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The use of interlayers in modeling interface crack propagation

Ninad Karkamkar, Biswarup Bose, Philip McLaughlin, Sridhar Santhanam *

Mechanical Engineering Department, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, United States

a r t i c l e i n f o

Article history:

Received 2 January 2008

Received in revised form 22 July 2008

Accepted 28 July 2008

Available online 3 August 2008

Keywords:

Interface fracture

Mixed-mode fracture

Delamination

Interlayer

a b s t r a c t

Delaminations are a common mode of failure at interfaces between two material layers

which have dissimilar elastic constants. There is a well-known oscillatory nature to the sin-gularity in the stress fields at the crack tips in these bimaterial delaminations, which cre-

ates a lack of convergence in the modewise energy release rates. This makes constructing

fracture criteria somewhat difficult. An approach used to overcome this is to artificially

insert a thin, homogeneous, isotropic layer (the interlayer) at the interface. The crack is

positioned in the middle of this homogeneous interlayer, thus modifying the original ‘bare’

interface crack problem into a companion ‘interlayer’ crack problem. Individual modes I

and II energy release rates are convergent and calculable for the companion problem

and can be used in the construction of a fracture criterion or locus. However, the choices

of interlayer elastic and geometric properties are not obvious. Moreover, a sound, consis-

tent, and comprehensive methodology does not exist for utilizing interlayers in the con-

struction and application of mixed-mode fracture criteria in interface fracture

mechanics. These issues are addressed here. The role of interlayer elastic modulus and

thickness is examined in the context of a standard interface fracture test specimen. With

the help of a previously published analytical relation that relates the bare interface crackstress intensity factor to the corresponding interlayer crack stress intensity factor, a suit-

able thickness and elastic modulus are identified for the interlayer in a bimaterial four-

point bend test specimen geometry. Interlayer properties are chosen to make the interlayer

fracture problem equivalent to the bare interface fracture problem. A suitable mixed-mode

phase angle and a form for the fracture criterion for interlayer-based interface fracture are

defined. A scheme is outlined for the use of interlayers for predicting interface fracture in

bimaterial systems such as laminated composites. Finally, a simple procedure is presented

for converting existing bare interface crack fracture loci/criteria into corresponding inter-

layer crack fracture loci.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

Interface fracture is a common mechanism of failure in many engineering applications including laminated fiber compos-

ites, thermal barrier coatings, and electronic packaging. A lot of attention has been devoted to the analytical characterization

of the mechanics of interface cracks as well as to the experimental determination of suitable fracture criteria. The analytical

mechanics of interface fracture in particular has received close attention [1–3]. Numerical estimation of suitable fracture

parameters for characterizing the stress field at interface crack tips has been the subject of numerous investigations [4–

11]. Experimental determination of critical fracture parameters using a variety of specimen geometries has also been an ac-

tive area of research [12–17].

0013-7944/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2008.07.006

* Corresponding author. Tel.: +1 610 519 7924; fax: +1 610 519 7312.

E-mail address: [email protected] (S. Santhanam).

Engineering Fracture Mechanics 75 (2008) 5087–5100

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c at e / e n g f r a c m e c h

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00137944

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The nature of the singularity at a bimaterial interface crack tip has been well documented [18,19]. The stresses and dis-

placements near the interface crack tip possess an oscillatory character quite unlike that of a crack in a homogeneous med-

ium. The near-tip asymptotic solution also predicts crack-face contact just behind the crack tip [2,20]. The normal and shear

stresses along the crack-line near the crack tip are coupled and the mode I and II stress intensity factors are not defined.

While the total strain energy release rate (SERR) is analytically defined, it cannot be broken into separable modes I and II

strain energy release rates (SERRs) [21].

Interface fracture is inherently mixed-mode in nature in spite of the difficulties associated with defining modes I and II

stress intensity factors or corresponding energy release rates. Characterizing the mixed-mode nature of the stress field with a

suitable mixed-mode parameter and developing suitable forms for mixed-mode fracture criteria has been the subject of sev-

eral investigations [1,2,22–24].

A technique that has been used frequently in the numerical calculation of fracture parameters for interface cracks is the

insertion of a thin, isotropic, homogeneous layer (interlayer) at the interface and the location of the crack within the inter-

layer, thus effectively converting the problem into a homogeneous crack problem [4,25–28]. This allows the modewise SERRs

to be calculated. The use of these interlayers has been justified particularly in laminated composites because of the physical

presence of a thin resin layer between adjacent plies [26]. In other instances, where such an interlayer is not physically pres-

ent, the interlayer has been artificially introduced primarily as a convenience that allows the calculation of individual mode

(I and II) energy release rates or stress intensity factors [4].

From a study of the literature, it appears that a theoretical foundation for the use of an interlayer has not been established.

No criteria have been established for the choices of interlayer elastic and geometric properties for modeling delamination

fracture. Such criteria would be useful in calculation of modewise SERRs for bimaterial interface cracks with the desired

accuracy that can be used in mixed-mode fracture criteria for efficient design of composite and other bimaterial structures.

In what follows, an attempt is made to develop guidelines for making appropriate choices of interlayer properties and for the

effective use of interlayers in interface fracture mechanics.

A brief overview of interface fracture mechanics in Section 2 is followed by a discussion on the use of an interlayer in

Section 3. A process to select a suitable thickness and elastic modulus for the interlayer is outlined with the help of an exam-

ple problem of a four-point bimaterial bend specimen in Section 4. A systematic and consistent approach for the use of inter-

layers in dealing with the interface fracture problem is outlined in Section 5. In Sections 5 and 6, a methodology is presented

for utilizing interlayers in constructing and applying mixed-mode fracture criteria for interface cracks.

