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Failure Criteria for FRP Laminates
CARLOSG. DA VILA,1,* PEDROP. CAMANHO2 ANDCHERYLA. ROSE3
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 associatedin situstrengths. The LaRC03 criteria are applied to a
few 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 THATlead 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 modes
explains the generally poor predictions by most participants in the World Wide Failure
Exercise [17] (WWFE). The results of the WWFE round-robin indicates that the predic-
tions of most theories differ significantly from the experimental observations, even when
analyzing 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 composites
suggests 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 ofCOMPOSITEMATERIALS, Vol. 39, No. 4/2005 323
0021-9983/05/04 032323 $10.00/0 DOI: 10.1177/0021998305046452 2005 Sage Publications
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new theories where necessary. The present paper describes a newly developed set of six
phenomenological criteria for predicting failure of unidirectional FRP laminates.
For simplicity, the development was based on plane stress assumptions. In the following
section, the motivation for developing nonempirical failure criteria to predict matrix
cracking under in-plane shear and transverse tension or compression is established by
examining several existing criteria. Next, new matrix failure criteria for matrix tension and
compression are proposed. Then, a new fiber kinking failure criterion for fiber compression
is developed by applying the matrix failure criteria to the configuration of the kink. The
resulting 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 are
compared 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 have
been derived to relate internal stresses and experimental measures of material strength to
the 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 Schu rmann [5,6] over
Hashins theories are examined.
Hashin Criteria 2D (1980)
Hashin established the need for failure criteria that are based on failure mechanisms. In
1973, 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. The
criteria 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 tension
and compression failure. Given the difficulty in obtaining the plane of fracture for the
matrix compression mode, Hashin used a quadratic interaction between stress invariants.
Numerous studies conducted over the past decade indicate that the stress interactions
proposed by Hashin do not always fit the experimental results, especially in the case of
matrix or fiber compression. It is well known, for instance, that moderate transverse
compression (22
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where is an experimentally determined constant and may be regarded as an internal
material friction parameter. The denominator SL22 can be considered an effectivelongitudinal shear strength that increases with transverse compression 22 (see sign
convention in Figure 1). As in Hashins theories, the angle of the fracture plane is not
calculated.
Pucks Action Plane proposal [5] accounts for the influence of transverse compression
on matrix shear strength by increasing the shear strength by a term that is proportional tothe normal stress n acting at the fracture plane. Pucks matrix failure criterion under
transverse compression is:
T
ST Tn
2
L
SL Ln
2 1 2
whereTandL are the shear stresses acting on the fracture plane defined in Figure 1. The
direct contribution of 22 has been eliminated by assuming that fracture initiation is
independent of the transverse compressive strength. Internal material friction ischaracterized by the coefficients L and T, which are determined experimentally.
A key element to Pucks proposal is the calculation of the angle of the fracture plane, ,
shown in Figure 1. Puck determined that matrix failures dominated by in-plane shear
occur in a plane that is normal to the ply and parallel to the fibers ( 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,Pucks predicted failure envelopes correlated reasonably well with the test results [6].
However, Pucks semi-empirical approach uses several material parameters that are not
physical and may be difficult to quantify without considerable experience with a particular
material system.
LaRC03 CRITERIA FOR MATRIX FAILURE
In this section, a new set of criteria is proposed for matrix fracture that is based on the
concepts proposed by Hashin and the fracture plane concept proposed by Puck. In the
case of matrix tension, the fracture planes are normal to the plane of the plies and parallel
to the fiber direction. For matrix compression, the plane of fracture may not be normal to
the ply, and Hashin was not able to calculate the angle of the plane of fracture. In the
present 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.
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Criterion for Matrix Failure Under Transverse Compression (22
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where the terms T and L are referred to as coefficients of transverse and longitudinal
influence, respectively, and the operand xh i 12x jxj:Matrix failure under compression
loading is assumed to result from a quadratic interaction between the effective shear
stresses acting on the fracture plane. The failure index for a failure mode is written as an
equality stating that stress states that violate the inequality are not physically admissible.
