Document 9JvdN00pQwEL9Kev1LnQ30DDV

MICROMECHANICAL CHARACTERIZATION OF SURFACE TREATED REINFORCING FIBERS IN DUCTILE POLYMER MATRICES Author: W. A. Fraser Supervisor: F. H. Ancker Project No.: 597M02 File No.: 4280 Research and Development Department Chemicals and Plastics Union Carbide Corporation Bound Brook, New Jersey , ..--iivr.o FEB 5 i3/o UCC 003936 1 Summary TABLE OF CONTENTS Page ........................................................................................................................................ 1 Introduction ............................................................................................................................. 2 1. Fiber Reinforcement Theory ........................................................................... 4 1.1 Composite Properties: "Thermoelastic" and "Strength" .................................................. 4 1.2 Continuous Fiber Reinforcement: Tensile Failure Modes ........................................................................... 6 1.3 Discontinuous Fiber Reinforcement .......................................... 7 2. Experimental Method ............................................................................................... 10 2.1 Constituents ........................................................................................................11 2.2 Procedure ............................................................................................................. 12 3. Experimental Data....................................................................................................... 13 3.1 Effect of Total Specimen Strain on 2,/d Distributions .............................................................................. 13 3.2 Fiber/Matrix Interactions ............................................................ 13 3.3 Thermal Stability of Fiber/Matrix Interactions .................................................................................................... 14 3.4 Pitch Based Graphite and Thornel 300 .................................. 15 3.5 Data Interpretation Guidelines .................................................. 16 4. Theoretical Analysis ............................................................................................ 16 4.1 Statistics of Brittle Fracture .................................................. 17 4.1.1 Weakest Link Strength Models ..................................... 17 4.1.2 The Weibull Distribution Function ....................... 19 4.2 The Assumed Fiber Strength Model ............................................. 22 4.3 Stochastic Fiber Fragmentation Model .................................. 25 4.3.1 Proof Testing and Truncated ' Breaking Strength Distributions ............................ 25 4.3.2 Computer Simulated Fiber Fragmentation .... 26 4.4 Parameter Estimation Procedure ................................................... 28 4.4.1 Objective Function & Search Method .................... 29 4.4.2 Parameter Constraints ........................................................ 29 4.5 Data Analysis .................................................................................................. 32 5. Conclusions .................................................................................................................. 33 Acknowledgements ....................................................................................................... 34 References ........................................................................................................................ 35 UCC 003937 TABLES AND FIGURES ii Page TABLE I: TABLE II: TABLE III: TABLE IV: TABLE V: Components of a Typical GlassTreatment Fiberglass Descriptions Properties of Thornel 300 Properties of Pitch BasedGraphite Matrix Resin Properties FIGURE 1: FIGURE 2: FIGURE 3: FIGURE 4: FIGURE 5: FIGURE 6a: 6b: 6c: FIGURE 7: FIGURE 8: FIGURE 9: FIGURE 10: FIGURE 11: FIGURE 12: FIGURE 13: FIGURE 14: FIGURE 15: FIGURE 16: FIGURE 17: FIGURE 18: FIGURE 19: FIGURE 20: FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE 21: 22: 23: 24: 25: 26: 27: 28a: 28b: 29: FIGURE 30: Stress Transfer to an Embedded Fiber Critical Fiber Length Concept Fiber Length Efficiency Factor Multiple Fiber Fracture Compression Mold and Fiber Fork Fragment l/d Cum. Dist'n. (OCF 885-PP) l/d Histogram l/d Frequency Curve Effect of Specimen Strain on l/d Dist'n. l/d Dist'ns.; OCF 885, OCF 415 in Nylon 6 OCF 885, OCF 415 in Polypropylene OCF 885, OCF 415 in HDPE Summary Plot/OCF 885 in Nylon 6,PP.,HDPE Summary Plot,OCF 415 in Nylon 6,PP.,HDPE Effect of Temperature, OCF 885 in Nylon 6 Effect of Temperature, OCF 415 in HDPE Pitch Based Graphite in Nylon 6, PP. Thornel 300 in Nylon 6, PP. Pitch Based Graphite in Epoxy (ERLA 4617) OCF 885, OCF 415, Pitch Fiber, Thornel 300 in PP. OCF 885, OCF 415, Pitch Fiber, Thornel 300 in Nylon 6 General Strength-Length Relationship for Glass Assumed Link Strength Dist'n. Function Hypothetical Fiber Strength Length Curves Proof Testing Effects Stochastic Fiber Fragmentation Strength-Length Behavior - Flaw Free Fiber Theoretical l/d Dist'n.'s -Flaw Free Fiber Strength-Length Behavior-Flawed Fiber Theoretical l/d Cum. Dist'n.-Flawed Fiber Theoretical l/d Frequency Curve Experimental and Optimized Theoretical l/d Cum. Dist'ns.-OCF 885 in PP. Computed Strength-Length CurveOCF 885 in PP. 38 38 39 39 40 41 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 UCC 003938 FIGURE 31: FIGURE 32: Experimental and Optimized Theoretical /d Cum. Dist'ns.-OCF 885 in Nylon 6 Computed Strength-Length CurveOCF 885 in Nylon 6 APPENDICES A. Data Smoothing and Differentiation B. Probabilistic Derivation for the Effect of Specimen Size on Strength C. The Mixed Element, Weakest Link, Fiber Strength Model D. The Sequential Fragment Proof Testing Equations E. Data Reduction Program Listing F. Data Analysis Program Listing Ill Page 73 74 75 77 79 81 84 98 UCC 003939 BUSINESS CONFIDENTIAL 1. PROJECT REPORT MICROMECHANICAL CHARACTERIZATION OF SURFACE TREATED REINFORCING FIBERS IN DUCTILE POLYMER MATRICES AUTHORS-' W. A. Fraser DATE; January 30 , 197 5 PROJECT NO.: 597M02 supervisor: F. H. Ancker FILE NO.: 4280 SUMMARY The present report discusses reinforcing fibers and fiber surface treatments in terms of the micromechanics theories of bulk composite strength. A new single filament technique is discussed which enables direct experimental determination of the fundamental micromechanics parameters which control the strength properties of a composite. A brief outline of this technique has been reported earlier in the form of a paper submitted to the 1975 SPI conference on Reinforced Plastics/Composites1. The present report expands on this subject in several distinct areas. (1) A review of pertinent fiber reinforcement theory is presented to illustrate the importance of coupling and sizing agents in enhancing bulk composite properties. (2) Additional data are presented including fragment SL/d distributions for an experimental sample of Carbon Products' new continuous filament pitch based graphite and "Thornel" 300 in both nylon 6 and polypropylene. Also discussed is a preliminary effort in extending the procedure to include thermoset resins. Jl/d distributions are presented for the pitch fiber in an epoxy matrix (ERLA 4617) tested at room temperature and 150'" C. (3) A more detailed and comprehensive discussion of the statistics of brittle fracture describes the development and evolution of the assumed probabilistic fiber strength model. (4) Detailed listings of the computer programs used for both data reduction and data analysis are included in an appendix. This report is intended to serve as a reference manual for future utilization of the fiber fragmentation, coupling/sizing agent evaluation procedure. Research and Development Department Chemicals and Plastics Union Carbide Corporation Bound Brook ,New Jersey UCC 003940 2. INTRODUCTION The U. S. market for fiber reinforced plastics is approaching two billion pounds per year with an above average projected rate of growth. Union Carbide Corporation supplies many components for the composites market, e.g. polymers and chemicals (C & P) , silane coupling agents (C & P), asbestos fibers (Mining & Metals), and graphite fibers (Carbon Products). Within the last decade, significant advances have been made in the theoretical analysis of composite properties. The initial incentive was the need for super performance materials for space and aircraft applications but, now, the mechanics theories are in creasingly being applied to the design of lower cosb mass pro duced composite parts. At the same time, however, these theories have received only limited use by raw material producers in op timizing the constituents of composite materials. The present report discusses current composite theories and applies them to the problem of optimizing fiber surface treatments. High strength, high modulus, reinforcing fibers such as glass or graphite are brittle materials which are suceptible to severe property deterioration due to surface damage in handling. Furthermore, when incorporated in a composite, efficient utilization of the fiber reinforcement requires high fiber/matrix adhesion. To overcome these problems, glass fibers for composites are.always surface treated with complex mixtures of sizing and coupling agents immediately after spinning. Similarly, asbestos.fibers with surface treatments have recently been introduced com mercially and surface treatments for graphite are now being re cognized as a necessary requirement for broad industrial acceptance. High efficiency of reinforcement depends on both the inherent fiber properties and the fiber surface treatment. The fiber treatments increase the reinforcement efficiency by two distinct mechanisms, namely: a. by improving the fiber/matrix adhesion - which is a principal function of the coupling agent, and b. by protecting the fiber from surface damage and thus assure retention of high fiber tensile strength - which is a principal function of the sizing agent. These improvements must further be maintained under varying environmental conditions, e.g. at elevated temperature (to assure high heat distortion temperature, i.e., low creep at elevated temperature), in water (to permit exposure to high humidity), etc. Also, depending on the ultimate application UCC 003941 3. of the fibers, other properties such as prevention and dis sipation of static charge build-up, fiber handling characteristics in filament winding and weaving processes, etc., may be additional requirements. Optimization of fiber surface treatments is inherently a difficult physical/chemical problem because individual com ponents of practical formulations often perform more than one function and in many instances may interfere with each other. Furthermore, traditional composite evaluation techniques mea sure only the overall fiber reinforcement efficiency and thus provide little insight into the micromechanics effects of the fiber treatments. Specialized evaluation techniques capable of distinguishing between the basic coupling agent effect (in terfacial adhesion) and the basic sizing agent effect (fiber protection) have been too tedious to find general utility for optimization of piactical fiber treatments. ' In recognition of these problems, it was the aim of this project to develop an improved technique for direct ex perimental determination of the fundamental micromechanics parameters characterizing both the adhesion promoting cap abilities and the fiber protection performance of a given fiber treatment formulation for a given fiber/polymer com posite. The method is based on a convenient experimental pro cedure and is, thus, practical for the routine screening and evaluation of new coupling/sizing agent systems. The procedure provides detailed insight into the micromechanical effects of the various physico-chemical fiber treatments and should facilitate the development of new and improved coupling/sizing systems and processes. Furthermore, by correlating fundamental fiber and in terface characteristics with other important composite properties such as fracture toughness, impact strength, fatique and creep resistance, significant contributions may be realized in under standing and eventually in optimizing these properties. The new single filament test and data analysis pro cedure developed during this program has been summarized in a recent report - r issued in the form of a paper submitted to the 1975 SPI Conference on Reinforced Plastics/Composites'. The present report covers in detail the evolution of this new technique, the underlying basic concepts, the experimental pro cedures, including illustrative examples of the application of the technique to specific fiber/polymer systems. Finally, a complete listing of the computer program package is included in an appendix. A cooperative program has been initiated with the silane coupling agent group at Tarrytown aimed at utilizing this technique for optimizing coupling/sizing agent systems for specific fiberglass/ polymer composite systems. UCC 003942 4. 