2. Mixed-mode interface fracture

For a crack tip located within a homogeneous, isotropic, elastic solid, with elastic modulus E and Poisson’s ratio m (see

Fig. 1) under a plane stress or plane strain loading, the stress fields in the vicinity of the tip have the following well-known

form [2]:

Nomenclature

E elastic modulusG total strain energy release rate

GI; GII modes I and II strain energy release ratesi

ffiffiffiffiffiffiffi

1p

K I homogeneous mode I stress intensity factor

K II homogeneous mode II stress intensity factorK Ic; K IIc critical modes I and II fracture toughness for cracks in homogeneous mediaK ¼ K 1 þ iK 2 complex interface stress intensity factor

l characteristic length parameter for interface cracksr ; h polar coordinates

a; b Dundurs parameters

bimaterial parameter

l shear modulus

m Poisson’s ratio

w mixed-mode loading phase angle for homogeneous cracks

w mixed-mode loading phase angle for interface cracksh total thickness of a bimaterial beam or strip

h1; h2 thickness of individual material layers in a bimaterial beam

hin

thickness of interlayer introduced into a bimaterial beamrmn stress tensor

ð Þin subscript ‘in’ indicates a quantity associated with the interlayer

ImðÞ; ReðÞ imaginary and real parts of a complex quantity

C function describing mixed-mode fracture

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rmn ¼ K Ið2pr Þ1=2rImnðhÞ þ K IIð2pr Þ1=2rII

mnðhÞ ð1Þ

where r and h are the polar coordinates centered at the crack tip, and K I and K II are the mode I and II stress intensity factors.

The h dependencies, rImn and r II

mn, are well-known [29].

The total SERR, G, is related to the stress intensity factors via:

G ¼ ðK 2I þ K 2IIÞ=E ð2Þ

where

E ¼ E =ð1 m2Þ for plane strain

E for plane stress

( ð3Þ

There is a clear separation between modes I and II in the calculation of the energy release rate with the modes I and II strain

energy release rates given by

GI ¼ K 2I =E

GII ¼ K 2II=E ð4Þ

There are a wide variety of numerical and analytical techniques for computing stress intensity factors (and energy release

rates) for such cracks in a homogeneous medium. There is also a lot of experimental data on critical conditions for crack

propagation under mixed-mode loading in a homogeneous medium. Several empirical correlations have been developed

as mixed-mode fracture criteria for cracks in homogeneous media relating stress intensity factors or energy release rates.

A common criterion used is the empirical elliptical criterion [30].

K IK Ic

2

þ K II

K IIc

2

¼ 1 ð5Þ

where the subscript c refers to critical values of the stress intensity factors.An alternative, but equivalent, manner of expressing mixed-mode fracture criteria [2] in homogeneous media is to define

a mixed-mode loading phase angle w and to then express the total energy release rate G at fracture as a function of the angle

w.

G ¼ CðwÞ ð6Þ

The mixed-mode loading phase angle w for cracks in homogeneous media is defined as

w ¼ tan1 K II

K I

ð7Þ

The function C in Eq. (6) is usually determined empirically.

Interface crack problems are more complex. The stress field at the tip of an interface crack ( Fig. 2) is given by

rmn ¼ Re½Kr i

ð2pr Þ1=2

rImnðh; Þ þ Im½Kr

ið2pr Þ

1=2

rIImnðh; Þ ð8Þ

Fig. 1. A semi-infinite crack in a homogeneous, isotropic, elastic, infinite two-dimensional medium.

N. Karkamkar et al. / Engineering Fracture Mechanics 75 (2008) 5087–5100 5089

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where K is the complex interface stress intensity factor, i ¼ ffiffiffiffiffiffiffi

1p

, and r and h are polar coordinates as shown in Fig. 2. The

functions rImn and rII

mn can be found in [18,19]. The bimaterial parameter is

¼ 1

2p ln

1 b

1 þ b

ð9Þ

b is the classical Dundurs parameter along with a.

a ¼

l1ðj2 þ 1Þ l2ðj1 þ 1Þ

l1ðj2 þ 1Þ þ l2ðj1 þ 1Þ

b ¼ l1ðj2 1Þ l2ðj1 1Þ

l1ðj2 þ 1Þ þ l2ðj1 þ 1Þ

ð10Þ

with l j ð j ¼ 1; 2Þ being the shear moduli and j j ¼ 3 4m j for plane strain and j j ¼ ð3 m jÞ=ð1 þ m jÞ for plane stress, m j being

the Poisson’s ratios.

The complex stress intensity factor, K ¼ K 1 þ iK 2, has both real and imaginary parts, K 1 and K 2, respectively. The patho-

logical nature of the singularity at the interface crack tip has been well discussed in the literature. The stresses in the vicinity

of the crack tip show an oscillatory singularity and there is also interpenetration of the crack faces just behind the crack tip.

Moreover, the normal and shear stress fields ahead of the crack tip are coupled and are modulated by the magnitudes of both

K 1 and K 2. Modes I and II energy release rates show non-convergence, though the total energy release rate G is convergent.

This lack of definition of the mode I and II energy release rates has been a problem for the fracture mechanics community as

it seeks to establish mixed-mode fracture criteria for interface fracture. A mixed-mode fracture criterion such as in Eq. (5),

developed for cracks in homogeneous media, is not applicable for interface cracks since modes I and II stress intensity factorsand energy release rates are not definable for interface cracks.

A key to describing the fracture of bimaterial interfaces is to obtain or define a measure of the mixed-mode character of

the applied loading, equivalent to w in Eq. (7). This would enable a mixed-mode fracture criterion definition of the kind illus-

trated in Eq. (6) to be applicable to interface fracture. Rice [1] provided a definition that has found traction in the fracture

mechanics community. He defined a loading phase angle w with the help of a ‘characteristic’ length l.

w ¼ tan1 ImðK liÞ

ReðK liÞð11Þ

The choice of the length l is arbitrary, but it is usually chosen to be a value corresponding to a material length scale such as

the size of the fracture process zone ahead of a crack tip. Once the mixed-mode nature of the loading is quantified with the

parameter w, the fracture toughness as measured by the total energy release rate G at fracture, can be expressed as a function

of the mixed-mode parameter w.

