Failure criteria

CHAPTER

Failure criteria

2

2.1 INTRODUCTION A composite bonded repair generally comprises three basic structural elements: a damaged skin laminate, a repaired patch laminate, and a layer of bonding adhesive. Furthermore, the skin damage is normally cleaned up before the repair to become a through or a part-through hole with a straight or scarf edge. Thus, a repair analysis process will require a margin safety check for each of these three basic elements based on the appropriate failure criteria. Due to the local increase in overall stiffness of the repair, the local joint of dissimilar materials at an angle or right corner, and the presence of various forms of geometric discontinuity such as holes and crack-like damages, high local stress concentrations, or even stress singularities appear in the adhesive, skin, and patch laminate. Various failure criteria for assessing strengths of adhesive and composites in those cases are therefore reviewed in this chapter.

2.2 ADHESIVE FAILURE CRITERIA Failure in the bond line of a composite joint or repair is normally characterized by one of the following three modes or their mixed combination: (a) Cohesive failure (b) Adhesive failure (c) Interlaminar failure A cohesive failure is characterized by failure of the adhesive itself. Traces of adhesive material can be found on both sides of the fracture surfaces of specimens that fail cohesively. In contrast, an adhesive failure is characterized by a failure of the joint at the interface between the adhesive and the composite adherent. This latter failure mode is usually a result of poor surface preparation and unsuitable surface qualities of the adherent. Traces of adhesive material can only be found on one side of the fracture surfaces of specimens with an adhesive failure mode. Finally, an interlaminar failure is characterized by a failure of the matrix of the adherent ply adjacent to the bond line. The interlaminar failure indicates that the adhesive is stronger than the adherent in the joint, a desirable situation in practical design. Because failure criteria for the adhesive and interlaminar failure modes are usually based on the interfacial fracture mechanics theory that will be delineated in Section 2.3.2 as well as in Bonded Joints and Repairs to Composite Airframe Structures. http://dx.doi.org/10.1016/B978-0-12-417153-4.00002-5 Copyright # 2016 Elsevier Inc. All rights reserved.

21

22

CHAPTER 2 Failure criteria

Chapter 5, only failure criteria for the cohesive failure will be presented in this section. Furthermore, because adhesive materials are generally divided into two broad categories such as brittle and ductile adhesive, appropriate failure criteria for each category will be discussed in the two subsections below.

2.2.1 FAILURE CRITERIA FOR BRITTLE ADHESIVES Brittle adhesives are characterized by their mechanical behavior that is linear elastic up to the final failure. As a result, these adhesives do not fail in a stably progressive manner. Rather, they fail by a sudden material rupture without any prior plastic deformation. Even though most epoxy adhesives are ductile, however, their mechanical behaviors at low operating temperature (e.g., a cold, dry condition) are still reasonably assumed to be brittle. There is a large variety of failure criteria in the literature for cohesive failure of brittle adhesives, and they are summarized below (Tong and Steven, 1999; Tong and Soutis, 2003): (a) Maximum stress or strain criterion (b) Fracture mechanics criterion (c) Corner singularity criterion According to the maximum stress or strain criterion, the adhesive will fail when its maximum stress or strain reaches the limiting or allowable value. As it is well known that stress and strain in the adhesive are singular at the ends of the overlap, application of these criteria to stress or strain there deserves special attention. Fortunately, because most closed-form methods are developed based on a beam or plate theory with the adhesive layer being modeled as continuous springs, stress and strain solutions from these methods do not exhibit stress or strain singularity at the overlap ends. It is therefore straightforward to apply the maximum stress or strain criterion to predict a cohesive failure of the adhesive in this case. However, a finite element or continuum-based solution will predict stress and strain singularity at the ends of the overlap. As a consequence, stresses and strains in the end region increase indefinitely as the mesh there continues to be refined. Peak values of stress or strain at the overlap ends therefore cannot be used in the maximum stress or strain criterion because they depend strongly on the element sizes. To overcome this mesh-dependent problem, the so-called zone-based criteria have been proposed, that couple the above maximum stress or strain criteria with a characteristic length or finite zone. The zonebased criteria assume that the cohesive failure will occur when the adhesive stress or strain exceeds its limiting value at a characteristic length or over a finite zone, that is, – Stress criterion: σ 1 jx¼l ¼ σ crit ð 1 l σ 1 dx ¼ σ crit l 0

(2.1)

2.2 Adhesive failure criteria

– Strain criterion: E1 jx¼l ¼ Ecrit ð 1 l E1 dx ¼ Ecrit l 0

(2.2)

where σ 1 and E1 can be either the maximum principal stress and strain, peel stress and strain, or shear stress and strain of the adhesive, σ crit and Ecrit are the corresponding critical (allowable) stress and strain, and l is the length of the critical zone or a characteristic length (Tong and Steven, 1999). The criterion given by the first equation of Equation (2.1) is also referred to as a point stress criterion while the criterion given by the first equation of Equation (2.2) is called a point strain criterion. Unlike the original criteria, the zone-based criteria introduces a new parameter, the size of the critical zone or a characteristic length, which is determined by correlation of analytical predictions with test data. All of the above criteria are broadly categorized as the strength of material-based approach because either the adhesive layer is assumed to be flawless or the effect of stress or strain singularity at the end of overlap is completely ignored. Furthermore, these criteria also predict that strength of an adhesive bond is proportional to the square root of the adhesive thickness. However, experimental data seems to show the opposite trend: A bond strength in general decreases with the increasing adhesive thickness (Tong and Soutis, 2003). A simple remedy to this discrepancy is to use a different allowable for a different adhesive thickness, that is, a lower stress or strain allowable for a thicker adhesive, or to recalibrate the characteristic length for different adhesive thicknesses. Figure 2.1

Apparent shear strength (psi)

6000

5000

4000

3000

2000

1000

Hysol EA9394 PTM&W ES6292 MGS A100/B100 Each data point is an average of 3-4 specimens.

0 0.00

0.02

0.04

0.08 0.10 0.06 Bondline thickness (in.)

0.12

0.14

0.16

FIGURE 2.1 Apparent shear strength of commercial adhesives as a function of bond line thickness (Tomblin et al., 2001).

