Evaluation of interfacial fracture toughness using cohesive zone model

Composites: Part A 33 (2002) 1433–1447 www.elsevier.com/locate/compositesa

Evaluation of interfacial fracture toughness using cohesive zone model Namas Chandra* Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University, Tallahassee, FL 32310, USA

Abstract Interfaces play a critical role in determining the stiffness, strength and fracture properties of polymeric, metallic, and ceramic matrix composites. In this paper, while comparing the origin of interfaces in the three systems, attention is focused on the metal – (intermetal – ) matrix composites. The roles of processing induced residual stresses, and the chemistry evolution during in service on the mechanical properties in general, and fracture properties in particular are delineated. Stress-based and energy-based failure criteria to model interfaces are described with examples drawn from Titanium matrix composites. Finally a detailed discussion on using cohesive zone models (CZMs) to describe fracture and failure of interfaces is presented. While it is contented that CZMs present the best alternative from physics and computational perspectives, it is emphasized that the choice of the specific form and parameters is very important. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Metal-matrix composites (MMCs); Polymer-matrix composites (PMCs); A. Ceramic-matrix composites (CMCs); B. Interface/interphase

1. Introduction Composites are materials with at least two chemically distinct constituents with a distinct interface separating them; one of the constituents, called the reinforcing phase is in the form of long or short fibers, whiskers, sheets, or particles, and embedded in the other constituent designated as the matrix phase. The reinforcing material and the matrix material can be metal, ceramic, or polymer. Typically, reinforcing material is strong with low densities (high specific stiffness) while the material of the matrix is usually ductile or tough. Some commonly used fibers are glass fibers, carbon fibers, silicon carbide and aramid fibers, while the matrix is either polymer, metal or ceramic. In composites, the constituents are arranged in a specific spatial manner (architecture) carefully designed in order to obtain the desired thermal, mechanical, optical, electrical, magnetic or electronic properties tailored for specific applications. The material systems and the architectural arrangements are thus engineered to achieve the properties not possible using any single material. For example, some metal based composite materials combine the best features of the constituents to offer better mechanical properties such as higher strength and higher stiffness to density ratios, higher fatigue life and better wear resistance and minimize others (e.g. weight and cost). When two different material * Tel.: þ1-850-410-6320. E-mail address: [email protected] (N. Chandra).

systems are brought together for the purpose of achieving a superior property, there exists a surface of separation called the interface and often impedes the success of the compositing process in all types of composites including polymer matrix, metal matrix, and ceramic matrix composites. Interface is an unavoidable and inherent product of a composite. Composite materials rely on the interface to transfer properties (e.g. stress) from the matrix to the reinforcement and thus the behavior of interfaces critically determines the performance of the composites themselves. For instance, in ceramic composites, interfaces must be sufficiently weak to achieve toughening through crack deflection and fiber pullout, but strong enough for load transfer. As interfaces arise in the region of transition between two different phases with different constitutive behaviors, mechanistic characterization of interfaces is a challenging task. The task is complicated by the fact that the interface region continues to evolve both temporally and spatially during the consolidation, storage and service phases of the composite. Formulating the problem of the mechanics of interfaces and developing methods to characterize their behavior are of great interest and importance. Accurate and standardized methods to characterize the complex interfacial interactions (adhesion, thermal stress, gradient of the elastic modulus, surface roughness, thickness of the interfacial region) is still an unresolved issue. This issue cuts across all types of composite material systems and has been the subject of

1359-835X/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 8 3 5 X ( 0 2 ) 0 0 1 7 3 - 2

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many specialized symposiums in the last decade and a half (IPCM series in Europe, and ICCM series in USA/Japan). Though we are developing tests and mechanical models for interfaces on a regular basis, we still have very little appreciation of how to design the optimum interface for a particular system that can be both strong and tough, and resistant to moisture and heat throughout the life of the composite. In this paper, we first discuss the common features that bind as well as separate interfaces in different composite systems. We then turn our attention on the evolution of interfaces with a special emphasis on titanium based metal matrix systems. The role of interfaces in determining the mechanical properties is then explored, with a focus on fracture and failure. Appropriate analytical/computational models that describe the interface mechanical properties are then presented. A novel approach called cohesive zone model that has a strong physics and computations bias is then described in detail with examples. Significant portion of the paper is devoted to CZMs. CZM is being increasingly used in describing fracture and failure behavior in a number of material systems. In a recent review of cohesive zone models for metal ceramic interfaces, Chandra et al. [1] provide a discussion of some of the numerous models that have been proposed for the cohesive zone: linear, polynomial, exponential, trigonometric, trapezoidal and tri-linear to name a few, see Table 1 in Ref. [1]. It is found that most of the CZMs are essentially ad hoc relationships between certain components of the traction vector and the components of certain kinematical quantities and thus cannot be viewed as a complete description of mechanics of cracks. Also it can be seen from the table that the magnitudes of parameters in CZMs vary widely ranging from MPa to GPa for tractions, joules to kilojoules for energy, nanometers to micrometers for separation distance for similar material systems. Any two of three parameters (cohesive energy, cohesive strength and displacement jump) are commonly believed to uniquely represent the fracture process through cohesive zone model regardless of the shape of CZMs. Obviously if one of the models is appropriate, the others cannot be and in fact not one of them is adequate. The shortcomings of the models can be traced to not incorporating the effects of the various micromechanisms that can have a profound influence on the initiation and propagation of cracks.

2. Formulation of interface problem The continuum models of materials are generally stated as boundary value problems in which the governing equations of the continuum are identified with the balance laws of classical physics, and a constitutive relationship between the kinematic quantity and the kinetic quantity is postulated. This constitutive equation is specific to the body under consideration within the range of temperature, rate of

