the material are not of essence for this study,although the properties were those of neat epoxy resin pla-ques considered in the next section.

The overstressed volume function v(q) normalized tospecimen volume for a typical open hole coupon under uni-axial loading with boundary conditions (18) is shown inFig. 2. Two scales are used: the solid line depicts the func-tion for the entire specimen (left scale), and the dashed lineshows the same function on the smaller scale (right-handside scale), which corresponds to higher stress regions nearthe hole. Step-like behavior in the proximity of q = 0.261means that a significant volume of the specimen is experi-encing a homogeneous state of stress. The fact that q islower than 1/3 reflects the finite geometry of the plate,resulting in stress concentration equal to KT = 3.84 forthe present geometry.

0

0.2

0.4

0.6

0.8

1

0

0.002

0.004

0.006

0.008

0.01

0 0.2 0.4 0.6 0.8 1

v(q)/V0

q

q=0.261

func

tion

v(q)

for

ent

ire s

peci

men

function v(q) for near hole region

v(q)/V0

Fig. 2. Function v(q) for an isotropic plate with a hole, overall shapeshown with solid line and the near-hole shown with dashed line.

Function f(qrm,v(q)) in Eq. (11) for the specimen athand is shown in Fig. 3 for two different values of Weibullmodulus a. For a = 5 (hypothetic material with very large23% coefficient of strength variation) qc = 0.261, whichcorresponds to the uniformly loaded large area away fromthe hole, meaning that most failures of the open hole spec-imen will not happen throughout the hole at all. The CFVcorresponding to this case is very large, and according toFig. 2 (solid curve) is comparable to the volume of the spec-imen. For a less variable material with a = 9(c.v. � 13.2%), we find qc = 0.838, which corresponds tothe near hole area. In this case Vc is on the order of lessthan 0.001V0, as can be seen in Fig. 2.

The average values of notched strength predicted byWeibull integral criterion (2) and CFV criterion (11) and(12) are compared in Fig. 4 as functions of Weibull modu-lus a. In both cases the average strength was calculated byusing Eq. (15) and normalized to average strength of theunnotched specimens. The error bars show +/� one stan-dard deviation bounds. The percentage values correspond-ing to selected values of Weibull modulus a are therespective values of the coefficient of variation. Asexpected, the Weibull integral (2) and the CFV criterionconverge to each other with increasing a (decreasing thec.v.). Only for the smallest a the predicted means are justoutside the one standard variation bounds from each other.The deterministic 1/KT estimate is approximately 20%more conservative than the statistics-based prediction evenfor a = 50 (c.v. = 2.5%).

4.2. Experimental data

Prior to considering the strength of laminated compos-ites, a set of experiments on neat epoxy plaques curedaccording to manufacturer’s recommendations was con-ducted. All specimens were cut out of 3.0 mm-thick panels.Two sets of 20 unnotched specimens having dimensions ofL = 5.08 cm long · W = 1.27 cm wide and L/2 · W/2 were

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

f(qσm,v(q)), α=9 (c.v.=13.2%)

f(qσm,v(q)), α=5 (c.v.=23%)

q

q=0.261

Pro

babi

lity

of fa

ilure

Fig. 3. Probability of finite volume failure function f(qm,v(q)) for twovalues of Weibull modulus.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

fc

F

1/KT

Mea

n N

otch

ed s

tren

gth

and

varia

tion

Weibull modulus - α

20% 10% 5% 2.5%

Fig. 4. Average failure load and standard deviation estimated by usingWeibull integral (2)—F and estimated by using CWV criterion (11), (12)—fc.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Pro

babi

lity

of F

ailu

re

Normalized Strength

F, βF

fc, βfc

XT/KT

β1

β=84.1MPa

notc

hed

str

engt

h

unno

tche

d st

reng

th, V

1

unno

tche

d st

reng

th, V

0

Fig. 5. Experimental data and predictions of tensile strength in neat resinplaques with and without open hole. Test points: j—control volumeunnotched specimens, m—small volume unnotched specimens, s—openhole specimens. All solid curves are Weibull distributions with a = 10.7.Strength is normalized by that of the unnotched specimens XT.

