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Modelling of the interphase in polymer-matrix composite material systems
Modelling of the interphase in polymer-matrix composite material systems K.L. REIFSNIDER (Virginia Polytechnic Institute and State University, USA) Received 24 February 1994; revised 14 March 1994 The interphase, or transition region between fibre and matrix in composite material systems, is known to play an essential role in the performance of composite components. In fact, it has recently been discovered that the interaction and cooperative action of the constituents in the system controlthe durability of such systems, in stark contrast to other behaviour characteristics such as stiffness and quasi-static (or 'instantaneous') properties wherein the constituents contribute to the properties, essentially, in proportion to their presence. Hence, this 'systems effect' has become the focus of much attention, and the object of modelling efforts. However, modelling the interphase with properly set boundary value problems is a considerable challenge, especially because the properties in these transition regions are often a function of spatial position, i.e., they are non-uniform as a function of position. The author has constructed models of the interphase region which admit this non-uniformity, and has achieved solutions to several classes of these problems. The present paper will describe the general problem of the 'systems effect', discuss the practical issues of how the interface enters micromechanical models of strength and durability, and present models for the interphase region that include the variability of material properties in those transition regions. Special attention will be given to the contrasts between these considerations and the rule-of-mixtures concepts that currently pervade the field, and to the importance of these differences to the practical opportunities offered by the ability to properly model and 'design' interphase regions in polymer-matrix composites. Key words: interphase; polymer-matrix composites; continuous fibres; micromechanical modelling; tensile strength; compressive strength; long-term behaviour; remaining strength; fife prediction It will be the premise of this paper that the interphase region between the fibres and matrix in continuous fibrereinforced composite material systems greatly influences the long-term performance of such systems, especially the remaining strength and life under combinations of cyclic loading by mechanical means and in the presence of aggressive environments such as high temperature and corrosive chemicals. Further, it will be shown that these variations in long-term performance can be well represented in certain instances by examining the influence of micro-constituent properties on strength of composite material systems as represented by micromechanical models, We will use the usual definition of the interphase region as the region that is formed as a result of the bonding between fibre and matrix which has significantly distinct
morphology or chemical composition compared with the bulk fibre or bulk matrix material. The interphase region may be a diffusion zone, a nucleation zone, a chemical reaction zone, and so forth, or any combination of the aboveL Attempts to modify both the static and longterm properties of composite materials by altering the interphase region between fibres and matrix have been made for many years by a myriad of investigators. Modifications to glass fibre/resin systems date back to the early 1940s, and, by the 1960s, it was clearly shown that the mechanical performance of reinforced plastics depends not only on the fibre and the matrix but also on the nature of the interphase region, which has the function of transferring stresses between the constituent materials2,3. Likewise, the influence of interface and interphase regions in ceramic- and metal-matrix composite materials has been strongly established and widely
001 0-4361/94/07/0461-09 0 1994 Butterworth-Heinemann Ltd COMPOSITES. VOLUME 25. NUMBER 7. 1994 461
Fig. 1 Low magnification scanning electron micrograph of the general grain morphology in a Nicalon/SiC composite processedat 1300°C (with permission of R. Lowden, Martin Marietta, Oak Ridge, TN, USA)
discussed for many years, and is the subject of a great body of recent literature and current research 4-6. Classical representations of the interphase region usually assign constant properties to finite regions of material for the purpose of setting 'simple' mechanics boundary value problems based on those discrete boundaries. However, these representations do not properly recognize the variation of both composition and microstructure (atomic and molecular morphology) from point to point that exists in virtually all 'real' composite material systems in which an interphase region occurs. Fig. 1 provides evidence of this local variability for a silicon carbide/silicon carbide system. This scanning electron micrograph, developed by Ric Lowden and the research group at Martin Marietta/Oak Ridge National Laboratory, clearly shows that the fibre is surrounded not only by a carbon interphase region in these as-processed materials,
Fig. 2 Low magnification scanning electron micrograph of a PVP sizing (dark region) and neighbouring affected region (adjacent ring) in a carbon/epoxy composite
beyond the coating in which the PVP material has diffused into the matrix to cause variations in composition as well as properties as a function of distance from the fibre into the matrix region. Here, again, we see that a region of influence has developed in a manner which is defined by local kinetics and chemistry, as a function of the processing of these composite material systems. From the standpoint of mechanical behaviour-especially from the standpoint of long-term remaining strength and life, usually discussed under the topics of 'durability' and 'damage tolerance'--how can we incorporate the observed properties and property variations that occur in these material systems into the mechanical models which anticipate the influence of these properties and variations on both static and long-term global performance characteristics?
but that there is also a radial variation in microstructure (and by implication, microproperties) in the surrounding matrix material as a function of distance from the centre
THE MICROMECHANICS OF STRENGTH AND LIFE
of each fibre. These spatial variations, or 'gradients' as they are sometimes called, introduce substantial variations in the mechanics required to describe the local stress and strain fields, and ultimately introduce substantial variations in the mechanical behaviour of the composite systems, especially in their durability. The nature of these variations and the mechanics formulations that are necessary to properly describe them are discussed in References 3, 6 and 7.
A large body of research, conducted by many researchers over the past 15-20 years, has clearly shown that the strength and long-term performance (and life)ofcontinuous fibre-reinforced composite material systems is, for the most part, determined by micro-events at the fibre/ matrix interphase leveP ~ ~3. In the next few paragraphs, we will examine just three examples of the manner in which this influence controls behaviour, and try to identify modelling approaches to the inclusion of those details in our estimates of remaining strength and life. We will begin our discussion with the question of tensile strength in the fibre direction and compression strength in the fibre direction, and then addresses the question of predicting remaining strength in the presence of damage induced by mechanical, thermal and chemical environments over long periods of time.