loading, type of loading and environment and is usually obtained phenomenologically. If the boundary value problem consists of domains, where one set of equations cannot be applied, then different sets are used to represent the individual domains. A case in point is the modeling of heterogeneous composites in which different forms of constitutive relations are used. For such bodies, there is a distinct separation between the various regions, where the usual theories of continuous media are valid. Such a separation leads to discontinuity in at least a few of the field quantities, e.g. strains. In the sense of continuum mechanics, this separation defines the interface that needs to be independently characterized to determine the response of bodies represented by regions of different materials. This prescription is even more critical, if the interfaces were to physically separate (open up or slide) during the loading or unloading process. 2.1. Generic formulations Consider the two solid bodies V1 and V2 separated by a common boundary S as shown in Fig. 1(a), where S can be considered as the same surface S1 [ V1 and S2 [ V2, in the initial configuration, i.e. S1 ¼ S2 ¼ S. Mathematically, we would like to define S as an infinitesimally thin 3D domain with surfaces S1 and S2 being the part of V1 and V2 before separation occurs. For all practical purpose, the surface S1 and S2 can be identified as a single surface, as a part of either of the domains. A material particle initially located (within
moves either of the domain V1 and V2) at some position X; to a new location x
; with a one to one correspondence
tÞ between x
and X
given by the equation of motion x
¼ xðX; or xi ¼ x(Xj,t ). In a generic sense, S defines interface between the two domains. If V1 is a metal and V2 a ceramic, then S represents a metal –ceramic interface; if V1 and V2 belong to the same material depicting grains of different orientations then S is a grain boundary, and if V1 and V2 represent the same domain V1 < V2 ¼ V, then S is an internal surface which is not yet separated. In any one of those cases, if S separates to S 1 and S 2 (fractures) as shown in Fig. 1(b), then the process creates new internal/external surface violating the fundamental laws of continuity. Obviously the newly formed region cannot be uniquely mapped from the deformed configuration. The equation of motion of the body xi ¼ x(Xj, t ) cannot identify the new region. This is the fundamental problem in modeling fracture (creation of internal/external surface) in the framework of the mechanics of continuous medium. The surface S represented by the unit normal N
(N
1 [ S1 ; N
2 [ S2 and N
¼ N
1 ¼ N
2 ) acting along the boundary separating the domain prior to deformation is as shown in Fig. 1(a). In the deformed configuration as shown in Fig. 1(c) n^ 1 and n^ 2 represents the unit normal of the surfaces (separated or otherwise).

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Constitutive equation is written in terms of the normal and tractions. For a narrow region (crack tip region) the directions of n^ 1 and n^ 2 are approximated to be same. A typical constitutive relation of Vp is given by T – d relations (see Fig. 1(d). if d
, d
sep ;

s~n^ ¼ T


ð2Þ

Beyond a separation distance of d
. d
sep ; the traction being identically zero within Vp

d
$ d
sep ;

Fig. 1. Conceptual frame work of cohesive zone model for interface.

For the domains V1 and V2 the constitutive equation can then be written as

sij ¼ Lijkl ðDkl 2 DIn kl Þ

ð1Þ

The elasticity tensor Lijkl is assumed to be isotropic; where Dkl , DIn kl are the total and inelastic parts of the rate of deformation tensor and sij is Jaumann rate of Cauchy stress. 2.2. Interface S If S continues to be a part of V1 and V2 (having points/particles common to both), then the motion of S can be uniquely defined by the motion of either V1 or V2 : Though the surface normal N^ would have rotated and deformed to n^ ; in the sense of the kinematics, N^ ¼ LF~ 21 n^ (L ¼ ds/dS ¼ stretch ratio, ðS; sÞ is length of a small segment in original and deformed configurations) which is unique. Thus we have one-to-one relationship between the deformed and undeformed configuration. However, if S is to be separated as shown in Fig. 1(b), then we have created a new surface in the traditional sense of the term. Consider the region bounded by S 1 and S 2 belonging to a new domain Vp : Assume that Vp is a 3D domain made of extremely soft glue, which can be shrunk to a surface but can expand to a 3D domain. The constitutive relation of Vp is expressed quite differently from that of a typical 3D solid (e.g. V1 or V2 ). The two surfaces that are initially part of V1 and V2 (S1 and S2 ) have normal N^ 1 and N^ 2 in the undeformed configuration; N^ 1 and N^ 2 are equal and opposite. During deformation the surfaces rotate to new normal n^ 1 and n^ 2 : As the surfaces separate we have two surfaces S 1 ðS 1 [ V1 > Vp Þ and S 2 ðS 2 [ V2 > Vp Þ:

s~n^ ¼ T
¼ 0

ð3Þ

It can also be construed that when d
. d
sep in the domain Vp ; the stiffness Lijkl ; 0: In order to implement the vectorial inequalities given in Eqs. (2) and (3), typically separate identities are postulated for the normal and tangential components with limits set for each of them. The formulation described above can be implemented in a computational scheme like FEM. The advantage of this formulation is that material separation is achieved without loss of continuity. By creating new surfaces, the traction and the stiffness of the cohesive zone elements connecting these newly created surfaces are made to vanish, but the displacements across them are still continuous. On the other hand, in other computational schemes like node releasing techniques, the new surfaces are created by use of ad hoc criterions and by altering the boundary conditions, which in turn modifies the stiffness iteratively. 2.3. Stress-intensity factors in dissimilar materials In this section, the mathematical details associated with stress-intensity factors in a bimaterial system are addressed for plane strain conditions. Also attention is restricted to small scale yielding, where the plastic zone at the crack tip is sufficiently small compared to the crack length itself. As shown in Fig. 2, with the interface on the x-axis, let E1 ; m1 and n1 be the Young’s modulus, shear modulus and Poisson’s ratio of elestic – plastic material 1 lying above the interface and similarly E2 ; m2 and n2 for elastic material 2 lying below the interface. The remote loading is prescribed using the stress intensity factors, K1 and K2 ; for the crack-tip stress field of the elastic bimaterial problem [2]. The tractions at a distance r ahead of the crack tip on the interface are found to be Kr i1 ðK þ iK Þr i1 syy þ isyx ¼ pffiffiffiffiffi ¼ 1 pffiffiffiffiffi2 2pr 2pr

ð4Þ

where K is the complex-valued stress intensity factors which can be determined by solving the full boundary-value problem for a given test-piece. 1 is so-called oscillation index given by   1 12b 1 ¼ ln ð5Þ 2 1þb where b is the second Dundurs’s mismatch parameter as

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species. Both phenomena can result in a degradation of mechanical properties of the interface and hence the composite itself. In this section, interfacial characteristics are discussed with respect to chemistry evolution, change in reaction thickness and effect of the above on interfacial properties. 3.1. Interfaces in MMC/CMC systems Based on the constituents in composites, interfaces can be classified into metal/ceramic, ceramic/ceramic or polymer/metal interfaces. In this section, the inherent characteristics of each type of interface and the difference between them in terms of thermal residual stress, chemical reaction products-interphase and matrix inelasticity are discussed.

Fig. 2. Geometry of interface crack in bimaterial.

shown below

b¼

1 m1 ð1 2 2n2 Þ 2 m2 ð1 2 2n1 Þ 2 m1 ð1 2 n2 Þ þ m2 ð1 2 n1 Þ

ð6Þ

and mi is the shear modulus, and ni is Poisson’s ratio, and subscripts 1 and 2 refer to the upper and lower materials, respectively. The remote energy release rate is ! 1 1 2 n21 1 2 n22 2 G ¼ ð1 2 b Þ ðK12 þ K22 Þ þ ð7Þ 2 E1 E2 When 1 – 0; the definition of mode mixity requires a choice of distance L ahead of the tip be made at which the relative amount of shear stress to normal stress acting on the interface is determined by tan c ¼

syx Im½ðK1 þ iK2 ÞLi1  ¼ syy Re½ðK1 þ iK2 ÞLi1 

ð8Þ

because Williams’ elastic solution describes the stress state outside the inelastic zone. It is sensible to specify L to be the order of the inelastic zone size.

