Failure Criteria for FRP Laminates by Carlos G Davila Et Al

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Journal of Composite

DOI: 10.1177/0021998305046452 2005; 39; 323 Journal of Composite Materials

Carlos G. Davila, Pedro P. Camanho and Cheryl A. Rose Failure Criteria for FRP Laminates

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Failure Criteria for FRP Laminates

CARLOS G. DAVILA,1,* PEDRO P. CAMANHO2AND CHERYL A. ROSE

3

1MS 240, NASA Langley Research Center

Hampton, VA 23681, USA2DEMEGI, Faculdade de Engenharia

Rua Dr. Roberto Frias,

4200-465 Porto, Portugal3MS 135, NASA Langley Research Center

Hampton, VA 23681, USA

ABSTRACT: A new set of six phenomenological failure criteria for fiber-reinforcedpolymer laminates denoted LaRC03 is described. These criteria can predict matrixand fiber failure accurately, without the curve-fitting parameters. For matrix failureunder transverse compression, the angle of the fracture plane is solved bymaximizing the Mohr-Coulomb effective stresses. A criterion for fiber kinking isobtained by calculating the fiber misalignment under load and applying the matrixfailure criterion in the coordinate frame of the misalignment. Fracture mechanicsmodels of matrix cracks are used to develop a criterion for matrix failure in tensionand to calculate the associated in situ strengths. The LaRC03 criteria are applied to afew examples to predict failure load envelopes and to predict the failure mode foreach region of the envelope. The analysis results are compared to the predictionsusing other available failure criteria and with experimental results.

KEY WORDS: polymer-matrix composites (PMCs), matrix cracking, crack, failurecriterion.

INTRODUCTION

THE MECHANISMS THAT lead to failure in composite materials are not yet fully under-stood, especially for matrix or fiber compression. The inadequate understanding of

the mechanisms, and the difficulties in developing tractable models of the failure modesexplains the generally poor predictions by most participants in the World Wide FailureExercise [1–7] (WWFE). The results of the WWFE round-robin indicates that the predic-tions of most theories differ significantly from the experimental observations, even whenanalyzing simple laminates that have been studied extensively over the past 40 years [3].

The uncertainty in the prediction of initiation and progression of damage in compositessuggests a need to revisit the existing failure theories, assess their capabilities, and to develop

*Author to whom correspondence should be addressed. E-mail: [email protected]

Journal of COMPOSITE MATERIALS, Vol. 39, No. 4/2005 323

0021-9983/05/04 0323–23 $10.00/0 DOI: 10.1177/0021998305046452� 2005 Sage Publications



new theories where necessary. The present paper describes a newly developed set of sixphenomenological criteria for predicting failure of unidirectional FRP laminates.For simplicity, the development was based on plane stress assumptions. In the followingsection, the motivation for developing nonempirical failure criteria to predict matrixcracking under in-plane shear and transverse tension or compression is established byexamining several existing criteria. Next, new matrix failure criteria for matrix tension andcompression are proposed. Then, a new fiber kinking failure criterion for fiber compressionis developed by applying the matrix failure criteria to the configuration of the kink. Theresulting set of six failure criteria is denoted LaRC03 and are summarized in the Appendix.Finally, examples of calculated failure envelopes are presented. The predicted results arecompared to the results of other available failure criteria and with experimental results.

STRENGTH-BASED FAILURE CRITERIA

Strength-based failure criteria are commonly used with the finite element method topredict failure events in composite structures. Numerous continuum-based criteria havebeen derived to relate internal stresses and experimental measures of material strength tothe onset of failure. In the following sections, the Hashin criteria [8,9] are briefly reviewed,and improvements proposed by Sun et al. [10], and Puck and Schurmann [5,6] overHashin’s theories are examined.

Hashin Criteria 2D (1980)

Hashin established the need for failure criteria that are based on failure mechanisms. In1973, he used his experimental observations of failure of tensile specimens to propose twodifferent failure criteria, one related to fiber failure and the other to matrix failure. Thecriteria assume a quadratic interaction between the tractions acting on the plane of failure.In 1980, he introduced fiber and matrix failure criteria that distinguish between tensionand compression failure. Given the difficulty in obtaining the plane of fracture for thematrix compression mode, Hashin used a quadratic interaction between stress invariants.

Numerous studies conducted over the past decade indicate that the stress interactionsproposed by Hashin do not always fit the experimental results, especially in the case ofmatrix or fiber compression. It is well known, for instance, that moderate transversecompression (�22<0) increases the apparent shear strength of a ply, which is not wellpredicted by Hashin’s criterion. In addition, Hashin’s fiber compression criterion does notaccount for the effects of in-plane shear, which reduce significantly the effectivecompressive strength of a ply. Several researchers have proposed modifications toHashin’s criteria to improve their predictive capabilities. Modifications proposed by Sunand Puck are described below.

IMPROVED CRITERIA FOR MATRIX COMPRESSIONSun et al. [10] proposed an empirical modification to Hashin’s 1973 criterion for matrix

compression failure to account for the increase in shear strength due to compressive �22.The Sun criterion is:

�22YC

� �2þ

�12SL � ��22

� �2

¼ 1 ð1Þ

324 C. G. DAVILA ET AL.



where � is an experimentally determined constant and may be regarded as an internalmaterial friction parameter. The denominator SL

��22 can be considered an effectivelongitudinal shear strength that increases with transverse compression �22 (see signconvention in Figure 1). As in Hashin’s theories, the angle of the fracture plane is notcalculated.