23

24

CHAPTER 2 Failure criteria

illustrates the effect of adhesive thickness on the adhesive shear strength for some commonly used adhesives (Tomblin et al., 2001). On the other hand, the fracture mechanics and corner singularity criteria use a strain energy release rate or a singularity parameter like the stress-intensity factor to characterize failure. In the fracture mechanics criteria, an arbitrary crack of a finite size must be introduced into a midplane of the adhesive layer. However, the crack size is normally chosen to be equal to the smallest bond line defect that can be detected by the nondestructive inspection techniques. The stress-intensity factor or strain energy release rate then can be computed by using a closed-form or a finite element method. The latter method, however, requires that a refined mesh must be employed at the crack tip vicinity in the analysis. To account for the interaction between the peel and shear stress of adhesive on its failure, a linear interaction fracture mechanics failure criterion is commonly employed in the prediction of cohesive failure. This criterion is given by the following equation: GI GII + ¼1 GIc GIIc

(2.3)

where GI and GII are the mode I and II strain energy release rate, respectively, while GIc and GIIc are mode I and II fracture toughness. As strain energy release rates are proportional to the square of the adhesive stresses and peak adhesive stresses are proportional to the square root of adhesive stress, fracture mechanics-based failure criteria will predict joint strength to be independent of bond line thickness. Thus, to account for the decreasing effect of bond strength on the increasing adhesive thickness as observed from experimental data, GIc and GIIc in Equation (2.3) are apparent (in situ measured) fracture toughness that may depend on the adhesive thickness. It is well known that stresses and strains in the adhesive are singular at the end of the overlap with a square edge even in the absence of a crack-like damage there. These stress and strain singularities are referred to in the literature as the corner singularity, and they were confirmed by both experiment and finite element analysis. As the corner stress-intensity factor uniquely characterizes the deformation at an interface corner, it would serve as a failure criterion for disbond free bonded joints provided that the size of the process zone is comparable to the corner singularity zone. For an elastic adhesive, through asymptotic analysis using finite element method, and by assuming that the adherent is several orders of magnitude stiffer than the adhesive so that it can be modeled as a rigid material in the asymptotic analysis, Wang and Rose (2000) found that the singular stress field at the end of the overlap can be represented by the following relations: σ ij ðr, θÞ ¼ Kr λ fij ðθÞ λ ¼ 1:29νa ð1  0:768νa Þ K ¼ ½Aðνa Þσ max + Bðνa Þτmax t2a Aðνa Þ ¼ 0:836  2:23νa + 6:29ν2a  9:64ν3a Bðνa Þ ¼ 3:12  15:8νa + 40:1ν2a  37:6ν3a

(2.4)

2.2 Adhesive failure criteria

where K is the stress-intensity factor, νa is the adhesive Poisson’s ratio, and ta is the adhesive thickness. In Equation (2.4), σ max and τmax are the maximum peel and shear stress of the adhesive at the overlap end obtained from a closed-form method. Equation (2.4) provides a practical way to estimate the stress-intensity factor of a corner singularity from adhesive stresses obtained by a closed-form method without performing a detailed finite element analysis. Not all bonded joints and repairs have an overlap end with a square edge as considered above. A spew fillet is sometimes formed at the overlap end, and studies have shown that spew fillets formed during bonding at the ends of overlap may have an important influence on the joint strength (Adams and Harris, 1987; Groth, 1998; Kairouz and Mathews, 1993; Adams et al., 1997). Formation of the spew fillet not only modifies the stress distribution along the midplane of the adhesive layer (Adams and Peppiatt, 1974), but may also eliminate the corner singularity. For a spew corner configuration shown in Figure 2.2 with the adhesive modulus far smaller than the adherent modulus, the order of the stress singularity at the corner A and B can be obtained from the Williams’s solution under plane strain condition, and it is given in Figure 2.3. It is clear that spew corner would be free of singularities when the fillet angle is less than a critical angle, which is dependent on the Poisson’s ratio of the adhesive. The value of this critical angle is shown in Figure 2.4. The absolute maximum angle at which no corner singularity would exist for any Poisson’s ratio is 45°. An example of a joint with a spew fillet end that would be free of corner singularity is shown in Figure 2.5. The end of adhesive layer in this case has been shaped so that all interface angles are less than 45°. Among the three criterion groups mentioned above, the maximum stress and maximum strain criteria coupled with the characteristic length concept are simplest. In particular, point stress and point strain criteria are the two most popular criteria for a cohesive failure of the adhesive, and they are widely used in practical applications due to their simplicity but sufficient accuracy. It is worth noting that because a brittle

Inner adherend Adhesive layer

y

tA x

C

Spew fillet

Outer adherend B

FIGURE 2.2 Configuration of a spew fillet.

A

q

25

CHAPTER 2 Failure criteria

Order of singularity (λ)

0.5

s∼r l

na = 0.45

0.4 0.3

na = 0.3 na = 0

0.2 0.1 0

0

30

60

90

120

150

180

Fillet angle q (°)

FIGURE 2.3 Order of corner singularity at spew fillet (points A and B in Figure 2.2). 90 Maximum angle q (°)

26

80 70 60 50 40

0

0.1

0.2 0.3 Poisson’s ratio na

0.4

0.5

FIGURE 2.4 Maximum fillet angle θ for zero corner singularity.

Inner adherend Adhesive layer

ta

45° No singularity

Outer adherend

FIGURE 2.5 Adhesive end free of corner singularity.

2.2 Adhesive failure criteria

FIGURE 2.6 Comparison of experimental results of joint strength and theoretical predictions. The variable t0 denotes a baseline adhesive thickness.

adhesive assumes to be linear elastic up to its final failure, these two criteria are essentially equivalent. On the other hand, because bond lines of most bonded joints and repairs are normally designed to be damage tolerant, too, a fracture mechanics criterion is therefore also used for predicting adhesive failure. The bond line in this case is assumed to contain a small crack-like defect (a disbond) in the middle layer of the adhesive at a critical stress or strain location, and its residual strength as determined by the fracture mechanics criterion must be higher than the design ultimate load. Maximum stress or strain criterion and the fracture mechanics criterion both fail to predict correctly the effect of adhesive thickness on bond strength when they are applied directly without calibration with a characteristic length and without usage of in situ adhesive allowable or apparent fracture toughness. In contrast, the corner singularity criterion has been found to be a promising criterion that can unify the failure loads of butt joints of varying bond line thicknesses (Reedy and Guess, 1997) and the failure loads of single-lap joints (Groth, 1998). Figure 2.6 shows comparisons between predictions from various strength-based, fracture-based, and corner singularity criteria for five sets of experimental results on single-lap joints and one set of data on single-strap joints (Tong and Soutis, 2003).

2.2.2 FAILURE CRITERIA FOR DUCTILE ADHESIVES Most epoxy adhesives are ductile. Ductile adhesives can be described to behave as an elastic-plastic material. Among all criteria mentioned in Section 2.2.1, only the

27

CHAPTER 2 Failure criteria

maximum strain criterion and its associated zone-based criteria are valid for ductile adhesives. For example, a stress criterion will not be appropriate for a ductile adhesive behaving like an elastic-perfectly plastic material because the stress in this case is not unique beyond yielding. In contrast, fracture mechanics and corner singularitybased criteria are derived strictly within a linear elasticity theory. The yielding of ductile adhesives has been reported to be affected by hydrostatic stress. In analyzing the stresses in bonded doublers or joints, often only two stress components are considered, the peel stress and the shear stress. However, due to the constraint imparted by the stiff adherents, the stress state in an adhesive layer is truly triaxial (Adams et al., 1997; Wang and Rose, 1997). The lateral stress components parallel to the interface give rise to high hydrostatic tension. This hydrostatic stress is important in determining the plastic yield behavior of adhesive (Wang and Chalkey, 2000), and affecting the crazing mechanism. As an example, the influence of hydrostatic stress on the plastic yielding behavior of a film adhesive FM73 is illustrated in Figure 2.7 (Wang and Chalkey, 2000). Equivalent shear stresses at yielding obtained from a series of experiments for FM73 adhesive under various combinations of tension, compression, and shear loading (Wang and Chalkey, 2000) are plotted in Figure 2.7 as a function of hydrostatic stress. It can be seen that when the hydrostatic stress is negative (in compression), the equivalent shear stress is approximately constant, indicating that the conventional von Mises yield criterionpisffiffiffi applicable as the equivalent shear stress relates to von Mises stress by a factor of 3. However, under tensile hydrostatic stress, the equivalent shear stress decreases rapidly as the hydrostatic stress increases. These data suggest that a new, modified von Mises yield criterion (Bowden and Jukes, 1972) must be used for positive hydrostatic stress,

Equivalent shear stress, t eq (MPa)

28

40 35 30

Shear tension Shear Neat compression compression Pure shear

25

Neat tension

20 15 10

t eq = 38.6 + 1.13*p Constrained tension

5 –30 –20 –10 0 10 20 30 Hydrostatic pressure (negative value of), –p, (MPa) Hydrostatic stress p (MPa)

FIGURE 2.7 Influence of hydrostatic stress on yield stress of FM73 adhesive.

