[American Institute of Aeronautics and Astronautics 47th AIAA/ASME/ASCE/AHS/ASC Structures,...

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, Rhode Island, 1-4 May 2006

Failure Modeling and Simulation of Composites Subjected toBypass and Bearing Loads

Vinay K. Goyal∗, Jacob I. Rome, John C. Klug,Dick J. Chang, Howard A. Katzman, and Franklin D. Ross

The Aerospace Corporation, El Segundo, CA 90245

An analytical model to predict the bypass and bearing strength of composites is developed. The modelincludes a contact algorithm, finite deformation theory, nonlinear material behavior, and the progressive dam-age of composites. The orientation and density of matrix-cracks and fiber damage are represented througha damage tensor, an internal state tensor that is thermodynamically consistent within the framework of con-tinuum damage mechanics. The damage tensor evolves as damage associated with different failure modesaccumulates. Matrix-cracks are predicted using a three-dimensional failure criteria, which is based on thestresses acting on a potential fracture plane. For use with the failure criteria, a new anisotropic shear consti-tutive law is postulated governing the transverse and the in-plane shear nonlinear mechanical behavior. Forthe prediction of initiation and progression of multiple delaminations through the thickness of the composite, acohesive-decohesive constitutive law is adopted. The analytical model is implemented in a finite element com-mercial code via user subroutines. The problem of a notched composite loaded in tension is examined. Thegeneral characteristics of the predicted failure modes are in good agreement with the experimental observa-tions obtained from the open literature. Numerical simulations are also conducted for the filled hole tension,double sided bypass and bearing load, and single sided bypass and bearing test configurations. The predictedstrength and the strain behavior is in good correlation with in-house experimental data. The predicted failuremodes are consistent with observations made in the open literature.

I. Introduction

A common challenge in the development of aircraft and spacecraft structures is maintaining structural integrityin the presence of mechanically fastened joints. This challenge is amplified when the structures include compositelaminates, which have shortcomings from a microscopical and macroscopical standpoint. In the micro-scale, stressconcentrations develop between stiff fibers and relatively compliant matrix material, and in the macro-scale stressconcentrations develop near discontinuities such as a drilled hole. These stress concentrations lead to failure modesthat can initiate at loads below the ultimate strength of the composite material.

The focus of this paper is to quantify joint strength via an improved simulation of failure modes in compositestructures. The design and analysis of bolted joints in composite structures is complex and uncertain because failureloads depend on a combination of factors such as material selection, stacking sequence, bolt clamping force, loadingvector, geometric configuration, and manufacturing defects. The contact between the bolt and the bolt hole may inducelarge strains and high stress concentrations in the vicinity of the bolt hole boundary. Eventually an accumulation oflocalized failures – such as fiber failure, delamination, and matrix-cracks – that propagate from the edge of the bolthole leads to the ultimate failure of the mechanically fastened joint. This complexity requires analytical procedures thatcan account for these variables in order to reasonably predict the response of new structures. Analytical work needs tobe developed based on lessons learned from the extensive experimental work characterizing failure mechanisms thatoccur in mechanically fastened composite joints.1–6

Failure modeling of composite joints is challenging due to the multi-scale damage mechanisms that occur in com-posite laminates: micro-cracks in the micro-scale, and delaminations and through-the-thickness cracks in the macro-scale. Significant work has been conducted in modeling and simulating the failure of composites using progressive

∗Corresponding author. Tel.:+1-310-336-6315. E-mail: [email protected] c© 2006 by The Aerospace Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

Covered by the Master Copyright License from The Aerospace Corporation to AIAA.

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American Institute of Aeronautics and AstronauticsAIAA-2006-2172

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

AIAA 2006-2172

Copyright © 2006 by The Aerospace Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

failure methodology (PFM). This follows from the pioneering work by Chaboche7 and Chang and Chang,8 who de-veloped a PFM within the context of continuum damage mechanics and incorporated it in the finite element method.PFM is capable of determining failure mechanisms, the path of sub-critical damage growth, the ultimate strength andthe residual strength. Within PFM, fiber failure, matrix-cracking, and delamination failure modes are predicted usingcriteria, and degradation rules are applied to simulate failure initiation and progression for these modes. An extensiveliterature review of investigations that have been made both experimentally and analytically on the stress and strengthanalysis of mechanically fastened joints in fibre-reinforced plastics is presented by Camanho and Matthews.9 PFM de-veloped prior to 199710–12and even more recently13,14 for the analysis of mechanically fastened joints consider onlyfailure mechanisms that occur in the plane of the laminate. These two-dimensional models only account for stressconcentration relief prior to failure in the plane of the laminate, and cannot model the three dimensional failure modesoccurring through-the-thickness of the laminate. However, their work proved sufficient in demonstrating that model-ing damage accumulation of various damage modes with PFM is critical in obtaining appropriate stress distribution inthe vicinity of the bolt hole boundary. Stress distribution at a pin-hole has been investigated with three-dimensionalmodels,15–17but few18,19 have devoted their efforts to develop a three-dimensional PFM that considers failure modesthat occur out-of-plane with the objective of predicting the ultimate strength of composite joints.

Authors have commented on the need to improve the approach on the prediction of the different failure mechanismswithin the PFM. Camanho and Matthews18 highlighted the importance of including a methodology to predict delam-ination in the PFM. A few authors13,14,20have demonstrated the need to include in-plane nonlinear shear in the PFMto obtain realistic strength predictions. For the prediction of matrix-cracks an in-plane failure criteria is needed. TheWorld-Wide Failure Exercise (WWFE)21 demonstrated that Puck-Schurmann22 in-plane failure criteria was amongthe best in predicted biaxial failure envelopes. Puck and Schurmann noted that their theory is three-dimensional butonly two dimensional stress situations were dealt with, and the potential of the new theory was not fully exploited forthree-dimensional problems. Three-dimensional failure criterion is essential in the prediction of potential transversematrix-cracks that generally occur in mechanically fastened joints.

In this paper, a PFM that includes a methodology for the prediction of discrete delamination events throughthe thickness of the composite, a failure criteria for the prediction of matrix-cracking based on the work of Puck-Schurmann, and a new nonlinear shear constitutive law that considers transverse shear nonlinearity is presented. APFM is efficiently formulated within the context of continuum damage mechanics and it includes geometric non-linearity, material nonlinearity, and full contact. The methodology is used in the strength prediction of various testconfigurations and compared to experimental data either available in the literature or developed in-house. The paper isstructured as follows: (1) Failure Mechanisms in Composite Joints, (2) Progressive Failure Methodology, (3) FailureModeling and Simulation, and (4) Concluding Remarks.

II. Failure Mechanisms in Composites Joints

There are three failure modes that typically occur in composite joints: net-tension, shear-out, and bearing. Net-tension and shear-out are usually associated with in-plane failure mechanisms while the bearing failure mode is as-sociated with out-of-plane failure mechanisms. The failure mechanism in the plane of the lamina are fiber breakage,matrix cracking, and fiber-matrix interface cracking; and the out-of-plane failure mechanisms are transverse matrixcracking and delaminations. The failure mode dealt with here is the net section failure mode, but multiple failuremechanisms in the plane and out of plane occur because bearing and bypass loads are applied.

Developing an accurate and computationally efficient numerical procedure to predict damage rests on the un-derstanding of the material microstructural changes that take place within the composite. This paper will focus onthose fracture criteria and degradation models that exist which are physics-based and simple enough for applicationin engineering design. The development and implementation of these models to predict the failure mechanisms incomposites is discussed in this section. An analytical approach for each of the failure modes will be incorporatedin a progressive damage analysis to predict damage accumulation. The first damage mechanism to be discussed isthe delamination failure mode. This is followed by a postulation of a nonlinear constitutive law that describes themechanical behavior of composites when subjected to shear loading. The section ends with a discussion on the failurecriteria for matrix-cracks and fiber failure.

A. Delamination Failure Mode

Delamination is a common failure mode associated with ply discontinuity that can strongly influence stiffness lossand generate local instabilities that eventually lead to compressive failure. At the ply level, laminated composites

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exhibit transverse shear and normal stress concentrations in the vicinity of material and geometric discontinuities. Ageometric discontinuity exists at a free edge, and a material discontinuity exists at the interface of two adjacent plieswith different fiber orientation angles. Both discontinuities are present at the bolt hole boundary, thereby exacerbatingthe interlaminar stress concentrations. Since interlaminar stresses are nonzero at the free edge, the stress field in thevicinity of a bolt hole in a composite is three-dimensional, especially in the case of a thick composite. The non-uniformcontact stress distribution that exists between the bolt and the bolt hole edge provides an additional catalyst to initiatedelaminations.