3. Evolution of interface and its effects on mechanical properties The interface between the matrix and the reinforcement is probably the most important element that determines many of the properties of composite materials but the least understood and the one which is controllable in the least. Fracture and failure of fiber-reinforced composites more often than not are initiated at the fiber –matrix interface or its immediate vicinity. Stress and strain discontinuity and singularity arising at the crack tip are the root causes for the failure. Also the thermomechanical properties of the interface region are for the most part unspecified. Another area unique to the environmental behavior of composites is the chemical interaction between the matrix and the reinforcement. Under service conditions, the nature of the interaction can vary with time either from chemical reactions between them or by permeation of environmental

3.1.1. Thermal residual stress Residual stresses are self-equilibrating internal stresses that exist in a body in the absence of any external force. In composites, thermal residual stresses (TRS) are primarily Type II (mesoscopic) internal stresses arising from the mismatch in the coefficient of thermal expansion, CTE between the constituents when the composite is cooled from the consolidation temperature, or thermal cycled during service. The build-up of TRS can be considered to be the net effect of two competing processes: generation and relaxation. The generation of TRS depends on the Young’s moduli and CTE mismatch between matrix and reinforcement, and the temperature differential between the consolidation and operating temperatures. The relaxation on the other hand is a function of the time dependent inelastic properties of the matrix. When high levels of TRS are produced (due to processing or in service), the local stresses may exceed either the yield or fracture stress causing crack initiation even before the application of any external load. In MMCs, the mismatch in the CTE is quite significant ðamatrix $ 2afiber Þ and also the temperature differential involved in the cooling process is high (about 800– 900 8C to room temperature) [3,4]. Therefore, the TRS under these conditions can reach such high levels to induce plastic deformation in the matrix, just from the cooling process. In CMCs, the CTE mismatch between the constituents are insignificant, for example, in SCS-6 fiber reinforced borosilicate glass matrix composite, a max ¼ 3.5 £ 1026 K21, afiber < 2.6 – 4.3 £ 10 26K 21, but the TRS induced is still high enough to influence the interfacial properties. In PMCs, the residual stress is introduced by chemical shrinkage and/or CTE mismatch. It has been shown that in the case of some polymer – matrix composites, such as polyester – matrix systems, the large thermal strain introduced during the curing cycle can cause fiber/matrix debonding in the composites [5]. Thus, TRS is a significant factor in MMCs compared to other systems. 3.1.2. Interfacial chemistry and its effect In the development of high temperature materials for

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aerospace applications, MMCs, especially SiC fiber reinforced titanium matrix composites (TMCs) have shown considerable promise. An important factor in the use of TMCs is the high reactivity between the two constituent phases to form reaction zone at the interface, which influence the interfacial properties and therefore the overall performance of the composites. In this section, the interfacial characteristics are discussed with respect to chemistry evolution. Studies of reaction zone formation in TMCs have been carried out by several researchers [6,7]. The common reaction products are TiC, Ti2AlC, Ti5Si3 and Ti3Al. In the composite systems (SCS-6/Ti metal 21s and SCS-6/Ti – 6Al –4V) studied by our group, the first three reaction products form with some additional titanium compounds with molybdenum, niobium and vanadium as ingredients. The effect of elevated temperature exposure on the interfacial fracture behavior of MMCs has been studied by subjecting SCS-6/Timetal 21s specimens to selected temperatures (450, 700 and 927 8C) in vacuum for varying periods of time (i.e. 25, 70 and 120 h). The change of reaction zone thickness is measured as a function of time for two temperatures 927 and 700 8C and compared with the reaction zone size for as processed conditions. The variation in the reaction zone and the coating thickness for specimens exposed to 927 8C for 25, 70 and 120 h were measured and plotted in Fig. 3 along with the as-processed sample as reference. It is found that the initial growth of the reaction zone for the 25 h case is by partial consumption of the coating and partial transformation of the matrix region. With extended exposure up to 120 h the reaction zone growth occurs more through transformation of the matrix than through consumption of the coating. The exposure studies conducted at 700 8C showed that the reaction zone growth with time and also the change in the thickness of the outer coating are significantly reduced. It is observed that the coating is not fully consumed even after being subjected to high temperatures for extended periods of time. Fiber push-out test was carried out on as-processed and heat-treated specimens, and the typical load displacement curves were shown in Fig. 4(a) and (b). It can be seen from Fig. 4(a) that for as-processed specimen, after attaining the peak load, there is a sharp load drop which corresponds to the complete debonding and the onset of the frictional sliding. Full debonding is also characterized by a sharp acoustic signal shown in the figure. For heat-treated specimen, it is found from Fig. 4(b) that the peak load has significantly increased, and there is no distinct sharp load drop associated with complete debonding. There is also no sharp acoustic signal, which marks the event of complete debonding as observed in the earlier case. The push-out results at 700 and 450 8C are observed to be similar to that of the as-processed case characterized by abrupt load drops after complete debonding accompanied by sharp acoustic signals. Numerical simulation of push-out tests representing the

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Fig. 3. Variation in reaction zone size with exposure time for SCS-6/ Ti metal 21S composite heat-treated at 927 8C.

conditions of specimens, exposed to 450, 700 and 927 8C for 120 h and as-processed specimens are carried out, and the calculated fracture toughness values are shown in Table 1. It is observed that the fracture toughness values for specimens heat-treated at 450 and 700 8C are not very different from that of the as-processed specimen and are in the range of 55 – 60 J/m2. However, for the specimen exposed at 927 8C,

Table 1 Computed strain energy release rates after slicing for different composite systems Material systems (vol. ratio ¼ 0.35)

SCS-6/Ti-15-3 [25] SCS-6/Ti-6-4 [26] SCS-6/Ti metal [27] SCS-6/CASI [28] SCS-6/Borosilicate [28]

Fiber residual stress Axial (MPa)

Radial (MPa)

2750 2858 2985 2220 20

2223 2272 2304 2100 8

Gir (J/m2)

37.6 42.3 47.6 7.86 0.19

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heat-treated specimens, the debonding initiates more in the interface region than in the coating. The cracks are seen to form in the middle of the reaction zone or near the reaction zone matrix interface, and the fracture surfaces are rough with the presence of large amount of debris. 3.2. Effect of interface on transverse and longitudinal properties

Fig. 4. Typical load displacement curve for a push-out test conducted on (a) 700 8C processed specimen and (b) 927 8C processed specimen.