2.3 Composite failure criteria

τeq  e μp ¼ τY a

(2.5)

where τeq and τY a denote the equivalent shear stress and the yield stress under shear (zero hydrostatic stress), respectively, p is the hydrostatic stress, and the experimen tal data shown in Figure 2.7 suggest that the coefficient μ is approximately 1.13 for FM73 adhesive. It is therefore important to model the elastic-plastic behavior of a ductile adhesive in the analysis using the modified von Mises yield criterion or an equivalent linear Drucker-Prager criterion. Positive hydrostatic stress affects not only the plastic yielding behavior of the adhesive, but also its strength via crazing mechanism. It was previously observed by Lietchti and Freda (1989) and Wang and Chalkey (2000) that epoxy adhesive under tensile loading tends to fail by crazing due to high triaxial tensile stress state at the crack tip. The crazing process involves the formation of microcracks ahead of the crack tip in the direction of the maximum principal strain. Similar to Equation (2.5), Wang (1997) has proposed the following modified strain failure criterion that accounts for the effect of pressure sensitivity: E1 +

p tan β ¼ Ecrit E

(2.6)

where E1 is the maximum principal strain, E is the Young’s modulus, tan β is the parameter reflecting the sensitivity of the material to hydrostatic stress, and p is again the hydrostatic stress. As discussed immediately below, Wang’s criterion can also be used with the characteristic length concept to predict the cohesive failure of ductile adhesives. Unlike brittle adhesives, failure in ductile adhesives is usually progressive. This is because of the material softening and load redistribution within a joint due to a large plastic deformation before final failure. As all criteria considered in this section do not specify a damage evolution law for a progressive failure of the bond line, it is necessary to couple the maximum strain criterion or Wang’s criterion with a characteristic length. In other words, a point strain criterion or a Wang’s criterion together with a characteristic length concept can be used in practice to predict the cohesive failure of ductile adhesives.

2.3 COMPOSITE FAILURE CRITERIA Failure of composites is normally divided into two broad failure modes: intralamina and interlaminar failures. Intralamina failure mode is an in-plane failure mode that involves fiber breakage, matrix failure, or disbond along the interface between matrix and fiber. In contrast, interlaminar failure involves interfacial failure between different plies, and it is also referred to as a delamination. It is well known that inplane matrix failure can lead to a delamination. Thus, it may be necessarily to include in the failure criteria for each of these two broad failure modes the cross-interaction terms between them. However, for simplicity, intralamina and interlaminar failure criteria are usually assumed to be uncoupled, and they are specified separately as delineated in the next two subsections.

29

30

CHAPTER 2 Failure criteria

2.3.1 INTRALAMINA FAILURE CRITERIA Numerous failure criteria for intralamina failure of composites have been proposed over the past five decades. Some of these criteria were evaluated thoroughly through a Worldwide Failure Exercise during a 10-year period from 1994 until 2004 (Soden et al., 1998; Kaddour et al., 2004). The objective of this exercise is to benchmark the status, accuracy, and bounds of validity of these criteria. In the exercise, originators or leading experts of various failure theories and criteria used their own theory and criteria without prior knowledge of the experimental data to predict the performance of specified carbon- and glass fiber-reinforced epoxy laminates subjected to a range of biaxial loads, using the same given material properties, laminate layup, and loading conditions. Several conclusions are drawn from the exercise (Soden et al., 2004). First, most criteria are unable to capture some of the trends in the failure envelopes of experimental results. Second, results from this exercise indicate that Tsai-Wu criteria (Tsai and Wu, 1971) along with Cuntze (Cuntze and Freund, 2004) and Puck (Puck and Schu¨rmann, 1998) failure theories did well overall on a lamina level. Third, Bogetti’s failure theory (Bogetti et al., 2004) seems to provide the best prediction of initial failure load in multidirectional laminates. Bogetti’s theory employs a maximum strain failure criterion within a three-dimensional form and accounts for both lamina progressive failure and nonlinear shear behavior. Lastly, Cuntze, Puck, Tsai-Wu, and Zinoviev (Zinoviev et al., 1998) failure theories achieved the highest score on the final strength prediction for multidirectional laminates. Even though some of the best and most promising failure theories and criteria are identified from the exercise, very few of them are available in a form that can be readily utilized in practical applications. In this regard, only criteria that are either easily implemented into a practical predictive tool or already implemented into the available commercial codes such as ABAQUS or MSC/NASTRAN will be discussed in this section. All failure criteria considered below have been used either in a nonprogressive failure analysis for a laminate strength check or in a progressive failure analysis of the commercially available software as damage initiation criteria (Lapczyk and Hurtado, 2007). When these failure criteria are used in a nonprogressive failure analysis, they are usually used in conjunction with the characteristic length as discussed in Section 2.2 to predict strength of a laminate with a geometric discontinuity such as a notch or a cutout.

2.3.1.1 Maximum stress or strain failure criteria The maximum stress/strain failure criteria state that failure occurs in a ply when any one of the fiber direction, matrix direction (transverse to fiber direction), or shear stresses/strains exceeds the corresponding allowable. These criteria can be expressed in terms of stresses or strains as follows (Jones, 1975): jσ 1 j jσ 2 j jτ12 j ¼ 1; ¼ 1; ¼1 jX j jY j jSj

(2.7)