Camanho et al.4 and Ireman et al.6 conducted a comprehensive experimental investigation using X-ray andsectioning at different loading percentages of the ultimate failure load in order to measure and characterize the failuremechanisms of mechanically fastened joints. Camanho et al. specifically studied double-lap joints with finger-tightwashers, and specimens that fail in either bearing, tension or shear-out modes; and Ireman et al. studied quasi-isotropicsingle bolted joints. In both works, the failure of the specimens occurred by a process of damage accumulation thatinvolved fiber fracture, delamination at the laminate loaded hole, matrix-cracks, fiber microbuckling and internaldelaminations. Ireman et al. demonstrated that at high load levels extensive delaminations were found through thethickness of the composite and extended as far as an inch from the hole boundary. Delamination was shown to havea strong influence on the joint strength for pin-loaded joints, because of high transverse tensile stresses present at thehole boundary. Park H.-J.5 used acoustic emission and load-displacement techniques to demonstrate that stackingsequence and clamping force influences the delamination failure mode and the ultimate bearing strength of compositejoints. Their work also demonstrates that delaminations are an important damage mechanism.

From the analytical standpoint, the onset of delamination has been classically predicted with stress methods usingfinite element methods. Ochoa and Engblom23 and Eason and Ochoa24 implemented delamination modeling in a pro-gressive failure methodology. The initiation and growth of delamination was predicted using a strength-based failurecriteria. Delamination progression was simulated by degrading the out-of-plane elastic properties of the elements rep-resenting the plies. This assumption may lead to an incorrect stress distribution through the thickness of the laminateand cannot explain scaling effects, where thicker laminates delaminate easier than thinner laminates. The onset ofdelamination may be inaccurate when geometric singularities are present and the solution can be mesh dependent.An alternative analytical approach is to predict delamination by degrading the material properties at the interfacesbetween the plies instead of the elements representing the plies. Such a technique can be incorporated with nodalrelease techniques, which have been used extensively to predict the progression of delaminations. This method has notbeen applied to complex problems involving joints.25–27 Despite the widespread use of the nodal release technique,the lack of material length scale has adverse implications in attempting to incorporate both the requirements of criticalstress and fracture energy for simulating delamination growth. The absence of a material length scale leads to strongdependence on the size of the finite elements near the stress raiser. To reduce scale dependence, an interface elementapproach is adopted in this paper.

A new computational fracture mechanics technology has been developed in recent years28–31 for the prediction ofdiscrete delaminations in composite materials via the use of interface elements, but has not yet been applied in the fail-ure prediction of mechanically fastened joints. The interface element formulation consists of a nonlinear distributionof continuous springs with a constitutive law that satisfies a strength based failure criterion and a fracture mechanicscriterion. The interfacial constitutive law for the formulation of the interface element governs the deformations of theseinterface layers and determines whether delamination growth occurs or not. This unified strength-fracture approachallows the prediction of the initiation and growth of delamination.32–34 Interface elements are positioned at interfaceswhere the delamination may potentially grow. The prediction of multiple delaminations and non-self similar delam-ination growth is then obtained automatically as an outcome of the nonlinear solution procedure without resorting tomanual node release.

Prior versions to ABAQUS 6.5-6 did not have the interface finite element formulation to predict delamination, andthus the element was previously developed and implemented35 in ABAQUS using a user element subroutine (UEL).The drawback with the UEL approach is that ABAQUS does not support visualization of the damage distribution atthe interface element level, so the shape and size of the delamination is not known, except for separations that arevisible from the deformation plot. To overcome this visualization difficulty, an effort was undertaken36 to predictdelamination using standard, but very thin, solid brick elements (C3D8I) available in the ABAQUS element library.The traction-separation law normally accompanying interface elements was transformed to a stress-strain law for aninterphasic element of very small thickness and implemented in ABAQUS with a user material subroutine UMAT.Although good predictions were obtained with this approach, severe convergence issues surfaced occasionally, mainlytriggered by excessive element distortion. The element distortion problem occurred more often under the initiationand growth of shear delamination. Moreover, the formulation of the C3D8I element was not meant to be used for

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Input Deck

*COHESIVE SECTION, RESPONSE=TRACTION SEPARATION, ELSET=elsetname, MATERIAL = materialname

*MATERIAL, NAME = materialname *USER MATERIAL, CONSTANTS = 8 GIc, GIIc, GIIIc, T1c, T2c, T3c, Kmax, α

User Material Subroutine

Update displacement jump field Update stresses based on the updated displacement jump Update the Jacobian matrix

Figure 1. Cohesive element implementation in the ABAQUS finite element software.

the prediction of delamination. As of the date of which this paper was written, the most current version of ABAQUS(6.5-6) has an added capability to predict delamination using interface elements. The interface element type usedin this paper is the COH3D8, an eight-noded element, three displacement degrees of freedom per node, and linearinterpolation. There are two convergence issues with the COH3D8. The first issue is that the convergence has beenfound to be excessively slow with the built-in bilinear constitutive law that accompanies this element formulation.The source of the slow convergence is the tangent stiffness discontinuity present in the bilinear constitutive law. TheNewton-Raphson nonlinear solver enters into nonconvergent oscillating cycles of the residual force while attemptingto converge to a solution. The second issue for slow convergence is related to numerical instabilities arising fromstiffness matrices with negative eigenvalues and ill-conditioned stiffness matrices that often accompany the type ofproblems involving softening behavior.

The first convergence issue is addressed by selecting a constitutive law with a smooth slope.35 The constitutive lawhas an exponential form, and it naturally satisfies a multi-axial quadratic stress criterion for the onset of delaminationand a mixed mode fracture criterion for the progression of delamination.Naturallymeans that the criteria do not needto be evaluated in the numerical algorithm, because the criteria and its corresponding failure degradation rule is built-inin the constitutive law. These constitutive equations are also thermomechanically consistent, and this is achieved byincluding a damage parameter to prevent the restoration of the previous cohesive state between the interfacial surfaces.The capability to predict delamination with this constitutive law has been demonstrated by simulating steady-statequasi-static delamination growth for loading–unloading cycles of various Mode I, Mode II, Mode III, and mixed modefracture test specimens.35 The finite element results are in good agreement with either experimental data available inthe literature or with linear elastic fracture mechanics analytical solutions. The second convergence issue is resolvedby setting negative diagonal terms in the material tangent stiffness matrix to zero. This modification does not affectthe solution, but it improves convergence efficiency.

The constitutive law implementation is conducted with a user material subroutine UMAT. The input deck is sup-plemented with the cards shown in Figure 1. Following are the material parameters to be defined in the input deck:Gic - critical energy release rate for Mode I, II, and III;Tic - interlaminar strengths;Kmax - an interpenetration factor;andα - defined by the following fracture criteria:(

GI

GIc

)α/2

+(

GII

GII c

)α/2

+(

GII

GII c

)α/2

= 1, (1)

whereGic is the Modei critical energy release rate, andi = I , II , III . Mode I corresponds to opening mode and ModeII and III correspond to sliding modes. The steps in the development of the subroutine are straightforward: update thedisplacement jump tensor, update the traction tensor, and update the Jacobian matrix. The material properties for theAS4 material used in this paper are as follows:GIc = 1.0 lb/in, GIIc = GIIIc = 1.7 lb/in, T1c = 8000.0, T2c = T3c =10000.0 psi,Kmax= 1.8×1012 lb/in3, andα = 2.0.

B. Anisotropic Nonlinear Shear Constitutive Law

Composite laminates typically exhibit a nonlinear response when subjected to shear loads either in the plane of thecomposite or transverse to the plane of the composite. Material nonlinearity in the plane of the composite is exhibited,

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y1212 /σσ

y1212 /εε

m1

1

1n

y1212 /σσ

y1212 /εε

m1

1

1n

Figure 2. In-plane nonlinear shear behavior observed in Iosipescu testing and Off-Axis Shear test methods.

for example, during off-axis testing or Iosipescu testing. A typical nonlinear curve is shown in Figure 2. Testing isgenerally used to extract the initial shear modulus and the shear strength. Define a coordinate system where materialaxes 1 and 2 are in the plane of the composite, and axes 3 is in the transverse direction of the composite. Hahn andTsai37 proposed a nonlinear shear stress-strain relationship of the form

ε12 =σ12

G12+ζσ3

12, (2)

whereε12,σ12, andG12 are the in-plane shear strain, in-plane shear stress, and the initial shear modulus, respectively.The material constantζ is determined experimentally. Chang et al.38 studied the failure strength of composite lami-nates with a pin loading the hole, and later used this nonlinear model in the study of notched composites subjected totensile loading.8 Dano et al.13 adopted the shear constitutive law postulated by Hahn and Tsai to predict the failure ofsingle mechanically fastened joint. He demonstrated that the nonlinear shear model plays an important role in accuratepredictions of the failure strength. Including this nonlinearity is of paramount importance to find the appropriate stressdistribution and concentration at the bolt hole.18,20,39 In-plane matrix-cracks are predicted prematurely when linearin-plane shear material properties are used. Including the nonlinear shear behavior of the composite in the failure mod-eling relieves the in-plane shear stress concentration at the hole boundary and delays the prediction of matrix-cracks.A shear stress concentration also exists in the transverse direction.

The focus of this section is to develop a constitutive law that effectively accounts for the nonlinear shear behavior inthe transverse direction. For the in-plane shear behavior of composites, complex polynomial forms and sophisticatednonlinear models have been proposed. In contrast, the goal here is to develop an exponential constitutive law withthe minimum number of material parameters which also takes into account the transverse nonlinear shear behavior.Material parameters characterizing the constitutive law can be informed by testing. To better understand the three-dimensional nonlinear shear constitutive law, the analytical development of the constitutive law in the plane of thelaminate will be addressed first.

