# Full text of "Probabilistic Fiber Composite Micromechanics"

## See other formats

NASA Contractor Report 198443 Probabilistic Fiber Composite Micromechanics Thomas A. Stock Cleveland State University Cleveland, Ohio January 1996 Prepared for Lewis Research Center Under Grant NAG3-550 National Aeronautics and Space Administration PROBABILISTIC FIBER COMPOSITE MICROMECHANICS THOMAS A. STOCK Bachelor of Science in Civil Engineering Northwestern University June, 1983 Submitted in partial fulfillnent of requirements for the degree MASTER OF SCIENCE IN CIVIL ENGINEERING at the CLEVELAND STATE UNIVERSITY March, 1987 This thesis has been approved for the Department of Civil Engin ee ring and the College of Graduate Studies by Prof. Paul X./tellini, Thesis Conmittee Chairman C/UlkaymU /j?-Z4-&7 Dr. C C. Chamis, Adjunct Associate Professor Dr. John H. Hemann, Department Chairman 11 ftCKNOU-EDGETEHIS This thesis is the result of work performs*! at NflSA- Lewis Research Center under grant MftG 3-550. I wish to express my gratitude to Dr. Christos Chamis, from the Structures Division at Lewis, for his insight, enthusiasm, patience, and wit, all of which contributed to my learning, of which this thesis is a small part. His example is an inspiration to ma, in school, work, and life. Dr. Bellini, my advisor, deserves thanks for patiently reading the many drafts, and supplying perspective and criticism to aid in clarifying my ideas. He has been a teacher and a friend. ill ABSTRACT Probabilistic composite micronEchanics nethods are developed that sinulate expected uncertainties in unidirectional fiber conposite properties. These netbods are in the form of computational procedures using Ifente Carlo simulation. The variables in vfcich uncertainties are accounted for include constituent and void volune ratios, constituent elastic properties and strengths, and fiber misalignment. A graphite/epoxy unidirectional conposite (ply) is studied to demonstrate fiber composite material property variations induced by random changes expected at the material micro level. Regression results are presented to show the relative correlation between predictor and response variables in the study. These computational procedures make possible a formal description of anticipated random processes at the intraply level, and the related effects of these on composite properties. xv PAGE 1 1 6 TABLE OF CONTENTS CHAPTER I. INTRODUCTION A. Background B. Purpose C. Formulation of the Model D. Method of Investigation 9 1. Brief Description of ICAN 2. Summary of Variables 3. Ifcnte Carlo Methods E. Brief Sunnary of Results II. METHODS OF CALCULATION A. Overall Plan 1. Input Structure for ICAN 2. Constituent Property Variations 3. Repeated Runs 4. Data Collection B. Generation of Pseudo Random Numbers 23 1. Uniform Distribution 23 2. Normal Distribution 25 3. Gamma Distribution 2S 4. Weibull Distribution 28 C. Property Distribution Assumptions 31 9 10 13 15 17 17 17 18 20 20 D. Use of ICAN 47 1. Composite Hicronechanics 47 2. Laminate Theory 48 3. Strength Theories 49 E. Review of Applicable Statistical Concepts 51 1. Sample Maans 51 2. Sample Standard Deviation 51 3. Confidence Interval Estimates 51 4. Multiple Linear Regression 53 III. RESULTS * 57 A. Property Histograms and Distributions 57 B. Fiber Strength Effect 73 C. Matrix Strength Effect 76 D. Fiber Orientation Effect 80 E. Fiber Stiffness Effect 80 F. Regression Msdels 1®1 IV. DISCUSSION 13® A. Overview 13® B. Histograms and Distributions 131 C. Confidence Curves 133 D. Examination of Regression Models 134 V. CONCLUSIONS 136 REFERENCES 138 VI APPENDIX A- SOURCE PROGRAM LISTINGS 14 ® APPENDIX B- ICflH PROGRAM DETAILS 154 1. Conposite Micronechanics 155 157 2. Laminate Theory 171 3. Strength Theories VI 1 LIST OF TABLES TABLE TITLE ?££ I INPUT DATA PCJR SAMPLING M II CASE 1 RESULTS 59 III LONGITUDINAL MODULUS (ECU), SIMPLE 104 IV TRANSVERSE MODULUS (EC22), SIMPLE 105 V SHEAR MODULUS (EC12), SIMPLE 106 VI POISSON'S RATIO, MBJOR (NUC12), SIMPLE * 107 VII POISSON'S RATIO, MINOR (NUC21), SIMPLE 108 VIII LONG. THERM. EXPANSION (CTE11), SIMPLE 109 IX TRANS. THERM. EXPANSION (CTE22), SIMPLE HO X LONG. TENSILE STRENGTH (SCX5TT), SIMPLE HI XI LONG. COMPRESSIVE STRENGTH (SCXXC), SIMPLE 112 XII TRANSVERSE TENSILE STRENGTH (SCYYT) , SIMPLE 113 XIII TRANSVERSE COMPRESSIVE STRENGTH (SCYYC), SIMPLE 114 XIV IN PLANE SHEAR STRENGTH (SCXYS), SIMPLE 115 XV LONGITUDINAL MODULUS (ECU), INTERACTION 118 XVI TRANSVERSE MODULUS (EC22), INTERACTION 119 XVII SHEAR HDDULUS (EC12), INTERACTION 120 XVIII LONGITUDINAL THERMflL EXPANSION 121 vm XIX TRANSVERSE THERMAL EXPANSION 122 XX POISSON RATIO, MAJOR (NUC12), INTERACTION 123 XXI POISSON RATIO, MINOR (NUC21), INTERACTION 124 XXII LONGITUDINAL TENSILE STRENGTH (SCXXT), INTERACTION 125 XXIII LONG. COMPRESSIVE STRENGTH (SCXXC), INTERACTION 126 XXIV TRANSVERSE TENSILE STRENGTH {SCTTT) , INTERACTION 127 XXV TRANSVERSE COMPRESSIVE STRENGTH (SCYYC), INTERACTION 128 XXVI IN PLANE SHEAR STRENGTH (SCXYS), INTERACTION 129 IX LIST OF FIGURES FIGURE TITLE Pi^ 1 Photomicrograph of Graphite/Epoxy cross 5 section showing variation in fiber content. 2 Conventional Ifedel 8 3 Substructure Hbdel 8 4 Coordinate Systems A1 5 Order of ICflN input data cards H 6 Command Input 19 7 Constituent Variation Input 19 8 Flow Chart of Probabilistic Integrated 22 Composites Analyzer 9 Uniform Distribution: general form 29 10 Normal Distribution 29 11 Gamma Distribution density functions 3© 12 Weibull Distribution function 3® 13 Normal Distribution Simulation 32 14 Gamma Distribution Simulation 33 15 Gamma Distribution Simulation 34 16 Gamma Distribution Simulation 35 17 Gamma Distribution Simulation 36 18 Ueibull Distribution Sinulation 37 Matrix Shear Strength 19 Ueibull Distribution Sinulation 38 Matrix Shear Strength 20 Ueibull Distribution Simulation 39 Matrix Tensile Strength 21 Ueibull Distribution Simulation 4® Matrix Tensile Strength 22 Ueibull Distribution Simulation 41 Matrix Tensile Strength 23 Ueibull Distribution Simulation 42 Matrix Compressive Strength 24 Ueibull Distribution Simulation * 43 Matrix Compressive Strength 25 Ueibull Distribution Sinulation 44 Fiber Tensile and Compressive Strength 26 Ueibull Distribution Simulation 45 Fiber Tensile and Compressive Strength 27 Ueibull Distribution Sinulation 46 Fiber Tensile and Compressive Strength 28 Typical Stress-Strain behavior of unidirectional 50 fiber composites 29 In-plane fracture nodes of unidirectional (ply) 5® fiber composites 30 Sampling results for Longitudinal Elastic Modulus 60 31 Sampling results for Transverse Elastic Modulus 61 32 Sampling results for In plane Shear Modulus 62 33 Sampling results for Poisson Ratio (major) 63 34 Sampling results for Poisson Ratio (minor) 64 35 Sampling results for Longitudinal Thermal Expansion 65 XI 48 Longitudinal Elastic Modulus, for various shape parameters of fiber orientation 49 Transverse Elastic Modulus, for various shape paraneters of fiber orientation 50 In-plane Shear Modulus, for various shape parameters of fiber orientation 51 Longitudinal Tensile Strength, for various shape paraneters of fiber orientation 52 Longitudinal Compressive Strength, for various shape parameters of fiber orientation 53 Transverse Tensile Strength, for various shape parameters of fiber orientation 68 69 36 Sampling results for Transverse Thermal Expansion 66 37 Sampling results for Thermal Expansion Coupling 67 38 Sampling results for Longitudinal Tensile Strength 39 Sampling results for Longitudinal Compressive Strength 40 Sampling results for Transverse Tensile Strength 70 41 Sampling results for Transverse Compressive Strength 71 42 Sampling results for In- plane Shear Strength 72 74 75 43 Longitudinal Tensile Strength, for various shape parameters of fiber strength 44 Longitudinal Compressive Strength, for various shape parameters of fiber strength 45 Transverse Tensile Strength, for various 77 shape parameters of matrix strength 46 Transverse Compressive Strength, for various 78 shape parameters of matrix strength 47 In-plane Shear Strength, for various 79 shape parameters of matrix strength 81 82 83 84 85 86 xn 87 64 Longitudinal Compressive Strength; for various shape paraneters of fiber nodulus 65 Transverse Tensile Strength; for various shape paraneters of fiber modulus 66 Transverse Conpressive Strength; for various shape paraneters of fiber nodulus 67 In Plane Shear Strength; for various shape paraneters of fiber nodulus 88 89 54 Transverse Conpressive Strength, for various shape paraneters of fiber orientation 55 In-plane Shear Strength, for various shape paraneters of fiber orientation 56 Poisson's Ratio (major); for various shape paraneters of fiber orientation 57 Poisson's Ratio (minor); for various 90 shape paraneters of fiber orientation 58 Longitudinal Elastic Ifcdulus; for various 91 shape paraneters of fiber nodulus 59 Transverse Elastic fbdulus; for various 92 shape paraneters of fiber nodulus 60 In Plane Shear ffodulus; for various - 93 shape paraneters of fiber nodulus 61 Poisson's Ratio (major); for various 94 shape paraneters of fiber nodulus 62 Poisson's Ratio (minor); for various 95 shape paraneters of fiber nodulus 63 Longitudinal Tensile Strength; for various 96 shape paraneters of fiber nodulus 97 98 99 100 Kill B.l Components of Stress acting on elenental unit cube 174 B.2 Rotation of coordinates from 1-2 to x-y. 174 B.3 Bending geometry in the x-z plane 175 B.4 Shearing force deformations on straight cross section 175 B.5 Stress and {foment resultants 176 B.6 Laminate index notation contention 176 xiv CHAPTER I IKTRODUCTION A. Background The diverse requirements of recent engineering applications have motivated designers to explore specialized structural and material systems. Ceramic materials, for example, have several attractive structural properties, such as their high stif fnessA*ight ratios, and low variation of stiffness and strength over wide ranges of environmental conditions. A significant disadvantage inherent to brittle structural materials is their vulnerability to failure due to cracks propagating from f lavs. The increased probability of a flaw occurring in a material as the volune increases leads to bulk strengths «hich are a fraction of the theoretical strength of the material. The size effect on material strength (Ref . 