G ¼ Cðw; lÞ ð12Þ

where the dependence of the function on the choice of the characteristic length l is explicitly indicated. This approach has

been used quite extensively [12,14–17]. For example, Wang and Suo [12] have experimentally determined the interface

toughness curves for several interfaces including aluminum/epoxy, brass/epoxy, and plexiglass/epoxy, using a single spec-

imen geometry, the Brazil Nut Sandwich.

Fig. 2. The interface crack problem: a semi-infinite interface crack between two distinct semi-infinite material planes.


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3. Interlayer-based interface fracture mechanics

An alternative approach to dealing with interface fracture initiated with the work of Atkinson [25]. Atkinson [25] pro-

posed that the ‘bare’ interface crack could be modeled as a crack enclosed within a thin elastic, homogeneous layer. This thin

interlayer is introduced at the interface, thus separating the original two materials forming the interface (Fig. 3). Since the

crack is located within the homogeneous thin layer, calculation of modes I and II stress intensity factors and corresponding

strain energy release rates is both theoretically and practically feasible. Using finite element (FE) analysis and the modified

crack closure (MCC) method [31], individual modes I and II strain energy release rates (SERR) can be calculated.There have been several numerical/analytical studies related to the use of an interlayer in interface fracture [4,11,26–

28,32] since Atkinson’s work [25]. For instance, Chakraborty and Pradhan [27] investigated the influence of an interfacial

resin layer on delamination initiation in broken ply graphite/epoxy composite laminates. They performed a full 3D finite ele-

ment analysis with each layer of the laminate modeled as homogeneous and orthotropic. They concluded that delamination

initiation was a mixed-mode phenomenon and the dominance of the mode of delamination was governed by the resin layer

stiffness, thickness, and lamina orientation at the interface. Davidson and Hu [28] investigated the effect of an adhesive

interlayer on the mode ratio and total energy release rate for delaminations in laminated composites. Two different geom-

etries were considered and for both geometries, it was found that the fracture mode ratio was extremely sensitive to inter-

layer modulus, whereas total energy release rate was not. The trends in mode ratio vs. resin modulus, i.e., increasing or

decreasing mode ratio with increasing modulus were found to be dependent on remote loading.

Mathews and Swanson [26] studied the effect of interlayer thickness, and to a very limited extent the interlayer modulus,

on numerically calculated modes I and II SERRs for an interlayer crack in an unsymmetrical composite laminate in two dif-

ferent specimen geometries. They found that as the interlayer thickness shrank, the total SERR for the interlayer crack prob-lem approached the total SERR for the ‘bare’ interface (no interlayer) crack problem. Other studies that have investigated the

use of an interlayer to calculate fracture parameters for the interface crack problem include Raju et al. [4] and Dattaguru et al.

[11].

There are several unaddressed issues in the literature related to the use of interlayers in interface fracture. Several mate-

rial systems with interfaces possess a natural thin interlayer at the interface. For example, some laminated composites have

a thin resin rich layer that separates the plies [26]. However, this resin rich layer is not necessarily of uniform thickness. Its

elastic properties are also not readily apparent. Several other material systems, such as coatings on a substrate, do not pos-

sess a clearly defined physical interlayer. Artificially introducing an interlayer into these systems changes the nature of the

problem as the compliance of the interlayer will be different from the material it replaces. A theoretical framework does not

exist that provides justification for the use of an artificial interlayer for interface crack problems where a real interlayer does

not exist. Criteria for selecting interlayer elastic properties (modulus and Poisson’s ratio) and thickness are not clearly estab-

lished in the literature.

A recent theoretical result [10] provides a foundation for developing a framework for the use of interlayers in interfacefracture problems. In [10], the problem of Fig. 2 is termed the ‘bare’ interface crack problem. Fig. 3 with an interlayer located

at the interface, is labeled the ‘companion’ interlayer crack problem. The interlayer is linear-elastic, homogeneous, and iso-

tropic. The modes I and II energy release rates as well as the corresponding stress intensity factors are well-defined for the

companion problem and are calculable by either analytical or numerical means. For a thin interlayer, with thickness much

smaller than any other length dimension in the companion problem (and also the original ‘bare’ interface crack problem), an

analytical, universal relation exists that ties the complex stress intensity factor of the interface crack problem, K ¼ K 1 þ iK 2,

and the stress intensity factors (K I and K II) of the companion problem. The relationship relies on quantities that depend on

Material 1

Material 2

Interlayer

hin

x1

x2

Fig. 3. The interlayer crack problem: a semi-infinite crack located within a thin, elastic, homogeneous layer between two distinct semi-infinite materialplanes.


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the elastic properties of all three materials involved (the two materials that form the ‘bare’ interface and the interlayer mate-

rial) as well as the thickness of the interlayer, h in, in Fig. 3. The universal relation has the form [10]:

K I þ iK II ¼ qei/K ðhinÞi ð13Þ

Here is the bimaterial parameter of Eq. (9). The parameter q is defined by [10]:

q2 ¼ 1

2ð1 b2

oÞ 1 au

1 þ au

þ1 al

1 þ al ð14Þ

The non-dimensional material parameters au, al, and bo that appear in Eq. (14) are Dundurs elastic mismatch parameters. All

pertinent Dundurs parameters for the problem of Fig. 3 are defined as follows:

bo ¼ 1

2

ð1 2m2Þ=l2 ð1 2m1Þ=l1

ð1 m1Þ=l1 þ ð1 m2Þ=l2

ð15Þ

au ¼ ð1 minÞ=lin ð1 m1Þ=l1

ð1 m1Þ=l1 þ ð1 minÞ=lin

bu ¼ 1

2

ð1 2minÞ=lin ð1 2m1Þ=l1


ð16Þ

al ¼ ð1 minÞ=lin ð1 m2Þ=l2


bl ¼ 1

2

ð1 2minÞ=lin ð1 2m2Þ=l2


ð17Þ

Elastic constants m1 and l1 are the Poisson’s ratio and shear modulus, respectively, for material 1. Similar definitions apply

for the other elastic constants seen in Eqs. (15)–(17). The parameter / in Eq. (13) is a function of the Dundurs parameters

au; bu; al; bl and is numerically calculated by solving a boundary value problem [10]. Tabulated values for / as a function

of the four Dundurs parameters are provided in [10].

