Damage and Damage Mechanics

Fatigue of Composite Materials edited by K.L. Reifsnider © Elsevier Science Publishers B.V.,

1990

Chapter 2

Damage and Damage

Mechanics

K E N N E T H L. R E I F S N I D E R Materials Response Group, Engineering Science and Mechanics Department, Institute and State University, Blacksburg, VA 24061-4899 (U.S.A.)

Virginia

Polytechnic

Contents Abstract 11 1. Introduction 11 2. The fatigue effect in composite materials 12 3. The fatigue process 17 4. The mechanics of damage development 26 4.1. Notched damage development 49 5. The mechanics of strength reduction 53 6. Concluding remarks 73 Acknowledgements 74 References 75

Abstract This chapter attempts to establish the fundamental nature of fatigue as a physical process in composite materials, and to identify some of the fundamental aspects of mechanics representations of that process. The chapter begins with a discussion of the physical details of " d a m a g e " , and uses those details to define "the fatigue effect" for composite materials, especially composite laminates. Requirements for mechanics representations of the fatigue process are discussed, and limitations of classical representations are indicated. Strength reduction is introduced as a method of esti­ mating remaining life, and the computational aspects of that approach are explored. A specific mechanistic example of such an approach, the "critical element method", is described, and some results of that modeling scheme are examined. Horizons for continuing work are identified and discussed. 1. Introduction The word damage, as such, seems to have appeared in early forms of French as a noun, based on the Latin " d a m n u m " for "loss or h u r t " . Verb forms appeared in early 11

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English writings, and the word was widely used and broadly interpreted by uncounted authors including Huxley, Byron, and Shakespeare. Throughout this period of development to the modern adoption by the scientific community, the word has maintained the general interpretation of harm or injury that diminishes the value or usefulness of the object which is " d a m a g e d " . Physical explanations of damage usually associate the concept with loss of integrity such as micro-crack formation, or with the degradation of properties such as loss of strength. In the present context, we will be concerned with damage as the physical consequence of "fatigue", or more particularly, as the measurable physical evidence of a fatigue process. It is the purpose of this chapter to examine the physical details of fatigue damage in composite materials, to identify the rudiments of mechanics representations of damage, and to examine a few engineering consequences of damage development and growth. To that end, sections will be dedicated to the "fatigue effect in composite materials", "the fatigue process", "the mechanics of damage development", and "the mechanics of strength reduction". A "closure" is added to identify continuing directions, needs, and opportunities.

2. The fatigue effect in composite materials We begin by recognizing that the "fatigue effect" is defined, in part, by the differing perspectives of individuals who investigate and characterize material behavior, those who design and build engineering components, and users who are concerned with the long-term performance of such engineering devices. We might expect a physical chemist or physicist to view the phenomenon of fatigue in terms of molecular or atomic motions which bring about a progressive change in the material properties during the application of some load history. The mechanicist might view the situation as the generation of micro-variations in geometry, especially the formation of cracks, and the association of stress concentrations with those micro-details. The designer/engineer is likely to view the situation at a somewhat more global level and be concerned only with phenomenological representations which help him to answer the question, " H o w long will this component last?" To begin our discussion, we will adopt the perspective of the investigator (scientist or engineer) who asks the fairly basic question, " W h a t is the fatigue effect in composite materials?" We further assume that the general objective of our inquiry is to reach a level of development of understanding and descriptive ability so that we can control, quantitatively describe, and predict the strength, stiffness, and life of engineering laminates under long-term loading in the presence of the fatigue effect. To this end, our approach will be primarily mechanistic and fundamental, but somewhat exclusive in the sense that we will choose the specific subjects of our discussion from an extremely large array of details and phenomena associated with the general subject, on the basis of our best estimate of the funda­ mental nature of their association with the long-term engineering behavior of com­ posite materials.

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We define fatigue damage as the cycle-dependent degradation of internal integrity. Although it is acknowledged that the degree of influence of these changes in integrity on the mechanical response of materials may vary greatly, our definition suggests that we will only be concerned with those events and processes which are directly related to strength, stiffness, or life in some engineering sense. Even with such restrictions, we must state that the micro-events which reduce the strength and stiffness, and deter­ mine the life of composite materials, are complex, various, and intricately related to a variety of failure modes in different circumstances. This is a consequence of the physical and mechanical nature of the micro events that contribute to degradation of integrity, and the complexity of the materials themselves, especially their inhomogeneity. Indeed, the types of events that occur under fatigue loading in composite materials are frequently associated with the details of that inhomogeneity. For the general class of composite materials that appear to be of interest for engineer­ ing applications, especially high-modulus fibrous composites, and particulate or fiber-reinforced metal, polymer, glass, or ceramic matrix composites, the types of micro-events that contribute to the process of damage development can usually be classified in categories such as those suggested in fig. 1. For such material systems, the reinforcement is usually chosen on the basis of its ability to contribute high strength or high stiffness to the composite, whereas the matrix material is chosen primarily on the basis of considerations such as its ability to transfer load at the micro-level, its resistance to environmental effects, and its ease of manufacture. F o r such material systems, as suggested by fig. 1, damage events include anelastic deformation (such as plastic flow, crazing, and viscoelastic behavior), micro-cracking of both the matrix and the reinforcement, debonding of the matrix and reinforcement phases, delamination of plies or laminae of a laminate, and various combinations and CONSTITUENT

DAMAGE

MICRO-CRACKING CHEMICAL PLASTIC

DEGRADATION DEFORMATION

CRAZING

BOUNDARY

SEPARATION

DEBONDING DELAMINATION INTERPHASE

INHOMOGENEOUS YIELDING

CRACKING

DEFORMATION OF BOUNDARY

MATERIAL

DISCONTINUOUS

DEFORMATION

DISCONTINUOUS

ROTATION

GRADIENTS

GRADIENTS

Fig. 1. Example categories of damage in composite materials.

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superimpositions of these events that create the damage accumulation and growth associated with loss of internal integrity. Although it may appear that these events are isolated and distinct, and indeed in the literature they are frequently discussed as such, it is the premise of the writer that the more c o m m o n circumstance of their occurrence is a causative sequence of events that are related mechanistically and mechanically. We will defer the discussion of such a sequence, to deal first with the more funda­ mental question, " W h a t is the basic nature and origin of the cycle-dependence of properties and behavior in a damage process?" We emphasize again that this volume is concerned primarily with cycle-dependent degradation. We admit that, although this is a reasonable pretence, it is not possible to separate completely the contributions or occurrences of cycle-dependent and time-dependent physical phenomena that are associated with fatigue behavior. Nevertheless, as we are concerned here with mechanistic aspects of the fatigue effect, an attempt will be made to provide some physical arguments for the occurrence of cycle-dependent damage development. The "fatigue effect", in general terms, is the reduction of remaining strength or stiffness, and possible failure, after a finite number of load cycles has been applied. These load cycles are smaller in magnitude than the load required to cause failure in a single cycle. Hence, some process of "fatigue d a m a g e " must lower the strength of the component under load. However, the damage process is not uniform, i.e. not all volume elements are equally reduced in strength. Instead, damage generally consists of discrete events which cause non-uniform or inhomogeneous material response. Perhaps the most universal characteristic of the "fatigue effect" is inhomogeneous deformation. As a result, there is frequently a significant micro-geometric contribution to the fatigue effect, e.g., micro-cracks cause local stress concentrations which cause further damage, etc. Although it is not always feasible or necessary to represent these individual geometric details in models of the fatigue process, in special cases they dominate the problem and provide a good basis for analysis. The use of linear elastic fracture mechanics ( L E F M ) to describe fatigue degradation as the propagation of a single dominant crack in homogeneous materials is an example of such a situation. That representation is not always adequate for homogeneous materials, and is often not appropriate for inhomogeneous composite materials. An apparent exception to this generalization may be the problem of delamination of laminar composites, a phenomenon which is mechanistically similar to self-similar single-crack propagation in homogeneous materials. The basic physical principle behind any cycle-dependent behavior is the occurrence of non-conservative deformation which changes the internal nature or geometry of the material and its ability to respond to continuing load histories. In the present context, non-conservative deformation means that some of the energy introduced into the material by the work done by applied tractions or displacements is not stored as strain energy, but is dissipated as the driving force for some internal process such as micro-crack formation or growth, thermodynamic events such as the rearrangement of molecular or atomic morphology (diffusion, grain boundary motion, etc.), chemi­ cal events such as stress-assisted corrosion, and a variety of other internal events. Although not all non-conservative processes produce fatigue in the sense that it

Damage and damage

(a)

(b)

mechanics

(c)

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(d)

Fig. 2. A conceptual hypothesis for the mechanism of crack extension in homogeneous materials in which ductile yielding at the crack tip causes a crack surface increase which forms a new increment or crack length upon unloading (I), and a generalization of that concept in which the plastic deformation is replaced by some more general process (II).

reduces the remaining stiffness, strength, or life, it is believed that all fatigue damage processes are non-conservative. F r o m the standpoint of mechanics, this means that the current state of the material and state of stress are dependent upon the history of prior loading and associated deformation. Although there are a myriad of circum­ stances and consequences associated with such non-conservative behavior, certain aspects are of particular interest to us. Perhaps the most commonly cited cause for cycle-dependent deformation is illustrated in fig. 2. It has to do with local geometry variation and a zone of behavior that develops at the tip of a single crack during cyclic loading in a crack-opening mode. The non-conservative deformation process is illustrated by fig. 2(d), which indicates that the loading history of an element of material in the neighborhood of the crack tip is different in the loading phase from the unloading phase so that a certain amount of energy is dissipated in the cyclic deformation process. If the material responds by plastic deformation near the crack tip, a scenario suggested by case I of fig. (2a-c) may account for cycle-dependent behavior. In that scenario it is imagined that the plastic deformation which occurs upon loading as shown in fig. 2(b) creates a new section of crack surface which collapses ahead of the crack upon unloading as indicated in fig. 2(c). This new crack length, δ advances the tip of the crack with each successive cycle. In actuality, then, the non-conservative material behaviour shown in fig. 2(b) (plastic deformation) combines with a local change in geometry as shown in fig. 2(c), to create a process which advances the crack and creates progressive degra­ dation with continued loading. Although the details of this scenario should not be taken too literally for a given material, the general philosophy and principles involved are widely cited for the case of single through-crack growth in common metals and other homogeneous materials. The second scenario illustrated in case II of fig. 2 is a generalization of the first. The figure suggests that upon loading [indicated by fig. 2(b)], a process zone develops

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ahead of the crack tip. Within that process zone, non-conservative deformation processes of a general nature are assumed to occur. When the loading of a given cycle is relaxed, as in fig. 2(c), the state of the material and state of stress in the process zone created by the prior loading are different from the initial states corresponding to the prior unloaded case [fig. 2(a)]. Hence, upon reloading, it is reasonable to expect that a new response to crack opening will occur, possibly causing further crack extension during the next cycle. The important generalization to be made here is that any non-conservative response of material in the vicinity of a crack tip will contribute a local load-history dependence of material behavior which can cause progressive growth of the crack, or in general, irreversible changes in the associated state of the material and state of stress. This is the essence of the damage process. Of course, it is clear that this non-conservative activity may be either cycle-dependent or timedependent (e.g. viscoelastic deformation). Although the geometry readjustment scenario described in the first part of fig. 2(a) is one of the easiest types of local behavior to visualize, it is not hard to imagine that other types of material behavior such as dislocation motion, polymeric crazing, or a variety of mechanical or chemical micro-structural rearrangements may also contribute to this type of damage process. It is also necessary to mention that a dependence on time may enter the problem in various ways. Two of the more important ones are illustrated in fig. 3. In fig. 3(a) is depicted the fracture of a fiber during the cyclic tensile loading of a fibrous composite material. The stress wave created by that fracture travels away from the fracture position at the speed of sound in the directions of wave propagation in that material. Information on the stress redistribution caused by that fiber fracture can travel to the neighboring material at a rate limited by that propagation speed. Hence, there is an inherent limitation in the rate at which damage in one position can cause damage in another position by stress redistribution. If we now consider the fact that the distribution of fiber strengths is statistical and, therefore, the distance to the next statistically stronger fiber may be several hundred fiber diameters or more, and consider the fact that the load may be diminishing, because of its cyclic nature, during

/

\

(a)

/

\

/

\

(b)

Fig. 3. Two examples of time-dependent processes: stress wave propagation from a local fracture event (a), and a local activation process such as plastic deformation (b).

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the time of travel, it is possible that the next fiber to break as a result of load redistribution will not fracture until the next loading cycle or until subsequent events provide a local driving force. A similar situation is shown in fig. 3(b). In that situation, a local rate process such as plastic deformation, a chemical process, or viscoelastic behavior progresses at a rate controlled by the local driving force available and the inherent activation energy required for the events involved. Hence, the degree and extent of development of such behavior is influenced by the a m o u n t of time spent at each load level during the cyclic loading process. For composite materials the considerations illustrated in figs. 2 and 3 must be superimposed because of the complexity of damage, which typically consists of various combinations of the events listed in fig. 1. In fact, it is u n c o m m o n for a single crack to dominate the damage development and fracture process in composite materials. F o r such inhomogeneous materials it is more common to speak of damage accumulation than of damage propagation in the simple sense associated with crack propagation in homogeneous metals, for example. Still, it is useful to identify many micro-damage events in terms of initiation and propagation, at least at the microlevel. The physical behavior associated with initiation-controlled damage develop­ ment and propagation-controlled damage development is somewhat distinct. The manner in which initiation and propagation are described and modeled are generally different. As indicated earlier, the primary focus of the present treatment of fatigue in composite materials is the response of laminated composite systems. This is largely because of the experience of the writer, which suggests that laminated composites will continue to form the basis for a majority of the engineering components which use composite materials. To generate a systematic treatment of the mechanics of the damage development process in composite laminates, we introduce the concept of damage as a causative sequence and discuss the generic physical aspects of damage modes and events as a background for analysis. 3. The fatigue process We will be concerned with load histories thought to be associated with c o m m o n engineering service requirements, and especially those associated with long but finite life. It is assumed that the load histories experienced by the composite laminates to be discussed include both tensile and compressive load excursions, and that the laminates include plies which have various orientations to a principle load direction, called the 0° direction. Under such conditions, we might expect some combination of damage modes to occur, as suggested in fig. 1. Perhaps it is well to note that the literature contains a profuse record of observations of damage results and damage events; that record can be consulted for specific and special interests of the reader. Starting points for such an effort include a number of Special Technical Publications published by the American Society for Testing and Materials, and a myriad of papers (cf., refs 1-10). The fatigue damage development process for many of the fibrous composite material systems commonly used in engineering components is illustrated in generic fashion by the events depicted in figs. 4-9. The degree of development

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of each of the stages shown in those figures will generally depend upon the stacking sequence, the ply thicknesses, and the material types in a specific laminate, but the basic nature of the damage is thought to be represented by the details in those figures. We will focus on the generic aspects of the details in these figures for the present discussion. A more complete discussion of specific details associated with some of the figures as well as a more comprehensive discussion of laminate fatigue performance in general can be found in the chapter on "Fatigue Behavior of Composite Laminates" in this volume. For laminates that have off-axis plies, such as the common [0, 90, ± 4 5 ] quasiisotropic stacking sequence, the first and most prolific damage mode observed is usually matrix cracking. Although the toughness and ductility of the matrix material may accelerate or retard the initiation of such cracks, even to the extent of suppressing matrix crack formation entirely for quasi-static loading in some cases, cyclic loading is known to cause matrix cracks in virtually every laminated high-modulus con­ tinuous fiber composite material system known to the writer (which has been carefully examined for such behavior) as of this writing. This includes metal matrix and ceramic matrix materials as well as polymer matrix systems (cf. refs. 1-11). These cracks usually form through the thickness of the plies in a direction parallel to the fibers and perpendicular (at least in transverse projection) to the dominant load axis (the " 0 ° " direction). Figure 4 shows a schematic diagram of matrix crack formation in a cross-ply laminate (a) and an edge replica of an actual specimen of that type showing such cracks in a graphite epoxy coupon (b). Figure 5 shows an edge replica of a specimen with multiple cracks in off-axis plies, some coupled together by edge delaminations. This type of transverse crack formation has received a great deal of attention and is, in comparison with other micro-events, relatively well described and understood. The threshold of initiation of such cracks can be reasonably well anticipated by laminate analysis coupled with a common "failure theory" such as the maximum strain, Tsai-Wu, or Tsai-Hill concepts for quasi-static loading. F o r cyclic loading, however, the use of such analyses to predict the incidence of first crack formation in the weakest plies (sometimes called first ply failure) is at best an engineering approximation. Although this situation does not appear to have been completely investigated experimentally, it does appear that one can say that matrix cracks will always form at cyclic load amplitudes which exceed the quasi-static load level known to cause first ply failure, but it is also possible that cracks will occur during cyclic loading at loads somewhat below those calculated for the quasi-static case. It has been shown that it is important to consider thermal and hygroscopic residual stress in attempts to calculate the cyclic or quasi-static load levels associated with such initiation events [12]. For certain metal matrix composites (or others that do not develop matrix cracks under quasi-static loading), initiation under cyclic loading can be expected to occur for stress amplitudes above the "shakedown limit", the threshold amplitude above which the hysteresis loop associated with cyclic loading does not stabilize [11]. Although the estimation of these stress levels associated with first ply failure can be a useful design exercise, the experience of the writer suggests that such cracks should be assumed to be present in any attempt to characterize or model the s