The matrix failure index (FIM) is
LaRC03#1 FIM Teff
ST
2
Leff
SLis
2 1 4
where ST and SLis are the transverse and longitudinal shear strengths, respectively. The
subscriptMindicates matrix failure. The subscript is indicates that for general laminates
thein situlongitudinal shear strength rather than the strength of a unidirectional laminate
should be used. The constraining effect of adjacent plies substantially increases the
effective strength of a ply. It is assumed here that the transverse shear strength ST is
independent of in situ effects. The calculation of the in situ strength SLis 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 22cos2 T 22sin cos L 12cos
5
Using Equations (3) and (5), the effective stresses for an angle of the fracture plane between 0 and 90 are
Teff 22cos sin T cos
Leff cos 12j j L22cos 6
CALCULATION OF COEFFICIENTST AND L AND STRENGTH ST
The coefficients of influenceT andL are obtained from the case of uniaxial transverse
compression (22
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Solving Equation 8 for T gives
T 1tan20
9
Puck [5] determined that when loaded in transverse compression, most unidirectional
graphiteepoxy composites fail by transverse shear along a fracture plane oriented at
0 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 0 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 transverse
shear strength to the transverse compressive strength:
ST YC cos 0 sin 0 cos 0tan20
10
For a typical fracture angle of0 53 gives ST 0.378 YC, as was shown in Figure 2.The transverse shear strength is often approximated as ST 0.5 YC, which would implyfrom Equations (7) and (9) that the fracture plane is at 0 45and thatT 0. Using thisapproximation, Hashins 1980 two-dimensional criterion for matrix failure in compression
becomes identical to his 1973 criterion.
The coefficient of longitudinal influence, L, can be determined from shear tests with
varying 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
SL
T
ST) L S
L cos20
YC cos2 011
DETERMINATION OF THE ANGLE OF THE FRACTURE PLANE
The 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. The
correct angle of the fracture plane is the one that maximizes the failure index (FI) in
Equation (4). In the present work, the fracture angle is obtained by searching for the
maximum of the FI (Equation (4)) within a loop over the range of possible fracture angles:
0 <
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Criterion for Matrix Failure Under Transverse Tension (22 > 0)
Transverse matrix cracking is often considered a benign mode of failure because it
normally causes such a small reduction in the overall stiffness of a structure that it is
difficult to detect during a test. However, transverse matrix cracks can affect the
development 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 transverse
tensile stresses, a failure criterion must be capable of calculating the in situ strengths. The
in 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 ply
when it is constrained by plies with different fiber orientations in a laminate, compared
with the strength of the same ply in a unidirectional laminate. The in situ strength also
depends on the number of plies clustered together, and on the fiber orientation of the
constraining plies. The results of Wangs [16] tests of [0/90n/0] (n 1, 2, 3, 4) carbonepoxylaminates shown in Figure 4 indicate that thinner plies exhibit higher transverse tensile
strength.
Accuratein situ strengths are necessary for any stress-based failure criterion for matrix
cracking in constrained plies. Both experimental [1618] and analytical methods [1921]
have been proposed to determine the in situstrengths. In the following sections, the in situ
strengths are calculated using fracture mechanics solutions for the propagation of cracks
in a constrained ply.
FRACTURE MECHANICS ANALYSIS OF A CRACKED PLY
The 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 slit
crack in a ply, as proposed by Dvorak and Laws [19]. The slit crack represents amanufacturing defect that is idealized as lying on the 13 plane, as represented in Figure 5.
0
20
40
60
80
100
-140 -120 -100 -80 -60 -40 -20 0
22
=0
=52
=30
=40
=45
, MPa
0
40
45
52
12, MPa
Figure 3. Matrix failure envelopes for a typical unidirectional E-glassepoxy lamina subjected to in-plane
transverse compression and shear loading.
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It has a length 20 across the thickness of a ply, t. Physically, this crack represents a
distribution of matrixfiber debonds that may be present in a ply as a consequence of
manufacturing defects or from residual thermal stresses resulting from the different
coefficients of thermal expansion of the fibers and of the matrix. Therefore, the slit crack is
an effective crack, representing the macroscopic effect of matrixfiber debonds that occur
at the micromechanical level [16].Plane stress conditions are assumed. The transverse tensile stress 22 is associated with
Mode 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 the
3-(transverse,T)-direction, or in both directions.
The components of the energy release rate for the crack geometry represented in Figure 5
were determined by Dvorak and Laws [19]. For mixed-mode loading, the energy release
rate for crack growth in theTand L directions,G(T) andG(L), respectively, are given by:
GT a02
2I 022222 2II044212
GL a04 2I022222 2II044212
12
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
Transversestrengthof90
ply,
GPa
Inner 90 ply thickness 2a, mm
Figure 4. Transverse tensile strength as a function of number of plies clustered together, with models from
Dvorak [19] based on experimental data from Wang [16].