Puck’s Action Plane proposal [5] accounts for the influence of transverse compressionon matrix shear strength by increasing the shear strength by a term that is proportional tothe normal stress �n acting at the fracture plane. Puck’s matrix failure criterion undertransverse compression is:

�T

ST � �T�n

� �2

þ�L

SL � �L�n

� �2

¼ 1 ð2Þ

where �T and �L are the shear stresses acting on the fracture plane defined in Figure 1. Thedirect contribution of �22 has been eliminated by assuming that fracture initiation isindependent of the transverse compressive strength. Internal material friction ischaracterized by the coefficients �L and �T, which are determined experimentally.

A key element to Puck’s proposal is the calculation of the angle of the fracture plane, �,shown in Figure 1. Puck determined that matrix failures dominated by in-plane shearoccur in a plane that is normal to the ply and parallel to the fibers (�¼ 0�). When thetransverse compression rises, the angle of the fracture plane � jumps to about 40�, andincreases with compression to 53� 2� for pure transverse compression. In the WWFE,Puck’s predicted failure envelopes correlated reasonably well with the test results [6].However, Puck’s semi-empirical approach uses several material parameters that are notphysical and may be difficult to quantify without considerable experience with a particularmaterial system.

LaRC03 CRITERIA FOR MATRIX FAILURE

In this section, a new set of criteria is proposed for matrix fracture that is based on theconcepts proposed by Hashin and the fracture plane concept proposed by Puck. In thecase of matrix tension, the fracture planes are normal to the plane of the plies and parallelto the fiber direction. For matrix compression, the plane of fracture may not be normal tothe ply, and Hashin was not able to calculate the angle of the plane of fracture. In thepresent proposal, Mohr-Coulomb effective stresses [11] are used to calculate the angle ofthe fracture plane.

Figure 1. Fracture of a unidirectional lamina subjected to transverse compression and in-plane shear.

Failure Criteria for FRP Laminates 325



Criterion for Matrix Failure Under Transverse Compression (�22<0)

The Mohr-Coulomb (M-C) criterion [11] is commonly used in applications wherefracture under tension loading is different from fracture under compression loading, suchas in soil mechanics or in the fracture of cast iron. The application of the M-C criterion tomultiaxial failure of epoxy resins was studied by Kawabata [12]. While studying the failureof chopped glass-fiber/epoxy mat laminates under confining pressures, Boehler and Raclin[13] found the Tsai–Wu criterion to be inadequate, and formulated a shearing criterionbased on the M-C criterion. Taliercio and Sagramoso [14] used the M-C criterion within anonlinear micromechanical model to predict the macroscopic strength properties of fibercomposites.

The M-C criterion is represented geometrically in Figure 2 by Mohr’s circle for a state ofuniaxial compression. The angle of the plane of fracture �0 is 53�, which is a typicalfracture angle for laminated composites under transverse compression loading (Puck [5]).The line AB is the tangent to Mohr’s circle at A and is called the Coulomb fracture line.The M-C criterion postulates that in a state of biaxial normal stress, fracture occurs forany Mohr’s circle that is tangent to the Coulomb fracture line. The effective stress �eff isrelated to the stresses �T and �n acting on the fracture plane by the expression�eff¼ �Tþ ��n. In the literature, tan�1(�) is called the angle of internal friction and it isassumed to be a material constant. When �¼ 0, the M-C criterion is equivalent to theTresca condition [11].

In general, the fracture plane can be subjected to transverse as well as in-plane stresses,in which case the effective stresses must be defined in both orthogonal directions as shownin Equation (3).

�Teff ¼ �T�� þ �T�n� �

�Leff ¼ �L�� þ �L�n� � ð3Þ

-Yc

τeff=0.378 Yc

η=

τT

τT

σn

σn τ

σ

σ22= -YC

A

B

0

0

α =0

α =530

53

-1tan(2 )α0

Figure 2. Mohr’s circle for uniaxial compression and the effective transverse shear.




where the terms �T and �L are referred to as coefficients of transverse and longitudinalinfluence, respectively, and the operand xh i ¼ 1

2ðxþ jxjÞ:Matrix failure under compression

loading is assumed to result from a quadratic interaction between the effective shearstresses acting on the fracture plane. The failure index for a failure mode is written as anequality stating that stress states that violate the inequality are not physically admissible.The matrix failure index (FIM) is

LaRC03#1 FIM ¼�TeffST

� �2

þ�LeffSLis

� �2

� 1 ð4Þ

where ST and SLis are the transverse and longitudinal shear strengths, respectively. The

subscript M indicates matrix failure. The subscript ‘is’ indicates that for general laminatesthe in situ longitudinal shear strength rather than the strength of a unidirectional laminateshould be used. The constraining effect of adjacent plies substantially increases theeffective strength of a ply. It is assumed here that the transverse shear strength ST isindependent of in situ effects. The calculation of the in situ strength SL

is is discussed in alater section.

The stress components acting on the fracture plane can be expressed in terms of the in-plane stresses and the angle of the fracture plane, � (see Figure 1).