1. Fiber Reinforcement Theory The experimental procedure and theoretical analysis scheme to be outlined in this report provide fundamental micro mechanics parameters which describe the specific adhesion pro moting capabilities and fiber protection performance of a given fiber treatment formulation. To understand the technique and appreciate the utility of the information it provides, one must have some insight into the fundamentals of fiber reinforcement theory. Several aspects of this subject will be discussed in this section. 1.1 Composite Properties; "Thermoelastic" and "Strength" The following brief review of the micromechanics of uniaxially aligned, fiber reinforced composites will be limited to the analysis of static properties. These properties, using a classification due to McCullough2, fall into two cate gories: "thermoelastic" and "strength". Thermoelastic pro perties, which include tensile modulus, shear modulus, thermal expansion etc., can usually be treated in terms of average pro perties and average responses of the individual constituents. The simple rule of mixtures has been found, in most cases, to be quite adequate for predicting these composite properties. Longitudinal responses are represented by the parallel reaction of the fiber and resin phase. pc - ? Pi v* X -1 U-i) where Pc = composite property P i = component property V component volume fraction n = number of components {two in simple systems) Transverse responses in the composite consider the series re action of the fiher and resin phase and are given somewhat less accurately by 1n ------- Z Pc i-1 Vi /Pi (1-2) The rule of mixtures has also been used to describe the strength characteristics of highly idealized composites containing uniform strength, unidirectional, continuous fibers. Implicit in this analysis is the assumption of "unit structural UCC 003943 5. behavior" among the constituent elements3. This concept implies perfect bonding between matrix and reinforcement, ensuring uniform force transfer between constituents and therefore, their deformation as a single structural unit. For fiber loadings greater than a certain critical value, the composite tensile strength is given by ac* = of* Vf + am (1-Vf) (1-3) where ac* = composite tensile strength Of* = breaking stress of fibers o = stress supported by matrix at of fibers Vf = volume fraction of fibers breaking strain With composite strain equal to both the fiber and matrix strain, failure of the fibers results in immediate failure of the composite unless the remaining matrix, (1-Vf), can support the full composite stress, i.e., 0f*Vf + om (1-Vf) > ora* (1-Vf) (1-4) where om is the ultimate tensile strength of the matrix. Equation (1-4) defines a minimum fiber volume fraction which must be exceeded for equation (1-3) to be applicable: a *--o v = m m___________ fmin Ofj.* + am * - am (1-5) At fiber loadings less than Vfmin, the fibers undergo multiple fracture, a phenomenon which forms the basis for the fiber treatment evaluation procedure to be described in Section 2. Provided that the yield elongation of the matrix is sufficiently large with respect to the fiber failure strain, the fibers will successively fracture into smaller and smaller fragments depending on the maximum shear stress, t, which the interface (or matrix) can sustain. The preceding composite strength model using thermo elastic property analysis procedures, while illustrative, is a vast oversimplification. Strength is a statistical phenomenon, strongly dependent on the nature of local environments and, consequently, cannot be accurately described in terms of average component responses. A prerequisite for the analysis of strength is the identification of the modes of composite failure, and the parameters which influence it. Attention must be focused on the UCC 003944 6. ways in which "unit structural behavior" breaks down. Coupling and sizing agents have a major influence on the stress limits for this breakdown and on the mechanisms by which it occurs. 1.2 Continuous Fiber Reinforcement: Tensile Failure Modes In composite systems consisting of strong brittle fibers in ductile and semi-brittle matrices, failure mechanisms can be quite complex with the mode and stress level at failure dependent on matrix, fiber, and interfacial characteristics. A number of failure modes have been observed and distinguished. Consider a composite reinforced with uniaxially aligned con tinuous filaments and subjected to a tensile load. With semibrittle resins, fiber breakage can initiate a shear crack which propagates along the fiber/matrix interface separating or de coupling the fiber from the matrix and reducing its effective ness over a substantial fiber length. Coupling agents, designed for specific fiber/resin systems,are intended to inhibit this type of failure. Another failure possibility is that following a single fiber fracture,a crack propagates through the matrix , parallel to the fibers or perhaps transversely across the com posite. These failure modes are influenced, in part, by the toughness of the matrix, i.e., its ability to deform and re distribute shear stresses. If the preceding failure modes are arrested, increasing the applied tensile load can result in progressive fracture of the fibers accumulating throughout the composite. Parratt1* and Riley and Reddaway* suggested that, in the extreme case, progressive fracture might reduce fiber length to the point that further increases in load cannot be transmitted to the fiber because the maximum interfacial shear strength is ex ceeded. In essence, what was originally a continuous fila ment reinforced composite has now been reduced to a system consisting of discontinuous fibers. Composite failure pro ceeds by shear failure of the matrix and fiber pullout. The concepts of fiber transfer length and critical fiber length are essential to understanding this failure mechanism and will be discussed in more detail in a subsequent section. In proposing this mechanism, the authors also recognized that the statistics of fiber strength must be considered. As the fibers become shorter, breaking at local defects, they become stronger {see Section 4) and can, therefore, sustain a larger load be fore fracture. Both coupling and sizing agents, affecting the fiber transfer length (fiber/matrix adhesion) and fiber strength (degree of fiber damage) respectively, influence the stress level at which this failure mode occurs. UCC 003945 7. Hale and Kelly6 have shown that the preceding failure mode is viable only at low fiber loadings. For higher fiber volume fractions Rosen7 has hypothesized that composite fail ure can take place before the conditions for fiber pullout are reached. In this mechanism, composite failure supposedly re sults from the weakening of a composite cross section by the statistical accumulation of fiber fractures with increasing load. The model considers that in the vicinity of an individual fiber break, a portion of the fiber is ineffective, i.e., it does not support the full composite stress. The composite is modeled as being composed of layers of dimension equal to the ineffective length (fiber transfer length). Any fiber fracturing within this layer is incapable of transmitting a load across the laye^ and the applied load at this cross-section is then uni formly distributed among the unbroken fibers that remain. When enough fiber fractures accumulate in any one layei; composite failure results. In developing this composite strength model, Rosen recognized the statistical nature of the phenomenon and tried to describe it quantitatively by incorporating brittle fiber, weakest-link fracture statistics. More will be said about statistical fiber strength models in Section 4 of this report. Sizing agents, which influence fiber strength character istics (magnitude and variability) clearly affect the ultimate strengths of composites subject to failure via the cumulative weakening mechanism. Before extending this discussion to include dis continuous fiber reinforced systems, it should be noted that composite properties other than tensile strength show depend encies on fiber coupling and sizing agents. Cooper9 has shown that both fiber/matrix adhesion and fiber flaw structure strongly influence the energy absorbing processes contributing to the work of composite fracture and hence composite tough ness. If fiber surface treatment effects could be accurately characterized, treatment modification might become an important design vehicle for meeting both composite tensile strength and fracture toughness performance requirements. 1.3 Discontinuous Fiber Reinforcement In a composite reinforced with discontinuous fibers, shear strains in the resin are the origin of the mechanism by which tensile loads on the composite are transferred to the fiber. Consider a fiber of radius r imbedded in a resin matrix which is subjected to a tensile load (Figure 1). Let t(x) be the value of the shear stress acting at the fiber matrix inter face. A force balance gives ,, rx a(x)irrz = 2TTr / t (x) dx Jo (1-6) UCC 003946 8. where: a(x) t(x) r x = Tensile stress transferred to = Interfacial shear stress = Fiber radius = Distance from fiber end fiber The distribution of tensile stress in the fiber is determined by the distribution of shear stress at the fiber/matrix interface. A number of models have been proposed for calculating the longitudinal tensile stress distribution in an embedded fiber. The models of Cox9, Dow`tt, and Rosen7 consider the case of an elastic fiber in an elastic matrix and all assume perfect bonding between matrix and fiber ("unit structural behavior"). This assumption is probably valid at low de formations but at high deformations where interfacial failure may take place the plastic flow model of Kelly and Tyson11 may be more appropriate. If we assume that when the interfacial region fails, it does so in shear, by yielding and flowing plastically, t(x) in eq. (1-6) can be considered constant. Thus, integration gives a(x) = (2/r)xx (1-7) This equation, indicating a linear build up of tensile stress from the fiber end, can be used to develop the concept of a critical transfer length. This is the length required to build the stress up to a level to break the fiber and is given by where: Xc = Vd/4T (1-8) Xc = critical transfer length = breaking stress of fiber d^ = fiber diameter t = interfacial shear strength X,, is observed to decrease as t increases. Since stress builds up from both ends of a fiber, a critical fiber length, lc, can be defined as the minimum length of fiber capable of reaching its ultimate strength. From symmetry ic = 2Xc = of*d/2t (1-9) UCC 003947 9. These ideas are illustrated in Figure 2. The concept of critical fiber length is essential to understanding the nature of reinforcement in composites containing discontinuous fibers. The rule of mixtures which has been used to describe the strength of ideal, uniaxially aligned continuous fiber reinforced composites, eq. (1-3), must now include a term which measures the efficiency of reinforcement for short fibers relative to that of continuous fibers. The tensile strength of a uniaxially oriented dis continuous fiber reinforced system is, thus, given by c = ei,af*vf + m U-Vf) (1-10) Where the meaning of the symbols is as for eq. (1-3). e^ is a fiber length efficiency factor li and is given by e= V2c for for a <c l >c (1-11) where is the fiber length When the composite contains a distribution of fiber lengths, the efficiency factor becomes cn e = 2 U./21 ) (Vi/Vf) + E (1-g/2Z.) (Vi/vf) (1-12) * i=o 1 G r i=c where V. = volume fraction of fibers of length i = total volume fraction of fibers Eq. (1-11), is shown graphically in Figure 3 (e^ vs. the reduced fiber length, l/lc). When the fiber length equals c, the re inforcing efficiency is observed to be only 50% of that at tained in continuous fiber systems. Increasing. %/Sic to 5, however, results in a reinforcing efficiency of 90%. The importance of coupling and sizing agents and the necessity for surface treatment optimization in short fiber reinforced composites can be discussed in terms of equations (1-10) and (1-11). In applications involving the injection molding of short UCC 003948 10. fiber filled plastics, the fibers must be relatively short in order to maintain good flow properties, especially when molding intricate parts. Two factors affecting the re duced fiber length and thus the reinforcing efficiency, eq. (1-11), in such a system are (1) the fiber/matrix adhesion which controls lc (a function of the coupling agent)and (2) the effectiveness of the fiber treatment in minimizing fiber attrition during molding (one function of the sizing agent). As seen from eq. (1-10) , sizing agents can also influence the strength properties of these systems by another mechanism i.e., by preserving the fiber strength, a * . These effects have recently been studied by Burns et al in chopped fiber glass reinforced nylon 66.13 This resin is fairly forgiving with respect to fiber/matrix adhesion in that conventional amino-silane treated glass filaments are thought to exhibit interfacial shear strengths approaching the resin shear strength. With adhesion close to optimal, the only mechanisms available for upgrading composite strength depend on sizing agent effectiveness. Relative to systems with fiber lengths tightly distributed about &c T very significant increases in composite tensile strength can be achieved with only a slight upward shift in average fiber length (see Figure 3). Burns, in fact, reports that substitution of a tough film forming size for the more traditional PVAc coating reduced fiber attrition during molding to the extent that composite strength increased 30%. A portion of this strength in crease, as discussed previously, can also be attributed to improved values. In many resin systems, fiberglass/matrix adhesion can be the strength limiting factor. Coupling agents must be specifically tailored for specific applications. Further more, since coupling and sizing agents can interfere with each other,their selection must be carefully coordinated if optimum composite strength is to be realized. Table 1 lists the com ponents of a typical glass treatment formulation. An example of component interference could be the adverse effect of a lubricant on adhesion vs. its effect in minimizing processing induced fiber damage. Another might relate to the protection capabilities of the film former vs. its compatability with the matrix resin. These are just two examples of optimization problems associated with fiber surface treatment formulation. 