1. In-Plane Shear Constitutive Law

Defineσ, ε, yσ, yε = yσ/G, G, and uσ as the shear stress, shear strain, yield stress, yield strain, shear modulus, andthe ultimate stress. The yield stress is defined as the threshold at which nonlinear shear damage accumulates and theultimate stress is defined as the threshold at which failure occurs. Below the yield stress value the stress-strain relationis linear and given by

σ = Gε, ε≤ yε. (3)

For yε < ε < uε, the nonlinear shear stress constitutive law can be written as:

σ(ε) = yσ+(uσ− yσ) f (ε), ε > yε, (4)

where f (ε) is an interpolating function. For example, the following function forf (ε)

f (ε) =ε− yεuε− yε

, (5)

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produces a bilinear stress-strain constitutive law. Since a discontinuous tangent stiffness can lead to oscillatory con-vergence difficulties,30 an interpolation function is chosen such that the tangent stiffness of the stress-strain curve iscontinuous. This imposes restrictions on the functionf (ε) as follows: (1) The yield strain must correspond to theyield stress; (2) The tangent stiffness must be continuous at the yield strain; (3) The ultimate strain must correspondto the ultimate stress; (4) The tangent stiffness at the ultimate strain is assumed to be zero; (5) The stress must alwaysincrease, or the tangent stiffness must always be nonnegative; and (6) The stress function must be concave down.These restrictions can be written mathematically as follows:

(a) f (yε) = 0; (b) f ′(yε) =G

uσ− yσ; f (uε) = 1; f ′(uε) = 0; f ′(ε)≥ 0; f ′′(ε) < 0, (6)

where f ′(ε) represents the derivative of the functionf with respect to the strainε. Although a polynomial of thirddegree seems to be a natural choice for the functionf (ε), it can be shown that there is no functionf (ε) that can satisfyall the restrictions established. An exponential function forf (ε) will be chosen instead, and it is derived as follows.To simplify the derivation, assume for the moment that the yield stress is zero,yσ = 0, that is, the stress-strain isnonlinear at a stress level above zero. It follows from Equation 4:

σ(ε) =u σ f (ε). (7)

An exponential function is chosen for the functionf (ε) that satisfies the last four restrictions in Equation 6:

f (ε) =εuε

exp

[1φ

(1−( ε

uε

)φ)]

, (8)

whereφ is a material parameter that will be used to fit the initial shear stiffness. Now, a non-zero yield stress will beestablished. The constitutive law in Equation 7 is translated from the origin to the yield state of the material as follows:

σ(ε) = yσ+(uσ− yσ) f (ε), (9)

where the translated functionf (ε) is obtained as:

f (ε) =ε− yεuε− yε

exp

[1φ

(1−(

ε− yεuε− yε

)φ)]

. (10)

With this stress function, the restrictionf (yε) = 0 now holds. The material parameterφ is obtained to satisfy thesecond restriction in Equation 6. This restriction translates to the following equation that must be solved forφ:

f ′(ε) =(

1uε− yε

)(1−(

ε− yεuε− yε

)φ)

exp

[1φ

(1−(

ε− yεuε− yε

)φ)]

, f ′(yε) =G

uσ− yσ, (11)

which leads to the expression forφ as follows:

1φ

= ln

( uε/ yε−1uσ/ yσ−1

). (12)

Substituting this expression forφ into Equation 11 gives the final stress-strain constitutive law that satisfies all restric-tions given in Equation 6:

s= 1+(n−1)(

e−1m−1

)(m−1n−1

)λ, λ = 1−

(e−1m−1

)φ, (13)

with the following definitions:

s=σyσ

; e=εyε

; n =uσyσ

; m=uεyε

;1φ

= ln

(m−1n−1

). (14)

Equation 13 represents the final form of the nonlinear constitutive law in the plane of the laminate. This constitutivelaw will be now expanded to a three-dimensional plane.

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2. Shear Strain Energy

The proposed constitutive law is postulated in the lamina material principal coordinate system. The following indicesare defined for simplicity:

i = 12, 23, 13. (15)

An energy functional will be used to develop the anisotropic constitutive law. The strain energy for the deformationfor each of the shear plane is postulated as:

Πi = yσiyεi

Zsi de, (16)

wheree, si are the effective strain and stress respectively. The effective straine defines the yield surface, wherebye≤ 1 corresponds to a linear behavior of the material ande> 1 corresponds to yielding or damage accumulation byshear. The effective strain consists of a quadratic function of the shear strains, individually normalized with respect tothe yield strain as follows:

e=

√(ε12yε12

)2

+(

ε13yε13

)2

+(

ε23yε23

)2

. (17)

With this equation and Equation 16, the shear stress for each shear plane is obtained as follows:

σi =∂Πi

∂e∂e∂εi

= yσ yεi si1e

1yε2

i

εi = Gi εisi

e, (18)

whereGi is the shear modulus for planei = 12,23,13.

3. Anisotropic Constitutive Law

To fully characterize the nonlinear shear behavior of the constitutive law, the following material parameters are spec-ified: (1) Ultimate strength and corresponding ultimate strain of each shear plane,uσi ,uεi ; (2) Shear moduli for eachshear plane,Gi ; and (3) The in-plane ultimate–strain to yield–strain ratio,m= uε12/

yε12. The ratiom will be consid-ered a material parameter and it is obtained from in-plane testing. This ratio further defines the ultimate-strain to yieldstrain ratio for the other shear planes as follows:

m=uε12yε12

=uε13yε13

=uε23yε23

, (19)

where this is assumed to reduce the number of material characterization parameters. Without test data, a reasonableassumption ism= 2. The effective strain in Equation 17 is modified to the following equations:

e=

1, e≤ 1;√(ε12yε12

)2+(

ε13yε13

)2+(

ε23yε23

)2, 1 < e< m;

m, e≥m.

(20)

These equations translate to the following transition stages: (1) Fore≤ 1 the material behavior is linear and nointeraction occurs between distinct shear planes; (2) For 1< e < m, the shear behavior is nonlinear and the strainis a function of the different shear planes; and (3) Fore≥ m, shear failure occurs. Modifying the one dimensionalnonlinear shear stress in Equations 13-14 to a three-dimensional form, one obtains the following form for the effectivestress:

si = 1+(ni −1)(

e−1m−1

)(m−1ni −1

)λi

, (21)

where λi = 1−(

e−1m−1

)φi

,1φi

= ln

(m−1ni −1

), i = 12,13,23,

with the stress-strain relationship defined by Equation 18. Only one parameter,ni , has not been defined. It will becomeevident, with further manipulations, thatni is a material parameter given by the ultimate–stress to yield–stress ratio

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for each shear plane. As defined earlier, shear failure occurs ate= m. Substitution of this into Equation 21 gives thefollowing effective stress:si = ni . Substitution of this effective stress into Equation 18 leads to

uσi = Giuεi

ni

m=

yσiyεi

uεini

m= yσi

uεiyεi

ni

m= yσi m

ni

m= yσi ni , (22)

hence,

ni =uσiyσi

, (23)

is the ultimate–stress to yield–stress ratio, and it is consistent with the one dimensional shear model. It can be shownthat the only restrictions on the use of these equations is given bym > 1,ni > 1, andni < m. This last inequalityis obtained by the fact that the shear stiffness at failure must be less than the initial shear stiffness. In summary,the nonlinear shear stress-strain relationship is given by Equation 18, and the effective strain and stress are defined inEquations 20 and 21, respectively. Refer to the earlier discussion in this subsection on the required material parameters.

4. Yield and Ultimate Surfaces

The nonlinear constitutive law can be shown to satisfy a yield quadratic surface criteria for damage accumulation anda ultimate quadratic surface criteria for failure. At the yield surface the effective strain is unity and from Equation 21the effective stress also becomes unity,si = 1. So at the yield surface the constitutive law in Equation 18 becomes:

σi = yσiεiyεi

, (24)

and so the following relationship for the yield surface holds:√(σ12yσ12

)2

+(

σ13yσ13

)2

+(

σ23yσ23

)2

=

√(ε12yε12

)2

+(

ε13yε13

)2

+(

ε23yε23

)2

= 1. (25)

The constitutive law can also be shown to satisfy an ultimate quadratic surface criteria. At the ultimate surface, theeffective strain ise= mand the corresponding effective stress becomessi = ni from Equation 21. The constitutive lawin Equation 18 then becomes:

σi = yσiεiyεi

ni

m. (26)

Squaring both sides of this equation and summing overi = 12,23,13, the ultimate surface is obtained as a quadraticinteraction of shear stresses as follows:√(

σ12uσ12

)2

+(

σ13uσ13

)2

+(

σ23uσ23

)2

=1m

√(ε12yε12

)2

+(

ε13yε13

)2

+(

ε23yε23

)2

= 1. (27)

5. Irreversible Damage

The constitutive law in Equation 18 is modified to account for the irrecoverable energy dissipated during damageaccumulation. The effective strain is tracked with the parameterθ as follows:

t+δtθ = max(

1, tθ, t+δte)