1(L) 2
2a0
2a0
t
2a 0
3 (T) 3 (T)
L
Figure 5. Slit crack geometry (after Dvorak [19]).
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where it can be observed that the energy release rateG(L) for longitudinal propagation is a
function of the transverse slit size a0 and that it is not a function of the slit length in the
longitudinal direction aL0 .
The parametersi,i I, II in Equation (12) are the stress intensity reduction coefficientsfor propagation in the transverse direction, and the parameters i, i
I, II are the
reduction coefficients for propagation in the longitudinal direction. These coefficients
account for the constraining effects of the adjoining layers on crack propagation: the
coefficients are nearly equal to 1.0 when 20 t, and are less than 1.0 when a0 t.Experimental results [18] have shown an increase in the in situ transverse tensile strength
of [ /90n]s, 0,30,60, laminates for increasing stiffness of adjoining sublaminates. This implies that the value of the parameter idecreases with increasing stiffness ofadjoining sublaminates. Considering that a transverse crack can promote delamination
between the plies, Dvorak and Laws [19] suggested that the effective value of ican be
larger than obtained from the analysis of cracks terminating at the interface, and suggested
the use ofi
i
1.
The parameters 0jjare calculated as [19]:
022 2
1
E2
221
E1
044
1
G12
13
The Mode I and Mode II components of the energy release rate can be obtained for the
T-direction using Equation (12) with i
1:
GIT a02
022
222
GIIT a0
2
044
212
14
The corresponding components of the fracture toughness are given as:
GIcT a02
022YTis 2
GIIcT a02
044SLis2
15
where YTis and SLis 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:
GIL a04
022
222
GIIL a04 044212 16
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and the components of the fracture toughness are:
GIcL a04
022YTis 2
GIIcL a
04
044SLis2 17
Having obtained expressions for the components of the energy release rate and fracture
toughness, a failure criterion can be applied to predict the propagation of the slit crack
represented in Figure 5. Under the presence of both in-plane shear and transverse tension,
the critical energy release rateGcdepends on the combined effect of all microscopic energy
absorbing mechanisms such as the creation of new fracture surface. Relying on
microscopic examinations of the fracture surface, Hahn and Johannesson [22] observed
that the fracture surface topography strongly depends on the type of loading. With
increasing proportion of the stress intensity factor KII, more hackles are observed in the
matrix, thereby indicating more energy absorption associated with crack extension.
Therefore, Hahn proposed a mixed mode criterion written as a first-order polynomial in
the stress intensity factorsKIand KII. Written in terms of the Mode I and Mode II energy
release rates, the Hahn criterion is
1 gffiffiffiffiffiffiffiffiffiffiffiffi
GIiGIci
s gGIi
GIci GIIi
GIIci 1, i T,L 18
where the material constant g is defined identically from either Equation (14) or (17) as:
g GIcGIIc
022
044
YTisSLis
219
A failure index for matrix tension can be expressed in terms of the ply stresses andin situ
strengths YTis and SLis by substituting either Equations (14), (15) or (16), (17) into the
criterion in Equation (18) to yield:
LaRC03#2 FIM 1 g22
YTis g 22
YTis 2 12SLis
2
1 20
The criterion presented in Equation (20), with both linear and quadratic terms of the
transverse normal stress and a quadratic term of the in-plane shear stress, is similar to the
criteria proposed by Hahn and Johannesson [22], Liu and Tsai [23] (for transverse tension
and in-plane shear), and Puck and Schu rmann [5]. In addition, if g 1 Equation (18)reverts to the linear version of the criterion proposed by Wu and Reuter [24] for the
propagation of delamination in laminated composites:
GI
GIc
GII
GIIc 1 21
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Furthermore, using g 1, Equation (20) reverts to the well-known Hashin criterion [8]for transverse matrix cracking under both in-plane shear and transverse tension, where the
ply strengths are replaced by the in situ strengths:
FIM 22YTis
2 12SLis
2 1 22
FAILURE OF THICK EMBEDDED PLIES
A 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-glassepoxy and carbonepoxy laminates, Dvorak
and Laws [19] calculated the transition thickness between a thin and a thick ply to be
approximately 0.7 mm, or about 56 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 the
energy release rate for the crack slit is twice as large in the transverse direction as it is in
the longitudinal direction. Since Equation (14) also indicates that the energy release rate is
proportional to the crack length 2a0, the crack will grow unstably in the transverse
direction. Once the crack reaches the constraining plies, it can propagate in the
longitudinal direction, as well as induce a delamination.