2. Experimental Method The experimental method comprises embedding a single continuous filament in a tensile specimen of a particular resin. UCC 003949 11. Since this is well below the minimum fiber loading for re inforcement, (eq. (1-5)), deformation of the tensile specimen results in the filament fracturing into small fragments as illustrated in Figure 4. Kelly14 illustrated this multiple fracture phenomenon in a system consisting of tungsten wires embedded in a copper matrix. After deformation, the copper matrix was dissolved and the wire fragments extracted. On analysis of the fragment length distribution, it was found that its bounds were approximately lc and Zc/2. lc in this case was calculated from eq. (1-9)using the appropriate tungsten wire values for aand d and substituting the ultimate shear strength of copper as a value for x. Re calling the definition of critical fiber length, this re sult is not unexpected. The longest fibers capable of surviving in this system would have a length just under l since longer fibers by definition, must fracture. The shortest fragments would arise with fracture of fibers just longer than Z , this occurring very near their midpoint. Analogous experiments were run by Wadsworth and Spilling15 for carbon fibers embedded in an epoxy resin and by Ongchin, Olender, and Ancker12 for fiberglass in high density polyethylene. In both these studies it was observed that the range of the fragment length distributions was very much larger than the expected 2:1 predicted from the critical fiber length concept. Both sets of authors recognized that this dispersion in the fragment length distribution is a consequence of the statistical aspects of fiber strength. Glass and carbon fibers are flaw sensitive materials, i.e., when fracture occurs, it is neither preceded nor accompanied by extensive plastic de formation. The appreciable variation in tensile strength characteristic of these materials can be described in terms of statistical strength models, the parameters of which relate to the level of fiber damage (sizing agent effective ness) . Section 4 discusses statistical strength models in more detail and outlines a procedure for determining model parameters from fiber fragmentation data. 2.1 Constituents The fibers which were investigated included two samples of continuous filament E glass roving prepared for us by Owens Corning Fiberglas in the form of fiber cakes These laboratory spun yarns were treated with two commercially used forming sizes designated OCF 885 and OCF 415. Their properties are listed in Table 2. Also included in the study were samples of PAN based graphite, Thornel 300 (grade WYP 30.' 1/0) UCC 003950 12. as well as an experimental sample of Carbon Products new continuous filament pitch based graphite (designated Parma280-22-9) . The Thornel 300 was coated with an epoxy-com patible size whereas the pitch fiber was untreated. The pro perties of these fibers are listed in Tables 3 and 4. The matrix resins studied were nylon 6 (Plaskon 8201, Allied Chemical), polypropylene (Shell 5524), and polyethylene (DMDJ-7008, Union Carbide). Resin properties are listed in Table 5. 2.2 Procedure Embedding a single continuous filament in a tensile specimen of a given resin was accomplished using the fiber fork 16 and compression mold shown in Figure 5. Single < filaments, carefully teased out of a given yarn, were mounted on the stainless steel fiber fork by epoxying their ends adjacent to positioning notches. The fiber loaded fork was then placed in the mold with each fiber sandwiched be tween a narrow strip of resin sheeting and a resin plaque overlay. The mold cover plate was carefully positioned on top of the resin plaque and the entire assembly placed in a laboratory scale hydraulic hot press. Molding temperatures were ap proximately 250C for nylon 6, 17 5C for polypropylene, and 150C for polyethylene. Molding pressure was of the order of 1000 psi for each resin. The mold cavity was designed to minimize both longitudinal channel flow, which could cause filament pretensioning or even fiber fracture, and transverse flow, which could result in embed ding a curvilinear fiber. The tensile specimen had a crosqsection of 0.25 in. x 0.075 in. and an effective gage length of 3.0 in. The tensile specimens are deformed on an Instron at a jaw speed of 0.02 in./min. to approximately their yield elongation. The specimens are then placed in pyrex glass petri dishes (one specimen per dish) and pyrolyzed on a hot plate at 350C for six hours. After the resin has been transformed into a carbonized char, the petri dishes are placed in a muffle furnace (1-1-1/2 hrs.) for final ashing. Furnace temperatures are typically 54 0 C for polyethylene and poly propylene and 600C for nylon 6. The recovered fiber frag ments are then measured using a Nikon comparator projection microscope. The covered petri dishes are placed on the traveling stage of this instrument and fiber length readings are taken directly from the projection screernat 20X mag nification using a pair of dial calipers. A fiber diameter measure- UCC 003951 13. merit was made for a single filament per dish using a conventional optical microscope at lOOx magnification. The ashing procedure does not degrade the fiber to the extent of biasing the fragment distribution (even in the case of graphite fibers) since ashing an und^ormed tensile bar (thermoplastic matrices) results in the recovery of the unbroken filament. 3. Experimental Data Fragments normalized by their diameter were ranked according to their size and a fractional cumulative frequency distribution evaluated. These data were then smoothed and differentiated using the moving strip, least squares polynomial method described by Hershey, Zakin and Simha17 to give an empirical probability curve of fragment aspect ratios. The smoothing and differentiation procedure is outlined in Appendix A and the data reduction and plotting computer programs listed in Appendix E. Figures 6a, b, c show a cumulative distribution curve (raw data), a fragment aspect ratio histogram, and a calculated probability density curve for OCF 885 m polypropylene. This fragment 2/d dis tribution is seen to be quite broad extending well beyond the 2:1 bounds predicted from the simple critical fiber length concept. It should be noted that in order to ac cumulate a statistically significant number of fiber frag ments for analysis, data from six tensile specimens were lumped together in calculating this distribution curve. 3.1 Effect of Total Specimen Strain on z /d Distributions The data which will be discussed in this and suc ceeding paragraphs will be presented in the form of smoothed cumulative fragment aspect ratio ( i/d) distribution curves. Figure 7 demonstrates that there is little effect of tensile elongation on the cumulative distribution curves of OCF 885 in polypropylene. Fragmentation is essentially complete after a total strain of 0.075 with the measured distribution repre senting the final curve i.e., the interface has reached its maximum stress limit and appears to be yielding plastically. The slight differences observed in the distribution curves are attributed, at least m part, to the limited amount of data (75-100 fragments) used in characterizing each curve. 3.2 Fiber/Matrix Interactions Nylon 6: Figure 8 shows the 2/d distribution curves for both OCF 885 and OCF 415 in nylon 6 after room temperature UCC 003952 14. deformation. OCF 885 fractured into much smaller fragments than OCF 415 indicating a considerably higher degree of fiber/ matrix interaction. This difference is primarily a reflection of sizing agent/matrix compatibility. The PVAc size of OCF 885 is expected to be considerably more compatible with nylon 6 than the olefinic size of OCF 415. The fact that OCF 415 fractures at all is ascribed to the action of the amine reactive coupling agent present in the treatment formulation. Polypropylene: As in nylon 6, Figure 9 shows that OCF 885 exhibits a higher degree of fiber/matrix interaction . than OCF 415; however, both distributions are shifted, to some extent, on the abscissa to higher &/d values. Inter facial shear strengths for both fibers are somewhat lower in polypropylene than in nylon 6. Both sizing agent/matrix com patibility and coupling agent effectiveness are operating here. Polyethylene: In HDPE there is a reversal in position of the fragment &/d distribution curves as shown in Figure 10. OCF 415 with its olefinic size exhibits a much higher degree of interaction with polyethylene than the PVAc sized OCF 885. The coupling agents of each treat ment formulation probably have little, if any, effect in this resin and thus the observed differences in interfacial shear strength can be interpreted primarily in terms of sizing agent/matrix compatibility. Summary plots for both glasses in each of the three resins are shown in Figures 11 (OCF 885) and 12 {OCF 415). In resins which show little response to coupling agents, such as HDPE, sizing agent/matrix compatibility becomes a primary factor controlling fiber/matrix interfacial shear strength. In more readily coupled resins such as nylon 6, lack of size/ matrix compatibility may be compensated for, somewhat, by in cluding a coupling agent in the treatment formulation. This is clearly demonstrated in Figure 12 where OCF 415 perform ance in HDPE is observed not to differ markedly from that in nylon 6. The amine reactive coupling agent in the 415 olefinic treatment - formulation apparently recoups much of the antici pated loss in interfacial shear strength expected on the basis of size/matrix (olefinic/nylon 6) incompatibility. 3.3 Thermal Stability of Fiber/Matrix Interactions The thermal stability of specific fiber/matrix in teractions can be studied by running the fiber fragmentation test at elevated temperatures. The tensile specimen containing the embedded filament is placed in the environmental test chamber UCC 003953 15. of a suitably equipped Instron, brought up to temperature and deformed while hot. Figure 13 shows how the fragment Si/d distributions for the OCF 885 - nylon 6 system shift to higher 1/d values as the temperature is raised from 25C to 100 C to 150C. In Figure 14, the OCF 415 - HDPE system is observed to be even more temperature sensitive. Two factors influence these shifts; the temperature dependence of the bulk resin shear strength and the thermal stability of the inter facial interaction. In a given resin, elevated temperature fragmentation data may well be quite useful in distinguishing between fiber treatments that rely merely on treatment/matrix compatibility to promote adhesion and those that provide true chemical coupling. 