The universal relation in Eq. (13) provides the basis for constructing a comprehensive methodology for using interlayers

in a consistent and sound manner in interface fracture mechanics. Such a methodology has not been described in the liter-

ature, even though several research papers have utilized interlayers in interface crack problems. The methodology includes

the calculation of interlayer-based interface crack fracture parameters, constructing interlayer-based interface fracture cri-

teria/loci, and transferring and applying the fracture criteria/loci to describe fracture in structural interface cracks.

On the issue of using interlayers in a consistent and sound manner for calculating interface crack fracture parameters, it is

proposed here that the interlayer modulus and thickness must be chosen such that Eq. (13) can be used to convert the result-

ing interlayer crack mode I and II SERRs ( GI and GII) into the complex stress intensity factor for the corresponding bare inter-

face crack. For this, the interlayer thickness must be small compared to other physical dimensions of the problem and the

total SERRs of the two problems (bare interface crack and the companion interlayer crack) must be nearly identical. The ac-

tual values of the interlayer thickness and modulus that achieve this are problem dependent. The process by which the inter-

layer properties are determined are shown next for an example test specimen geometry of a four-point bimaterial bend

specimen.

4. Interlayer for a bimaterial four-point bend specimen

Charalambides et al. [13] have developed a standard test specimen geometry for characterizing interface fracture for

bimaterials. Their bimaterial notched four-point bend fracture specimen is shown in Fig. 4. A delamination (interface crack)

extends out symmetrically from the central, vertical notch. They obtained an analytical expression for the total SERR ðGbareÞfor this geometry using Euler–Bernoulli beam theory (see Appendix Eq. (A.1)).

2LP/2b

P/2b

h1

h2

h

l

Material 1

Material 22a

Fig. 4. Bimaterial four-point bend ‘bare’ interface crack problem of Charalambides et al. [13].


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Total strain energy release rate ðGbareÞ for the geometry in Fig. 4 can also be computed accurately using the finite element

(FE) method. However individual modes I and II cannot be determined either analytically or numerically since this is an

interface crack [21]. In order to allow calculation of individual mode SERRs, a very thin, elastic interlayer is inserted at

the interface of the bimaterial four-point bend specimen (see Fig. 5). The crack is located in the middle of this interlayer.Thus a companion interlayer crack problem is created in Fig. 5 to the original bare interface crack problem of Fig. 4. An ana-

lytical expression for the total SERR ðGinÞ for the companion problem of Fig. 5 has been determined in the present work using

Euler–Bernoulli beam theory, assuming plane strain (see Appendix Eq. (A.3)). The introduction of the interlayer for the com-

panion problem facilitates the numerical calculation of the individual mode (modes I and II) energy release rates, GI and GII.

GI and G II will depend on the choice of interlayer elastic modulus and thickness.

To calculate GI and GII, the bimaterial notched four-point bend specimen with an isotropic interlaminar layer (Fig. 5) was

modeled using the FE software ABAQUS, under plane strain conditions. The effect of interlayer properties on total and mode-

wise energy release rates was examined. Interlayer material elastic modulus and thickness were varied. Modewise and total

SERRs were computed using the modified crack closure (MCC) method [31]. Numerical total energy release rate results were

compared with the predictions of Eqs. (A.1) and (A.3). When convergence is achieved in the FE simulations of the four-point

bend specimen with the interlayer, the numerical total SERR should be comparable to the predictions of Eq. (A.3), since this

equation is a result of modeling the four-point bend specimen with the interlayer. However, an objective of this exercise is to

identify suitable interlayer properties that would ensure a numerical (FE based) total SERR equal or close to that predictedanalytically by Eq. (A.1), which was determined by Charalambides et al. [13] for the bare interface problem. This would en-

sure that the overall mechanics of crack propagation are similar for the two problems. For engineering purposes, a difference

of less than 2% between the total energy release rates of the two problems would be considered acceptable. The scatter in

experimentally determined fracture toughness results can often exceed this metric.

Elastic and geometric properties for the upper (1) and lower (2) material layers were chosen to resemble those of com-

posite plies. Both material layers (1 and 2) were chosen to be of equal thickness, equal to a standard ply thickness

ðh1 ¼ h2 ¼ 0:125 mmÞ of a T300/5208 graphite/epoxy (Gr/Ep) composite laminate. The four-point bend specimen used for

the study had a length 2L ¼ 8 mm, dimension l = 1 mm, and half-crack length a = 1 mm (see Fig. 5). h is the total thickness

of the beam. Since the analyses leading to Eqs. (A.1) and (A.3) are limited to isotropic properties, the full orthotropic elastic

constant set of T300/5208 Gr/Ep was not used. To get the effect of widely different material stiffnesses, one isotropic material

(termed ‘‘Quasi 0”) was assumed to have a Young’s modulus equal to that of T300/5208 Gr/Ep in the axial (fiber) direction,

and the other isotropic material (termed ‘‘Quasi 90”) had its Young’s modulus set to that of the transverse modulus of a uni-

directional T300/5208 Gr/Ep ply. The effect of the interlayer modulus was explored by varying the Young’s modulus of theinterlayer, E in. Poisson’s ratio of 0.30 was assumed for all layers. Table 1 gives the numerical values for elastic properties

used.

In addition to varying interlayer elastic modulus, multiple interlayer thicknesses were studied. An analysis of stress and

deformation patterns in multilayer laminates by Pipes and Pagano [33] has shown that much of the shearing deformation

between layers occurs in a region encompassing the interface having a thickness of about one-tenth the layer thickness.