180 E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185

tested in tension in the direction of length. In addition a setof 20 notched specimens with in-plane dimensions L · W,containing a D = 5.08 mm center hole were also tested.In all cases the length given is equal to the gage sectionlength, whereas the specimens contained additional2.54 cm tabs on each end. The elastic stiffness propertieswere E = 3.79 GPa and m = 0.39. The average tensilestrength of the control volume V0 = 1.93 cm3 (L · W) spec-imens was XT = 80 MPa with a coefficient of variation(c.v.) of 13.4% (corresponds to a = 9). The second set ofunnotched specimens of smaller dimension and volumeV1 = V0/4 had an average strength of xT = 91 MPa andc.v. of 12% (corresponds to a = 10). Three panels were pro-duced and showed similar coefficient of strength variationand average strength. A particular difficulty in definitionof the unnotched strength distribution is defining the Wei-bull shape parameter, which reflects the intrinsic variationof strength rather than that caused by experimentation. Inpresent work we are following the methodology describedby Bazant [27] and determine a from the average strengthvalues of the two sets of specimens of different volumes as

a ¼ lnðX T=xTÞ= lnðV 1=V 0Þ.

It results in a = 10.7, which corresponds to c.v. of 11.3%and is quite close to the c.v. of the experimental valuesfor either set of unnotched data. The strength of the openhole specimens was predicted by using the unnotchedstrength distribution with b = 84.1 MPa, a = 10.7 andV0 = 1.93 cm3, where the parameter b was calculated fromEq. (14a) based on the average strength value of XT. Theaverage open hole strength predicted by using Weibull inte-gral method and CFV was rave,F = 44.7 MPa and rave,fc =52.4 MPa, respectively, whereas the average value of exper-imental data is rave = 46.9 MPa.

The experimental data and predictions are shown inFig. 5. The solid squares and triangles correspond to exper-imentally measured strength of unnotched specimens of

control volume and small volume specimens respectively.The open circles show experimentally measured strengthof the specimens with open holes. The continuous curvesin Fig. 5 are Weibull distributions with the same shapeparameter a = 10.7. The one closely fitting the control vol-ume unnotched data has b = 84.1 MPa, and the one closelyfitting small volume unnotched data has b1 = bXT/xT. Thepredicted open hole specimen strength distributions(bF = 47.5 MPa, bfc = 53.7 MPa) appear to bound theexperimental data.

A note must be taken that the results of strength predic-tion are highly sensitive to values of Weibull modulus a. Anaccurate determination of the Weibull modulus requires asignificant number of unnotched specimens to be tested,avoiding strength values, which result from grip failures.In principle, however, the question of predicting notchedstrength based on testing exclusively unnotched samplesis answered. Overall, the Weibull statistics-based strengthprediction provides a physically sound indisputableimprovement upon the stress concentration factor-basedvalue of XT/KT, which is also shown in Fig. 5.

5. Notched strength prediction in quasi-isotropic laminates

The present section is devoted to evaluation of probabil-ity of failure in quasi-isotropic laminates with open holes.Experimental data reported by Whitney and Kim [26] wereused in the present study. The T300/934 composite and twostacking sequences, [±45/0/90]s and [90/0/±45]s, were con-sidered. Unnotched strength, open hole strength with threedifferent hole sizes of D = d, D = 3d and D = 6d, whered = 2.54 mm (0.100, 0.300 and 0.600), and strength with threesizes of through-the-thickness cracks were evaluated on a

Table 1Material stiffness properties used in the analysis

[0] [90/0/�45/45]s

EL (GPa) 146.7 54.9

ET (GPa) 11

Ez (GPa) 11vLT 0.3 0.3

vLz 0.3vzT 0.33GLT (GPa) 4.82 20.7

GLz (GPa) 4.82GzT (GPa) 4.82

The data reported in Ref. [26] is typed in bold, and the data added for 3Danalysis is typed in regular font.