3.4 Pitch Based Graphite and Thornel 300 Figure 15 shows the fragment SL/d distribution curves of untreated, experimental samples of Carbon Pro ducts' new continuous filament pitch based graphite fiber in nylon 6 and polypropylene after room temperature de formation. Corresponding Thornel 300 Si/d curves for these same two resins are shown in Figure 16. Interfacial shear strength levels for both fibers are observed to be higher in nylon 6 than in polypropylene. This behavior probably re flects the differential in bulk resin shear strength between these matrices in addition to any difference in degree of specific fiber/matrix interaction. A very preliminary effort was initiated in extending the fiber fragmentation procedure to include thermoset resins. The performance of the pitch based graphite fiber was evaluated in a UCC epoxy matrix (ERLA-4617). Since considerable bulk composite property data existed at Parma on this resin system, it was a natural candidate for study. The ERLA-4617 resin, cured with a near stoichiometric equivalent quantity of methylenediamaline (MDA),was found to have an elongation of 4.5 to 5.0% at a strain rate of 6.7 x 10 3 min~`. Since this elongation was greater than the 1-2% rupture elongation reported for the fiber, utilization of the fiber fragmentation procedure seemed appropriate. Figure 17 shows fragment Z/d distribution curves after room temperature and 150C deform ation. As with the thermoplastic resins, the fiber fragments defining these curves were recovered by ashing the matrix resin. On examining this procedure, however, it was found that the distributions may be somewhat biased. With this resin matrix,even undeformed tensile specimens were found to generate a significant number of fragments on ashing. Apparently the thermal expansion coefficient differential existing between fiber and matrix causes thermal stresses in the initial stages UCC 003954 16. of ashing of sufficient magnitude to fracture the fiber. This was not a problem in thermoplastic matrices where resin yield strength falls off rapidly with increasing temperature. In spite of this undesired spurious frag mentation, the curves of Figure 17 serve to illustrate the sensitivity of the technique for studying fiber/matrix interfacial characteristics in thermoset systems. In future studies involving these resins, alternate fragment recovery procedures will be used. A possible method is acid digestion of the matrix, a technique currently in use at Parma18 to measure resin content in resin-graphite composite materials. 3.5 Data Interpretation Guidelines Composite plots of i/d distribution curves for OCF 885, OCF 415, Carbon Products' new pitch based graphite fiber and Thornel 300 in both polypropylene and nylon 6j respectively are shown in Figures 18 and 19. These figures are presented in order to illustrate the potential hazards which exist in making simple interpretations based on raw fiber fragmentation data. Analysis discussed exclusively in terms of relative degrees of fiber/matrix adhesion can be quite misleading. It must be recognized that two factors control the position of the i/d distribution curves: increased interfacial shear strength drives the distributions to lower i/d values as does increased fiber damage (i.e.> a weaker fiber will fragment into smaller pieces). The pitch based graphite fiber appears to exhibit excellent inter facial characteristics in both resins. Note, however, that its i/d distribution as measured in polypropylene is quite dispersed, an indication, perhaps, of a highly damaged fiber. This is a reasonable explanation since this fiber has no protective surface coating (sizing agent). This complication in the data interpretation is resolved via the theoretical analysis scheme to be described in the following section. 4. Theoretical Analysis The sensitivity of the experimental procedure to specific fiber/matrix interactions was demonstrated in Section 3. Furthermore the experimental fragment /d distribution curves were observed to be quite broad extending well beyond the 2:1 bounds predicted from the simple critical fiber length concept. This dispersion has been attributed to the statistical aspects of brittle fiber strength, a consequence of the dis tribution of local flaws of varying severity along the length of the fiber. Sizing agents, as discussed previously, are intended to minimize defects and consequently their capabilities UCC 003955 17. in this respect must be contained in this dispersity of the fragment ,/d distribution curves. 4.1 Statistics of Brittle Fracture Brittle fracture is a much less reproducible phenomenon than fracture preceded by extensive plastic deformation. As Freudenthal19so aptly describes,"The process originates in highly localized regions of the material where inherent defects in the microstructure, or defects produced in the course of permanent deformation lead to localized intensities of tensile stresses so high that some of the existing defects are converted into flaws at which the cohesion of the material has actually been destroyed." The strength of a brittle material is a function of its de fect structure, and it is this concept that explains the large differences between measured strength values and theoretical values based on calculations of the energy needed to break interatomic bonds. If it is accepted that fracture at any stress level is dependent on the statistical expectation of encountering a critical or fatal flaw, we see immediately that this prob ability increases with specimen size. This behavior can be rationalized from purely probabilistic reasoning without making any assumptions concerning the physical nature of material inhomogeneities. A more formal derivation of the functional dependence of fracture probability on specimen size is given in Appendix B. 4.1.1 Weakest Link Strength Models Assuming flaws to be distributed at random with a certain concentration per unit volume, then the strength of a given specimen is determined by its most severe flaw, i.e., its weakest point. F. T. Pierce20 in his study of cotton yarns was first to describe this phenomenon mathematically with the introduction of the chain-type weakest link model for fiber strength. This model assumes a fiber to consist of n fiber elements linked together in series. The stat istical problem becomes one of relating the distribution of chain strength to the distribution of largest flaws appearing in each of the fiber elements. Pierce recognized the close relation of this model to the generalized asymptotic theory of extreme values developed by Tippet21, Fisher and Tippet22, and Frechet23. This theory is concerned with finding the distribution of largest (or smallest) values in a sample of size n. UCC 003956 18. A number of assumptions are required in modeling brittle fiber fracture as a weakest link phenomenon (1) The total fiber is subdivided into segments of equal length, each containing a number of distributed flaws of varying sizes. (2) No interaction exists between flaws. {3) The strength of a fiber element is uniquely determined by the largest flaw present in the element. (4) The strength of any bulk specimen is uniquely defined by the strength of that segment that contains the most severe flaw. The mathematical formulation of the weakest link model requires the definition of several terms. a -- A random variable representing strength, either the strength of a fiber element or the strength of an entire fiber. f {cr) -- The probability density function (p.d.f) of element strength, i.e. the "underlying" distribution. The fraction of elements having a strength in the interval a to 0 + do is given by f{o)da. F{o)" The cumulative distribution function (c.d.f.) of element strengths given by jaf(a)do. The probability of encountering an element of strength 40. g (o}'-- The p.d.f. of fiber (chain) strength. The fraction of fibers having a strength between a and O+da is given by g (a)do. 0- G(a)--The c.d.f. of fiber (chain) strength given by fg(0)do. The probability of encountering a fiber of strength {o ' The desired expression is g(a) in terms of f(a), thus the; probability that a chain of n links will break at a stress, 0, is simply the probability, f (o), that one link will fail at stress a, times the probability that the (n-1) links are still surviving, l-F(0jT] , times the number of combinations, n, in which this may occur. n-1 g(0) = nf (0) [l-F(c)] (4-1) This is the probability density function of the distribution of smallest values in samples of size n. Integrating equation (4-1) and rearranging yields UCC 003957 19. 1-Gto) = [>F(0j] n (4-2) This is the probability that a fiber of n links will survive up to a stress level 0. This equation emphasizes the assumed statistical independence of the elements in that the joint prob ability of survival equals the product of the individual prob abilities of survival for each link. A number of researchers have described brittle fracture as a weakest link process and have used a variety of distribution functions to characterize f(a), the under lying fiber element p.d.f. Pierce20, in his original work on cotton yarns assumed a Gaussian form of f(a) as did Kontorova and Frenkel21* in their work on the brittle strength of crystals. Chechulin25 developed the equations for a gamma distributed elemental strength. The consequence of these different forms of f(a) is that the moments of g(a) show different dependencies on n. The true form of f(a) for a particular fiber could conceivably be determined by the accumulation of statistically significant amounts of tensile strength data at a variety of gage lengths. By examining the variation of the expected value of g(c) with the fiber length (n), the actual form of f(a) could be inferred. Generally speaking, however, only a moderate number of test replications is available at any one gage length and consequently discrimination between statistical distribution functions for f(a) on this basis warrants only a limited degree of confidence. Experimental research in the field of brittle fracture statistics, as Freudenthal`3 points out, is in herently susceptible to degenerating into indiscriminate data accumulation and curve fitting. A more rational ap proach is to develop physically relevant probability models which simulate observed phenomenon and can therefore be used for extrapolation of results beyond the range of observation or simply to characterize the brittle fracture characteristics of a given material. It is this latter application which is of concern in this report. 4.1.2 The Weibull Distribution Function In 1939 Weibull26, using elementary probability theory, derived an expression for the probability of rupture, S, of an isotropic brittle material of volume, V, subject to any given distribution- of stress, cr. This expression is given by S = 1-exp -Jn(a)dv (4-3) UCC 003958 20. where n(<?) = a function characteristic of each particular material. This equation was generalized to model systems in which the material function may not be constant throughout the whole volume of the solid (e.g. materials with distinct surface flaws and interior flaws). In this case the equation becomes S = 1- exp - {a)dv-J\\2 (a)dv (4-4) Based on purely heuristic reasoning, Weibull proposed an expression for n{<?) of the form a-a n (<?) = m (4-5) the scale parameter, ' c* the location parameter and m, the shape parameter are unknowns, characteristic of the particular material. Equation (4-5) was justified purely on the grounds that it was the simplest mathematical expression satisfying the con ditions that n(<?) be a positive, nondecreasing function vanishing at some stress level, ac, which is not necessarily zero. Con sidering its origins, the distribution function defined by sub stituting equation (4-5) into (4-3) has been critized as being devoid of any physical meaning. However, as Weibull27 points out, this same objection could be applied against all other probability functions used to characterize real populations. Furthermore, when applied to statistical strength problems, the parameters of this distribution can, at least, be as sociated with true material characteristics. Implicit in Weibull's analysis is the principle that it is the weakest link which determines fracture strength. Thus, if F(a), representing the probability that a unit volume (fiber element) fractures at a stress level a, is given by 'CT-cr m F( c) = 1-exp- (4-6) then the probability of failure of a chain of n fiber elements is, on substitution of this expression into equation (4-2) -P?)G(c) = 1 - exp The associated p.d.f. is g (a) = nm <? (4-7) m (4-8) UCC 003959 21. The mean chain or fiber strength, ^a>, is given by the expected value or first moment of g(a) about the origin. <a> ag(a)da (4-9) Evaluation of this definite integral gives <a> = a + a /n1//m Co Vm / (4-10) Where T = the complete gamma function This equation indicates that average chain strength varies as n~l/ra The variance of a i.e. the square of the standard deviation s is given by and the coefficient of CV = -------"'TmUT+1 m variation, 1/2 ^m' cv=s/a, becomes (4-12) From this expression it is clear that cv is a function only of the Weibull shape parameter, m. In