1) can be explained by the "weakest link" concept. Griffith ( Ref. 2) reasoned that very small solids, for example wires or fibers, might be expected to be stronger than large ones, due to the additional restriction on the size of the flaws. In the limit, a single line of nolecules mast possess the theoretical molecular tensile strength of a material. A consequence of the size effect on strength was the development of fiber composite materials which consist of thin, strong fibers bound together by a ductile matrix. The advantages of fine, strong fibers can explain the current trend toward increased use of fiber conposite materials in denanding aerospace applications. Properties of a composite laminate depend on the properties of the constituent materials, their distribution, and orientation. Laminates are composed of layers of unidirectional ly reinforced plies (laminae). The lamina is typically considered the basic unit of material in a composite structural analysis, which requires knowledge of the material properties of each individual lamina and its geometric orientation. The branch of conposite mechanics that predicts ply material properties based on the properties, concentration, and orientation of its constituents is knowi as conposite micronBchanics , and frequently incorporates the traditional ttschanics of Materials assumptions. The desired laminate is created by stacking of plies in specific directions. The integration of ply properties to yield laminate properties is called laminate theory. Laminate variables such as ply orientation and stacking sequence can be tailored to yield a laminate with the desired material properties. Thus, the laminated conposite is a suitable material for component design. Analysis of fiber composite structures is currently performed using a variety of computer codes. From the original codes based on classical micronechanics and laminate theory, recent codes (Bef . 3,4) have been developed which incorporate the current state of the art. Complete 3 nEchanical, thermal , and hygral properties are calculated, and can be used to compute response, Advanced failure criteria are used to calculate composite strengths. Environmental effects are also quantified. The usefulness of these codes has been deimnstrated by comparison with experimental and finite element results (Ref. 5,6). The analytical capability of many codes is limited by the deterministic nature of the computations. Specifically, fixed values for constituent material properties, fabrication process variables (i.e. constituent volume ratios) and internal geometry must be used as input. Ifevjever, random variations in these parameters are not -only expected, but easily observed experimentally. (See Fig. 1) The analysis of composite structures requires reliable predictive ntxlels for material properties and strengths. However, the prediction efforts have been complicated by inherent scatter in experimental data. Since uncertainties in the constituent properties, fabrication variables, and internal geometry would lead to uncertainties in the measured composite properties, the question arises: How much of the "statistical" scatter of experimentally observed composite properties can be explained by reasonable statistical distribution of input parameters in composite microraechanics and laminate theory predictive models? The increasing use of probabilistic methods in structural mechanics has been shown to provide a more realistic depiction of structural response due to load variations. (Ref. 7) The recognition that material parameters are characterised by a spectra of values ( that is, are statistical in nature ) rather than by a unique set of values, points to probabilistic nethods as a logical analysis approach. Fig. 1- Photonlcrograph of Graphite/Epoxy cross section showing variation in fiber content. (Ref. 19) B. Purpose The aim of this thesis i» to develop a oonputational oapability to simulate the probabilistic variations in the irechanical behavior of unidirectional fiber composites. The Ifonte Carlo ■ethod is used to simulate a variety of random processes, to quantify fiber coaposite material variations induced by random changes in composite fiber alignnent, constituent properties, and fabrication process variables. This random process description is an attempt to more accurately predict the behavior of manufactured materials, %fcich inherently include these random variations. The characterization of fiber reinforced composites through simulation of local nonuniformities provides an economical alternative to experimentation to measure material properties. C. Formulation of the Model Tte mxiel comronly used in characterizing fiber composites is based on the calculation of properties of the basic unit of an orthotropic ply. The layup geonetry is then used in laminate equations to calculate opposite prooerites (See Figs. 2a, 2b). In this «*, hoover , the basic unit is taken as the sub-ply. which consists of only one fiber-matrix level in the material, nicronechanics theory is used to calculate the properties of the assumed orthotropic sub-ply, each with randomly distributed fabrication variables and material properties. Distributed fiber directions, due to possible misalignment within the ply, are then used in the laminate equations to calculate ply properties. T*is substr^turing of the composite ply represents a novel attempt at characterization of fiber composite material properties based on probabilistically distributed constituent properties, individual fiber misalignment, and fabrication process variables (See Figs. 3a,3b). ^is formulation is particularly well suited to the probabilistic description of fiber composite material properties. Since the ^cromechanios and laminate equations can be used to calculate ply properties at any number of points in a ply, a tractable finite element structural analysis based only on simple distributional assumptions for physical parameter variations can be perform*. T*is nodel supplies a rational procedure for composite material property assessment, because it treats the material as the result of a series of random processes which occur at the intraply level. 8 ^: V .... ^^•tDpOtV' . . . -.._.. ___— Xm}ji ssaas XL -L\ $r (a) orthotropic ply (b) laminate Fig. 2- Conventional Model fiber misalignment (a) subply (b) ply Fig. 3- Substructure Model D. Method of Investigation 1. Brief Description of ICAN The Integrated Composite Analyzer (ICflN) is a computer program for comprehensive linear analysis of multilevel fiber composite strictures. The program contains the essential features required to effectively design structural components made from fiber composites. It now represents the culmination of research conducted since the early 1970»s, at the National Aeronautics and Space Administration (NASA) Lewis Research Center (LeRC), to develop and code reliable composite mechanics theories. This user friendly, publicly available code incorporates theories for 1. conventional laminate analysis 2 intraply and interpiy hybrid composites 3. hygral, thermal, mechanical properties and response 4 ply stress-strain influence coefficients 5*. microstresses and microstress influence coefficients 6. stress concentration factors around a circular hole 7. predictions of delamination locations around a circular hole 8. Poisson's ratio mismatch details near a straight free edge g. free edge inter laminar stresses 10. laminate failure stresses 11. normal and transverse shear stresses 12 explicit specification of matrix-rich interpiy layers \l\ f£ite element material cards for NASTOAN, MARC A detailed description of ICAN can be found in Reference (3). The ICAN code and documentation are available through COSMIC, the Computer Software Management and Information Center, Suite 112, Barrow Hall, Athens GA, 30602. 10 2. Sunirary of Variables The variables studied in this *ork csan be separated into two categories. The independent variables to be simulated using random sampling consist of the following (see Fig. 4a for fiber coordinate system) : Geometry: fiber orientation angle Fabrication variables: fiber volume ratio void volume ratio Fiber properties longitudinal elastic modulus transverse elastic modulus shear modulus, 1-2 plane shear modulus, 2-3 plane fiber tensile strength fiber compressive strength ffetrix properties elastic modulus matrix tensile strength matrix compressive strength matrix shear strength (THETA) (FVR) (WR) (EFP1) (EFP2) (GFP12) (GFP23) (SFPT) (SFPC) (OF) (SfPT) (SMPC) (SIPS) The dependent variables to be calculated using ICAN consist of the following ply properties, measured about the material axes (see Fig. 4b): normal modulus in 1-1 direction normal modulus in 2-2 direction shear modulus in 1-2 plane Poisson's ratio for strains in 2 by stresses in 1 direction Poisson's ratio for strains in 1 by stresses in 2 direction Coefficients of thermal expansion in 1-1 direction in 2-2 direction coupling coefficient (sen) (EC22) (EC12) direction induced (NUC12) direction induced (N0C21) (CTE11) (CTE22) (CTE12) 11 (a) fiber i yV_ _£_ M .