the fracture toughness is observed to actually increase significantly to about 70 J/m2. For better understanding, the mechanics of the failure process, the fracture surface of the failed push-out specimens were examined by SEM. It is observed that in all the cases, debonding initiated from the bottom of the specimen. This is consistent with theoretical predictions [6] and experiments, where the thermal residual stress are present [8,9]. On the other hand, it is found that the locations for the crack initiation are different for different heat-treated specimens. For the 700 8C specimen, the fracture surface reveals that the cracks initiate more frequently in the middle and in the region near the coating reaction zone interface indicating a sharp abrupt fracture process. The fracture surfaces for the as-processed and those subjected to 450 8C heat treatments are observed to be similar to that of the 700 8C case. But the fracture surface of the 927 8C heattreated specimens are quite different from that of the 700 8C

To study the effect of interface on mechanical response of composites, it is necessary to understand the various types of interfacial bonding. The bonding at the interface is achieved through mechanical and/or chemical means. Mechanical bonding is induced by the interlocking at the asperities and corresponds to the roughness of the mating surfaces, and the process of mechanical locking can result in load transfer across the interface. Chemical bonding may be introduced through the dissolution of fiber into the matrix, or through a diffusional process. The latter may result in a reaction interphase zone at the interface in composites, especially in MMCs. Extensive experimental investigation of the transverse and longitudinal behavior [10] clearly indicates that the transverse mechanical behavior is influenced by interfacial bonding conditions. Debonding at the interface is a significant deformation mechanism in the case of transverse loading in SCS-6/Ti-15-3 MMC. Inelastic deformation mechanisms causing the non-linear response could be distinctly different in longitudinal and transverse loadings. Longitudinal deformation is dominated by plasticity of the matrix while transverse deformation is controlled by both damage, e.g. interfacial debonding (Fig. 5(a), matrix cracking) and plasticity. The transverse deformation is shown to have a three-stage behavior (Fig. 6) with damage dominating stage II and plasticity dominating stage III. Stage I exhibits almost elastic behavior with traces of plasticity. The presence of plastic deformation is evidenced by the presence of slip bands on etched microstructure, and interfacial debonding is determined by in situ surface replica studies using acetate tapes. The change in the slope of transverse stress – strain response while unloading from stage II to stage III also is a clear indication of interfacial debonding. Interfacial debonding results in sharply reduced slope on unloading caused by the closure of debonded surface as can be seen in Fig. 5(b). The strain offset is extremely small at zero load during stage II and strain offsets increase as the specimen is loaded into stage III (Figs. 5 and 6).

4. Mechanical characterization of interface An interface is a distinct 2D or 3D surface or zone of transition separating the matrix and fibers, with a distinct discontinuity in physical, chemical, or mechanical characteristics [8]. Stiffening and strengthening rely on the load

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Fig. 6. Experimental transverse response of 08 SCS-6/Ti-15-3 MMC [10].

Fig. 5. Surface replica of transversely loaded SCS-6/Ti-15-3 MMC specimen [10]. db, debonded zone; sb, slip bands.

transfer across the interface; toughness is influenced by the propagation of cracks at the interface, and ductility is affected by the relaxation of peak stresses at the interface. The transverse stress – strain behavior in particular is affected significantly by the interfacial bonding conditions. Despite this crucial role played by interfaces in controlling the mechanical behavior of the composites, it is rather difficult to offer reliable generalizations regarding the optimal design of interfacial structure in composites. Nevertheless, it is possible to identify certain principles for designing specific types of the composite material. As a prelude to this, a brief outline of technical procedures commonly used to study the mechanical characterization of interface is discussed in this section. Mechanical characterization is useful in obtaining quantitative information about the strength and toughness of the interface. Currently no test methods are available to

directly measure the normal strength of interface [11], but several techniques such as push-out test are being increasingly used in the recent years to measure the interfacial shear properties for various material systems [9,12,13]. These include fiber push-out, fiber pull-out, fiber push-in (or fiber indentation), fiber fragmentation, and thermal cycling. Of these, only push-out and push-in directly measure in situ a fiber’s response under load in a manufactured composite material. Test can also be used as an analysis tool for composite parts to evaluate the cause for failure. Most other test methods employ model materials that are representatives of the real composite. Since push-out test provides both bond strength and sliding stress data, the push-out method develops more information about the micromechanical interfacial behavior than the push-in test. For these reasons, the push-out method was chosen for the current study.

5. Computational simulation of interface The strength of the fiber/matrix interface affects overall composites properties significantly, both in unidirectional and laminated forms. Most methods developed for measuring these properties require the use of a micromechanical model to interpret the results of the experiments in terms of so-called interface properties. In the past few years, several interface models have been used in our group, i.e. stressbased spring layer model, energy-based equivalent domain integral (EDI) model and cohesive zone model.

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5.1. Spring layer model In this model, the fiber – matrix interface is modeled using a contact-friction formulation, and the bonding at the interface of negligible thickness possessing the required strength is simulated using a number of springs with specified stiffnesses. Therefore, the interface in this case represents the border separating distinct phases such as fiber and matrix. As long as perfect bonding conditions exist across the interface, traction and displacements will be continuous across the interface. Chandra and coworkers [14 –18] incorporated this model in a micromechanical analysis for predicting the transverse behavior of MMCs and IMCs. A stress based criterion for debonding, and frictional resistance based criterion for interfacial sliding are used in those works to capture debonding and sliding, which are presumed to occur in a chronological sequence, with increasing loading. Debonding is postulated to occur under the combined action of normal tensile stress leading to mode I failure and shear stress leading to mode II failure at the interface. A quadratic stress based failure theory !2 !2 sr t þ f $1 f tr sr is applied [19,20], where sfr is the strength in the normal direction resisting crack opening, and tfr is the shear stress. Subsequent to the debonding process, interfacial frictional sliding occurs in the mating debonded surfaces according to Coulomb’s law. In the above equation, the effect of sr is included only when it is positive. It is found that in TMCs, the residual stresses decrease with the increase in temperature resulting in a decrease in the axial strain mismatch thus reducing the residual shear stress at the interface. The push-out test is numerically simulated at 23 and 400 8C assuming a constant value of interfacial shear strength (t p ¼ 400.0 MPa) in this temperature range. The observations in Figs. 7 and 8 show that the peak load Pmax increases with temperature. This is because the reduction in shear stresses at elevated temperatures requires a larger load for failure initiation from the support end. It is also shown in Fig. 8 that the load for the initiation of debonding increases with the temperature. The debonding sequence indicates that the failure under these conditions of temperature and thickness, still initiates from the support end. The difference between Pmax and Pi decreases with increase in temperature. This can be attributed to the reduction in residual shear stresses at the loading end at elevated temperature, which causes the resistance of further debonding to decrease as the crack propagates towards the loading end. Apart from the shear stresses, radial clamping stresses at the interface also decrease with increase in temperature, which cause the significant reduction in tfr from 23 to 400 8C. (The coefficient of friction m is taken as 0.25 in these calculations.)

Fig. 7. Predicted push-out behavior at T ¼ 23 and 400 8C; t p ¼ 400.0 MPa.