, 0θ = 1. (28)

The damage is tracked via history dependent damage parametersωi and these evolve withθ as follows:

ωi = 1− si(θ)θ

, (29)

so that the constitutive law becomes:σi = Giεi(1−ωi). (30)

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6. Shear Tangent Stiffness Matrix

The constitutive law is implemented in the finite element commercial software ABAQUS via user subroutine UMAT.The subroutine requires the definition of the Jacobian (tangent stiffness matrix) and the stress-strain constitutive law.Since an analytical expression exists for the stress-strain law, the components of the tangent stiffness matrix can beobtained with the following equation:

Di j =∂σi

∂ε j, (31)

wherei, j = 12,13,23. These components are defined as piecewise functions in the range of the effective strain as itwill be shown. Fore< 1, the tangent stiffness matrix is a diagonal matrix and the diagonal components are constantwith respect to the strain:

Di j = Giδi j , (32)

For 1≤ e< m, the components of the tangent stiffness matrix can be obtained as follows

Di j = Giδi jsi

e+Gi

εi ε jyε2

j e2

(∂si

∂e− si

e

), (33)

where it can be shown:

∂si

∂e= λi

(m−1ni −1

)λi−1

, λi = 1−(

e−1m−1

)φi

,1φi

= ln

(m−1ni −1

). (34)

For the case in whiche≥m, the components of the tangent stiffness matrix are:

Di j = Giδi jni

m. (35)

Subsequent to accumulated damage, shear unloading may occur. The tangent stiffness matrix for this case becomes:

Di j = Giδi j (1−ωi). (36)

7. Shear Material Properties

The material properties used in this paper are as follows: (1) Ultimate strength for each shear planeuσ12 = uσ13 =uσ23 = 12.0 ksi, ultimate strain for each shear planeuε12 = uε13 = uε23 = 0.04; (2) Shear moduli for each shear plane,G12 = 0.89 msi,G13 = 0.89 msi,G23 = 0.55 msi; and (3) The in-plane ultimate–strain to yield–strain ratio,m= 2.

C. Intraply Damage Mechanisms

1. Matrix Cracking

Hashin40 developed strength-based failure criteria based on experimental observations of unidirectional compositessubjected to tensile loads. The failure criteria by Hashin do not always fit experimental data as shown by numerousstudies, specially in the case of matrix compression. Hinton et al.21 conducted the World- Wide Failure Exercise

σnn σnt

σnl

β

Potential fracture plane 1

2

3

Figure 3. Potential fracture plane is defined byβ, and it is measured with respect to Plane 1-3.

9 of 26


(WWFE) whereby the leading theories for predicting failure in composite laminates were compared against exper-imental evidence in order to assess the qualitative and quantitative robustness and accuracy of each failure theory.The comparisons included biaxial strength envelopes for a range of unidirectional and multi-directional composites.Parıs41 conducted a detailed and critical review and evaluated many of the failure theories tested in the WWFE. TheWWFE results indicated that even for simple layups and loadings, many of the failure criteria that have been proposedto date fall short of predicting the failure strength accurately. In the WWFE, the numerical predictions with the fail-ure theory by Puck and Schurmann22 showed good agreement with experimental results for the prediction of biaxialfailure envelopes. The application of their criteria was limited to in-plane failures, and since in mechanically fastenedjoints multiple failures occur through the thickness of the laminate, the three-dimensional version must be considered.Their failure theory follows the work of Hashin. Hashin’s failure criteria for matrix-cracks consists of a quadraticinteraction of stress invariants, whereby the criteria is made independent of the stress parallel to the fiber, based on thefact that any possible failure plane is parallel to the fibers.

The failure criteria proposed by Puck and Schurmann is essentially an extension of Hashin’s postulate, wherebythe fracture plane is unknown and it must be determined. This is accomplished by maximizing the failure index withrespect to the fracture plane angle. Since matrix-cracking occurs in a plane parallel to the fibers, the fracture planeoriented at an angleβ with respect to the global coordinate system must be obtained. The tractions acting on potentialfracture plane are the traction normal to the planeσnn, the shear traction in the direction transverse to the fibers andparallel to the fracture planeσnt, and the shear tractionσnl in the direction parallel both to the fiber and to the fractureplane. The potential fracture plane is defined byβ, and it is measured with respect to Plane 1-3, where the 1-axisdefines the orientation of the fibers and the 3-axis defines the stacking direction. For reference, the tractions are shownin Figure 3. The tractions in a potential fracture planeβ are given by force equilibrium as follows:

σnn =σ22+σ33

2+

σ22−σ33

2cos(2β)+σ23sin(2β) (37)

σnt = −σ22−σ33

2sin(2β)+σ23cos(2β)

σnl = σ12cos(β)+σ13sin(β)

whereσi j is the local state of stress that rotates with the lamina orientation.σ11 is the stress parallel to the fiber,σ33 isthe stress in the stacking direction,σ12 is the in-plane shear stress, andσ13,σ23 are the transverse shear stresses.

The matrix-cracking failure criteria for a tensile traction,σnn > 0, is as follows:(σnn

Y

)2+(

σnl

Sl

)2

+(

σnt

St

)2

= 1 (38)

whereY,Sl , andSt are the normal, longitudinal shear, and transverse shear strength allowables, respectively. Under acompressive traction,σnn < 0, the strength allowables in the failure criteria in Equation 38 are modified to predict theapparent increase in strength of a lamina subjected to lateral compression. Such modification involves the usage ofinternal friction coefficients as follows:(

σnl

Sl −ηnlσnn

)2

+(

σnt

St−ηntσnn

)2

= 1 (39)

where the contributionσnn has been eliminated to indicate that forσnn > 0 fracture is promoted, while forσnn < 0shear fracture is impeded. In compression there is additional resistance to shear fracture. This resistance increaseswith increasing compressive traction. This effect is incorporated with internal material friction coefficientsηnt,ηnn.The denominators in this failure index can be regarded as effective shear strengths.

The potential fracture plane oriented at an angleβ is obtained by maximizing the failure index in Equation 38if the normal traction is positive or maximizing the failure index in Equation 39 if the normal traction is negative.Although this angle can be obtained analytically, it is time consuming, and the angle is rather obtained by using asearch algorithm. This simple algorithm consists of determining the failure index at each potential fracture angle, andselecting the one that yields the maximum value for the failure index.

The strength allowables for AS4-3502 are used as a reference in all the analysis conducted in this paper:Y = 8.0ksi, Sl = St = 12.0 ksi, andηnl = ηnt = 0.3. The basis for the selection of this friction coefficient is based on thework of Puck. He determined that when composites are loaded in transverse compression, the fracture plane is usuallyoriented at 53±2 degrees with respect to the transverse plane. The friction coefficient was estimated to be in the orderof 0.3.

10 of 26


2. Fiber Breakage

For fiber failure, the critical strain will be used as the criterion to predict whether fiber breakage occurs as follows:

ε11 < cε11, ε11 < 0 (40)

ε11 > tε11, ε11 > 0

wherecε11 is the strain allowable in compression andtε11 is the strain allowable in tension. A typical strain allowable,tε11 = 0.015, for AS4 material system is used in the analysis. In compression the strain allowable used iscε11 = 0.012.Stress concentrations at the hole boundary may lead to premature failure by fiber breakage. The critical damage areaAf postulated by Chang and Chang8 will be used as the criterion instead.Af represents the area where the tensilestrain allowable has exceeded the average tensile strain allowable of the composite. The postulate is based on thefiber bundle theory, whereby there exists a fiber interaction zone. This zone consists of an average distance betweenbreaks of neighboring fibers.Af = 0.02in2 is used in all the analyses presented in this paper, since it produces the bestnumerical correlation with experimental data.

III. Progressive Failure Methodology

Refer to Figure 4 for the following discussion on progressive damage philosophy and its relation to validation. Thegoal is to develop a reliable and practical analytical tool capable of simulating the response of composite joints andpredicting their structural margins. This effort can be broadly divided into three sections: coupon testing, componenttesting and analysis. Coupon testing is essential in the damage modeling of composites since it is used to determinematerial properties, fracture properties and failure criteria. For example, all the fracture properties can be determinedfrom the double cantilever beam test, end crack torsion, end load split, and mixed mode bending; shear modulus andshear strength data can be determined either from[−45/+45]ns off-axis loading test, short beam shear test, or torsiontest; and tensile strength data can be determined from uniaxial testing. Coupon tests can also give insight into theprogression of damage and response of the structure based upon observations and measurements. Refer to Figure 5 forthe following discussion. The type of tests shown in this figure are the simple tensile test, Iosipescu testing, and theopen hole strip testing. From the tensile test, one can determine the Young modulus and the ultimate tensile strength inthe fiber direction. From the Iosipescu testing, one can determine the shear stiffness, the yield stress, and the ultimateshear strength in the plane of the laminate. For the open hole strip testing, one can determine the critical area for whichneighboring fibers fail.