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Fig. 4. Schematic diagram of matrix crack formation in a cross-ply laminate (a), and an edge replica of an actual specimen of that type showing matrix cracks in a graphite epoxy coupon (b).

fatigue behavior of a composite laminate, especially as even a single cycle of tensile overload will generally introduce such cracks in significant numbers. The mechanics of matrix crack formation constrains the number of cracks that form in a specific type of laminate (i.e. a specific material system and geometry). In fact, a specific pattern of cracks known as the "characteristic damage state" for matrix cracking provides an upper limit or "saturation state" for this type of matrix cracking [7, 10, 12]. We will discuss the characteristic damage state in more detail in the next section, which deals with the mechanics of these damage processes. Throughout our discussion here, we will refer to the type of matrix cracks shown in fig. 4. as "primary cracks". Although it is true that these cracks generally form under either quasi-static or cyclic loading, subsequent damage micro-events which are influenced or caused

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Fig. 5. Edge replica showing multiple cracking of off-axis plies with crack coupling by edge delamination.

by the presence of primary cracks differ considerably under those two types of loading. Under cyclic loading which includes both tensile and compressive load excursions, the formation of primary matrix cracks is the beginning of a causative sequence of micro-damage events which ultimately determines the residual strength and life of a given laminate. This subsequent damage sequence (or process) is responsible for strength reduction during fatigue loading, as the primary cracks themselves occur under quasi-static loading to failure in a test of initial strength. The most immediate cyclic consequence of these cracks is the occurrence of fiber fractures which begin to form in adjacent plies near the crack tips, even within the first third of life [13-15]. Figure 6(a) is a schematic diagram of such fiber fractures near primary matrix

(a)

(b)

(c)

Fig. 6. Schematic diagram of fiber fracture patterns associated with matrix cracks (a), the distribution of fiber fractures through adjacent plies near a primary matrix crack (b), and local debonding near the tips of the broken fiber ends (c).

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Fig. 7. Line of fiber fractures in the 0° ply of a quasi-isotropic graphite epoxy laminate near the tip of a primary matrix crack in a 90° ply identified by the deposit of gold particles (white specks in the photograph) along the line of fiber fractures carried to that position by a doped penetrant material introduced into the matrix cracks.

crack tips. T h a t figure also includes schematic diagrams of the internal distribution of fiber fractures through 0° plies near a primary matrix crack (b), and the local debonding that occurs near the tips of the broken fiber ends (c). It should be emphasized that these fiber fractures have been observed near the edge of a speci­ men, preferentially located near crack tips; they have also been observed in the interior of laminate specimens, where they form periodic arrays of fiber fracture lines which have the same spacing as the primary matrix cracks. The density of fiber fractures drops off as a function of distance from the matrix crack tips either in the direction of the thickness of the adjacent ply or in any direction away from the crack tip in the planar interface between the broken and unbroken plies. Figure 7 shows such a line of internal fiber fractures in the 0° plies of a quasi-isotropic graphite epoxy laminate, near a primary matrix crack in an adjacent 90° ply. The position of the matrix crack has been marked by gold particles (white "dust-like" particles in the figure), which were carried to that position by a liquid penetrant introduced into the matrix cracks. These fiber fractures are especially significant, as fibers control the residual strength of essentially all of the engineering laminated composite systems in use today. Determination of the precise physical and micro-mechanical nature of the initiation, growth, and consequence of fiber fracture

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Fig. 8. Radiographs showing primary and secondary cracks in a cross-ply laminate (a), and a [0/ ± 45] laminate (b) after cyclic loading. Arrow shows a locally delaminated region in the interface between two plies near the point at which matrix cracks cross.

s

in composite laminates presents one of the most challenging problems in the field of the mechanical response of composite laminates. The third major type of micro-damage event which occurs in the causative sequence of damage development in laminates is represented in fig. 8, which shows two radiographic photographs of primary and secondary cracks in a cross-ply laminate (a) and a [0, ± 45] specimen (b). These secondary cracks generally form in the latter part of the life of a specimen or component. They are initiated because of the tensile normal stress field which is created parallel to the crack front of the primary matrix cracks, as described in the next section. In some cases, these cracks extend through the thickness of the adjacent ply and for a considerable distance across the tips of primary cracks in the adjacent ply, as shown in fig. 8(a). However, it is more common for these cracks to be limited in extent, extending only a short distance away from the primary crack tip in the direction of the plane between the plies which contain the primary and secondary cracks, and only a short distance into the thickness of the ply s

Damage and damage

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J& 4 P

~

•

: · · ' ;

!~5

>

WÈÈÈÊÈÊm

Fig. 9 . Radiograp h showin g primar y crack s (horizontal ) an d secondar y crack s (vertical ) whic h cros s a t a n interface betwee n th e plies , causin g loca l delaminatio n i n th e regio n indicate d b y th e arrow .

which i s sustainin g secondar y cracking , a s illustrate d i n fig. 8(b) . I n polyme r matri x materials, i t i s commo n t o find thes e secondar y crack s i n grea t number s ubiquitousl y distributed throughou t th e laminate . The pl y interfac e positio n wher e primar y an d secondar y crack s cros s create s a region o f specia l interest . W e wil l se e i n th e nex t sectio n tha t th e stresse s i n tha t regio n are peculia r t o th e loca l microstructura l geometr y create d b y th e interactio n o f th e stress fields o f th e crosse d crack s a t th e intersection . I n fact , a possibl e consequenc e of suc h stres s fields i s a loca l delaminate d regio n whic h develop s nea r th e crosse d crack position , a s illustrate d i n fig. 9 . Tha t figure show s a n X-ra y radiograp h whic h indicates severa l crosse d crack s (dar k lines ) i n a cross-pl y laminate , an d a loca l internal delaminate d regio n (show n b y th e arrow ) a t th e pl y interface . A simila r internal delaminatio n i s show n b y th e arro w i n fig. 8(a) . W e hav e bee n abl e t o establish tha t thes e locall y delaminate d region s are , i n fact , associate d wit h th e incidence o f th e crossin g o f primar y an d secondar y crack s [13-15] .

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Fig. 10. Schematic diagram of (half of the thickness) of a cross-ply laminate, showing a regular array of primary cracks, and examples of secondary matrix cracking and local delamination forming near the primary crack tips at the interface between the 0° and 90° plies. The fiber fractures (not shown) would also accumulate in the localized regions near the crossed cracks.

Hence, the causative sequence of micro-damage development in composite laminates under combined tensile and compressive loading (excluding dominant edge effects, which we have not yet discussed) can be thought of as a process of initiation and successive localization which includes initiation of primary matrix cracks, fiber fractures, local debonding, secondary cracking, and possible local delamination which becomes more and more concentrated in a small region around the intersection of primary and secondary cracks, as suggested by the schematic diagram in fig. 10. Although it is possible that some local propagation may be occurring (between broken fibers, for example), it is important to notice that this causative process primarily involves initiation in contrast to propagation, and localization in contrast to dispersed, independent accumulation of damage events. These are essential dis­ tinctions because the mechanics of propagation, nucleation, localization, and accumulation are generally different. It is also important to notice that localization involves the interaction of several different types of damage. If valid mechanistic models of the strength and life of composite laminates are to be developed, it is imperative that such details be included in the representations. In an earlier paragraph we mentioned that we would delay the discussion of edge-induced delamination as a special case of damage development. Although we will deal with some of the details of the mechanical situation responsible for this particular damage mode in the next section, it should be mentioned here that it is well established that, in general, laminates which are fabricated from anisotropic plies of material that terminate at a common edge generate highly three-dimensional stress states at those positions which may introduce interlaminar stresses that are sufficiently high to cause interlaminar failure initiation and propagation beginning at those locations. The mechanics of this situation is closely related to the mechanics of the crossed-crack problem that we mentioned in association with the internal delamination noted above as a generic damage mode. However, the practical

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b) 50 Cycles Fig. 11. Delaminations at the edge of a [0/ ± 45/90] graphite epoxy laminate under static loading (a), and after 50 cycles of fatigue loading (b). s

consequences of edge delamination are more easily recognized as edges are common to many engineering structures, not only at the termination of the geometry of a given component but also as boundaries to cut-outs, holes, fasteners, joints, and attach­ ments. F o r this reason, a great deal of attention has been focused in the engineering community on this particular damage mode [16]. Figure 11 shows the damage pattern at the edge of a graphite epoxy laminate which includes regions of delamination. The patterns shown in the figures suggest a relationship between the development of edge delamination and the occurrence of transverse matrix cracks which terminate at the edge of the specimen. Such a relationship is common and was also noted earlier for the internal delaminations discussed above. Although the delaminations shown in fig. 11 are fairly limited in extent, it is possible for these delaminations to grow in a self-similar manner into the interior of a laminate, propagating as a large, single, dominant interlaminar crack in a manner which is reminiscent of through-crack propagation in common homogeneous materials such as metals. Figure 12 shows a radiograph which reveals several large delaminated regions in a coupon specimen. It is easy to imagine that such a delamination could grow to dimensions which are sufficiently large to influence the mechanical response of an engineering component. This is especially likely to occur if the global loading spectrum includes both tension and compression load excursions, as micro- or macro-buckling can provide a major driving force for the propagation of such delaminated regions. In the opinion of the writer, the propagation of such delamination is much more thoroughly understood and completely described than the initiation of such a damage

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Fig. 12. Radiograph showing large delaminations near the edge of a quasi isotropic graphite epoxy laminate after cyclic loading.

mode. Although some authors appear to have been successful in designing special specimens in which delamination initiates as a " p u r e " damage mode, it is much more common in virtually all of the composite material systems familiar to the writer to find delamination initiating at locations where matrix cracks terminate at a ply boundary, or where that termination also involves a transverse crossing-crack perpendicular to a primary matrix crack, in the nature described earlier. The general nature of the mechanics of both the initiation and propagation events will be described in the next section, based on fundamental concepts, with references given to more specialized and less generic cases of the development of this particular type of damage mode [6, 16-20]. 4. The mechanics of damage development We begin by defining the mechanics of damage development to be those phil­ osophies and analytical formulations based on principles of mechanics which are associated with states of stress and states of material that are peculiar to high-modulus fibrous composite laminates which endure loading histories known to cause changes in laminate strength, stiffness, and life because of micro-failure events collectively called " d a m a g e " . We will be concerned with load histories thought to be associated with common engineering service requirements and especially those associated with long (but finite) life. We will attempt to build on the concept, introduced in the previous section, that damage is a causative sequence. Hence, we will make a special effort to develop systematic rational philosophy which emphasizes the association between events, rather than indulge in a desultory recitation of observations and listing of events. To that end, the experimental data that will be presented have been carefully chosen to represent generic behavior (in the opinion of the writer) whenever possible. However, we are reminded that this field is at present incompletely formalized; it is not possible to give a general axiomatic development of the mechanics of damage development based on first principles which accurately describes the strength, stiffness, and life of continuous-fiber laminated composite materials as a function of time (or number of applied cycles) at the time of writing. Hence, our approach to presentation will be by example based on experience. In that context, the reader should be forewarned that, although this chapter will address the collective experience of the technical community, the first-hand experience of the writer will form the context for the development presented and the perspectives aired. We will

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LAMINATE RESPONSE

INHOMOGENEOUS

ANISOTROPIC

CONSTITUENT-LEVEL DAMAGE EVENTS

COMPLEX MACRO-STRESS STATES

DAMAGE GROWTH BARRIER

COUPLING OF RESPONSE

GENERIC DAMAGE PATTERNS

COMPLEX INTERNAL STRESS

DAMAGE LOCALIZATION

COMPLEX STRENGTH CONCEPTS

COMPLEX MICRO-STRESS STATES Fig. 13. Listing of events, characteristics, and responses influenced by inhomogeneity and anisotropy of composite materials.

attempt to build a systematic development based on what are thought to be the central and most important general issues. There are two special characteristics of composite laminates which have a major influence on the manner in which mechanics representations of damage development are set. As suggested by fig. 13, those characteristics are inhomogeneity and aniso­ tropy. Of the two, inhomogeneity has the greater influence on laminate response to cyclic loading. F r o m the behavioral standpoint, inhomogeneity presents a dichotomy. In many respects, it is the single most important factor contributing to the generally superior resistance of laminated composite materials to fatigue damage development. At the same time, it is almost certainly the greatest contributing factor in the initiation of damage at the micro-level. In many respects, the mechanics of fatigue damage development (for homogeneous as well as inhomogeneous materials) can be con­ sidered to be concerned with a description of progressive deformation localization. It is not difficult to imagine that inhomogeneity contributes greatly to this localization process. First, as fig. 13 suggests, local damage events play the role of damage initiators. As we have seen earlier, matrix cracking, debonding, and ply separation are examples of these events. However, damage growth is severely inhibited by the presence of adjacent plies or adjacent constituent phases, particularly by their different properties and response, and also because of micro-separations between constituent materials in some cases. In fact, this constraint on damage growth forms the basis for a progressive localization and intensification of damage in a successively smaller volume of material as load cycling continues. However, it should be empha­ sized that this localization process is repeated (with variations in details peculiar to local situations) in patterns of damage that are widely distributed throughout the specimen. This widely distributed damage development process and the resulting damage patterns (which are often associated with the geometric patterns created by the arrangement of constituent materials) are a characteristic of laminated composite

28

K.L.