The interphase regions in polymer-based composite systems can be just as important and distinctive as those observed in ceramic- and metal-matrix composites. An extensive set of investigations has shown that variations of as much as 50% in compressive strength and as much as two orders of magnitude in notched fatigue life can be caused by altering just the interphase region, which may be as little as 1% of the total composite by weight8 ~0. These remarkable effects can be caused, for example, by fibre sizing such as the use of polyvinylpyrrolidone (PVP) when carbon fibres and epoxy matrix materials are used. Fig. 2 shows an example of fibres that have been sized with PVP. In addition to a dark region which indicates the PVP coating, the figure shows there is a 'gradient' region
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Fig. 3 indicates the physical nature of the problem associated with the determination of tensile strength in a continuous fibre-reinforced composite in which the fibres control the strength of the material in the fibre direction. In that situation, fracture of the fibres is regarded as the primary event which controls the process of damage development and eventual accumulation of damage to
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cause final failure of the global components. As the figure shows, when a fibre fails, there is a 'region of influence' created around the fibre failure position in which a stress concentration exists and the surrounding material transfers stress to unbroken parts of the composite. At some distance away from the end of the broken fibre, the axial stress in the broken fibre returns to its normal value and the stress field in the composite is 'undisturbed'. This distance of reduced stress in the fibre is called the 'ineffective length' because the ends of the fibre within that region are ineffective in supporting the applied load in the fibre. If the matrix around the fibre (and the interphase region) as well as the composite itself is very stiff, then stresses transfer back into the broken fibre very quickly, and the ineffective length is small in size. In that situation, the local stress concentration near the broken fibre is quite high because of the rapid nature of stress transfer in that region. However, if the material surrounding the broken fibre is compliant, the ineffective length and the corresponding region of influence is quite large, since a large distance is required to transfer stress back into the broken fibre, These two contrasting situations create a very great difference in the response of the material to multiple fibre breaks. If the local region of influence is quite small and the stress concentrations are high, the tendency for neighbouring fibres to break and for a fracture to propagate across the specimen in the region of the first fibre fracture is very great. On the other hand, if the surrounding material is very compliant and the region of influence is quite large, then fibre fractures create a large region of influence which will involve a large volume of material around each fibre break, and those large regions have a greater tendency to interact with other regions created by fibre failures to cause a progressive failure consisting of the interactions of those regions at different points along
the length of different fibres. The classical philosophy associated with the latter sequence of fracture development is called bundle theory, and has been described in considerable detail in the literature by numerous authors 14. Hence, if we were to 'design' a composite for unidirectional tensile strength, we would be motivated by bundle strength theory to use a very stiff matrix and interphase region to reduce the ineffective length, to reduce the possibility of interaction of fibre fractures at different positions along the length of neighbouring fibres. Conversely, if we consider the local stress concentration effects and the corresponding tendency for 'brittle fracture' by a single propagating flaw in the direction transverse to the fibre axes to be controlling, we would be motivated to make the surrounding matrix and interphase region very compliant to reduce that local stress concentration. As is frequently encountered in nature, the correct answer to the design problem for tensile strength in such materials is an optimum choice of properties of the fibre, matrix and interphase region to balance the tendency for rupture associated with local stress concentration against the tendency for rupture due to statistical accumulation of defects in the sense of a bundle strength concept. However, this fact presents a considerable challenge to efforts to model the tensile strength, which are necessary if we have any hope of conducting a systematic and rigorous optimization or providing guidance to the materials development community in their efforts to create composite materials with optimum tensile strengths. Fig. 4 illustrates an example of an attempt to establish a micromechanical representation of the local process that causes failure in the fibre direction under tensile loading for continuous fibre composites in such a way that the
COMPOSITES
. NUMBER
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two dominant physical situations described above can be properly balanced. The tensile model illustrated in Fig. 4 was developed by Gao and Reifsnider ]5, and features a core of broken fibres, an adjacent region of non-linear deformation (which may be plasticity or fibre/matrix sliding), a first unbroken fibre, and an adjacent region of composite material. Equilibrium equations are written (for an axisymmetric representation) for each of those discrete regions, and boundary conditions are matched at the boundaries of those regions. The problem is solved iteratively so that adjacent fibres may fracture, the plastic deformation regions may grow, and statistical criteria of final instability may be accessed to determine the final strength of the global composite 16. This model features a calculation of the local stress concentration in the first adjacent unbroken fibre by various means to determine if fracture progresses in the direction transverse to the loading axis, and assesses bundle strength concepts to determine the stability of the fracture process of damage accumulation, When such a philosophy is constructed, one obtains a predicted maximum in the composite strength relationship as a function of several micro-constituent properties. Fig. 5 illustrates such a phenomenon for the case of tensile strength vs. the yield strength of the matrix material for the interphase region immediately adjacent to a fibre fracture. As we would expect from our earlier discussion, a clear maximum in that relationship is indicated, suggesting that there is a 'correct way' to design the composite material to optimize tensile strength. Hence, using such a model, we can suggest how a composite material should be made to complement the efforts of the materials community to establish how composite specific value of yield strength of the matrix can be specified by the model, theInpreviousthe composite manufacturera materialclearlySyStems can be and made. presentparagraphseXample, can adjust the matrix properties to match that value as well as possible, It is demonstrated in the that the interphase region can have a major influence on the tensile strength o f continuous fibre composites in the
fibre direction. In particular, it is possible to choose an interphase region that yields readily, and provides some
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C O M P O S I T E S . N U M B E R 7 . 1994