�n ¼ �22 cos2 �

�T ¼ ��22 sin � cos�

�L ¼ �12 cos �

ð5Þ

Using Equations (3) and (5), the effective stresses for an angle of the fracture plane �between 0� and 90� are

�Teff ¼ ��22 cos� ðsin �� T cos�Þ� �

�Leff ¼ cos� ð �12j j þ �L�22 cos�Þ� � ð6Þ

CALCULATION OF COEFFICIENTS �T AND �L AND STRENGTH ST

The coefficients of influence �T and �L are obtained from the case of uniaxial transversecompression (�22<0, �12¼ 0). At failure, the in-plane compressive stress is equal to thematrix compressive strength, �22¼�YC. The effective transverse shear stress at failure is

�Teff ¼ ST ¼ YC cos� ðsin �� T cos�Þ ð7Þ

Under uniaxial transverse compression, fracture occurs at a fracture angle �0 thatmaximizes the effective transverse shear. Taking the derivative of the transverse shearstress, Equation (7), with respect to � gives

@�Teff@�

¼ 0 ¼ � sin �0 ðsin �0 � �T cos�0Þ þ cos�0 ðcos�0 þ �T sin �0Þ

) cos2 �0 � sin2 �0 þ 2�T sin �0 cos�0 ¼ 0

ð8Þ




Solving Equation 8 for �T gives

�T ¼�1

tan 2�0ð9Þ

Puck [5] determined that when loaded in transverse compression, most unidirectionalgraphite–epoxy composites fail by transverse shear along a fracture plane oriented at�0¼ 53� 2�. Therefore, the coefficient of transverse influence is in the range0.21� �T� 0.36. Note that if the fracture plane were oriented at �0¼ 45�, the coefficientof transverse influence would be equal to zero.

The transverse shear strength ST is difficult to measure experimentally. However,substituting Equation (9) into Equation (7) gives an expression relating the transverseshear strength to the transverse compressive strength:

ST ¼ YC cos�0 sin �0 þcos �0

tan 2�0

� �ð10Þ

For a typical fracture angle of �0¼ 53� gives ST¼ 0.378YC, as was shown in Figure 2.

The transverse shear strength is often approximated as ST¼ 0.5YC, which would imply

from Equations (7) and (9) that the fracture plane is at �0¼ 45� and that �T¼ 0. Using thisapproximation, Hashin’s 1980 two-dimensional criterion for matrix failure in compressionbecomes identical to his 1973 criterion.

The coefficient of longitudinal influence, �L, can be determined from shear tests withvarying degrees of transverse compression. In the absence of biaxial test data, �L can beestimated from the longitudinal and transverse shear strengths, as proposed by Puck:

�L

SL¼

�T

ST) �L ¼ �

SL cos 2�0

YC cos2 �0ð11Þ

DETERMINATION OF THE ANGLE OF THE FRACTURE PLANEThe angle of the fracture plane for a unidirectional laminate loaded in transverse

compression is a material property that is easily obtained from experimental data.However, under combined loads, the angle of the fracture plane must be determined. Thecorrect angle of the fracture plane is the one that maximizes the failure index (FI) inEquation (4). In the present work, the fracture angle is obtained by searching for themaximum of the FI (Equation (4)) within a loop over the range of possible fracture angles:0<�<�o. A graphical representation of matrix failure envelopes at various fractureangles is shown in Figure 3 for a unidirectional E-Glass–epoxy composite subjected totransverse compression and in-plane shear loading. As seen in the figure, the fracture anglethat maximizes the FI for small transverse stresses is �¼ 0�. When the applied transversestress �22 has a magnitude equal to approximately 2/3 of the transverse compressivestrength, YC, the angle of the critical fracture plane switches from �¼ 0� to �¼ 40�,and then rapidly increases to �¼ �0, the angle of fracture for uniaxial transversecompression.




Criterion for Matrix Failure Under Transverse Tension (�22>0)

Transverse matrix cracking is often considered a benign mode of failure because itnormally causes such a small reduction in the overall stiffness of a structure that it isdifficult to detect during a test. However, transverse matrix cracks can affect thedevelopment of damage and can also provide the primary leakage path for gases inpressurized vessels.

To predict matrix cracking in a laminate subjected to in-plane shear and transversetensile stresses, a failure criterion must be capable of calculating the ‘in situ’ strengths. Thein situ effect, originally detected in Parvizi et al.’s [15] tensile tests of cross-ply glass fiber-reinforced plastics, is characterized by higher transverse tensile and shear strengths of a plywhen it is constrained by plies with different fiber orientations in a laminate, comparedwith the strength of the same ply in a unidirectional laminate. The in situ strength alsodepends on the number of plies clustered together, and on the fiber orientation of theconstraining plies. The results of Wang’s [16] tests of [0/90n/0] (n¼ 1, 2, 3, 4) carbon–epoxylaminates shown in Figure 4 indicate that thinner plies exhibit higher transverse tensilestrength.

Accurate in situ strengths are necessary for any stress-based failure criterion for matrixcracking in constrained plies. Both experimental [16–18] and analytical methods [19–21]have been proposed to determine the in situ strengths. In the following sections, the in situstrengths are calculated using fracture mechanics solutions for the propagation of cracksin a constrained ply.

FRACTURE MECHANICS ANALYSIS OF A CRACKED PLYThe failure criterion for predicting matrix cracking in a ply subjected to in-plane shear

and transverse tension proposed here is based on the fracture mechanics analysis of a slitcrack in a ply, as proposed by Dvorak and Laws [19]. The slit crack represents amanufacturing defect that is idealized as lying on the 1–3 plane, as represented in Figure 5.

0

20

40

60

80

100

-140 -120 -100 -80 -60 -40 -20 0

σ22

α=0

α=52

α=30

α=40

α=45

, MPa

0˚

40˚

45˚

52˚

τ12, MPa

Figure 3. Matrix failure envelopes for a typical unidirectional E-glass–epoxy lamina subjected to in-planetransverse compression and shear loading.