2.3 Composite failure criteria

j E1 j j E2 j jγ j ¼ 1; ¼ 1; 12 ¼ 1 jXE j jY E j jSE j

(2.8)

where σ 1, σ 2, and τ12 are the fiber direction, matrix direction, and shear stresses in a ply; X, Y, and S are the longitudinal, transverse, and shear stress allowables; and the rest are defined similarly for ply strains and strain allowables. The longitudinal and transverse stress or strain allowables, X and Y or XE and YE, are tension allowable if the stresses or strains are greater than zero and compression allowable if otherwise. The maximum stress or strain failure criteria are noninteractive failure criteria with a rectangular failure envelope in 2D space and they are applied at a ply or lamina level. As pointed out by Hart-Smith (1990, 1998), the original maximum strain failure criteria contain two deficiencies for fiber-dominated laminate with carbon fiberreinforced plastic composites, which needs further improvements. The first deficiency is on the treatment of the transverse tension failure in a ply. Ply transverse tension failure is basically a matrix tension failure. Because most lamina materials are highly orthotropic, the allowable for a ply transverse tension failure is usually very low for an isolated lamina. As a consequence, application of the original maximum strain failure criteria usually leads to an unrealistic prediction of low transverse tensile strength in a laminate because the above criteria do not account sufficiently for the constraining effect of adjacent orthogonal plies on delaying initiation and arresting propagation of matrix cracks in each lamina of the laminate under transverse loads. As an example, while the transverse strength of an isolated 0° ply is typical very low, the corresponding transverse strength of a 0/90 laminate can be as high as its longitudinal strength. Matrix cracks in the 0° ply of a 0/90 laminate are unlikely to initiate or propagate because its orthogonal ply (i.e., a 90° ply with high stiffness and high strength in transverse direction to 0° ply) will effectively restrain the opening of these cracks. Thus, to remedy this deficiency, the tensile transverse strength of each lamina in a laminate will be set minutely above its longitudinal strength, so that the theory will not predict unrealistically a premature matrix tension failure in a well-designed laminate with a sufficient number of orthogonal plies in both longitudinal and transverse directions. In contrast, the second deficiency of the original maximum stress/strain failure criteria is due to their overprediction of the fiber-dominated laminates strengths for biaxial stresses of opposite sign. The original criteria do not account for the effect of fiber shear off in this special loading case. The most widely used modification to address this latter deficiency is to truncate the original failure envelope with a 45° sloping line passing through the greater (tensile or compressive) strain-to-failure for unidirectional lamina subjected to uniaxial stress. It turns out that this truncation in effect also imposes a limit on the in-plane shear strength of matrix-dominated laminates. Figure 2.8 shows the strain envelope of a lamina with the two abovementioned modifications incorporated when XtE is numerically greater than XcE . From Figure 2.8, the vertical strength limits are characterized by the following simple formula: XEc  E1  XEt

(2.9)

31

32

CHAPTER 2 Failure criteria

e2

45° (1+n12)Xet

C Xet

B

D –Xec

Xet

45°

e1

45°

A –(1+n12)Xet

(1+n12)Xet

–Xec E

45°

F

Xet

n12 Xet

–(1+n12)Xet

FIGURE 2.8 Strain failure envelope for a lamina based on Hart-Smith modified (truncated) maximum strain failure criterion. In this illustration, X tE is assumed to be numerically greater than X cE .

where the superscripts c and t signify compression and tension allowable, respectively, while the rest are defined previously. On the other hand, the horizontal cutoffs are likewise defined by the similar equation, noting that transverse strain limits are set equal to those in the longitudinal direction XEc  E2  XEt

(2.10)

Finally, the 45° sloping cutoffs are simply expressed as jE1  E2 j  ð1 + ν12 ÞXE

XcE

XtE

(2.11)

in which XE is the numerically greater of and and ν12 is the Poisson’s ratio. Another approach to improve the original maximum stress or maximum strain failure criteria is based on the noninteractive maximum laminate strain failure criteria. These criteria are governed by the same Equation (2.8) but with appropriate laminate strain components and laminate strain allowable, rather than ply strains and lamina allowable. The maximum laminate strain criteria are found to work well for carbon fiber-reinforced laminates with traditional and proper layup (consisting of only 0, 90, 45, 45 ply orientation, well-dispersed plies, and each fiber orientation has an appropriate ply percentage for load carrying capability). Under such limitation, a laminate will fail predominantly by a fiber failure mode, and the laminate strain approach works well. However, as the approach uses laminate strain allowable, it will require extensive testing for each laminate layup configuration used in the

2.3 Composite failure criteria

design envelope. Moreover, for a general laminate (nontraditional, insufficient plies in orientation parallel to loading directions) and for other fiber-reinforced laminates such as glass-fiber composites, matrix failure and its interaction with fiber failure become important. In this case, the failure characteristic of a laminate test coupon under a uniaxial loading and that of a same or similar laminate under multiaxial loading may be different. Thus, a generalization of the results from coupon tests to a design laminate may require further validation with test.

2.3.1.2 Tsai-Hill and Tsai-Wu failure criteria Tsai-Hill failure criterion is based on Hill’s proposed yield criterion for anisotropic materials, and it is given by (Tsai, 1965) σ 2 1

X



σ 1 σ 2 σ 2 2 τ12 2 + + X2 Y S

(2.12)

Tsai-Hill criterion is applied at a lamina level, and it describes a smooth elliptical failure envelope. To include stress and strength interactions in the fiber and matrix directions, Tsai-Hill criterion has been generalized into a Tsai-Wu criterion. TsaiWu criterion is also a generalization of the tensor failure criterion. In general terms, the Tsai-Wu criterion is (Tsai and Wu, 1971) F1 σ 1 + F2 σ 2 + F3 τ12 + F4 σ 21 + F5 σ 22 + F6 τ212 + 2F7 σ 1 σ 2 ¼ 1

(2.13)

1 1 1 1 + ; F2 ¼ t + c ; F3 ¼ 0; Xt Xc Y Y 1 1 1 F4 ¼  t c ; F5 ¼  t c ; F6 ¼ 2 ; XX YY S i 1 h 2 F7 ¼ t 1  ðF1 + F2 ÞY t  ðF4 + F5 ÞðY t Þ 2Y

(2.14)

where F1 ¼

It is clear from the above equations that Tsai-Hill and Tsai-Wu criteria give identical results when tension allowable are equal to the compression allowable, Xt ¼ Xc and Yt ¼ Yc.

2.3.1.3 Hashin failure criteria Hashin failure criteria distinguish among the various different failure modes of the unidirectional lamina: tensile and compressive fiber failures and tensile and compressive matrix failure, and they are given separately for each of these failure modes as follows (Hashin, 1980): Tensile Fiber Mode: σ 1 > 0 σ 2 τ 2 1 12 + ¼1 Xt S

(2.15)

Compressive Fiber Mode: σ 1 < 0 σ 1 ¼ Xc

Tensile Matrix Mode: σ 2 > 0

(2.16)

33

34

CHAPTER 2 Failure criteria

σ 2 τ 2 2 12 + ¼1 Yt S

Compressive Matrix Mode: σ 2 < 0  σ 2 2 2ST

"

Yt 2ST

+

#

2

1

σ 2 τ12 2 + ¼1 Yc S

(2.17)

(2.18)

where ST is the transverse or out-of-plane shear strength while S is the longitudinal or in-plane shear strength, and the rest are defined previously. Similar to Tsai-Hill failure criterion, Hashin failure criteria are also quadratic polynomials of stresses. However, they are derived from general polynomials of the first four transversely isotropic stress invariants with the highest order term for each of these four invariants being selected in such a way that results in a highest order of stress to be quadratic. Furthermore, Hashin failure criteria assume that two distinct fiber and matrix failure modes occur in two different fracture planes and only relevant stress components on the associated fracture plane of each mode will contribute to the failure criteria for that failure mode. As a consequence, the failure envelope described by Hashin failure criteria is only piecewise smooth, with each smooth branch modeling a distinct failure mode. Hashin criteria provides an improvement for prediction of intralamina failures. However, as pointed out by Hashin, the quadratic failure criteria for matrix mode imply that the fracture plane is the maximum transverse shear plane and this may not be always true. A more general approach based on the Mohr-Coulomb failure theory for matrix failure is also suggested by Hashin to overcome this limitation. Puck and Schu¨rmann (1998), and more recently Davila et al. (2005) have taken some form of this general approach in their development of matrix failure criteria.