fact Irwin28 suggested that cv may be closely approximated by cv = 1.2/m (4-13) Thus, in essence, the parameter m is an inverse measure of the coefficient of variation. Using this result we can also gain some insight into the physical significance of the other Weibull parameters, ao and ac. In the limit as m becomes very large (m->->) the statistical characterization of strength re duces to the classical theory of fracture, i.e., failure occurs when the material reaches its "fracture strength" a material constant. From equation (4-10) when m-+^i.e.; cv-*o 0 = ao + ac (4-14) where ac is a lower limiting material strength, possibly zero, and cl+ol represents the ultimate strength of the material. The Weibull distribution function has been applied to a wide variety of problems not directly related to the statistics UCC 003960 22. of brittle strength, the distribution of particle sizes of fly ash 26, the fatigue of single fibers2*, the abrasion resistance of yarns ^etc. Its wide applicability is due, in part, to the flexibility of the expression. When m=l, the probability density function becomes the exponential function, with iA= 2, the Rayleigh distribution and with m=3.25 most of the Weibull curve is identical with the normal dis tribution function31. Brittle fiber fracture applications of Weibull type statistics have been quite numerous. Wallhaus32 used it to investigate the effects of time, moisture, static load, and mechanical damage on the strength of single glass filaments. Kies33 used a modified Weibull distribution in this studies on the strength of glass. Metcalfe and Schmitz34 investigated the effect of gage length in the tensile testing of E and S glass fibers. Their studies indicated logarithmic strengthlength plots were not linear over the entire gage length range investigated (.025-30 cm). For a given fiber, a slope change was observed at some critical gage length. This effect was attributed to the presence of a mixed population of flaws. One type of flaw is severe, has wide spacing and governs failure of long fibers. The other type is less severe, has narrow spacing and determines failure at short gage lengths. Their data suggested that greater fiber damage causes increasing loss of strength at long gage lengths, increasing the slope of the logarithmic strength-length curve in this region,and that it also tends to move the position of the slope change to shorter gage lengths. These trends are illustrated in Figure 20. This bilinear form of the logarithmic strengthlength plot indicates that a single exponent Weibull dis tribution function is inadequate to describe failure pro cesses in fiberglass. The more complex Weibull form of equation (4-4) may therefore be more appropriate. 4.2 The Assumed Fiber Strength Model Metcalfe and Schmitz3* demonstrated that a plot of fiber strength vs. fiber length was a very effective means of characterizing law structure and assessing the damage level exisitllng in'fiberglass. This concept forms the basis of our analysis scheme for evaluating the fiber protection capabilities of specific sizing agent systems. The first step in this procedure is the development of a mathematical model which can simulate the strength-length trends actually observed in the tensile testing of fibers. A simple Weibull distribution function for fiber fracture was found to predict, unsatisfactorily, a linear logarithmic plot UCC 003961 23. of mean fiber strength vs. fiber length (equation 4-10). The bivariate Weibull distribution function, equation (4-3) , proposed for materials with a mixed flaw population has the form G(c) = 1-ex p-n m2 (4-15) Oi o2 This six parameter probability model should have enough flexibility to-'accommodate observed bilinear fiber strengthlength trends but unfortunately its functional form pre cludes evaluation of an analytic expression for the ex pected value (i.e. , mean fiber strength). Numerical integration to compute <a>, while a possibility, was not considered viable because of the excessive expansion in computation time this would necessitate in other aspects of the overall analysis procedure. This limitation of the bivariate Weibull expression rendered it an unacceptable model. The bilinear logarithmic strength-length be havior observed for fiberglass has been simulated by Rosen35 in terms of the weakest link model by assuming a double rec tangular distribution function for f(a), the underlying fiber element p.d.f. The following analysis is a permutation of Rosen's model. Using the weakest link, chain type representation of a fiber, the concept of a mixed flaw population can be modeled by assuming that two types of links exist in the fiber chain: virgin or "as spun" links and severely flawed or damaged links. The virgin or "as spun" links, modeling the fiber as it comes out of the spinneret, were assumed to be represented by a simple Weibull expression f(o)v-(l-P)Voo(^!)m`1 exp-^m (4-16) = (l-P)jl-exp- ^ -oc\m F(c), (4-17) where ac = a lower limiting virgin link strength not equal to zero. aQ = a characteristic link strength related to the flaw free material strength. UCC 003962 24. = an inverse measure of the dispersion in link strength . P = a weighting factor indicating the fraction of the link population with very severe flaws. The subscripts v on f(c) and F(a) in these equations refer to virgin element expressions. These virgin or "as spun" links were assumed to exhibit some dis persion in strength simply because of the presence of unavoidable, microscopic surface imperfections. A small portion of the links were assumed to contain very severe flaws, a consequence of handling or perhaps environmentally induced damage. These weakened elements were modeled, for simplicity, as possessing a simple rectangular distribution function, f(a)f isolated at low stress levels (the sub script f refers to flawed elements}: f(0)p = p/(crb-CJa) (4_18) where H&)pF(o)r <4'19) a = lower limiting flawed link strength a 0 = upper bound on flawed link strength b The composite elemental strength p.d.f, i.e., equations (4-16) and (4-18) is plotted in Figure 21 for some assumed model parameter values. In this figure SA=cra., SB=cb, SC=cc SO=0 . o Equations (4-16) - (4-19) can now be substituted into the generalized weakest link expressions (equations 4-1 and 4-2) to give the p.d.f. and c.d.f. expressions for a fiber chain of n links. This analysis is detailed in Appendix C as is the calculation of the expected value of g(a) (i.e. the mean fiber strength, <c>). The mean strength of a fiber composed of n links drawn from the population characterized by equations (4-16) and (4-18) is given by <c> = aa. + (c^-a,) [j.-(l-P)n+1] /P(n+1) + (ac-ab) [l-0n + fl-p] n (a /n1/m) r LJ o m (4-20) UCC 003963 25. where a.a, cr , a c, oQ , m , and P are defined as before and r = complete gamma function This expression is plotted in Figure 22 as Ln <a> vs. L n. L, where L=*i (n is the number of fiber elements, and & , the fiber element length is assumed to be .001 in. Several strength-length curves are shown for different values of P, the fraction of the fiber element population containing severe flaws. Arbitrary values were assumed for the other parameters of the fiber element strength distribution function. As the fiber length increases (i.e., n increases) <a>, the mean fiber or chain strength, falls off rapidly depending on the value of P. For P = .0005 and len gths greater than 0.5 in. fiber strength is essentially con trolled by the very few weaker strength elements which are en countered more and more frequently. As P is decreased the strength- length dependence is dimished. The curves of Figure 20 might, thus, be interpreted as characterizing fibers exhibiting varying levels of damage. The preceding six parameter statistical strength model seems capable of simulating observed mean fiber strength-length behavior while avoiding the complications associated with the' bivariate Weibull expression, equation (4-15). The next step requires utilizing experimentally determined fiber fragmentation data to back out values of the parameters associated with this model and thus define characteristic Ln<o>-Ln L curves for a given fiber systems. These curves will then, he used to discriminate between fiber treatments providing various degrees of fiber protection (i.e., sizing agent effectiveness). 4.3 Stochastic Fiber Fragmentation Model The parameter estimation procedure developed above is based on a coupling of the statistical fiber strength model with a computer simulated stochastic model of the fiber frag mentation process. Discussion of this computer simulation re quires an understanding of one additional concept: that of mechanical proof testing. 4.3.1 Proof Testing and Truncated Breaking Strength Distributions The fracture strength of brittle fibers is character istically spread over a very wide range. Curve I in Figure 23 might represent a typical cumulative strength distribution curve, G(c). It is bounded on the left by a , the zero probability UCC 003964 26. strength {^0} and on the right by a a value related to the theoretical, flaw free strength of ^he given material. The expected value or mean strength associated with this initial distribution is <u>_. Curve II in Figure 22 is the strength distribution curve obtained after eliminating all specimens of strength less than the proof stress up = <a>1. Assuming that the surviving elements were all unaffected by the proof test, the truncated cumulative distribution curve, Gp(u) can be re lated to the original distribution, G(u) by an expression due to Weibull26'. Gp(0) G(u) - G (Up) 1-G(up) at =0 a < Up (4-21) The expected value characterizing this new distribution is <u>j. The dispersion of the truncated distribution is ob served to be considerably less than the original, in fact, as Up increases, approaching, in the limit, the 100% probability stress, u , the surviving elements define a deterministic material, i.e., they all fail at the stress level up. The generalized eq uations governing proof testing affects in terms of our assumed fiber strength model are outlined in Appendix D. 4.3.2 Computer Simulated Fiber Fragmentation The following detailed description of the fiber fragmentation model is presented in terms of Figure 24. Con sider a single filament of diameter d, 3 inches in length, em bedded in some given resin matrix. The mean strength of this fiber, u,,.* can be computed using the statistical fiber strength model (equation 4--2b) fr some arbitrary set of model parameter estimates. On tensile deformation of the resin matrix, the model requires that the first fiber break occur at the stress level u *. Although fiber fragmentation is stochastic in nature there are restrictions on where the break can take place. These arise as a consequence of the mechanism through which stress is trans ferred from the matrix to the fiber. Recalling the concept of a critical transfer length, i.e., the length required to build the stress up to a level to break the fiber, (equation 1-8), a por tion of the fiber end does not realize the stress up^* and con sequently the fiber cannot break in these regions. These inef fective regions of the fiber, given by A c/2 = up*d/4T (4-22) depend both on the breaking strength of the fiber and on the UCC 003965 27. interfacial shear strength, t, another model parameter. As suming some reasonable value for x, the fiber strength, ap,*, is substituted into equation (4-22) and ilc/2 computed. Ex cluding these ineffective fiber ends as potential fracture sites, the computer selects a random number which determines the first fiber break location. The resulting two fragments are now stronger than the initial fiber for two reasons: (1) probabilistically they are stronger because of their reduced length and (2) both fragments have essentially been proof tested to the stress level * (they are at least this strong). A new fragment breaking stress can now be computed which takes into account both strengthening mechanisms. The probabilistic strength increase is based on the length of the longer of the two fragments. The fragment strength equation, as shown in Appendix B, assumes different forms depending on the value of the proof testing stress, and its relation to the fiber model parameters. The general form of this equation is cfFi* = F(Li, oFi_1( ua, afa, ac, ao-, m, P) (4-23) where at_r 3.. * = current fragment breaking stress Li = length of longest surviving fragment aFi-l* = stress level at which previous break took place (i.e., proof testing stress) The new fragment breaking stress a_,2* is computed from equation (4-23) and substituted into equation (4-22) (keeping t constant) to determine a new and somewhat longer critical transfer length. These ineffective, unbreakable regions are subtracted from each fragment end andva* new random number is generated. This time, however, the random number is weighted such that the longer