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2Analysis of Composite Materials

2.1 Constitutive Relations

Laminated composites are typically constructed from orthotropic plies(laminae) containing unidirectional fibers or woven fabric. Generally, ina macroscopic sense, the lamina is assumed to behave as a homogeneousorthotropic material. The constitutive relation for a linear elastic ortho-tropic material in the material coordinate system (Figure 2.1) is (Danieland Ishai 2006, Gibson 2007, Herakovich 1998, Hyer 2009, Jones 1999,Reddy 1997)

=

S S S

S S S

S S S

S

S

S

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1

2

3

23

13

12

11 12 13

12 22 23

13 23 33

44

55

66

1

2

3

23

13

12

(2.1)

where the stress components i, ijare defined in Figure 2.1, and the Sijareelements of the compliance matrix. The engineering strain components i, ijare defined as implied in Figure 2.2.

In a thin lamina, a state of plane stress is commonly assumed by setting

= = = 03 23 13 (2.2)

For Equation (2.1), this assumption leads to

= + S S3 13 1 23 2 (2.3a)

23= 13= 0 (2.3b)

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12 Experimental Characterization of Advanced Composite Materials

Thus, for plane stress the through-the-thickness strain 3is not an indepen-dent quantity and does not need to be included in the constitutive relation-

ship. Equation (2.1) becomes

0

0

0 0

1

2

12

S S

S S

S

1

2

12

11 12

12 22

66

=

(2.4)

The compliance elements Sijmay be related to the engineering constants(E1, E2, G12,12,21),

1/ / /11 1 12 12 1 21 2S E S E E= = = (2.5a)

1/ , 1/22 2 66 12S E S G= = (2.5b)

The engineering constants are average properties of the composite ply. Thequantities E1and12are the elastic stiffness and Poissons ratio, respectively,corresponding to stress 1(Figure 2.2a):

/1 1 1E = (2.6a)

/12 2 1 = (2.6b)

E2and21 correspond to stress 2(Figure 2.2b):

/2 2 2E = (2.7a)

/21 1 2 = (2.7b)

2

1

3

3

1

212

31

23

31

12

23

FIGURE 2.1Definitions of principal material directions for an orthotropic lamina and stress components.

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It is often convenient to express stresses as functions of strains. This isaccomplished by inversion of Equation (2.4):

=

Q Q

Q Q

Q

1

2

12

11 12

12 22

66

1

2

12

0

0

0 0

(2.10)

where the reduced stiffnesses Qijcan be expressed in terms of the engineer-ing constants:

/(1 )11 1 12 21Q E= (2.11a)

/(1 ) /(1 )12 12 1 12 21 21 1 12 21Q E E= = (2.11b)

/(1 )22 2 12 21Q E= (2.11c)

66 12Q G= (2.11d)

2.1.1 Transformation of Stresses and Strains

For a lamina whose principal material axes (1,2) are oriented at an angle with respect to the x,ycoordinate system (Figure 2.3), the stresses and strainscan be transformed. It may be shown (see, e.g., Hyer 2009) that both thestresses and strains transform according to

[ ]

=

T

1

2

12

x

y

xy

(2.12)

z, 3

2

y

1x

FIGURE 2.3Positive (counterclockwise) rotation of principal material axes (1,2) from arbitrary x,y-axes.

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and

/2 /2

1

2

12

T

x

y

xy

[ ]

=

(2.13)

where the transformation matrix is (Hyer 2009)

[ ]

T =

m n mn

n m mn

mn mn m n

2

2

2 2

2 2

2 2

(2.14)

and

m =cos, n =sin (2.15a,b)

From Equations (2.12) and (2.13), it is possible to establish the laminastrainstress relations in the (x,y) coordinate system:

=

S S S

S S S

S S S

x

y

xy

y

xy

11 12 16

12 22 26

16 26 66

x

(2.16)

The Sij terms are the transformed compliances defined in Appendix A.Similarly, the lamina stressstrain relations become

11 12 16

12 22 26

16 26 66

Q Q Q

Q Q Q

Q Q Q

x

y

xy

x

y

xy

=

(2.17)

where the overbars denote transformed reduced stiffness elements, definedin Appendix A.