From component testing, the second leg in the model development, the structure can be evaluated on a macro-scaleto evaluate margins. However, insight into the progression of failure is more difficult since failure may be unstableand the testing cost is considerably higher for flight components than it is for coupons. Finally, analysis is done tovalidate the material model. Since the coupon tests are simpler than the flight component, it follows that the analyticalmodel will first be used to predict the response of the coupon tests. A finite element model is created to reflect theexperimental set-up; material properties are carried over from coupon testing or found in the literature, as appropriate,and care is taken to ensure that the loads and boundary conditions reflect the actual test configuration. The next step inthe process is to compare predictions with test observations – including the measured strain, delamination formationand growth, shear damage, matrix cracking or fiber failure – in order to correlate the relevant material parameters.This is especially important in the modeling of damage, whereby friction coefficients and even strength propertiesneed to be calibrated. Anchoring the finite element modeling to the testing may require several iterations and involvesinvestigating sensitivities to the mesh, load, boundary conditions, failure criteria, and even the material parametersused in the failure criteria.

Once there is agreement between the finite element results and the test data, the model can be used to analyzea subscale structural component and compared to experimental results. If large differences exist between the finiteelement results and the test data, then the model needs to be re-examined. Typically, the material, boundary conditions,and failure criteria would be revised at the coupon level. This refinement continues until the subscale analysis showsgood agreement with the subscale test data. The analytical tool is considered to be reliable once the model canaccurately predict the response of both the coupon and component tests.

No definitive method exists to predict joint strength, and most methods based on strength failure criteria underes-timate joint strength. Although an underestimate is obtained in some cases, these methods may predict some aspectsof damage growth observed in testing. In this paper, the strength-only approach is replaced by a strength-fracture ap-proach for the prediction of delamination and matrix damage. The damage is modeled within the context of continuumdamage mechanics and it is applied to the homogenized material description of each material ply. The orientation of

11 of 26


Goal

Reliable Analytical Tool

Improve Model

Fidelity

No

Material & Loads Modeling

Material Characterization

Coupon Testing

Failure Criteria

Finite Element of Coupon Test

Margins Test

Flight Component

Testing

Flight Component Finite Element

Analysis

Margins Model

No

YesMargins Agree?

FEM & Testing Agree?Yes

Goal


Improve Model

Fidelity

No



Coupon Testing

Failure Criteria


Margins Test

Flight Component

Testing


Analysis

Margins Model

No

YesMargins Agree?


ConvergedSolution?

LoadPr+1 = Pr + ∆Pr

∆Pr ≠ 0?σi, εi, ∆εi

σi+1, K, D

Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)

Update Stiffness Matrix

kl

ij

εσ

K∂∂

=

Update Damage Tensor D

EquilibriumIterations

ABAQUS Pre-Processing

ABAQUS Post-

Processing

ABAQUS Processing Initial Load

∆Po, Po

Failure Load

IterationsExceeded?

Decrease ∆Pr

YesNo

NoYes

Yes

No

ConvergedSolution?



σi+1, K, D

Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)


kl

ij

εσ

K∂∂

=


Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)


kl

ij

εσ

K∂∂

=




ABAQUS Post-

Processing


∆Po, Po

Failure Load

IterationsExceeded?

Decrease ∆Pr

YesNo

NoYes

Yes

No

Figure 4. Flowchart demonstrating the strategy used to develop the modeling technique and adjust for specific applications. Both couponand component testing are required, along with corresponding finite element analyses.

Test Coupon Stiffness Strength ASTM Standard

Tensile E11 X D3039/3039M-95a

Iosipescu Shear G12 sy, su D5379/D5379M-98

Open Hole Strip Af D5766/D5766M-95

1.5-inch 0.75-inch

1-inch

Coupon Width

D5766/D5766M-95 d --- D5379/D5379M-98 SA G12 D3039/3039M-95a XT E1

Test Method (ASTM Standard)

Strength Stiffness

Figure 5. Tensile testing, Iosipescu shear testing, and open hole strip testing are typical coupons used to characterize stiffness and strength.

Goal


Improve Model

Fidelity

No



Coupon Testing

Failure Criteria


Margins Test

Flight Component

Testing


Analysis

Margins Model

No

YesMargins Agree?


Goal


Improve Model

Fidelity

No



Coupon Testing

Failure Criteria


Margins Test

Flight Component

Testing


Analysis

Margins Model

No

YesMargins Agree?


ConvergedSolution?



σi+1, K, D

Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)


kl

ij

εσ

K∂∂

=




ABAQUS Post-

Processing


∆Po, Po

Failure Load

IterationsExceeded?

Decrease ∆Pr

YesNo

NoYes

Yes

No

ConvergedSolution?



σi+1, K, D

Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)


kl

ij

εσ

K∂∂

=


Update Stresses

εi+1 = εi + ∆εi, σi+1 = f(εi+1)


kl

ij

εσ

K∂∂

=




ABAQUS Post-

Processing


∆Po, Po

Failure Load

IterationsExceeded?

Decrease ∆Pr

YesNo

NoYes

Yes

No

Figure 6. Flowchart describes the implementation of the damage mechanics approach using ABAQUS.

12 of 26


the damage and its relation to the elastic constants is determined by this approach. The following description of thedamage pertains to the lamina only.

The distributed microscopic damage will be quantified by the use of a second order symmetric damage tensor fieldD that describes the orientation and density of cracks.42 The principal directions ofD are assumed to coincide withthe principal material directions of the lamina in order to simulate orthotropic damage. Define the eigenvalues of thetensorD asDi , i = 1,2,3. The eigenvalues of this damage tensor define the reduction in load carrying capacity onplanes that are perpendicular to the ith principal direction. The eigenvalues are such that 0≤ Di ≤ 1, whereDi = 0represents a virgin state andDi = 1 represents complete loss of stiffness in the plane perpendicular to the i-th principalmaterial direction. These damage eigenvalues will be related to the material stiffness reduction via the effective stressconcept proposed as follows:

σi =σi

1−Di, i = 1,2,3. (41)

The effective strain tensorεi can be obtained from the principle of equivalent elastic strain energy density, which statesthat the elastic strain energy density is the same regardless whether it is computed based on an apparent stress-strainor based on an effective stress-strain. The effective strain tensor is obtained as follows:

εi = εi(1−Di), i = 1,2,3. (42)

The elastic strain energy density can also be used to define the constitutive stress-strain relationship for a damagedmaterial as follows:

Ψ(ε,D) =12

εiσi =12

ε jCi j εi , (43)

whereCi j is the damage elasticity tensor. Substitution of Equation 42 into Equation 43 leads to the following expres-sion:

12

ε jCi j εi =12

ε j

1−D jCi j

εi

1−Di=

12

ε jCi j εi , (44)

from which the damage elasticity tensor components become:

Ci j = Ci j (1−Di)(1−D j). (45)

Let i = 1,2,3 correspond to the direction of the fiber, to the direction perpendicular to the fiber in the plane of thelaminate, to the direction perpendicular to the fiber transverse to the laminate, respectively. Fiber breakage correspondsto D1 = 1 and the components of the damage elasticity tensorC1 j ,Cj1 become zero. The physical interpretation of thereduction in stiffnessC13,C31,C21,C12 is that if the fiber fails, there is no load transfer from the fiber to the matrix. Fromthe engineering standpoint, this reduction represents that the Poisson ratioν12,ν13 no longer affects the mechanicalbehavior of the composite. For matrix cracking in the plane of the laminate and transverse to the laminateD2 = 1, andthe components of the damage elasticity tensorC2 j ,Cj2 become zero. If matrix-cracking occurs, the shear stiffnessfor each shear plane is also degraded using the damage parameterD2. For delamination, the energy dissipated by thisfailure mechanism will be taken into account using interface elements, hence the damage variableD3 will be held aszero.

These damage parameters are used in the progressive damage problem, which can be divided in two parts: onerelates to the failure prediction at the lamina level, and second relates to the prediction of damage initiation at thelaminate level, which can then progress and lead to ultimate failure. The progressive damage implementation isdepicted in Figure 6. The damage evolution equations are solved at the Gaussian integration points within each materialply, thus explicitly accounting for damage variation through the thickness and in the plane of each material ply. Theflowchart is color-coded as follows: Green represents the stage at which the user interfaces directly with the finiteelement software ABAQUS; Blue represents the user material subroutine UMAT; and Peach represents the ABAQUSstandard nonlinear incremental procedure. First, the ABAQUS input file needs to be prepared in a typical fashion withadditional parameters relating to the strength and fracture of the composite. Then, the analysis is initiated by applyingan initial load to the finite element model. A converged solution must be obtained because the structure is no longer inequilibrium due to differences between the internal loads and the external loads. The stress, strain, and stiffness matrix(Jacobian) are updated. At each load step, the failure criteria presented in the previous section are applied at eachGaussian point. If failure occurs, the damage state is obtained and the components of the stiffness matrix are reducedusing Equation 45. The components are reduced to a small value instead of zero, because a reduction to zero leads tonumerical difficulties in the nonlinear procedure. In this paper, the damage mode related to fiber failure,D1, is chosenasD1 = 1−

√E22/E11. If matrix-cracking occurs in tension, the damageD2 is chosen as 0.9 and in compression is

chosen as 0.7.18 Following the application of the damage tensor, the modified Newton-Raphson nonlinear iterativeprocedure is conducted to achieve equilibrium prior to the next load step.