It has a length 2�0 across the thickness of a ply, t. Physically, this crack represents adistribution of matrix–fiber debonds that may be present in a ply as a consequence ofmanufacturing defects or from residual thermal stresses resulting from the differentcoefficients of thermal expansion of the fibers and of the matrix. Therefore, the slit crack isan ‘effective crack,’ representing the macroscopic effect of matrix–fiber debonds that occurat the micromechanical level [16].

Plane stress conditions are assumed. The transverse tensile stress �22 is associated withMode I loading, whereas the in-plane shear stress �12 is associated with Mode II loading.The crack represented in Figure 5 can grow in the 1-(longitudinal, L)-direction, in the3-(transverse, T )-direction, or in both directions.

The components of the energy release rate for the crack geometry represented in Figure 5were determined by Dvorak and Laws [19]. For mixed-mode loading, the energy releaserate for crack growth in the T and L directions, G(T ) and G(L), respectively, are given by:

GðTÞ ¼�a02

ð�2I�022�

222 þ �2II�

044�

212Þ

GðLÞ ¼�a04

ð�2I�022�

222 þ �2II�

044�

212Þ

ð12Þ

Thin ply model

[ 25/90 ]s± n

[0/ /0]90n

[25 /-25 /90 ]s2 2 2

[90 ]s8

Onset ofdelamination

Thin Thick

Thick ply model

Unidirectional

T300/944

0

0.1

0.2

0.3

0.4 0.8 1.2 1.6 2.00

Tran

sver

se s

tren

gth

of 9

0 p

ly, G

Pa

˚

Inner 90 ply thickness 2a, mm˚

Figure 4. Transverse tensile strength as a function of number of plies clustered together, with models fromDvorak [19] based on experimental data from Wang [16].

1 (L ) 22a 02a 0t

2a 0

3 (T) 3 (T)

L

Figure 5. Slit crack geometry (after Dvorak [19]).




where it can be observed that the energy release rate G(L) for longitudinal propagation is afunction of the transverse slit size a0 and that it is not a function of the slit length in thelongitudinal direction aL0 .

The parameters �i, i¼ I, II in Equation (12) are the stress intensity reduction coefficientsfor propagation in the transverse direction, and the parameters �i, i¼ I, II are thereduction coefficients for propagation in the longitudinal direction. These coefficientsaccount for the constraining effects of the adjoining layers on crack propagation: thecoefficients are nearly equal to 1.0 when 2�0� t, and are less than 1.0 when a0 � t.Experimental results [18] have shown an increase in the in situ transverse tensile strengthof [� �/90n]s, �¼ 0�,30�,60�, laminates for increasing stiffness of adjoining sublaminates��. This implies that the value of the parameter �i decreases with increasing stiffness ofadjoining sublaminates. Considering that a transverse crack can promote delaminationbetween the plies, Dvorak and Laws [19] suggested that the effective value of �i can belarger than obtained from the analysis of cracks terminating at the interface, and suggestedthe use of �i¼ �i ¼ 1.

The parameters �0jj are calculated as [19]:

�022 ¼ 2

1

E2��221E1

� �

�044 ¼

1

G12

ð13Þ

The Mode I and Mode II components of the energy release rate can be obtained for theT-direction using Equation (12) with �i ¼ 1:

GIðTÞ ¼�a02

�022�

222

GIIðTÞ ¼�a02

�044�

212

ð14Þ

The corresponding components of the fracture toughness are given as:

GIcðTÞ ¼�a02

�022ðY

Tis Þ

2

GIIcðTÞ ¼�a02

�044ðS

LisÞ

2ð15Þ

where YTis and SL

is are the in situ transverse tensile and shear strengths, respectively.For propagation in the longitudinal direction, the Mode I and Mode II components of

the energy release rate are:

GIðLÞ ¼�a04

�022�

222

GIIðLÞ ¼�a04

�044�

212

ð16Þ




and the components of the fracture toughness are:

GIcðLÞ ¼�a04

�022ðY

Tis Þ

2

GIIcðLÞ ¼�a04

�044ðS

LisÞ

2ð17Þ

Having obtained expressions for the components of the energy release rate and fracturetoughness, a failure criterion can be applied to predict the propagation of the slit crackrepresented in Figure 5. Under the presence of both in-plane shear and transverse tension,the critical energy release rate Gc depends on the combined effect of all microscopic energyabsorbing mechanisms such as the creation of new fracture surface. Relying onmicroscopic examinations of the fracture surface, Hahn and Johannesson [22] observedthat the fracture surface topography strongly depends on the type of loading. Withincreasing proportion of the stress intensity factor KII, more hackles are observed in thematrix, thereby indicating more energy absorption associated with crack extension.Therefore, Hahn proposed a mixed mode criterion written as a first-order polynomial inthe stress intensity factors KI and KII. Written in terms of the Mode I and Mode II energyrelease rates, the Hahn criterion is

ð1� gÞ

ffiffiffiffiffiffiffiffiffiffiffiffiGIðiÞ

GIcðiÞ

sþ g

GIðiÞ

GIcðiÞþ

GIIðiÞ

GIIcðiÞ¼ 1, i ¼ T ,L ð18Þ

where the material constant g is defined identically from either Equation (14) or (17) as:

g ¼GIc

GIIc¼

�022

�044

YTis

SLis

� �2

ð19Þ

A failure index for matrix tension can be expressed in terms of the ply stresses and in situstrengths YT

is and SLis by substituting either Equations (14), (15) or (16), (17) into the

criterion in Equation (18) to yield:

LaRC03#2 FIM ¼ ð1� gÞ�22YT

is

þ g�22YT

is

� �2

þ�12SLis

� �2

� 1 ð20Þ

The criterion presented in Equation (20), with both linear and quadratic terms of thetransverse normal stress and a quadratic term of the in-plane shear stress, is similar to thecriteria proposed by Hahn and Johannesson [22], Liu and Tsai [23] (for transverse tensionand in-plane shear), and Puck and Schurmann [5]. In addition, if g¼ 1 Equation (18)reverts to the linear version of the criterion proposed by Wu and Reuter [24] for thepropagation of delamination in laminated composites:

GI

GIcþ

GII

GIIc¼ 1 ð21Þ




Furthermore, using g¼ 1, Equation (20) reverts to the well-known Hashin criterion [8]for transverse matrix cracking under both in-plane shear and transverse tension, where theply strengths are replaced by the in situ strengths:

FIM ¼�22YT

is

� �2

þ�12SLis

� �2

� 1 ð22Þ

FAILURE OF THICK EMBEDDED PLIESA thick ply is defined as one in which the length of the slit crack is much smaller than

the ply thickness, 2a0� t, as illustrated in Figure 6. The minimum thickness for a thick plydepends on the material used. For E-glass–epoxy and carbon–epoxy laminates, Dvorakand Laws [19] calculated the transition thickness between a thin and a thick ply to beapproximately 0.7mm, or about 5–6 plies.

For the geometry represented in Figure 6, the crack can grow in the transverse or in thelongitudinal direction. Comparing Equations (14) and (16), however, indicates that theenergy release rate for the crack slit is twice as large in the transverse direction as it is inthe longitudinal direction. Since Equation (14) also indicates that the energy release rate isproportional to the crack length 2a0, the crack will grow unstably in the transversedirection. Once the crack reaches the constraining plies, it can propagate in thelongitudinal direction, as well as induce a delamination.

Crack propagation is predicted using Equation (20), and the in situ strengths can becalculated from the components of the fracture toughness as:

YTis ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GIcðTÞ

�a0�022

s

SLis ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GIIcðTÞ

�a0�044

s ð23Þ

It can be observed from Equations (23) that the in situ strengths of thick plies YTis and

SLis are functions of the toughnesses GIc(T ) and GIIc(T ) of the material and the size of the

material flaw, 2a0. Therefore, the in situ strengths for thick plies are independent of the plythickness, as has been observed by Dvorak and Laws [19] and Leguillon [25], and as wasshown in Figure 4.

FAILURE OF THIN EMBEDDED PLIESThin plies are defined as having a thickness smaller than the typical defect, t<2a0, so the

slit crack extends across the entire thickness t of the ply, as represented in Figure 7.

2a0

YT

tτ12, σ22, SL

is is

Figure 6. Geometry of slit crack in a thick embedded ply subjected to tension and shear loads.




In the case of thin plies, crack defects can only grow in the longitudinal (L) direction,or trigger a delamination between the plies. The in situ strengths can be calculated from thecomponents of the fracture toughness as:

Thin ply: YTis ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8GIcðLÞ

�t�022

s

SLis ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8GIIcðLÞ

�t�044

s ð24Þ

where it can be observed that the in situ strengths are inversely proportional toffiffit

p. The

toughnesses GIc (L) and GIIc (L) can be assumed to be the values measured by standardFracture Mechanics tests, such as the double cantilever beam test (DCB) for Mode I andthe end-notch flexure test (ENF) test for Mode II. Using Equations (23) and (24), Dvorakand Laws [19] obtained a good correlation between the predicted and experimentallyobtained in situ strengths of both thick and thin 90� plies in [0/90n/0] laminates, as shownin Figure 4.

FAILURE OF UNIDIRECTIONAL LAMINATESThe fracture of a unidirectional specimen is taken as a particular case of a thick ply [19].

The defect size is 2a0, as for thick embedded plies. However, in the absence of constrainingplies, the critical initial slit crack is located at the surface of the laminate. For tensileloading, the crack can be located at the edge of the laminate, which increases the energyrelease rate when compared with a central crack. In the case of shear loading, there is nofree edge, so the crack is a central crack, as shown in Figure 8. Crack propagation forunidirectional specimens subjected to tension or shear loads is obtained from the classicsolution of the free edge crack [26].

GIcðTÞ ¼ 1:122�a0�022ðY

T Þ2

GIIcðTÞ ¼ �a0�044ðS

LÞ2

ð25Þ

2a0

Y T

2a0 τ12, SL

σ22,

Figure 8. Unidirectional specimens under transverse tension and shear.

σ22, YTτ12, SLis is

Figure 7. Geometry of slit crack in a thin embedded ply.




where YT and SL are the material tensile and shear strengths as measured fromunidirectional laminate tests. Note that the in situ strengths for thick plies can be obtainedas a function of the strength of unidirectional laminates by substituting Equations (25)into (23):

Thick ply : YTis ¼ 1:12

ffiffiffi2

pYT

SLis ¼

ffiffiffi2

pSL

ð26Þ

The failure criterion for unidirectional plies under in-plane shear and transverse tensionis represented in Equation (20). The toughness ratio g for a thick laminate can also becalculated in terms of the unidirectional properties by substituting Equations (25) intoEquation (19) to yield:

g ¼GIc

GIIc¼ 1:122

�022

�044

YT

SL

� �2

ð27Þ

LaRC03 CRITERIA FOR FIBER FAILURE

Criterion for Fiber Tension Failure

Based on the results of his experimental investigations, Puck and Schurmann [5]recommends the use of a fiber failure criterion that is based on the ‘effective strain’ actingalong the fibers. However, Puck also reports that the difference between using the effectivestrain and the longitudinal stress is insignificant [6]. The LaRC03 criterion for fibertension failure is a noninteracting maximum allowable strain criterion that is simple tomeasure and is independent of fiber volume fraction and Young’s moduli. Consequently,the LaRC03 failure index for fiber tensile failure is:

LaRC03#3 FIF ¼"11"T1

� 1 ð28Þ

Criterion for Fiber Compression Failure

Compressive failure of aligned fiber composites occurs from the collapse of the fibers asa result of shear kinking and damage of the supporting matrix (See for instance therepresentative works of Fleck and Liu [27] and Soutis et al. [28], and Shultheisz and Waas’[29] review of micromechanical compressive failure theories). Fiber kinking occurs as sheardeformation, leading to the formation of a kink band.