Reifsnider

materials (and composite materials in general), and set apart the behavior of these material systems from homogeneous materials. Characteristic distributed-damage development is the basis for the widely accepted approach to this subject known as damage accumulation. Finally, inhomogeneity contributes to the complexity of the stress states associated with damage development. This is true in the simple sense, as the extreme complexity of damage caused by inhomogeneity can be expected to contribute a highly non­ uniform complex micro-stress state to the problem. However, it is also true in a less direct sense as, even before damage develops, the material inhomogeneity creates a complex internal stress state associated with internal load-sharing among constituent materials that generally involves strongly two- or three-dimensional stress fields with complex coupling, even for unidirectional global loading. The dimensional scale of these inhomogeneities is often critically important. As their scale becomes very small so that the internal structure causes inhomogeneity over very small distances, the material becomes more "homogeneous". Indeed, many common metals which are inhomogeneous below length scales of the order of 1 0 in. (the order of 1 μιη) are often assumed to be " h o m o g e n e o u s " for purposes of analysis. Highly homogeneous materials (down to a very fine scale) are most resistant to the initiation of inhomogeneous deformation (and subsequent fatigue), whereas highly inhomogeneous materials are most resistant to the growth of inhomogeneous deformation (such as cracks). Composite materials, in general, fit into the latter category and are very "fatigue resistant" in that sense. However, the " o p t i m u m inhomogeneity" is highly dependent upon comparative constituent stiffnesses and strengths, constituent geometries, and active damage and failure modes for specific loading situations. This is a very complex and incompletely understood topic. It plays a critical role in many cases, especially for high-temperature ceramic composites and particulate-reinforced materials. 4

It is also clear that anisotropy has a major influence on damage development. The most obvious source of this influence is the complex stress state associated with reinforcement phases (particles or fibers) that may be aligned or arranged in some geometric fashion which results in a directional dependence of the mechanical properties of the composite system. A m o n g other things, these complex stress states may result in coupling; for example, coupling of axial extension to shear behavior. However, the most challenging effect of anisotropy has to do with the complex internal stress redistribution that is associated with damage. This aspect of the problem is especially challenging when (as is usually the case) the synergistic effects of anisotropy and inhomogeneity are present. We noted earlier that damage develops in a manner which is peculiar to the inhomogeneity of most composite systems. It is also true that damage develops in a manner which is peculiar to the anisotropy (global and local) in composite systems. As we have mentioned in the previous paragraphs, cracks develop in directions parallel to fibers in continuous-fiber-reinforced materials. However, when the plies of an inhomogeneous laminate meet at an interface, these cracks oppose one another in their directions as described above. The stress fields associated with these types of defect interactions are distinctly and uniquely threedimensional and present a very great challenge to efforts to achieve an analytical

Damage and damage

mechanics

29

DAMAGE MODES DURING FATIGUE LIFE I-Matrix Cracking Fiber Breaking

3-Delamination Fiber Breaking

PERCENT OF LIFE

5-Fracture

100

Fig. 14. Schematic representation of the development of damage during the fatigue life of a composite laminate.

representation of the consequences of cumulative damage in composite laminates. The anisotropy and inhomogeneity of such a composite system are essential ingredients to any valid representation of the mechanics of damage development; the degree to which they are properly represented is a primary factor in the determination of the accuracy of such a representation. Our approach to the discussion of the subject of damage mechanics can be outlined with the help of fig. 14. It is assumed that the load histories experienced by the composite laminates to be discussed include both tensile and compressive load excursions, and that the laminates include plies which have various orientations to a principle load direction (the 0° direction). Under such conditions, we might expect some combination of damage modes to occur as suggested by the schematic diagrams in fig. 14. Although the number of possible damage modes and combinations of damage modes is large, the number of failure modes is comparatively small. We will present our subject chronologically, as suggested by fig. 14. We will begin with a discussion of primary crack formation followed by a discussion of fiber fracture, the mechanics of secondary cracking, and finally a discussion of internal delamination. Separate sections will be dedicated to the discussion of edge delamination, and to an extension of the development given below for continuous-fiber laminated composite systems to other composite systems. Finally, a major section will be dedicated to the mechanics of strength reduction as a natural application of the ideas and information to follow.

30

K.L.

Reifsnider

Ρ

Fig. 15. Schematic diagram of "first ply failure" in a cross-ply laminate, and the associated change in stiffness of the laminate under uniaxial loading.

As we noted in the previous discussion, matrix cracks can be expected to initiate in plies which experience tensile stress excursions in directions perpendicular to the fibers in those plies, if those excursions exceed amplitudes which are sufficient to cause failure of the matrix material between the fibers or separation of the fibers and the matrix phase. These cracks generally occur in composite systems which have brittle matrix materials (such as graphite epoxy and glass epoxy systems), but also occur under cyclic loading in composite systems which have ductile matrix components such as the metal matrix class of composites. Figure 15 is a schematic representation of the situation for quasi-static loading normally referred to as "first ply failure". That terminology is commonly used to describe the conceptual situation wherein the first (primary) cracks in off-axis plies form. As indicated by the figure, the stiffness of the laminate is diminished by the reduction in stiffness of the plies, which crack because the load originally carried by those plies is redistributed to some extent into the remaining undamaged material. Although the concept of first ply failure is certainly a clear and reasonable one, the corresponding physical phenomenon is not well defined and is not nearly as distinct as suggested by the abrupt change in slope of the stress-strain curve indicated schematically in fig. 15. In fact, the detection of the first transverse crack to occur in a large specimen is a matter of diligence and interpretation experimentally, and the corresponding change in slope of the stressstrain curve is generally a reasonably gradual one as the formation of a single crack or a small number of cracks in a large volume of specimen material will not cause a large change in the total stiffness of a laminate. Nevertheless, an estimate of the load or strain level at which primary matrix cracking will begin to occur is useful for at least two major reasons. First, as we have seen in the previous section, primary matrix cracks are the source of subsequent damage development under cyclic loading, and form the basis not only for damage development localization under tensile load excusions, but also for the development

Damage and damage

El

mechanics

SPECIMEN A2 0 0 , 9 8 , 9 8 , 9 0 3 ·

0.00

0.S0

1.00

1.50

31

El

2.00

2 . 5 0 E3

LOAD CLBS) Fig. 16. Change in crack density in the 90° plies of a glass epoxy specimen under uniaxial quasi-static loading and the associated change in laminate stiffness.

of localized buckling and the growth of delamination under compressive load excursions. And second, matrix cracking is observed to occur at similar load or strain levels in cyclic loading as is observed for quasi-static loading situations, although this comparison is clouded by the fact that under cyclic loading crack densities multiply at a given fixed load amplitude, whereas under quasi-static loading it is very difficult to identify the lowest load level at which first ply failure (the first crack) occurs. For this reason as much as any other, it is difficult to compare experimental observations of first ply failure with estimates of the quasi-static load level at which that failure should begin. "Failure theories" are commonly used to estimate the load level which causes the initiation of primary cracks; the T s a i - W u theory and the maximum strain theory are common examples. Attempts to use a Griffith-type energy balance criterion to predict the onset of cracking are also common (cf. refs. 21 and 22). With cyclic load, the reduction in stiffness happens continuously over a period of time (cycles of loading), as indicated in fig. 16, rather than at a specific load and strain value (or range of values), as suggested by fig. 15. However, the association between the number of cracks that form and the compliance of the specimen is just as direct and absolute, as indicated by the proportionality between those two quantities shown by the typical curve in fig. 16. The mechanics of this situation can be recovered from a representation of the stress relaxation around the primary cracks that form in the off-axis plies. Even a simple one-dimensional shear lag scheme captures many of the main features of the total laminate behavior as suggested by fig. 17 [7, 23]. Of course, more complex and complete representations can be constructed; in fact, Laws and Dvorak have recently used an effective modulus scheme to obtain an excellent representation of the data shown in fig. 17 [24]. Attempts have also been made to generate representations of the formation of multiple primary cracks in off-axis plies during cyclic loading [25]. Chou and Wang postulated the existence of a "damage function" which maps the quasi-static crack

KL.

32

Reifsnider

[0,90,90,90] 1.25

—

r

s

E-GLASS

I

Δ

ι

EXPERIMENTAL (SPECIMEN THEORETICAL

*> ω

1

0

(SHEAR

Α2)

LAG)

0

LU

ζ Ll

t 0.75 μ­ α) û UJ Ν

-j

-

Δ Δ

0.50

^"LAMINATE

<

Δ

& ANALYSIS

Έ

ce

9

0.25

-

0.00

ι

0.00

0.50

1

1.00

1

1.50

2.00X10"

CRACK DENSITY ( C R A C K S / I N . ) Fig. 17. Comparison of calculated stiffness change in a glass epoxy specimen as a function of crack density with observed data, and the limit on that change of stiffness calculated by a discount method using laminate analysis.

density curve as a function of load level onto a corresponding crack density curve as a function of the number of cycles at a corresponding load. They then assumed a statistical distribution of inherent material micro-flaws which grow continuously during time with fatigue loading. When the individual size of those flaws reaches a characteristic length, a primary crack is formed. This provides the necessary criterion for a flaw to become a crack, and defines the time (cycles) under load required to form a crack with the aid of a flaw growth equation. The flaw growth equation is assumed to be a power law in the strain energy release rate associated with crack growth in the specific plies of a given laminate, which can be estimated using numerical analysis schemes. Figure 18 provides examples (taken at r a n d o m from the data in ref. 25) of the types of results produced by such an exercise. Actually, the prediction of the occurrence (or absence) of such primary cracks is of relatively little consequence in the engineering sense. However, the stress fields around these cracks are of considerable consequence in the causative sequence of damage that develops during fatigue loading. It is possible to anticipate the number and arrange­ ment of such cracks, and this information can be used for subsequent analysis of behavior. However, the mechanics of this situation is controlled by the details of the multiple crack pattern and not by the individual stress fields around each of the cracks acting alone. This problem was made tractable by the startling discovery that primary cracks in off-axis plies do not continue to initiate monotonically throughout a quasi-static or cyclic loading test. Instead, the number of cracks in a given off-axis ply "saturates" for a given cyclic load level, and thus creates a stable pattern of regularly spaced

Damage and damage

mechanics

33

30

S

I 2 25h~ 3 4 5 20

5 3 ksi ( 3 6 5 . 4 M P a ) - d a t a 4 3 ksi ( 2 9 6 . 5 M P a ) - d a t a 4 3 ksi - simulation 3 8 ksi - simulation 3 8 ksi ( 2 6 2 M P a ) - d a t a [0 ,90 ] 2

2

y

σ

10

/

2

y /

s

(from Chou, et. al.) ι. Ο

I I

/

/

/

-

10

I 1

0

1 2

Ί

^ 3

1 4

1

1

5

6

7

Cycles (log n) Fig. 18. Predicted and observed crack densities generated by Chou and Wang [25].

10.0

Ε CD Ζ Ο

100 ~

1

APPLIED STRESS (ΜΡα) 200 300 400 500 1

8.0

600 1

700

Ο FATIGUE DATA • QUASISTATIC DATA

6.0

< û_ 4.0

CO

ο

< 2.0

Δ

οor

0

(

• I 1 0.2 0.4 0.6 0.8 NO. OF CYCLES (MILLIONS)

1.0

Fig. 19. Crack spacing in 45° plies of a [0/90 + 45] graphite epoxy laminate as a function of increasing quasi-static load or cycles. s

cracks which remains essentially undisturbed throughout the remainder of the life of the specimen. In our 1977 paper announcing this discovery, this generic pattern of cracks was named the "characteristic damage state" (CDS) of the laminate [26]. The formation of such a pattern is indicated by fig. 19, which shows the spacing between cracks in the —45° plies of a [0, 90, ± 4 5 ] AS-3501-5 graphite epoxy laminate as a function of quasi-static load level and cycles of loading at about two-thirds of the ultimate strength (R = 0.1). Cracks develop quite early in the life and quickly stabilize to a very nearly constant pattern with a fixed spacing. However, the same behavior occurs for quasi-static or cyclic loading, in the sense that crack development occurs s

K.L.

34

Reifsnider

N O R M A L S T R E S S IN BROKEN PLY J

*

•/

/-STRESS LEVEL WHICH FORMED PRIOR CRACK

^ D I R E C T I O N O F LOADING

\

0°^ SPECIMEN EDGE

> HHAALLFF THICKNESS J

S H O R T E S T DISTANCE AT WHICH N E W CRACK C A N F O R M ^ 'CHARACTERISTIC DAMAGE STATE

II

Fig. 20. Diagram of the shear lag concept for stress distribution in cracked plies of a laminate which can be used to explain the "characteristic damage state" for matrix cracking.

over a small range of load and quickly stabilizes into a pattern with the same spacing as the fatigue crack pattern. In fact, the two patterns are essentially identical regular crack arrays in a given ply regardless of load history, as shown in fig. 19. Similar behavior is observed for other off-axis plies. We have determined that the C D S is a laminate property, that is, it is completely defined by the properties of the individual plies, their thickness, and the stacking sequence of the variously oriented plies. The C D S is independent of extensive variables such as load history and environment (except as the ply properties are altered) and internal affairs such as residual or moisture-related stresses [1, 2, 7, 10, 27, 28]. The C D S is most important in the context of the analysis of the damage develop­ ment process in laminated composite materials. As it is a well-defined laminate property, it is possible to set boundary value problems properly, based on the pattern geometry, and expect the results to be useful for the prediction of subsequent laminate behavior. A clear demonstration of that fact is provided by a prediction of the pattern itself. The formation of the C D S can be anticipated using the simple concept indicated in fig. 20. If the tensile normal stress in the off-axis ply of a laminate reaches a level sufficiently high to cause a crack to form, the stress will be relaxed to zero at the crack face and will rise as a function of distance from the crack at a rate which depends solely on the rate at which the laminate can transfer stress back into that ply, i.e. which depends on the stiffness of the broken ply, the stiffness of the neighboring plies, and their relative thicknesses. The distance required for that stress to reach the original (undisturbed) level is the shortest distance from the original crack at which another crack can form. This minimum crack spacing is the characteristic spacing (the CDS) which is a "stable state", a laminate property that is unique for each specific laminate. Using this simple concept, it is possible to predict the C D S with reasonable accuracy, as we have reported for several dozen cases elsewhere. An example of such a pre­ diction and the observed pattern is shown in fig. 21. There is an analogy between the significance of this concept and the significance of the single crack problem in homogeneous materials. The C D S is a well defined damage state which can be accurately described and predicted, and which is the

Damage and damage

1 '1

I

1

(

ι

ι

I

1

I ' I

t

)

I

ι

Ί

Ί

I

mechanics

35

I' 1

—r

Fig. 21. Predicted (top) and observed (bottom) characteristic crack spacings in a [0/90 ± 45] graphite epoxy laminate, using the shear lag model illustrated in fig. 20.

"starting point" for those processes which control the strength, stiffness, and life of a laminate. The C D S for matrix cracking has become a universally accepted starting point for the description of fatigue damage in continuous-fiber multi-axial laminates, and has been described and predicted by a number of investigators. One of the most notable examples of a different approach to the prediction of such states is provided by the efforts of A.S.D. W a n g and co-workers, who have used an energy method for multiple transverse crack analysis to predict the equilibrium spacing [29]. Although it does not apply to our discussion of fatigue, a point of interest in the formation of the C D S under quasi-static loading is worth noting. We argued that the shortest distance from a single crack in a cracked ply at which the normal stress just reaches the original value that caused the crack to form is the closest distance at which a new crack can form, so that the crack pattern will be stable when all cracks have reached that spacing. When two cracks form with that spacing the normal stress in the broken ply is much reduced over the distance between the cracks, as the local reduction in stress for both cracks is superimposed. For quasi-static loading, the load increases until laminate failure occurs so that the reduced stress between the cracks could eventually reach a sufficiently high value to form a crack at the mid-point between each pair of cracks, where the stress is maximum. In actuality, the local reduction of stress is so large that the laminate breaks before the second C D S forms, in every material system and laminate examined by the writer. N o second C D S appears to have been reported. Hence, the C D S is an essentially stable condition for both quasi-static and cyclic loading, although subsequent C D S formation with spacings equal to half the value of the first (second, third, etc.) C D S are theoretically possible with increasing quasi-static loading. Of course, any change in local constraint (such as local delamination) will also alter the C D S as it will change the rate at which stress is transferred back into the broken ply. The reduction in laminate stiffness caused by primary matrix crack formation in common engineering (fiber-dominated) laminates is usually small, of the order of about 10%. These changes alone are generally not of great engineering consequence except

K.L.