^flKJt SiCAJt (b) material Fig. 4- Coordinate Systems Fig. 5- Order of ICAN input data cards 12 Ply strengths in material directions longitudinal tensile JSCJOCT) longitudinal compressive (SCXXC) transverse tensile (SCYYT| transverse compressive (SCYYC) in-plane shear (SOWS) The descriptions above should be consulted periodically for the definitions of variables that henceforth will be referred to symbolically. 13 3. Monte Carlo Methods Complicated stochastic processes can be simulated by a variety of numerical methods generally referred to as Monte Carlo raathods (Ref . 8). The term refers to that branch of experimental mathematics concerned with experiments on random numbers. Since the advent of high speed computers, they have found extensive use in most fields of science and engineering, in analyzing many physical processes of a statistical nature, or where direct experimentation is not feasible. In general, they can be economically used to achieve a level of precision between 9® and 95 percent. ft Monte Carlo experiment refers to the procedure of randomly assigning a value to an independent random variable in a chosen itodel, and observing the dependent variable at the conclusion of the process being modeled. A Monte Carlo procedure is composed of n such independent experiments. When n is sufficiently large, the observations will yield, by virtue of the laws of large numbers, a statistically meaningful description of the physical problem. Tne form of Monte Carlo used in this study is as follows: 1. Define the system model by assuming a. model regression function b. method of error incorporation c. probability distributions of all errors (for all independent variables) . d. any equations used to model the phenomena of interest 2. Use the computer and random sampling techniques to select values of the independent variables. 3. Calculate dependent (output) variables using the prescribed 14 equations. 4. Estinate regression paraneters for the assoned radel. 5. Replicate the experiment, each tiws with a new set of input values. 6 Use appropriate statistical nethods to calculate properties of the distribution of paraneter estimates. 15 E. &-ief SuraiBry of Results ft ply made from the flS-Q-aphite /I1K epo«y composite system i. staled. The sonte carlo scheme is used to generate . nu*«r of response results, Wich are analyzed in graphical and nu^rical form, to 5u pply a random process description cf composite ply elastic constants, therm,! expansion coefficients, and strengths. Histogram and distribution plots of results for assumed narrow and wide variations in input properties are compared with a deterministic base case for an aligned ply. The figures demonstrate the range of values that response tables assume for the example data under consideration. Confidence intervals are calculated for response variables in subsequent samples, which are normalized with respect to an appropriate independent variable, to yield plots of normalized response as a firo tion of fiber volun* ratio, for various values of distribution parameters for the related independent variable. These plots demonstrate the sensitivity of ply properties to randomly selected uncertainties in constituent and fabrication variables. Several multiple line-- regression models were calculated for response variables. The relative correlation of predictor (independent, tables with response is staled for all output properties considered. Varying levels of significance were achieved in the regression .guations, due to the differences in complexity of response variables. Elastic constants can be described adequately with simple r^n-essor fusions, a~l generally explain bet««. »^" P—" «* tte observed response variations about a «. The regression .-dels 16 studied far strength, although achieving better reliability with higher order regressor functions, demonstrate suoh low significance as to be practically useless for predictive purposes. This is not an unexpected result, because of the coup lex nature of strength behavior in cowposite neterials. CHAPTER II PETHODB OF CALCULATION A. Overall plan 1. Input stricture for ICAN The input data for a typical execution of the available ICAN program consists of (see Fig 5) 1. header card 2. control cards 3. ply data cards 4. imterial system cards 5. load cards For repeated use of the ICAN program, input data files must be created and used one at a tine. Each successive run of the master program (of ^ich ICAN is made a subroutine) vrites the input file from user-supplied parameters and calls ICAN. The ply data cards contain randomly generated fiber orientation angle values. The material system cards contain randomly generated values for fiber and void volune ratios. 17 18 2. Constituent Property Variations Each siEoessive execution of ICfiH uses » distinct set of nmterial properties for fiber and matrix. The random number generation is performed with user-supplied parameters which are stored in a separate file. The options of using either generated properties or using the values contained in the resident data bank are available. Any subset of the parameters described may be generated or held constant with proper specification of the Boo leans which control the input to the ICAN program, (see Figs. 6,7) i 19 FIBER STRENGTH VARIES; CONSTANT FIBER VOLUTIE RATIO OF 0.30; TAPE 03131 STDATA 13 1 15 T T 50 T F T T F 000.0 10.0 0.300 0.200 3.00 3 F F p 1Y 70.00 70.00 .0 -000 WATCRDAS-IOTHS AS_1I ? H ? -° '" PLOAD 10. 0.0 0.0 0.0 PLOAD 0.0 . 0.0 0.0 PLOAD 0.0 0.0 OPTION Fig. 6- Command Input i. 3100E 08 0.3000E 07 EFP1 I MoOOE 07 0.2000E 06 EFP2 I S 2000E 07 0.2000E 06 GFP12 5 B1000E 07 0.1000E 06 GFP23 I S'loOOE 06 0.1000E 02 SFPT I 4000E 06 0.1000E 02 SFPC I 5000E 06 0.5000E 05 ET1P I I i5ool 05 0.1000E 02 S"PT I 03500E 05 0.1000E 02 SriPC J •.}»« 5| dOOOE 02 Fig. 7- Constituent Variation Input. Example for AS ; J graphite fiber and IMHS Epoxy matrix, with wide variations of stiffnesses and strengths. 20 3. Repeated runs The user* must specify the number of I CAN runs desired in a given sample. In this study, fifty (50) runs were used throughout, to take advantage of the simplification in statistics by using suitably large samples. From elementary statistics, it is known that any process that is the result of the combined interaction of several probabilities can be assumed to approximate a normal distribution. For phenomena that are assumed to approximate a normal distribution, the simplest forms ior calculating statistics apply to suitably large samples (usually greater than thirty). The sample size of fifty was chosen to supply a practicably large amount of data, within the restrictions imposed on computation time. The data generated by repeated execution of the ICAN routines is stored in a sequential access dataset, where the 50 output files are separated by end of file markers. This arrangenent allows a single Fortran unit to be used for output throughout. A simple flowchart of the data generation routines is shown in Fig. 3(a). 4. Data collection The ICAN output files are searched to locate the specific material properties and strengths of interest in this study. The flowchart of data collection routines is shown in Fig. 8(b). After obtaining the sample of ICAN output, the investigator may choose to scrutinize parameters or calculate statistics aside from those chosen in this study. This is likely, in light of the large quantity of data available and the need for limiting the scope of this particular study to 21 representative properties. The user „ld have to supply additional code or adapt existing code to suit his purposes in this case. The coded mxlif ioations to ICAN used in this study are included in Append A. [ 22 C " m ) CKU sumoutihe CALL JUMOUTIIrt UMAT H 6ENERATE. WtlTC MHtXM DATA / Mil 7 Air I MRS CALL SJMOUT1IC ICAMI CALL SUMOUTIRE ICAIM lEVINO DATA ■ARK /MUTE 7 dutwt ml UNIT « / TOFItf UNIT C @ (a) data generation program / 28 / CALL STATISTICS SWaOUTIHCS HOT MSTDSMS MB diotuwtjojb HOT CORFI DCIICC CURVES tOKSSlO* AMLTSIS (b) analysis procedures F1g. 8- Flow chart of Probabilistic Integrated Composites Analyzer 23 B. Generation o£ Pseudo Random Numbers An integral part of any nonte carlo simulation is the use of random numbers having a specified distribution *hich is assuned to characterize the process under study. Indeed, many statistics textbooks carry tables of random numbers as appendices. Situations using large samples require many repeated calculations, each with different "random- numbers. Since filling of a computer mznory with a large table of random numbers is wasteful, algorithms have been developed (Ref . 9) to generate streams of random numbers whenever needed in the process of calculations. The numbers used are usually obtained using sone form of a recursion relation, hence the sequence is terned pseudo-random. 