Argon [30] was the first to analyze the kinking phenomenon. His analysis was based onthe assumption of a local initial fiber misalignment. The fiber misalignment leads toshearing stresses between fibers that rotate the fibers, increasing the shearing stress andleading to instability. Since Argon’s work, the calculation of the critical kinking stress hasbeen significantly improved with a more complete understanding of the geometry of thekink band as well as the incorporation of friction and material nonlinearity in the analysismodels [27–29, 31].




Several authors [32,33] have considered that misaligned fibers fail by the formationof a kink band when local matrix cracking occurs. Potter et al. [34] assumed thatadditional failure mechanisms that may occur under uniaxial longitudinal compres-sion, such as crushing, brooming, and delaminations, are essentially triggered by matrixfailure.

In the present approach, the compressive strength XC is assumed to be a knownmaterial property that can be used in the LaRC03 matrix damage criterion (Equation (4))to calculate the fiber misalignment angle that would cause matrix failure under uniaxialcompression.

CALCULATION OF FIBER MISALIGNMENT ANGLEThe imperfection in fiber alignment is idealized as a local region of waviness, as shown

in Figure 9. The ply stresses in the misalignment coordinate frame m shown in Figure 9 are

�m11 ¼ cos2 ’ �11 þ sin2 ’ �22 þ 2 sin ’ cos ’j�12j

�m22 ¼ sin2 ’ �11 þ cos2 ’ �22 � 2 sin ’ cos ’j�12j

�m12 ¼ � sin ’ cos ’ �11 þ sin ’ cos ’ �22 þ ðcos2 ’� sin2 ’Þj�12j

ð29Þ

At failure under axial compression, �11¼�XC, and �22¼ �12¼ 0. Substituting thesevalues into Equation (29) gives �m

22 ¼ � sin2 ’CXCand �m12 ¼ sin ’C cos ’CXC, wherethe angle ’C is the total misalignment angle for the case of pure axial compressionloading.

To calculate the FI for fiber kinking, the stresses �m22 and �m12 are substituted into the

criterion for matrix compression failure (Equation (4)). The mode of failure in fiberkinking is dominated by the shear stress �12, rather than by the transverse stress �22. Theangle of the fracture plane is then equal to 0�, and �Teff ¼ 0. The matrix failure criterionbecomes

�Leff ¼ XCðsin ’C cos ’C � �L sin2 ’CÞ ¼ SLis ð30Þ

where SLis is the in situ longitudinal shear strength defined in Equation (23) for

thick plies and in Equation (24) for thin plies. Solving for ’C leads to the quadraticequation:

tan2 ’CSLis

XCþ �L

� �� tan ’C þ

SLis

XC

� �¼ 0 ð31Þ

11

22

m

m

−XC−XC

ϕ σ11

σ22 σ

σ

Figure 9. Imperfection in fiber alignment idealized as local region of waviness.




The smaller of the two roots of Equation (31) is

’C ¼ tan�11�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4 SL

is=XC þ �L

SLis=X

C q

2 SLis=X

C þ �L

0@

1A ð32Þ

Note that if �L were neglected and ’ were assumed to be a small constant angle,Equation (30) would give ’C � SL

is=XC, which is the expression derived by Argon [30] to

estimate the fiber misalignment angle.The total misalignment angle ’ can be decomposed into an initial (constant)

misalignment angle ’0 that represents a manufacturing imperfection, and an additionalrotational component ’R that results from shear loading. The angles ’0 and ’R can becalculated using small angle approximations and Equations (29)

’R ¼�m12G12

¼�’�11 þ ’�22 þ j�12j

G12

’0 ¼ ’C � ’R��XC¼ ’C �

�m12G12

��XC

) ’0 ¼ 1�XC

G12

� �’C

ð33Þ

where the absolute value of the shear |�12| is used because the sign of ’ is assumed to bepositive, regardless of the sign of the shear stress. Recalling that ’¼ ’0þ ’R, it is nowpossible to solve Equations (33) for ’ in terms of ’C.

’ ¼j�12j þ ðG12 � XCÞ’C

G12 þ �11 � �22ð34Þ

Fiber compression failure by the formation of a kink band is predicted using the stressesfrom Equation (29) and the failure criterion for matrix tension or matrix compression. Formatrix compression ð�m

22 < 0Þ, the criterion is the Mohr-Coulomb criterion given inEquation (4), with �¼ 0� and �Teff ¼ 0. The criterion for fiber kinking becomes:

LaRC03#4 FIF ¼j�m12j þ �L�m

22

SLis

� �� 1 ð35Þ

For fiber compression with matrix tension, the transformed stresses of Equation (29) aresubstituted into the matrix tensile failure criterion given in Equation (20) to yield thefollowing criterion for fiber kinking:

LaRC03#5 FIF ¼ ð1� gÞ�m22

YTis

� �þ g

�m22

YTis

� �2

þ�m12SLis

� �2

� 1 ð36Þ

It is interesting to note that for �22¼ 0, the fiber failure criterion in Equation (35) becomes