2.1.2 Hygrothermal Strains

If fibrous composite materials are processed at elevated temperatures, ther-mal strains are introduced during cooling to room temperature, leading toresidual stresses and dimensional changes. Figure 2.4 illustrates dimen-sional changes of a composite subjected to a temperature increase of T fromthe reference temperature T. Furthermore, polymer matrices are commonlyhygroscopic, and moisture absorption leads to swelling of the material. Theanalysis of moisture expansion strains in composites is mathematicallyequivalent to that for thermal strains.

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The constitutive relationship, when it includes mechanical-, thermal-, andmoisture-induced strains, takes the following form (Hyer 2009):

=

+

+

S S

S S

S

1

2

12

11 12

12 22

66

1

2

12

1T

2T

1M

2M

0

0

0 0 0 0

(2.18)

where superscripts T and M denote temperature- and moisture-inducedstrains, respectively. Note that shear strains are not induced in the principalmaterial system by a temperature or moisture content change (Figure 2.4).Equation (2.18) is based on the superposition of mechanical-, thermal-, andmoisture-induced strains. Inversion of Equation (2.18) gives

=

Q Q

Q Q

Q

T M

T M

0

0

0 0

1

2

12

11 12

11 12

66

1 1 1

2 2 2

12

(2.19)

Consequently, the stress-generating strains are obtained by subtraction of thethermal- and moisture-induced strains from the total strains. The thermal-and moisture-induced strains are often approximated as linear functions ofthe changes in temperature and moisture concentration,

=

T1T

2T

1

2 (2.20)

=

M1M

2M

1

2

(2.21)

3

2

T

T + T

T + T

T2

1

1

3D View Top View

FIGURE 2.4

Deformation of a lamina subject to temperature increase.

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where T and M are the temperature change and moisture concentrationchange, respectively, from the reference state.

The transformed thermal expansion coefficients x y xy, , are obtainedfrom those in the principal system using Equation (2.13). Note, however, thatin the principal material coordinate system, there is no shear deformationinduced, that is, = = 016 16 ,

= + m nx

21

22 (2.22a)

= + n my

21

22 (2.22b)

= mnxy 2 ( )1 2 (2.22c)

The moisture expansion coefficients x, y, xy are obtained by replacingwith in Equations (2.22).

The transformed constitutive relations for a lamina, when incorporatingthermal- and moisture-induced strains, are

+

+

=

S S S

S S SS S S

x

y

xy

x

y

xy

xT

y

T

xyT

xM

y

M

xyM

11 12 16

12 22 26

16 26 66

(2.23)

=

Q Q Q

Q Q Q

Q Q Q

x

y

xy

11 12 16

12 22 26

16 26 66

x xT

xM

y yT

yM

xy xyT

xyM

(2.24)

2.2 Micromechanics

As schematically illustrated in Figure 2.5, micromechanics aims to describethe moduli and expansion coefficients of the lamina from properties of thefiber and matrix, the microstructure of the composite, and the volume frac-tions of the constituents. Sometimes, it is also necessary to consider the smalltransition region between bulk fiber and bulk matrix (i.e., interphase) (Drzalet al. 2000). Much fundamental work has been devoted to the study of thedistributions of strain and stress in the constituents and the formulationof appropriate averaging schemes to allow definition of macroscopic engi-neering constants. Most micromechanics analyses have focused on unidi-rectional continuous-fiber composites (e.g., Hashin 1983, Christensen 1979,

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Hyer and Waas 2000), although properties of composites with woven fabricreinforcements can also be predicted with reasonable accuracy (Byun andChou 2000).

The objective of this section is not to review the numerous microme-chanics developments. The interested reader can find ample informa-tion in the previously referenced review articles. In this section, we limitthe presentation to some commonly used simple estimates of the stiff-

ness constants E1, E2 ,12 ,21, and G12and thermal expansion coefficients1and 2required for describing the small strain response of a unidirec-tional lamina under mechanical and thermal loads (see Section 2.1). Suchestimates may be useful for comparison to experimentally measuredquantities.