13 of 26


Ptot

Pb

Pbp

Single-Sided

Pb

Pbp

Ptot Byp

ass

Stre

ss

Bearing Stre

WtPultbp

DtPultb

Double-SidedSingle-Sided

Net sectionfailure

Bearingfailure

Byp

ass

Stre

Bearing Stress

WtPultbp

DtPultb

Double-SidedSingle-Sided

Net sectionfailure

Bearingfailure

Figure 7. Bypass and bearing failure envelope.

IV. Failure Modeling and Simulation

The ultimate goal of this paper is to develop the analytical capability to predict the bearing and bypass strengthof a composite system. A typical failure envelope is shown in Figure 7. Net section failure typically occurs fora width-to-diameter less than 6, otherwise a bearing failure mode occurs. This paper will only address specimenconfigurations that lead to a net section failure mode. The first problem that is addressed is the strength prediction ofnotched composite laminates. The second problem that will be addressed is the strength prediction of the followingthick composite joint configurations: filled hole loaded in tension (FHT), double side bypass load and bearing load(DSJ), and single side bypass and bearing load (SSJ). Each numerical simulation is qualitatively and quantitativelycompared to experimental data.

The geometry and mesh for all analyses presented in this paper are generated with commercial software packages.The finite element analyses are performed in ABAQUS 6.5-6 with an IBM P5 - 1.6 GHz machine and AIX 5.2 UNIXoperating system. Elements used in the analyses are based on a geometrically nonlinear formulation. The simulationsare conducted using displacement control since it provides the basis for numerical stability as the composite undergoesextensive damage growth. The stabilize parameter in ABAQUS is used to add stability in the nonlinear convergenceprocedure. The nonlinear procedure is complex due to the multiple failure modes, extensive contact, and their interac-tions. Four indicators are used to assess whether catastrophic failure of the test configuration has been predicted: (1)Negative eigenvalues, which are effective indicators of instability, (2) Damage that has reached the edge of the com-posite, (3) Significant increase in damage growth in consecutive load increments, and (4) Drop in strain in a regionwhere a sustained strain increase is expected.

A. Notched Composite

Qualitative and quantitative results are presented here for a notched composite to demonstrate that the modeling ap-proach appropriately captures the general characteristics of the different failure modes observed in the testing. Thenotched composite problem is applicable to general joint loading in composites, since a notch is essentially the limitingcase of a hole. Similar failure modes found in a composite with a hole subject to tensile loading – such as single-sidedor double-sided bearings – are also exhibited in a notched composite. These failure modes are more pronounced in anotched composite and include delamination formation and matrix cracking due to stress concentrations at the cracktip.

Vaidya et al.43 conducted experiments to study the effects on unnotched and notched strength of fiber dominatedlaminates. The geometry of the notched composite is shown in Figure 8. Samples were cut from 12 inch square platesof AS4/3501-6 graphite epoxy with a typical thickness of 0.5 inches. The test specimens were 10 inches long by 1.5inches wide, and included 1.5-inch long end tabs. The samples were notched by drilling a starter hole, cutting witha waterjet, and extending the notch with a jeweller’s saw blade to minimize delaminations. The samples were testedquasi-statically at a rate of 1-inch per minute on an MTS machine while recording load and displacement. The crack-tip damage was studied at various loads with x-ray and microscopic inspection techniques. The crack length selectedfor the study presented in this paper is 0.5-inch.

Damage simulation will be demonstrated with the notched example. The material properties of the AS4/3501-6are as follows:E11 = 21.0 msi, E33,E22 = 1.5 msi, ν12 = ν13 = 0.31, andν23 = 0.4. Other material and strength

14 of 26


(a) Geometry of the end notch, geometry of the quarter symmetry model with the appropriate boundary conditions

(b) Mesh density utilized for the end notch progressive failure analysis

(c) Delamination distribution in the notched specimen

P P

Figure 8. A schematic of the notched composite experiment is shown on the left. The model is simplified by employing symmetry boundaryconditions and using a quarter-model for the analysis, as shown on the right.

(a) Geometry of the end notch, geometry of the quarter symmetry model with the appropriate boundary conditions

(b) Mesh density utilized for the end notch progressive failure analysis

(c) Delamination distribution in the notched specimen

P P

Figure 9. The mesh used in the notched composite analysis is shown above, where the thick line represents the crack. The mesh density ismuch higher near the crack tip, but is otherwise regular with only rectangular elements.

(a) Delamination Prediction (b) Matrix-cracking Prediction

Figure 10. The analytical model predicts an elliptical delamination and matrix-cracks initiating and progressing from the notch parallelto the tensile loading condition. The color red indicates full damage state while blue indicates an undamage state.

15 of 26


properties were listed in previous sections of this paper. The mesh is shown in Figure 9. The layup in this analysisis [902/02/902/02/902]. Delaminations are consistently predicted through the thickness of the composite at the 0/90interface. The delaminations are predicted to initiate in the vicinity of the crack tip at 85 percent of the failure load andextend axially from the crack tip (in the direction of the load). See Figure 10a. These delaminations are triggered by 65percent shear mode and 35 percent opening mode. The shape of the delamination front is predicted to be elliptical innature. In general, these numerical predictions are consistent with x-ray observations of notched composites.43 Matrix-cracking is predicted to initiate at 72 percent of the load and progress until the specimen fails by fiber breakage in anet section failure mode. Fiber breakage occurred predominantly in the 0 -degree plies. Significant matrix damage ispredicted transverse to the laminate thickness and several matrix cracks were also predicted parallel to the 0-deg plies.As shown in Figure 10b, matrix-cracking emanated from the crack-tip and primarily extended in the axial direction.The shape of the damage is irregular. This is attributed to the computational procedures, and this numerical effect canbe minimized by using a tighter controlled damage approach. In general the numerical predictions are consistent withthe experimental X-ray examination reported in the paper by Vaidya et al.43

As shown, the finite element simulations for this specimen configuration demonstrated extensive damage in thevicinity of the crack tip prior to ultimate failure. Matrix-cracks and delaminations alleviated the high stress concentra-tion in the vicinity of the crack tip. A linear elastic model predicted notch strength of 45 ksi, a 35 percent underpredic-tion compared to the experimental strength of 68.8 psi. In contrast, the predicted strength using the damage approachpresented in this paper is 61.7 ksi, which is an 11 percent underprediction with respect to the experimental data. Thisdemonstrates the need to incorporate damage modeling in the analysis of notched composites.

B. Composites Subject to Bearing and By-pass Loads

The objective of this section is to demonstrate the capability of the analytical model to predict different failure mech-anisms of composites subjected to combined bypass and bearing loads. This section will not address the experimentaltesting in detail, but some data will be used to validate the modela. A schematic of the composite dogbone specimenis shown in Figure 11. The composite laminate consists of AS-4 fibers and Fibercote E-765 matrix constituent. Thefiber volume fraction varies from 50 percent to 60 percent. The dogbone composite plate consists of 16 sublaminates,and each sublaminate has the following stacking sequence:[0/− 45/90/45]s. The laminate stacking sequence is[0/−45/90/45]16s and the total thickness of the 128-ply sample is 0.75-inch. This is the as-designed thickness, butthe as-built thickness varies as much as 5 percent. A typical sample is 3.0-inch wide and contains a 0.5-inch diameterhole drilled through the center of the 8.5-inch gage section. A 0.5-inch steel bolt is placed through the hole in thesample. The shaft of the bolt is 1.5-inch long, with a 0.30-inch long and a 0.75-inch diameter head. Between the bolthead and the sample sits a loading washer. Refer to Figure 11. The opposite face of the sample has either an identicalload washer or a dummy washer, depending on the load case. The bolt is secured using a 0.075-inch thick, 1-inchdiameter flat washer and a 0.6-inch wide nut and is tightened to a set torque. This paper will not address the effects oftorque on the failure load.

The three different loading configurations analyzed are shown in Figure 12. The first loading case is filled-holetension (FHT), where no bearing load is applied to either load washer. The bolt constrains the deformation of thehole perpendicular to the tensile loading. In the second loading case, the bypass load was applied in the same manner,while a bearing load was applied to both the upper and lower load washers. This configuration is called the doublesided bearing and bypass load case, and it will be denoted with DSJ throughout the rest of the paper. In the finalconfiguration, the single sided bearing and bypass load, denoted as SSJ, is obtained by loading only the upper loadwasher. This is achieved by replacing the lower load washer with a dummy. For the double- and single-sided bearingand bypass load cases, initially the bearing and bypass loads are applied simultaneously. When the bearing loadreaches a predetermined value it is held fixed while the bypass load is increased to failure. See Figure 13.