0.5

0.75

[90/0/45/-45]s[45/-45/0/90]s

z/H0 - ply

0 - ply

top surface

2-D solution

1

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 181

total of 296 tests. Both stacking sequences showed verysimilar notched strength with 10% difference in the unnot-ched strength due to reportedly extensive delamination infinal stages of failure of the unnotched [±45/0/90]s lami-nates. The stress–strain curves reported for the unnotched[90/0/±45]s laminate were linear until failure, and that iswhy this stacking sequence was chosen for the present anal-ysis. The ply and laminate stiffness properties used for plylevel analysis and for lamination level analysis are shown inTable 1. The data reported in Ref. [26] are shown in a boldfont, and the additional data added by the authors for stiff-ness properties in the out-of-plane z-direction are shown ina normal font. The geometry of the specimens had a gagelength of L = 22.86 cm in the x-direction and a width ofW = w, W = 1.5w and W = 2w in the y-direction, wherew = 2.54 cm for hole sizes of d, 3d and 6d, respectively.All notched strength data in the cited report were reportedafter the finite width correction factor was applied to exper-imental data. For the purposes of present comparison theactual experimental values were restored by using the cor-rection factors from Ref. [2] and dividing the reported cor-rected values of strength by 1.01, 1.05 and 1.12 for the holediameter to width ratios of D/W = 0.1, 0.2 and 0.3,respectively.

In order to perform the strength predictions one needsto define the Weibull parameters for strength in the fiberdirection. The experimental data from Whitney and Kim[26] for unnotched and notched strength relevant to presentanalysis is summarized in Table 2. As in the previous sec-tion the difficulty is in obtaining the shape parameter ofthe strength distribution, which characterizes the intrinsicstrength variation. As seen in Table 2 the coefficient of var-

Table 2Experimental data from Ref. [26] on unnotched and uncorrected notchedstrength for [90/0/�45/45]s laminate

ravg (MPa) c.v. (%) a

Unnotched 500.2 7.8 14.70.100 Hole 315.8 5.6 20.10.300 Hole 259.2 4.2 27.20.600 Hole 208.0 6.2 18.1

iation of strength obtained experimentally for unnotchedand notched specimens with different hole sizes varies sig-nificantly. There is no monotonic trend, which can beestablished for the coefficient of variation as a function ofhole size, except that it is lower than for unnotched speci-mens. Based on the range given in Table 2, the analysiswas carried out for the following values of a = 14.7, 18,24 and 30.

The failure prediction was performed by using maxi-mum fiber direction strain. Eq. (1) was replaced with

f ðV ; eÞ ¼ 1� e� V

V 0

ebe

� a

. ð22ÞWeibull parameter be was obtained as be = b/Equasi, whereEquasi is the laminate tensile modulus given in Table 1. Thevalue of b was calculated for each value of a by using Eq.(14a) for ra = 499.2 MPa. The dimensions of the unnot-ched specimens were assumed to be the same as in Ref.[2] and were L = 10.125 cm, W = 1.905 cm.

Three-dimensional distribution of the stress–strain fieldsat the hole edge has been extensively studied analytically[21–23] and experimentally Mollenhauer et al. [24]. Exper-imental investigation of the strain distribution at the holeedge was performed by means of Moire interferometrydeveloped by Mollenhauer and Reifsnider [25]. Ply level(3D) stress analysis based on the displacement splineapproximation approach Iarve [22] described above wasperformed for the [90/0/ ± 45]s laminates with all threehole sizes. Fig. 6 illustrates the nonuniformity of the fiberstrain distribution at the hole edge at the h = 90� location,where the angle h was defined by Eq. (18) and shown inFig. 6. At this location the fiber direction strain coincideswith the circumferential strain ehh. The strain value is nor-malized to the specimen elongation e0 defined in Eq. (16).A significant strain increase in the 0� plies for both

2.5 3 3.5 4 4.5 50

0.25

Normalized circumferential strain

midsurface

Fig. 6. Through the thickness distribution of the circumferential straincomponent at open hole edge of quasi-isotropic laminates with D = d holeat h = 90� circumferential location.

182 E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185

quasi-isotropic stacking sequences over the lamination the-ory-based 2D constant value of 3.02 can be seen. At thesame time the strain distributions in the 0� plies of thetwo laminates are the same, which is in agreement withthe practically equal notched strength values reported forthe two laminates in Ref. [26].