Crack propagation is predicted using Equation (20), and the in situ strengths can be
calculated from the components of the fracture toughness as:
YTis ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GIcTa0022s
SLisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2GIIcTa0
044
s 23
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) andGIIc(T) of the material and the size of the
material flaw, 2a0. Therefore, thein situstrengths for thick plies are independent of the ply
thickness, as has been observed by Dvorak and Laws [19] and Leguillon [25], and as was
shown in Figure 4.
FAILURE OF THIN EMBEDDED PLIES
Thin plies are defined as having a thickness smaller than the typical defect,t
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In the case of thin plies, crack defects can only grow in the longitudinal (L) direction,
or trigger a delamination between the plies. Thein situstrengths can be calculated from the
components of the fracture toughness as:
Thin ply: YTisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8GIcLt022
s
SLisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8GIIcLt044
s 24
where it can be observed that the in situ strengths are inversely proportional toffiffi
tp
. The
toughnesses GIc (L) and GIIc (L) can be assumed to be the values measured by standard
Fracture Mechanics tests, such as the double cantilever beam test (DCB) for Mode I and
the end-notch flexure test (ENF) test for Mode II. Using Equations (23) and (24), Dvorak
and Laws [19] obtained a good correlation between the predicted and experimentally
obtainedin situ strengths of both thick and thin 90 plies in [0/90n/0] laminates, as shownin Figure 4.
FAILURE OF UNIDIRECTIONAL LAMINATES
The 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 constraining
plies, the critical initial slit crack is located at the surface of the laminate. For tensile
loading, the crack can be located at the edge of the laminate, which increases the energy
release rate when compared with a central crack. In the case of shear loading, there is no
free 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 classic
solution of the free edge crack [26].
GIcT 1:122a0022YT2
GIIcT a0044SL2 25
2a0
YT
2a0
12,S
L
22,
Figure 8. Unidirectional specimens under transverse tension and shear.
22,YT
12,SL
is is
Figure 7. Geometry of slit crack in a thin embedded ply.
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where YT and SL are the material tensile and shear strengths as measured from
unidirectional laminate tests. Note that the in situstrengths for thick plies can be obtained
as a function of the strength of unidirectional laminates by substituting Equations (25)
into (23):
Thick ply : YTis 1:12ffiffiffi
2p YT
SLisffiffiffi
2p
SL26
The failure criterion for unidirectional plies under in-plane shear and transverse tension
is represented in Equation (20). The toughness ratio g for a thick laminate can also be
calculated in terms of the unidirectional properties by substituting Equations (25) into
Equation (19) to yield:
g GIc
GIIc 1:122
022
044
YT
SL 2 27
LaRC03 CRITERIA FOR FIBER FAILURE
Criterion for Fiber Tension Failure
Based on the results of his experimental investigations, Puck and Schu rmann [5]
recommends the use of a fiber failure criterion that is based on the effective strain acting
along the fibers. However, Puck also reports that the difference between using the effective
strain and the longitudinal stress is insignificant [6]. The LaRC03 criterion for fibertension failure is a noninteracting maximum allowable strain criterion that is simple to
measure and is independent of fiber volume fraction and Youngs moduli. Consequently,
the LaRC03 failure index for fiber tensile failure is:
LaRC03#3 FIF"11
"T1 1 28
Criterion for Fiber Compression Failure
Compressive failure of aligned fiber composites occurs from the collapse of the fibers as
a result of shear kinking and damage of the supporting matrix (See for instance the
representative 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 shear
deformation, leading to the formation of a kink band.
Argon [30] was the first to analyze the kinking phenomenon. His analysis was based on
the assumption of a local initial fiber misalignment. The fiber misalignment leads to
shearing stresses between fibers that rotate the fibers, increasing the shearing stress and
leading to instability. Since Argons work, the calculation of the critical kinking stress has
been significantly improved with a more complete understanding of the geometry of the
kink band as well as the incorporation of friction and material nonlinearity in the analysismodels [2729, 31].
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Several authors [32,33] have considered that misaligned fibers fail by the formation
of a kink band when local matrix cracking occurs. Potter et al. [34] assumed that
additional failure mechanisms that may occur under uniaxial longitudinal compres-
sion, such as crushing, brooming, and delaminations, are essentially triggered by matrix
failure.