fragment is given a higher probability of fracturing but never theless a finite probability exists that the smaller fragment breaks. In Figure 24 fragmentation now produces three fibers. The longest is again evaluated and its strength, <?,,-*, is determined from equation (4-23) where the proof testing stress is now 0-2** On calculation of the new (longer) critical trans fer length, it is observed (Figure 24) that the middle fragment is now shorter than the current critical fiber length and, there fore, by definition can undergo no further fragmentation. It is essentially decoupled from the system. Stochastic fragmentation continues until, eventually, all fibers are shorter than the current critical fiber length. At this point, fragmentation ceases and the program counts the total number of fiber frag ments, assessing whether this constitutes enough for statistical analysis. If not, the process is initiated on a new three inch fiber with the model parameters aa^, a , a , m, P, and x un changed. Fragments are accumulated untilca theoretical fragment UCC 003966 28. length or l /d distribution is adequately defined (usually 150 or 200 fragments). ' The fundamental characteristics of the theoretical fragmentation model can be demonstrated by several examples. Consider, first, the fragmentation behavior of a flaw free fiber, i.e., one which exhibits a strength independent of length. In terms of our statistical fiber strength model, this implies that a , a. , and P = 0 (i.e., no flawed elements) and as discussed inasecion 4.1.2, that the Weibull shape parameter, m, is very large. With cc = 0 and m = 100, the fiber breaking strength becomes 0Q ( assumed in this case to be 500,000 psi). The slight negative slope of the strengthlength curve shown in Figure 25 is a consequence of assuming a finite value for m. These flaw free fiber parameters were substituted into the stochastic fragmentation model to generate > theoretical fragment .i/d distribution curves at several levels of t. Figure 26 shows smoothed cumulative 9./d curves for inter facial shear strength values of 300, 500, and 700 psi. As the interfacial shear strength is increased, the distribution curves, as expected, shift left to lower aspect ratios. In addition, note that the bounds on these distributions, as predicted from the simple critical fiber length concept, are in a 2:1 ratio. Now consider a flawed fiber exhibiting a mean strength significantly dependent on length and characterized by the model parameters listed in Figure 27. Substituting these values into the stochastic fragmentation model and assuming an interfacial shear strength of 1000 psi resulted in the cumulative fragment length distribution curve of Figure 28a. Smoothing and differentiating, these points numerically gave the associated frequency curve (Figure 28b). Note the considerable dispersion in this distribution, i.e., the bounds are in a 4:1 ratio in contrast to the 2:1 of Figure 26. Introduction of the concept of a length dependent fiber strength into the stochastic fragmentation model, ex plains the dispersion observed in measured fragment i/d dis tribution curves. The next step in the analysis procedure involves the reverse process of using these experimentally determined 1/d distribution curves back out values for the interfacial shear strength, t, and the statistical fiber strength parameters which define a characteristic strengthlength function for a given fiber/treatment system. The analysis thus provides, simultaneously, the specific adhesion promoting and fiber protection capabilities (coupling and sizing agent effects) of a particular fiber treatment for mulation. 4.4 Parameter Estimation Procedure The parameter estimation procedure is essentially one of constrained nonlinear optimization. The constraints imposed UCC 003967 29. on the values of the independent search variables arise from physical considerations of brittle fiber tensile test results and from restrictions implicit in the mathematical formulation of the statistical strength model. The nonlinearity of the problem is obvious. 4.4.1 Objective Function and Search Method The first step in the estimation process required de fining an objective function which measures the goodness of fit between the experimentally determined l/d distribution and that generated theoretically. The function which was devloped op erates on the cumulative distribution curves, the smoothed form in the case of the data and the raw curve of the theoretical model. Numerical smoothing of the theoretical points was not done in order to reduce computation time. Each curve was broken into twenty equally spaced, cumulative cells and an average frag ment Vd value computed for each cell. The objective function was defined as the sum of the squares of the deviations in mean SL /d values between corresponding cells. Cells at the extremes (tails) of the distributions could be excluded, if desired, in cases where the data in these regions was of questionable accuracy. Function minimization, was accomplished using the Nelder-Mead simplex algorithm*6. As McMaster 37 points out, comparative studies indicate that this direct search method is the most effective algorithm not requiring derivative evaluation for finding the minimum of a function of several variables. Gradient techniques requiring numerical evaluation of derivatives were not used because of the difficulty in obtaining accurate gradient estimates in systems subject to stochastic fluctuation such as this one. The simplex algorithm, while always convergent to a local optimum if the convergence criterion is strict enough, suffers a serious deficiency (as do other existing algorithms) in that this optimum may or may not be the global optimum. This search code seeks an optimum which, unfortunately, is a strong function of the initial starting point. To obtain a so called, "best" optimum (not necessarily the global optimum) a coarse grid presearch, as recommended by McMaster37, was incorporated into the algorithm. By using large initial step sizes in de fining the simplex of independent variables, one may simulate the preferred but inefficient case of starting a search from many different initial points. 4.4.2 Parameter Constraints The use of most optimization codes requires that the independent variables be free to move from - to + . As stated UCC 003968 30. previously, however, a number of physical, intuitive, and mathe matical constraints on the model parameters distinguish between feasible and non-feasible regions in vector space. These variable restrictions can be expressed as inequality constraints of the form A - Xi - U (4-24) where Z= a lower bound on the variable x^ u = an upper bound on x^ Constrained search variables can be transformed into unconstrained variables before the search is carried out. A trigonometric transformation originally suggested by Box38 and found by McMaster35 to be highly efficient in handling boundary constraints is given by xi = & + (u-A ) sin2Y'i (4-25) The constrained optimization is performed in the variable Y.. This transformation is fairly linear near the constraint boundaries and, therefore, eliminates the simplex step size problems characteristic of certain other transformation forms on approaching this region. The parameters which must be determined include the six variables which define the fiber strength characteristics aa' b' c' In' ant^ P (s^Z3-n9 agent performance) and the interracial shear strength T (coupling agent performance). Since, in essence, it is the quotient of fiber strength and interfacial shear strength which determines the location of experimental .-/d distribution curves, some fundamental infor mation is required if meaningful, unconfounded performance evaluations are to he achieved. This information may be, for example, a knowledge of the mean fiber strength, <a>, at two different, gage lengths as determined experimentally from single filament testing. One of these lengths, Ll, is three inches, the length of the initial unfragmented fiber. Substituting the mean strength values and their associated fiber lengths [i.e., number of fiber elements or links) into equation (4-20) defines two equality constraints of the form FL1(<0>1' V V <JC, V m' p) " 0 (4-26) FL2(<0>2' V V ac, m, P) =0 (4-27) where <u>1 and <o>2 are the experimentally determined mean UCC 003969 31. strength values at gage lengths Ll and L2, respectively. These constraints force the computed fiber strength-length curve to pass through the experimental points and on rear rangement can be used to reduce the number of search variables by two. In addition to the equality constraints defined by equations (4-26) and (4-27), several inequality constraints also restrict the values available to certain statistical strength parameters. From physical intuition, a , the lower bound on the severely flawed link strength must Be some value greater than zero. Fibers of near zero tensile strength will not be realized because of the unconscious proof testing which accompanies sample preparation. These fibers are eliminated simply because they cannot be prepared for testing. This lower stress limit was somewhat arbitrarily assumed to be 10,000 psi which corresponds to a working tensile load of the order of 1 gram for a typical 0.6 mil diameter fiber. This value may be further justified in fiberglass evaluations because it is generally considered to represent the strength of bulk (^ infinite size) plate glass. From Figure 21 it is observed that the upper bound on <j , is c. , however, a more critical constraint is da < <d>^. Thus from equation (4-25) c?a = 10- + (<a>! - 10k) sin2 Y1 (4-28) The Weibull distribution shape parameter, m, was constrained between 1.0 and 50.0 m = 1.0 + 49.0 sin2 Y2 (4-29) The inequality constraint on d^ is expressed as da < and the upper bound on dc is dQ, thus b- c and ab = aa + {ac ' <V sin' Y3 (4-30) ac = ab+ to"(,b) sin2 Y4 (4-31) Equations (4-26) and (4-27) can now be used to solve for P and aQ, respectively. P = El1(<o>1, ca, <Jb, ac, aQt m) (4-32) ao = EL2(<a>2' V V ac' m' P) (4-33) UCC 003970 32. On evaluation of expressions (4-28) and (4-29), equations (4-30) - (4-33) constitute four equations in four unknowns. Since equations (4-32) and (4-33) have nonlinear forms, the set was solved by an iterative process. The interfacial shear strength, t , is bounded by zero (no adhesion) and t the shear strength of the polymer itself (optimum adhesionj3 ' T T sin2 Yc max 5 (4-34) The Nelder-Mead simplex algorithm operates on the unconstrained, five dimensional vector y. Listings of the everall analysis programs are given in Appendix F. It is recognized that while the parameter constraints impose severe restrictions on permissible parameter values, the search algorithm is still susceptible to converging to a local optimum. Furthermore, the absolute parameter values which are computed, even if optimal, may not define a unique solution. Experience indicates, however, that the character istic strength-length curve which they define is unique for a given set of fragment S/d data. 4.5 Data Analysis Two sets of smoothed cumulative S/d data (OCF 885 in polypropylene and OCF 885 in nylon 6) were processed by the preceding analysis scheme. The fiber strength constraints were determined by tensile testing single filaments of OCF 885 at gage lengths of 1.0 and 3.0 inches. Approximately 100 breaks were made at each gage length with each load value normalized in terms of stress by measuring specific individual filament diameters after rupture. Mean fiber breaking stresses were evaluated and used as the constraints expressed in equations (4-26) and (4-27). Figure 29 shows the results of an optimization run performed on OCF 885 - polypropylene fragment S/d data. Both the experimental and optimized theoretical, smoothed cumulative S( /d distribution curves are plotted. The agreement is observed to be quite good over the major portion of the distribution with the deviation in the lower extremum attributable, in part, to data inaccuracies (i.e., only a few data points characterize this region). The optimized interfacial shear strength, 1200 psi, is about 46% of the von Mises' shear strength value for polyprolylene, 2600 psi (i.e., tensile yield strength/ilT) , thus the general purpose surface treatment of OCF 885 is apparently not optimal for polypropylene. The fiber strength-length curve generated from this optimization is shown in Figure 30. The UCC 003971 33. crosses represent experimental, mean fiber strength values determined by single filament tensile testing. The computed mean strength-length curve is observed to pass through the constrained points at 1.0 and 3.0 inches and to approximate, reasonably well, the strength-length behavior at other gage lengths. Figure 31 shows the experimental and optimized theoretical l/d distributions for OCF 885 m nylon 6. 