2.2.1 Stiffness Properties of Unidirectional Composites

Although most matrices are isotropic, many fibers, such as carbon andKevlar (E.I. du Pont de Nemours and Company, Wilmington, DE) have direc-

tional properties because of molecular or crystal plane orientation effects(Daniel and Ishai 2006). As a result, the axial stiffness of such fibers is muchgreater than the transverse stiffness. The thermal expansion coefficientsalong and transverse to the fiber axis also are quite different. It is commonto assume cylindrical orthotropy for fibers with axisymmetric microstruc-ture. The stiffness constants of the fibers required for plane stress analysisof a composite are EL, ET,LT, and GLT, where Land Tdenote the longitudinaland transverse directions of a fiber, respectively. The corresponding thermalexpansion coefficients are Land T , respectively.

The mechanics of materials approach (Hyer 2009) yields

= +E E V E V Lf f m m1 (2.25a)

=

+

E E E

E V E V

Tf m

Tf m m f 2

(2.25b)

Matrix properties

Fiber properties

Micromechanics

Lamina Properties

FIGURE 2.5

Role of micromechanics.

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= + V VLTf f m m12 (2.25c)

=

+

G G GG V G V

LTf m

LTf m m f 12

(2.25d)

where subscripts fand mrepresent fiber and matrix, respectively, and thesymbol Vrepresents volume fraction. Note that once E1, E2 , and12are cal-culated from Equation (2.25a),21 is obtained from Equation (2.8). Equations(2.25a) and (2.25c) provide good estimates of E1 and 12. Equations (2.25b)and (2.25d), however, substantially underestimate E2 and G12 . More realisticestimates of E2 and G12are provided in the work of Hyer (2009) and Rosen

and Hashin (1987).Simple, yet reasonable, estimates of E2 and G12 may also be obtained from

the Halpin-Tsai equations (Halpin and Kardos 1976):

=

P

P (1 + V )

1 V

m f

f

(2.26a)

=

P P

P + P

f m

f m

(2.26b)

where Pis the property of interest (E2or G12), and Pf and Pmare the corre-sponding fiber and matrix properties, respectively. The parameter is calledthe reinforcement efficiency; =E( ) 22 and =G( ) 112 , for circular fibers.

2.2.2 Expansion Coefficients

Thermal expansion (and moisture-swelling) coefficients can be defined byconsidering a composite subjected to a uniform increase in temperature (ormoisture content) (Figure 2.4).

The thermal expansion coefficients 1and 2 of a unidirectional compos-ite consisting of transversely isotropic fibers in an isotropic matrix may bedetermined using the mechanics of materials approach (Hyer 2009):

= +

+

E V E V

E V E V

Lf Lf f m m m

Lf f m m1 (2.27a)

= + V VTf f m m2 (2.27b)

Predictions of 1using Equation (2.27a) are accurate, whereas Equation (2.27b)underestimates the actual value of2. An expression derived by Hyer and Waas(2000) provides a more accurate prediction of 2:

= + +

+

V V E E

E V E V V VTf f m m

Lf m m LTf

Lf f m mm Lf f m

( )( )2 (2.28)

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2.3 Laminated Plate TheoryStructures fabricated from composite materials rarely utilize a single com-posite lamina because this unit is typically very thin and highly ortho-tropic. To achieve a thicker cross section and more balanced properties,plies of prepreg or fiber mats are stacked in specified directions. Such astructure is called a laminate (Figure 2.6). Most analyses of laminated struc-tures are limited to flat panels (see, e.g., Daniel and Ishai 2006 and Hyer2009). Extension to curved laminated shell structures may be found in thework of Herakovich (1998), Hyer (2000), and Vinson and Sierakowsky (2002).

In this section, attention is limited to a flat laminated plate under in-planeand bending loads. The classical theory of such plates is based on the assump-tion that a line originally straight and perpendicular to the middle surfaceremains straight and normal to the middle surface, and that the length of theline remains unchanged during deformation of the plate. These assumptionslead to the vanishing of the out-of-plane shear and extensional strains:

= = =xz yz z 0 (2.29)

where the laminate coordinate system (x, y, z) is indicated in Figure 2.6.Consequently, the laminate strains are reduced to x, y, and xy. The assump-tion that the cross sections undergo only stretching and rotation leads to thefollowing strain distribution (Hyer 2009):

= + z

x

y

xy

x0

y0

xy0

x

y

xy

(2.30)

z

y

x

FIGURE 2.6Laminate coordinate system.