FIF ¼��11XC

þj�12j

kSL� 1 with k ¼

1

1� ’2 � 2�L’> 1 ð37Þ




The linear interaction between �11 and �12 in Equation (37) is identical to the form used byEdge [7] for the WWFE. For T300/914C, Edge suggests using an empirical value ofk¼ 1.5. However, Edge also indicates that other researchers have shown excellentcorrelation with experimental results with k¼ 1. Using the WWFE strength values ofXC

¼ 900MPa, SL¼ 80MPa, YC

¼ 200MPa, an assumed fracture angle in transversecompression of 53�, and Equations (11) and (32), we get: �L¼ 0.304, ’C¼ 5.3�. With theapproximation ’ ffi ’C, Equation (37) gives k¼ 1.07.

LaRC03 CRITERION FOR MATRIX DAMAGE IN BIAXIAL COMPRESSION

In the presence of high transverse compression combined with moderate fibercompression, matrix damage can occur without the formation of kink bands or damageto the fibers. This matrix damage mode is calculated using the stresses in the misalignedframe in the failure criterion in Equation (4), which gives:

LaRC03#6 FIM ¼�mTeff

ST

� �2

þ�mLeff

SLis

� �2

� 1 ð38Þ

where the effective shear stresses �mTeff and �mL

eff are defined as in Equation (6), but in termsof the in-plane stresses in the misalignment frame, which are defined in Equations (29).

�mTeff ¼ ��m

22 cos� ðsin �� T cos�Þ� �

�mLeff ¼ cos� ð �m12

�� þ �L�m22 cos�Þ

� � ð39Þ

As for all matrix compressive failures herein, the stresses �mTeff and �mL

eff are functions ofthe fracture angle �, which must be determined iteratively.

VERIFICATION PROBLEMS

Example 1 Unidirectional 0� E-glass/MY750 epoxy

All of the failure modes represented by the six LaRC03 criteria (Equations (4), (20),(28), (35), (36) and (38), and summarized in the Appendix) can be represented as a failureenvelope in the �11–�22 plane. An example is shown in Figure 10 for a 0� E-glass/MY750epoxy lamina. Puck’s WWFE analysis results [6] are shown for comparison. In the figure,there is good agreement between LaRC03 and Puck in all quadrants except biaxialcompression, where LaRC03 predicts an increase of the axial compressive strength withincreasing transverse compression. Testing for biaxial loads presents a number ofcomplexities, and experimental results are rare. However, Waas and Schultheisz [35]report a number of references in which multiaxial compression was studied by superposinga hydrostatic pressure in addition to the compressive loading. For all materials considered,there is a significant increase in compressive strength with increasing pressure. Inparticular, the results of Wronski and Parry [36] on glass/epoxy show a strength increaseof 3.3MPa per MPa of hydrostatic pressure, which gives an actual strength increase of




4.3MPa per MPa of applied transverse biaxial stress. More recently, Sigley et al. [37]found 32–71% increase in compressive strength per 100MPa superposed pressure. Theresults of Wronsky et al. is comparable to the 4.3MPa/MPa increase in compressivestrength for the plane stress failure envelope in Figure 10.

Example 2 Unidirectional composite E-Glass/LY556

A comparison of results from various failure criteria with the experimental results inthe �22–�12 stress plane obtained from the WWFE [38] is shown in Figure 11. It can beobserved that within the positive range of �22, all the quadratic failure criteria andLaRC03 give satisfactory results. Since the Maximum Stress Criterion does not prescribeinteractions between stress components, its failure envelope is rectangular. The mostinteresting behavior develops when �22 becomes compressive. Hashin’s 1973 criteriongives an elliptical envelope with diminishing �12 as the absolute value of compressive �22increases, while the experimental data shows a definite trend of shear strength increase as�22 goes into compression.

The envelope for Hashin’s 1980 criteria was calculated using a transverse strengthobtained from Equation (10), and it provides a modest improvement in accuracycompared to the 1973 criterion. Of the criteria shown in Figure 11, Sun (Equation (1)),LaRC03#1 (Equation (4)), and Puck (Equation (2)) capture the shear strength increase atthe initial stage of compressive �22. The failure envelope for Sun’s criterion (Equation (1))was calculated using �L¼ 0.336 from Equation (11). The results indicate a significantimprovement over Hashin’s criteria. An even better fit would have been achieved using ahigher value for �L. Puck’s envelope, which was extracted from the 2002 WWFE [6],appears to be the most accurate, but it relies on fitting parameters based on the same testdata. The LaRC03 curve uses the stiffnesses and strengths shown in Table 1, an assumed

1500 500 0 1000 1500

50

50

100

150

5001000

Puck

FI ,M LaRC03 #1

FI ,M LaRC03 #6

FI ,M LaRC03 #2

FI ,F LaRC03 #3

FI ,F LaRC03 #4

FI ,F LaRC03 #5

σ11, MPa

σ =−11 Yc

σ22, MPa

σ =022m

Figure 10. Biaxial �11–�22 failure envelope of 0� E-glass/MY750 epoxy lamina.




�0¼ 53�, and no other empirical or fitting parameter. In the case of matrix tension, Puck’spredicted failure envelope is nearly identical to LaRC03#2.