36

Reifsnider

TABLE 1 Stresses in individual plies of a sample laminate [0, 90, + 45] T300-5208. s

Ply

0 90 + 45 -45 a

σ (MPa)

o

o (MPa)

a

χ

xy

y

(MPa)

Before

After

Before

After

Before

After

2631 167 600 600

2993 0 503 503

-2.3 -796 400 400

-4.7 -1000 503 503

0 0 417 -417

0 0 503 -503

Applied stress σ = 1000 MPa.

as they affect vibration frequencies. However, the stress redistributions which accom­ pany the matrix cracks are, indeed, consequential, a fact that is easily demonstrated. When calculating the quasi-static strength of an unnotched laminate, a common scheme is to calculate the ply stresses using laminate analysis, invoke some failure criterion to predict first ply failure (usually matrix cracking), reduce the moduli in the broken ply (usually E perpendicular to the fibres and the in-plane shear stiffness, G), recalculate ply stresses, test for second ply failure, and so on until "last ply failure" is predicted. This scheme, commonly referred to as the ply discount method, has been widely used over a period of at least 15 years, and is known to provide good engineering estimates of laminate strength when edge effects do not dominate the failure process. Table 1 shows the stresses in the individual plies of a sample laminate before and after matrix cracks form in the 90° and ± 4 5 ° plies as determined from a laminate analysis using the discount method. The stress in the fiber direction of the 0° plies (which controls final fracture) is increased from 2631 to 2993 M P a , a j u m p of 14%, which is then used in a failure analysis of some sort to predict the "correct" strength of the laminate. This is a typical result, and it is especially important to note that the increase in the fiber direction stress in the 0° plies is generally very nearly the same as the (per cent) decrease in the Young's modulus of the laminate in that direction, a fact that is of very great help to design-level engineering estimates. 2

It is important to remember, however, that these stress redistributions (and the stiffness reductions that cause them) are not, in reality, uniform, they exist only near the matrix cracks in the off-axis plies. The details of the stress field created by a crack in an inhomogeneous material play important roles in the development of the specific damage events discussed in the previous section. Figure 22 shows a schematic diagram of a crack that has formed in material Β (taken to be an off-axis ply with an orientation such that primary matrix cracking occurs during cyclic tensile load excursions) sandwiched in a laminate with properties characterized by material A in the diagram. A magnified view of the local stress field region near the tip of a crack which stops at the interface between materials A and Β is shown in the right-hand part of fig. 22, along with a material element at the crack tip acted on by only those components of stress of special interest to us in the present discussion. Also shown in the figure is a listing of the order of influence of those stress components. This order is suggested by the chronology of the

Damage and damage

mechanics

37

Fig. 22. Diagram of the stress components which play a major role in the development of damage near primary crack tips in composite laminates.

subsequent events after primary cracks have formed and by the relative importance and intensity of those events for a given generic situation. The reader is reminded that this listing is entirely an artifice for our convenience of discussion as these stress components and the remaining components not shown act together on material elements near the crack tip, and their individual influence (or the material resistance to that influence) cannot be completely separated. Nevertheless, the presence or absence of certain of these components and their general magnitude is clearly associated with certain physical events which define the subsequent damage process. Perhaps the most widely discussed influence of such a crack is the concentration of normal stress in the crack opening direction immediately ahead of the crack tip, σ in fig. 22. Although this particular aspect of the problem has been widely discussed for homogeneous materials, it is much less common to find descriptions of that situation in composite materials. Figure 23, taken from ref. 30, shows an example of the increase in σ in the neighboring plies as a result of a crack in the 90° plies of a quasi-isotropic graphite epoxy laminate calculated using a quasi-three-dimensional finite difference scheme [31]. The direct proof of the existence of these concentrated stresses was provided in refs. 32 and 33, wherein a very-high-resolution moiré dif­ fraction device was constructed and used to resolve strain distributions in the 0° ply of several different laminates in regions near cracks in adjacent off-axis plies during quasi-static loading. An example of those results is shown in fig. 24. Figure 24(a) was produced by the interference between a reference beam and a beam that was incident on a diffraction grating which had about 800 lines per millimeter and was bonded to the specimen surface. The cracks in the off-axis 90° plies of the [0, 9 0 ] glass epoxy specimen can be seen as white horizontal bars with a spacing of about 4 m m in the original photograph. The constant-displacement diffraction lines are denser in the region of the off-axis cracks, which indicates a strain concentration in the 0° ply surface which is being observed. The strain distribution calculated from that pattern between two of those cracks is shown in fig. 24(b), along with strain plots calculated χ

χ

3

s

K.L.

38

Reifsnider

K - 9 0 ° - K -45°-r— 4 5 ° - * f * - 0 ° ^

—

—

175 150 125 100 _ σ Q_

75 ~ 50 25

0

0.25

0.50 z/t

0.75

0 1.00

Fig. 23. Through the thickness variation of σ for a quasi-isotropic laminate of graphite epoxy with 90° and — 45° ply cracking. χ

from a simple shear lag model for three choices of the value of the parameter G\b, the only parameter in that model. The stress concentration in the crack opening normal stress component is first in our list of "the order of influence" because of its importance to the fiber failure damage mode. As we mentioned in the previous section, the first (chronologically) and possibly most significant effect of the elevation of this stress component is the nucleation of fiber fractures in adjacent plies in regions next to the crack tips. As fibers ultimately define the strength of these laminates, this is in a sense "the beginning of the end" of the damage process. It is surprising that, as we noted earlier, this fiber fracture initiation begins so early in the sequence of damage development, usually in the first one-third of the life of a laminate. The second stress component in our "order of influence" is the normal stress parallel to the crack front, σ . To appreciate the physical situation which gives rise to this stress, it is important to recall that the primary matrix cracks generally extend through the thickness of a ply, and run parallel to the fibers over large distances in the j-direction, commonly across the entire width of a test specimen. Hence, from the standpoint of the local crack opening behavior, we can assume that the deformation near the tip of the crack occurs under plane strain conditions in the j-direction. The local increase in the normal stress σ causes a corresponding region of increased local strain in that direction, which causes the material element in that state of stress to contract in the j-direction (in an attempt to maintain its volume) by an amount which is greater than the surrounding material further away from the crack tip. This contraction is, of course, resisted by the surrounding region, which results in the creation of a tensile normal stress, a , in the material along the edge of the crack. This behavior is not peculiar to the stress fields around primary cracks in composite ν

χ

y

Damage and damage

mechanics

39

4.0 m m .

0.0 m m .

(b)

STRAIN

0.8

DISTRIBUTION

~i

BETWEEN

CRACKED

1 Δ

:

SECTIONS

1

EXPERIMENTAL (G/b) = 600 M S I / I N .

(G/b)=200 M S I / I N .

0.6

(G/b)=50 M S I / I N .

LU Ο

or Δ

Λ

Δ

< or

0.2

0.0 POSITION

(mm)

Fig. 24. Moiré diffraction pattern showing increased strain in 0° plies in regions near cracks in the underlying 90° plies of a [0, 9 0 ] glass epoxy laminate (a), and a comparison of the measured values of those strains between two such cracks with analytical predictions of those strains using a shear lag model with three different values of the parameter (G/b) in that model (b). 3

s

K.L.

40

Reifsnider

laminates, but is well established in homogeneous materials where fracture mechanics solutions can be used to obtain estimates of the magnitude of this stress component. This stress component is a primary contributor to the occurrence of secondary cracking as demonstrated in fig. 8 and discussed earlier. It is listed as the second most important stress component in our order of influence, as secondary cracking behavior is ubiquitous in our observations of damage development in composite laminates which have fairly brittle matrix materials such as epoxy (cf. refs. 33 and 34). This is true even in situations where the global ply stress in the ^-direction (calculated from laminate analysis) is compressive as a result of Poisson mismatches between the individual plies. Hence, although the strains associated with this stress component have not been accurately measured, its influence on the development of damage is thought to be large and significant based on the high density of secondary cracks that is observed in so many situations. The next two components of stress to be discussed are the interlaminar stresses σ and τ . These two components will be discussed together because they act to produce similar damage events. Examples of these stress components as noted by Talug and Reifsnider [30, 31] are shown in figs. 25 and 26. The physical situation associated with these stress components can be appreciated by considering the deformations around the crack shown in fig. 22. These are two physical tendencies of particular interest. First, the material directly above and below the free faces of the crack in material Β will tend to translate in the load direction (opening the crack), and is prevented from doing so (in part) by shear forces which develop along the interfaces of the A and Β materials which keep the fractured portions of the Β ply from "slipping out". That physical situation is associated with the τ components shown on the material element in the right-hand side of figs. 22 and 25. Second, as the material along the crack face translates in the load direction, it tends to pull the crack tips inward towards the center of the ply in the sense that the tension in a string is increased by ζ

χζ

χζ

6 4 2

ο -2 -4 -6 0

0.25

0.50

0.75

1.00

Z/t Fig. 25. Calculated distribution of interlaminar normal and shear stress though the thickness of an eight-ply quasi-isotropic laminate with a crack in the 90° ply next to the center line.

Damage and damage

mechanics

41

k - 9 0 ° - ( * - 4 5 ^ t — 4 5 ° - r — 0° - I o

0 CRACK —ι TIP

0.5 z/t

1.0

Fig. 26. Calculated contours of the interlaminar shear stress (in psi units) for a graphite epoxy quasiisotropic laminate with a 90° crack as shown in fig. 25. The plot is in a plane which contains the thickness dimension and the length dimension of the laminate.

pulling a point in the center of the string in the direction transverse to its axis. This physical situation is reflected in the stress component σ , an interlaminar normal stress which acts to pull the interface between plies A and Β apart near the crack tips. The consequences of these interlaminar stresses are pervasive and complex. We will discuss them in some detail. To set the stage for that discussion, it should be mentioned that the general consequence of these stress components is the introduction of delamination as a damage mode. However, this can, and does, occur in a large variety of ways, several of which have special importance to the damage development process. We begin with a surprising generality. F o r common laminates of engineering interest (made from material systems available at present) subjected to cyclic loading, delamination is usually initiated by primary cracks in the matrix material. In the case where the global stress field of a laminate is such that delamination is likely to initiate and grow (and possibly control life), primary matrix cracks provide a preferential starting point for their initiation. In the instance when the global stress field does not support delamination initiation and growth at that level, primary matrix cracks may still cause delamination in the region of such cracks or in the region where primary cracks and secondary cracks cross. These are the two cases of greatest interest to us and we will discuss them separately. The most c o m m o n situation for which global delamination is induced is the occurrence of (theoretically) singular three dimensional stress fields at the geometric ζ

K.L.

42

Reifsnider

edge of a laminate. The mechanics of this situation are more widely and thoroughly discussed than possibly any other single problem in composite materials. The literature is profuse, extensive, and detailed (cf. refs. 19-21). In another chapter of this book, O'Brien discusses delamination initiation and growth from the free edges of straight-sided specimens, from notches and holes, in bonded joints, and in impacted specimens. These treatments are assisted by a myriad of analytical schemes to estimate local stress states and calculate the release of strain energy. In the opinion of the writer, the most consistently successful approach to the prediction of edge-related delamination initiation and growth is the use of strain energy release rate concepts, either the total strain energy release rate or the release of individual components of strain energy thought to be associated with crack opening, sliding, and shearing motions. O'Brien has shown, for example, that various forms of the simple equation G = y

(Ε* -

EJ

(1)

U

can be used for this purpose, where G is the total strain energy release rate, ε is the global laminate strain, t is the thickness of the specimen, E\ is the undelaminated laminate modulus, and E* is the modulus of a totally delaminated specimen. A description of stress fields in the neighborhood of propagating delaminations has been fairly well formalized (cf. refs. 35-37). F o r present purposes we need to discuss the mechanics of delamination initiation as part of the causative damage sequence, a subject which is not well developed in the literature. Figure 27 shows an example of this sequence for the edge delamination case. Primary cracks which form in the off-axis plies are seen to act as initiation points for delamination development. The mechanics of this situation is suggested by figs. 25 and 26, and have been discussed by a variety of authors [19, 37, 38]. Although this kind of behavior is commonly observed and widely recognized for tensile fatigue loading of composite laminates, it is less well appreciated that it is most severe and am

Fig. 27. Association of edge delamination with matrix cracking (arrow) in a graphite epoxy laminate with many off-axis plies under cyclic loading.

Damage and damage

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43

pronounced for combinations of cyclic tension and compression. F o r example, a typical quasi-isotropic graphite epoxy laminate may be able to sustain cyclic amplitudes of 6000 με if only tensile loading or compressive loading is applied, whereas fully reversed tension-compression loading with an amplitude of 4500 με may cause failure in 100000 cycles. Under those conditions, the delamination appears to initiate immediately and grow rapidly as soon as primary matrix cracks are observed [39]. Although edge delamination is arguably a special case, it is of practical importance and will be discussed further in the next section when failure of the laminate is addressed. The mechanics associated with the physical relationship between primary cracking and delamination can be discussed on a more general level. Indeed, this association is an important part of the causative damage sequence in the more general case when edge-dominated damage development is not an important part of the problem. The edge problem is a special case of this situation. We noted in our discussion of figs. 9 and 10 that it is possible for delaminations to develop in the interior of a laminate, especially in regions where primary and secondary cracks cross at an interface between plies. We note here that such behavior occurs preferentially at positions through the thickness which are closer to the outside surfaces, as we might expect, as out-of-plane motion is not constrained there. Although the discovery of this physical behavior by Jamison is fairly recent, and the degree of development of this damage mode varies widely (from unobservable to easily observable), the tendency for development is ubiquitous in the presence of primary and secondary cracks, and the stress redistributions peculiar to these internal delaminations are thought to be significant in the context of failure initiation. To demonstrate the mechanics of this situation as quickly and clearly as possible, we will rely heavily on the work of Highsmith [40]. In a dissertation on the topic, Highsmith considered the problem of determining the local stress around the crossing point of a transverse crack in the 90° plies with a longitudinal split in the 0° plies within a [0, 90 ] graphite epoxy laminate. This so-called crossed crack problem in that particular laminate was picked for consideration because of physical observations which indicated a high tendency for internal delamination formation in that situation. The idealized damage state that was analyzed is shown in fig. 28(a). It was assumed that the damage state consisted of uniformly spaced transverse cracks in the 90° plies and uniformly spaced longitudinal splits in the 0° plies. In the early stages of damage development, the longitudinal splits may be limited to short distances in the x-direction, and may not extend through the total thickness of the 0° plies. However, when internal delaminations begin to form, the damage idealization shown in fig. 28 is a reasonable approximation of a typical state of damage. Figure 28(b) shows a schematic representation of the representative quadrant from the idealized damage state that was analyzed. Cracks in the 90° and 0° plies are treated as free faces, a situation which contributes greatly to the eventual consequences of the internal delamination. Highsmith used a structural theory for the mechanical response of a laminate developed by Pagano which invokes Reissner's variational principle, a variational form which reduces to the governing equations of elasticity. Pagano's structural theory is based upon the assumption of linear variations, of 2

s

44

K.L.

Reifsnider

Fig. 28. Schematic diagram of a crossed-crack pattern in a cross-ply laminate, showing a quadrant of a representative volume that was analyzed by Highsmith (a), and the equilibrium element used for the analysis with corresponding coordinate systems and dimensions (b).

in-plane stresses through the thickness of a layer of material. The theory produces a system of 23 equations per layer that describes the equilibrium conditions, con­ stitutive behavior, and interlayer continuity conditions imposed on each layer of the laminate. The reader is referred to ref. 40 for the details of this treatment.

Damage and damage

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45

The method was validated using experimental measurements of actual deformation fields from the surface of a specimen with crossed-crack patterns similar to those analyzed. A sample of the results of particular interest to us here is shown in figs. 29-31. Figure 29 shows the predicted distribution of axial stress σ in the 0° ply at the 0°/90° interface where the cracks cross. The crossing point of the cracks is in the rear corner of the three-dimensional plot. The two cracks in the 90° and 0° plies run along the back faces of the diagram. The diagram clearly shows a stress concentration in the axial normal stress caused by the 90° crack, as we might expect. There is a small but perceptible increase in this stress component near the position where the cracks cross, but there is no concentration caused by the longitudinal crack in general, a result that is also consistent with expectations. Figure 30 shows the predicted distribution of interlaminar normal stress, σ at the 0°/90° interface over the same region. Again, the stress component is elevated near the transverse crack in the 90° plies, as we might expect from our observations of fig. 25 for the two-dimensional analysis. However, there is also a distinct synergism between the crossed cracks, shown by the elevated value of σ at the crossing point. These elevated values of tensile interlaminar normal stress are a major driving force for the formation of the local delaminations in that region that we observed earlier. Figure 31 shows a plot of the other interlaminar stress of primary concern to us. This shear stress is also greatly elevated at the position of the transverse crack and is relatively uninfluenced by the presence of the longitudinal crack in this case. χ

ζ

ζ

K.L.

46

Reifsnider

0 Fig. 30. Contour plot of the interlaminar normal stress at the interface between the 0° and 90° plies near the intersection of cracks in those plies for the laminate shown in fig. 28(b).