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where [ , , ] and [ , ]0 0 0

x ,x y xy y xy are the midplane strains and curvatures,respectively, andzis the distance from the midplane.

Force and moment resultants, [ , , ]N N Nx y xy and [ , , ]M M Mx y xy , respec-tively, are obtained by integration of the stresses in each layer over the lami-nate thickness h:

=

N

N

N

dz

x

y

xyk

x

y

xyh

h

2

2

(2.31)

=

M

M

M

zdz

x

y

xy h/ 2

h/ 2

k

x

y

xy

(2.32)

where the subscript krepresents the kthply in the laminate. Combinationof Equations (2.24) with Equations (2.302.32) leads to the following con-stitutive relationships among forces and moments and midplane strainsand curvatures:

N N N

N N N

N N N

A A A

A A A

A A A

B B B

B B B

B B B

x xT

xM

y yT

yM

xy xyT

xyM

11 12 16

12 22 26

16 26 66

x0

y0

xy0

11 12 16

12 22 26

16 26 66

x

y

xy

+ +

+ +

+ +

=

+

(2.33)

M M M

M M M

M M M

B B B

B B B

B B B

D D D

D D D

D D D

x xT

xM

y yT

yM

xy xyT

xyM

11 12 16

12 22 26

16 26 66

x0

y0

xy0

11 12 16

12 22 26

16 26 66

x

y

xy

+ +

+ +

+ +

=

+

(2.34)

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where the Aij, Bij, and Dij are called extensional stiffnesses, coupling stiff-

nesses, and bending stiffnesses, (Hyer 2009), given by

( ) ( )11

A Q z zij ij k k k k

N

= =

(2.35a)

1

2( ) 2 1

2

1

B Q z zij ij k k k k

N

( )= =

(2.35b)

( ) 3 13

1

D =13

Q z zij ij k k k k

N

( ) =

(2.35c)

The so-called ply coordinateszk(k=1, 2N), where Nis the number of pliesin the laminate, indicate the location of the ply boundaries (see Figure 2.7).zkmay be calculated from the following recursion formula:

/2 00z h k= = (2.36a)

1, 2, ,1z z h k Nk k k= + = (2.36b)

in which hkis the ply thickness of the kthply. Normally, all plies are of the

same thickness.For situations involving a temperature change T, it is common to consider

only the steady-state condition at which the temperature is uniform throughoutthe laminate. However, in a transient situation, the transfer of heat by conduc-tion (Ozisik 1980) or moisture diffusion (Jost 1960) has to be considered (see Pipeset al. 1976). For steady state, the thermal force resultants are determined from

1

1

N

N

N

=

Q Q Q

Q Q Q

Q Q Q

(z z ) T

xT

yT

xyT

k

11 12 16

12 22 26

16 26 66 k

x

y

xy

k k

k

N

=

(2.37)

z

zNh2

h

N

2

1z0

z1

FIGURE 2.7Definition of ply coordinateszk.

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The moisture-induced force resultants [ ]N , N ,NxM

yM

xyM are obtained in the

same manner as the thermal force resultants, by replacing [ , , ]x y xy with

[ , , ]x y xy and Twith M in Equation (2.37).

The thermal moment resultants [ ]M ,M ,MxT

yT

xyT are determined from

1 21

2

1

M

M

M

=2

Q Q Q

Q Q Q

Q Q Q

z z T

xT

yT

xyT

k

11 12 16

12 22 26

16 26 66 k

x

y

xy

k k

k

N

( )

=

(2.38)

The moisture-induced moment resultants [ , , ]M M MxM

yM

xyM

are obtained byreplacing the values with values and Twith Min Equation (2.38).

For laminates with the plies consisting of different materials, the mois-ture concentration may vary through the thickness in a stepwise man-ner. At steady state, this is simply incorporated into the analysis by letting = M M k( ) (Carlsson 1981).

Equations (2.33) and (2.34) may conveniently be written as

=

N

M

A B

B D

0

(2.39)

where [N] and [M] represent the left-hand side of Equations (2.33) and (2.34),that is, the sum of mechanical and hygrothermal forces and moments,respectively. Equation (2.39) represents the stiffness form of the laminateconstitutive equations.