2LP/2b P/2b

h1

h2

h

l

Material 1

Material 22a

Interlayer hin

Fig. 5. Model of companion problem for Fig. 4 with an interlayer inserted at the interface.

Table 1

Elastic properties of layer materials for the problem of Fig. 5

Layer Young’s modulus (GPa) Poisson’s ratio

Quasi 0 E 0 ¼ 180 m0 ¼ 0:30

Quasi 90 E 90

¼10:3 m90

¼0:30

Interlayer E in ¼ variable min ¼ 0:30


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Mathews and Swanson [26] cite evidence for interlayer thicknesses of the order of 1/100 of material layer thicknesses (h1 or

h2). Hence, interlayer thickness to material layer thickness ratios ðhin=h1Þ between 0.1 and 0.01 were considered in this study.

Two material configurations were analyzed. The first configuration, termed a Quasi-[0/90] layup, had layer 1 (top) with

Quasi 0 properties and layer 2 (bottom) with Quasi 90 properties. In the second configuration, termed Quasi-[90/0], layer 1

(top) had Quasi 90 properties and layer 2 (bottom) had Quasi 0 properties. Fig. 6 shows the effect of varying elastic modulus

and thickness of the interlayer on the total SERR for the crack in the companion problem of Fig. 5. The interlayer SERRs were

determined numerically (FE). The bare interface SERR results in Fig. 6 (horizontal lines) were computed using Eq. (A.1). All

SERRs were non-dimensionalized according to Eqs. (A.1) and (A.3). For both configurations, Quasi-[0/90] and Quasi-[90/0], it

is seen that the total SERR depends on both elastic modulus and thickness of the interlayer. For the total SERR of the inter-

layer based companion problem to be within 2% of the SERR of the bare interface problem for both material configurations,

the results in Fig. 6 indicate that the interlayer thickness ratio, hin=h1, should be 0.01 (or less), and the interlayer elastic mod-

ulus should be close to that of the more compliant material (Quasi 90). Hence in subsequent analysis to assess modes I and IISERRs, GI and GII, the interlayer thickness ratio, hin=h1, was chosen to be 0.01 and the interlayer modulus, E in, was varied from

that of the Quasi 90 to an order of magnitude lower.

Fig. 7 shows the effect of varying the elastic modulus of the interlayer on modes I and II SERRs of the crack in the com-

panion problem (Fig. 5) for the Quasi-[0/90] configuration. The interlayer thickness ratio, hin=h1, is 0.01. GI and GII are com-

0 1 2 3 4 5 6 7

x 104

4

6

8

10

12

14

16

18

Interlayer modulus, MPa

N o n − d i m e n s i o n a l S E R R

SERR: [0/90]; hin

=1% h1

SERR: [0/90] Bare Interface

SERR: [0/90]; hin

=10% h1

SERR: [90/0]; hin

= 1% of h1

SERR: [90/0]; hin

= 10% of h1

SERR: [90/0] Bare Interface

Fig. 6. Effect of elastic modulus and thickness of interlayer on SERR for [0/90] and [90/0] four-point bend specimens of Fig. 5.

0 2 4 6 8 10 122

4

6

8

10

12

Elastic modulus of interlayer, GPa

N o n d i m e n s i o n a l S t r a i n E n e r g y R e l e a s e R a t e

FE GI

FE GII

FE Gin

Analytical Gin

Analytical Gbare

Fig. 7. Effect of interlayer modulus on energy release rates for interlayer crack in the Quasi-[0/90] layup: interlayer to material layer thickness ratio = 0.01.


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puted numerically. It is observed that, with the reduction of interlayer modulus, the FE mode I SERR ðGIÞ increases, FE mode II

SERR ðGIIÞ decreases. Comparison of the numerically computed total SERR, Gin, with the analytical interlayer based total SERR

(Eq. (A.3)) shows that the numerical results for all cases are within a tenth of one percent of the analytical results. Clearly, the

FE analysis is producing very accurate results.

For the second configuration, Quasi-[90/0], the effect of interlayer modulus on modewise SERRs is shown in Fig. 8. Again,

FE and analytical results for G in are virtually identical. Gin is almost independent of interlayer modulus, E in, for the range of

values considered. Analysis was also conducted to determine the influence of the interlayer thickness ratio, hin=h1, on GI and

GII. These results are shown in Fig. 9 for the Quasi-[90/0] configuration. With the reduction of interlayer thickness, both G I

and G II decrease. In Fig. 9, the interlayer modulus was set equal to the modulus of the compliant layer.

Figs. 7 and 8 indicate that choosing the interlayer thickness ratio, hin=h1, to be 0.01 and the interlayer modulus close to the

modulus of the compliant layer ensures that the total SERR of the companion problem of Fig. 5 is within 2% of the bare inter-

face problem of Fig. 4. Individual mode SERRs, G I and G II, are still dependent on the choice of the interlayer modulus. Thequestion of choosing an appropriate value for the interlayer modulus is addressed next with the help of Eqs. (4) and (13).

Using Eqs. (4) and (13), the modes I and II SERRs of Fig. 8 for the companion interlayer crack problem (Fig. 5) can be con-

verted to interface stress intensity factors for the equivalent bare interface crack problem (Fig. 4). These results are shown

0 2 4 6 8 10 120

1

2

3

4

5

6

Elastic modulus of interlayer, GPa

N o n d i m e n s i o n a l S t r a i n E

n e r g y R e l e a s e R a t e

FE GI

FE GII

FE Gin

Analytical Gin

Analytical Gbare

Fig. 8. Effect of interlayer modulus on energy release rates for interlayer crack in the Quasi-[90/0] layup: interlayer to material layer thickness ratio = 0.01.

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

Interlayer thickness/Ply thickness

N o n − d i m e n s i o n a l S t r a i n E n e r

g y R e l e a s e R a t e

FE GI

FE GII

FE Gin

Analytical Gin

Analytical Gbare

Fig. 9. Effect of interlayer to material layer thickness ratio on energy release rates for interlayer crack in the Quasi-[90/0] layup: E in ¼ 10:3 GPa.