5.2. Energy-based equivalent domain integral method Fracture toughness is considered as a more appropriate measure of the bond strength at the interface. The interface failure process is modeled using a strain energy-based failure criteria given by Gi $ Gic where Gi is the strain energy release rate for the interface crack and Gic is the critical value of the strain energy release rate. Several methods are available to compute the energy release rate ðGÞ for an existing crack, viz. virtual crack closure technique, contour integral (CI) and the equivalent domain integral method. Among these techniques, the EDI

Fig. 8. Variation of failure initiation load with temperature; t p ¼ 400.0 MPa.

N. Chandra / Composites: Part A 33 (2002) 1433–1447

method is ideally suited to investigate the crack propagation since it does not require the use of singular elements for calculating energy release rates; the CI method is unable to study the propagation of crack due to the requirement of a focused mesh, i.e. a ring of singular elements around the crack tip, although it is relatively independent of mesh density. As well known, J-integral is a path-independent parameter and is equivalent to the rate of change of total potential energy with reference to the crack length. To facilitate the numerical implementation, the J-integrals are converted to equivalent area/domain integrals [14 – 16]. The conversion of line integrals to area integrals is very advantageous because all the quantities necessary for computation of the domain integrals are readily available in finite element analyses. It is indicated [21] that the EDI method can give accurate results for the J-integral for mode I, mode II and mixed mode problems. As shown in Fig. 9(a), for an arbitrary closed contour G around the crack tip, the Jintegral is defined in the absence of any body force as  ð  ›ui J xk ¼ Wnk 2 sij n dG ›x k j G where k ¼ 1; 2 and W is the total strain energy density defined as: ð1ij W¼ sij d1ij 0

The integral Jx1 and Jx1 are two path-independent integral that compute the total amount of the energy flux leaving the contour G in the two directions x1 and x2 ; respectively. This approach was formulated by Chandra and coworkers [22] and implemented through user subroutines in the finite element code MARC [23]. The formulations and implementations were validated with typical problems for different composite systems including ones with cracks along bimaterial interfaces subjected to remote and crack face normal and shear tractions. The mode I and mode II contributions were also examined since the decomposition method allows separation of individual modes. It was observed that the individual modes were path dependent (variation of 4% between first and last paths) even though the total integral was path-independent. This is consistent

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with the data in Ref. [21]. The individual modes even though path dependent can still provide rough estimates of the mode I and mode II contributions. The EDI method above is employed to simulate the push-out test for different composite systems, and the finite element description is shown in Fig. 9(b). Modeling such a test involves three major steps (i.e. cooling, slicing and loading) which is described in detail elsewhere [14,16]. Slicing of a push-out specimen from a bulk composite (step 2) induces the redistribution of the residual stresses near the end of specimen. The observations in Table 2 show that the levels of the axial residual stress in the fiber predicted by the simulation for Ti-15-3/SCS-6 (fiber volume fraction is 35%) is of the order of 750 MPa. This compares reasonably well with the values of 749 ^ 47 MPa measured experimentally by Pickard et al. [24] using the dissolution technique. The values obtained for other composite systems are also shown in Table 2. In terms of energy release rates, it is found that symmetric values for energy release rates are introduced at both the ends of the push-out specimen. This also indicates the tendency of the fiber to protrude out of the matrix at either end. The values of Gir (the value of Gi after slicing and before the application of indenter load; purely due to residual stresses) listed in Table 2 serve as a lower bound for Gir of these MMC systems. The processing simulations are also performed on typical CMCs such as CAS I/SCS-6 and borosilicate/SCS-6 to compare energy release rate values. As expected the Gir values for the CMCs were at least one order less than those for the MMCs since the residual stresses generated in the CMCs are much smaller compared to those in MMCs (see Table 1). Using the simulation, the critical values of energy release rate Gic for MMCs are found to be considerably higher than the those for CMCs. It can be seen from Table 3 that the predicted fracture toughnesses are about 10 J/m2 higher than the values of Gir, the energy obtained after slicing of the MMCs. This indicates that most of the energy contribution to the failure process comes from the residual stresses. Also the interfacial fracture energies of the three systems are not far apart being in the range of 50– 60 J/m2. These values are on the lower side as they are based on the minimum energies required for fiber push-out assuming that there are no pre-existing flaws such as matrix cracks in the vicinity Table 2 Interface fracture toughness for different composite systems

Fig. 9. Contours used in the equivalent domain integral method.

Material systems

Sample thickness (mm)

Fiber volume fraction

Peak load (exp.) (Ref.)

Gir (simulation) (J/m2)

SiC/RBSiC SCS-6/Ti-15-3 SCS-6/Ti-6-4 SCS-6/Timetal 21 s

1.27 0.45 0.50 0.32

0.46 0.35 0.35 0.35

4.750 [29] 20.78 [30] 28.00 [31] 11.26 [32]

2.30 50.0 52.5 54.0

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N. Chandra / Composites: Part A 33 (2002) 1433–1447

Table 3 The values of parameters used for simulation

25 8C 500 8C 650 8C

dmax

tmax (MPa)

Dct (m)

Dcn (m)

ft (J/m2)

1.0 1.0 1.0

200 100 90

3.5 £ 1025 1.3 £ 1025 1.42 £ 1025

3.5 £ 1025 1.3 £ 1025 1.42 £ 1025

3000 650 640

of the interface. The model used in this work does not take into account the mechanical locking effects due to asperities at the interface, which may increase the apparent Gic values. 5.3. Cohesive zone model In the cohesive zone approach an independent specification of a local constitutive relation for the interface is prescribed. This approach is quite different from all others presented earlier in that interface is treated as a separate material with its own constitutive relationship. This interface constitutive relationship reflects the coupling of stresses (or tractions) at the separating surfaces of the interfaces, to the displacements the surfaces suffer. One class of interfacial constitutive model is based on the existence of a potential fðDÞ that measures the energy cost to displace the adjacent planes across an interface by a relative displacement D [33]. The resulting tractions are given by T¼

›fðDÞ ›ðDÞ

5.3.1. Application of cohesive zone model In this work, two of the popular CZMs (exponential model [34] and bilinear model [35]) are used to simulate the push-out test in MMC systems. For the exponential CZM, the interfacial potential is given by

FðDn ; Dt Þ ¼ Fn þ Fn expð2Dn =dn Þ{½1 2 r þ Dn =dn  £ ½ð1 2 qÞ=ðr 2 1Þ 2 ½q þ ½ðr 2 qÞ=ðr 2 1Þ £ Dn =dn expð2D2t =d2t Þ}