Sometimes, it is more convenient to express the midplane strains and cur-vatures as a function of the forces and moments. This represents the com-pliance form of the laminate constitutive equations, which is obtained by

inversion of Equation (2.39):

=

a b

c d

N

M

0

(2.40)

Expressions for the compliance matrices [a], [b], [c], and [d] are given inAppendix A.

2.4 St. Venants Principle and End Effects in Composites

In the testing and evaluation of material properties, it is generally assumedthat load introduction effects are confined to a region close to the grips orloading points, and a uniform state of stress and strain exists within the

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test section. The justification for such a simplification is usually based on theSt. Venant principle, which states that the difference between the stresses

caused by statically equivalent load systems is insignificant at distancesgreater than the largest dimension of the area over which the loads are acting(Timoshenko and Goodier 1970). This estimate, however, is based on isotro-pic material properties. For orthotropic composite materials, Horgan (1972,1982), Choi and Horgan (1977), and Horgan and Carlsson (2000) showed thatthe application of St. Venants principle for plane elasticity problems involv-ing orthotropic materials is not justified in general. For the particular prob-lem of a rectangular strip made of highly orthotropic material and loaded atthe ends, Choi and Horgan (1977) demonstrated that the stress approached

the uniform St. Venant solution much more slowly than the correspondingsolution for an isotropic material.

The size of the region where end effects influence the stresses in a rectan-gular strip loaded with tractions at the ends may be estimated by

( / )1 12

1 2b

2E G

(2.41)

where bis the maximum dimension of the cross section, and E1and G12are

the longitudinal elastic and shear stiffnesses, respectively.In this equation,is defined as the distance over which the self-equilibrated

stress decays to 1/eof its value at the end ( )=e 2.718 . When the ratio E1/G12is large, the decay length is large, and end effects are transferred a consider-able distance along the gage section. Testing of highly orthotropic materi-als thus requires special consideration of load introduction effects. Arridgeet al. (1976), for example, found that a very long specimen with an aspectratio ranging from 80 to 100 was needed to avoid the influence of clamp-ing effects in tension testing of highly orthotropic, drawn polyethylene film.

Several other cases are reviewed by Horgan and Carlsson (2000).

2.5 Lamina Strength Analysis

When any material is considered for a structure, an important task for thestructural engineer is to assess the load-carrying ability of the particularmaterial/structure combination. Prediction of the strength of compositematerials has been an active area of research since the early work of Tsai(1968). Many failure theories have been suggested, although no universallyaccepted failure criterion exists (Hinton et al. 2004). As pointed out by Hyer(2009), however, no single criterion could be expected to accurately predictfailure of all composite materials under all loading conditions. Popularstrength criteria are maximum stress, maximum strain, and Tsai-Wu criteria

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(Sun 2000). These criteria are phenomenological in the sense that they do not

rely on physical modeling of the failure process. The reason for their popu-larity is that they are based on failure tests on simple specimens in tension,compression, and shear (Chapters 810) and are able to predict load levelsrequired to fail more complicated structures under combined stress loading.

In the following presentation, failure of the lamina is first examined, andthen failure of the laminate is briefly considered. It is assumed that the lam-ina, being unidirectional or a woven fabric ply, can be treated as a homo-geneous orthotropic ply with known, measured strengths in the principalmaterial directions. Furthermore, the shear strength in the plane of the fibers

is independent of the sign of the shear stress. The presentation is limited toplane stress in the plane of the fibers. Table 2.1 lists the five independent fail-ure stresses and strains corresponding to plane stress.

Notice here that superscripts T and C denote tension and compression,respectively, and that strengths and ultimate strains are defined as positive;that is, the symbols indicate their magnitudes. For example, a composite plyloaded in pure negative shear ( 0)12 < would fail at a shear stress = S12 6

and shear strain = e12 6 .