1.0 and 3.0 inch tensile test data was again used as parameter constraints in this optimization. The agreement between the experimental and theoretical curves is, again, quite good over the central portion of the distribution. The optimized interfacial shear strength, 4130 psi, is about 60% of the von Mises' shear strength value for nylon 6, 6800 psi, in dicating that the general purpose size of OCF 88 5 is somewhat better for nylon 6 than for polypropylene. There is, however, still room for improvement. The strength-length curve generated from this analysis and shown in Figure 32 is observed to pass through the 1.0 and 3.0 gage length strength values and to approximate, quite well, the shorter gage length strength-length behavior. ' In analyzing OCF 885 fragmentation data in two different resin matrix systems, the stochastic fiber frag mentation model is seen to be quite successful in simulating the experimental data. Furthermore, the back calculated strengthlength curve computed for OCF 885 was found to be essentially independent of the resin m which the fiber was embedded. This was a critical test m assessing the validity of the analysis scheme. The dispersity of experimental fragment H/d distribution curves is, thus, truly characteristic of the degree of damage inherent in a given fiber and can, via this analysis procedure, be translated into a more readily mterpretable and useful form, i.e., a mean fiber strength-length curve. 5. CONCLUSIONS The embedded single filament test is highly sensitive to specific fiber/matrix interactions (coupling agent effect). The dispersity of the fiber fragmentation distributions is characteristic of the damage level of the fiber and thereby of the protection afforded by the fiber treatment (sizing agent effect). The strength/length dependency of fiberglass, within the range of observed fragment lengths, is accurately described by a mixed element, weakest link fiber strength model. UCC 003972 34. A stochastic model of the embedded fiber fracture sequence using the above fiber strength model simulates observed trends in fiber fragment distributions. Computer analysis of fiber fragment data in terms of the above models result in reasonable interfacial shear strength values for the individual fiber/polymer systems. The computed fiber strength/length relations are in good agreement with independently determined glass fiber strength data. Furthermore, the computed strength/length curve is independent of the matrix resin in which the fiber is embedded The embedded single filament technique and associated analysis is valuable in elucidating the principal coupling the sizing effects of fiber surface treatments. This is of particular value in the optimization of commercial fiber treatment for mulations. ACKNOWLEDGEMENTS The author wishes to express his appreciation to Mr. John Stenstrom for his efforts in conducting much of the experimental work and to Mr. Gideon Salee of the University of Connecticut for performing the single filament tensile tests. Thanks is also extended to Professor A. T. DiBenedetto {Univ. of Conn.) and Mr. Fred H. Ancker for their assistance and contributions concerning the theoretical aspects of this project. WAF:PLK Attachments: References Tables Appendices U-J. W. A. Fraser UCC 003973 REFERENCES 35. 1. Fraser, W. A. and Ancker, F. H., "A Computer Modeled Single Filament Technique for Measuring Coupling and Sizing Agent Effects in Fiber Reinforced Composites," Project Report File #4208, November 11, 1974. 2. McCullough, R. L., "Concepts of Fiber Resin Composites," Marcel Dekker, Inc., New York, (1971). 3. Corten, Herbert T., in "Modern Composite Materials," Broutman, Lawrence J. and Krock, Richard H., Editors, Addison-Wesley, Inc., Reading, Massachusetts, (1967). 4. Parratt, N. J., "Defects in Glass Fibers and Their Effect on the Strength of Plastic Mouldings," Rubber and Plastics Age, Vol. 41, p. 263, (1960). 5. Riley, V, R. and Reddaway, J. L., "Tensile Strength and . Failure Mechanics of Fiber Composites," J. Mat. Sci., Vol. 3, p.41 (1968). 6. Hale, D. K. and Kelly, A., Strength of Fibrous Composite Materials in Annual Review of Materials Science, Vol. 2, Huggins, R. A., ed., Annual Reviews, Inc., Palo Alto, Calif. (1972) . 7. Rosen, B. W., "Tensile Failure of Fibrous Composites," AIAA J., Vol. 2, No. 11, pt 1985-1991, (1964). 8. Cooper, G. A., "The Fracture Toughness of Composites Rein forced with Weakened Fibers," J. Mat. Sci., Vol. 5, p. 645 654, (1970). 9. Cox, H. L., "The Elasticity and Strength of Paper and Other Fibrous Materials," Brit. J. Appl. Phys., Vol. 3, p. 72, (1952) . 10. Dow, N. F., "Study of Stresses Near a Discontinuity in a Filament Reinforced Composite Metal," General Electric Report TIS R635D61, AD-414673 (1963). 11. Kelly, A. and Tyson, W. R., "Tensile Properties of Fiber Reinforced Metals: Copper/Tungsten and Copper/Molybdenum," J. Mech, Phys. Solids, Vol. 13, p. 329 (1965). 12. Ongchin, L., Olender, W. K., and Ancker, F. H., "Fiber/ Matrix Adhesion and the Fracture Behavior of Glass Reinforced High Density Polyethylene," Proc. 27th Conf. SPI Reinforced Plastics Division, February, 1972, Sect. 11-A. UCC 003974 36. 13. Burns, R., Hankin, A. G., and Johnson, A. E., "Glass Fiber Reinforcement of Polyamide Polymers," Proc. 29th Conf. SPI Reinforced Plastics Division, February, 1974, Sect. 20-A. 14. Kelly, A., "The Strengthening of Metals by Dispersed Particles," Proc. Roy Soc. (London) A282, p. 63-79, (1964). 15. Wadsworth, N. J. and Spilling, I., "Load Transfer From Broken Fibers m Composite Materials," Brit. J. Appl. Phys., (J. Phys. D.) Ser 2, Vol. 1, p. 1049-1058, (1968). 16. Brady, W. C., Tiede, R. L., and Veagie, F. M., "Glass Fiber Manufacturing Process Development," AFML-TR-68-361, Dec., 1968. 17. Hershey, Harry C., Zakin, Jacques L., and Simha, Robert, "Numerical Differentiation of Equally Spaced and Not Equally Spaced Experimental Data," I & EC Fund., Vol. 6, No. 3, p. 413-421, (1967). , 18. Parma Standard Test Method - PSM 101, "Gravimetric Determin ation of Resin Content in Resin-Graphite Composite Materials," Revised 4/26/73. 19. Freudenthal, Alfred M., "Statistical Approach to Brittle Fracture," in Fracture, Vol. II, Ed. H. Liebowitz, New York, Academic Press, (1968). _ 20. Pierce, F. T., "Tensile Tests for Cotton Yarns Part V: The Weakest Link," J. Text. Inst., Vol. 17, p. 355 (1926). 21. Tippet, L. H. C., "On the Extreme Individuals and the Range of Samples Taken From a Normal Population," Biometrika, Vol. 17, p. 364 (1925). 22. Fisher, R. A. and Tippet, L. H. C., Proc. Cambridge Phil Soc. Vol. 24, p. 180, (1928). 23. Frechet> M., Ann. Soc. Polon. Mat. Cracow, Vol. 6, p. 93, (1927). 24. Kontorova, T. A., "A Statistical Theory of the Brittle Strength of Real Crystals," J. Tech. Phys. (USSR), Vol. 10, p. 886, (1940). 25. Chechulin, B. B., "Statistical Theory of Brittle Strength," Zhm. Tekh. Fiz., Vol. 24, p. 292, (1954). 26. Weibull, W., "A Statistical Theory of the Strength of Materials," Proc. Roy. Swedish Inst. Engr. Res., Ing. Vetenskaps Acad. Handl Vol. 151, (1939). UCC 003975 37. 27. Weibull, W., "A Statistical Distribution Function of Wide Applicability/' J. Appl. Mech., Vol. 18, p. 293, (1951). 28. Irwin, G. R., "Relatively Unexplored Aspects of Fracture Mechanics," Univ. of Illinois, TAM Report No. 240, Sect. V. , Feb., 1963. 29. Prevorsekf Dusan; Lyons, James, and Whitwell, J.C., "Statistical Treatment of Data and Extreme Value Theory in Relation to Fatigue in Textiles," Text. Res. J., Vol. 33, p. 963-973, (1963)-. 30. Barella, A., "On Certain Applications of Weibull*s Distribu tion to Fatigue Phenomena m Yarns," Text. Res. J., Vol. 35, p. 1051-3, (1965). 31. Steiger, F. H., "Practical Applications of the Weibull Distribution Function," Chem. Technol. Vol. 1, p. 225-231, (1971). 32. Wallhaus, R. A., "A Statistical Study of the Factors Influ encing the Strength of Glass Fibers," TAM Report No. 217, May, 1962, Dept, of Theoretical and Appl. Mech., Univ. of Illinois. 33. Kies, J. A., "The Strength of Glass," NRL Report 5908, Naval Res. Lab., Washington, D. C., April 3, 1958. 34. Metcalfe, A. and Schmitz, G. K., "Effect of Length on the Strength of Glass Fibers," ASTM, Vol. 64, p. 1075-1093, (1964). 35. Rosen, B. W., "Mechanics of Composite Strengthening in Fiber Composite Materials," ASM, Metals Park, Ohio, (1965). 36. Nelder, J. A. and Mead, R., "A Simplex Method for Function Minimization," Computer J., Vol. 7, p. 308, (1965). 37. McMaster, L. P., "Modifications of the Optimization Algorithm and the. Integration Procedure for the Mathematical Models of the Styrene Reactors," UCC R&D Report, File No. 2775, March 30, 1971. 38. Box, M. J., "A Comparison of Several Current Optimization Methods and the Use of Transformations in Constrained Problems," Computer J., Vol. 9, p. 67, (1966). 39. Knox, L. J., "Studies of Breaking Stress Distributions of Viscose Filaments," Ph.D. Thesis, Dept, of Chem. Eng. Princeton University, 1970. 40. Abramowitz, M., Stegun, I. A., ed. "Handbook of Mathematical Functions," N.B.S. Applied Mathematics, Series 55, (1964). UCC 003976 38. TABLE I COMPONENTS OP A TYPICAL GLASS TREATMENT FORMULATION Coupling Agent - Promotes fiber/matrix adhesion Sizing System Film Former - Usually applied as an aqueous emulsion, protects filaments from damage and gives some integrity to the bundles of indivi dual glass filaments, gives a stable forming package (a fiber cake) Lubricant - Lubricates filaments while being drawn and aids passage over guide points. Antistat - Prevents build-up of static charges due to friction in winding Micellaneous - TABLE II FIBERGLASS DESCRIPTIONS DESIGNATION Glass Type Treatment Coupling Agent Sizing System Treatment Loading Mean Fiber Diameter (50 Filaments) Tensile Modulus OCF 885 E Roving UCC A-1100 PVAc + Lubricants 1.6 Wt. % 0.58 mils 10.5 x 106 psi OCF 415 E Roving Amine Reactive 0.7 Wt. % 0.59 mils 10.5 x 106 psi UCC 003977 TABLE III PROPERTIES OF THORNEL 300 (Grade WYP 30 1/0) No. of Filaments Filament Diameter Density Strand Tensile Strength Young's Modulus 2000 0.3 mils 1.70 g/cc 325 kpsi 34 mpsi 39. TABLE IV PROPERTIES OF CARBON PRODUCTS' PITCH BASED, CONTINUOUS FILAMENT GRAPHITE Experimental Yarn (280-22-9) - untreated. No. of Filaments Filament Diameter Density Strand Tensile Strength Young's Modulus 120 0.43 mils 2.08 g/cc 226 kpsi 40 mpsi UCC 003978 40. TABLE V MATRIX RESIN PROPERTIES MATERIAL NYLON 6 (DRY) POLYPROPYLENE HDPE Density 1.13 Tensile Yield Strength 11,800 psi Elongation at Yield 10% Secant Modulus 380,000 psi 0.905 4,450 psi 15% 200,000 psi 0.962 ' 4.000 psi 15% 150.000 psi UCC 003979 T I TTT 1 1 1 1 1 1 1 L i........ rrrr i i i i i r 41. K ' W' 1 'I ' I I'l.T-l /.T7 /:/ ./..I.,i. i i l\xx a (*) FIGURE 1: Stress Transfer to a Uniaxially Loaded Fiber in a Matrix V FIGURE 2: (a) Critical Transfer Length-Length Required to Build the Stress up to a Level to Break the Fiber, (b) Critical Fiber Length-Minimum Length of Fiber Capable of Reaching its Breaking Strength UCC 003980 FIGURE 3 : F ib e r L e n g th E f f ic ie n c y F a c to r 42. cn co c* <> X H Oz 111 in _J IT Ui CO O 111 ro O Z> O UJ cr OJ 001 H010V3 A0N3l91dd3 H19N31 UCC 003981 FIBER FRAGMENTATION 43. a FIGURE 4: Multiple Fiber Fracture (a) Generating Shortest Fragments ^i.c/2 (b) Generating Longest Fragments UCC 003982 44 FIGURE 5: Compression Mold and Fiber Fork for Molding Single Filament Embedded Tensile Specimens UCC 003983 45. FRACTION CUMULATIVE FREQUENCY 0*90*B0*70*9- 0*4- 0*3- 0*3- 0*1-- OCF 885 IN POLYPROPYLENE 0*01------^ t-----------f------1------1------1------1------1------1------1------1------1------1------1-- 0 100 300 300 FRAGMENT ASPECT RATIO H------1------1------ r 400 FIGURE 6a: Fragment A/d Cumulative Distribution (OCF 885 in Polypropylene)-Raw Data UCC 003984 46. EO18- OCF 885 IN POLYPROPYLENE 16" 14h- .. LlJ y ac LlI CL io QL UJ i0 6-- 4-- 2 ------1------1------1------I------1------1------1------1------1------H- 100 200 300 FRAGMENT ASPECT RATIO 4------I------h 400 FIGURE 6b: Fragment l/d Histogram (OCF 885 in Polypropylene) UCC 003985 R E LA TIV E FREQUENCY 47. FIGURE 6c: Numerically Smoothed Fragment i/d Frequency Curve {OCF 885 in Polypropylene) UCC 003986 48 AJNBIXGdJ 3AIlV"nwrD NDIIZJVUJ UCC 003987 T3 (fl P <L) Cc (1) fl) S fa fa >1 pa aj o cu U X& fa f>a1 co o cu cH fac <0 P Lf> P oo CO 00 s fa Si u S *O" " o a) w a, a) co > Vi fa 3 ni u -p oC E-* O fa 4-1 -p 03 Up QJ 4J fa 10 fa fa fa P t D OM fa i 0 49. a in co oo Cm U O 1 UH o 0*9t* FRAGMENT ASPECT RATIO ADN3T1D3U3 3AI1VTWD NDI13V3d UCC 003988 g h Eam ^c 0) OiT3 ><; c w nJ 00 8 H Cm 50. in 00 00 tn u o UJ a o H +> p XHt u -P UJ -H Q T3 >1 a. o 4J u aa <u >1 etJ> O <0 0, p4 -tHS nj in P r-l a 0) 5 b uO a) CL, nd xa w nj DO H ooooooooo ADN3nD3dJ SAIlVTVrO N0I13VH3 UCC 003989 -G*0 51. UCC 003990 l'C H FRAGMENT ASPECT RATIO  52. A3N3nD3dJ 3AIlV~inm3 NDIlDVad co C ow rH ft -P Q 3 EC Jft H T3 ue +j rd CO H CD Qa (D TJ rH ft o I--i Sh Ld ft flj ft -P >i c < CD EH fot P- I-- <D <0 ft Z XC wo Ll) rH ~ >1 LD P SC 0 < rH C ft *H IZ Ll >i m lH 00 <TJ 00 1 ft 9u cn o I--I f< D U H ft UCC 003991 09f 53 ADN3nD2dJ 3AI1VTWD NQI13Vy3 UCC 003992 m o M OW 'H flj -P P 3X ja H, T3 P -P W H P TS >1 a 0 rH P id Oi -P >i a 0) O eH P' QJ W> ft XC WO tH " >t -P TZ 0 i--I c fl( -r-l >,LT) P rH id't 1P U WO (N P P H P 10*- 54. Da^lG8S P -H -H 2Z jHN II II H in M (D I" H H b 111 H<mu o4 +J g Q) id PGm 0G i--I G >i 0 13 0C G -H 4GJ in n] oo S4 oo 0 &fa 18 4-4 44 0 M 0G E-I OH 4-1 +4 oG ,0 44 'H uG 0 44 4-1 cn 4-1 H W a o H fa 0*9f UCC 003993 55. A3N3fi03dd BAiivnwra Naiiavad UCC 003994 T) -P 01 ns M Cm WOj CQ 0 H3 0) C Mi -H 3 jj i/"> ITS rH Ml'd* as 01 Cm B$ Uo Eh 4h +) O 10 a) c Eh O 4H '4HJ O3 A -M> H UH <u -M> 4H 10 <4H -H mo 3 D o H P*4 0*91- 1 * 01- UCC 003995 56 . xocM> ,Uc Hoca) m jj >i U -H <X pH 0 4M P 0*0, qi, C h^ c (u o OH fEi3 P* P H >0 0 -H fi XHJ Cm (0 p W VO P 0 W0C -h e o a a >h H >i \TJ -CP 2 oe +i U 'H e -e QJ T3 QJ e o J3 tji P -H z efO na>J &4 <LjJ 2 U hpo H pfl'Hp dDX P Cm C - (0 (X owe gpu Ll H U P 3 T3 0) TS O Du O W wx p cm pma LTl s DU H Cm 57. :: hi ;ss i-- L<D DC Ll p o ,g W G Or--1 -P G A H P -P U] H Qa o T3 P \ CM >i 4J O G CM 0) E T3 tn G rtf rtf P &4 U5 rP G rtf O P <H <GU 2>i G p -P aa) o x <=> W f*l vo D U M [l. UCC 003996 58. "4" Tfr- "4 ftr rl o* o A3N3no3aj 3AiivnnmD NDiiDvad Z UJ 2 tD < QL 3 U. 8 H Pi (U A -rH T3 Ph T3 0 \ 4J 0) rtf +J ID -H -M U & 3 -P .