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relaxation of the local stress concentration near a fibre fracture, but does not substantially reduce the stiffness of the material in the transverse direction which, as we will see, is important to the enhancement of compressive strength. Moreover, this kind of a 'local design' can be done effectively with fibre coatings which involve small amounts of material and small variations in processing. This inherent efficiency in the manufacturing of such material systems is a major feature and motivation for the design and application of interphase regions as a method of altering strength, remaining strength and life of composite systems. Fig. 6 suggests a comparable sequence of events that contribute to the compressive failure of many continuous fibre composites. Fig. 6(a) shows the first event in the process when microbuckling eventually develops as a failure mode. The reader should be advised that other failure modes are possible, and, in composite materials that have very large fibre diameters, a crushing process may dominate the compressive failure. However, many
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composites form 'kink bands' upon compressive loading, although the maximum load during the process of that kink band formation may occur at different points in the sequence of events that causes the final geometry to form. The question of when the maximum load occurs is a critical one, since that maximum load defines and controls the global compression strength. The process of forming a kink band is inherently an unstable one, and after a maximum load is reached, the local behaviour becomes unstable and forms the final geometry in an essentially instantaneous manner. Hence, we have several questions that need to be addressed in order to solve this problem correctly. Currently, answers are not known to many of the important questions associated with this behaviour, but a considerable amount of literature is available when the physical situation is more precisely defined by experimental work 17,18. In Fig. 6(a), we see the first event in the process of microbuckle or 'kink band' formation. Fig. 6(a) suggests that a local region of fibre begins to buckle over a 'critical length' that is determined by the stiffness of the surrounding material and the degree to which the fibres are bonded to the surrounding matrix and interphase regions. Once again, if the local stiffness of the material is great, and there is a strong bonding between the fibre and the surrounding matrix and interphase regions, the critical microbuckling length becomes small, and the subsequent strength of the composite is generally large. In fact, we have shown in our laboratory that there is a strong correlation between the length of this initial microbuckling region and the final compressive strength of the material ~9. Hence, this initial elastic instability has a major influence on the subsequent events which may be combinations of elastic and plastic behaviour. A second stage in the process is shown in Fig. 6(b) which suggests that the initial elastic buckling process transforms to a shear process and begins the formation of a local kink band by subsequent shearing to form the geometries shown in Fig. 6(c). In Reference 20, Xu and Reifsnider present an analysis of the microbuckling process which incorporates the transverse stiffness of the surrounding matrix and interphase region and also includes the influence of the strength of the interphase on subsequent buckling strength and global compressive strength. The specific inclusion of the interphase region in that micromechanics model provides an opportunity to assess the importance of the interphase to compression strength. Perhaps the most surprising result revealed by that analysis is illustrated in Fig. 7. Therein, the compression strength predicted by the model is plotted as a function of the length of sliding between the fibre and the surrounding matrix or interphase region during the process of fibre/matrix debonding or local plasticity in the interphase region. As can be seen from the figure, the influence of any sliding or plastic deformation between the fibre and the matrix for the interphase is quite large; as that sliding distance becomes even a small fraction of the total critical microbuckling length, the compressive strength of the component drops by as much as 40%. This is, perhaps, one of the most significant indications of the importance of the interphase region to fibre direction strength. However, we have also found that the interphase can greatly enhance compressive strength of continuous fibre composites. In fact, we have shown in
our laboratory that the addition of coatings on carbon fibres in epoxy matrix systems can cause increases of as much as 50% in compressive strength. We have not yet been able to fully understand or to correctly represent the influence of such coatings, but it is clear that several aspects of this problem are distinctive. It is also clear that there is a remarkable opportunity for altering the compressive strength of continuous fibre composites by altering their interphase regions. The previous two examples illustrate the importance of interphase regions to fibre direction tensile and compressive strength. The fact that an interphase, which may involve only a few percent of the total weight or volume of the composite material, can cause large variations in tensile andcompressivestrengthin the directionofloading that is clearly controlled by fibre behaviour is quite remarkable, and quite surprising. Of course, it is even more easily shown that so-called 'matrix controlled' properties such as transfer tension and shear strength are greatly influenced by the interphase region ~9.21. Therefore, we can conclude that strength, in general, is greatly influenced by interphase regions for continuous fibrereinforced composite material systems. This provides an opportunity for us as designers of material systems and as designers of composite components that is unparalleled.
EFFECT OF INTERPHASE REGIONS ON LONGTERM BEHAVIOUR Our previous discussion has indicated that the interphase region between the fibres and matrix in continuous fibrereinforced systems can alter the strength of the composite substantially, even for failure modes that are 'fibre controlled'. Obviously, this is an important finding for engineering applications. However, our experience also suggests that the influence of interphase regions is even greater over long periods of loading (and exposure to aggressive environments), i.e., the durability' and damage tolerance of composites are remarkably sensitive to interphase regions, to the extent that long-term composite
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performance can often be altered by orders of magnitude by alterations of the properties of the interphase regions, Our Materials Response Group has developed an approach to the modelling of long-term behaviour, using simulation of damage and failure events based on mechanistic models 22-24. This is a complex subject that is beyond the available space; we will attempt only to indicate the concepts here and identify how the models of quasi-static strength discussed above can be used in such an approach, to predict remaining strength and life. Fig. 8 illustrates the basic approach to performance simulation that we use in the MRLife T M code series that we have developed. As the figure shows, we begin by identifying failure modes in the laboratory. Each independent and distinct failure mode (such as tensile bundle failure or micro-kink formation, etc.) is modelled separately. Our approach is distinctive in the fact that we set a boundary value (local stress) problem on the basis of the state of the material, and local geometry and loading associated with the fracture event. This is done by defining a 'representative volume' which includes all of the local information that influences the failure event for a given failure mode. This working volume is 'representative' in the sense that we assume, in general, that damage is distributed in a statistically uniform manner, and that failure is caused by damage accumulation. Conceptually, the representative volume is further divided into two parts: a 'critical element', which defines the initiation of the final failure event (i.e., global failure occurs when the critical element fails), and 'subcritical
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C O M P O S I T E S . N U M B E R 7 . 1994