Although these results have been developed for a specific laminate as a matter of expediency, they are by no means peculiar to this situation. In a limited study of the parameters which influence the results of the analysis, Highsmith has shown that whereas the amplitudes of these stresses are influenced by geometry effects such as ply thicknesses and material properties, the general nature of the results is unchanged for other crossed-crack situations. This is as we might expect, as the physics of the situation, and in particular the boundary value problem of mechanics, is fixed by the definition of the crossed-crack problem, whereas parameters such as the material properties and ply dimensions influence only the magnitudes of the local effects. F r o m these results it is clear that the internal stresses near a matrix crack which stops at a ply interface tend to cause separation of that interface, and that this tendency is significantly elevated when two such cracks cross at the interface as is common in many of the composite laminates in use today. F r o m the standpoint of the mechanics of this situation, we are left with the question of how that local stress state changes if such a delamination forms. It is clear that if a delamination formed in the interior of a specimen between two plies in the absence of any matrix cracks, continuity conditions at the boundary of such a delamination would force the defor­ mations in that region to be the same as the surrounding undelaminated material. However, the presence of the matrix cracks (which cause the delaminations to form) relaxes the constraint of the boundary conditions around the delamination and

Damage and damage

[0,90 ] 2

mechanics

47

s

Fig. 31. Contour plot of interlaminar shear stress at the interface between 0° and 90° plies at the intersection of matrix cracks in those plies for the laminate shown in fig. 28(b).

contributes greatly to the nature of the local stress state in the region of the delami­ nation, especially at the delamination front. This discovery was also noted by Highsmith [40]. Figure 32, taken from that reference, is a portion of a penetrant-enhanced X-ray radiograph taken of a [0, 9 0 ] specimen which had been subject to tensile fatigue loading. The dark blotches which look like ink spots are actually dye penetrant which seeped into the moiré grid that was bonded to the specimen surface. We can see evidence of several transverse cracks and two longitudinal cracks in the radiograph. To the right of the centrally located longitudinal spilt there is a shadowy area that is semi-elliptical in shape, indicated by an arrow. This area is, in fact, a small internal delamination which was found to be at the 0°/90° interface nearest the specimen grating. Using the high-resolution moiré diffraction scheme mentioned earlier, the deformations in that region were analyzed. Figure 33 shows a typical result of that analysis (further details are included in ref. 40). The longitudinal split is located at the origin of the plot in fig. 33. The delamination front occurs at a distance of 0.035 in. from the split. Within the delamination, there is a plateau of nearly constant transverse strain of about — 0.28%. F o r linear elastic response of that laminate a longitudinal strain of 0.728% is expected. The corresponding Poisson's ratio is 0.385, which is quite large. At the delamination tip, the transverse strain is concentrated, with a value of —0.65% indicated. Beyond that position, the transverse strain returns to the global value that we would expect for the laminate. Highsmith has reasoned that, as the average axial 2

s

K.L.

48

Reifsnider

Fig. 32. Radiograph of crossed-crack pattern showing a region of internal delamination near the vertical secondary cracks, indicated by an arrow.

stress σ along a specimen section passing through the delamination must be the same as the applied stress to satisfy equilibrium, and as the delaminated material is less stiff than its undelaminated surroundings, there must be a region of stress concentration to ensure equilibrium. This concentration of stress is observed to occur at the delami­ nation front, not only in the direction transverse to the load axis but in other directions including the load axis direction. Although this problem still requires a great deal of further study, it is now clear that internal delaminations which form around matrix cracks can cause significant elevations in the in-plane values of normal stress in the plies surrounding the delamination. We might expect these elevated stress values to contribute to further damage and possibly to the failure of the laminate. Quite recently the writer has found that fiber fracture occurs preferentially in regions of internal delamination. It is not clear whether this fracture occurs at the front of the delamination as it grows over the χ

Damage and damage

E-1

Ρ ζ

H-III-J-2-6

8.00

LJ Ο

S

SPECIMEN

49

mechanics

A

Δ

Δ Δ

A

~

A

-2.00

Η

α -4.00

Δ

Δ

"Δ:

Δ

(0

£ -6.00 Lu > (0

£

0.00

0.25

0.50

0.75

1.00

1.25

1 . 5 0 Ε-1

TRANSVERSE P O S I T I O N C I N . ) Fig. 33. Measured values of transverse normal strain across the delamination boundary shown in fig. 32.

region as part of the growth process, or if the fibers break in the delaminated region after delamination has occurred. Earlier it was mentioned that the damage development process discussed above can be greatly accelerated in certain circumstances by combined tension-compression loading. This is certainly the case with delamination, either internal delamination or edge delamination. A discussion of the failure modes associated with such an acceleration will be deferred to the next section. The nature of this acceleration is not entirely intuitive. Although it is clear that such things as local instability (buckling) may cause local delaminations to grow rapidly, it is less obvious that other types of damage such as matrix cracking may also be accelerated by combinations of tension and compression. Schultz and Reifsnider have found, for example, that secondary cracks in 45° plies which form along the tip of primary cracks in the 90° plies of a quasi-isotropic graphite epoxy laminate can initiate and grow during compressive load excursions under cyclic tension-compression loading [41, 42], Although the mechanics of that situation is not different from the crossed-crack problem described above, the interpretation and consequences of the results requires special attention. 4.1. Notched damage

development

Another problem which requires special attention is the development of damage in the region around a stress concentration or notch in composite laminates. This subject is very much on the frontier of understanding, and it is an area of current research. Our present purposes will be served by noting only a few salient features of the mechanics of this problem which are of particular significance to the observed mechanical response. Certainly the most prominent feature of the fatigue response of a notched composite laminate is the magnitude of the stress redistribution in the vicinity of the notch as a result of damage development during long-term loading.

50

K.L.

Reifsnider

Obtaining accurate estimates of the stresses in the individual plies of a laminate near the boundary of a notch is a substantial challenge even in the undamaged condition. The local stress fields, especially in the boundary layer near the surface of the notch, are greatly complicated by the fact that the notch is both a stress concentration and a free edge of the laminate. Although this aspect of the problem has been discussed in considerable detail in the literature, discussions of the details of the stress state in the presence of damage introduced by cyclic loading are sparse [43-47]. Figure 34 shows a moiré diffraction pattern of axial displacement fields around a notch in a quasi-isotropic graphite epoxy laminate after stress redistribution caused by fatigue damage development taken from ref. 44. We can see a significant disturbance of the pattern in the region around the hole, especially associated with the longitudinal splits that have developed in the 0° ply on the outside surface of the specimen. As damage develops in this region, the constraint afforded by the hole (which causes the stress concentration) is relaxed in a

Fig. 34. Moiré diffraction pattern of displacement fields around a notch in a quasi-isotropic graphite epoxy laminate after stress redistribution caused by fatigue damage development.

Damage and damage

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51

manner whic h i s simila r t o a chang e o f geometr y whic h woul d mak e th e hol e mor e elliptical i n th e directio n o f loading . A s th e axi s o f th e effectiv e ellips e " g r o w s " i n th e loading directio n correspondin g t o suc h damag e developmen t (o r th e axi s transvers e to th e loa d directio n contracts) , th e coupo n become s mor e lik e tw o ligament s no t associated b y th e constrain t o f th e hol e whic h the y bound . Thi s i sconsisten t wit h bot h the stres s distribution s an d th e strengt h behavio r observed , i n th e region s whic h control th e strengt h an d lif e o f th e notche d laminates . O f course , th e transvers e constraint o f th e notc h regio n i n th e area s coinciden t wit h th e elongate d axi s o f th e effective ellips e i s no t correctl y represente d b y thi s analogy . Nevertheless , som e o f th e important genera l feature s o f th e physica l proble m ca n b e deduce d fro m thi s typ e o f concept. Figur e 3 4 suggests , fo r example , tha t th e damag e aroun d th e hol e fo r th e cas e shown ma y influenc e a regio n whic h i s roughl y twic e th e lengt h o f a hol e diamete r i n the loa d axi s direction , whic h suggest s tha t a n effectiv e geometr y correspondin g t o a n ellipse wit h a semi-majo r axi s o f twic e th e lengt h o f th e semi-mino r axi s ma y b e appropriate. Figur e 3 5 show s severa l plot s o f th e normalize d axia l norma l stres s nea r the hol e alon g th e y-axi s whe n th e ellipticit y o f th e hol e i s increase d alon g th e x-axi s to ratio s o f 2 : 1 , 3 : 1 , an d 4 : 1 .W e ca n se e tha t th e magnitud e o f thi s stres s componen t near th e hol e i s greatl y influence d b y thi s effectiv e ellipticity . ( A mor e thoroug h discussion o f thes e change s appear s i n th e chapte r o n "Fatigu e Behavio r o f Com posite Laminates". ) I t i s importan t t o not e tha t al l o f th e stres s component s an d thei r distributions ar e influence d greatl y b y change s i n th e loca l geometr y o r constrain t caused b y damag e development , a fac t whic h wil l b e o f considerabl e utilit y t o u s i n the nex t section . Although suc h a chang e i n th e globa l stres s field i s eas y t o discus s an d visualize , th e associated damag e developmen t an d failur e processe s ar e controlle d b y event s whic h occur a t th e pl y levels . Ther e appear s t o b e n o direc t evidenc e tha t a correc t des cription o f th e pl y stresse s i n th e presenc e o f th e comple x damag e state s commo n J o notche d laminate s ha s ye t bee n achieved . Indeed , th e mos t difficul t aspec t o f thi s problem i s th e judgemen t tha t i s require d t o interpre t th e proble m i n suc h a wa y tha t we nee d no t analyz e al l o f th e detail s i n suc h a comple x damag e state , bu t ca n b e concerned onl y wit h th e dominan t aspect s o f th e mechanic s proble m tha t contro l th e final strengt h an d lif e o f th e laminate . Th e difficulty , o f course , come s fro m th e nee d to obtai n sufficien t locaï^xperimenta l informatio n an d t o achiev e adequat e under standing s o tha t a n appropriat e mechanic s proble m ca n b e clearl y identifie d an d properly set . The mechanic s o f damag e developmen t tha t w e hav e discusse d abov e ha s bee n limited t o th e cas e o f damag e developmen t i n laminate d continuous-fibe r composit e materials. Thi s situatio n i s th e mos t commo n an d typica l on e tha t scientist s an d engineers hav e t o dea l wit h a t present . However , man y othe r aspect s o f thi s proble m are o f technica l an d academi c interest . Th e mechanic s o f micro-plasticit y ca n b e use d to establis h fatigu e threshold s i n meta l matri x composites , fo r exampl e [11] . A description o f thi s approac h ca n b e foun d i n anothe r chapte r o f thi s book . Another subjec t no t discusse d i s th e mechanic s o f fatigu e damag e developmen t i n particulate composities . Althoug h thi s proble m i s widel y allude d t o an d discusse d i n phenomenological terms , th e mechanic s o f thi s situatio n i s no t wel l establishe d

52

0

1

2

K.L.

Reifsnider

3

4

5

(V/b) Fig. 35. Plots of the normalized axial normal stress near the boundary of a center hole in a laminate along the transverse axis as the ellipticity of the hold changes.

(cf. refs. 48 and 49). It is clear that cumulative damage does occur in these materials under cyclic loading and there are some distinct peculiarities associated with this phenomenon. One particularly interesting aspect of the problem appears to be associated with the behavior of these materials in repeated cyclic compression, a loading mode which is otherwise thought to be "safe" for such materials. Because of the indefinite nature of available information, a presentation of an approach to the mechanics of this situation will not be made here. It should be noted that a rigorous formalism has developed around efforts to " t o u g h e n " ceramics. When particulate reinforcement (especially at low volume fractions) is used for this purpose, the fatigue process may be controlled by crack propagation, and the damage accumulation process and associated mechanics are usually concerned with diverting or inhibiting crack growth by altering the stress or strain field ahead of the crack or by bridging

Damage and damage

mechanics

53

the crack with secondary phases [50, 51]. However, rigorous understandings of the fatigue behavior of these materials (in the opinion of the writer) have not yet appeared. 5. The mechanics of strength reduction The experimentalist who attempts to understand the micro-behavior associated with the long-term response of laminated composite materials quickly finds that it is impossible to account for all of the details and relationships between all of the individual damage events that occur in such a process. When representations of this process are sought, the greatest challenge is not necessarily how to solve the resulting problem, but rather how to identify and see the proper problem. Part of the answer to that question must come from the objectives of such a representation. The most common objective is to describe the strength, stiffness, and life of composite laminates. We have mentioned some approaches to the stiffness problem. We make the claim that the problem of life prediction is really a subset of the problem of strength prediction. Hence, our last major topic in this chapter will be the mechanics of strength reduction due to damage development in composite laminates. Our approach to the subject will be organized along the lines suggested by fig. 36. We make the claim that there are three major issues to be dealt with. The first of these is the effect of the damage process and the individual events in that process on the nature of internal load-sharing among the constituent materials and plies. We should remark in passing that if we were to allow a sufficiently wide range of observational levels (macro-, micro-, submicro-, . . .) the mechanics of the subject of fatigue as a phenomenon could be discussed almost entirely in terms of internal local geometry adjustments. Certainly this matter is basic to the discussion of fatigue in composite materials. The second topic has to do with the progressive localization of damage development and the consequent inhomogeneity of the deformation process which results from that localization. In a very direct sense, if localization did not occur, fatigue failure could not occur, as some localized process must be responsible for the failure of a material component or element at a general level of stress which is significantly lower than the undamaged ultimate strength of the material. Finally, we will discuss failure of "critical elements", a rational scheme for extending the mech­ anics of damage development to a description of the failure modes in composite laminates in such a way that residual strength and life can be predicted. The matters of interest associated with the effect of damage on internal load distribution are categorized in fig. 37. It is certainly clear that a change of geometry at the ply level, usually caused by matrix cracking, fiber fracture, delamination, and MECHANICS

OF STRENGTH

EFFECT

OF DAMAGE

PROGRESSIVE FAILURE

OF

REDUCTION PROCESS

LOCALIZATION " C R I T I C A L

ON AND

INTERNAL

LOAD

INHOMOGENEITY

ELEMENTS"

Fig. 36. Issues which control the mechanics of strength.

D I S T R I B U T I O N

54

K.L.

EFFECT

O F DAMAGE

PROCESS

CHANGE

O F GEOMETRY

CHANGE

OF PLY

CHANGE

OF

AT

ON

INTERNAL

PLY

Reifsnider

LOAD

D I S T R I B U T I O N

LEVEL

MODULI

CONSTRAINT

Fig. 37. Nature of the effect of the fatigue damage process on the mechanics of strength.

other associated events is of primary importance to this situation. In the previous section we discussed a number of attempts to characterize this part of the phenomenon from the standpoint of local stress redistribution. The principle influence of such a geometry change on strength is associated with the stress con­ centration caused by it. A major consequence of that stress concentration, in turn, is an increasing tendency for damage to form in regions near such local geometry disturbances, a tendency which contributes greatly to the localization process mentioned in fig. 36. In many respects, this is the heart of the fatigue problem in composite laminates. A proper understanding and description of the effect of local geometry changes, however, depends greatly upon context, especially upon the region of influence of that local change compared with the nature of the failure event which controls the final strength or life of the laminate for a specific type of remote loading. We can solve a mechanics problem which indicates that there is a very high stress concentration associated with a local crack, for example, but if the volume of material associated with that high stress concentration is small compared with the volume of material associated with the final fracture process, the consequences of that stress concentra­ tion are small or even negligible. Hence, the proper interpretation of such an analysis is entirely dependent upon our understanding of the precise nature of the failure event that is of interest. Based on some 15 years of experience of the composites community with the prediction of engineering strength of composite laminates, we can say for quasi-static loading that the region of influence of matrix cracking can be considered to be a region with a characteristic dimension that is of the order of magnitude of one ply thickness from the tip of a matrix crack; there is at present very little evidence that the local stress singularity near the tip of the crack controls attendant damage development. In an earlier section we discussed the "discount m e t h o d " whereby the quasi-static strength of a laminate strength was calculated by reducing the transverse and in-plane shear moduli of each off-axis ply that is known to crack during quasi-static loading to failure, to allow the load carried by those plies to be redistributed to the 0° plies in the laminate analysis calculation of strength. When edge delamination does not dominate the problem, experience indicates that this scheme yields good engineering values for the laminate tensile strength. It is easy to forget, however, that these stress redistributions (and the stiffness reductions that caused them) are not, in reality uniform. They exist only in the neighborhood of the matrix cracks in the off-axis plies, as suggested by fig. 38 and discussed in the previous section. The ply stresses calculated from a laminate analysis using the stiffness reduction (discount) scheme mentioned above are actually equal to the average net-section stresses in the

Damage and damage

/

mechanics

55

\

f i

\

/ /

s' \ι

^\

<_> \

0°

/

—• 0°

θ°

1

/

0°

Fig. 38. Schematic diagram of the regions of influence of stress fields near the tip of primary cracks in a composite laminate. A locally delaminated region is also shown near the tip of the right-hand crack.