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in Table 2. The calculated bare interface stress intensity factors (K ) are presented in a non-dimensional form, (Khiðbh

3=2=PlÞ),

which makes it suitable for comparison with the numerical results of Charalambides et al. [13]. As can be seen from Table 2,

the interlayer method produces reasonably consistent values for the calculated interface stress intensity factors, largely

independent of the value of the interlayer modulus E in. The last row in Table 2 provides a comparable numerical result from

Charalambides et al. [13]. This result was obtained by Charalambides et al. [13] using a numerical method distinct from the

interlayer method used here. Hence, it provides an independent check on the effectiveness of the interlayer based results in

the first four rows of Table 2.

Similarly, for the Quasi-[0/90] configuration, the modes I and II SERRs of Fig. 7 for the companion interlayer crack problem

can be converted to interface stress intensity factors for the equivalent bare interface crack problem. These results are shown

in Table 3. In Table 3, the interlayer method produces consistent values for the bare interface stress intensity factor, K , for all

interlayer modulus E in values considered, except for the case E in ¼ 1:0 GPa. This deviant result for E in ¼ 1:0 GPa is to be ex-

pected as the compliance of the interlayer is made significantly larger than both material layers. Unfortunately, an indepen-

dent check for the interlayer based K reported in Table 3 is not available from Charalambides et al. [13], since they did not

consider E 2=E 1 ratios less than 1. However, the results in Table 3 appear to be consistent with the trends exhibited by the

results in [13].

For both configurations it is clear that even though the modes I and II SERRs for the interlayer crack problem vary with

interlayer modulus, the corresponding bare interface stress intensity factor, K 1 þ iK 2, calculated from the numerical G I and

GII, is independent of the interlayer modulus (except for the very compliant value of E in ¼ 1 GPa). Although previous work on

interlayers in the literature [4,26,27,34] has shown that modes I and II energy release rates varied with interlayer thickness

and modulus, no clear answer was available regarding the appropriate thickness and modulus to use. The results in Tables 2

and 3 show that, for the bimaterial four-point bend specimen, if the interlayer is sufficiently thin (0.01 of smallest physical

dimension in problem), choosing any interlayer modulus reasonably close to the compliant layer modulus will ensure that

the interlayer does not perturb the original bare interface problem. These choices for the interlayer modulus and thickness

pertain only to the bimaterial four-point bend problem considered. For other bimaterial problems, a similar analysis would

have to be conducted to establish suitable values for the interlayer modulus and thickness.

For appropriately chosen interlayer thickness and modulus, the bare interface crack problem is completely equivalent to

the corresponding companion interlayer crack problem, since the overall energy release rates of the two problems are about

the same (within 2%) and the stress intensity factors of the interface crack problem can be computed from that of the com-

panion problem using the universal relation, Eq. (13). Once interlayer-based fracture parameters can be calculated in a con-

sistent and sound manner, interlayer-based interface fracture criteria/loci can be formulated and applied to solve structural

interface crack problems. This is described next.

5. An approach for interface fracture prediction using interlayers

Fracture mechanics relies on the use of fracture parameters to characterize fracture (mixed-mode or otherwise) in labo-

ratory test specimens and the transferability of the parameters and the results from these test specimens to larger (or smal-

ler) scale practical structures. Fracture criteria are usually constructed for bimaterial interfaces by utilizing laboratory test

Table 2

Non-dimensionalized bare interface stress intensity factor K calculated from modes I and II SERRs of interlayer crack problem, using universal relation of Eq.

(13)

E in (GPa) FE GI (N/mm) FE GII (N/mm) ReðKhiÞ h3=2b

Pl ImðKhiÞ h3=2b

Pl

1.0 4.23 0.85 0.48 0.60

3.0 4.59 0.49 0.50 0.58

6.5 4.75 0.32 0.49 0.59

10.3 4.81 0.26 0.50 0.58

Bare (no interlayer) [13] 0.50 0.57

Quasi-[90/0] configuration, hin=h1 ¼ 0:01, ¼ 0:083. The last row provides an independent numerical result from Charalambides et al. [13] for comparison.

Table 3

Non-dimensionalized bare interface stress intensity factor K calculated from modes I and II SERRs of interlayer crack problem, using universal relation of Eq.

(13)

E in (GPa) FE GI (N/mm) FE GII (N/mm) ReðKhiÞ h3=2 b

Pl ImðKhiÞ h3=2b

Pl

1.0 4.23 7.53 4.38 2.15

3.0 3.05 8.66 4.27 2.30

6.5 2.46 9.20 4.30 2.27

10.3 2.20 9.40 4.28 2.28

Quasi-[0/90] configuration, hin=h1 ¼ 0:01, ¼ 0:083.


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specimens such as the bimaterial four-point bend specimen of Charalambides et al. [13] (see Fig. 4) and, as described by Rice

[1], take the form of Eq. (12). These criteria are then used to predict fracture in practical structures prone to failure due to

bimaterial interface cracks.

An approach is advocated here for interface fracture mechanics that relies on the use of interlayers. In the analytical/

numerical models used to determine the fracture parameters for the laboratory test specimens, a thin interlayer is intro-

duced creating companion interlayer problems. Modes I and II SERRs, GI and GII, are computed for these companion problems

along with the total energy release rate, Gin. Corresponding modes I and II stress intensity factors, K I and K II are computed

from GI and G

II using Eq. (4). A mixed-mode parameter is defined in the following manner:

win ¼ K II

K I

in

ð18Þ

The subscript, in, refers to the use of the interlayer. Laboratory testing coupled with the results from the analysis of these

companion problems can be used to generate a fracture criterion of the form:

Gin ¼ Cinðwin; E in;hinÞ ð19Þ

The presence of E in and hin in Eq. (19) indicates the dependence of the form of the function Cin on the specific values chosen

for the interlayer modulus, E in, and thickness, hin. G in in Eq. (19) represents the total energy release rate at failure.