In the present approach, the compressive strength XC is assumed to be a known
material 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 uniaxial
compression.
CALCULATION OF FIBER MISALIGNMENT ANGLE
The imperfection in fiber alignment is idealized as a local region of waviness, as shown
in Figure 9. The ply stresses in the misalignment coordinate framemshown in Figure 9 are
m
11 cos2
11 sin2
22 2sin cos j12jm22 sin2 11 cos2 22 2sin cos j12jm12 sin cos 11 sin cos 22 cos2 sin2 j12j
29
At failure under axial compression, 11 XC, and 22 12 0. Substituting thesevalues into Equation (29) gives m22 sin2 CXCand m12 sin C cos CXC, wherethe angle C is the total misalignment angle for the case of pure axial compression
loading.
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. The
angle of the fracture plane is then equal to 0, and Teff 0. The matrix failure criterionbecomes
Leff XCsin C cos C L sin2 C SLis 30
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 quadratic
equation:
tan2 C SLis
XC L
tan C S
Lis
XC
0 31
11
22
m
m
XC
XC
11
22
Figure 9. Imperfection in fiber alignment idealized as local region of waviness.
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The smaller of the two roots of Equation (31) is
C tan11
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4 SLis =XC L
SLis =X
C
q
2 SL
is
=XC
L
0@
1A 32
Note that if L were neglected and were assumed to be a small constant angle,
Equation (30) would give C SLis =XC, which is the expression derived by Argon [30] toestimate 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 additional
rotational component R that results from shear loading. The angles 0 and R can be
calculated using small angle approximations and Equations (29)
R m
12G12
11 22 j12jG12
0 C R
XC C
m12
G12
XC
) 0 1 XC
G12
C
33
where the absolute value of the shear |12| is used because the sign of is assumed to be
positive, regardless of the sign of the shear stress. Recalling that 0 R, it is nowpossible to solve Equations (33) for in terms ofC.
j12
j G12
XC
C
G12 11 22 34
Fiber compression failure by the formation of a kink band is predicted using the stresses
from Equation (29) and the failure criterion for matrix tension or matrix compression. For
matrix compression m22 1 37
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The linear interaction between11and 12in Equation (37) is identical to the form used by
Edge [7] for the WWFE. For T300/914C, Edge suggests using an empirical value of
k 1.5. However, Edge also indicates that other researchers have shown excellentcorrelation with experimental results with k 1. Using the WWFE strength values ofXC
900 MPa, SL
80 MPa, YC
200 MPa, an assumed fracture angle in transverse
compression of 53, and Equations (11) and (32), we get: L 0.304, C 5.3. With theapproximation ffi C, Equation (37) gives k 1.07.
LaRC03 CRITERION FOR MATRIX DAMAGE IN BIAXIAL COMPRESSION
In the presence of high transverse compression combined with moderate fiber
compression, matrix damage can occur without the formation of kink bands or damage
to the fibers. This matrix damage mode is calculated using the stresses in the misaligned
frame in the failure criterion in Equation (4), which gives:
LaRC03#6 FIM mTeff
ST
2
mLeff
SLis
2 1 38
where the effective shear stressesmTeff andmLeff are defined as in Equation (6), but in terms
of the in-plane stresses in the misalignment frame, which are defined in Equations (29).
mTeff m22cos sin T cos
mLeff cos m12
Lm22cos
39
As for all matrix compressive failures herein, the stresses mTeff and mLeff are functions of
the 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 failure
envelope in the 1122plane. An example is shown in Figure 10 for a 0 E-glass/MY750
epoxy lamina. Pucks WWFE analysis results [6] are shown for comparison. In the figure,
there is good agreement between LaRC03 and Puck in all quadrants except biaxial
compression, where LaRC03 predicts an increase of the axial compressive strength with
increasing transverse compression. Testing for biaxial loads presents a number of
complexities, and experimental results are rare. However, Waas and Schultheisz [35]
report a number of references in which multiaxial compression was studied by superposing
a hydrostatic pressure in addition to the compressive loading. For all materials considered,
there is a significant increase in compressive strength with increasing pressure. In
particular, the results of Wronski and Parry [36] on glass/epoxy show a strength increaseof 3.3 MPa per MPa of hydrostatic pressure, which gives an actual strength increase of