elements', which cause local stress redistribution and geometry changes, but do not cause the failure event, directly. Generally speaking, damage(usually associated with events in the subcritical elements) changes the local state of stress. The condition of the critical element controls the state of the material (which may also degrade). As Fig. 8 shows, we attempt to use micromechanics to represent the changes caused by damage events such as matrix cracking, delamination, debonding and local plasticity (in the subcritical elements, for the most part). Modelling the 'state of material' in a mechanistic sense is more difficult. If we then choose a failure criterion that is appropriate for the failure mode concerned, we can compare the state of material and state of stress to estimate the remaining (global) strength. Comparing that remaining strength to the applied conditions can provide an estimate of remaining life as a function of loading history. The question of how to model changes in the state of material is pertinent to our present discussion. We would like to consider the changes in remaining strength and life of a continuous fibre composite caused by variations in the interphase region between the fibres and matrix material. The changes in the interphase may be hypothetical (as part of a design parameter study) or historical (caused by long-term exposure to mechanical, thermal and chemical 'loadings'). In the context of the critical element scheme illustrated by Fig. 8, the failure criterion selected for a given failure mode will, generally, have a form that compares the local stresses to the current value of material strengths, of the general form
constituent (and interface/ interphase)material parameters appearing in t h e tensile and compressionmodels T a b l e 1. List o f
r t local stresses I (Fractional chance of failure) = ~ l o c a l strengths] (l) where the local material strengths may be a tensor array of numbers for anisotropic materials. As damage develops, the local stresses change because the distribution of local stresses is altered by such things as cracks, that relax the stresses in some elements of volume (near a stress-free crack face, for example) and increase stresses in other areas (such as near a crack tip). The material strengths also change, due to degradation driven by mechanical, chemical and thermodynamic processes, Physical ageing may, for example, alter the free volume of polymer-matrix materials and alter the strength of that constituent. At elevated temperatures, oxidation may weaken carbon fibres, or diffusion may alter composition from point to point, etc. However, we can use the micromechanical representations of strengths discussed earlier to address these changes. Table 1 is a partial listing ofthe parameters that appear in the micromechanical models. The elastic stiffness and strength of each of the constituents, including the interphase regions, appears independently in the micromechanical tensile and strength models. (In Table 1, 'T' indicates that tensile strength (as we have modelled it)
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is affected, and 'C' indicates that the parameter appears in the compression model.) Hence, if matrix cracking or viscoelastic deformation alters the matrix stiffness, for example, that change (which can be precisely and independently evaluated in the laboratory) can be entered directly into the micromechanical model, and the composite response assessed directly. Table 1 also indicates that the geometry and arrangement of the constituents, the yield or debonding strength of the interphase region or the matrix near the fibre/matrix interphase, and the
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COMPOSITES . NUMBER 7 . 1994
467
location and shape parameters of the fibre strength distribution enter the strength models. Moreover, the parameter, 77, which is associated with the degree to which the fibre and matrix are in initial contact, and the parameter, s, which denotes the length along the fibre over which fibre/matrix debonding and sliding occurs, also appear. Each of these parameters may be altered by long-term exposure to mechanical, chemical or thermal environments, and those alterations, and their interactions, are 'automatically integrated' and interpreted into global remaining strength by the micromechanical strength representations and the simulation code. Fig. 9 shows a simulation from our MRLife code using the compression model mentioned above. Several simulations were made of the damage development process in a critical element having dimensions of about 4.6 mm, located next to the centre hole boundary at 90 ° to the loading direction (which is vertical in the figure). In one simulation, only matrix cracking was assumed in that region; in another, both matrix cracking and debonding between the fibres and matrix was allowed, progressively as a function of the number of cycles of loading. The predicted lives are seen to vary between about 940 000 cycles and less than 50 000 cycles, when the compression failure mode controls life. (Predictions using an assumed tensile failure mode were not so widely spaced.) The simulations were run in an attempt to represent the effect of coating carbon fibres with polyvinylpyrrolidone (PVP), as discussed earlier. In the present case of notched fatigue, that coating enables longitudinal splitting to relax the stress concentration near the centre hole. The longitudinal splits are clearly shown in the radiograph at the top right of Fig. 9. Under these conditions, the observed life is over one million cycles, compared with the estimate of about 940 000 cycles to failure. When a 'standard' epoxy sizing is used, the observed life is about 10 000 cycles, nearly t w o o r d e r s o f magnitude less (and less than the 50 000 or so cycles predicted by the model). The radiograph in the bottom left of Fig. 9 shows the corresponding damage state, which is much more severe in the 'critical element' which controls the final fracture of the component.
SUMMARY It is clear from the experience reported here, and from much additional information in the literature, that the interphase region between the fibre and matrix in a continuous fibre-reinforced composite can greatly alter strength and life, even for polymer-matrix composites, and even for properties normally thought of as 'fibre controlled'. An approach to the representation of these effects has been suggested, which is based on the use of micromechanical strength models to 'assemble' and integrate the influences of the constituents and the interfaces and interphase regions between them. Moreover, it would appear that this approach provides a method of 'designing' composite material systems for specific properties and for specific long-term behaviour, i.e., to specify the properties, geometry and arrangement of the constituents and interphase regions to achieve specified durability and damage tolerance performance. Although we have travelled only a short distance down this road
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and much is yet to be done, initial results are encouraging, and the MRLife simulation tool is already capable of serving the applied community well.
ACKNOWLEDGEMENTS The author gratefully acknowledges the financial support of the NSF Science and Technology Centre for High Performance Polymeric Adhesives and Composites at Virginia Tech under contract DMR-8809714, and the Virginia Institute for Material Systems. The author also acknowledges with thanks the assistance of Shelia Collins in the preparation of the manuscript.
REFERENCES