The fracture criterion of Eq. (19) is used to predict bimaterial interface crack propagation in real, practical structures.

Again, in the analytical/numerical model used to calculate fracture parameters for the structure, an artificial interlayer is

deliberately introduced. The key here is that the choice of the interlayer modulus, E in, and thickness, h in, for the model of

the structure must be identical with the E in and hin used for the test specimen modeling that generated Eq. (19). Doing this

ensures the comparability of the stress fields in the vicinity of the crack tips of the companion interlayer problems for thetest specimen and the practical structure. If the interlayer properties for the structure are chosen to be different from that of

the test specimen, then the results, in the form of Eq. (19), generated from the laboratory testing of specimens, cannot be

applied to assess interface crack propagation in the structure.

This interlayer approach is equivalent to the approach of Rice [1]. In Rice’s approach, the fracture criterion (Eq. (12)) is

developed from laboratory testing of specimens and is based on the choice of a characteristic material length scale l. For this

fracture criterion to be transferable to structural interface crack problems, an identical value of l must be used to assess the

mode-mix parameter, w, for the interface crack in the structure [1]. The use of identical l for the test specimen and the struc-

ture ensures the comparability of the local stress states at the interface crack tips of the two problems. Similarly, for the

interlayer approach, the use of identical interlayer modulus, E in, and thickness, hin, in the companion problems for the test

specimen and the structure ensures the comparability of the local stress states of the two problems.

The universal relation of Eq. (13) [10] provides the theoretical justification for the use of the interlayer approach, symbol-

ized by Eqs. (19) and (18). For suitably chosen interlayers, fracture parameters, GI and GII, calculated for the companion inter-

layer crack problem can be transformed into the fracture parameters, K 1 and K 2, for the corresponding bare interface crackproblem, using Eqs. (13) and (4). This was shown in Tables 2 and 3. Hence a Gin and win calculated for a companion interlayer

crack problem can be converted to a G and w for the corresponding bare interface crack problem, for preset choices of inter-

layer properties, E in and hin, and the Rice length l. Therefore, the interlayer based fracture criterion of Eq. (19) is completely

equivalent to the Rice fracture criterion of Eq. (12). In fact, given a fracture locus of the form Eq. (12), it is relatively easy to

convert it to an equivalent interlayer based fracture locus of the form Eq. (19). This is shown next.

6. Constructing interlayer-based interface fracture criteria/loci

Several standard fracture specimens have been devised to measure the fracture resistance of bimaterial interfaces. For

instance, Wang and Suo [12] have used a modified version of the Brazilian Disk test to determine energy release rate, G,

at fracture as a function of loading phase angle for a variety of interfaces. These fracture loci were constructed using the ap-

proach of Rice [1], by defining the loading phase angle, w, utilizing a fixed material length scale, l. The fracture locus for an

aluminum–epoxy interface (from [12]) is shown in Fig. 10 as a dashed line. The loading phase angle, w, for this locus was

calculated by Wang and Suo [12] using Eq. (11), with l ¼ 100 lm.

To obtain the toughness curve for an interlayer based approach, an interlayer of thickness hin is assumed to exist at the

aluminum (E = 70 GPa, m = 0.35)–epoxy (E = 4 GPa, m = 0.34) interface, thereby creating the companion interlayer crack prob-

lem. A thickness of hin ¼ 20 lm is chosen which is more than an order of magnitude smaller than the smallest physical

dimension of the Brazilian Disk Sandwich of Wang and Suo [12]. The modulus E in of the interlayer is chosen to be 5.0 GPa.

The universal relation, Eq. (13), relates the stress intensity factor of the interlayer crack problem with the stress intensity

factor of the interface crack problem. With a little algebra, it can be shown that the phase angle win for the interlayer problem

(Eq. (18)) is related to the phase angle w (Eq. (11)) by

win ¼ w þ / þ ln hin

l

ð20Þ

As indicated in Section 3, / is obtained from tabulated results in Santhanam [10], and is a function of the Dundurs

parameters. Since the interlayer thickness is extremely small, it is assumed that the energy release rate Gin for the inter-


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layer problem is identical with the energy release rate G for the bare interface crack problem. The plot of the ‘interlayer’

fracture locus (Gin vs. win) for the aluminum–epoxy interface is shown as a solid line in Fig. 10. What is clearly seen is

that the interlayer fracture locus is phase shifted with respect to the bare interface crack fracture locus. The overall

shape remains the same. A very similar transformation can be performed of bare interface fracture loci results in the

literature such as in [15] where the interface fracture toughness of a glass epoxy system is reported based on tests con-

ducted with the Brazilian Disk specimen. An l ¼ 600 lm was used by Banks-Sills et al. [15]. The ‘interlayer’ fracture locus

(Gin vs. win with E in ¼ 5:0 GPa, min ¼ 0:3, hin ¼ 100 lm) for the glass–epoxy interface is shown as a solid line in Fig. 11

while the bare interface locus of Banks-Sills et al. [15] is shown as a dashed line. The overall shape of the locus remains

the same, but the locus is ‘phase-shifted’.

Thus, it is relatively easy to transform existing fracture loci in the literature for bare interface cracks to corresponding locifor ‘interlayer’ cracks, for an assumed interlayer thickness hin and modulus E in.

−0.5 0 0.5 1 1.5 20

10

20

30

40

50

60

70

Phase Angle (radians)

I n t e r f a c e T o u g h n e s s ,

G ( J / m 2 )

Interlayerbased Toughness Curve

Bare Interface Results from [12]

Fig. 10. Fracture toughness as a function of phase angle for an aluminum/epoxy interface. Toughness results of Wang and Suo [12] are reported as a

function of angle ^w. The ‘interlayer’ locus uses angle

^win. E in ¼ 5:0 GPa, min ¼ 0:30, and h in ¼ 20 lm.