2.5.1 Maximum Stress Failure Criterion

The maximum stress failure criterion assumes that failure occurs when anyone of the in-plane stresses 1, 2, or 12 attains its limiting value independentof the other components of stress. If the magnitudes of the stress componentsare less than their values at failure, failure does not occur, and the elementor structure is considered safe. For determining the failure load, any of thefollowing equalities must be satisfied at the point when failure occurs:

=XT

1 1 (2.42a)

= XC

1 1 (2.42b)

=XT

2 2 (2.42c)

= XC

2 2 (2.42d)

TABLE 2.1

Basic Strengths of Orthotropic Plies for Plane Stress

Direction/Plane Active Stress Strength Ultimate Strain

1 1 XT1, X

C1 e

T1, e

C1

2 2 XT2, X

C2 e

T2, e

C2

1,2 12 S6 e6

Note: All strengths and ultimate strains are defined by theirmagnitudes.

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=S12 6 (2.42e)

= S12 6 (2.42f)

For unidirectional and fabric composites, Equations (2.42a) and (2.42b) indi-cate failure of fibers at quite high magnitudes of stress, whereas Equations(2.42c)(2.42f) indicate matrix or fibermatrix interface-dominated failuresat much lower magnitudes of stress for unidirectional composites. For fab-ric composites, however, Equations (2.42c) and (2.42d) indicate failure of thefibers oriented along the 2-direction.

2.5.2 Maximum Strain Failure Criterion

The maximum strain criterion assumes that failure of any principal planeof the lamina occurs when any in-plane strain reaches its ultimate value inuniaxial tension, compression, or pure shear.

=eT

1 1 (2.43a)

=

e

C

1 1 (2.43b)

=eT

2 2 (2.43c)

= eC

2 2 (2.43d)

=e12 6 (2.43e)

= e12 6 (2.43f)

In these expressions, the symbol e represents the magnitude of the ultimatestrain. If any of these conditions becomes satisfied, failure is assumed to betriggered by the same mechanism as uniaxial loading or pure shear loading.Similar to the maximum stress criterion, the maximum strain criterion hasthe ability of predicting the failure mode.

2.5.3 Tsai-Wu Failure Criterion

Tsai and Wu (1971) proposed a second-order tensor polynomial failure cri-terion for prediction of biaxial strength, which takes the following form forplane stress:

+ + + + + =F F F F F F2 11 1 2 2 11 12

22 22

66 122

12 1 2 (2.44)

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Failure under combined stress is assumed to occur when the left-hand sideof Equation (2.44) is equal to or greater than one. All of the parameters of the

Tsai-Wu criterion, except F12, can be expressed in terms of the basic strengths(Table 1.1),

( )

( )

= =

= =

=

F X X F X X

F X X F X X

F S

T C T C

T C T C

1 1 1

1 1 1

1

1 1 1 11 1 1

2 2 2 22 2 2

66 62

(2.45)

F12 is a strength interaction parameter that has to be determined froma biaxial experiment involving stresses 1 and 2. Such experiments are,unfortunately, expensive and difficult to properly conduct. As an alterna-tive, Tsai and Hahn (1980) suggested that F12be estimated from the follow-ing relationship:

= F F F

1

212 11 22 (2.46)

The Tsai-Wu criterion has found widespread applicability in the com-posite industry because of its versatility and that it provides reasonablepredictions of strength. It does not, however, predict the mode of failure.Furthermore, it must be pointed out that the World-Wide Failure Exercisefound that none of the current failure models provides accurate predictions(Hinton et al. 2004).

2.6 Laminate Strength Analysis

Analysis of failure and strength of laminated composites is quite differ-ent from the analysis of strength of a single ply. Failure of laminates ofteninvolves delamination (i.e., separation of the plies), which is discussed inChapters 14 and 17. This failure mode is influenced by the three-dimensionalstate of stress that develops near free edges in laminated specimens (Pipeset al. 1973). Furthermore, multidirectional composite laminates are com-

monly processed at elevated temperatures, and the mismatch in thermalexpansion between the plies leads to residual stresses in the plies on cool-ing (Hahn and Pagano 1975, Weitsman 1979, Jeronimidis and Parkyn 1988).Exposure of the laminate to moisture will also influence the state of residualstress in the laminate (Springer 1981).