2.3.1.4 Larc03 criteria Larc03 criteria are plane stress criteria that are developed by Davila et al. (2005). Extension of Larc03 criteria to include a general three-dimensional loading and in-plane shear nonlinearity has been carried out by Pinho et al. (2005). Similar to Hashin criteria, Larc03 criteria consist of a set of six failure criteria for different failure modes of composites. Larc03 criterion for a matrix failure under transverse compression ðσ 2 < 0Þ is based on the Mohr-Coulomb failure theory, and it is given by the following equation:  T 2  L 2 τeff τ + eff ¼ 1 ST Sis

where

(2.19)

 

τTeff ¼ σ 2 cos α sinα  ηT cos α  

τLeff ¼ cos α jτ12 j  ηL σ 2 cosα S cos 2α0 Y c cos 2 α0 ST ηT ¼ ηL S

ηL ¼ 

(2.20)

2.3 Composite failure criteria

1 hxi ¼ ðx + jxjÞ; α0 is the orientation angle of a fracture plane failed by transverse 2 shear when loaded in purely transverse compression and it is given by α0 ¼ 53  2° for most unidirectional graphite-epoxy composites, α is the angle of the fracture plane under combined loads σ 2 and τ12, and the rest are defined previously. In Equation (2.19), the in situ strength of the shear strength Sis is specified rather than the lamina strength value S to account for the constraining effect of adjacent plies on the substantial increase in individual lamina shear strength, and its calculation will be given later. For a given combination of σ 2 and τ12, α must be determined iteratively between a range of 0 and α0 so that the combined terms  T 2  L 2 τeff τ + eff in Equation (2.19) is at maximum. T S Sis On the other hand, Lar03 criterion for a matrix failure under transverse tension ðσ 2 > 0Þ is derived from fracture analysis of a slit crack in a ply of a laminate as illustrated in Figure 2.9. An effective slit crack with dimensions of 2a0  2aL is used in the model to represent the macroscopic effect of matrix-fiber debonds that occur in a laminate at the micromechanical level due to manufacturing defects or residual thermal stresses. Appropriate choices for a0 and aL to be used in the analysis are dependent of the material considered. The criterion for a matrix failure under transverse tension then can be expressed as  2  2 σ2 σ2 τ12 ð1  gÞ t + g t + ¼1 Yis Yis Sis

where g¼

(2.21)

  GIc ðLÞ GIc ðTÞ Λ022 Yist 2 ¼ ¼ 0 GIIc ðLÞ GIIc ðTÞ Λ44 Sis   1 ν221  Λ022 ¼ 2 E2 E1 1 Λ044 ¼ G12

(2.22)

(2.23)

GIc(L) and GIIc(L) are mode I and II fracture toughness in the longitudinal (fiber) direction while GIc(T) and GIIc(T) are the similar fracture toughness but in 3 (T)

3 (T)

2a0

1 (L)

2

t

2aL

FIGURE 2.9 A slit crack model for predicting matrix tension failure in Larc03 criteria.

2a0

35

36

CHAPTER 2 Failure criteria

transverse direction; Ytis and Sis are in situ transverse tensile and longitudinal shear strengths and their expressions are dependent on three idealized configurations of a slit crack in a ply of a laminate; all are found to be independent of the longitudinal slit size aL: Case 1: A slit crack that spans only a few plies of a large number of cluster plies in a laminate ð2a0 ≪ tÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GIc ðTÞ ¼ πa0 Λ022 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GIIc ðTÞ Sis ¼ πa0 Λ044 Yist

(2.24)

where a0 is the half length of a transverse slit size. Case 1 represents the macroscopic effect of matrix-fiber debonds in a thick laminate where the crack will grow in the transverse (thickness) direction. Case 2: A slit crack that spans across all cluster plies of a laminate ð2a0 ¼ tÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8GIc ðLÞ ¼ πtΛ022 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8GIIc ðLÞ Sis ¼ πtΛ044 Yist

(2.25)

Case 2 represents the macroscopic effect of matrix-fiber debonds in a thin laminate where the crack already extends across the entire thickness of the cluster plies and subsequently will grow in the longitudinal direction. Case 3: A slit crack in a thick unidirectional laminate pffiffiffi Yist ¼ 1:12 2Y t pffiffiffi Sis ¼ 2S

(2.26)

In contrast, Larc03 criterion for fiber tension failure is simply given by E11 ¼1 XEt

(2.27)

However, the fiber compression failure is governed by three separate criteria, depending on the level of shear kinking and damage of the supporting matrix to cause fiber collapse. Furthermore, because the imperfection in fiber alignment plays a critical role in compressive strength of composites, ply stresses in the misalignment coordinate frame of an idealized local region of waviness as shown in Figure 2.10 must be used in the prediction of a composite longitudinal compressive strength. In other words, criteria for fiber compression failure must be expressed in terms of these local ply stresses. The ply stresses in the misalignment coordinate frame are given by

2.3 Composite failure criteria

sm 22

s 22 Xc

s 11

j

sm 11

Xc

FIGURE 2.10 Fiber misalignment due to ply waviness in laminate. 2 2 σm 1 ¼ σ 1 cos φ + σ 2 sin φ + 2cos φ sinφjτ 12 j 2 2 σm 2 ¼ σ 1 sin φ + σ 2 cos φ  2cosφ sin φjτ12 j  2  2 σm 12 ¼ σ 1 sinφ cos φ + σ 2 sinφ cos φ + cos φ  sin φ jτ12 j

(2.28)

where jτ12 j + ðG12  Xc Þφc G12 + σ 1  σ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0   1 Sis Sis L 1  1  4 + η C B Xc Xc C B C   φc ¼ tan 1 B C B Sis L A @ 2 c +η X φ¼

(2.29)

Fiber compression failure by the formation of kink band with matrix compression is given by an equation similar to Equation (2.19) but with α ¼ 0 and τTeff ¼ 0 as fiber kinking is dominated by the shear stress τ12, not by the transverse stress σ 22. Thus, a criterion for fiber compression failure by the deformation of kink band with matrix compression can be expressed by the following equation: *   τ m  + ηL σ m 12

2

Sis

+

¼1

(2.30)

When the level of fiber compression is moderate while the matrix compression level is high, matrix damage can occur without the formation of kink bands or fiber damage. A failure criterion in this case must take a full form of Equation (2.19), that is, 

τmT eff ST

2  mL 2 τ + eff ¼1 Sis

m  

T τmT eff ¼ σ 2 cos α sinα  η cos α  m  

L m   τmL eff ¼ cos α τ 12  η σ 2 cos α

(2.31)

(2.32)

In contrast, a fiber compression failure by kink band with matrix tension is governed by a failure criterion similar to Equation (2.21), that is, ð1  gÞ

 m 2  m 2 σm σ2 τ 2 + g + 12 ¼ 1 Yist Yist Sis

(2.33)

37

38

CHAPTER 2 Failure criteria

In summary, Larc03 criterion consists of a set of six criteria for six different failure modes, and they are given respectively by Equations (2.19), (2.21), (2.27), (2.30), (2.31), and (2.33).