The comparison of the average strength values predictedby using ply level (3D) analysis and lamination theorybased (2D) analysis with the experimentally obtained aver-age values is shown in Fig. 7. All strength values are nor-malized by average unnotched strength value ofra = 499.2 MPa. Thick solid lines connect the experimentaldata and the thin lines connect the predicted values. Thepredictions based on Weibull integral (2) are marked withF on the left and contain triangle symbol pointing up,whereas the CFV prediction contains triangle symbolpointing down. Four pairs of curves of predicted values

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6

Nor

mal

ized

Str

engt

h

Normalized Hole diameter D/d

F

fc

F

F

F

fc

fc

fc

α=14.7

α=18α=24

α=30

2D stress analysis

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6

Nor

mal

ized

Str

engt

h

Hole diameter D/d

F

fc

F

F

F

fc

fc

fc

α=14.7

α=18

α=24

α=30

3D stress analysis

1 3 5 7

1 3 5 7(a)

(b)

Fig. 7. Weibull integral and CFV based predictions of average strength ofquasi-isotropic laminates for four values of a. Solid curves correspond toa = 18 and 30, whereas dashed lines to a = 14.7 and 24.

corresponding to different values of a are shown. Thetrends which can be seen on the two graphics in Fig. 7aand b show that for larger hole sizes the 3D and 2D anal-ysis based strength predictions are very close especially forlow values of a. The second trend, namely that the Weibullintegral and CFV based predictions converge with increas-ing a can also be seen. However, the more important trend,which is unambiguously observed, is that all predictionssignificantly underestimate the hole size effect on strength.

This observation can be explained by examining the sizeof the CFV as a function of hole diameter and a. We shallintroduce a linear measure lc of the CFV by assuming it tohave a square in-plane cross-section and span through thethickness of the ply so that

lc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV c=h=2

p. ð23Þ

Estimation of the linear dimension of the CFV is importantin order to compare it with micromechanical scale param-eters such as fiber diameter and ineffective length intro-duced by Rosen [20]. The size lc, defined by Eq. (23) fordifferent holes sizes is shown in Table 3, where the fist num-ber corresponds to 2D analysis and the second (after for-ward slash) to 3D analysis. The values of lc decreasefrom around 0.8 mm to as small as 0.01 mm for small holesand large values of a. The latter value is significantly belowthe estimate of so called ineffective length d [20], where

d ¼ 1:14d f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffiffiffiffimfp � 1

s ffiffiffiffiffiffiffiEf

Gm

r. ð24Þ

For the T300/934 material we estimate the value ofd = 0.0538 mm for fiber modulus Ef = 206.8 GPa (T300),matrix shear modulus Gm = 1.31 GPa (934 Epoxy), fibervolume fraction vf = 0.6 and fiber diameter of df = 7 lm.The strength of the unidirectional composite cannot bescaled by using Eq. (22) down to dimensions of less than2d, because for such ultra short composites the process offiber macrocrack formation will no longer be self similarto larger composites. However, even 2d appears to be be-low the limit of applicability of Eq. (22) . A Monte-Carlosimulation and shear lag stress analysis based study of limitsizes for Weibull scaling of unidirectional compositestrength was recently performed by Landis et al. [31]. Itwas shown that for composites with fiber strength scattercorresponding to Weibull modulus of a = 10 (carbon fibers5–6) and length l0 = 3d significantly over predict thestrength of longer composites with l0 = 6d and l0 = 9d,when Weibull scaled. The intermediate value of l0 = 6d(�46df) was considered as the minimum scalable length.It is also worth mentioning that Landis et al. [31] investi-gated the size of the minimum Weibull scalable cross-sec-tion, which was on the order of 30 · 30 fibers. Thus theseare the particular dimensions in the fiber direction and inthe direction perpendicular to fibers which matter ratherthan the volume. The CFV method allows in principle to

Table 3Linear size in mm of CFV for notched laminates (2D/3D)

a = 14.7 a = 18 a = 24 a = 30

0.100 Hole 0.140/0.0813 0.106/0.056 0.0933/0.0424 0.0718/0.01050.300 Hole 0.422/0.263 0.353/0.183 0.289/0.137 0.220/0.03200.600 Hole 0.848/0.707 0.707/0.462 0.572/0.299 0.420/0.079

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 183

obtain its individual dimensions in different directions.However, in the present study we are using a simple esti-mate given by Eq. (23) and comparing it to the largest ofthe scalable volume dimensions from Landis et al. [31].