1 Swain, R., Reifsnider, K.L., Jayaraman, K. and EI-Zein, M. 'Interface/interphase concepts in composite materials systems' J Thermoplastic Composites3(1990)pp 13-23 2 Plueddemann, E.P. (Ed) Interfaces in Polymer Matrix Composites (Academic Press, New York, 1974) 3 Jayaraman, K., Reifsnider, K.L. and Swain, R.E. "Elastic and thermal effectsin the interphase: Part 1. Comments on characterization methods' J Composites Technol and Res 15 (1993)pp 3 13 4 Brennan, J.J. 'Interracial characterizationof glass and glass-ceramic matrix/Nicalon SiC fiber composites' in Tailoring Multiphase and Composite Ceramics edited by R.E. Tressler, G.L. Messing, C.G. Pantano and R.E. Newnham (Plenum Press, New York, 1985) pp 549 560 5 Petrasek, D.W. and Weeton, J.W. 'Effects of alloying on roomtemperature tensile properties of tungsten-fiber-reinforcedcopper-alloy composites' Trans AIME 230 (1964) pp 977 990 6 Jayaraman, K. and Reifsnider, K.L. 'Residual stresses in a composite with continuously varying Young's modulus in the fiber/ matrix interphase' J Composite Mater 26 (June 1992) pp 770-791 7 Jayaraman, K. and Reffsnider, K.L. 'Local stress fieldsin a unidirectional fiber-reinforced composite with a non-homogeneous interphase region: formulation' Adv Composites Left l (1992) pp 54 57 8 Ho, H., Lesko, J., Morton, J., Wilkinson, S., Ward, T. and Reifsnider, K.L. 'Effect of fiber treatment on the in-plane shear properties of composite materials" J Adhesion 42 (1993) pp 39-53 9 Lesko, J., Fogg, B., Miller, W., Jengsankar, A., Reifsnider, K.L and Claus, R. 'Embedded Fabry Perot fiber optic strain sensors in macromodel composites' Optical Engng 31 No 1 (January 1992) p 13 10 Lesko, J., Swain, R.E., Cartwright, J.M., Chin, J.W., Reifsnider, K.L., Dillard, D.A. and Wightman, J.P. 'Interphases developed from sizings and their chemical structural relationship to composite compressive performance' submitted to J Adhesion 11 Reifsnider, K.L. 'Interpretation of laboratory test information for residual strength and life prediction of composite systems' in Cyclic Deformation, Fracture, and Nondestructive Evaluation and Advanced Materials, A S T M STP 1157 edited by M.R. Mitchell and O. Buck (American Society for Testing and Materials, Philadelphia, PA, 1992) pp 205-223 12 Fatigue ofCompositeMaterialseditedbyK.L. Reifsnider(Elsevier Science Publishers, New York, 1991) 13 Chou, T.W. Microstructural Design of Fiber Composites (Cambridge University Press, New York, 1992) 14 Tsai, S.W. and Hahn, H.T. Introduction to Composite Materials (Technomic Publishing Co, Westport, CT, 1980) 15 Gao, Z. and Reifsnider, K.L. 'Micromechanicsof tensile strength' A S T M STP from 4th Syrup on Composite Materials." Fatigue and Fracture (American Society for Testing and Materials, Philadelphia, PA, in press) 16 Batdod, S.B. 'Tensilestrength of unidirectionally reinforcedcomposites' J Reinforced Plastics and Composites I (1982) p 153 17 Stief, P.S. 'An exact two-dimensionalapproach to fiber microbuckling' lnt J Solids and Structures 23 (1987)pp 1235 1246 18 Wins, A.M., Babcock Jr, C.D. and Knauss, W.G. 'A mechanical model for elastic fiber microbuckling' J Appl Mech 57 (1990) pp 138-149 19 Lesko, J.J., Swain, R.E., Cartwright, J.M., Chin, J.M., Reifsnider, K.L., Dillard, D.A. and Wightman, J.P. 'Interphase developed
20 21 22 23
from sizings and their chemical structural relationship to cornposite performance" J Adhesion (in press) Xu, Y.L. and Reifsnider,K.L. 'Micromechanical modeling ofcomposite compressive strength" J Composite Mater 27 (1993) pp 572 587 Case, S.W. "Micromechanics of strength-related phenomena in composite materials' M S Thesis (Virginia Polytechnic Institute and State University, Blacksburg, VA, May 1993) Subramanian, S. "Effect of fiber matrix interphase on the long term behavior of cross-ply laminates' PhD Dissertation (Virginia Polytechnic Institute and State University, Blacksburg, VA, 1994) Reifsnider, K.L. 'Use of mechanistic life prediction methods for the design of damage tolerant composite material systems" in Advances in Fatigue L(/btime Predictive Techniques, A S T M STP
1211 edited by M.R. Mitchell and R.W. Landgraf (American Society for Testing and Materials, Philadelphia, PA, 1993) pp 3 18 24 Reifsnider, K.L. "Performance simulation of polymer based composite systems' Proe Int Symp on DurabiliO, o[ Polymer Based Composite Systemsjbr Structural Applications (Elsevier Applied Science, New York, 1991) pp 3 26
AUTHORS
Dr Reifsnider is with the Materials Response Group at Virginia Polytechnic Institute and State University, Blacksburg, VA, USA.
COMPOSITES. NUMBER 7. 1994
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morphology or chemical composition compared with the bulk fibre or bulk matrix material. The interphase region may be a diffusion zone, a nucleation zone, a chemical reaction zone, and so forth, or any combination of the aboveL Attempts to modify both the static and longterm properties of composite materials by altering the interphase region between fibres and matrix have been made for many years by a myriad of investigators. Modifications to glass fibre/resin systems date back to the early 1940s, and, by the 1960s, it was clearly shown that the mechanical performance of reinforced plastics depends not only on the fibre and the matrix but also on the nature of the interphase region, which has the function of transferring stresses between the constituent materials2,3. Likewise, the influence of interface and interphase regions in ceramic- and metal-matrix composite materials has been strongly established and widely
001 0-4361/94/07/0461-09 0 1994 Butterworth-Heinemann Ltd COMPOSITES. VOLUME 25. NUMBER 7. 1994 461
Fig. 1 Low magnification scanning electron micrograph of the general grain morphology in a Nicalon/SiC composite processedat 1300°C (with permission of R. Lowden, Martin Marietta, Oak Ridge, TN, USA)
discussed for many years, and is the subject of a great body of recent literature and current research 4-6. Classical representations of the interphase region usually assign constant properties to finite regions of material for the purpose of setting 'simple' mechanics boundary value problems based on those discrete boundaries. However, these representations do not properly recognize the variation of both composition and microstructure (atomic and molecular morphology) from point to point that exists in virtually all 'real' composite material systems in which an interphase region occurs. Fig. 1 provides evidence of this local variability for a silicon carbide/silicon carbide system. This scanning electron micrograph, developed by Ric Lowden and the research group at Martin Marietta/Oak Ridge National Laboratory, clearly shows that the fibre is surrounded not only by a carbon interphase region in these as-processed materials,
Fig. 2 Low magnification scanning electron micrograph of a PVP sizing (dark region) and neighbouring affected region (adjacent ring) in a carbon/epoxy composite
beyond the coating in which the PVP material has diffused into the matrix to cause variations in composition as well as properties as a function of distance from the fibre into the matrix region. Here, again, we see that a region of influence has developed in a manner which is defined by local kinetics and chemistry, as a function of the processing of these composite material systems. From the standpoint of mechanical behaviour-especially from the standpoint of long-term remaining strength and life, usually discussed under the topics of 'durability' and 'damage tolerance'--how can we incorporate the observed properties and property variations that occur in these material systems into the mechanical models which anticipate the influence of these properties and variations on both static and long-term global performance characteristics?
but that there is also a radial variation in microstructure (and by implication, microproperties) in the surrounding matrix material as a function of distance from the centre
THE MICROMECHANICS OF STRENGTH AND LIFE
of each fibre. These spatial variations, or 'gradients' as they are sometimes called, introduce substantial variations in the mechanics required to describe the local stress and strain fields, and ultimately introduce substantial variations in the mechanical behaviour of the composite systems, especially in their durability. The nature of these variations and the mechanics formulations that are necessary to properly describe them are discussed in References 3, 6 and 7.