1. Uniform Distribution The starting point for many random number schemas is the uniform random number generator, which simulates a sample from the uniform distribution. A continuous random variable has a uniform distribution over an interval a to b ( b > a ) if it is equally likely to take on any value in this interval. The probability density function is thus constant over ( a,b ) and has the form = elsewhere The probability distribution function is, on integrating F(x) =0 x < a x - a _ a < x < b b - a 24 = 1 x > b The uniform distribution is shown in density and distribution form in Figs. 9a and 9b. Lehmer (Ref. 1©) proposed the congruent ial nethod of generating pseudo random numbers conforming to the uniform distribution. The recurrence relation takes the form: x. = (ax. . + b ) modulo m x v i-i where the notation signifies that x A is the remainder when (ax i _ 1 ♦ b) is divided by m. The multiplier a, increment b, and modulus m are integers. The starting value x_ must be assumed, and is known as the "seed" of the generator. Generators for which b = © are known as multiplicative. They are called mixed when b is nonzero. Because selection of the multiplier a and modulus m strongly influence the generator, most generators in use are of the multiplicative form. A discussion of the choice of parameters, maximum period, and degree of correlation of this generator is available (Ref. 11). For a given uniform random number u on the interval (0, 1) a random number x having a desired distribution F(x) is often obtained by solving the equation u = F(x) for x (Ref. 12). Since the process requires the determination of the inverse distribution function F (x), its use depends on the ease of deriving the expression or some approximation. The following sections describe the distributions used, and methods for generating random numbers on those distributions. 25 2. lfarmal (Gaussian) Distribution The most ooimon distribution is the familiar normal distribution, with the "bell shaped" density function, given by f(x;A*,a 2 ) = 1 f (x-/0 2 1 ■ ■ exp : — -«• < x < oo, ft < », and o > with nean // and standard deviation a. The distribution function is written F(x) = JTaa 1 exp (u-fi)' -2^~ du «hich cannot be expressed in closed form analytically but can be nunerically evaluated at any value of x. The Box-fcller or "Polar" method (Ref . 13) is most comronly used for generating random deviates from a nean to approximate the normal distribution. If x A and x 2 are independent uniform random variables, then 0.5 y t = o{-2 In x 1 ) W ' cos 2nx 2 + ft 0.5 y = o{-2 In x x r sin ani 2 + * are independent random variables with the standard normal distribution having nean fi and standard deviation a. 26 3. Gamma Distribution The gamrn distribution is a two-parameter distribution whieh is flexible in fitting * variety of random processes. It is a one sided distribution in that physical quantities that are limited to values in the positive range are frequently modeled by it. Its density function is given by A -Xx k-1 f(x) = e x /•(k) Were x, X, k > «, and k is an integer. The parameters X and k nay be interpreted as scale and shape parameters, respectively. T(k) is the well known gamma function, r<» k-i -u... *"( k ) = Jo u e du ' which is widely tabulated. The gamra distribution function is given by X k f (*> = T(krj0 u r(k,xx) x k-1 -Xu du x > r(k f «l = I x" _ e"*dx = elsewhere where T(k,u) is the incomplete gamra function . T U k-1 J® which is also widely tabulated. For integer values of k, T(k) = (k-1)! and the gaima distribution is known as the Erlangian distribution after A. K. Erlang, who introduced it in the theory of queues and !farkov processes. 27 Garni* variates are generated using the sapence satisfying the uniform distribution on the interval (€>,i). The recursion relation is 1 y L = - — ln u i« -hi-- — 1 "^ vtere x is a gamma var x i=l iate having parameters X and k (Ref. 14)- 28 4. Vfeibull Distribution The Ueibull distribution (Ref . 15) is mist popular **en modeling problems of reliability, material strength, and fatigue. The Ifeibull density function is given by f (x;a,fl) = o^~ 1 exp(-<iir) < x < co, a > 0, £ > 1 where a and fi are the shape and scale parameters, respectively. The cumulative distribution function y = F(x) = 1 - exp[-(x/0) a ] leads iimediately to the inverse relationship p-^y) = x = - fi[ ln(l-y) ] 1/a as the desired ifeibull random generator i*en y is a uniform random variable. Figures 9-12 show the above distributions in analytical form. 29 fx<*> Fx<x) £T- ■»■« ■♦•* (a) density (b) distribution Fig. 9- Uniform Distribution: general form. -4 -3 -: 2-101 2 (a) density -a -a -i (b) distribution Fig. 10- Normal Distribution uo Hounj uou^Mlsta lin<H 3 « "21 ' 6 * J a -nam amptij T 1 T I SCO -\ oso «■ - sio -»oi SUOl* 3un J- /Citsuap uo^nq^siQ euiueg -U *6W w*t OC 31 C Distribution Assumptions n. tables chosen for — ««- « tto ~ «- * iCh "— »•• .ssv^tions can be mede to describe their distribution. The fiber » _i«, r.snect to ply axes is assuned to follow a geometric conf iguration with respect to p y mml distribution with mean - — (d*«) - — ™ 1 ' •*-« delation, to be specified. T** fiber .1- ratio is assumed to be „. lly distributed about some mean bet«en ... a* ».7. The -id „« ratio, *i=h is ideally -11. L — < » «»— •— distribution s„ewd toward ~~. («■ «« »» «- «" °^«~"°" ..sed, a value of ze~ has a probability of zero. This model L ^sen hecause the state of —t present manufacturing technology precipes the fabrication of a fiber composite completely free of void.) The properties of individual fibers and matrix are varied. The „! and shear moduli are assumed to follow the normal distribution, and the straths are assumed to be tkitall distributed. Figs. 13-27 show the results of random, numoer generation in each distribution studied. The density (or histogram, and curative distribution plots are sho». Several -ibull and ga™e distribution simulations are — . to demonstrate the effects of assu»d parameter variations on the distribution sampling. 32 HISTOGRAM FOR NORMAL GENERATOR Standard Deviation 1IOOO 14000 1*000 12000 w £10000 j- « -I x 0000 s u 4000 ♦000 *000 RANCE < t (a) histogram DISTRIBUTION OF NORMAL GENERATOR Mean « 0.0 Standard Deviation 1.0 jo io in ifl ifl » ° RAN6E C E -01) o.jr^o- (b) cumulative distribution Fig 13- Normal Distribution Simulation with mean of 3 n n and standard deviation of 1.0. 33 HISTOGRAM FOR GAMMA GENERATOR ♦500 r 4000 X= 3.0 »500 k » 1 3000 «2500 bi S 5?000 at it. 1S00 - 1000 - "-| soo i i*— i i ' ' i - -Jr- i I fc '5 lOiii A ft W •ANCE < C -01> (a) histogram J5 ♦ ~« DISTRIBUTION OF GAMMA GENERATOR ■ a A A A A A ■ AN8C < t -01> (b) cumulative distribution Fig. 14- Gamma Distribution Simulation 34 HISTOGRAM FOR GAMMA GENERATOR X - 5.0 k « 3 i A riT-A A A RAN6E < £ -01> (a) histogram DISTRIBUTION OF GAMMA GENERATOR "A — A A fr A A A A A A A - A CAN6C < t -01 > (b) cumulative distribution Fig. 15- Gamma Distribution Simulation 35 H1ST06RM* FOR GAMMA GENERATOR ■ANSI « « -01> (a) histogram DISTRIBUTION OF 6AHMA GENERATOR < t -on • AN6C * l "* uo^einuits uoiVKH-nsKl w««9 "LI * 6 U uoiVKUJ*sip aAH«inuna (<1) (to- a > 39Nva V* fl * V n . r V Q »* tf qt ' * ° 9-1 0*E - X dQiVa3N39 VWHV9 jo NQiinaiaisia uieuBois.m (e) (10- 3 > 3SNV8 V <VLV V V ^ V °'' ^ S * I dQ!Va3N33 VHWV3 SQi HVdSQiSIH oooc - 000% - 0009 - ooot * - ooooi- - 00021 000%t ooott J OOOtl J\ 001 002 oot a oo*: -I OOSo 009 004 oot J 00* 9C CUMULATIVE rteouENCv u -o 38 HISTOGRAM FOR HE1BULL GENERATOR *— ,, l'i l 1 * l'l A A RANtc < e oo) (ksi) (a) histogram nooo r 14000 1*000 12000 £10000 « f tooo 9 u 4000 ♦000 *ooo DISTRIBUTION OF HE1BULL GENERATOR 6 « 13 ksi a - 20 A \\ A l'l A A MNte < e oo> (icsO (b) cumulative distribution Fi" 19- Weibull Distribution Simulation s * Matrix Shear Strength 39 HISTOGRAM FOR HE1BULL GENERATOR ISO 6 « 15 ksi. 140 a ' 5 1« 120 - S JOO Ml 3 S to m ib 40 ♦0 20 J- 1 Tt * * f • • j__ ( L >, — 1% — fa — r* — Ti - 11 & RAxte < c oo> (a) histogram (Its DISTRIBUTION OF HEIBULL GENERATOR (b) cumulative distribution welbull Distribution Simulation " Matrix Tensile Strength Fin 20- WeiDU.ii J " Matrix 4iJ *so r %00 »so 300 2(0 ■» 200 ... . 100 - to e »r- . HISTOGRAM FOR HEIBULL GENERATOR 6-15 ksi. a - 10 ti Y ^ro — rV — r* — rs ij *■ d KAHte < c oo> Usi; (a) histogram • V .DISTRIBUTION OF HEIBULL GENERATOR 6 --15 ksi . a -10 "J ■ 10 I* IT I* ••..«» «» ■AMtC < C 00) (ksi) (b) cumulative distribution Fig. 21- Weibull Distribution Simulation Matrix Tensile Strength 41 itooo r 1*000 1*000 12000 ~ 10000 ? tooo h 9 U 4000 ♦000 J- 2000 HISTOGRAM FOR HEIBULL GENERATOR .. .- T* — rr rank < t oo> (ksD (a) histogram DISTRIBUTION OF HEIBULL GENERATOR B - 15 ksi. a ■ 15 o V 'i I'a^rt tV— rt — 1% A A ... c c oo> ri^i) (b) cumulative distrlbuYTon Flo 22- Wei bull Distribution Simulation Fig- " £v"«„ T. n< n» Strenqth •Ante MeiDUII UlStriuw..— - Matrix Tensile Strength 42 . HISTOGRAM FOR HEIBULL GENERATOR %S0 i» ♦00 B - 35 ksl . sso a - 10 soo - £2S0 w 9 Szoo is \ ISO r A 100 - so - ~lj °ir* **¥ j — n — rt — » w *s *fl *» •< *ANSC < C 00) (kSl) (a) histogram itooo r 1*000 1*000 12000 £10000 S tooo 9 u AOOO %000 *000 . .DISTRIBUTION OF HEIBULL GENERATOR B - 35 ksl. o - 10 o» V i ' o l 2C A A rti A ri* rt i'o RANte < e oo> (ksl) (b) cumulative distribution Fig. 23- Welbull Distribution Simulation Matrix Compressive Strength ^WM^IP •M^in«3 (q) ( V S*) tOO 3 > Hy« w n, q u of y y *fr aoiva3N39 nnai3M jq NOiinaiaisto' uibj6oisim («?} (ts^) coo a > >•»*■ o * y ",* V ^ 02 tsi se aoiva3N39 nnai3M aoj uvasoisiH e* : 44 HISTOGRAM FOR NEIBULL GENERATOR rance < e on (ksi) (a) histogram DISTRIBUTION OF HEIBULL GENERATOR ltOOO IAOOO B - 400 ksi . 1*000 a « 10 12000 - Itl > — 10000 ^~ « ml / W tooo 9 o I 4000 I ♦000 I >000 ■hi — ^U -\ — A — Jm — A — A — ztr -t>. ~t» €9 *W •» ■"• '• " — w - ■ansc < c oi> (ks1) (b) cumulative distribution FIq 25- Welbull Distribution Simulation Fiber Tensile and Compressive Strength 45 HISTOGRAM FOR HEIBULL 6ENERAT0R too • 00 6 « 400 ksl 700 a « 15 400 "500 2»oo m f too 200 - 100 *AN6C c I oi> (ksi) (a) histogram DISTRIBUTION OF HEIBULL GENERATOR 9 *° " *° »*y«£™ (i ei>(Icsi} (b) cumulative dlstrtbut on c . , 6 uei bull Distribution Simulation F1g. 26- *™^ en$11 , and Compressive Strength 46 HISTOGRAM FOR HEIBULL GENERATOR A A A rancc c e on (ksi) (a) histogram DISTRIBUTION OF HEIBULL GENERATOR B . 