The CFV method can be readily modified to accountfor additional physical considerations, such as presenceof a limit scaling volume and define the probability of fail-ure as

f ðl0Þ ¼fc; lc P l0;

f ðvðq0Þ; q0Þ; l0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðq0Þ=h=2

p; lc < l0.

(ð25Þ

So that when CFV has a linear dimension larger then l0then the probability of failure coincides with Eq. (11) andelse with that of the volume corresponding to dimensionl0. In the present study we will consider the same valuesof l0 = 3d, 6d, and 9d as in Landis et al. [31].

The evaluation of f(l0) is illustrated in Fig. 8 showing theprobability of local failure f(v(q), q)) and the linear size ofthe corresponding volume l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðqÞ=h=2

pas functions of

q. The hole diameter is 7.62 mm and a = 24. The originalCFV is defined on the bell-shaped f(v(q),q) curve by itsmaximum and corresponds to qc = 0.8775, which yieldsfc = 0.34 and the average failure stress of 216.5 MPa. Thesize of the CFV can be obtained from the l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðqÞ=h=2

pcurve at q = qc and is equal to lc = 0.137 mm, which isapparently smaller then l0 = 6d (6d = 0.323 mm). Accord-ing to Ref. [31] the strength of such a small volume cannotbe estimated based on Weibull scaling and is in fact higher.The best one can do without precise knowledge of strength

0

0.4

0.8

1.2

1.6

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

l=(v(q)/h/2)^0.5

f(v(q),q)

q-argument

l0=0.323mm

fc

f(lo)

Line

ar R

epre

sent

atio

n of

Vol

ume

(mm

)

Probability of F

ailure

lc=0.137mm

FailureProbability Difference

Fig. 8. Modification of the CFV to take into account micromechanicsbased limit of Weibull scaling.

values of such ultra small volumes is to evaluate the prob-ability of failure of a larger volume, which has a lineardimension of l0 = 6d (6d = 0.323 mm). Now we followthe dashed horizontal line from left to right and find avalue of q = 0.78, which corresponds to l0 = 6d and thenthe associated probability of failure f(l0) = 0.175, whichyields the average failure strength value of 241.3 MPa.

Revised predictions by using the criterion (25) for allhole sizes and values of a are shown in Fig. 9a–c. Theresults obtained by using 2D analysis (up pointing trian-gles, dashed line) and 3D ply level analysis (down pointingtriangles, solid lines) are shown for l0 = 3d, 6d, and 9drespectively. In all cases the Weibull modulus values ofa = 14.7, 18, 24 and 30 were examined. It is apparent thatthe introduction of l0 = 6d as a limiting size of Weibullscaling yields results that capture the magnitude of experi-mentally observed strength increase with the hole sizereduction. The values of a in the mid 20s range, are alsoconsistent with the 23–29 values cited for similar materialsby Wisnom [30]. It is not surprising that criterion (25)yields practically similar results for stress analysis per-formed by using lamination theory as well as ply level anal-ysis, since 6d � 3h. It is also important to point out that nosubcritical damage such as matrix cracking and delamina-tions were taken into account. In most laminates, however,modeling of such damage is critical and can only be accom-plished by using 3D ply level stress analysis.

It is worth mentioning that similar correction of the pre-dictions based on Weibull integral (2) by taking intoaccount micromechanical process zone is not equallystraight forward and leads to considerations such as non-local Weibull theory by Bazant [27], which was proposedin the 1990s.