(i <u 3 oj E 3> pi tn CJ (0 - Pi TS IX +J 0 U If! 3 3 (0 X O t3 (0 *H on <u U X +> Pi Pj u n) 3 +> 3 --HP E 3 0 lx U X) X 0P o IX 3 4J L E U 3 0) 00 E-t 4-i E 3 O 3 (0 4J 1H ' ifi in H 0 3 P4 i"- o *H in y-t +J 3 rr O3 +J O XI -H Pi 3 *C SS 0 +J 4J W m in 3 m -H O 3 WQU r-~ S3 8 uM b O UCC 003997 09f 59 UCC 003998 M-) Ti OG rfl W GOO H O +J'f1 X33 i--1 H <15 UG +J U UJ 0 H JG QH T) ' \ 4<JD 'H H JG 3 ft U (0 QJ cue Q) O O EH <-) >i u oj ft a) w o ft rO P x w ft ft >1 JG .H -O 0 -p +J ft 0 -H H ft G ft -H >tLD IT) P 00 H <0 CO E ft ft 3UU mCO oo D U H ft 1*0 60. LH T3 oq (T3 to qOo r-t O +J ro q i Xl >H H <U uq 4J H 03 O H A Q E-i \T3 LT*l a? 00 CO ADN3nD3dJ 3AIiVim"0 NDiiavaj UCC 003999 Ol o H fcj UCC 004000 61. to w ITJ pH U p o <P ft -- H -r A > 0] IS) f? -P O -H 51 (0 H too 0) K T) CJ -P cn <u C ft Q) H A ia io .c -p -P Q) tn C <U g Po P u CO 4-1 n) w p0) p<]) C , Q) H U ft o CN g D 0 H ft 62 H in CL in i * * o & m in UJ CL ti--n O-OT*) AlIlIBVBOBd UCC 004001 FIGURE 2 1 : A ssum ed F ib e r E le m e n t (L in k ) S tre n g th D is t r ib u t io n F u n c tio n iO *G 63 (31V3S 301) (ISd) S53dS UCC 004002 10**0 10* 10**3 . 0**3 LENGTH (MILS) (LOG SCALE) FIG U R E 2 2 : H y p o th e tic a l F ib e r F u n c tio n o f P , th e S tre n g th -L e n g th F ra c tio n o f th e C urves as a L in k P o p u la tio n C o n ta in in g S e ve re F la w s 10**4 64 AiniBvsmd baiivtwd UCC 004003 i i--i in fd H -H QU 0 A +J 4J (fl D> G 0) 4-> CO <D A + C (0 ou H tn a. C >1 H E-< +> n nj 0 Eh H4 O o o Vh ft <4~l 0 4J U 0 UH MH +> M A0 0 AU H + co CN g g H ft 0*9 STOCHASTIC FRAGMENTATION MODEL o 65. a a FIGURE 24: The Stochastic Fiber Fragmentation Model Flow Sheet UCC 004004 *0T 66. UCC 004005 10*1 10* *E 0**3 LENGTH (MILS) (LOG SCALE) 10* /Ovoncoaj 3aiivtwo NDiiDvaj fa 01 c H e 3 tn tn < tn C o -p A2 H -P W H Q T3 \ -p C O' nJ M u <u fa ,Q H >H fa nl O Q) H <U -P P <U fa P O> <U <0 XH Eh fa \o r-j g D U H fa UCC 004006 10**6 68. UCC 004007 10**1 10**2 . 10**3 LENGTH (MILS) (LOG SCALE) 10**4 69. FRACTION CUMULATIVE FREQUENCY FIGURE 28a: Theoretical Fragment i/d. Cumulative Distribution Assuming a Flawed Fiber UCC 004008 70. RELATIVE FREQUENCY FIBER LENGTH (MILS) FIGURE 28b: Theoretical Fragment i/d Distribution Assuming a Flawed Fiber (Numerically Smoothed Frequency Curve) UCC 004009 FRACTION CUMULATIVE FREQUENCY FIGURE 29: Experimental and Optimized Theoretical A/d Distribution Curves for OCF 885 in Polypropylene UCC 004010 lG 72 't * t o (S-**0TIScn SSBdlS 00 * o LlJ _<J o V) oo e o i-i CO 03 4-1 T! XH (D O 4J U) 3 OJ ft G CS -4-1 0 <S -U U G CO <U w CO 4 4 ow _1 M 5 W 4 o fc; t-i x: O' 4-> G 4-> O' U G G ft d) d) e G d> G 1 GG XI a> -h 4-1 i--1 ft Oi G ft a> 0) O r-4 O >-i O' 4-4 ft G Ul >l-H rH U3 MO d> ft 4-) JQ N G H LO cu ft 00 w CD 0) G G ft u ft 0) U d) S O PS o m o D X> U rd -H H -p in ft G G d> dJ .o ^6 UCC 004011 10 T fr ac tio n cum ulative frequency FIGURE 31: Experimental and Optimized Theoretical l/d. Distributions for OCF 885 in Nylon 6 UCC 004012 UCC 004013 74 . -p c GJ 0) p a; a; P K p IP V) u Ifl M +J ro (0 <--I -P G 0 tfl co > n <d p GG U O -P 'H U) .c -p o p nl E-t D1 -P G G <U (D 0 H JP I tn tfl 18 fl -p p a) CP[m G (1) U0 P P G WO P >ir-l d)2'H ! "n. IP -H LO Ip 00 0) CO i-t G C' 18 P C mtj'H sow CM m s p o H IP 75. APPENDIX A Data Smoothing and Differentiation Fragments, normalized by their diameter, are ranked according to their size and a fractional cumulative frequency distribution is evaluated. These data points are smoothed and differentiated using the moving strip polynomial method of Hershey, Zakin and Sihma17 to give an empirical probability density curve of fragment aspect ratios. An odd number of consecutive data points (3, 5, 7, . ..) are con sidered to constitute a strip and a least squares polynomial is fitted to these points. to represent the data. The following model is assumed m F (x) = l~ 0j j =o J +s (A-l) where m = degree of the polynomial Bj= power coefficients e = error The least squares power coefficients, 8-, were solved for by Gauss-Jordan elimination. In data smoothing, F (x^) , the data value at the midpoint of the strip is replaced by its approximation, Y(x^), as calculated from mj Y{xi) = I 8- x. j=o J (A-2) The strip then advances along the abscissa eliminating the leftmost point and adding a new point to the right-hand side. In the tails of the distribution, points between the strip midpoint and the extremes are approximated by off center formulas. The amount of smoothing increases with the number of points in the moveable strip and decreases with increasing m, the degree of the polynomial. When m=n-l, where n equals the number of points in the strip, the polynomial fits the data exactly (i.e., no smoothing). To differentiate the data, it is generally smoothed first, then the differential of the least squares polynomial {equation A-2) is evaluated at the strip midpoint. UCC 004014 76. dY(xn dx. 1 m j-l "j=l j Bi Xi (A-3) Ordinarily five or seven point strips are used however, in the central portion of a distribution these points could be nearly coincident and appear to define a steeply inclined line. Derivative evaluation could result in a series of "phantom peaks" appearing in the empirical probability density curve. This effect is particularly troublesome as the total number of data points increases. One approach to elimin ating this effect is simply to increase the number of points in the strip. N, thus becomes an empirical variable, the best value is a function of the data being analyzed. Another approach, as suggested by Knox3*, is to define a constant, K. When the ratio of a data point and the current reference point {i.e., last value of x at which Y(x) was evaluated) is less than 1.0 + K, Y(x) is not evaluated at this point. Thus, K is also a variable which must be determined empirically. The current data analysis program 'uses a second degree polynomial for both smoothing and differentiation. Multiple smoothing passes (*'4) generally precede differentiation and "phantom peaks" are eliminated by the variable strip size technique. UCC 004015 77. APPENDIX B Probabilistic Derivation for the Effect of Specimen Size on Strength The following derivation is due to Freudenthal19. Consider that inhomogenerties of critical severity are uni formly distributed over the volume V {or length , or area) of the considered body. Let P*{V) = the probability of nonoccurrence of the critical flaw in volume V. P*(Vi) = the probability of nonoccurrence in volume Vi where does not include V. The probability of nonoccurrence in V + V! is P*(V + Vi) = P*(V) . P* {V i. (B-l) assuming that the probabilities P*(V) and P*(Vi) are independent. Differentiating with respect to V gives P* (V+Vi) = P* {V i) + P*{V) (B-2) Dividing equation B-2 by B-l gives lnP*(V+V!) = n P* (V) (B-3) This equation equals a constant because it must hold for any arbitrary value of V j -- In P*(V) = C (B-4) Integrating with the boundary conditions P*(0) = 1 P* () = o UCC 004016 78. produces the general expression P* (V) -cv e (B-5) Where c is related to the flaw concentration {Weibull26 ) and has the units V"1. The probability of fracture, thus becomes Pf(V) = 1 - e~cV (B-6) For the same concentration of flaws, the probability of fracture increases rapidly with increasing volume. UCC 004017 79. APPENDIX C The Mixed Element, Weakest Link/ Fiber Strength Model f (cr) , the fiber element probability density function (p.d.f.) is given by f(o) = 0 for o < a f(0) = p/(ab-aa) f<0) = O f(0) = (1-P) (m/oo)[ o~0 \ m-1 c\ /i o-o^c \m exp- o/ \o for 0_ < 0 < 0. cl -- -- D for b a, < 0 < 0c,, for a > o,, --c (C-l) The fiber element cumulative distribution function (c.d.f.), F<0) = Jaf(o) da, is given by F(o) = 0 F{0) = p(a-CTa)/(<Vaa* F (a) = p F(0) = l-(l-P) exp-( o-oc \m for a < a. for 0a. --< 0 --< 0bU for 0. < 0 < 0,, bc for 0 > 0 --c (C-2) By employing the weakest link equations, expressions can be derived for the strength distribution functions characterizing a fiber or chain of n fiber elements. The chain p.d.f. is given by g(0) = n f(0) {1-F<0)]n"1 The chain c.d.f. is given by G(0) = 1 - [1 - F(a)]n UCC 004018 Substituting (C-2) into (C-4) yields G(a) 0 G(a) 1 G (a) 1 for a < a a for ao,-- <a <-- a.d for bo.< o < 0,,c G(o) 1 for 0 --> c The average fiber or chain strength can be calculated from Therefore <cr> = f So [1 - G(a) ] da 80. (C-5) (C-6) which yields <a> - aa + (ob-a) [} ~ (1 " P)n+1]/p (w+1) where ' + (oc-ob) [1 - P]n + [1 - Pln (VnV") r T r = complete gamma function (c-8) UCC 004019 APPENDIX D 81. The Sequential Fragment Proof Testing Modifications The c.d.f. for fibers which have been proof tested to the stress level dp can be expressed in terms of the original distribution by the relation G la) = 0 for a < a G(a) - G(a ) G_ (a) = -------------------1 - G(ap) for a > a -P (D-l) Gp{a), in terms of our fiber strength model, thus assumes different forms depending on the value of dp and its relation to the model parameters, a&, d^, and o,,. If aa --< ap --< a.b 1 - Gp (a) =1 for d < a_ $r/t - W1 - P for d < d < a. P- - b 1-Pn . a -aa\ i n 1 - pf-E. a ab-aa ,, *"(/a---d,, 1-P for d.b < d < c (D-2) n for d --> o c If d.b < dp '< dc 1 - Gp(d) - 1 ^d-dc \ m exp-n for d < a. for d --> dc,, (D-3) If dp -> dc 1 - G (d) = 1 = exp-n for d < d. for d > o -P (D-4) UCC 004020 82. The average fiber or chain strength can be evaluated from eouation (c-6) for a a --< a p --< a,b (0b"aa) {<a> = aP + pTH+TT l -p a -a _E__ a ah~an _Ella 1-P iab~aa- -n n+1 (1-P) + (0c-ab) (1 ~ P) n 1 - Pf aj--* -n + (1 - P) n !e3 1-P ffb"a -n (VB 1/m l'here f = complete gamma function for a, < a < a bpc = a + (a /n1/m) <a> c \ o' / r 2i m for ap -> a c (D-5) (D-6) <a> =a + P exp where y = incomplete gamma function m' V a y (D-7) The incomplete gamma function, y , in equation (D-7) was evaluated using an identity relation with the chi-square probability function. (See Abramowitz1*" pg. 941, eq. 26.4.19). Y(a,x) = T(a) P(xVv) (D-8) where v=2a=2/mi . \m '--Fa P(X2/v) is the probability that the random variable x, distributed according to the chi-square distribution with v degrees of freedom, is less than or equal to x2* It was computed using the IBM SSP subroutine, CDTR. UCC 004021 83. Since this program is restricted to values of v > 0.5 (i.e., a > 2.5) the recurrence formula yta+l,x ) = a-Y (a, 3$ - jfe x (D-9) was also used in the calculation (see Abramowitz 5 pg. 262, equation 6.5.22). UCC 004022 DISTRIBUTION BOUND BROOK Ancker, F. H. (6) Azrak, R. G. Baker, L. M. Bertolucci, M. D. Brown, A. Carrick, W. L. Fan, Y. L. Fraser, W. A. (8) Gorham, W. F. Hale, W. F. Johnston, N. W. Leung, M. Lynn, J. W. Marsh, B. D. McKenna, L. A. Merriam, C. N. *Michno, M. J., Jr. Nicholas, J. S. Ongchin, L. Pan, F. Y. Reinking, N. H. Sauers, M. Staub, R. B. Whelan, J. M. White, Dr. C. E. Technical Reports Center (10) NIAGARA FALLS Byrne, R. E. Myers, J. L. Appendicies E and F included. NEW YORK Burnett, L. D. Ford, C. E. Forsyth, R. B. Hiler, D, C. Hutchinson, K. J Jellinek, M. H. Lutz, A. W. Shechter, L. v Szabo, T. T, ^Phurber, W. C. Tomfohrde, H. F. Townsend, H. N. Zutty, N. L. III TARRYTOWN Marsden, J. G. Orenski, P. Ranney, M. W. Sterman, S. Tinsley, S. W. Welch, T. H. Williams, T. C. SOUTH CHARLESTON Atkins, K. E. Bryant, G. M. Kayser, R. F. Koleske, J. V. Smith, J. J. Smith, P. L. Walter, A. T. Whitehurst, W. E PARMA Bacon, R. Chambers, W. E. Chwastiak, S. Didchenko, R. Eckstein, B. H. Kampe, D. J. Singer, L. Stroup, R. C. Volk, H. F. UCC 004023