2.3.2 INTERLAMINAR FAILURE CRITERIA A most commonly used interlaminar failure criterion is based on the linear elastic fracture mechanics (LEFM) concept. However, there are several basic issues associated with LEFM as it applies to interlaminar failure or interfacial fracture. First, the singular stress field at the tip of an interfacial or delamination crack is oscillatory for most bimaterial systems (Williams, 1959). Second, as a result of this stress oscillation, the decomposition of the total strain energy release into separate fracture modes I, II, and III is not possible without introducing an arbitrary length parameter (Rice, 1988; Tay, 2003). Thus, there is a lack of uniqueness for the strain energy release rate components, and fracture characterization under mixed-mode condition by material testing may not be physically based. Another consequence of the oscillatory behavior of the crack tip stress is that the corresponding crack tip displacement field implies interpenetration of the crack surfaces (Comninou, 1977). Third, the delamination or interfacial crack may not necessarily propagate along the initial interface, depending on the mode mixity and the relative fracture toughness between the interface and the adjacent bonding plies. This phenomenon is also known as crack jumping (Tay, 2003). Finally, the effect of fiber bridging is not accounted for in LEFM (Daridon et al., 1997). Nevertheless, for simplicity, the failure criterion considered in this section will not address the issues of crack jumping and fiber bridging. Furthermore, the present discussion is also limited to the case of an interfacial crack between two orthotropic materials with material principal axes parallel to the reference axes. From LEFM, the singular stress field ahead of the tip of an interfacial crack along the crack plane shown in Figure 2.11 is given by (Suo, 1990) σ3

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi H33 =H11 + iτ13 ¼ Kr iE = 2πr

(2.34)

X3

Material 1 r X1 Material 2

FIGURE 2.11 An interfacial crack in bimaterials.

2.3 Composite failure criteria

where E ¼ ð1=2π Þ ln ½ð1  βE Þ=ð1 + βE Þ h i h i  βE ¼ ðs11 s33 Þ1=2 + s13  ðs11 s33 Þ1=2 + s13 ðH11 H33 Þ1=2 2 1 h i h i H11 ¼ 2nλ1=4 ðs11 s33 Þ1=2 + 2nλ1=4 ðs11 s33 Þ1=2 1 2 h i h i H33 ¼ 2nλ1=4 ðs11 s33 Þ1=2 + 2nλ1=4 ðs11 s33 Þ1=2 1

2

(2.35)

λ ¼ s11 =s33

1=2 1 n ¼ ð1 + ρÞ 2 1 ρ ¼ ð2s13 + s55 Þðs11 s33 Þ1=2 2

K is the complex stress-intensity factor, r is the distance ahead of the crack tip, i is the imaginary number, subscripts 1 and 2 of a square bracket indicate material 1 and 2, respectively, and sij are material compliances that are defined through the stressstrain relation as εi ¼ sij σ j in contract notation form. βE is a generalization of one of the Dundurs’ parameters for isotropic materials and E is referred to as a bimaterial constant. For plane strain, sij will be replaced by sij0 where s0ij ¼ sij  si2 sj2 =s22 . For a crack in a homogeneous isotropic solid, from Equation (2.35), λ ¼ 1, βE ¼ E ¼ 0, H11 =H33 ¼ 1; and Equation (2.34) is then reduced to the classical result pffiffiffiffiffiffiffi σ 3 + iτ13 ¼ K= 2πr

(2.36)

The complex stress-intensity factor in this case can be expressed as K ¼ KI + iKII

(2.37)

where KI and KII are respectively the real and imaginary parts of the complex stressintensity factor, as well as mode I and mode II components of the complex stressintensity factor. The physical significance of KI and KII as fracture parameters to characterize pure mode I and mode II of fracture in the homogeneous isotropic material stems from their relationship with the crack opening normal stress σ 3 and crack sliding shear stress τ13 at the crack tip as r ! 0 along the crack plane via Equation (2.36). Specifically, from Equations (2.36) and (2.37), it follows that pffiffiffiffiffiffiffi σ 3 ¼ KI = p2πr ffiffiffiffiffiffiffi τ13 ¼ KII = 2πr

(2.38)

Thus, the ratio of the crack sliding shear stress to the crack opening normal stress is constant, independent of the distance r ahead of the crack tip, and it is equal to KII/KI. In that context, KII/KI is the measurement of the ratio of two crack tip stress components as r ! 0. The real and imaginary parts of the complex stress-intensity factor K then correspond to mode I and II components of fracture. For future discussion, a mode mixity ψ that is defined by ψ ¼ tan

is introduced.

1

   

τ13 1 KII 1 ImðK Þ ¼ tan ¼ tan σ 3 r¼0 KI ReðK Þ

(2.39)

39

40

CHAPTER 2 Failure criteria

When the interfacial crack lies between two different orthotropic materials with a combination of their material constants resulting in βE ¼ 0 or E ¼ 0; KI and KII defined by Equation (2.37) still retain their physical interpretation as the mode I and II stressintensity factors because KII/KI, except for a proportional constant, is again a measurement of the ratio of two crack tip stress components as r ! 0 as shown below: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi H33 =H11 ¼ KI = 2πr pffiffiffiffiffiffiffi τ13 ¼ KII = 2πr τ13 ðr Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KII ¼ H11 =H33 lim r!0 σ 3 ðr Þ KI σ3

(2.40)

In contrast, for an interfacial crack that lies between two general orthotropic materials with βE 6¼ 0 and E 6¼ 0, a separation of the complex stress-intensity factor into mode I and II components cannot be done without ambiguity. This can be seen by noting r iE ¼ eiE ln r and rewriting Equation (2.34) as σ3

or

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi H33 =H11 + iτ13 ¼ KeiE ln r = 2πr

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H33 =H11 ¼ ðReðK ÞcosE lnr  ImðK ÞsinE lnr Þ= 2πr pffiffiffiffiffiffiffi τ13 ¼ ðReðK ÞsinE lnr + ImðK Þcos E lnr Þ= 2πr

σ3

(2.41)

(2.42)

τ13 ðr Þ will not converge because the term σ 3 ðr Þ E ln r become undefined for r ¼ 0 unless E ¼ 0. Also, the real and imaginary parts of the complex stress-intensity factor no longer can be interpreted as mode I and mode II components of fracture as σ 3 and τ13 contain both real and imaginary parts of the complex stress-intensity factor in their expression. Furthermore, the stress ratio τ13/σ 3 of two stress components ahead of the crack tip along the crack plane will vary with the distance r. Thus, the mode mixity cannot be selected uniquely in this case and it must be specified for a particular distance r. Following Rice’s suggestion (Rice, 1988), ψ will be redefined similar to Equation (2.39) as It is clear from Equation (2.42) that lim r!0