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A common failure mode in laminates containing unidirectional plies ismatrix cracking, which is failure of the matrix and fibermatrix interface in a

plane perpendicular to the fiber direction (Figure 2.8). Such a failure is calledfirst-ply failure and occurs because of the presence of a weak plane transverseto the fiber axis in such composites. In fabric composites, no such weak planesexist perpendicular to the xy plane, and failure initiates locally in fiber towsand matrix pockets before ultimate failure occurs (Alif and Carlsson 1997).At any instant, local failures tend to arrest by constraint of adjacent layers ortows in the laminate before the occurrence of catastrophic failure of the lami-nate. Wang and Crossman (1980), Crossman et al. (1980, 1982), and Flaggs andKural (1982) found a very large constraint effect in composite laminates with

unidirectional plies. They examined matrix cracking in a set of laminatescontaining unidirectional 90 plies grouped together and found that the insitu strength depends strongly on the number of plies of the same orientationgrouped together and on the adjacent ply orientations. Consequently, thereare a host of mechanisms influencing failure of laminates, and as a result,accurate failure prediction is associated with severe difficulties.

Various methods to predict ply failures and ultimate failure of compositelaminates are reviewed by Sun (2000). A common method to analyze lam-inate failure is to determine the stresses and strains in plies in the lami-nate using laminated plate theory (Section 2.3) and then examine the loadsand strains corresponding to the occurrence of first-ply failure as predicted

by the failure criterion selected. The ply failure mode is then identified.Swanson and Trask (1989), and Swanson and Qian (1992), performed biaxialtensiontension and tensioncompression testing on several carbon/epoxylaminate cylinders made from unidirectional plies. Ply failures were identi-fied using the strength criteria mentioned in Section 2.5. Final failure of the

z

0

45

45

45

x

45

0

90

90

FIGURE 2.8Matrix crack in a unidirectional ply in a laminate (first-ply failure).

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cylinders was predicted by using a ply property reduction method (ply-by-ply discount method) by which failed plies are identified, and the transverse

and shear stiffnesses E2and G12, respectively, of the failed plies are assignednumbers very close to zero. The laminate with reduced stiffness is thenagain analyzed for stresses and strains. Comparison of the predictions withmeasured ultimate failure data of the cylinders revealed good agreement forall criteria. It was concluded that ultimate failure predictions based on themaximum stress and maximum strain criteria are quite insensitive to varia-tions in the ply transverse failure strengths XT2 and S6 . This is an advantage

because, as discussed, these strengths are difficult to determine in situ.

2.7 Fracture Mechanics Concepts

The influence of defects and cracks on the strength of a material or struc-ture is the subject of fracture mechanics. The object of fracture mechanicsanalysis is the prediction of the onset of crack growth for a body containinga flaw of a given size. To calculate the critical load for a cracked composite,

a common assumption is that the size of the plastic zone at the crack tip issmall compared to the crack length. Linear elastic fracture mechanics hasbeen found useful for certain types of cracks in composites (i.e., interlaminarcracks) (Wilkins et al. 1982) or matrix cracks in unidirectional brittle matrixcomposites (Wang and Crossman 1980).

The equilibrium of an existing crack may be judged from the intensityof elastic stress around the crack tip. Solutions of the elastic stress field inhomogeneous isotropic and orthotropic materials show that stress singu-larities are of the r 1/2 type, where ris the distance from the crack tip (see

Anderson 2005). Stress intensity factors may be determined for crack prob-lems in which the crack plane is in any of the planes of orthotropic materialsymmetry (Sih et al. 1965). It is possible to partition the crack tip loadinginto the three basic modes of crack surface displacement shown in Figure 2.9.Mode I refers to opening of the crack surfaces, mode II refers to sliding, andmode III refers to tearing.

It has, however, become common practice to investigate interlaminarcracks using the strain energy release rate G. This quantity is based onenergy considerations and is mathematically well defined and measur-able in experiments. The energy approach, which stems from the originalGriffith (1920) treatment is based on a thermodynamic criterion for frac-ture by considering the energy available for crack growth of the system onone hand and the surface energy required to extend an existing crack onthe other hand. An elastic potential for a cracked body may be defined as

H =W U (2.47)

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where Wis the work supplied by the movement of the external forces, andUis the elastic strain energy stored in the body. If Gcis the work required tocreate a unit crack area, it is possible to formulate a criterion for crack growth:

H G Ac (2.48)

where A is the increase in crack area.A critical condition occurs when the net energy supplied just balances the

energy required to grow the crack; that is,

H = G Ac (2.49)

Equilibrium becomes unstable when the net energy supplied exceeds therequired crack growth energy,

H > G Ac (2.50)

The strain energy release rate Gis defined as

G =

H

A (2.51)

In terms of G, the fracture criterion may thus be formulated as

G Gc (2.52)

This concept is illustrated for a linear elastic body containing a crackof original length a. Figure 2.10 shows the load P versus loading point

y

x

z

(a) (b) (c)

FIGURE 2.9Modes of crack surface displacements. (a) mode I (opening), (b) mode II (sliding), and (c) mode III(tearing).