−1.5 −1 −0.5 0 0.5 1 1.5 20

5

10

15

20

25

30

Phase Angle, radians

C r i t i c a l E n e r g y R e l e a s e R a t e ,

( N / m )

Bare Interface Results from [15]

Interlayerbased Toughness Curve

Fig. 11. Fracture toughness as a function of phase angle for a glass/epoxy interface. Toughness results of Banks-Sills et al. [15] are reported as a function of angle w. The ‘interlayer’ locus uses angle win. E in ¼ 5:0 GPa, min ¼ 0:3, and hin ¼ 100 lm.


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7. Conclusions

Delaminations are a common mode of failure at interfaces between two material layers which have dissimilar elastic con-

stants. It is well known that total energy release rates are well defined for such interface fracture problems, but individual

modes (I and II) do not exist. Hence fracture criteria constructed with the help of modes I and II energy release rates (GI and

GII), for mixed-mode crack propagation of cracks in homogeneous media appear not to be feasible for bimaterial delamina-

tions. Instead, a common approach chosen by the fracture mechanics community is to construct the fracture locus as the

total energy release rate, G, at fracture as a function of a loading phase angle. The phase angle is defined with an arbitrarilychosen length scale. Another approach used in the literature is to artificially insert a thin, isotropic, elastic layer (interlayer)

at the interface, thereby creating a companion interlayer crack problem to the original bare interface crack problem. Modes I

and II energy release rates are well defined and calculated relatively easily for the companion problem using FE analysis.

However, a consistent and systematic way of utilizing the interlayer approach to describing interface fracture and construct-

ing fracture loci has not been offered in the literature. This gap has been addressed in the present work.

A consistent method has been described for using interlayers to calculate interface crack fracture parameters. Interlayer

modulus and thickness are to be chosen such that Eq. (13) can be used to convert the resulting interlayer crack mode I and II

SERRs (GI and GII) into the complex stress intensity factor for the corresponding bare interface crack. For this, the interlayer

thickness must be small compared to other physical dimensions of the problem and the total SERRs of the two problems

(bare interface crack and the companion interlayer crack) must be nearly identical (within 2%). Suitable interlayer thickness

and elastic modulus were determined for an example problem of a bimaterial four-point bend test specimen with an inter-

face crack. Once an appropriate interlayer has been defined, modes I and II stress intensity factors calculated from the com-

panion problem can be used to define a mixed-mode phase angle, ^

win. Data obtained from testing of suitable interfacefracture test specimens can then be used to construct interlayer fracture loci as total energy release rate, G , at fracture as

a function of the phase angle, win. In fact, existing fracture loci reported in the literature as energy release rate as a function

of the Rice phase angle w can be easily converted to ‘interlayer’ fracture loci as was shown in Section 6. These ‘interlayer’

fracture loci can be used to predict fracture for interface cracks in structures with the same bimaterial combinations as

the test specimens. The key to the consistent use of the interlayer approach is to ensure that the companion problem used

to calculate the fracture parameters for the structure has an interlayer with identical thickness hin and modulus E in as was

used for the test specimen. Only this will ensure that the interlayer fracture locus derived from the test specimen is appli-

cable to the interface crack in the structure. It is therefore imperative that ‘interlayer’ fracture loci also report the interlayer

modulus E in and thickness h in used in their construction.

Acknowledgement

This work was funded by a Research Grant from Materials Sciences Corporation, Horsham, PA.

Appendix

Charalambides et al. [13] developed an analytical expression, using Euler–Bernoulli beam theory, for the SERR for a

delamination crack in a bimaterial, notched, four-point bend test specimen (Fig. 4). They assumed both materials to be iso-

tropic. In Fig. 4, P is the applied load, l is the spacing between inner and outer loading lines, b is the specimen width, 2a is the

crack length, h1 and h2 are the thicknesses of the two material layers, h ¼ h1 þ h2, E 1 and E 2 are the Young’s moduli of the

upper layer and lower layer, respectively, and m1 and m2 are the corresponding Poisson’s ratios.

When the half-crack length a is sufficiently greater than the thinner of the two material layers (h1 or h2), the energy re-

lease rate Gbare approaches a steady state value given by [13]:

E 2h3b

2Gbare

ð1 m22ÞP 2

l2

¼ 3

2

½1=ðh2=hÞ3 k=fðh1=hÞ3 þ kðh2=hÞ3 þ 3kðh1h2=h2

Þ½h1=h þ kh2=h1g ðA:1Þ

where

k ¼ E 2ð1 m2

1Þ

E 1ð1 m22Þ

ðA:2Þ

The expression on the left hand side of Eq. (A.1) is non-dimensional.

The companion problem to Fig. 4 is shown in Fig. 5. An analytical expression for the total SERR for the companion problem

of Fig. 5 has been determined in the present work, assuming plane strain. The thickness of the interlayer is h in. E in is the

Young’s modulus of the interlayer and min its Poisson’s ratio. The total strain energy release rate Gin for the crack embedded

in the interlayer, under steady state conditions, was calculated analytically using Euler–Bernoulli beam theory to be:

E 2h3b

2Gin

ð1 m22ÞP 2l2

¼ 1

8k2h

3 1

I seff

1

I eff ðA:3Þ


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where

I seff ¼ ðhin=2Þ3

12

" #þ k2

h32

12

" #þ

1

4

hin

2 h2k2

hin

2 þ h2

h i2

hin

2 þ h2k2

ðA:4Þ

I eff ¼ 1

12

ðh

3in þ k1h

31 þ k2h

32Þ þ

1

4

hink2h2ðhin þ h2Þ2 þ k1h1k2h2ðh1 þ h2 þ 2hinÞ2 þ k1h1hinðh1 þ hinÞ2

hin þ k1h1 þ k2h2

ðA:5Þ

k1 ¼ E 1=ð1 m2

1ÞE in=ð1 m2

inÞ ðA:6Þ

and k2 is defined in a similar manner with E 2 and m2 replacing E 1 and m1, respectively. The expression on the left hand side of

Eq. (A.3) is non-dimensionalized in a manner identical with Eq. (A.1).

References

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