400 ksi . o - 20 A A A A A *'» ■ance < c on (ksi) (b) cumulative distribution f<« 27- Welbull Distribution Simulation F1g. 27- y^eVVensile and Compressive Strength 47 D. Use of ICfiH TOis section describe, «. — «« theories - — *— asperated in «- K- »«— * ^ UC ~ titi0n 00n ~ nti0OS ' *r-l«ti— . - definitions are included in SpP^i" B. 1. Composite Kicronechanics «. „ra~h of coeposite ,-chanics «*» relate, ply properties to ^stituent properties is »— as «^site -erodes. * inputs ^sist not oniy - constituent -terial P-P-ties (fiber an- -«-). « gec-tric elation and fabrication pr-ess <*** includes pi r byural, «— ». - — ^ PrcPertie " ^ a5SU " n0nS EOr equation development are: (Ref . 16) ^ .fccbanics - ^-^"^SirSn^ieT" 005 ' ^T^^tf^i^lotdsl^i Jto tne scbe-tic STSxf ^?it. ( cLtlt— t. behaue in , linear elastic »nner S^M e^at'the fiber^tri, interface. „. direction conventions an- ter-inolooy used in «. *-*«. are! .. Properties assured .Ion, fiber direction are call- longitudinal. , h-„,verse to fiber direction are called 2 Properties neasured transverse » 1. 2. 3. 4. 5. 6. 3. 4. 48 2. Laminate Theory Classical laminate theory supplies a convenient procedure to predict the response of a laminate to external load. The theory uses anisotropic elasticity to obtain the stress-strain relationship for the basic lamina. The stress-strain relations of individual laminae are transform* to coincide with a global set of reference axes. The stress-stain law of the laminate in terms of the properties and distribution of individual laminae are calculated using a summation. Resultant forces and moments are defined by integrating the stresses through the thickness of the laminate. The plate constitutive equation is inverted, giving midplane strains and plate curvatures in terms of applied forces and moments. These strains and curvatures are substituted into the lamina stress-strain equation to obtain lamina stresses in the global system. The stresses obtained are then transformed into the principal material system of the lamina in question and compared with ultimate stresses obtained using failure criteria. 49 3. Strength Theories Trow *ke use of several assumptions. The strength theories in ICPM make use ot . that there are five characteristic values of First, it is assumed that there arc strength of a unidirectional composite: 1 longitudinal tensile strength 2. U^lt-i-1 c^pressi^ .tr-^h (3 separate criteria) a. rule of mixtures b. fiber microbuckling c. de lamination 3. transverse tensile strength 4. transverse compressive strength 5 in-plane or intralaminar shear strength -«*.« usually associated vith these strengths are shov* The fracture nodes usually assw-* schematically in Fig. 29. 0^ ply straths are calculate* (in «. ,1T — *— sy.te«), g eo»etric transforations are usee to calculate oolite failure — Tte process us«l i. briefly describe below. , Calculate 1— (A- — »« — > ~— " 1 ~ ^ """ to ply st«noths (in ply system) »- •** -*• 2 Calculate .tain of *•«— «-* t0r e ~ h PlY - 3 calculate -ni- - — U-. - •" P»". - " «" , ~ <J ' as tbe failure street* of the co^site for a particular failure node. 5© B iilT- W LONCnUOMAl \ \ N. MATRIX <1M \ \>raotn \-rurui MIXAUUMNA* S»CAR stram Fia 28- Typical Stress-Strain behavior of unidirectional fiber composites. I MUnfRuAMl fltOt COMf«SSI0H BCLMMNATION/ swttimc FIBER MCtWUCUINC « lantfeioWl eomr tsslon. f Id TnnswM tanstan MTnntvtrM W trmaalmr *!»•'. Fia 29- In-plane fracture modes of unidirectional s * folvl fiber composites. (ply) ttbe r compos 31 E . Review of applicable Statistical Concepts Cc^osit. property a~ calculate Tor 1^ sables — a ^ ic set - -i.tr— - — — "- " ^ -— *" OTl l sa^lio, «-, does -t appl*. *— - — — " sufficiently large. 1. Sample Ifeans Calculation of toe -an sample -I- P~~«s * «»— n nean = x = n vtere n = sanple size x .= sample data values 1 , « mn the sanple nean is assumed to be the Tte population mean is unknot, so the sanp best estimator of the population mean. 2. Sample Standard Deviation te e.ti«te of «. POPUl-ion — ~*«- - —— using the statistically efficient estimtor n ,1/2 f n * («.-«)* 1 • niy > o - [ n - 1 i=l V * J 3. Confidence Interval Estimates .. iB the area of statistical inference is the ftn important problem m the area A * ,_ .station Of Elation palters ,- » — — ■ *~* *~ _*. statistics. Peters i -.-«. — *— 52 nation of the sanpling distribution of a statistic S. 11* sailing distribution of S is assumed as approximately nornal (whioh is true for .any statistical distributions if n > »). Confidence interval estates are constructed for the statistic S. Thus, intervals are identified for wMch it can be asserted with a reasonable degree of certainty that they contain the paraneter considered. Obviously, the degree of certainty (or confidence level) will vary with the size of the interval chosen. Values of confidence coefficients, = c , are associated with confidence levels. For example, an actual sample, statistic S is expected to be found lying in the interval (x - V ) to (x ♦ zj) (where a is the unknown population standard deviation) sone percent of the tine. L* the z c value in this example be 1. Assuming a normal sanpling distribution, (with * c = 1) the normal distribution area function specifies that S falls between (x - a) and (x ♦ a) about 68.277. of the tine. Similarly, the confidence of x lying in the interval (S - «) to (S ♦ a) is about 68. 2T/.. The endpoints of the intervals are known as confidence limits. Various confidence coefficients « , corresponding to frequently used confidence levels, have been tabulated. In this work, the confidence interval for neans is given in terms of the sample statistics by x + z - c itere ^ is the confidence coefficient, which takes on values of 1.645, i!*», and 2.58* for the 90, 95, and 997. confidence levels, S3 respectively. 4. Regression The ter. -r^ssion- as used in the are. of statistic refer, to the process of formating a ..the^tical — .1 to explain ranoo-ly ohserved phe^ena. So.* rational f— for the »y each triable enters the -de! _t be asswed. Co^arison or the decree of fit of differ-nt assu«d — •»• »*-»»* leadS t0 * tetMr -* 1 ' "" to5i ° regression strategy used here consists of; X. «,su»e a -ItipU linear regression -*1. 11- nor«i equations for such a nodel are: {Y} = [X)ifi + (*) inhere (Y) = sector of dependent variable values [K] = mtrix of functions of independent variable {0} = regression "true" values {«?} = errors The nom*l equations can be solved as follow: [X ] T {Y} = [X] T [XK/3> + [X] T ( e > {b} = [xVWoo Were /b} = paraneter estiBBtes 2 . „. , st«^rd statistical p«*ao, (™- 17) to esti-t. r^r^.ion paraneters . 54 3. Calculate properties of regression parameter distributions to assess nodel precision. In the event that [X T X] is singular, implying that some of the normal equations are linearly dependent, [X T X] _1 does not exist. The nodel should be expressed in terms of fe«sr parameters, or should include assumed restrictions on the parameters. The square of the multiple correlation coefficient, R , is usually calculated for each regression nodel, and supplies a convenient measure of the degree of fit between data values {Y} and values {Y> predicted by the regression equation. It is defined by Sum of Squares due to regression nodel 2 . , - R = Total Sum of squares about mean Y T (Y. - Y)* Z (Y i - Y) : Frequently, it is necessary to determine if inclusion of particular terms in a regression nodel is worthwhile. To this end, the extra portion of the regression sum of squares which arises due to the terms under consideration is calculated. The mean square (defined as the sum of squares divided by the corresponding degrees of freedom) derived from this extra sum of squares can be compared with s*, the estimate of a*, to see if it appears significantly large. If it does, the terms under 35 consideration should be inclined. The statistic is frequently compared to the appropriate percentage point of the F- distribution, %faich is tabulated. Supopose the extra sum of squares due to a parameter, given that a number of other paraneters are already in the nodel, is calculated. Symbolically, SS(b.|b ,b 1 ,...,b._ 1 ,b. +1 ,...,b k ) i = l,2,...,k represents a one degree of freedom ( 1 df ) sum of squares which neasures the portion of the regression sum of squares due to the coefficient b.. This is a measure of the value of adding a ^ term to the nodel which previously did not include 0.. The corresponding nean square, equal to the SS (since it has one df ) can be compared by an F- test to s 2 . This is know* as a partial F- test for the single parameter 0., which is a special case of the F- test described earlier. The stepwise regression procedure (Ref . 18) is a structured way to insert variables in order of correlation until the regression equation is satisfactory. The partial correlation coefficient measures the relative importance of terms not yet in the wodel, to choose the next candidate for entry. The analagous statistic, F- to enter (or F- to renove) is usually evaluated for each predictor at every stage as though it were the last term to enter the model, to determine if terms retained at a previous step have become superfluous, because of soma linear dependence with terms now in the nodel. The largest F- statistic calculated at each step is compared with the appropriate percentage point of the F- distribution, and the predictor variable is entered (or 56 renewed) based on the significance of this F- test. Testing of the least useful predictor is performed at every step. The R* statistic is calculated, to provide a measure of the value of the regression at each step. This stepwise linear regression schene is used in this work because of its conputational economy, and because it allows the analyst to assess the relative influence (or correlation) betwaen individual predictor variables of a selected model and response for a particular data sample. Other sche.es are available (Ref. 18), such as backward elimination. The stepwise procedure is recoimended for its direct nature in testing the nodel with only significant predictor terms. CHAPTER III RESULTS A. Property Histograms and Distributions In this work, fiber and matrix properties are .IImnI to assune a range of val«s to assess the sensitivity of the composite ply properties to constant perturbations. Graphite fiber and epoxy matrix are used as the constants. Initially, t. separate samples of output data are generated and studied to demonstrate the effects of input parameter changes on composite material properties. These two cases are compared with a deterministic base case with no random input property generation. The data for all three cases is given in Table I. The results of cases 2 and 3 are shown in histogram and cumulative distribution form in Figs. 30 - 42. The results of the deterministic case 1 are summarized in Table II, and can be easily compared with the histograms and distributions. 