ð9Þ

where smax and tmax are interface normal and tangential strength, respectively; Dcn and Dct are the critical normal and tangential separations at which complete separation is assumed. Details of the models are available in Ref. [1]. 5.3.2. Finite element approach The thin-slice push-out test is modeled as an axisymmetric problem [36], since the loading (both thermal and mechanical) produces axially symmetric displacement fields. The discretized mesh has 4800 axisymmetric 4-node elements to model the fiber and the matrix. Duplicate nodes are created at the interface on the fiber and matrix sides. 240 axisymmetric cohesive elements with each having 4 nodes and zero thickness in the direction normal to the interface are used to model the interface behavior. The general-purpose commercial code ABAQUS [37] is employed to carry out the analysis due to its capability of handling non-linear problems and, also because of its flexibility in allowing user-defined subroutines to be linked to the main program. The cohesive element model is input as a user-defined element subroutine UEL into ABAQUS. The mesh used in the simulation is shown in Fig. 10. To simulate the single fiber push-out test, the elastic constitutive behavior is assumed for the fiber, and the matrix is assumed to be a rate independent elastic – plastic material. The temperature dependency of the elastic and inelastic properties [25,38] of the constituent phases is included in a piece-wise linear manner. The material properties used in the analysis are given in Fig. 11. The analysis is done in the three steps described in detail elsewhere [14,16]. 5.3.3. Results and discussion 5.3.3.1. Comparison between experimental results and numerical predictions. The non-aged Ti metal-21S/SCS-6 composite system was fabricated and push-out tested as a part of this work. The reinforcing fibers are made of SCS-6 materials with 142 mm diameter, with the volume fraction of fiber being 35%. The following three cases selected from a set of experimental data of Osborne et al. [39], are used in the current study:

with q ¼ Ft =Fn

r ¼ Dpn =dn

where Fn and Ft are work of normal and shear separations, respectively; Dn, and Dt are normal and tangential displacement jumps, respectively; dn and dt are interface characteristic-lengths; Dpn is the value of Dn after complete shear separation under the condition of normal tension being zero, i.e. Tn ¼ 0. On the other hand, for bilinear CZM, the normal (Gn) and tangential (Gt) works of separation per unit area of interface are given by

Gn ¼ smax Dcn =2;

Gt ¼ tmax Dct =2

ð10Þ

† Test temperature: 25 8C; sample thickness: 0.63 mm. † Test temperature: 500 8C; sample thickness: 1.07 mm. † Test temperature: 650 8C; sample thickness: 1.10 mm. Force –displacement data obtained from the experiments are shown in Fig. 12. The experimental data represents the average of at least two sets of experiments in each case. Finite element simulation was carried out and the discretized mesh is shown in Fig. 10. The bilinear CZM for the interface is characterized by four parameters: dmax, tmax, Dct and Dcn : Since shear failure is by far the dominant failure mechanism in current study, it is assumed that Dcn is equal to Dct : Therefore, only three parameters need to be

N. Chandra / Composites: Part A 33 (2002) 1433–1447

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Fig. 10. The model used for push-out test simulation and details of interface element.

determined in the simulation. To determine tmax and Dct ; the work of shear separation (ft) is first estimated based on the area under the force – displacement curves obtained from the experiment (Fig. 12). The values of tmax for

Fig. 11. Variation of material properties with temperature.

various temperatures are chosen so that the experimental response is closely matched. And then Dct is determined by using Eq. (27) in Ref. [1]. The values used in the simulation are listed in Table 3. The experimental and computational results for various temperatures (i.e. 25, 500 and 650 8C) are shown in Fig. 12 and the zero displacement is shifted relative to the starting position in the experiments. As noted above, the samples for 500 and 650 8C are approximately of the same thickness and can be compared directly. However, due to maximum force limitations, the sample for room temperature is thinner. First, it can be seen from both experiments and simulations

Fig. 12. Experimental measurement and numerical predictions, at different temperatures. Note: Displacement for each temperature is relative to the starting position.

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that the peak load decreases with the increase in temperature due to the relaxation of compressive residual stresses that act on the fiber as a clamping force at elevated temperatures [40]. The simulation response for various temperatures matches the experimental results very well. Also, it can be seen that it is linear for room temperature until the maximum load is reached. This is followed by a catastrophic and complete debonding. For elevated temperatures, each curve has an initial linear portion, and as the load continues to increase, a reduction in slope is observed (especially for the plot at 650 8C). It was reported that the change in slope from linearity for elevated temperatures corresponds to the initiation of the interfacial debonded crack [41]. However, from the present results, it is clearly seen that the debonding does not take place until the peak load is reached. It was reported [40] that the matrix material behaves elastically at lower temperatures. At higher temperatures (. 400 8C), the ability of the matrix phase to flow under the applied stress is enhanced. It is seen in the simulation that plastic deformation in the matrix indeed takes place along the entire interface during the loading process at elevated temperatures. Even at the ambient temperature, plastic deformation occurs near the bottom along the interface. However, it is just not enough to cause the slope change in macroscopic force– displacement curve. Hence, it should be reasonable to conclude that the change in slope during loading at elevated temperatures is due not to the initiation of interfacial debonding, but due to plastic deformation of the matrix material at the interface. It is clear that the choice of CZM is very critical to reproduce the macroscopic mechanical response of the material system. 5.3.3.2. Comparison between two cohesive zone models. At the outset it appears that the CZM for the interface and conventional field equations with appropriate constitutive equations for the bulk (matrix and fiber) will reproduce the test results. Consequently, the agreement leads to a set of cohesive zone parameters (see Table 3) that can then be used for other geometric and loading configurations. Before such a positive assertion can be made, it is worthwhile to analyze the uniqueness of the shape of the CZM and the sensitivity of the chosen parameters on matching the experimental data. For the purpose of examining whether the form of the CZM is important if the energy (area under force– displacement curve) is maintained at the same level, two forms (exponential and bilinear) are chosen. An energy value of fn ¼ ft ¼ 3000 J/m2 was used for the two forms. The simulation results for bilinear and exponential forms are shown in Fig. 13(a) and (b), respectively. The value of tmax was selected as 200 MPa (as in Table 3 for ambient temperature case), and then altered to 300 and 400 MPa, which corresponds to curves 1, 2 and 3 (Fig. 13(a) and (b)), respectively. It should be noted that a bilinear CZM with tmax ¼ 200 MPa matches the experiment very well.

Fig. 13. Load– displacement plots simulated using exponential and bilinear CZMs. (a) bilinear model [35], (b) exponential model [34]. Note: Pd, plastic deformation; Db, debonding.

Fig. 13(a) and (b) bring home some very important conclusions regarding a number of issues. Fig. 13(a) shows that even though the cohesive energy is the same, increasing the interfacial shear strength changes the response of the push-out test. This result is in contrast to the common belief that the separation process is governed mainly by the energy of separation. Again, focusing on tmax ¼ 300 and 400 MPa in Fig. 13(a), there is a non-linear response when the force is above 50 N. On closer examination, it is observed that matrix yields under those conditions, leading to the slope change in the macroscopic response. Arrows in the Fig. 13(a) indicate the loading at which the initial plastic deformation occurs. The initial debonding for all the three cases, however, takes place at the peak load. Hence, the average shear stress required for failure tavg ¼ Ppeak =2pdL is different in the three cases. The present discussions again clearly reveal that selection of tmax is critical in determining the macroscopic mechanical response.