"GLOBAL"

STRESS PERCENT

PLY

REDISTRIBUTION INCREASE

OF

FIBER

STRESS

CRACKING

1 6

[tt

DELAMINATION

6 3

(AE

(TOTAL

EXAMPLE

SEPARATION)

CALCULATIONS

X

*

22%)

= 39*)

3 4

FOR A

GRAPHITE/POLYIMIDE

=

1

-

TOTAL -

INCREMENT

Q U A S I - I S O T R O P I C

LAMINATE

Fig. 39. "Global" stress redistribution at the "net section" level.

individual plies at the position of the matrix cracks. Hence, we conclude that ply-level net-section stresses (not the highly localized stress concentrations near the crack tips) control the quasi-static strength of composite laminates. Some important insights can be gained from a consideration of regions of influence at that level. Figure 39 shows an example of ply-level net-section stress calculations for two types of damage known to occur during the fatigue loading of composite laminates. The calculations were made for a typical quasi-isotropic graphite polyimide laminate. If a discount scheme is used to reduce the moduli of all of the off-axis plies (assuming that cracks have formed in all 90° and 45° plies of the laminate) a change in the fiber-direction stress in the 0° plies of 16% is predicted. It is interesting to note that the change in axial modulus of that laminate would be about 22%, a number which is similar to the change in normal stress in the fibers. If that 16% increase in the fiber stress is included in an analysis of strength of that laminate, a relatively accurate prediction of the engineering strength is obtained, as has been verified by experiments in the author's laboratory for this case. However, we have mentioned that delamination is also possible, both at the edge of the specimen or in the interior.

K.L.

56

Reifsnider

log Ν Fig. 40. Schematic representation of the reduction of strength by a fatigue damage process to the level of applied cyclic loading, resulting in failure which defines the "life locus" of specimens or components subjected to cyclic loading at various levels.

N o w we raise a basic and essential issue, illustrated by fig. 40. We ask the question, "Exactly how is the strength of a composite laminate reduced during cyclic loading so that it fails at a stress level that is dramatically lower than the quasi-static strength?" As we have noted, it is common for composite laminates to fail after, say, 1 000000 cycles at a stress level of the order of 6 0 % of the static ultimate strength. (Of course, the specific numbers may vary greatly depending upon material systems, laminate stacking sequences, and experimental details.) The fundamental question is, " W h a t local situation controls this failure process?" Returning to our ply level arguments and to fig. 39, we can make an interesting observation on this point. If we assume for the moment that we are concerned with tensile strength, and if we further assume that the fiber stress in the 0° plies controls that strength, then we are led to ask, "Is there any damage process that occurs during subsequent fatigue loading that can cause the net-section stress in the 0° plies to increase by an amount which is sufficient to lower the strength to about 6 0 % of the static ultimate value?" F r o m fig. 39 we notice that if the laminate were totally delaminated and the off-axis plies were completely cracked (so that only the 0° plies were sustaining and supporting the load with absolutely no contribution being made by the other plies), the fiber direction stress would increase by 6 3 % , or an increment 3 4 % over the value cal­ culated for the threshold of quasi-static failure without delamination. Although it is clear that we must not overextend the conclusions we can make from such an observation, it is also clear that the matrix cracking and delamination process discussed above has a potential for stress redistribution, even at the net-section ply-level, which is sufficiently large to explain the strength reduction observed in typical fatigue tests. Another point of importance in this argument has to do with the fact that a total relaxation of constraint associated with delamination is rarely possible. By definition, a delamination is always constrained by the boundary conditions associated with the

Damage and damage

mechanics

57

response of the laminate around its boundary, unless the specimen has totally disintegrated. As we noted earlier, for an internal delamination, if no other damage were present the boundary constraints on a small delamination would prevent virtually all of the stress redistribution noted in fig. 39 from occurring. However, we have tried to make a very clear case in this chapter for the argument that delamination is nearly always associated with other damage - especially matrix cracks - in compo­ site laminates commonly used for engineering purposes during fatigue loading. This damage serves to relax the constraining effects of the boundaries of the delamination and to allow stress redistribution to occur. The second major point to be made at the ply-level brings us to the second major topic in fig. 36, and to a transition to more local considerations. We have advanced the argument that the fatigue damage process in composite laminates can be charac­ terized by progressive localization and inhomogeneity of the damage details, which results in a causative sequence of events that creates a generic pattern of damage in situations where a general spectrum of combined tension and compression load excursions is experienced. We have suggested such a generic pattern of localized damage in fig. 9. The value of this observation relates to the most difficult step for any mechanics analysis of this type, the transition from the micro-behavior to the macroresponse of the laminate. If such a generic pattern exists, and if it controls the strength of the laminate, we can formulate a well-set boundary value problem which describes the behavior of a "representative volume" of material known to be associated with the fracture event. Having done so, we can make the claim that the behavior of this representative volume is a sufficient representation of the remainder of the highly distributed damage state without requiring a specific treatment and subsequent integration of all of the effects of all of the individual damage events in this complex situation. This is, of course, not a new idea; it has been used for many years, especially for effective modulus theories. Its utility here, however, is especially important because it allows us to make mechanistic arguments at the micro-level and then to use those arguments for phenomenological behavior predictions at the global level, the transition required by the applied community. Hence, for computations of strength after significant fatigue damage has occurred, we make the claim that a representative volume (such as a region around the crossing point of a primary and secondary crack with the attendant local delamination as shown in fig. 9) may be analyzed. However, we do not have a single constant state of stress and state of material to be analyzed. Instead, our analysis must account for the fact that both the state of stress and state of the material in the representative volume are changing as a function of the number of applied cycles, so that our analysis must be one which continuously changes as well. How, then, are we to achieve a systematic representation of this problem? The nature of the problem to be addressed is illustrated by fig. 4 1 . As the damage process alters the local geometry as a function of the number of applied cycles of loading, the internal stress redistributes continuously in time, and gives rise to further damage development in a region or volume of material which becomes progressively smaller. Ultimately, this localization process creates a sufficiently intense local region of damage and stress which nucleates fracture of the entire laminate. To achieve a

K.L.

58

Reifsnider

Process of localization for tension-tension fatigue loading. Activity In: Critical Element Geometry Change |Stress State Change

1

I

"

Increase o f O in 0 degree plies, local stress concentration x

Subcritical Element Geometry Change Stress State Change Matrix crackings CDS formation ' ι Local variation of 0 . in 0 plies

*

{

Local fiber failures near matrix cracks Stress concentrations near fiber failures

ι

'

Increase interface stresses t

ι

'

Initiation event

*

Change of a l l local stresses

' j•

Fiber direction cross-cracking

'

i

l

'

Local delamination

Increase interface stresses Change of a l l local stresses

* — S

Local overload or c r i t i c a l stress localization

Specimen fracture Fig. 41. Process of localization for cyclic loading which involves tensile load excursions at the ply level.

rational approach to the representation of this localization process and the prediction of strength at any point in the process, the author and associates have postulated the approach suggested by figs. 41-44. The approach is mechanistic and depends on the "critical element" concept; the associated model for residual strength and life pre­ diction has become known as the critical element model. As indicated in fig. 42, critical elements are defined by the failure mode of a given laminate under a specific loading condition. They are critical in the sense that failure of these elements defines failure of the laminate. For the tensile failure of a quasi-isotropic laminate, for example, the critical elements may be the 0° plies, as fracture of those plies is the final event in the damage process. Other examples will be discussed below. Subcritical elements are defined by the damage modes that occur during cyclic loading. They are subcritical in the sense that failure of these elements defines the mechanics of local stress redistribution, but does not cause failure of the laminate in a direct way. The definition of these two elements is directly related to the two major tasks associated with the determination and prediction of residual strength as a function of the number of cycles of loading. These tasks are indicated in fig. 43. To discuss properly the mechanics of strength during cyclic loading it is necessary to determine the state of stress and the state of the material in a continuous fashion.

Damage and damage

C R I T I C A L

mechanics

59

ELEMENTS

FAILURE

OF

THESE

FAILURE C R I T I C A L

ELEMENTS

OF T H E

ELEMENTS

FAILURE

SUBCRITICAL

DEFINES

LAMINATE

DEFINED

BY

MODES

ELEMENTS

FAILURE

OF THESE

MECHANICS

ELEMENTS

OF LOCAL

DEFINES

STRESS

D I S T R I B U T I O N SUBCRITICAL BY

ELEMENTS

DAMAGE

DEFINED

MODES

Fig. 42. Definition of critical and subcritical elements.

STATE

O F :

STRESS

—

CONTOLLED

DAMAGE

—

το

BY

SUBCRITICAL

ELEMENTS

WHICH

CAUSES

STRESS R E D I S T R I B U T I O N

MATERIAL

- -

CONTROLLED

DEGRADATION CHANGES

—

BY

OF C R I T I C A L

ELEMENTS

WHICH

CAUSES

I N

PROPERTIES

Fig. 43. Approach to analysis of continuous fatigue damage.

As fig. 43 indicates, the state of stress is controlled by " d a m a g e " to subcritical elements which introduces changes in the internal geometry of the material. If these changes are properly analyzed by well-set mechanics boundary value problems, the attendant stress "redistribution" can be calculated as a function of damage develop­ ment. The state of material is controlled by "degradation" of critical elements, i.e. a change in the "properties" of the material associated with the elements which define failure of the laminate. It is essential to notice the relationship between these two states and their changes, as depicted in the flow diagram in fig. 44. The local stress state in which a critical element operates is defined by the stress redistribution associated with subcritical element damage. If, as we suggest, the material degradation is represented by some phenomenological constitutive information (such as a stress-life (S-N) curve of unidirectional material or a chemical rate equation), the rate of degradation may be accelerated by increases in the amplitude of the local stress state caused by matrix cracking, delamination, or other types of subcritical element damage.

K.L.

60

"Ί

Γ" Damage Modes, Patterns

Reifsnider

Subcritical Elements

Micromechanics

Global Geometry

1 Failure Mode

Global Properties

Local State of Stress

Critical Elements

Property Evolution

I I Representative I Volume

Constitutive Relationships

I

I

Local State of Material

Strength Concept

Remaining Strength Life

1

Fig. 44. Flow diagram of the "critical element model" for the calculation of the remaining strength and life of composite materials.

In addition to providing a proper mechanistic approach to the mechanics of this problem, the motivations for choosing the critical element approach are numerous. Whereas the number of damage modes in a composite material or component is commonly large, the number of failure modes and corresponding critical elements is generally much smaller. (The subdiscipline of fracture mechanics grew out of the definition of just three such failure modes for homogeneous materials.) Many continuous-fiber laminates are designed to be fiber-controlled, so that the local initiation of unstable growth of a region of fiber fracture may define the critical elements and the fundamental failure mode for a great variety of laminates. F o r reversed loading of laminated materials, the stability of undelaminated ligaments may define the failure mode. Although the literature is replete with examples of hundreds of damage modes, the number of fundamentally distinct failure modes of engineering interest may be an order of magnitude smaller. This results in a great convenience to the computational process associated with this model. The state of material associated with the critical elements defined by the failure mode enters the problem through constitutive relationships which must be obtained from experimental data. As the number of critical elements (and failure modes) is generally comparatively small, the amount of experimental support required for the modeling process is minimized. Several parameters influence the manner in which models in general and critical element concepts in particular are applied to the prediction of the residual strength and life of composite materials and components. Five of these will be mentioned below. With a few notable exceptions, the fatigue behavior of materials is commonly discussed in terms of one-dimensional stress. A great majority of the literature on this subject is so constrained. Such an assumption is wholly inadequate for composite materials as the inhomogeneity and anisotropy of such materials generally creates multiaxial stress fields (and strength fields) which cannot be effectively characterized in such a simple way. Hence, it is essential that a model be stated in such a way that the consequences of this multiaxiality can be fully incorporated into the strength

Damage and damage

mechanics

61

prediction and consequent life prediction appropriate for the failure modes and damage modes concerned. Another issue of primary concern is the possible non-uniformity of stress states, especially as it results from geometric features such as cut-outs, thickness variations, joints, and other common engineering circumstances. Again, the volume of general fatigue literature on this subject is rather paltry, although fracture mechanics approaches to the problem of a single flaw which grows from notches, holes, and other stress concentrators have received considerable attention. Comparable concepts for composite materials are incompletely developed, although a great deal of information on damage development around such discontinuities has appeared in the literature [2, 44, 47, 52]. It is essential that such non-uniformity be incorporated into the critical element model, or any other model of the behavior. The third parameter is really a plural subject that we shall call, collectively, cycle-dependent processes. This is the most familiar topic of fatigue discussions. Peculiar aspects associated with composite materials include the distributed nature of damage and the damage accumulation usually observed in such systems. A residual strength model must be able to receive and use understandings and representations (and improved versions thereof) of cycle-dependent processes as they become avail­ able for the prediction of strength in the presence of damage. The development of such understandings and representations is a continuing process driven not only by improve­ ments in the information, understandings, and analysis of existing material systems, but also by the development of new material systems for various engineering appli­ cations. A modeling philosophy must be sufficiently general so that these improve­ ments can be incorporated without significant conceptual or operational changes. The fourth parameter is also a plural subject which we shall categorize as timedependent processes. Classically, this subject has been separated from considerations of cycle-dependent processes normally called "fatigue". It is the opinion of the writer that this separation is artificial at best and an obstacle to progress in this field at worst. Although it is optimistic (and possibly unreasonable) to entertain the notion of combining the consequences of cycle-dependent and time-dependent processes in a micro-mechanical model of strength, it must be, nonetheless, a long-term objective of such a modeling effort. The final parameter to be mentioned consists of the general group of statistical variations that influence this problem. These variations and their sources are numerous. This general subject area is certainly a frontier which is in great need of rational development using mechanistic rather than empirical approaches, yet any cogent theory of residual strength and life must include such variations if the reliability, safety, and certifiability of composite structures is to be successfully addressed. Another chapter of this book discusses this topic. Although no-one appears to have successfully considered and represented all of the complexity associated with these parameters in a residual strength model, it does appear fair to suggest that model development with a narrower perspective would be more of an academic amusement than an engineering development. One approach to combining these considerations into a computation of residual strength as a function of the number of cycles of loading has been suggested by

62

K.L.