57 TABLE I- INPUT DATA FDR SflfTLING 58 IHPOT THETA (degrees) A* a FVR P a WR X k EFPi(ksi) a EFP2(ksi) a CFP12(ksi) a GFP23(ksi) H a SFPT(ksi) fi a SFFC(ksi) fi a ETF(ksi) A* a SlfT(ksi) fi a SrPC(ksi) fi a SfFSfksi) fi a CASE 1 0.0 0.50 0.01 31000 2000 2000 1000 400 CASE 2 CA8E3 400 500 15 35 13 0.0 0.0 5.0 10.0 0.5 0.5 0.1 0.2 3.0 3.0 3 5 31000 31000 1500 3000 2000 • 2000 100 200 2000 2000 100 200 1000 1000 50 100 400 400 20 10 400 400 20 10 500 500 25 50 15 15 20 10 35 35 20 10 13 13 20 10 59 toctf II- CASE 1 RESULTS PROPERTY CTE12 saocr scxxc SCYYC SCXYS VALUE ECU 1575 ° ksi BC22 1065 ksi EC12 516 ksi HUC12 0.275 NUC21 0.O18 CTEli ®' 775 x 10 " 8 -4 CTE22 0.181 x 10 O.OOO 203 ksi 165 ksi SCYYT ll - 74 ksi 27.41 ksi 10.01 ksi 69 HISTOGRAM FOR ECU LONGITUDINAL MODULUS HISTOGRAM FOR ECU L0N6ITUO1NAL MODULUS 10 r o * 5 a S *h or w 3 - 2 - 1 - tt» rrs . rs 770" RANGE < £ OS) (a) case 2 histogram I H ll III "h — ttt^ rh Hi ".fl iT3 TT5 TI7 .T^ BAN6E < E OS) (b) case 3 histogram DISTRIBUTION OF ECU L0N6ITU0INAL MODULUS DISTRIBUTION OF EC1 2 LONGITUDINAL MOOULUS ■AN6E (c) case 2 distribution • AN6E (d) case 3 distribution Fig. 30- Sampling results for Longitudinal Elastic Modulus 61 HISTOGRAM FOR EC22 TRANSVERSE MODULUS »o r ■ 7 u Z 51- 9 ItJ T a *■ 3 2 a HISTOGRAM FOR EC22 TRANSVERSE MODULUS RANGE < E 07) (a) case 2 histogram RANGE < E ° 7) (b) case 3 histogram DISTRIBUTION OF EC22 TRANSVERSE KOOULUS SO *s ♦0 35 «30 ho 9 °»s 10 DISTRIBUTION OF EC22 TRANSVERSE HOOULUS 9oV? — n TtT— H- RANGE < E 07> (c) case 2 distribution RANGE < E 07> (d) case 3 distribution F1,. 31- Sanpims r-.lt. for Transvers* EU.t1c r*du,us 62 HISTOGRAM FOR EC12 SHEAR HOOULUS < E 0*> (a) case 2 histogram HISTOGRAM FOR EC12 SHEAR MODULUS 10 r u ' • , IT III i i O.oT .A* .08 ."2 Ttt 72 BAN6E < E 07) (b) case 3 histogram USTR1BUT10N OF EC12 SHEAR MODULUS DISTRIBUTION OF EC12 SHEAR MOOULUS SO r 45 40 35 (to 9 »!S 10 A °.to — .d» .At — : RAN6E < E 04) (c) case 2 distribution 7TT— Tic rfr- RANOC < E 07) (d) case 3 distribution F1g. 32- Sampling results for In- plane Shear Modulus 63 HISTOGRAM FOR NUC12 POISSON RATIO- MAJOR .% •& •* RAN6E < E 00) (a) case 2 histogram HISTOGRAM FOR NUC12 POISSON RATIO- MAJOR »o r ■ 7 * RAN6E C E 00) (b) case 3 histogram HSTR1BUT10N OF NUC12 POISSON RATIO- "AJOR DISTRIBUTION OF NUC12 POISSON RATIO- «*JOR RAN6E < E ° 0> (c) case 2 distribution ■o4* — vb — vb — rii RANGE C E 00) (d) case 3 distribution Fig. 33- Sampling results for Polsson Ratio (major) 64 HISTOGRAM FOR NUC21 POISSON RATIO- MINOR HISTOGRAM FOR NUC21 POISSON RATIO- MINOR 10 r t nl If n If '?& '.sW .3V« ,1*75 '.? ' » — Tt RAN6E < E -01) (a) case 2 histogram TTT2 6T* 0.* o.b RANGE ( E -01) (b) case 3 histogram Ho DISTRIBUTION OF NUC21 POISSON RATIO- MINOR DISTRIBUTION OF NUC21 POISSON RATIO- MINOR ■hs — rsVs — rafc — rift RAN6E < c -on (c) case 2 distribution RAN6E < E -01> (d) case 3 distribution Fig. 34- Sampling results for Poisson Ratio (minor) 65 HISTOGRAM FOR CTE11 IONS. THERMAL EXPANSION 10 • 7 ». * u 2 5 at «" 3 2 1 rr -?h P IU ft H rrrr • in. n HISTOGRAM FOR CTE11 LONG. THERMAL EXPANSION io r TT55 — BTffr ( E -06) TJT15 RAN6E (a) case 2 histogram TO • ff.fc -B ' RANGE < E -06) (b) case 3 histogram TTT BTT" T.o 1STR1BUTION OF CTE1 1 LONG. THERMAL EXPANSION DISTRIBUTION OF CTE1 1 LONG. THERMAL EXPANSION RANGE < E -04) (c) case 2 distribution -0.4 -D'.2 RANGE ( E •04) (d) case 3 distribution F1g. 35- Sampling results for Longitudinal Thermal Expansion 66 HISTOGRAM FOR CTE22 TRAN. THERMAL EXPANSION 10 N u . ; IT .17 It ■rh- TT1 RANGE < E -04) (a) case 2 histogram HISTOGRAM FOR CTE22 TRAM. THERMAL EXPANSION »o r J r ill 1 J 4 3 d .it — rr* rrs m ^iu rs; RANGE < E -0*> (b) case 3 histogram HSTR1BUTI0N OF CTE22 TRAN. THERMAL EXPANSION DISTRIBUTION OF CTE22 TRAN. THERMAL EXPANSION 7*B zii RAN6E < C -Q*> (d) case 3 distribution F1g. RAN6E < E -0*> (c) case 2 distribution 36- Sampling results for Transverse Thermal Expansion 67 HISTOGRAM FOR CTE12 CROSS THERMAL EXPANSION 6.0 *.S *.o 3.5 **.0 o *2.5 9 Sa.o a ^1.6 1.0 0.6 0.0. I HISTOGRAM FOR CTE12 CROSS THERMAL EXPANSION JL 10 r TITT 72 DTD DV7 RAN6E < E -05) (a) case 2 histogram — irlfc Got* j. JL RAN8E (b) case 3 histogram BT? BT8" i E -05) -V.i DISTRIBUTION OF CTE12 CROSS THERMAL EXPANSION DISTRIBUTION OF CTE12 CROSS THERMAL EXPANSION RAN6E < t "OS) »*"« « E - 05) . o 1. * -k ♦<„« (d) case 3 distribution (c) case 2 distribution v°; Fig. 37- Sanplinn results for Thermal Expansion Coupling 68 HISTOGRAM FOR SCXXT L0N6. TENSILE STREN6TN 10 ■ 7 - - * u * 5 3 at ■" 3 - 2 t ■ D, 1 1 .1 J i im \ i- RANGE ( E 03) (a) case 2 histogram HISTOGRAM FOR SCXXT LDN6. TENSILE STRENGTH T2J s.o \ ™ ».S 4.0 m m 3.5 ,.3.0 u J2.5 22.0 ac "•I. 5 i n J J 1.0 JU ■ u 0.5 - i i 1— 0.0 #| , .68 .1 2 .6 .*u RANGE < E 03) (b) case 3 histogram DISTRIBUTION OF SCXXT L0N6. TENSILE STRENGTH DISTRIBUTION OF SCXXT L0N6. TENSILE STRENGTH ft TTT5 TT7 RANGE < E 03) 72. * 7T2 .1* 7JB T2* RANGE < t 03) (d) case 3 distribution (c) case 2 distribution Fig. 38- Sampling results for Longitudinal Tensile Strength 69 HISTOGRAM FOR SCXXC LONC. COMPRESS STRENGTH 772 RAN6E < E 03> (a) case 2 histogram HISTOGRAM FOR SCXXC LON6. COHPRESS STRENGTH S.O r ♦ .5 |- 4.0 3.5 „3.0 u *2.S 3 S2.0 at W 1.S 1.0 o.s tU — :h — ri*°-°.6T I —rU — tU — ris RANGE ( E 03> (b) case 3 histogram DISTRIBUTION OF SCXXC L0N6. COHPRESS STRENGTH DISTRIBUTION OF SCXXC LONG. COHPRESS STRENGTH F1g 1 .12 •.»* RANGE < E 03) (c) case 2 distribution 39- Sampling results for Longitudinal Compressive Strength RANGE < t 03) (d) case 3 distribution 70 HISTOGRAM FOR SCYYT TRAN. TENSILE STRENGTH HISTOGRAM FOR SCYYT TRAN. TENSILE STRENCTH to r at Ti ot B2 nnl h -L vrur RAN6E UTTff < E 02> bt¥? — it'ib (a) case 2 histogram .'02 " 6.02 — DTK — OTTT RANGE C E 02> (b) case 3 histogram .1* DISTRIBUTION OF SCYYT TRAN. TENSILE STRENGTH DISTRIBUTION OF SCYYT TRAN. TENSILE STRENGTH iO f0 -J ♦5 - / »5 I 40 40 - J 35 35 / 5! 30 $30 - f 525 525 .j 1 §20 9 °IS ™ §20 9 °I5 J 10 - 10 | s J 1 i • i ii ' 5 B.« -for ■"8.TJ4 0.10 0.14 TTTli *. (U -.02 0.02 0.04 o.io I RANGE (.E 02) RANGE ( E 02) (c) case 2 distribution (d) case 3 distribution n* Fig. 40- Sampling results for Transverse Tensile Strength ?i HISTOGRAM FOR SCYYC IRAN. COMPRESS STRENGTH 20 11 1* 1* o JlO S W ■ Or *■ 4 * 2 0, « HISTOGRAM FOR SCYYC TRAN. COMPRESS STREMSTH io r 7 . * u * 5 srre — DT2T RANGE < E 02> (a) case 2 histogram TJ^T5 — BTVs 3 2 I 1 T7 1 1 I ■oir ■BTT "6T1 RANSE < E 02) (b) case 3 histogram ttU 1STR1BUT10N OF SCYYC TRAM. COMPRESS STREMSTH DISTRIBUTION OF SCYYC TRAN. COMPRESS STRENGTH TJ^?T TJ7T5 0.25 0" RANGE < E 02> RANGE < E 02> (d) case 3 distribution Fig. (c) case 2 distribution 41- Sampling results for Transverse Compressive Strength 72 HISTOGRAM FOR SCXYS IN-PLANE SHEAR STRENGTH RANGE (a) case 2 histogram HISTOGRAM FOR SCXYS IN-PLANE SHEAR STRENGTH -.00 _L TJT55 oTTO" RANGE < E 02) (b) case 3 histogram bTT5 — oT20 DISTRIBUTION OF SCXYS IN-PLANE SHEAR STRENGTH SO ♦5 ♦0 95 Sfso ho 9 W IS 10 S 6T DISTRIBUTION OF SCXYS IN-PLANE SHEAR STRENGTH tw — :*! — rfe — rfr ^o RANGE < C 02) 85 :W B.B5 O.'lfl B.'lS U.'z o RAN6E < e 02) (c) case 2 distribution (d) case 3 distribution F1g. 42- Sampling results for In-plane Shear Strength 73 B. Fiber Strength Effect To show the effect of fiber strength changes on the longitudinal strengths of the conposite, several shape paraneters of the weibull Jistribution for fiber strength are assumed. The nonte carlo procedure is then conducted at several fiber volume ratio values. All properties are varied, except fiber volume ratio. The distribution paraneters of all properties except fiber strengths are held constant. The curves generated are showi in Figs. 43 and 44. In the figures the solid lines and symbols show the means of the 95V. confidence interval estimates for the sanple size of 5© chosen at each point. The points on both sides of each curve locate the upper and lowar bounds of the confidence intervals. The convention described is intended to provide a convenient indication of the dispersion of the sanple values at each point. 74 LONG. TENSILE STRENGTH 65 r CM A a « 20 ?60 D a « 15 UJ ^55 V a = 10 £50 a. u. 5*5 h- >c £f0 - <n ^ =.35 UJ »»J _ — -30 *^F^r -j c 525 e . Z 20 ^ • , 3 .* • 6 .7 FIBER VOLUME RATIO Fig 43- Longitudinal Tensile Strength; for various shape parameters cf fiber strength. 75 LONG. COMPRESS. STRENGTH 38 r FIBER VOLUHE RATIO .7 Fig 44- Longitudinal Compressive Strength; for various 9 shape parameters of fiber strength. 