The appearance of an additional parameter, such as thelength scale l0 = 6d in the present analysis is certainly dis-appointing from the stand point that its direct measure-ment is problematic. It is however, interesting to compareits magnitude l0 = 0.323 mm with the size of the character-istic distance d0 = 1.016 mm in the point stress failure crite-rion applied by Whitney and Kim [26] to the hole size effectanalysis of the same problem. The latter dimension is notunexpectedly higher because it combines the effects of Wei-bull scaling of unidirectional composite strength as well asmicroscopic fiber failure process zone effect. The proposedmethod in some way separates the ply level effects and thefiber level effects, which are responsible for collective ornonlocal type of failure, requiring certain finite volume tobe overstressed in order to occur.

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6

Nor

mal

ized

Str

engt

h

Normalized hole diameter D/d

α=14.7

α=18

α=24

α=30

l0=3δ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Nor

mal

ized

Str

engt

h

Normalized hole diameter D/d

α=14.7

α=18

α=24

α=30

l0=6δ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6

Nor

mal

ized

Str

engt

h

Normalized hole diameter D/d

α=14.7

α=18

α=24

α=30

l0=9δ

1 3 5 7

1 3 5 7

(a)

(b)

(c)

Fig. 9. CFV based prediction of tensile average strength in quasi-isotropiclaminates with micromechanics based limit of Weibull scaling. Solid linescorrespond to predictions made based on 3D stress analysis and thedashed lines to those based on lamination theory.

184 E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185

6. Conclusions

The critical failure volume method is proposed. The CFVwas defined as a subvolume, which has the highest probabil-ity of local failure. The failure in this case is defined as lossof load carrying capacity of this subvolume, rather than apoint failure event. An algorithm for identifying this region,based on isostress surface parameterization was proposed.It was shown that the probability of the subvolume failureat a given load is always lower then the probability of failurepredicted by Weibull integral criterion for the same subvo-lume. The two coincide for uniform stress fields and con-verge to each other when the scatter of strength reduces.

Notched strength of homogeneous neat epoxy resin pla-ques was studied experimentally and predicted based onWeibull integral and CFV methods. The open hole strengthwas predicted exclusively based on strength data measuredon unnotched specimens. The average strength of open holespecimens predicted by the two methods differed by approx-imately 11% and provided a good agreement with averagestrength measured experimentally. Experimental data wastightly bounded by two predictions, with the tail of distribu-tion being closer to CFV method and the high probabilityof failure data to that predicted by Weibull integral. Overallin the case of homogeneous material the traditional Weibullintegral method performed very satisfactory and the CFVprovided a tight lower bound on the failure probability.

The effect of the open hole size on strength of quasi-iso-tropic graphite-epoxy laminates was considered. The Wei-bull integral method and classical Weibull strength basedCFV showed that both methods are not capable of describ-ing the experimentally observed magnitude of the hole sizeeffect on strength. However, the analysis based on CFVidentified the size of the most likely failure region for smallhole sizes to be smaller then the estimated lower limit oflength scale at which the unidirectional composite strengthcan be scaled. The stress contour based framework of CFVmethod was modified to include such a limit scaling volumeand the revised results showed good correlation with exper-imental data. Similar modifications of the Weibull integralmethod are not equally straight forward and lead to non-local formulations, see Bazant [27].

Acknowledgments

This work was sponsored by the Air Force ResearchLaboratory, under contract number F33615-00-D-5006.The authors are grateful to Prof. O. Ochoa for considerablerevision of the manuscript and Dr. N. Pagano and Dr. M.Gurvich for comments and discussion.

References

[1] Waddoups ME, Eisenmann J, Kaminski BE. Macroscopic fracturemechanics of advanced composite material. J Compos Mater1971;5:446–55.

E.V. Iarve et al. / Composites: Part A 38 (2007) 174–185 185

[2] Nuismer RJ, Whitney JM. Uniaxial failure of composite laminatescontaining stress concentrations. Fract Mech Compos, ASTM STP1974;593:117–42.

[3] Lee J. Three-dimensional finite element analysis of damage accumu-lation in laminated composite. In: Sih GC, Tamuz VP, editors.Strength and fracture in composite materials. Latvia: Zinatne; 1983.p. 175–86.

[4] Talreja R. A continuum mechanics characterization of damage incomposite materials. Proc R Soc Lond 1985;399A:126–216.