Reifsnider

Stinchcomb and Reifsnider [39]. They postulated an integral form which can be used to receive and combine the results of micro-mechanical calculations of stress fields and constitutive information which defines the local state of the materials in such a way that residual strength is specified. Such an equation is given below. (2) All quantities are normalized by the initial quasi-static strength. The calculated quantity on the left is the current residual strength of a component or laminate normalized by the initial ultimate strength for that failure mode. The ratio R on the right-hand side of the equation is defined as

The function F with various subscripts is a generalized failure function which is defined by the failure mode to be analyzed and the concept of failure thought to be appropriate for that failure process. It may have the general character of a failure criterion such as the Tsai-Hill equation (to the half-power), or for one-dimensional stress states it may simply be a ratio of current stress in the critical element to ultimate strength of that element for uniaxial loading. Strain in the direction which controls failure of the critical element is represented by ε in eqn. (3). The subscript e indicates the value of that failure function evaluated in the critical element as defined above, and the subscript L indicates the value of that failure function in the laminate at some position remote from the location of the incipient failure event. Hence, the ratio R defined by eqn. (2) can be read as the ratio of the initial stress concentration to the current stress concentration in a representative volume (indicated by the subscript rv) of material associated with the fracture process in a globally non-uniform stress state such as a notched component. F o r uniform stress states, R takes on the value of unity. The quantity τ in eqn. (2) is a generalized time parameter which has the character of current time divided by time-to-failure, or the current number of cycles divided by the number of cycles to failure, etc. F o r more general situations, it may be thought of as the integral of entropy. The quantity k is a material parameter which usually has values close to unity for the cases examined by the writer. All quantities in the integral in eqn. (2) are evaluated in the critical element. The failure function, F, is written as a function of time as the stress state in a critical element changes as damage develops in the subcritical elements around it. The multiaxiality of the stress state is accounted for by the micro-mechanical calculations of the local stress state which results from the subcritical element damage, and the anisotropic material properties which enter into a computation of the failure function. The origin of the form of eqn. (2) is an identity which can be illustrated by reducing that equation to the form appropriate for a very simple situation. If we assume: that the stress state is uniform so that R = 1; that the applied stress (amplitude) is constant, and the stress in the critical element does not change as a function of time because of damage in subcritical elements so that the time-dependence in the

Damage and damage

mechanics

63

integrand disappears; that the stress state which controls the degradation and failure of the critical element is one dimensional so that the failure function F becomes simply the ratio of the one-dimensional stress in the critical element to the ultimate strength of the critical element; and that we are concerned only with cycle-dependent degra­ dation processes so that the normalized time-variable τ becomes equal to the ratio of the current number of cycles to the number of cycles of life at the constant amplitude being applied; then eqn. (2) can be written in simplified form as (4) It can be seen that, for this very simple situation, when the number of current cycles equals the life of the critical element, N, the residual strength reduces to the applied load level, the definition of life that we described earlier. The constant k accounts for the nonlinearity of the residual strength reduction curve as shown in fig. 40 between the two pinning points A and Β which are reproduced by eqn. (4) for the cases of η = 0 and η = N. The use of eqn. (2) and the concepts indicated in figs. 42-44 can be illustrated by a simple one-dimensional analysis of tension-tension fatigue loading of an angle-ply laminate which is constructed in such a way that no significant edge delamination occurs. Figure 45 indicates the nature of the interpretation of the terms in eqn. (2) for that situation. F o r this case, the life locus described by the function TV is a phenomenological representation of the life of the critical element, taken to be the 0° plies in this case. The equation is written as a function of the applied unidirectional fiber-direction stress, S(n), normalized by the ultimate strength of the element, S , which is the unidirectional initial strength of the material in the fiber direction. The material constants, A, B, and χ are determined by fitting the data obtained from fatigue testing of unidirectional material of the type from which the laminate was constructed. Because, in this case, we are concerned only with the unidirectional performance of the 0° plies (the critical elements), one such relationship will suffice for all laminates regardless of their construction (stacking sequence, etc.). As it is recognized that the 0° plies in the laminate may carry different amounts of the total load as the damage development in non-critical elements redistributes the stress and alters internal geometry, the applied stress on the critical element, S(n), is stated as a continuous function of the number of applied cycles, n. This quantity can be determined from measurements of changes in laminate stiffness which has been found, by experience and through a number of mechanics models, to be related to internal stress redistributions [23, 40, 53, 54]. The variable of integration, n/N, is a continuous function, even in circumstances when the applied loading spectrum is continuously varying in time. Hence, the residual strength equation can be used to determine the effect of cumulative damage under spectrum loading. As indicated by fig. 45, the degradation of the material in the critical elements enters the computation through the phenomenological characterization of the uniaxial unidirectional fatigue behavior of the 0° plies, represented by the S-N curve used to compute TV as a function of the local stress amplitude in the critical elements. The choice of the failure function is dependent upon the anticipated failure mode of the laminate itself and the nature of the participation of the critical elements in that failure event. u

64

K.L.

Reifsnider

SCENARIO = T E N S I O N - T E N S I O N , N O SIGNIFICANT DAMAGE A N D Δ Ε => LOCAL STRESS S T A T E * ANTICIPATED FAILURE

DELAMINATION EFFECT ;

Λ Λ

MODE

NORMALIZED CHANGE IN RESIDUAL STRENGTH

CRITICAL ELEMENT DEGRADATION EQUATION LOCAL STRESS STATE FROM Δ Ε M E A S U R E M E N T *

(l^-fO-FSnlMfl'-'d® LIFE RELATIONSHIP FOR Ο DEGREE PLIES

FAILURE FUNCTION FOR CRITICAL ELEMENT Ο DEGREE PLIES (

\

σ

ι ^

ζ^

σ

_B_"T'x

S(n) N = log"' " L S„ "

_ Ot(n)
.

T (n) 2

S2

*FROM

MECHANICS

SHAPE OF RESIDUAL STRENGTH CURVE FOR Ο DEGREE PLIES

MODELS OF DAMAGE

log

Π

Fig. 45. An interpretation of the terms in the residual strength equation for tension-tension fatigue loading of a composite laminate.

For the case of tensile failure of a laminate following a loading spectrum dominated by tensile load excursions, fracture initiation is usually controlled by fiber failure in the 0° plies (or the plies most nearly aligned with the load axis). The exact nature of this phenomenon has not been precisely established. Although some broken fibers can be found near primary matrix cracks quite early in the life of a fatigue specimen or component, as indicated in an earlier section, there are a greater number of fractures sites and a greater number of broken fiber groups in the final 10% or so of the damage-fatigue life curve. By examining deplied layers of graphite fibers from specimens cycled to this final damage acceleration stage, Jamison found groups of broken 0° fibers associated with matrix cracks in adjacent plies [55]. The number of broken fibers in a given group is small, usually four or five. These observations suggest that the locations of fiber fractures are not entirely controlled by the locations of statistically weaker regions in the fibers, but are biased by other damage modes which cause local stress concentrations and non-uniform stress along the length of the fibers. The small number of broken fibers in a heavily damaged but not fractured laminate suggests the premise that a limited or critical number of broken fibers in a small neighborhood triggers the fracture event. Such a conclusion is supported by the experimental and analytical work of Tamuzs [56], and the statistical theories of Harlow and Phoenix [57] as well as Batdorf [58]. As suggested earlier, the writer

Damage and damage

mechanics

F3-I

F5-5

10

65

F4-6

08

S

• Δ

06

F5-5 F2-2 F3-I FI-9 F4-6

ο

Iω

Predicted Observed Observed -Predicted - Predicted

0.4

ω

rr

life life residual strength residual strength with biaxial correction residual strength without correction

71.7 ksi 71 ksi 6 5 ksi 6 4 ksi 5 7 ksi

, 10 kn , , 2l.3kn , 2 0 0 kn , 2 9 0 kn , ^ 3 0 kn

RS 7 8 ksi life , R S = 8 7 ksi life , RS= 81.8 ksi

0.2 h

3.0

4.0

50

6.0

Log Cycles Fig. 46. Predicted and observed strength and life data for a graphite epoxy coupon specimen under tension-tension loading.

believes that this critical fracture event occurs as a result of increased fiber fracture in the generic region of highly concentrated and localized stress and damage such as those depicted in fig. 9, known to occur near the intersection of primary and secondary cracks at ply interfaces. Figure 46 from ref. 60 shows an example of the typical results obtained for the tension-tension loading of a 48-ply graphite epoxy laminate compared with pre­ dictions from eqn. (2). That figure also indicates the sensitivity of the model and the results to the biaxiality of stress in the critical elements (the 0° plies in this case) shown by the shift from the predicted curves indicated by dotted lines to the solid curves and the corresponding improvement in the agreement with data. Two other parameters should be mentioned. Time-dependent degradation pro­ cesses enter eqn. (2) as a reduction in the constitutive properties of the critical element or as a time-dependent stress redistribution as a result of the degradation of the subcritical elements. F o r example, if chemical degradation entered the problem illustrated in fig. 45, the constitutive information which is used to estimate the life of the critical element in the stress state defined by the subcritical elements would be influenced, presumably in a manner which would indicate a reduction of life, N, at a given point in time or number of cycles. That influence would be reflected in increased values of τ in eqn. (2) and a more rapid increase in the value of the calculated integral which reduces the residual strength. Another parameter to be considered is the non-uniformity of stress. T o predict the residual strength and life of notched components, a representative volume and included critical element must be chosen in a region which is associated with

66

KL.

Reifsnider

the failure event. That is likely to be a region of increased stress because of the concentration associated with the notch. The nature of that concentration is represented by the ratio of the value of the failure function in the critical element to the value at some remote point in the laminate, as indicated in eqn. (3). As all the factors in eqn. (2) are normalized, that ratio does not influence the calculation of the residual strength unless the global stress concentration changes as a function of time because of damage in subcritical elements in that region. If that happens, it is equivalent to a change in the level of stress applied to the representative volume, which will certainly influence the rate at which the critical element degrades. Such a change is accounted for by the ratio R in eqns. (2) and (3). If, as is commonly the case for many laminate types, the damage development around a notch or hole relaxes the geometric constraint and therefore reduces the global stress concentration on the representative volume which controls failure, the ratio R will increase, to reflect the observed increase in the strength of the laminate under those conditions [43, 47]. Such an increase is represented quite naturally by the model in a manner which directly reflects the physics and mechanics of the situation and the consequences of those events. Of course, as that global stress relaxation around a notch diminishes and stabilizes, and as damage continues to develop throughout the rest of the represen­ tative volume, a degradation in strength will eventually occur, as suggested by the curve in fig. 47. When that strength reduction reaches the applied stress level, fracture will occur in the notched component. This increase and subsequent decrease in residual strength is impossible to recover from a model which does not properly represent mechanistically the physics and mechanics of the situation, but has been recovered from applications of the critical element model to the prediction of the residual strength and life of notched coupons. We will provide an example of this application in subsequent paragraphs. The last parameter to be discussed has to do with statistical variations. This is a very challenging frontier, which is incompletely developed. It will serve present purposes

σι c

(7i> Ό Ο

Ε

Load History Fig. 47. Variation of residual strength for a notched composite laminate, showing an initial increase due to the relaxation of the stress concentration associated with damage development, and the subsequent decline in strength due to material degradation.

Damage

and damage

67

mechanics

to illustrate a simple example of how statistical variations can enter this residual strength calculation. We assume that the reliability distribution of static strength, F, of a laminate which fails by the failure mode under consideration is given by a Weibull function as illustrated in eqn. (5), and that the corresponding distribution of life, L, for that laminate with that failure mode is given by a Weibull distribution as shown in eqn. (6): P,(il) = exp[-(f /F )]" L

P (L) = f

(5)

L

exp[-(LILf]

(6)

where F and L are characteristic values and α and af are shape factors of the distributions. If we now assume that eqn. (2) controls the degradation of strength over a period of time, and make the equal-rank assumption of H a h n and others, we can write the probability of surviving life L as a Weibull function with the form L

JY

Fl{L)

P (L) = exp t

R ( L ) { l - t f

[ L - F

E

(7)

( T ) ] * ( T ) * - ' D T } / i 7

where Fl(L) is the applied failure function value in the critical element at life L. We can see that the effective characteristic strength and shape parameter of the distribution of life is different from the distribution of static strength, and is influenced by the mechanisms of degradation and damage that enter the denominator of the ratio in the exponent of eqn. (7). Hence, the shape factor and characteristic life of the component is dependent upon the nature of eqn. (2). In addition, it is clear that any distribution in initial static strength that may be associated with such things as material variability or manufacturing details is reflected in the characteristic strength and shape factor of that distribution as it appears in eqn. (7). A different treat­ ment of statistical effects is suggested by Sendeckyj in another chapter of this book. The critical element model has been used for a variety of tension-tension, tensioncompression, and block loading spectra and has been compared with the results of over 400 tests. The correlation between predictions and observations suggests that the model in its present form is useful for engineering calculations, and that the critical element concept and approach is valid [59-61]. Improvements in representations of the micro-mechanics of the behavior will improve the inputs into the model and the subsequent accuracy of the predictions. If delamination is a major damage mode, and the laminate is subjected to combined tension and compression loading cycles of comparable amplitudes, then compression failure modes can be expected. M a n y of these failure modes involve buckling of some sort, a fact which greatly complicates the analysis of such behavior as structural stability is very sensitive to geometry, stiffness, and boundary conditions, all of which can be significantly influenced by the development of damage during cyclic loading. One representation of this phenomenon can be achieved by writing eqn. (2) in the form (8) 0

K.L.

68

Reifsnider

where E* is the stiffness of a fully delaminated laminate, E is the undelaminated laminate stiffness, a is the length of the edge delamination (measured in the width direction) and b is the half-width of the laminate coupon. Although this equation is written for the delamination of a strip of material, it will serve our purpose of illustrating the nature of the mechanics that is usually involved for such a situation. It is generally assumed that the delamination length a is determined from an integral of a power law relationship between the rate of delamination propagation and the strain energy release rate, G: L

a = a(n ) = α x

j

G dn

(9)

fi

ο

The quantities α and β in that equation are assumed to be material constants [19, 62, 63]. If it is assumed that the strain energy release rate includes all modes of crack propagation, then we can use an expression introduced by O'Brien [62] to write G = G(n) = y

( £ - E*)

(10)

L

where ε is the applied laminate strain and t is the laminate thickness, as discussed in another chapter of this book. We can also use an expression introduced by O'Brien for the laminate stiffness as a function of the length of the delamination to write the laminate strain as a function of the number of applied cycles for a given value of applied stress σ . Ά

(£* - E )(a/b) L

+ E

L

Hence, all the quantities which are needed to make the computation suggested by eqn. (8) are known. However, there is still the question of what criterion to use to predict the failure event for a coupon or any other delamination situation. F o r the present case, we could make the following simple argument. We begin by writing eqn. (8) in the form (1 - S ) = (1 - σ /σ„) = (1 - E*lE )(alb) r

Γ

L

(12)

Next we ignore, for the moment, stress redistribution, in the sense that we assume that the strain energy release rate, G, in eqns. (9) and (10) is independent of the number of cycles of load application. Then we can also write a /a T

u

= (E*/E

L

- l)(a/b) + 1

(13)

where s is the critical buckling strain. The critical strain s should be determined from a proper analysis of the global or local stability considerations for a given case. If eqn. (13) is rearranged as c

c

Or = £

[(E* - E )(alb) L

+ E ] = s E(n) L

{u

(14)

Damage and damage

mechanics

69

the resulting expression can be read as stating that the residual strength of a laminate which is delaminating is equal to the reduced stiffness times the critical strain of that laminate. F o r edge delamination, O'Brien [64] has suggested that eqn. (14) is a proper representation of the residual strength of a laminate when the critical strain is equal to the quasi-static buckling strain for the laminate before delamination began. Prinz [65] has attempted to describe the residual strength of a delaminated specimen by obtaining an approximate solution to the buckling strength for a coupon with delaminated edges. Although a number of restrictive assumptions were made in the analysis, satisfactory correlation with data was achieved for the circumstances examined. The problem of determining a proper failure criterion and a corresponding well-defined critical element for the delamination-dominated situation is clearly a frontier in this field. Although success has been obtained for a number of individual cases for highly specific situations, there does not appear to be a systematic approach which yields consistently accurate predictions of the compression strength in the presence of delamination. Whereas comparatively little investigative effort has been dedicated to the problem of properly defining compression failure in the presence of delamination and other damage, the description of delamination propagation and initiation has received a great deal of attention in the literature. Of special interest is the work of O'Brien [18, 62, 64, 66], who has addressed initiation, growth, strength, and both fatigue and quasi-static loading situations, and Whitcomb [67-70] who has used a variety of approximate stress analysis techniques to address the question of strain energy release rate for edge and internal delaminations of various geometries, sublaminate buckling, and instability-related growth. Also of interest are a variety of publications generated by the applied community, such as ref. 71. One statement from that report sums up the present failure criteria situation by observing that "each contractor views the treatment of failure criteria from its own background, experience, and prejudice. Most agree on procedures for satisfying requirements for simple behavior such as unnotched in-plane tension, but most disagree on practically all other possible failure modes". For well-controlled situations such as combined tension-compression loading of coupon specimens, engineering-level predictions with reasonable accuracy can be made using simple failure criteria such as a critical delamination length. Examples of such predictions using eqns. (3)—(14) appear in tables 2 and 3. Table 2 shows predicted and observed strength and life data for quasi-isotropic graphite epoxy laminates subjected to tension-compression loading with three different stress ratios. Table 2 also demonstrates the ability of the critical element model to predict the results of block loading of a laminate with a stacking sequence which was completely different from the laminate for which the model was originally developed, one of a number of validation tests to which the model was subjected. The model is sensitive to the history and sequence of loading; the predicted and observed lives of laminates are different when the order of application of blocks of loading is reversed. Because of the mechanistic nature of the model, this sequence-dependence arises in a very natural way. Table 3 shows experimental data and model predictions for two different sets of block loading. Set 2 consisted of one block of loading of 150 000 cycles with a fully reversed strain amplitude of 3500 με, followed by a second block of fully reversed

K.L.