76 C. Matrix Strength Effect The effects of changes in matrix strength on oonposite strengths are studied by suitable variation of the shape paraneters governing the imtrix strength distributions. Analagous to the plots given for fiber strength effects, the matrix effects are showi in Figs. 45 - 47. 77 TRANS. TENSILE STRENGTH 1 10 r .3 £ a ' 20 □ a - 15 y a = 10 FIBER VOLUME RATIO .7 Fig. «- Transverse Tensile Strength; for various shape parameters of matrix strengths. 78 TRANS. COMPR. STRENGTH A a M 20 D a m 15 V a s 10 3 .+ .5 FIBER VOLUME RATIO Fig. 46- Transverse Compressive Strength; for various shape parameters of matrix strengths. 79 IN-PLANE SHEAR STRENGTH .3 .* .5 FIBER VOLUME RATIO .6 .7 Fi' 47- In-plane Shear Strength; for various shape parameters of matrix strengths. 90 D. Fiber Orientation Effect Assuned valtes of the fiber orientation angle distribution paraneter are consecutively used in the «onte carlo procedure to assess the effects on several conposite properties. These plots are sbowi in Figs. 48 - 57. E. Fiber Stiffness Effect Assuned values of the fiber modulus distribution paraneter are used in the simulation to similarly assess the effects on the related conposite properties. The plots thus generated are show* in Figs. 58-67. 81 LONG. ELASTIC MODULUS 95.0 r .3 . <r .5 .6 FIBER VOLUHE RATIO .7 Fig. 48- Longitudinal Elastic Modulus; for various 3 shale oarameters of fiber orientation. 82 TRANS. ELASTIC MODULUS A o - io° D o « 5° v o « r .3 .<*• .5 FIBER VOLUME RATIO .6 .7 Fig. 49- Transverse Elastic Modulus-, for various shape parameters of fiber orientation. 83 IN PLANE SHEAR MODULUS 100 r eg o 90 h 10 A ° ■ 10° D o ■ 5° V o « 1° .3 .* -5 FIBER VOLUME RATIO • 6 Fiq 50- In-plane Shear Modulus; for various 9 ' shane oarameters of fiber orientation. .7 shape parameu 84 LONG. TENSILE STRENGTH 65 r ? 60 UJ £50 o. u. 05 - M-0 - 35 - = 30 25 20 .3 .* -5 FIBER VOLUME RATIO • 6 Fiq 51- Longitudinal Tensile Strength; for va shaoe Daraineters of fiber orientation .7 nous 85 LONG. COMP. STRENGTH 38 r .3 • * .5 .6 FIBER VOLUHE RATIO .7 Fia 52- Longitudinal Compressive Strength; for various M3 - «h,L narameters of fiber orientation. shape parameters 86 TRANS. TENSILE STRENGTH 1 10 c4 A o « io° ?100 Do» 5° liJ Z 90 V c = 1° £ 80 Q. K 4 • K 70 ■^^^^"^^^ ^> >■ * ^^*^^^^&^^^^- • £ 60 ^^^^^^^^ cn ^ ^^^^^^^. * o B0 ID = "tO _J « a 30 - o * 1 1 i 20 • 3 .<f .5 .6 FIBER VOLUME RfcTIO • F-Ig. 53- Transverse Tensile Strength; for various shape parameters of fiber orientation. 87 TRANS. COMP. STRENGTH 1 10 r <u <?100 UJ 90 *^ 80 a. . X to \ 70 o >■ >- o 60 «/> %^ o 50 IU rw *r0 _J « X ec 30 20 £» o = 10° V o - 1° .3 .* ' -5 FIBER VOLUME RATIO • 6 .7 Fig 54- Transverse Compressive Strength; for various Mg - ch,np narameters of fiber orientation. shape parameters 88 IN PLANE SHEAR STRENGTH .3 .* .5 FIBER VOLUME RATIO Fig. 55- In-plane Shear Strength; for various shape parameters of fiber orientation. 89 PDISSON'S RATIO (MAJOR) Fig. .3 .* FIBER VOLUME RATIO 56- Polsson's Ratio (major); for various shape parameters of fiber orientation. 9© POISSON'S RATIO (MINOR) .3 .* -5 FIBER VOLUME RATIO .6 .7 F1g. 57- PoUson's Ratio (minor); for various shape parameters of fiber orientation. 91 LONGo ELASTIC MODULUS 70 r ^ o = los FIBER VOLUME RATIO .7 Fig 58- Longitudinal Elastic Modulus; for various 9 shape parameters of fiber modulus. 92 TRANS. ELASTIC MODULUS FIBER VOLUME RATIO Fig. 59- Transverse Elastic Modulus; for various shape parameters of fiber modulus. 93 IN PLANE SHEAR MODULUS 70 A a « 10% ^ <\J 1 = 65 - D o « 5* bJ .60 - V o - 1« Sr'' £55 ~* O. r^ % £50 \ • ^r^T CM oM-5 s\S UJ ^r ^^ • ^ o*° - UJ = 35 - ^^^1 « £30 If O 1 z 25 • 1 f • _J 3 - A .=> .6 . 7 FIBER VOLUME RATIO F1a 60- In Plane Shear Modulus; for various shape parameters of fiber modulus. 94 POISSON'S RATIO (MAJOR) 68 A o » 10« 66 D a « 5X 6f V o « 1% ^ S62 i • "60 ; >■* • ^^ * ^^^^^ 58 * (M |56 - . * i ^ . • • 5<r , • 52 - • ^ ^% i • 1 50 • 3 .* .5 • 6 FIBER VOLUME RATIO Fig. 61- Polsson's Ratio (major); for various shape parameters of fiber modulus. 95 POISSON'S RATIO (MINOR) . H- .5 FIBER VOLUME RATIO Fig. 62- Poisson's Ratio (minor) for various shape parameters of fiber modulus. .7 96 . LONG. TENSILE STRENGTH 65 r .7 FIBER VOLUME RATIO f<o 63- Longitudinal Tensile Strength; for various Fl9 * sXa?e parameters of fiber modulus. 97 LONG. COMP. STRENGTH 38 r ?36 UJ £32 in ^30 h S28 26 22 20 A a s 10% D s 5* V a X IS .3 .* - 5 FIBER VOLUME RATIO • 6 .7 Fig 64- Longitudinal Compressive Strength; for various shape parameters of fiber modulus. 98 TRANS. TENSILE STRENGTH 1 dU A a « 10% +*> ?1 10 D a * 5X UJ J00 V o ■ 1$ C 90 a. s <: so H- >- ^^^^^^ . £. 70 tn ^ » 60 UJ - 50 _i c « M-0 - o ac i • • 30 • 3 FIBER VOLUME RATIO .6 .7 Fin. 65- Transverse Tensile Strength; for various shape parameters of fiber modulus. 9» TRANS. COMPo STRENGTH 120 A o « 10X ^ CM °110 D o - 5% "100 V o « IX G 90 ■ a. X 1 *C 80 _^^^ o ^^^^^&% >- o 70 <n ^ c 6° ^^^^^ UJ - 50 —i « £ M-0 - o * i i « 30 • 3 .*(- .5 .6 FIBER VOLUME RATIO .7 Fig. 66- Transverse Compressive Strength; for various shape parameters of fiber modulus. 100 IN PLANE SHEAR STRENGTH FIBER VOLUME RATIO F1g. 67- In Plane Shear Strength; for various shape parameters of fiber modulus. .7 101 G. Regression Ifadels The output data of cases 2 thrown 11 are used as successive i«P«" to the regression schene. The goal of stepwise regression, as used here, is to neasure the degree of correlation between a dependent and a set or independent variables for a given -t of data. The outputs of the reoressions conduced show the independent variables accepted into the audel <bas«i on F-«es, criteria) in order of degree of correlation with the dependent liable of interest, along with the final B statistic. (The B 2 values represent t» scuare of the. nultiple correlation coefficient, a convenient measure of the fit hetween data values and values predicted by the regression equation.) The ordering of predictor variables by stepwise regression has several important uses. In this study, the sohe- facilitates easy investigation of the effects of -aterial changes on composite preperties. Si«* the nonte carlo sche- permits generation of large „ou»t, of data, the regression is easy, inexpensive, and can provide insigh t coaming the »nsitivity of dependent -riable, for ,s.»ed distribution, of predictor triable.. « variety of mteri.1 .^figurations and constituent distributions are enamined, and a -del constructed for each dependent (or response, liable. It t be noted that the relative correlations of predictor tables with response cables -ill he functions of the assu«. distributions, the p~ticular ■- _ .™i the functional manner in which the predictor data sanple considered, and the function. -iables are incorporated into the mel. . .Upl. regression -del was assu-d for each response triable. van 102 The first set of "simple" regression models uses as predictor functions only the independent variables as individual terms. To be more precise, the predictor variables used are not simply the independent variable values, for there are 15 of these for each layup. The arithmetic mean of independent variable values is thus used as the predictor variable in the first set of regression models. The only exception to this is the use of the sin* of the average of the fiber orientation angles as the angular dependence predictor, denoted by THETft in the tables to follow. The simpler response variables can be adequately described using the linear function forms in the regression models. The simple variables include the elastic constants, (ECU, BC22, EC12, NUC12, NUC21) and coefficients of thermal expansion (CTE11, CTE22). The results of the regressions performed in the "simple" manner are given in Tables III - XIV. In the tables the input labeled with HI through N5 and Wi through U5 represent narrow and wide distributions of all properties. Input labeled N6 through N10 and U6 through W10 describe the same distributions, except that the composite is assumed unidirectional, i.e. no angular variation. The distinction shows the reduction in predictive capability induced by deviations of the fibers from aligned orientation. The models assumed for the response (output) variables are of the form Y = B + B!X, + BjX, + B3X3 + . • • + B n X n vfcere Y = response variable (ECU, EE22, BC12, etc.) B = regression parameters to be obtained n 103 X = average of independent variable values through the thickness of the ply (TlErft, FVR, WR, etc.) Each nodel postulated contains all independent variables that appear in the equations for the related ply property (see Appendix B). 