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displacement ufor the cracked body where crack growth is assumed to occureither at constant load (fixed load) or at constant displacement (fixed grip).

For thefixed-loadcase,

=

U

P u

2 (2.53a)

W =Pu (2.53b)

Equation (2.47) gives

H =Pu Pu/2 =Pu/2 (2.54)

and Equation (2.51) gives

G =

P

2

u

A (2.55)

For thefixed-gripcase, the work term in Equation (2.47) vanishes, and

U =

u P

2 (2.56)

Note that P is negative because of the loss in stiffness followed by crackextension, and Gis

G =

u

2

P

A (2.57)

For a linear elastic body, the relationship between load and displacementmay be expressed as

u =CP (2.58)

Loa

d,

P

P

P P a a + a

u u + uDisplacement, u

FIGURE 2.10Load-displacement behavior for a cracked body at crack lengths aand a+a.

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expressions form the basis for the virtual crack closure (VCC) method forseparation of the fracture modes using finite element solutions of crack prob-

lems (Rybicki and Kanninen 1977, Krueger 2004).

2.8 Strength of Composite Laminates Containing Holes

Structures made from composite laminates containing cutouts or penetra-tions such as fastener holes (notches) offer a special challenge to the designer

because of the stress concentration associated with the geometric disconti-nuity. The strength can be substantially reduced compared to the strength ofthe unnotched specimen. In laminates containing notches, a complex fiber-

bridging zone develops near the notch tip (Mandell et al. 1975, Aronsson1984). On the microscopic level, the damage appears in the form of fiber pull-out, matrix microcracking, and fibermatrix interfacial failure. The type ofdamage and its growth depend strongly on the laminate stacking sequence,type of resin, and the fiber. As a consequence of the damaged material, theassumptions of a small process zone and self-similar growth of a single

crack, inherent in linear elastic fracture mechanics, break down.Because of the complexity of the fracture process for notched com-posite laminates, the methods developed for prediction of strength aresemiempirical. Awerbuch and Madhukar (1985) review strength modelsfor laminates containing cracks or holes loaded in tension. In this text,only the technically important case of a laminate containing a circularhole is considered.

A conservative estimate of the strength reduction is based on the stressconcentration factor at the hole edge for a composite laminate containing a

circular hole,

=

KN 1

0

(2.63)

where Nand 0are the notched and unnotched ultimate strengths of thelaminate, respectively, and K is the stress concentration factor. The stressdistribution can be obtained in closed form only for infinite, homogeneous,orthotropic plates containing an open hole (Lekhnitskii 1968). The stress con-

centration factor Kfor an infinite plate containing a circular hole loaded inuniaxial tension (Figure 2.11) is given in terms of the effective orthotropicengineering constants of the plate (Lekhnitskii 1968):

( )( )= + +K E E E Gx y xy x xy1 2 2 (2.64)

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Notice here that the stress concentration factor is defined as the stress atthe hole edge divided by the applied far-field stress. Notice further that xand

yare coordinates along and transverse to the loading direction (Figure 2.11).The stress concentration factor for finite-width plates containing holes Kis

larger than K (see, e.g., Konish and Whitney 1975, Gillespie and Carlsson1988). Plates where the width and length exceed about six hole diameters,however, may be considered as infinite, and Equation (2.64) holds to a goodapproximation.

It can be easily verified from Equation (2.64) that the stress concentration fac-tor for an isotropic material is 3. For highly orthotropic composites, the stressconcentration factor is much greater (up to 9 for unidirectional carbon/epoxy).

x

y R

x

FIGURE 2.11Infinite plate containing a circular hole under remote uniform tension.

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