70

Reifsnider

TABLE 2 Summary results for variable R series R

Specimen

Max ε

Min ε

Observed

Strength (%)

Life (10 ) 3

Predicted -0.5 -0.5 -0.5

C6-8 C8-6 C8-12

6000 5000 4500

-3000 -2500 -2250

34 269 1000 +

9.8 240

-1 -1 -1 -1

C7-1 C6-2 C5-11 C8-8

5000 4500 4500 4000

-5000 -4500 -4500 -4000

18 56 77 328

12.5 77.4 77.4 325

-2 -2 -2

C8-10 C7-9 C5-3

2500 2250 2000

-5000 -4500 -4000

45.6 232 1000 +

3

Predicted

13

Observed

Predicted

0.96

0.96

1.0

1.0

4.6 226

120 440

19 139

Predicted from stiffness change data. Predicted from delamination equations alone. b

TABLE 3 Results of block loading tests and predictions Set

Specimen

5-6 7-6 8-7

Block

Cycles (thousands)

150 57 150 32 150 11

Average life (thousands)

8-5 7-4 7-8

150 327 150 313 150 232

b

c

c

c

% Error

a

5.4

b

(222)

451 486

474

(600) a

173 195

183

(250)

Predicted life (thousands)

c

11

c

18

a

b

(490)

Delamination law driven calculation. Measured stiffness change driven calculation. Block 2 without previous block 1; calculation using estimates of stiffness change data.

loading type of loading loading

at 4500 με until failure occurred. Table 3 shows some typical results of that loading. For the three tests in that table, the average life observed for set 2 was 183 000 cycles. Set 3 loading consisted of a block of tension-tension with an R value of 0.1 and a maximum strain level of 4000 με for 150000

Damage and damage

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cycles, followed by fully reversed tension-compression loading at a strain amplitude of 4000 με. Table 3 indicates that the average life for the three tests shown there was 474000 cycles. The delamination propagation model described above was applied to these block loading situations. The initial value of the strain energy release rate, G, was calculated from the initial strain (determined by dividing the applied stress by the initial modulus), the laminate thickness, and the difference between the fully delaminated modulus and the initial modulus of the laminate as indicated in eqn. (10). Thereafter, as the number of cycles was incremented, the strain was increased according to eqn. (11), based on the calculations of current crack length from eqn. (9). The summation of the change in residual strength was determined from eqn. (8), using an appropriate computer code. Equation (9) was integrated numerically. The results of those calculations are also shown in Table 3. F o r set 2 loading, the predicted life is about 173 000 cycles, compared with the observed average life of 183 000. The difference of 5.4% is reasonably small. For Block 3 loading, the calculated life of 451000 cycles compares well with the observed average life of 474000 cycles, a difference of 4 . 8 % . Hence, based on these limited results (and others for which references are given above), the model appears to be self-consistent and to produce reasonable predictions, even for block loading situations. Two points regarding this application are worth special note. First of all, the value of strain used in eqn. (10) to calculate the strain energy release rate is the total strain range, not the strain amplitude. We can justify this choice on the basis of a variety of philosophies. The principle motivation for the author was provided by the apparent importance of the shear stresses (and strains) in the delamination process. If the inter­ laminar shear stresses are, indeed, a major part of the driving force for delamination initiation and propagation, then a strain range (or stress range) is a more appropriate quantity to use in the propagation equation than a strain (or stress) amplitude as the sign of the shear stress is immaterial to the process. Ultimately, the most convincing argument for the use of the strain range is the success and utility of the idea. The second important matter to be mentioned is that the block loading was handled in the calculations mentioned above by using the delaminated crack length obtained in the first block of loading as a starting point for the second block of loading, an initial crack length concept. Although this is consistent with the physical idea of the mechanism involved, various other choices are certainly possible. Of course, damage modes are sometimes combined. This is especially true for notched materials where various combinations of planar tension-compression, shear, and highly three-dimensional stress fields are present even for unidirectional loading. The modeling of residual strength after fatigue loading of notched composite laminates involves globally non-uniform stress fields. As the strength evolution equation [eqn. (2)] is written in terms of the critical element state of stress and state of material, special care must be used in applying the model to situations which involve non-uniform stress states at the global level, when the critical element strength evolution is interpreted in terms of the global strength of a component. This situation is best described by an example; we choose the problem of the strength evolution of a notched coupon laminate for this illustration.

72

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Reifsnider

The notched problem is of particular interest because the residual strength increases before decreasing because local damage relaxes the geometrical constraint of the notch, as noted earlier. Because of that, the residual strength typically rises over a larger fraction of the total life of the laminate or component, as suggested by fig. 47, before decreasing rapidly in the last small portion of the life. This interaction of damage modes and their consequences is represented by eqn. (2), as the relaxation of the local stress concentration in the representative volume as damage develops around the notch drives the ratio R to values greater than 1, which causes an increase in residual strength in the early part of the test. As damage develops further, the fraction τ or n/Nin the integrand of eqn. (2) becomes large and the damage begins to dominate the strength change. A n example of the application of eqn. (2) to the prediction of notched strength is shown in fig. 48. That example represents the behavior of a C300/PMR-15 woven material in plate form, with a stacking sequence of (0, 45, 0, — 45, 0, 0, — 45, 0, 45, 0 ) . Coupon specimens 25 m m wide with a 9.375 m m diameter outer hole were cyclically loaded at 9 4 % of their tensile ultimate strength at a stress ratio of 0.1. The residual strength of one such specimen measured by inter­ rupting such a test is also shown in fig. 48. T h a t figure also shows a prediction which neglects the effect of stress relaxation associated with damage development in the region near the hole; a clearly discrepant result is produced. It is clear that stress relaxation plays a key role in the physical situation, and in the model. Table 4 shows similar results generated by Wagnecz [72], who demonstrated an experimental method of characterizing damage which can be used in association with an approximate stress analysis scheme to estimate the value of R in eqn. (2). Other experience suggests that these results are typical, and that this approach is viable for many situations which involve globally non-uniform stress states. 2

Current research in this area suggests that proper representations of this phenomenon require accurate estimates of both the global stress field in the region of the stress concentration (the representative volume) as well as the local micro-stress fields that are associated with damage and failure events [72-74].

Δ

.0

PREDICTION WITH RELAXATION WITHOUT RELAXATION

0.8

NORMALIZED RESIDUAL STRENGTH

0.6

0.4

0.2

5

10

15

20

CYCLES ( 1 0 ) 3

Fig. 48. Residual strength prediction using eqn. (2).

25

Damage and damage mechanics

73

TABLE 4 Comparison of experimental and theoretical notched residual strengths of a 22-ply 0° laminate Specimen

a (in.)

b (in.)

Experimental strength

Theoretical strength

112 104 107 113

112 107 109 113

3

1B-4 1B-7 1B-9 1B-12

0.886 1.04 0.669 1.02

0.221 0.276 0.217 0.236

5

Tercentage of notched tensile strength of similar specimens. Percentage of original notched tensile strength.

6. Concluding remarks This chapter has presented only a small sampling of the mechanics associated with damage development in composite materials in a limited number of circumstances. Although limited space is a contributor to the incompleteness of this treatment, ignorance and uncertainty are major limitations as well. Although important progress in this area has been made by a variety of investigators, and some systematic approaches to the subject have been suggested, technical horizons surround us. Mechanistic models are only as good as our understanding of the physical damage events induced by fatigue loading in composite materials. If progress is to continue in the area of mechanistic modeling or cumulative damage, progress must continue in the development of understanding of these phenomenological events. There is also a need for more thorough and complete analysis of the internal stress states that exist in the neighborhood of damage events. This is especially true of events which involve or induce strongly three-dimensional and interactive stress states, such as transverse cracks which cross at the interface of two plies which have different orientations. Mechanistic modeling to date has concentrated on the development of damage. The coalescence and localization (interaction) of damage has not received sufficient attention, in the opinion of the writer. If accurate predictions of the fracture strength (or residual fracture strength) of laminates is to be obtained from mechanistic modeling, it is essential that additional attention be given to the determination of the precise nature of the fracture event itself, and to those events which precipitate the fracture process. Other horizons are abundant. Systematic attempts to understand and represent damage development and to predict residual strength for non-uniform stress states are only just beginning. There is a continuing need to develop non-destructive testing techniques and associated damage parameters that can be used for mechanistic modeling purposes, as well as for field interrogations for routine inspection purposes, and for quality assurance during manufacturing operations. There is great need for experimental investigation of the internal stress states associated with damage events, and of the pre-fracture states of stress and states of material. In the past few years a number of experimental techniques such as moiré diffraction have been perfected, which are capable of measuring very small displacements and displacement gradients

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associated with small damage events such as matrix cracks, fiber fractures, and local debonds or delaminations. It is essential that these techniques and others be developed and applied to fatigue damage development in composite laminates to provide further guidance for mechanics formulations. And finally, there is a very great need to bring together the mechanics of cycle-dependent damage development (fatigue) and the mechanics of time-dependent damage development usually dealt with under topic categories such as creep, stress rupture, and durability. Physically, these are frequently not separate types of behavior. For example, we can use a kinetic theory of crack growth to predict the residual strength and life of polymeric materials under creep-rupture conditions [75-77]. In the context of the discussion above, this type of behavior should be included in the "material degradation" part of a mech­ anistic model as suggested by fig. 43. There is no technical exclusiveness or conflict in these philosophies. Their separation has only to do with the distinction of academic disciplines and the unfamiliarity of technical communities one with the other. The study of damage accumulation in high-temperature composites is providing strong motivation for the development of a rational approach to the combination of these subdisciplines. It may be well to close this chapter by emphasizing the need for the development of the rational philosophy of this subject. Representations of damage which use fundamental concepts of mechanics are a recent phenomenon, generally associated with the (monotonie) creep deformations of homogeneous materials [78-80]. Cyclic (fatigue) deformations involve an evolution of properties and performance that is more complex and generally more severe than that which occurs for quasi-static or sustained loading. To model the remaining strength in such situations, it may be essential that the geometry of micro-events be correctly represented in the mechanics analysis of this process. Continuum (averaging) schemes suffer from an insensitivity to such specific micro-details, especially if the interaction of damage modes and events plays a significant role in defining the remaining strength. When geometry (at any dimensional level) is important, it must be included in the associated mechanics problem. Hence, we need to develop sufficient understanding of the physical behavior to choose what details can be "averaged" in a continuum representation (as phenomenological constitutive behavior) and what details must be explicitly represented. Damage and damage mechanics is not just a subject that we need to study and understand; it is an engineering reality that we must control and exploit to achieve safety, reliability and efficiency of composite components. Our mastery of the subject will help us determine what we can do with composite materials, what we should do with composite materials, and how to design composite materials to do those things well. Acknowledgements Many of the experimental data which appear in this chapter were developed under research grant AFOSR-85-0087 from the United States Air Force Office of Scientific Research. Some of the fiber fracture studies were supported by

Damage and damage

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75

grant N62269-85-C-0234 from the Naval Air Development Center. The author acknowledges that support, and various other contract and grant support as noted in the text with gratitude for the opportunity it provided for investigation of this subject. The author also acknowledges the assistance of Barbara Wengert, who typed the manuscript and provided invaluable help and expertise in the preparation of this book. Finally, the author acknowledges the contributions of his colleagues, Professors Henneke, Stinchcomb, and D u k e in the Materials Response G r o u p at Virginia Tech, as well as those of innumerable students who have contributed so much diligence and excellence to our research efforts. References 1 I.M. Daniel (Ed.), Composite Materials: Testing and Design A S T M STP 787, Philadelphia, PA, 1982. 2 K.L. Reifsnider (Ed.), Damage in Composite Materials: Basic Mechanisms, Accumulation, Tolerance, and Characterization, A S T M STP-775, Philadelphia, PA, 1982. 3 K.N. Lauraitis (Ed.), Fatigue of Fibrous Composite Materials, A S T M STP 723, Philadelphia, PA, 1981. 4 J.T. Fong (Ed.), Fatigue Mechanisms, A S T M STP-675, Philadelphia, PA, 1979. 5 H.T. Hahn, in S.W. Tsai (Ed.), Composite Materials: Testing and Design (Fifth Conference), A S T M STP 674, Philadelphia, PA, 1979, pp. 383-417. 6 T.K. O'Brien, in K.L. Reifsnider (Ed.), Damage in Composite Materials: Basic Mechanisms Accumulation, Tolerance, and Characterization, A S T M STP 775, Philadelphia, PA, 1982, pp. 140-168. 7 K.L. Reifsnider and A.L. Highsmith, Materials: Experimentation and Design in Fatigue, Westbury House, Guildford, Gt. Britain, 1981, pp. 246-260. 8 J.F. Mandell, D . D . Huang and F.J. McGarry, Compos. Technol. Rev. 3(3) (1981) 96. 9 M.G. Bader, J.E. Bailey, P.T. Curtis and A. Parviz, Proc. 3rd Int. Conf. on Mechanical Behavior of Materials, Cambridge, Gt. Britain, 1979. 10 K.L. Reifsnider, E.G. Henneke, II and W.W. Stinchcomb AFML-TR-76-81, Wright-Patterson A F B , Oh, 1979. 11 G.T. Dvorak and J.D. Taru, Interim Tech. Rep. 1, U.S. Army Grant DA-AROD-31-124-71-G103, Duke University, 1973. 12 R.D. Kriz, W.W. Stinchcomb and D R . Tenney, VPI-E-80-5, 1980. 13 R.D. Jamison and K.L. Reifsnider, Interim Rep. to A F Wright Aeronautical Laboratories, A F W A L TR-82-3103, 1982. 14 K.L. Reifsnider, W.W. Stinchcomb, E.G. Henneke, II and J.C. Duke, AFWAL-TR-83-3084, Vol. 1, Wright-Patterson A F B , O H , 1983. 15 K.L. Reifsnider and R.D. Jamison, Int. J. Fatigue, 4 (4) (1982), 187. 16 W.S. Johnson (Ed.), Delamination and Debonding of Materials, A S T M STP 876, Philadelphia, PA, 1985. 17 K.L. Reifsnider, E.G. Henneke and W.W. Stinchcomb, Composite Materials: Testing and Design (4th Conf.), A S T M STP 617, Philadelphia, PA, 1977. 18 T.K. O'Brien N A S A Tech. Mem. 85768, 1984. 19 G.E. Laws and D.J. Wilkins, N A V - G D - 0 0 5 3 , Naval Air Systems Command, 1984. 20 J.D. Whitcomb, N A S A Tech. Mem. 84598, 1983. 21 R.J. Nuismer and S.C. Tan, in Z. Hashin and C.T. Herakovich (Eds.), Mechanics of Composite Materials: Recent Advances, Pergamon, Oxford, 1982, pp. 437-448. 22 A H . Nayfeh, Fibre Sci. T e c h n o l , 10 (1977) 195. 23 A.L. Highsmith, M.Sc. Thesis, College of Engineering, Virginia Polytechnic Institute and State University, 1981. 24 N . Laws, G.J. Dvorak and M. Hejazi, Mech. Mater. 2 (1983) 123.

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