failure criteria.pdf

8/12/2019 failure criteria.pdf

1/23

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


2/23

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


3/23

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.

Failure Criteria for FRP Laminates 325


4/23

Criterion for Matrix Failure Under Transverse Compression (22


5/23

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


6/23

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 <


7/23

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.



8/23

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]).

330 C. G. DA VILA ET AL.


9/23

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



10/23

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



11/23

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


12/23

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.



13/23

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].



14/23

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.



15/23

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



16/23

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



17/23

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.



18/23

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


19/23

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



20/23

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


21/23

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

REFERENCES

1. Soden, P.D., Hinton, M.J. and Kaddour, A.S. (1998). A Comparison of the PredictiveCapabilities of Current Failure Theories for Composite Laminates, Composites Science andTechnology, 58: 12251254.

2. Soden, P.D., Hinton, M.J. and Kaddour, A.S. (2002). Biaxial Test Results for Strength andDeformation of a Range of E-glass and Carbon Fibre Reinforced Composite Laminates: Failure

Exercise Benchmark Data, Composites Science and Technology, 62: 14891514.3. Hinton, M.J., Kaddour, A.S. and Soden, P.D. (2002). A Comparison of the Predictive

Capabilities of Current Failure Theories for Composite Laminates, Judged against ExperimentalEvidence, Composites Science and Technology, 62: 17251797.

4. Hinton, M.J. and Soden, P.D. (1998). Predicting Failure in Composite Laminates: TheBackground to the Exercise, Composites Science and Technology, 58: 10011010.

5. Puck, A. and Schu rmann, H. (1998). Failure Analysis of FRP Laminates by Means of PhysicallyBased Phenomenological Models, Composites Science and Technology, 58: 10451067.

6. Puck, A. and Schu rmann, H. (2002). Failure Analysis of FRP Laminates by Means of PhysicallyBased Phenomenological Models, Composites Science and Technology, 62: 16331662.

7. Edge, E.C. (1998). Stress-based Grant-sanders Method for Predicting Failure of CompositeLaminates, Composites Science and Technology, 58: 10331041.

8. Hashin, Z. and Rotem, A. (1973). A Fatigue Criterion for Fiber-reinforced Materials,Journal ofComposite Materials, 7: 448464.

9. Hashin, Z. (1980). Failure Criteria for Unidirectional Fiber Composites, Journal of AppliedMechanics, 47: 329334.

10. Sun, C.T., Quinn, B.J. and Oplinger, D.W. (1996). Comparative Evaluation of Failure AnalysisMethods for Composite Laminates DOT/FAA/AR-95/109.

11. Salencon, J. (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity,Springer, New York.

12. Kawabata, S. (1982). Strength of Epoxy Resin under Multiaxial Stress Field, ICCM-IVConference, Tokyo, Japan.

13. Boehler, J.P. and Raclin, J. (1985). Failure Criteria for Glass-fiber Reinforced Composites

Under Confining Pressure, J. Struct. Mech., 13: 371392.



22/23

14. Taliercio, A. and Sagramoso, P. (1995). Uniaxial Strength of Polymeric-matrix FibrousComposites Predicted through a Homogenization Approach, International Journal of Solids andStructures, 32: 20952123.

15. Parvizi, A., Garrett, K. and Bailey, J. (1978). Constrained Cracking in Glass Fibre-reinforcedEpoxy Cross-ply Laminates,Journal of Material Science, 13: 195201.

16. Wang, A.S.D. (1984). Fracture Mechanics of Sublaminate Cracks in Composite Materials,Composites Technology Review, 6: 4562.

17. Chang, F.-K. and Chen, M.-H. (1987). The In Situ Ply Shear Strength Distributions inGraphite/Epoxy Laminated Composites, Journal of Composite Materials, 21: 708733.

18. Flaggs, D.L. and Kural, M.H. (1982). Experimental Determination of the In SituTransverse Lamina Strength in Graphite/Epoxy Laminates, Journal of Composite Materials,16: 103116.

19. Dvorak, G.J. and Laws, N. (1987). Analysis of Progressive Matrix Cracking in CompositeLaminates: II First Ply Failure, Journal of Composite Materials, 21: 309329.

20. Laws, N. and Dvorak, G.J. (1988). Progressive Transverse Cracking in Composite Laminates,Journal of Composite Materials, 22: 900919.

21. Tan, S.C. and Nuismer, R.J. (1989). A Theory for Progressive Matrix Cracking in CompositeLaminates, Journal of Composite Materials, 23: 10291047.

22. Hahn, H.T. and Johannesson, T. (1983). In: Dvorak, G.J. (ed.), Mechanics of CompositeMaterials, pp. 135142, AMD.

23. Liu, K.-S. and Tsai, S.W. (1998). A Progressive Quadratic Failure Criterion for a Laminate,Composites Science and Technology, 58: 10231032.

24. Wu, E.M. and Reuter, R.C.J. (1965). Crack Extension in Fiberglass Reinforced Plasticsand a Critical Examination of the General Fracture Criterion, Theor. and Appl. Mech.Report No. 275.

25. Leguillon, D. (2002). Strength or Toughness? A Criterion for Crack Onset at a Notch,EuropeanJournal of Mechanics A/Solids, 21: 6172.

26. Tada, H., Paris, P.C. and Irwin, G.R. (2000).The Stress Analysis of Cracks Handbook,3rd edn,

American Society of Mechanical Engineers, New York.27. Fleck, N.A. and Liu, D. (2001). Microbuckle Initiation from a Patch of Large Amplitude

Fibre Waviness in a Composite under Compression and Bending, European Journal ofMechanics A/Solids, 20: 2337.

28. Soutis, C., Smith, F.C. and Matthews, F.L. (2000). Predicting the Compressive EngineeringPerformance of Carbon Fibre-reinforced Plastics, Composites Part A: Applied Science andManufacturing, 31: 531536.

29. Schultheisz, C.R. and Waas, A.M. (1996). Compressive Failure of Composites, Part 1: Testingand Micromechanical Theories, Progress in Aerospace Sciences, 32: 142.

30. Argon, A.S. (1972). Fracture of Composites, Vol. 1. Treatise of Materials Science andTechnology Academic Press, New York.

31. Budiansky, B., Fleck, N.A. and Amazigo, J.C. (1998). On Kink-band Propagation in FiberComposites, Journal of the Mechanics and Physics of Solids, 46: 16371653.

32. Berg, C.A. and Salama, M. (1973). Fatigue of Graphite Fibre-reinforced Epoxy in Compression,Fibre Science and Technology, 6: 79118.

33. Schapery, R.A. (1995). Prediction of Compressive Strength and Kink Bands in CompositesUsing a Work Potential, International Journal of Solids and Structures, 32: 739765.

34. Potter, D.S., Gupta, V. and Hauert, S. (2000). Effects of Specimen Size and Sample AspectRatio on the Compressive Strength of Graphite/Epoxy Laminates, Composites Science andTechnology, 60: 25252538.

35. Waas, A.M. and Schultheisz, C.R. (1996). Compressive Failure of Composites, Part II:Experimental Studies, Progress in Aerospace Sciences, 32: 4378.

36. Wronski, A.S. and Parry, T.V. (1982). Compressive Failure and Kinking in Uniaxially Aligned

Glass-Resin Composite under Superposed Hydrostatic Pressure, Journal of Material Science,17:36563662.



23/23

failure criteria.pdf

Documents