A large body of research, conducted by many researchers over the past 15-20 years, has clearly shown that the strength and long-term performance (and life)ofcontinuous fibre-reinforced composite material systems is, for the most part, determined by micro-events at the fibre/ matrix interphase leveP ~ ~3. In the next few paragraphs, we will examine just three examples of the manner in which this influence controls behaviour, and try to identify modelling approaches to the inclusion of those details in our estimates of remaining strength and life. We will begin our discussion with the question of tensile strength in the fibre direction and compression strength in the fibre direction, and then addresses the question of predicting remaining strength in the presence of damage induced by mechanical, thermal and chemical environments over long periods of time.
The interphase regions in polymer-based composite systems can be just as important and distinctive as those observed in ceramic- and metal-matrix composites. An extensive set of investigations has shown that variations of as much as 50% in compressive strength and as much as two orders of magnitude in notched fatigue life can be caused by altering just the interphase region, which may be as little as 1% of the total composite by weight8 ~0. These remarkable effects can be caused, for example, by fibre sizing such as the use of polyvinylpyrrolidone (PVP) when carbon fibres and epoxy matrix materials are used. Fig. 2 shows an example of fibres that have been sized with PVP. In addition to a dark region which indicates the PVP coating, the figure shows there is a 'gradient' region
462
C O M P O S I T E S . N U M B E R 7 . 1994
Fig. 3 indicates the physical nature of the problem associated with the determination of tensile strength in a continuous fibre-reinforced composite in which the fibres control the strength of the material in the fibre direction. In that situation, fracture of the fibres is regarded as the primary event which controls the process of damage development and eventual accumulation of damage to
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cause final failure of the global components. As the figure shows, when a fibre fails, there is a 'region of influence' created around the fibre failure position in which a stress concentration exists and the surrounding material transfers stress to unbroken parts of the composite. At some distance away from the end of the broken fibre, the axial stress in the broken fibre returns to its normal value and the stress field in the composite is 'undisturbed'. This distance of reduced stress in the fibre is called the 'ineffective length' because the ends of the fibre within that region are ineffective in supporting the applied load in the fibre. If the matrix around the fibre (and the interphase region) as well as the composite itself is very stiff, then stresses transfer back into the broken fibre very quickly, and the ineffective length is small in size. In that situation, the local stress concentration near the broken fibre is quite high because of the rapid nature of stress transfer in that region. However, if the material surrounding the broken fibre is compliant, the ineffective length and the corresponding region of influence is quite large, since a large distance is required to transfer stress back into the broken fibre, These two contrasting situations create a very great difference in the response of the material to multiple fibre breaks. If the local region of influence is quite small and the stress concentrations are high, the tendency for neighbouring fibres to break and for a fracture to propagate across the specimen in the region of the first fibre fracture is very great. On the other hand, if the surrounding material is very compliant and the region of influence is quite large, then fibre fractures create a large region of influence which will involve a large volume of material around each fibre break, and those large regions have a greater tendency to interact with other regions created by fibre failures to cause a progressive failure consisting of the interactions of those regions at different points along
the length of different fibres. The classical philosophy associated with the latter sequence of fracture development is called bundle theory, and has been described in considerable detail in the literature by numerous authors 14. Hence, if we were to 'design' a composite for unidirectional tensile strength, we would be motivated by bundle strength theory to use a very stiff matrix and interphase region to reduce the ineffective length, to reduce the possibility of interaction of fibre fractures at different positions along the length of neighbouring fibres. Conversely, if we consider the local stress concentration effects and the corresponding tendency for 'brittle fracture' by a single propagating flaw in the direction transverse to the fibre axes to be controlling, we would be motivated to make the surrounding matrix and interphase region very compliant to reduce that local stress concentration. As is frequently encountered in nature, the correct answer to the design problem for tensile strength in such materials is an optimum choice of properties of the fibre, matrix and interphase region to balance the tendency for rupture associated with local stress concentration against the tendency for rupture due to statistical accumulation of defects in the sense of a bundle strength concept. However, this fact presents a considerable challenge to efforts to model the tensile strength, which are necessary if we have any hope of conducting a systematic and rigorous optimization or providing guidance to the materials development community in their efforts to create composite materials with optimum tensile strengths. Fig. 4 illustrates an example of an attempt to establish a micromechanical representation of the local process that causes failure in the fibre direction under tensile loading for continuous fibre composites in such a way that the
COMPOSITES
. NUMBER
7. 1994
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two dominant physical situations described above can be properly balanced. The tensile model illustrated in Fig. 4 was developed by Gao and Reifsnider ]5, and features a core of broken fibres, an adjacent region of non-linear deformation (which may be plasticity or fibre/matrix sliding), a first unbroken fibre, and an adjacent region of composite material. Equilibrium equations are written (for an axisymmetric representation) for each of those discrete regions, and boundary conditions are matched at the boundaries of those regions. The problem is solved iteratively so that adjacent fibres may fracture, the plastic deformation regions may grow, and statistical criteria of final instability may be accessed to determine the final strength of the global composite 16. This model features a calculation of the local stress concentration in the first adjacent unbroken fibre by various means to determine if fracture progresses in the direction transverse to the loading axis, and assesses bundle strength concepts to determine the stability of the fracture process of damage accumulation, When such a philosophy is constructed, one obtains a predicted maximum in the composite strength relationship as a function of several micro-constituent properties. Fig. 5 illustrates such a phenomenon for the case of tensile strength vs. the yield strength of the matrix material for the interphase region immediately adjacent to a fibre fracture. As we would expect from our earlier discussion, a clear maximum in that relationship is indicated, suggesting that there is a 'correct way' to design the composite material to optimize tensile strength. Hence, using such a model, we can suggest how a composite material should be made to complement the efforts of the materials community to establish how composite specific value of yield strength of the matrix can be specified by the model, theInpreviousthe composite manufacturera materialclearlySyStems can be and made. presentparagraphseXample, can adjust the matrix properties to match that value as well as possible, It is demonstrated in the that the interphase region can have a major influence on the tensile strength o f continuous fibre composites in the