Example 3 Cross-ply Laminates

Shuart [39] studied the compression failure of [� �]s laminates and found that for�< 15�, the dominant failure mode in these laminates is interlaminar shearing; for15<�<50�, it is in-plane matrix shearing; and for �>50�, it is matrix compression. Fiberscissoring due to matrix material nonlinearity caused the switch in failure mode fromin-plane matrix shearing to matrix compression failure at larger lamination angles. Thematerial properties shown in Table 2 were used for the analysis. The angle �0 was assumedto be a typical 53�. For other lamination angles, the fracture angle was obtained bysearching numerically for the angle that maximizes the failure criterion in Equation (4).The in situ shear strength did not need to be calculated since it is given by Shuart [39] asSLis ¼ 95:1MPa.The results in Figure 12 indicate that the predicted strengths using LaRC03 criteria

correlate well with the experimental results. The compressive strength predicted usingHashin 1973 criteria is also shown in Figure 12. For �<20�, the Hashin criteria result in anoverprediction of the failure load because the criterion does not account for the effect ofin-plane shear on fiber failure. For lamination angles near 70�, the use of the Hashin

-150 -50 0 50

50

100Puck ‘02

WWFE test ‘02

-100

LaRC03 #1

LaRC03 #2

Hashin ‘73

Sun ‘96

Hashin ‘80Hashin

Max. stress

τ12

σ22

, MPa

, MPa

Figure 11. Failure envelopes and WWFE test data [2] for unidirectional composite E-Glass/LY556.

Table 1. Properties of E-Glass/LY556 (MPa) from [38].

E11 E22 G12 �12 YT Y C SL

53,480 17,700 5830 0.278 36 138 61




criteria results is in an underprediction of the failure load because the criteria do notaccount for the increase in shear strength caused by transverse compression. All of theseeffects are represented by the LaRC03 criteria, which results in a good correlation betweenthe calculated and experimental values.

CONCLUSIONS

The results of the recently concluded World Wide Failure Exercise indicate that theexisting knowledge on failure mechanisms needs further development. Many of theexisting failure models could not predict the experimental response within a tolerable limit.In fact, differences of up to an order of magnitude between the predicted and experimentalvalues were not uncommon. In this paper, a new set of six phenomenologicalfailure criteria is proposed. The criteria for matrix and fiber compression failure arebased on a Mohr-Coulomb interaction of the stresses associated with the plane offracture. Failure envelopes for unidirectional laminates in the �11–�22 and �22–�12 stressplanes were calculated, as well as the compressive strength of cross-ply laminates.The predicted failure envelopes indicate good correlation with experimental results. Theproposed criteria, referred to as LaRC03, represent a significant improvement overthe commonly used Hashin criteria, especially in the cases of matrix or fiber failure incompression.

Com

pres

sive

str

ess,

MP

a

Lamination angle, degrees

Test, Shuart ‘89

Hashin ‘73

LaRC03

α0=530α=440α=00

Figure 12. Compressive strength as a function of ply orientation for [� �]s AS4–3502 laminates.

Table 2. Properties of AS4/3502 (MPa) from [39].

E11 E22 G12 �12 X C Y C SLis

127,600 11,300 6000 0.278 1045 244 95.1




APPENDIX: SUMMARY OF LARC03 FAILURE CRITERIA

Matrix Cracking

Fiber Failure

Required Unidirectional Material Properties: E1, E2, G12, �12, XT, XC, YT, YC, SL,

GIc(L), GIIc(L)Optional Properties: �0, �

L

The in situ strengths for a thin embedded ply are:

YTis ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8GIcðLÞ

�t�022

sand SL

is ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8GIIcðLÞ

�t�044

swhere �0

22 ¼ 21

E2��221E1

� �and �0

44 ¼1

G12

Toughness ratio g is obtained from fracture mechanics test data or from the unidirectionalproperties:

For a thin embedded ply, g ¼GIcðLÞ

GIIcðLÞ. Otherwise, g ¼ 1:122

�022

�044

YT

SL

� �2

The effective shear stresses for matrix compression are

�Teff ¼ ��22 cos� ðsin �� T cos�Þ� �

�Leff ¼ cos� ðj�12j þ �L�22 cos�Þ� �

(

The transverse shear strength is

ST ¼ YC cos�0 sin �0 þcos �0

tan 2�0

� �

Matrix tension, �22 � 0 Matrix compression, �22 < 0

FIM ¼ ð1� gÞ�22YT

is

� �þ g

�22YT

is

� �2

þ�12SLis

� �2

�11 < YC �11 � YC

FIM ¼�mTeff

ST

� �2

þ�mLeff

SLis

� �2

FIM ¼�TeffST

� �2

þ�LeffSLis

� �2

Fiber tension, �11 � 0 Fiber compression, �11 < 0

FIF ¼"11"T1

�m22 < 0 �m

22 � 0

FIF ¼j�m12j þ �L�m

22

SLis

� �FIF ¼ ð1� gÞ

�m22

YTis

� �þ g

�m22

YTis

� �2

þ�m12SLis

� �2




�T ¼�1

tan 2�0and �L � �

SLis cos 2�0

YC cos2 �0or from test data.

�0¼ 53� or from test data.

The stresses in the fiber misalignment frame are

�m11 ¼ cos2 ’ �11 þ sin2 ’ �22 þ 2 sin ’ cos ’ �12

�m22 ¼ sin2 ’ �11 þ cos2 ’ �22 � 2 sin ’ cos ’ �12

�m12 ¼ � sin ’ cos ’ �11 þ sin ’ cos ’ �22 þ ðcos2 ’� sin2 ’Þ �12

where

’ ¼j�12j þ ðG12 � XCÞ’C

G12 þ �11 � �22and ’C ¼ tan�1

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4 SL

is=XC þ �L

SLis=X

C q

2 SLis=X

C þ �L

0@

1A

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