ψ ¼ tan

1

"  #     iE τ13 1 Im KL 1 KII   ¼ tan ¼ tan σ 3 r¼L KI Re KLiE

(2.43)

where L is a reference length whose choice will be discussed shortly, while KI and KII are now related to the complex stress-intensity factor by KLiE ¼ KI + iKII

(2.44)

instead of Equation (2.37). The above definitions for ψ, KI, and KII reduce to Equations (2.39) and (2.37) when βE ¼ 0 or E ¼ 0. In Equations (2.43) and (2.44), the choice of the reference length L is arbitrary. However, because the underlying of the fracture mechanics assumes crack tip behav^ ior to be independent of specimen size, the choice of a fixed reference length (e.g., L) that is based on the material length scale such as the fracture process zone is normally a favorite choice for presenting fracture toughness data and thus also for developing

2.3 Composite failure criteria

interlaminar failure criterion. In contrast, elastic solutions for bimaterial interfacial cracks are normally expressed based on a reference length called L that is scaled with a specimen geometry length scale such as a crack length or a specimen thickness. This latter choice of the reference length is useful when comparing biomaterial elastic solution from one cracked geometry to another. For example, for a central biomaterial interfacial crack in an infinite plate under far field applied stresses, the mode mixity ψ from elastic solutions will be the same for all plate thickness when the reference length is chosen to be the plate thickness. Thus, the elastic solution for a bimaterial interfacial crack needs to be obtained for only one plate thickness as this same solution can also be used conveniently for other plate thickness. From Equations (2.43) and (2.44), the mode mixity ψ based on a reference length L that is scale with the specimen geometry length scale and ψ^ based on a fixed material length L^ are related by the following transformation:     L^ ψ^ L^ ¼ ψ ðLÞ + E ln L

In the delamination and disbond analysis, most failure criteria are given in terms of the total strain energy release rate and its mode I and II components defined, respectively, by Irwin’s crack closure integrals as (Irwin, 1957) G ¼ GI + GII 1 Δa!0 2tΔa

GI ¼ lim

GII ¼ lim

1

ð Δa

σ 3 ðr, 0ÞΔwðΔa  r,π Þdr

0

ð Δa

Δa!0 2tΔa 0

(2.45)

σ 13 ðr, 0ÞΔuðΔa  r,π Þdr

where t is the thickness of the plate, and the rest is previously defined. For a general bimaterial interfacial crack with βE 6¼ 0 and E 6¼ 0, the relationship between the total energy release rate and the modulus of the complex stress-intensity factor is given by (Suo, 1990) G¼

H11 jK j2 4 cosh 2 πE

(2.46)

It is clear from Equation (2.46) that the total energy release rate G is well defined for both E ¼ 0 and E 6¼ 0. However, like ψ, due to the oscillatory nature of the stress field ahead of the crack tip, mode I and II components of the total strain energy release rate (i.e., GI and GII) fail to converge as Δa ! 0. Thus, the decomposition of the total energy release rate experiences the same difficulty of nonuniqueness as that of the complex stress-intensity factor into KI and KII discussed earlier. Consequently, several different definitions of GI and GII have been proposed in the literature. For example, Davidson et al. (1995) proposed GI and GII to be defined in a similar manner to the total energy release rate according to the following equations: H11 KI2 4 cosh 2 πE H11 KII2 GII ¼ 4 cosh 2 πE GI ¼

(2.47)

41

42

CHAPTER 2 Failure criteria

where KI and KII are again mode I and II components of the complex stress-intensity ^ The factor that are defined based on a specific reference length like, for example, L. sum of GI and GII defined by Equation (2.47) will yield the same result for the total energy release rate as that given by Equation (2.46). It should be emphasized that even though GI and GII given by Equation (2.45) fail to converge individually as Δa ! 0, however, their sum will converge in the limit as evidence from Equation (2.46). Other definitions for GI and GII are based on a finite crack extension Δa (Yang et al., 2000). GI and GII in this case are still given by Equation (2.45) but without taking the limit of Δa going to zero, and they are defined with respect to a specific length of Δa in the same way as the usage of the reference length L or L^ discussed above to define KI and KII. As pointed out earlier, the decomposition of the complex stress intensity or total strain energy release rate when E 6¼ 0 is not unique and quite cumbersome to implement. Fortunately, most bimaterial systems considered in practical applications have a very small value of E, and the zone of stress oscillation is limited to a very small region compared to the ply thickness. Several approaches have been proposed to simplify the fracture analyses in such case. The first approach is to employ an isotropic resin interlayer between two materials and assume the crack is embedded within this interlayer. The crack is therefore no longer an interfacial crack, and it becomes an internal crack within a homogeneous isotropic material. A second approach is to modify slightly the material properties of the sublaminates adjacent to the interfacial crack such that they will result in βE ¼ 0 and thus E ¼ 0. The final approach is to apply the finite crack extension concept described above to the evaluation of GI and GII. Raju et al. (1988) have found that when GI and GII are evaluated numerically by a finite element method using virtual crack closure technique (VCCT) (Rybicki and Kanninen, 1977), values of GI and GII are nearly constant for an element size ranging between 0.25 and 0.5 of the ply thickness. In practice, a larger element size sometimes on the order of several ply thickness is also acceptable for use. The VCCT method will be described in detail in Chapter 5. Because evaluations of GI and GII by the VCCT method are performed using an Δa equal to the element size at the crack tip, a decomposition of the total strain energy release rate in this case is therefore associated with a finite crack extension Δa that is between 0.25 and several ply thicknesses. Because of the relatively mesh independence of FE results for this range of element size and Δa, the difficulty of the uniqueness in the mode decomposition is therefore mostly ignored in the literature on delamination or disbond analyses by a FE method. With that, the interlaminar failure criterion for a bimaterial interfacial crack can be expressed alternatively in the generalized functional form as   G ¼ Gc ψ^ , L^

(2.48)

where G and Gc are the (applied) total strain energy release rate and the interface toughness, respectively. According to Equation (2.48), the interface toughness is a ^ Thus, for a given function of both mode mixity and the fixed reference length L.

2.4 Summary

cracked geometry under prescribed loads, both the total strain energy release rate and mode mixity ψ^ based on a fixed L^ must be determined. The obtained total strain energy release rate is then compared with the interface toughness for that particularly obtained mode mixity. In Equation (2.48), the interface toughness is determined to be a function of the mode mixity ψ^ that is defined based on a decomposition of the complex stressintensity factor. However, because GI and GII in many applications are evaluated directly from numerical methods such as FE method without recourse to the stress-intensity factor components KI and KII, it is more convenient to express the interface toughness in terms of the mode mixity that is based on the decomposition of the total strain energy release rate, that is,   G ¼ Gc ψ^ G , L^ ψ^ G ¼ tan 1

GII GI

(2.49)

One common choice of interface toughness Gc for delamination in composites is the Benzeggagh-Kenane law that is given by (Benzeggagh and Kenane, 1996) Gc ¼ GIc + GIIc  GIc ðtan ψ^ G Þη  η GII ¼ GIc + GIIc  GIc GI

(2.50)

where η is the mode mixity exponent. Another common form of the interlaminar failure criteria used in delamination analysis of composite is the linear interaction law given by Equation (2.3) described previously in Section 2.2.1. Rather than using the total strain energy release rate and the mode mixity, Equation (2.3) expresses the failure criterion in terms the mode I and II components of the total strain energy release rate. Thus, Equations (2.3) and (2.49) are functionally equivalent. It should be emphasized that ψ^ in Equation (2.48) and ψ^ G in Equation (2.49) will be independent from the choice of the reference length when a resin interlayer between two delaminated plies is modeled, when material properties are modified to yield βE ¼ 0, or when strain energy release rate modes are evaluated by using the finite crack extension via VCCT with an element size ranging between 0.25 to several ply thicknesses. This is because either the stress oscillatory nature no longer exists in these approaches or the choice of the reference length is already embedded in the calculation of the mode mixity by a VCCT method.