fibre direction. In particular, it is possible to choose an interphase region that yields readily, and provides some
464
C O M P O S I T E S . N U M B E R 7 . 1994
0.49
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~0 ~5 20 25 30 35 40 4:5 50 Interfacial Strength (ksi)
Fig. 5 Variation of composite tensile strength as a function of the fibre/matrix interface (or interphase) strength, predicted by the model
relaxation of the local stress concentration near a fibre fracture, but does not substantially reduce the stiffness of the material in the transverse direction which, as we will see, is important to the enhancement of compressive strength. Moreover, this kind of a 'local design' can be done effectively with fibre coatings which involve small amounts of material and small variations in processing. This inherent efficiency in the manufacturing of such material systems is a major feature and motivation for the design and application of interphase regions as a method of altering strength, remaining strength and life of composite systems. Fig. 6 suggests a comparable sequence of events that contribute to the compressive failure of many continuous fibre composites. Fig. 6(a) shows the first event in the process when microbuckling eventually develops as a failure mode. The reader should be advised that other failure modes are possible, and, in composite materials that have very large fibre diameters, a crushing process may dominate the compressive failure. However, many
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composites form 'kink bands' upon compressive loading, although the maximum load during the process of that kink band formation may occur at different points in the sequence of events that causes the final geometry to form. The question of when the maximum load occurs is a critical one, since that maximum load defines and controls the global compression strength. The process of forming a kink band is inherently an unstable one, and after a maximum load is reached, the local behaviour becomes unstable and forms the final geometry in an essentially instantaneous manner. Hence, we have several questions that need to be addressed in order to solve this problem correctly. Currently, answers are not known to many of the important questions associated with this behaviour, but a considerable amount of literature is available when the physical situation is more precisely defined by experimental work 17,18. In Fig. 6(a), we see the first event in the process of microbuckle or 'kink band' formation. Fig. 6(a) suggests that a local region of fibre begins to buckle over a 'critical length' that is determined by the stiffness of the surrounding material and the degree to which the fibres are bonded to the surrounding matrix and interphase regions. Once again, if the local stiffness of the material is great, and there is a strong bonding between the fibre and the surrounding matrix and interphase regions, the critical microbuckling length becomes small, and the subsequent strength of the composite is generally large. In fact, we have shown in our laboratory that there is a strong correlation between the length of this initial microbuckling region and the final compressive strength of the material ~9. Hence, this initial elastic instability has a major influence on the subsequent events which may be combinations of elastic and plastic behaviour. A second stage in the process is shown in Fig. 6(b) which suggests that the initial elastic buckling process transforms to a shear process and begins the formation of a local kink band by subsequent shearing to form the geometries shown in Fig. 6(c). In Reference 20, Xu and Reifsnider present an analysis of the microbuckling process which incorporates the transverse stiffness of the surrounding matrix and interphase region and also includes the influence of the strength of the interphase on subsequent buckling strength and global compressive strength. The specific inclusion of the interphase region in that micromechanics model provides an opportunity to assess the importance of the interphase to compression strength. Perhaps the most surprising result revealed by that analysis is illustrated in Fig. 7. Therein, the compression strength predicted by the model is plotted as a function of the length of sliding between the fibre and the surrounding matrix or interphase region during the process of fibre/matrix debonding or local plasticity in the interphase region. As can be seen from the figure, the influence of any sliding or plastic deformation between the fibre and the matrix for the interphase is quite large; as that sliding distance becomes even a small fraction of the total critical microbuckling length, the compressive strength of the component drops by as much as 40%. This is, perhaps, one of the most significant indications of the importance of the interphase region to fibre direction strength. However, we have also found that the interphase can greatly enhance compressive strength of continuous fibre composites. In fact, we have shown in
our laboratory that the addition of coatings on carbon fibres in epoxy matrix systems can cause increases of as much as 50% in compressive strength. We have not yet been able to fully understand or to correctly represent the influence of such coatings, but it is clear that several aspects of this problem are distinctive. It is also clear that there is a remarkable opportunity for altering the compressive strength of continuous fibre composites by altering their interphase regions. The previous two examples illustrate the importance of interphase regions to fibre direction tensile and compressive strength. The fact that an interphase, which may involve only a few percent of the total weight or volume of the composite material, can cause large variations in tensile andcompressivestrengthin the directionofloading that is clearly controlled by fibre behaviour is quite remarkable, and quite surprising. Of course, it is even more easily shown that so-called 'matrix controlled' properties such as transfer tension and shear strength are greatly influenced by the interphase region ~9.21. Therefore, we can conclude that strength, in general, is greatly influenced by interphase regions for continuous fibrereinforced composite material systems. This provides an opportunity for us as designers of material systems and as designers of composite components that is unparalleled.
EFFECT OF INTERPHASE REGIONS ON LONGTERM BEHAVIOUR Our previous discussion has indicated that the interphase region between the fibres and matrix in continuous fibrereinforced systems can alter the strength of the composite substantially, even for failure modes that are 'fibre controlled'. Obviously, this is an important finding for engineering applications. However, our experience also suggests that the influence of interphase regions is even greater over long periods of loading (and exposure to aggressive environments), i.e., the durability' and damage tolerance of composites are remarkably sensitive to interphase regions, to the extent that long-term composite
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COMPOSITES.