N. Chandra / Composites: Part A 33 (2002) 1433–1447

Fig. 13(b) shows the overall response of the push-out test for the exponential model keeping the cohesive energy and interfacial strength identical to those of Fig. 13(a). It is immediately clear that the predicted response is in no way similar to the experimental observations. Arrows in the Fig. 13(b) indicate the loading at which the initial debonding occurs. The important question is why there is such a drastically different response. In fact, no amount of parametric variations on the exponential CZM can make it match with experimental results making it unsuitable for modeling the interface in this problem. Focusing on Fig. 13(b), for tmax ¼ 200 MPa, the peak load (about 53 N) does not correspond to either the initial debonding or the matrix yield stress. As the push-out load is increased, the interface responds to the external force according to its constitutive equation. When the peak stress in each element is reached (Fig. 13(b)), the slope is reversed and the interface becomes more and more compliant. Thus the overall stiffness of all the interfaces in tandem is continually altered with different cohesive elements changing their stiffness as the push out displacement increases. Since the separation in this model takes place with a significantly large displacement Dct ; the first debonding does not occur until very late in the process. Thus it is clear that only bilinear model is suitable for simulating the behavior of bimaterial interfaces in composite systems, while the exponential model does not match the push-out experimental data. However, the exponential model may be valid very well, when the wake regions of a crack is considerably large. This feature is addressed in great length in Ref. [42]. 5.3.3.3. Effect of interface characteristic-length, dmax. In this section, the effect of interface characteristic-length, dmax, on the force– displacement curve is studied and the results are shown in Fig. 14. The values of dmax are selected to be 0.1, 0.4, 0.7 and 1.0; the corresponding F –DU curves are I, II, III and IV, respectively. It is interesting to find that the shape of the force– displacement curve is dependent on the value of interface characteristic-length, which determines the shape of shear traction vs. shear separation. The arrows in the figure indicate the loading at which debonding occurs at the interface. Comparing the numerical results with the experimental data, it can be concluded that for the current material system, the suitable value for dmax is 1.0. Therefore, it is reasonable to conclude that the shape of the traction-separation curve for the CZM is an important factor in determining the macroscopic mechanical behavior of the interface in addition to the other interfacial parameters such as the cohesive energy. Thus, it is obvious that setting dmax ¼ 1.0 fixes the shape of the CZM; in other words, any other values of dmax corresponding to different shapes will yield unacceptable results for the present problem of metal – ceramic interfaces. This is an important point to be considered while selecting specific shapes of CZM for modeling metal ceramic interfaces.

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Fig. 14. Effect of interface characteristic-length dmax on load– displacement plots (I, II, III and IV represent corresponding force– displacement curves to different values of characteristic-lengths 0.1, 0.4, 0.7 and 1.0, respectively).

6. Concluding remarks Though composites in general has found increasing use in all walks of life, understanding interfaces will accelerate their applications especially in critical high temperature products. Specific issues addressed in this paper are follows: 1. The role of metal/ceramic and polymer/metal interfaces is crucial to the thermo-mechanical behavior of MMC/CMC systems. The stress –strain response, stiffness, strength, fatigue, fracture, failure behavior are strongly influenced by the interfacial bonding conditions. The bond strength of an interface is the net effect of the contributions from chemical and mechanical components. The former is due to the reaction at the interface during the compositing process and depends on the constituent material properties and in service conditions. The mechanical component is due to the sliding resistance at the interface and depends on the presence of residual stresses and interfacial surface irregularities. 2. In MMC/IMC systems, the interfacial chemistry evolves with time of exposure to elevated temperature and/or environment and influences both the size and phases within the reaction zone of the interface. Titanium alloys and intermetallics are very highly reactive at elevated temperatures and produce non-stoichiometric carbides and silicates within the interphases. 3. Residual stresses develop when the MMCs and IMCs are cooled after processing from the consolidation temperature at which they are stress free to room temperature. This is mainly due to the difference in the CTE of fiber

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4.

5.

6.

7.

8.

9.

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and matrix materials. These stresses reach considerable levels in all systems and more so in MMCs and IMCs. Residual stresses are affected by the evolution chemistry and by the inelastic properties of the matrix material. Microstructural studies reveal that reaction growth is due to transformation of both coating and the matrix. In TMCs the growth of the reaction zone into the matrix is much more than the growth towards the fiber, as carbon is able to diffuse much farther into the matrix than the diffusion of Ti into the fiber. Reaction zone sizes increase rapidly with time at 9278C. At lower temperatures such as 700 8C, no significant increase in reaction zone size is observed. Push-out tests conducted on heat-treated specimens reveal that the effect of thermal exposure on interfacial properties is predominantly due to exposure temperature. Time of the exposure is found to have only a secondary effect. Fracture surfaces of the specimens exposed at 927 8C show that debonding takes place in the reaction zone. However, in the as-processed and those heattreated at the lower temperature ranges debonding was initiated in the coating. This is because the weak region in the coating is replaced by the stronger interfacial region. The mechanical response of the interfaces is usually measured using a thin slice push-out test and transverse tensile testing. However, extracting quantitative interfacial properties from the push-out test is not trivial. A comprehensive process model to understand the interfacial failure process during push-out test is needed for better interpretation of the test results. In the numerical simulation, either a stress based failure criterion or an energy-based criterion can be used to model the failure process. The concept of continuum mechanics is extended to include a zone of discontinuity modeled by cohesive zones. A comprehensive analysis of some of the popular CZMs has been presented in the form of equations and the magnitude of the parameters. We illustrate some of the outstanding issues, e.g. physical interpretation of the area under the curve, form of the equations (shape of the traction curves), cohesive strength and scales in CZMs. Two CZMs (exponential and bilinear) have been employed to model the interface failure in metal matrix composites. The bilinear CZM is suitable to simulate the push-out test process and the calculated results match the experiment data very well. It is our view that the CZM represents the physics of the interface separation process and hence the shape of the CZM should in some sense depend on the inelastic processes occurring at the micromechanical level. When using CZMs to model separation in a given material system, an appropriate shape (form), depending on the type of material system and the inelastic micromechanical processes, should be used. Otherwise,

the CZM based modeling and simulation will not yield meaningful results.

Acknowledgements The author wishes to express his sincere gratitude to his past students who have generated most of the results cited in this work. The author is especially thankful Dr H. Li, the research associate who has spent significant effort in the presentation of the work.