The finite element model for the SSJ configuration is shown in Figure 14. Half symmetry is employed and theplane of symmetry is transverse to the coupon test configuration. The model represents the sample as four separateparts: (1) the bolt, nut and washer; (2) the upper load washer; (3) the dummy load washer; and (4) the dogbone sampleitself. The dogbone is divided into 3 sections–a fine mesh region near the bolt and coarse mesh regions at both ends.Since no critical stresses occur at the ends of the specimen, the coarse mesh region is used to reduce the number ofdegrees of freedom. In the fine mesh region, 16 elements through the thickness and approximately 19000 elementsare used. For the coarse mesh region, only 4 elements through the thickness for 320 elements in total are used. The

aAn extensive experimental testing program was conducted at The Aerospace Corporation Space Materials Laboratory to characterize the failuremechanisms that occur in filled hole tension, combined double sided bypass and bearing loads, and combined single sided bypass and bearing loadstest configurations. Design of the experimental setup, the test configurations, test coupon configuration, and experimental data will be published ina future paper.

16 of 26


24.0”

4.0” 8.5”

3.0”

6.0”

Thru holeD=0.50”

24.0”

4.0” 8.5”

3.0”

6.0”

Thru holeD=0.50”

0.75”

Do=2”

DI=1.35”

0.75”

Do=2”

DI=1.35”

1.125”

Figure 11. Schematic of the Dogbone Composite Specimen

Bypass loadBypass load

Bypass load

Bearing load

Bearing load

Bypass load

Bearing load

Bearing load

Bypass load

Bearing load

Bypass load

Bearing load

(a) Filled hole tension (FHT)

(b) Double sided bearing and by-pass load (DSJ)

(c) Single sided bearing and by-pass load (SSJ)

Figure 12. Loading Configurations

Load

Time

Bypass

Bearing

Figure 13. Bearing and Bypass Load as a Function of Time

17 of 26


Upper load washer

Lower load washer

Bolt, nut and washer

Upper load washer

Lower load washer

Bolt, nut and washer

Composite fine-mesh region

Composite coarse-mesh region

Figure 14. Finite element model for the single sided bearing load

Washer-composite contact

Bolt- bolthole contact

Bolt – washer contact

Bolt shank– washer contact

Figure 15. Contact Interfaces

Interface –1

Interface –2

Interface –3

Figure 16. Position of the Interface Layers

18 of 26


nodes of the fine mesh are tied to the surface of the coarse mesh using the TIE command available in ABAQUS. Thebolt has a mesh density along its axis to match that of the composite, and a total of 6720 elements. The upper loadand dummy load washers have 420 and 180 elements, respectively. Contact is modeled between each of the surfacesshown in Figure 15.

Potential delamination planes were identified through the thickness of the composite and the interface elementsare positioned in these critical locations where experience has shown that delamination is likely to occur. Interfaceelements are positioned at three separate planes of the composite sample in the finely meshed region. For reference,there are 127 ply interfaces, and each ply interface is numbered from bottom to top. The interface elements werepositioned between 0-deg plies at ply interface numbers 32, 64, and 96. See Figure 16 for reference. These elementsare only activated when the strength-based failure criteria for delamination is satisfied. Afterwards, delamination ispredicted to propagate with a fracture based criterion. A contact algorithm is activated to prevent penetration of thedelaminated faces.

Appropriate boundary conditions are enforced to obtain half-symmetry. The nodes on the right edge are held fixedand the nodes on the left edge are free to move in the through-thickness direction but are fixed in the plane of thelaminate as seen in Figure 13a. The bypass load is applied as a fixed displacement at the left edge, and the bearingload is applied as concentrated forces acting on the outer nodes of the load washer(s). A preload to simulate the torqueapplied to the bolt is created by applying a negative temperature gradient to the bolt. The temperature is effectivelyapplied in the axial direction by using an anisotropic coefficient of thermal expansion. This temperature is applied inorder to obtain a bolt tension corresponding to the torque that has been applied. In this case the torque applied in thetest specimens was on average 50 lb-in. The tensile force in the bolt can be obtained as

Fz =T

KDb,K = 0.2, (46)

whereFz is the tensile force in the bolt,K is a torque coefficient, andDb is the bolt diameter. In the analysis thathas been conducted, the delamination algorithm does not take into account the increased strength that results from theclamping force. This assumption may lead to premature delamination predictions. Because non destructive evaluationtechniques were not used to study the extent of damage progression, it is not possible to verify this statement.

The material properties of the composite system consisted of typical AS-4 material properties:E11 = 17.0 msi,E33 = E22 = 1.6 msi,ν12 = ν13 = 0.31, andν23 = 0.4. Note thatE11 = 21.0 msi is typically used for AS4 compositesystems, and this modulus corresponds to a fiber volume fraction corresponding to 60 percent. A thermal de-ply andother supporting data suggested that the test specimens were being fabricated with a lower fiber volume fraction, andin some cases below 50 percent. TheE11 = 17.0 msi value corresponds to a fiber volume fraction of 0.49, and it wasused here as a first estimate. This estimate led to good correlation between predicted and measured strains, which willbe shown.

The bypass stress – bearing stress failure points obtained from the finite element model are compared to the testresults in Figure 17. The bypass stress and the bearing stress are obtained from the bypass load and the bearing loadas follows:

σbp =Pbp

Wt, σb =

Pb

Dt(47)

whereσbp,σb,Pbp,Pb,W,D, t is the bypass stress, bearing stress, bypass load, bearing load, width, diameter, andthickness, respectively. The bypass strength, which is the total load for the FHT, is within 5 percent of the averageof the experimental strength prediction. Holding the bearing stress fixed to 61.6 ksi, the predicted bypass strength isalso in very good agreement with experimental data. For the SSJ, the bearing stress was held fixed at 57 ksi, and thebypass strength prediction was obtained as a range instead of a single value: 33 ksi – 39 ksi. The low end of the range,33 ksi, was selected on the basis that the predicted fiber strain dropped in a region where the strain was expectedto increase with increasing load. The high end of the range, 39 ksi, was selected on the basis that damage growthincreased drastically. At this bypass strength, more than half of the 0 degree plies failed. For this test configuration,SSJ, the average over the range of the predicted failure loads is within±8 percent of any of the experimental data.

Refer to Figure 18 for the following discussion on the evolution of damage and the type of damage that occurs inFHT, DSJ, and SSJ test configurations. The volumetric matrix damage (left vertical axis) and the total area of fiberdamage (right vertical axis) are shown as a function of total load (horizontal axis). For each of the test configurations,the corresponding load at which the initiation of delamination occurs is also identified in the plot by a red filled circleplaced on the matrix damage curve. For the FHT, the total load is equivalent to the bypass load; and for the DSJ andSSJ the total load is the sum of the bearing and bypass loads. The evolution of damage is shown from initiation toultimate failure. For all analyses, matrix damage initiated in the off-axis plies and it propagated gradually outwards

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0

10

20

30

40

50

60

0 15 30 45 60 75

Bearing Stress (ksi)

Bypa

ss S

tress

(ksi

)

FHT Test FHT ModelDSJ Test DSJ ModelSSJ Test SSJ Model

Figure 17. Predicted bypass-bearing failure envelope compared to test data for the filled hole tension (FHT), double sided joint (DSJ), andsingle sided joint(SSJ)

0

0.05

0.1

0.15

0.2

0 50 100 150Total Load (kips)

Mat

rix D

amag

e (in

3 )

0

0.5

1

1.5

2

Fiber Dam

age (in2)

FHT DSJ SSJFHT DSJ SSJ

MatrixFiber

Delam Initiates

Figure 18. Predicted volumetric matrix damage and accumulation of fiber failure as a function of total load for FHT, DSJ, and SSJ.

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from the bolthole. Extensive in-plane and transverse matrix-cracks were predicted for the SSJ configuration, while forthe FHT and DSJ the matrix-cracks were more localized. Accumulation of fiber failure occurred rapidly for the FHTand DSJ test configurations, and it initiated in the 0 -degree plies at the bolt hole. This damage led to global failure ofthe FHT and DSJ. The accumulation of fiber failure was more gradual and localized for the SSJ, and it initiated in theoff axis plies closest to the top surface of the composite in the region of the bolt hole. Fiber failure initially propagatedonly in off axis plies, and it occurred very gradually towards the bottom of the composite surface. Failure eventuallymoved to the 0-degrees plies, causing global failure.

Delamination was predicted to initiate before fiber failure for the FHT and DSJ, while it was predicted to occurafter the initiation of fiber failure for the SSJ. Delamination formation relieved the stress concentration at the bolt hole.