NUMBER
7 . 1994
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performance can often be altered by orders of magnitude by alterations of the properties of the interphase regions, Our Materials Response Group has developed an approach to the modelling of long-term behaviour, using simulation of damage and failure events based on mechanistic models 22-24. This is a complex subject that is beyond the available space; we will attempt only to indicate the concepts here and identify how the models of quasi-static strength discussed above can be used in such an approach, to predict remaining strength and life. Fig. 8 illustrates the basic approach to performance simulation that we use in the MRLife T M code series that we have developed. As the figure shows, we begin by identifying failure modes in the laboratory. Each independent and distinct failure mode (such as tensile bundle failure or micro-kink formation, etc.) is modelled separately. Our approach is distinctive in the fact that we set a boundary value (local stress) problem on the basis of the state of the material, and local geometry and loading associated with the fracture event. This is done by defining a 'representative volume' which includes all of the local information that influences the failure event for a given failure mode. This working volume is 'representative' in the sense that we assume, in general, that damage is distributed in a statistically uniform manner, and that failure is caused by damage accumulation. Conceptually, the representative volume is further divided into two parts: a 'critical element', which defines the initiation of the final failure event (i.e., global failure occurs when the critical element fails), and 'subcritical
466
A A
C O M P O S I T E S . N U M B E R 7 . 1994
elements', which cause local stress redistribution and geometry changes, but do not cause the failure event, directly. Generally speaking, damage(usually associated with events in the subcritical elements) changes the local state of stress. The condition of the critical element controls the state of the material (which may also degrade). As Fig. 8 shows, we attempt to use micromechanics to represent the changes caused by damage events such as matrix cracking, delamination, debonding and local plasticity (in the subcritical elements, for the most part). Modelling the 'state of material' in a mechanistic sense is more difficult. If we then choose a failure criterion that is appropriate for the failure mode concerned, we can compare the state of material and state of stress to estimate the remaining (global) strength. Comparing that remaining strength to the applied conditions can provide an estimate of remaining life as a function of loading history. The question of how to model changes in the state of material is pertinent to our present discussion. We would like to consider the changes in remaining strength and life of a continuous fibre composite caused by variations in the interphase region between the fibres and matrix material. The changes in the interphase may be hypothetical (as part of a design parameter study) or historical (caused by long-term exposure to mechanical, thermal and chemical 'loadings'). In the context of the critical element scheme illustrated by Fig. 8, the failure criterion selected for a given failure mode will, generally, have a form that compares the local stresses to the current value of material strengths, of the general form
constituent (and interface/ interphase)material parameters appearing in t h e tensile and compressionmodels T a b l e 1. List o f
r t local stresses I (Fractional chance of failure) = ~ l o c a l strengths] (l) where the local material strengths may be a tensor array of numbers for anisotropic materials. As damage develops, the local stresses change because the distribution of local stresses is altered by such things as cracks, that relax the stresses in some elements of volume (near a stress-free crack face, for example) and increase stresses in other areas (such as near a crack tip). The material strengths also change, due to degradation driven by mechanical, chemical and thermodynamic processes, Physical ageing may, for example, alter the free volume of polymer-matrix materials and alter the strength of that constituent. At elevated temperatures, oxidation may weaken carbon fibres, or diffusion may alter composition from point to point, etc. However, we can use the micromechanical representations of strengths discussed earlier to address these changes. Table 1 is a partial listing ofthe parameters that appear in the micromechanical models. The elastic stiffness and strength of each of the constituents, including the interphase regions, appears independently in the micromechanical tensile and strength models. (In Table 1, 'T' indicates that tensile strength (as we have modelled it)
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is affected, and 'C' indicates that the parameter appears in the compression model.) Hence, if matrix cracking or viscoelastic deformation alters the matrix stiffness, for example, that change (which can be precisely and independently evaluated in the laboratory) can be entered directly into the micromechanical model, and the composite response assessed directly. Table 1 also indicates that the geometry and arrangement of the constituents, the yield or debonding strength of the interphase region or the matrix near the fibre/matrix interphase, and the
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COMPOSITES . NUMBER 7 . 1994
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location and shape parameters of the fibre strength distribution enter the strength models. Moreover, the parameter, 77, which is associated with the degree to which the fibre and matrix are in initial contact, and the parameter, s, which denotes the length along the fibre over which fibre/matrix debonding and sliding occurs, also appear. Each of these parameters may be altered by long-term exposure to mechanical, chemical or thermal environments, and those alterations, and their interactions, are 'automatically integrated' and interpreted into global remaining strength by the micromechanical strength representations and the simulation code. Fig. 9 shows a simulation from our MRLife code using the compression model mentioned above. Several simulations were made of the damage development process in a critical element having dimensions of about 4.6 mm, located next to the centre hole boundary at 90 ° to the loading direction (which is vertical in the figure). In one simulation, only matrix cracking was assumed in that region; in another, both matrix cracking and debonding between the fibres and matrix was allowed, progressively as a function of the number of cycles of loading. The predicted lives are seen to vary between about 940 000 cycles and less than 50 000 cycles, when the compression failure mode controls life. (Predictions using an assumed tensile failure mode were not so widely spaced.) The simulations were run in an attempt to represent the effect of coating carbon fibres with polyvinylpyrrolidone (PVP), as discussed earlier. In the present case of notched fatigue, that coating enables longitudinal splitting to relax the stress concentration near the centre hole. The longitudinal splits are clearly shown in the radiograph at the top right of Fig. 9. Under these conditions, the observed life is over one million cycles, compared with the estimate of about 940 000 cycles to failure. When a 'standard' epoxy sizing is used, the observed life is about 10 000 cycles, nearly t w o o r d e r s o f magnitude less (and less than the 50 000 or so cycles predicted by the model). The radiograph in the bottom left of Fig. 9 shows the corresponding damage state, which is much more severe in the 'critical element' which controls the final fracture of the component.
SUMMARY It is clear from the experience reported here, and from much additional information in the literature, that the interphase region between the fibre and matrix in a continuous fibre-reinforced composite can greatly alter strength and life, even for polymer-matrix composites, and even for properties normally thought of as 'fibre controlled'. An approach to the representation of these effects has been suggested, which is based on the use of micromechanical strength models to 'assemble' and integrate the influences of the constituents and the interfaces and interphase regions between them. Moreover, it would appear that this approach provides a method of 'designing' composite material systems for specific properties and for specific long-term behaviour, i.e., to specify the properties, geometry and arrangement of the constituents and interphase regions to achieve specified durability and damage tolerance performance. Although we have travelled only a short distance down this road
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and much is yet to be done, initial results are encouraging, and the MRLife simulation tool is already capable of serving the applied community well.
ACKNOWLEDGEMENTS The author gratefully acknowledges the financial support of the NSF Science and Technology Centre for High Performance Polymeric Adhesives and Composites at Virginia Tech under contract DMR-8809714, and the Virginia Institute for Material Systems. The author also acknowledges with thanks the assistance of Shelia Collins in the preparation of the manuscript.
REFERENCES
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AUTHORS
Dr Reifsnider is with the Materials Response Group at Virginia Polytechnic Institute and State University, Blacksburg, VA, USA.
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