References [1] Chandra N, Li H, Shet C, Ghonem H. Some issues in the application of cohesive zone models for metal–ceramic interfaces. Int J Solids Struct 2002;39:2827–55. [2] Rice JR. Elastic fracture concepts for interfacical cracks. J Appl Mech 1988;55:98–103. [3] Arnold SM, Arya VK, Melis ME. Elastic/plastic analysis of advanced composites investigating the use of the compliant layer concept in reducing residual stresses resulting from processing. Technical report NASA TM 103204, NASA; 1990. [4] Arnold SM, Melis ME. Influence of engineered interfaces on residual stresses and mechanical response in metal matrix composites. Technical report NASA TM 105438, NASA; 1992. [5] Bailey JE, Parvizi A. J Mater Sci 1981;16:649. [6] Hall EL, Ritter AM. Structure and behavior of metal/ceramic interfaces in Ti alloy/SiC metal matrix composites. J Mater Res 1993;8(5):1158 –68. [7] Pang CH, Jeng SM, Yang JM. Interface control and design for SiC fiber reinforced titanium aluminate composites. J Mater Res 1993; 8(8):2040–53. [8] Jayaraman K, Reifsnider KL, Swain RE. Elastic and thermal effects in the inter-phase. Part II. Comments on modeling studies. J Compos Technol Res 1993;15(1):14– 22. [9] Clyne TW, Withers PJ. An introduction to metal matrix composites. UK: Cambridge University Press; 1993. [10] Majumdar BS, Newaz GM. Inelastic deformation of metal matrix composites: plasticity and damage mechanisms. Philos Mag A 1992; 66(2):187–212. [11] Jayaraman K, Reifsnider KL, Swain RE. Elastic and thermal effects in the inter-phase. Part I. Comments on characterization methods. J Compos Technol Res 1993;15(1):3 –13. [12] Eldridge JI, Bhatt RT, Kiser JD. Investigation of interfacial shear strength in SiC/Si3N4 composites. NASA TM 103739; 1991. [13] Bechel VT, Sottos NR. Application of debond length measurements to examine the mechanics of fiber pushout. J Mech Phys Solids 1998;46: 1675–97. [14] Ananth CR, Chandra N. Numerical modeling of fiber push-out test in metallic and intermetallic matrix composites—mechanics of the failure processes. J Compos Mater 1995;29:1488– 514. [15] Ananth CR, Chandra N. Evaluation of interfacial shear properties of metal matrix composites from fiber push-out tests. Mech Compos Mater Struct 1995;2:309–28. [16] Chandra N, Ananth CR. Analysis of interfacial behavior in MMCs and IMCs using thin-slice push-out tests. Compos Sci Technol 1995;54: 87– 100. [17] Ananth CR, Chandra N. Elevated temperature interfacial behavior MMCs: a computational study. Composites 1996;27:805– 13. [18] Ananth CR, Mukherjee S, Chandra N. Evaluation of fracture toughness of MMC interface using thin-slice push-out tests. Scripta Mater 1997;36:1333–8.

N. Chandra / Composites: Part A 33 (2002) 1433–1447 [19] Hashin Z. Failure criteria for unidirectional fibre composites. J Appl Mech 1980;47:329– 34. [20] Wisnom MR. Micromechanical modeling of the transver tensile ductility of unidirectional silicon carbide/6061 aluminum. J Compos Technol Res 1992;14:61– 9. [21] Raju IS, Shivakumar KN. An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems. Engng Fract Mech 1990;37(4):707–25. [22] Ananth CR, Mukherjee S, Chandra N. Effect of residual stresses on the interfacial fracture behavior of metal–matrix composites. Compos Sci Technol 1997;57:1501–12. [23] MARC analysis corporation, MARC Version: K5, User Manuals; 1992. [24] Pickard SM, Miracle DB, Majumdar BS, Kendig KL, Rothenflu L, Coker D. An experimental study of residual fiber strain in Ti-15-3 continuous fiber composites. AMET 1995;333–40. [25] Nimmer RP, Bankert RJ, Russel ES, Smith GA, Wright PK. Micromechanical modeling of fiber – matrix interface effects in transversely loaded SiC/Ti-6-4 metal matrix composites. J Compos Technol Res 1991;13(1):3–13. [26] Deng X. An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial interfaces and in homogeneous solids. Int J Solids Struct 1994;31:2407–29. [27] Grande DH, Mandell JF, Hong KCC. Fiber–matrix bond strength studies of glass, ceramic, and metal matrix composites. J Mater Sci 1988;23(1):311–28. [28] Majumdar BS, Newaz ZM. Isothermal fatigue mechanisms in Ti-based metal matrix composites. Technical Report NASA CR 191181; 1993. [29] Fox KM, Bowen P. Interface properties and fatigue crack growth in Ti-b 21S/SCS-6 composites. In: Fujishiro S, Aylon D, Kishi T, editors. Metallurgy and technology of practical titanium alloys. The Minerals, Metal and materials Society; 1994. p. 333–40. [30] Majumdar BS, Miracle DS. Interface measurements and applications in fiber reinforced MMCs. J Key Engng Mater—Special Issue Interf Compos Laminated Struct 1996;116/117:153–72.

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[31] Eldridge JI, Ebihara BT. Fiber-out testing apparatus for elevated temperatures. J Mater Res 1994;9(4):1035–42. [32] Roman I, Jero PD. Interface shear behavior of two titaniumbased SCS-6 model composites. In: Miracle DB, Anton DL, Grave JA, editors. Intermetallic matrix composites II. MRS Symposium Proceedings, 273. Materials Research Society; 1992. p. 337– 42. [33] Needleman A. Micromechanical modeling of interfacial decohesion. Ultramicroscopy 1992;40(3):203–14. [34] Xu XP, Needleman A. Void nucleation by inclusion debonding in a crystal matrix. Modelling Simul Mater Sci Engng 1993;1(2): 111–32. [35] Geubelle PH, Baylor J. Impact-induced delamination of laminated composites: a 2D simulation. Compos, Part B: Engng 1998;29(5): 589–602. [36] Kallas MN, Koss DA, Hahn HT, Hellman JR. Interfacial stress state present in a thin-slice fibre push-out test. J Mater Sci 1992;27(14): 3821– 6. [37] ABAQUS Manual, Version 5.8, Hibbit, Karlsson and Sorensen, Inc, USA; 1998. [38] Kroupa JL, Neu RW. The non-isothermal viscoplastic behavior of a titanium–matrix composite. Compos Engng 1994;4(9):965–77. [39] Osborne D, Ghonem H, Chandra N. Interface behavior of Ti matrix composites at elevated temperature. Compos, Part A: Appl Sci Manufact 2000;32(3–4):545–53. [40] Davidson DL. The micromechanics of fatigue crack growth at 25 8C in Ti –6A1–4V reinforced with SCS-6 fibers. Metall Trans 1992;23A: 865–79. [41] Kerans RJ, Parthasarathy TA. Theoretical analysis of the fiber pullout and push out tests. J Am Ceram Soc 1991;74(7):1585–96. [42] Li H, Chandra N. Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models. Int J Plast 2002; in press.