[5] Lee JW, Allen DH, Harris CE. Internal state variable approach forpredicting stiffness reductions in fibrous laminated composites withmatrix cracks. J Compos Mater 1989;23:1273–91.

[6] Nguyen BN. Three-dimensional modeling of damage in laminatedcomposites containing a central hole. J Compos Mater 1997;31(17):1672–94.

[7] Tan SC. A progressive failure model for composite laminatescontaining openings. J Compos Mater 1991;25:556–77.

[8] Camanho PP, Matthews FL. A progressive damage model formechanically fastened joints in composite laminates. J Compos Mater1999;33(24):2248–80.

[9] Shahid I, Chang F-K. Accumulative damage model for tensile andshear failures of laminated composite plates. J Compos Mater1995;29(7):926–81.

[10] Weibull W. A statistical theory of the strength of materials. IngeniorsVetenskaps Akademien Handlingar 1939(151):45.

[11] Wu EM. Failure analysis of composites with stress gradients. 1978,UCRL-80909.

[12] Tsai SW, Wu EM. General theory of strength of composite materials.J Compos Mater 1971;5:58–80.

[13] Wetherhold RC, Whitney JM. Tensile failure of notched fiberreinforced composite materials. Polym Compos 1981;2(3):112–5.

[14] Wetherhold RC. The effect of stress concentration on the reliability ofcomposite materials. J Compos Mater 1985;19:19–28.

[15] Wetherhold RC, Pipes RB. Statistics of fracture of compositematerials under multiaxial loading. Mater Sci Eng 1984;68:113–118.

[16] Gurvich MR, Pipes RB. Reliability of composites in a random stressstate. Compos Sci Technol 1998;58:871–81.

[17] Gurvich MR. Strength size effect for anisotropic brittle materialsunder random stress state. Compos Sci Technol 1999;59(11):1701–11.

[18] Kortshot MT, Beaumont PWR. Damage mechanics of compositematerials: II – A damage-based notched strength model. Compos SciTechnol 1990;39:303–26.

[19] De Morais AB. Open-hole tensile strength of quasi-isotropic lami-nates. Compos Sci Technol 2000;60(10):1997–2004.

[20] Rosen BW. Tensile failure of fibrous composites. AIAA J1964;2:1985–91.

[21] Iarve EV, Mollenhauer DH, Kim R. Theoretical and experimentalinvestigation of stress redistribution in open hole composite laminatesdue to damage accumulation. Composites Part A 2005;36:163–71.

[22] Iarve EV. Spline variational three-dimensional stress analysis oflaminated composite plates with open holes. Int J Solids Struct1996;44(14):2095–118.

[23] Iarve EV, Pagano NJ. Singular full-field stresses in compositelaminates with open holes. Int J Solids Struct 2001;38(1):1–28.

[24] Mollenhauer D, Iarve EV, Kim R, Langley B. Examination of plycracking in composite laminates with open-holes: a Moire interfer-ometric and numerical study. Composites: Part A 2006;37(2):282–94.

[25] Mollenhauer DH, Reifsnider KL. Measurements of interlaminardeformation along the cylindrical surface of a hole in laminatedcomposite materials by Moire interferometry. J Compos Sci Technol2000;60(12–13):2375–88.

[26] Whitney JM, Kim RY. Effect of stacking sequence on the notchedstrength of laminated composites. 1976, AFML-TR-76-177.

[27] Bazant ZP. Scaling of structural strength. London: Hermes PentonScience (Kogan Page Science); 2002.

[28] Gurvich MR, DiBenedetto AT, Ranade SV. A new statisticaldistribution for characterization the random strength of brittlematerials. J Mater Sci 1997;32(10):2559–64.

[29] Beremin FM. A local criterion for cleavage fracture of a nuclearpressure vessel steel. Metall Trans A 1983;14:2277–87.

[30] Wisnom MR. Size effects in the testing of fibre-composite materialsreview. Compos Sci Technol 1999;59:1937–57.

[31] Landis CM, Beyerlin IJ, McMeeking RM. Micromechanical simula-tion of the failure of fiber reinforced composites. Mech Phys Solids2000;48:621–48.