The predicted delaminations at failure loads are shown in Figure 19a and 19b for the FHT and DSJ, respectively.The predicted delaminations at 60 percent and 100 percent of the failure load are shown in Figure 20a and 20b,respectively. The predicted delamination is shown at three different locations through the thickness of the composite,where color code red represents damage and color code blue represents an undamaged state. The interface positionsare shown in Figure 16 for reference. For the FHT, delamination initiation is predicted to be caused by the free edgeat the bolt hole and it is predicted to propagate to the extent that matrix cracks propagate. Delamination initiated inthe middle interface first and it propagated less then a tenth of an inch from the bolt hole before the delaminationmoved to the outer interfaces. For the DSJ, the predicted delamination initiated in the outer interface first before thedelamination initiated in the middle interface. There are two regions where delamination is predicted in these outerinterfaces. Away from the bolt hole, the predicted delamination is caused by the shear loading generated by the washer;and at the bolt hole boundary, the predicted delamination is caused by the bearing load exerted by the bolt shaft. Thepredicted delamination at the middle interface is caused by the free edge effect present at the bolt boundary. The rateof delamination growth was predicted to be gradual compared to that of the matrix damage. For the SSJ, the predicteddelamination initiated at the outer interfaces first, followed by initiation in the middle surface. The delaminations inthe outer surfaces were caused by shear mode and generated by the washers. No noticeable delaminations causedby bearing damage were predicted for the SSJ. The predicted delamination in the middle surface was caused bya combination of bending and the free edge effect. For all cases (FHT, DSJ, SSJ), free edge delamination at theouter edges of the composite, normally observed in thick composites coupons, was not predicted. Not capturing thedelamination at the free edges may be related to the coarse mesh used in these regions. Since the potential delaminationinterfaces are between 0 degree plies, delamination is less likely to occur than if the potential delamination interfaceswere placed between a 45 degree ply and a -45 degree ply. In general, the predicted delaminations are in agreementwith experimental findings of similar tested configurations.4–6

The locations of eight strain gages placed on each test coupon are shown with their corresponding label (SG-n) inFigure 21. The top surface corresponds to the composite face closest to the applied load for the single-sided joint case.Strain gages 1-4 are located on the top surface and strain gages 5-8 are located on the bottom surface. These resultsare presented in Figure 22 and they are compared to strains at the equivalent locations in the model. The measured orpredicted strains are on the vertical axis, while the total load (sum of bearing and bypass) is on the horizontal axis.

For the FHT case, only two strain gages are shown, since the other strain gages can be shown to be identicalvia symmetry. The analysis and test results were each reviewed to confirm that the results were in fact symmetricwhere expected; the experimental strain data exhibited up to 5 percent variation. As seen in Figure 22a, there is goodagreement between the experimental and analytical results. In both cases, the strain along the centerline (SG-1) waslower than the strain towards the outer side (SG-2) of the sample. At the onset of failure, a drop in the strain along thecenterline (SG-1) was measured, and the analytical model correctly predicted this same response.

The comparison between the experimental measurements and analytical predictions for the DSJ are shown in Fig-ure 22b. At lower loads, the strains are not expected to correlate since different loading paths were used experimentallyand analytically. Once the experimentally applied bearing load has leveled off, the predicted strains show strong cor-relation with the experimental results. For example, the predicted strains for SG-1 and SG-2 are higher than they arefor SG-3 and SG-4, and the predicted strains along the centerline (SG-1 and SG-3) exhibit a change in slope near theonset of failure that was also observed during testing. The strain-load curve remains mostly linear near the edge of thesample (SG-2 and SG-4). The predicted strains are typically within a few percent of the measured strains, except theyare about 10 percent lower than the measured strains for SG-2.

The model also predicted the strains quite well for the SSJ loading case, with some differences that will be ex-plained below. Refer to Figure 22c and 22d. Two strain gages (SG-7, SG-8) located in the area with the most bendingexhibited some problems during testing. The predictions for SG-1, SG-2, SG-5 and SG-6 were very close to the mea-surements. The model correctly predicted that the strains would be slightly higher near the edge of the sample (SG-2)than they are along the centerline (SG-1) on the top surface. There was no predicted difference between the centerline

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(b) DSJ (a) FHT

Interface - 1

Interface - 2

Interface - 3

Direction of the bypass and/or bearing load

Figure 19. Predicted delamination progression at the failure load for the filled hole tension (FHT) and the double sided bypass and bearingload (DSJ) test configurations.

(b) 100% of failure load

Interface - 1

Interface - 2

Interface - 3

(a) 60% of failure load

Figure 20. Predicted delamination progression at 60 percent of the load and at the failure load for the single sided joint test configuration

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SG-2

SG-4

0.5” 1.0” 1.75”

SG-6 SG-5

SG-8 SG-7

0.5” 1.0”1.75”

Pbypass Pbypass

Pbearing

SG-1

SG-3

(a) Top Surface (b) Bottom Surface

Figure 21. Strain gage placement and label system.

0.0

0.2

0.4

0.6

0.8

1.0


Stra

in (%

)

SG-1 SG-1SG-2 SG-2

Test Model

0.0

0.2

0.4

0.6

0.8

1.0


Stra

in (%

)SG-1 SG-1SG-2 SG-2SG-3 SG-3SG-4 SG-4Test Model

-0.4

0.0

0.4

0.8

1.2

1.6


Stra

in (%

)

SG-1 SG-1SG-2 SG-2SG-3 SG-3SG-4 SG-4

Test Model

-0.4

0.0

0.4

0.8

1.2

1.6


Stra

in (%

)

SG-5 SG-5SG-6 SG-6SG-7 SG-7SG-8 SG-8

Test Model

(b) Double sided joint (a) Filled hole tension

(c) Single sided joint top composite surface (d) Single sided joint bottom composite surface

Figure 22. Predicted strain response for all test configurations compared to the experimental measured strain response.

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(SG-5) and near-edge (SG-6) strain gages on the bottom surface; this is completely consistent with what was observedduring the test. There is a difference between the predicted and measured strains for both the top and bottom surfaces(SG-3, SG-4, SG-7 and SG-8). For the four strain gage locations, the predicted strain curves are shifted compared tothe measured strain curves, all by a similar amount but in opposite directions on the two surfaces, indicating that morebending is predicted than was observed. The strains are shifted higher on the bottom surface (SG-7 and SG-8) andlower on the top surface (SG-3 and SG-4) by roughly the same amount. The additional bending of the composite islikely due to inappropriate boundary conditions near the applied load. One possibility for this discrepancy between thetest data and the numerical predictions is now considered. Analytically, the bearing load is applied uniformly alongthe thickness of the load washer and that washer is allowed to rotate. Experimentally, the loads were applied in sucha way that this washer was not allowed to rotate substantially. In the tests, any rotation of the load washer results inthe load being applied closer to the composite surface; this non-uniform load differs from the uniform load appliedanalytically, and results in a smaller moment arm. Since the moment arm is larger in the model than it is in the test,more bending is to be expected in the model which is reflected in the results shown. Referring back to Figure 4, thisfinding would lead one to re-examine the boundary conditions to better model this condition. In this case, one couldconsider either modeling a rigid load washer or applying the loads closer to the composite face.

In summary, the strain-load slope is predicted correctly for 13 of the 14 strain gages. At onset of failure, thepredicted strains decrease with increasing loading, a response that is also seen in the experimental data. The followingtwo indicators demonstrate that the damage model developed here captures the critical phenomena properly for FHT,DSJ, and SSJ: (1) The predicted failure load shows good agreement with the measured failure load; and (2) Thepredicted strains are consistent with measured strains.

V. Concluding Remarks

A novel analytical model to predict the bypass and bearing strength of composites is developed. The model in-cludes a contact algorithm, finite deformation theory, nonlinear material behavior, and progressive damage of compos-ites. The progressive failure methodology (PFM) incorporates the following failure modes: delamination, nonlinearshear damage, matrix-cracking, and fiber breakage.

The benefits of each of the failure modes implemented in the PFM are realized: First, a model to predict delamina-tions, using a hybrid of a strength and fracture based approaches, is incorporated in ABAQUS 6.5-6 as a user routine.This significantly improves both the ABAQUS implementation and the typical industry methods by speeding conver-gence an order of magnitude. Second, academia and industry usually only consider the nonlinear shear behavior in theplane of the lamina. In this paper, a new shear constitutive law is postulated that accounts for the nonlinear behaviorof the matrix transverse to the lamina as well as in the plane of the lamina. This correctly accounts for the energydissipated under shear loading in all shear planes. Third, a failure criterion for matrix-cracking is expanded to accountfor the three dimensional state of stress that normally exists in thick, mechanically fastened composite joints. Thiscriterion incorporates state-of-the-art composite failure analysis by using a search algorithm to determine the mostlikely fracture plane. Fourth, fiber failure is predicted only when a critical area of fibers have exceeded their strainallowables. This is in-line with current methods.

All four of these failure modes are predicted to initiate and propagate based on degradation rules that are derivedfrom continuum damage mechanics. This approach makes use of the effective stress concept, the principle of equiva-lent elastic strain energy density, and the damage eigenvalues. The orientation and density of matrix-cracks and fiberdamage are represented through a damage tensor, an internal state tensor that is thermodynamically consistent withinthe framework of continuum damage mechanics. The damage tensor evolves as damage associated with differentfailure modes accumulates.

The capability of the damage model was demonstrated with the following test configurations: notched composite,filled hole tension (FHT), double bearing sided bypass and bearing load (DSJ), and single sided bypass and bearingload (SSJ). The strength prediction, the characteristics of the different failure modes that were predicted, and thepredicted strain were all in general agreement with experimental data. The initiation and progression of multipledelaminations through the thickness of the composite, the accumulation of matrix cracks, and fiber breakage werepredicted for all test configurations. Since slight discrepancies were obtained in the failure modeling of the singlesided joint (SSJ), the next step is to explore the loading and boundary conditions of the SSJ model. This may alsorequire further refinement of the damage model.

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