2.4 SUMMARY A comprehensive review of the most common or recently developed failure criteria for composites and adhesive is delineated in this chapter. The review provides a basis for future application and discussion in latter chapters on the evaluations of the static strength and damage tolerance of bonded repairs or bonded joints in composite airframe structures.

43

44

CHAPTER 2 Failure criteria

REFERENCES Adams, R., Harris, J., 1987. The influence of local geometry on the strength of adhesive joints. Int. J. Adhes. Adhes. 7, 69–80. Adams, R., Peppiatt, N., 1974. Stress analysis of adhesive-bonded lap joints. J. Strain Anal. Eng. Des. 9, 185–196. Adams, R., Comyn, J., Wakes, W., 1997. Structural Adhesive Joints in Engineering. Chapman & Hall, London. Benzeggagh, M., Kenane, M., 1996. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode benidng apparatus. Compos. Sci. Technol. 49, 439–449. Bogetti, T., et al., 2004. Predicting the nonlinear response and progressive failure of composite laminates. Compos. Sci. Technol. 64, 329–342. Bowden, P.B., Jukes, J.A., 1972. The Plastic Flow of Isotropic Polymers. J. Mater. Sci. 7, 52–63. Comninou, M., 1977. Interface crack. ASME J. Appl. Mech. 44, 631–636. Cuntze, R., Freund, A., 2004. The predictive capability of failure mode concept-based strength criteria for multidirectional laminates. Compos. Sci. Technol. 64, 343–377. Daridon, L., Cochelin, B., Potier-Feryy, M., 1997. Delamination and fiber bridging modeling in composite samples. J. Compos. Mater. 31, 874–888. Davidson, B., Hu, H., Schapery, R., 1995. An analytical crack tip element for layered elastic structures. ASME J. Appl. Mech. 62, 294–305. Davila, C., Camanho, P., Rose, C., 2005. Failure criteria for FRP laminates. J. Compos. Mater. 39, 323–345. Groth, H., 1998. Stress singularities and fracture at interface corners in bonded joints. Int. J. Adhes. Adhes. 8, 107–113. Hart-Smith, L., 1990. A scientific approach to composite laminate strength prediction. In: Presented to 10th ASTM Symposium on Composite Materials: Testing and Design, San Francisco, California, April 24-25. Hart-Smith, L., 1998. Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates. Compos. Sci. Technol. 58, 1151–1178. Hashin, Z., 1980. Failure criteria for unidirectional fiber composites. J. Appl. Mech. 47, 329–334. Irwin, G., 1957. Analysis of stresses and strains near end of crack traversing plate. ASME J. Appl. Mech. 24, 361–364. Jones, R., 1975. Mechanics of Composite Materials. McGraw-Hill, Washington, DC. Kaddour, A., Hinton, M., Soden, P., 2004. A comparison of the predictive capabilities of current failure theories for composite laminates: additional contributions. Compos. Sci. Technol. 64, 449–476. Kairouz, K., Mathews, F., 1993. Strength and failure modes of bonded single lap joints between cross-ply substrates. Composites 24, 475–484. Lapczyk, I., Hurtado, J., 2007. Progressive damage modeling in fiber-reinforced materials. Compos. Part A 38, 2333–2341. Lietchti, K., Freda, T., 1989. On the use of laminated beams for the determination of pure and mixed-mode fracture properties of structural adhesives. J. Adhes. 28, 145–169. Pinho, S., et al., 2005. Failure Models and Criteria for FRP Under In-plane or ThreeDimensional Stress States Including Shear Non-Linearity. NASA Langley Research Center, Hampton, VA. NASA/TM-2005-213530.

References

Puck, A., Schu¨rmann, H., 1998. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos. Sci. Technol. 58, 1045–1068. Raju, I., Crew, J.J., Aminpour, M., 1988. Convergence of strain energy release rate components for edge-delaminated composite laminates. Eng. Fract. Mech. 30, 383–396. Reedy, E., Guess, R., 1997. Interface corner failure analysis of joint strength: effect of adherend stiffness. Int. J. Fract. 88, 305–314. Rice, J., 1988. Elastic fracture mechanics concepts for interfacial cracks. ASME J. Appl. Mech. 55, 98–103. Rybicki, E., Kanninen, M., 1977. A finite element calculation of stress intensity factors by a modified crack closured integral. Eng. Fract. Mech. 9, 931–938. Soden, P., Hinton, M., Kaddour, A., 1998. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos. Sci. Technol. 58, 1225–1254. Soden, P., Kaddour, A., Hinton, M., 2004. Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos. Sci. Technol. 64, 589–604. Suo, Z., 1990. Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. Lond. A 427, 331–358. Tay, T., 2003. Characterization and analysis of delamination fracture in composites: an overview of developments from 1990 to 2001. ASME Appl. Mech. Rev. 56, 1–32. Tomblin, J., Yang, C., Harter, P., 2001. Investigation of Thick Bondline Adhesive Joints. Office of Aviation Research, Washington, DC. DOT/FAA/AR-01/33. Tong, L., Soutis, C., 2003. Recent Advances in Structural Joints and Repairs for Composite Materials. Kluwer Academic Publishers, Boston. Tong, L., Steven, G., 1999. Analysis and Design of Structural Bonded Joints. Kluwer Academic Publisher, Boston. Tsai, S., 1965. Strength characteristics of composite materials. NASA CR-224. Tsai, S., Wu, E., 1971. A general theory of strength for anisotropic materials. J. Compos. Mater. 5, 58–80. Wang, C., 1997. Fracture of interface cracks under combined loading. Eng. Fract. Mech. 56, 77–86. Wang, C., Chalkey, P., 2000. Plastic yielding in a film adhesive under multiaxial stresses. Int. J. Adhes. Adhes. 20, 155–164. Wang, C., Rose, L., 1997. Determination of triaxial stresses in bonded joints. Int. J. Adhes. Adhes. 17, 17–25. Wang, C.H., Rose, L., 2000. Compact solutions for the corner singularity in bonded lap joints. Int. J. Adhes. Adhes. 20, 145–154. Williams, M., 1959. The stress around a fault or crack in dissimilar media. Bull. Seismol. Soc. Am. 49, 199–204. Yang, Z., Sun, C., Wang, J., 2000. Fracture mode separation for delamination in platelike composite structures. AIAA J. 38, 868–874. Zinoviev, P., et al., 1998. Strength of multilayered composites under plane stress state. Compos. Sci. Technol. 58, 1209–1224.

45