338 C. G. DA VILA ET AL.
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4.3 MPa per MPa of applied transverse biaxial stress. More recently, Sigley et al. [37]
found 3271% increase in compressive strength per 100 MPa superposed pressure. The
results of Wronsky et al. is comparable to the 4.3 MPa/MPa increase in compressive
strength 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 in
the 2212 stress plane obtained from the WWFE [38] is shown in Figure 11. It can be
observed that within the positive range of 22, all the quadratic failure criteria and
LaRC03 give satisfactory results. Since the Maximum Stress Criterion does not prescribe
interactions between stress components, its failure envelope is rectangular. The most
interesting behavior develops when 22 becomes compressive. Hashins 1973 criterion
gives 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 Hashins 1980 criteria was calculated using a transverse strength
obtained from Equation (10), and it provides a modest improvement in accuracy
compared 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 at
the initial stage of compressive 22. The failure envelope for Suns criterion (Equation (1))
was calculated using L 0.336 from Equation (11). The results indicate a significantimprovement over Hashins criteria. An even better fit would have been achieved using a
higher value for L. Pucks 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 ,MLaRC03 #1
FI ,MLaRC03 #6
FI ,M LaRC03 #2
FI ,FLaRC03 #3
FI ,FLaRC03 #4
FI ,F LaRC03 #5
11, MPa
=11
Yc
22
, MPa
=022
m
Figure 10. Biaxial1122 failure envelope of 0 E-glass/MY750 epoxy lamina.
Failure Criteria for FRP Laminates 339
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0 53, and no other empirical or fitting parameter. In the case of matrix tension, Puckspredicted 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; for1550, it is matrix compression. Fiberscissoring due to matrix material nonlinearity caused the switch in failure mode from
in-plane matrix shearing to matrix compression failure at larger lamination angles. The
material properties shown in Table 2 were used for the analysis. The angle 0was assumed
to 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] as
SLis 95:1 MPa.The results in Figure 12 indicate that the predicted strengths using LaRC03 criteria
correlate well with the experimental results. The compressive strength predicted using
Hashin 1973 criteria is also shown in Figure 12. For
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criteria results is in an underprediction of the failure load because the criteria do not
account for the increase in shear strength caused by transverse compression. All of these
effects are represented by the LaRC03 criteria, which results in a good correlation between
the calculated and experimental values.
CONCLUSIONS
The results of the recently concluded World Wide Failure Exercise indicate that the
existing knowledge on failure mechanisms needs further development. Many of the
existing 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 experimental
values were not uncommon. In this paper, a new set of six phenomenological
failure criteria is proposed. The criteria for matrix and fiber compression failure are
based on a Mohr-Coulomb interaction of the stresses associated with the plane of
fracture. Failure envelopes for unidirectional laminates in the 1122 and 2212 stress
planes were calculated, as well as the compressive strength of cross-ply laminates.
The predicted failure envelopes indicate good correlation with experimental results. The
proposed criteria, referred to as LaRC03, represent a significant improvement over
the commonly used Hashin criteria, especially in the cases of matrix or fiber failure incompression.
Co
mpressivestress,
MPa
Lamination angle, degrees
Test, Shuart 89
Hashin 73
LaRC03
0=530=44
0=0
0
Figure 12. Compressive strength as a function of ply orientation for [ ]sAS43502 laminates.
Table 2. Properties of AS4/3502 (MPa) from [39].
E11 E22 G12 12 XC YC SLis
127,600 11,300 6000 0.278 1045 244 95.1
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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:
YTisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8GIcLt022
s and SLis
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8GIIcL
t044
s where 022 2
1
E2
221
E1
and 044
1
G12
Toughness ratiog is obtained from fracture mechanics test data or from the unidirectional
properties:
For a thin embedded ply, g GIcLGIIcL
. Otherwise, g 1:122022
044
YT
SL
2
The effective shear stresses for matrix compression are
Teff 22cos sin T cos
Leff cos j12j L22cos
(
The transverse shear strength is
ST YC cos 0 sin 0 cos 0tan20
Matrix tension, 22 0 Matrix compression, 22
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T 1tan20
and L SLiscos 20
YC cos2 0or from test data.
0 53 or from test data.The stresses in the fiber misalignment frame are
m11 cos2 11 sin2 22 2 sin cos 12m22 sin2 11 cos2 22 2 sin cos 12m12 sin cos 11 sin cos 22 cos2 sin2 12
where
j12
j G12
XC
C
G12 11 22 and C
tan1
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
4 SLis =X
C
L SLis =XC q
2 SLis =XC L 0@ 1A
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