104 TUB SirPLE HDDH- uptrr FVR TEBIB ACCEPTED JuU * Ml 9.3 FVR.EFP1 N2 0.4 FVR,EFP1,1HEIA K3 0.5 FVR.EFPl.THETA M4 0.6 FVR.EFPl.THETA N5 0.7 FVR.EFPl Ul 0.3 FVR.THETA.EFPl U2 0.4 FVR.EFPl.THETA U3 0.5 FVR.THETA.EFPl W4 0.6 FVR.THETA.EFPl U5 0.7 FVR.EFPl.THETA N6 0.3 FVR.EFPl.HF H7 0.4 FVR.EFPi N8 0.5 FVR.EFPI N9 0.6 FVR.EFPl.EM 1 NiO 0.7 FVR.EFPI U6 0.3 FVR.EFP1.WR W7 0.4 FVR.EFPI UB 0.5 FVR.EFPi U9 0.6 FVR.EFPI wio 0.7 FVR.EFPI R* 83.17 92.63 94.02 94.59 84.00 64.49 89.88 72.85 65.37 57.83 99.83 99.81 99.69 99.74 99.77 99.13 98.40 98.90 99.59 99.34 105 tqct g TU- TRAtB VEBSE 1C0ULPS (EC22) SIMPLE IEDEL INPUT FVR TESTE ACCEPTED FVR,EFT>2 FVR FVR,EFP2 FVR.EFP2 FVR,EFP2,THETA FVR,THETA,EFP2 FVR f THETA,EFP2 FVR,THETA,EFP2 FVR,THETA,EFP2 FVR > THETA J EFT2 FVR,EFF2 FVR f EFP2 FWR.EFP2 FVR.EFP2 FVR.EFP2 FVR.EFP2 FVR,EFP2 FVR.EFP2 FVR.EFF2 FVR.EFP2 Nl 0.3 N2 0.4 K3 0.5 N4 0.6 N5 0.7 Ul 0.3 U2 0.4 U3 0.5 W4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 N10 0.7 U6 0.3 W7 0.4 UB 0.5 W9 0.6 W10 0.7 R 2 83.50 85.23 91.83 93.26 93.06 78.36 90.73 80.15 86.05 87.14 87.13 86.15 90.97 93.47 92.05 79.72 70.71 81.92 88.62 84.05 106 TABLE V- SHEAR MODULUS (EC12) SIMPLE MODEL INPUT FVR TEBM3 ACCfcyiM) HI 0.3 N2 0.4 M3 0.5 N4 0.6 N5 0.7 Ul 0.3 U2 0.4 U3 0.5 W4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 H9 0.6 H10 0.7 W6 0.3 W7 0.4 UB 0.5 W9 0.6 W10 0.7 THETA.FVR.GMP 97.01 THETfl I FVR l GrP,Gn'23 98.85 THETA,FVR f GMP,GFP12 97.50 THETA.FVR.GMP 98.01 THETA.FVR.GMP, GFF12 98.42 THETA.FVR.GMP 91.79 THETA.FVR * 94.27 THETA,FVR,GFP23 93-71 THETA.FVR 95.62 THETA.FVR, GMP.GFP23 96.67 FVR.GMP 97.66 FVR,GMP,GFP12 98.02 FVR,GMP,GFP23 96.65 FVR,GMP,GFP12 97.11 FVR,GMP,GFP12 98.55 FVR.GMP.GFP12 96.93 FVR.GMP.GFP12 92.45 FVR,GMP,GFFi2 95.16 FVR.GMP 97.18 FVR,GMP,GFP12 96.90 107 Tpra f VI- POISSON'S RATIO. MAJOR fNUC12) SIMPLE MDDEL INPUT FVR TERTE ACUfaKm) THETA, EFP1 THETA,FVR THETA, FVR Kl 0.3 H2 0.4 N3 0.5 N4 0.6 N5 0.7 Ui 0.3 U2 0.4 W3 0.5 U4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 Hie 0.7 U6 0.3 W7 0.4 UB 0.5 U9 0.6 uie 0.7 THETA, EFF1 THETA, FVR THETA THETA, WR THETA, FVR FVR FVR FVR FVR FVR FVR FVR.WR FVR FVR,GFF12,EFP2 FVR,EFP2 96.39 97.88 96.60 THETA, FVR,EFP1 98.32 THETA,FVR,En , 2 96.62 88.43 84.62 89.48 84.05 92.05 97.83 98.48 97.77 98.40 99.17 97.32 96.45 96.38 98.34 96.96 108 TABLE VII- POISSOH'S RATIO, MIHOR (HUC21) SIMT-E MODEL INPUT FUR 0.3 TERM3 ACCEPTED Nl THETA, FVR N2 0.4 THETA, FVR, EFP1 N3 0.5 THETA, FVR N4 0.6 THETA, FVR, EFP1 ,EFP2 N5 0.7 THETA,FVR > EFP1 Wl 0.3 THETA, FVR U2 0.4 THETA, FVR, EFP2 W3 0.5 THETA, FVR U4 0.6 THETA U5 0.7 THETA,FVR,EFF1 N6 0.3 FVR,EFP1,EFP2 H7 0.4 FVR,EFP1,EFP2 N8 0.5 FVR,EFP1,EFP2 N9 0.6 FVR,EFP1,EFP2 N10 0.7 FVR,EFP1,EFP2 U6 0.3 FVR,EFP1 I GFP12 W7 0.4 FVR,EFP1,EFP2 W8 0.5 FVR,EFP1,EFP2 W9 0.6 FVR,EFP1,EFP2 uie 0.7 EFP1,FVR,EFP2 R» 91.15 94.78 94.31 97.18 95.87 90.87 89.86 91.93 92.57 94.78 95.64 94.90 95.40 93.12 91.83 87.73 85.06 84.29 90.37 91.42 109 TABLE VIII- LCHG. THERM. EXPANS ION (CTE11) S TITLE KXEL INPUT FVR TERTE ACCEPTED FVR, THETA, EFP1 THETA, FVR.EFPI , WR FVR.THETA.EFPl ,WR FVR, THETA, EFP1.WR THETA, FVR THETA.FVR.EFFl THETA, FVR THETA, FVR THETA, FVR, WR THETA FVR,EFPi,WR FVR,EFPi,WR FVR,EFP1,WR FVR.EFPI FVR,EFPi FVR.EFPI FVR.EFP1 FVR.EFPi FVR,EFPi,WR FVR.EFPi Nl 0.3 N2 0.4 K3 0.5 M4 0.6 H5 0.7 Ul 0.3 U2 0.4 W3 0.5 U4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 NIC 0.7 U6 0.3 W7 0.4 U8 0.5 W9 0.6 W10 0.7 R* 90.29 94.46 95.72 95.23 87.63 80.53 78.91 84.77 74.37 80.50 97.21 96.96 96.53 96.60 96.24 91.60 90.88 91.55 96.03 94.13 lie TABLE IX- TRAMS. THERM. EKPAKSIOH (CTE22) SIICLE MXEL INPUT FVR TER»E ACCEPTED Nl 0.3 H2 0.4 H3 0.5 N4 0.6 M5 0.7 Wl 0.3 U2 0.4 V3 0.5 W4 0.6 U5 0.7 N6 0.3 H7 0.4 N8 0.5 N9 0.6 N10 0.7 US 0.3 M7 0.4 UB 0.5 W9 0.6 U10 0.7 FVR.THETA, WR 99.60 FVR.THETA.WR 99-21 FVR.THETA.WR 99.46 FVR.THETA 99.69 FVR.THETA 99.79 FVR.THETA 95.04 FVR.THETA, EFP1.WR 98.60 FVR.THETA 95.19 FVR.THETA 94.84 FVR.THETA 97.98 FVR.WR.EFP1 99.70 FVR.WR 99.53 FVR.WR 99.65 FVR 99.67 FVR 99.75 FVR.EFP1 99.15 FVR 98.81 FVR 98.88 FVR 99.47 FVR 99.22 Ill •ran F X- LOHG. TOBILE STRENGTH (SOgfll SIMPLE HDDEL INPUT FVR TERTE QCLkVlkl) FVR FVR.SFPT FVR FVR.SFPT.THETfl FVR FVR.SFPT SFPT.FVR EFP1.SFPT FVR.EfP FVR.SFPT FVR.SFPT FVR.SFPT.EFP1 FVR.SFPT FVR.SFPT SFPT.FVR SFPT.FVR FVR FVR.SFPT FVR.SFPT SFPT.FVR Nl 0.3 N2 0.4 K3 0.5 N4 0.6 N5 0.7 yi 0.3 U2 0.4 U3 0.5 m o.e U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 N10 0.7 U6 0.3 U7 0.4 U8 0.5 W9 0.6 WIO 0.7 12.25 43.72 21.68 43.68 40.97 33.37 39.02 26.13 42.27 33.55 52.12 68.43 34.89 49.00 24.00 46.61 19.33 33.13 34.40 37.65 112 Tfiaj: XI- LONG. COMPRESSIVE STREHCTH (SCXXC) SIMPLE MODEL input FVR TERTE flCUsKlkl) Nl 0.3 FVR N2 0.4 FVR N3 0.5 NONE N4 0.6 SFPC M5 0.7 FVR Ul 0.3 WR U2 0.4 THETA U3 0.5 GMP,SMPC W 0.6 THETA U5 0.7 NONE N6 0.3 SFPC N7 0.4 NONE N8 0.5 NONE N9 0.6 NONE N10 0.7 GFP12 U6 0.3 FVR W7 0.4 WR U8 0.5 WR W9 0.6 GFP12 W10 0.7 NONE 12.25 18.23 8.52 8.08 8.02 9.29 20.59 9.18 11.30 12.01 9.40 10.76 9.85 8.87 113 TABLE XII- TRANSVERSE TBSIIE S TRENGTH (SCYYT) SOTLE H3CEL INPUT FVR TER1E ACCEPTED Nl 0.3 FVR N2 0.4 FVR N3 0.5 SIFT N4 0.6 FVR HS o.7 HONE Ul 0.3 FVR.WR.STfT U2 0.4 FVR U3 0.5 FVR W4 0.6 NONE U5 0.7 FVR.SMPT N6 0.3 FVR N7 0.4 FVR N8 0.5 FVR.EFP2 N9 0.6 NONE N10 0.7 NONE W6 0.3 FVR W7 0.4 FVR US 0.5 SMT U9 0.6 FVR U10 0.7 FVR 27.03 32.91 8.10 41.92 26.89 41.43 14.74 31.05 9.43 8.19 15.58 33.87 13.39 8.62 27.85 32.77 114 TP»g XIII- TRflNSVESSE COWRESSIVE STRENGTH (SCYYC) SIPPLE IEOEL INPUT FVR TER1E flCCKFlED FVR.SWC 33 - 17 pyjj 30.10 NONE pyjj 38 . 93 NONE FVR.WR 28.19 Nl 0.3 N2 0.4 N3 0.S N4 0.6 N5 0.7 Ul 0.3 U2 0.4 U3 0.5 U4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 N10 0.7 US 0.3 W7 0.4 W8 0.5 U9 0.6 Ul® 0.7 FVR 43.26 FVR.SWC 19 - 57 NONE pyjj 15.85 NONE NONE NONE NONE NONE pyjj 28 . 68 FVR n ' S * NONE FVR 31.97 pyjj 33.05 115 TPPi F XIV- IN PLANE SHEAR STRENGTH (SOWS) SIIFLE MODEL INPUT FVR TERTE ACCEPTED FVR,THETA,GFP12 28.51 Nl 0.3 N2 0.4 H3 0.5 N4 0.6 NS 0.7 wi 0.3 V2 0.4 W3 0.5 U4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 N10 0.7 U6 0.3 W7 0.4 U8 0.5 U9 0.6 W10 0.7 FVR THETA FVR THETA THETA NONE NONE NONE NONE srps WR 8.74 14.96 THETA,GFP12, FVR, SITS 31.84 NONE THETA.WR.SIPS.FVR 48.16 43.26 8.40 14.75 8.25 8.53 FVR,S*FS,Gtt> 13.Q& NONE NONE GFP12.FVR 22.20 SMPS 17.73 116 Further regression node Is were studied, in an attempt to improve the predictive capability of the models, especially For the strengths. These models, incorporating higher order functions and combinations of predictor variables used in the simple node Is, show some improvement over the sinple models, proving the value of including the "interaction" effects of predictor variables in the regression wodels. In addition, the higher order interaction models can fit response functions over a wider range of fiber volume ratio, with associated improvements in the R 2 statistics. The data cases C0H1 and C0N2 contain selected points from the entire range of fiber volume ratios, to supply the samples for these runs. Furthermore, since higher order models are postulated, THETA is taken to be the cosine of the average of fiber orientation angles. The variable IWR is a "dummy" variable, that is a function of other variables in the model. It is defined as MVR = 1 - FVR - WR and is intended to represent an "average" matrix volume ratio over the thickness of the ply. The interaction models are shown in Tables XV - XXVI. The general form of the postulated models now includes higher order terms, so the predictor variables are tested up to the fourth power. Symbolically, Y = B„ + B, (THETA) + B 2 (FVR) + B 3 (WR) + B„(EFP1) + B 5 (EM») + B,(MVR) ♦ B 7 (THETA) 2 ♦ B, (THETA) (FVR) + B, (THETA) (WR) B l0 (THETA) (EFP1) + ... + B, , (THETA) 2 (FVR) (EFP1) + ... B 12 (THETA) ,, + B 13 (FVR) 2 + ... etc. ♦ 117 The number of tern* possible in a conplete fourth powar polynomial expansion beocmes unwieldy for the cases studied. Considering the limitation of the size of the predictor matrix in the regression package used (100 x 100), the terms are intuitively grouped in the hope of eliminating large groups at one tine. The regressions are conducted using "unlikely" candidates for admission into a particular model, and if no terms are entered, subsequent regressions are conducted without those terms. The justification for this approach is not a statistical argument, rather an interpretation of the physical principles active in any chosen model. The regressions to eliminate terms are merely used as a check on what seems intuitively reasonable. TABLE XV- LONGITUDINAL HDCULOS (ECU) 118 INPUT FVR INTERACTION MODEL TERH5 ACCEPTED Nl 0.3 THETA , '«FVRJtEFPl N2 0.4 THETA ,, *FVR*EFP1 N3 0.5 THETA ,, *FVR*EFP 1 N4 e.6 THETA ,, *FVR*EFP1 N5 0.7 THETA* , *FVR*EFP1 wi 0.3 THETA M *FVR*»EFP 1 U2 0.4 THETA U *FVR*EFF 1 W3 0.5 THETA ,| *FVR*EFT , 1 W4 0.6 THETA^FVRwEFP 1 U5 0.7 THETA a *FVR*EFPl N6 0.3 FVR*EFP1 , EMP 2 «MVR N7 0.4 FVR*EFPl,FVR a N8 0.5 FVR*EFP1 N9 0.6 FVR*EFT»i f EMP 2 *WR,WR g N10 0.7 FVR«EFPi , EMP*MVR U6 0.3 FVR*EFF1,WR W7 0.4 FVR*EFFi , 1»R 2 *FVR UB 0.5 FVR*EFF 1 , MVR 2 *EMP U9 0.6 FVR*EFP1 wie 0.7 FVR^EFPl , EHP*MVR COK1 VARIES THETA ,| *FVR*EFP 1 COH2 VARIES FVR*EFF1,WR' 1 84.50 92.66 93.76 94.24 85.08 63.84 89.86 71.79 64.37 55.68 99.82 99.83 99.72 99.79 99.79 99.17 98.53 98.99 99.58 99.38 96.48 99.92 Tarm XVI- TRANSVERSE ICOULOS (EC22) 119 INPUT FVR Nl 0.3 N2 0.4 H3 0.5 N4 0.6 N5 0.7 Ul 0.3 U2 0.4 U3 0.5 W4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 Ni0 0.7 U6 0.3 W7 0.4 UB 0.5 U9 0.6 wi0 0.7 C0K1 VARIES C0N2 VARIES INTERACTION 1EDEL TERES ACCEPTED FVR*EFP2*EWSEFP2 2 *FVR,THETA*FVR*IWR FVR*EFP2«H'F,EFP2 2 *FVR,THETA FVR*EFP2*EI"F , THETA 2 *EFP2 , WR* FVRwEFP2*ErP , TRETA 2 *EFP2 , BFP2 2 *FVR *** NEARLY SINGULAR THETA 2 *HUR , ETP 2 *EFP2 , 1WR 2 FVRJtEfP , THETA , EFP2 2 *FVR FVR*EFP2*Et-F , THETA , FV^EFP^WR THETA 2 *WR, rNR 2 *EFP2 , EWP*WR *** NEARLY SINGULAR FVR*EFP2*ErF , EFP2 2 *FVR, ETP«WR FVR*EFP2*EfF , ETP 2 FVR FVR*EFP2»EfF , FVR*EFP2«IWR FVR*EFP2*ErF , FVR a «E»* FVR* 1 > EFP2*EMP FVR*EFP2*E«> , EFP2 , EFP2 2 *Etf» , FVR*EFP2 FVR*EFP2*EMP , FVR*EFP2*IWR , FVR*WR FVR*EFP2*ErP , FVR*EFP2*WR FVR«EFP2*E3"P , EFP2 2 *rWR FVR*EFP2*EMP , WR 2 *FVR *w» NEARLY SINGULAR FVR*EFP2*EfF , FVR*EFP2*1WR 99.19 99.55 98.92 99.22 93.26 96.79 93.49 88.35 99.22 99.07 98.89 99.14 99.23 98.62 98.28 97.93 98.44 97.86 99.79 tohit VUTT- IN PLflNE SHEfiR IBDOLUS (EC12) 120 INPUT FVR Ni 0.3 N2 0.4 N3 0.5 N4 0.6 N5 0.7 Ul 0.3 U2 0.4 U3 0.5 W4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 Ni® 0.7 U6 0.3 W7 0.4 UB 0.5 U9 0.6 U10 0.7 CGN1 VARIES CON2 VARIES INTERACTION 1CDEL TERTE ACCEPTED THETA , FVRHGrF ,THETA' , *FVR»««f > *** NEARLY SINGULAR THETA 2 , FVR 2 «GM» , GFP12«GtP THETA, FVR 2 *IWR THETA , FVR 2 «Gtt> , GFP 1 2 THETA" 1 , FVR M «GMP , FVR 2 *** NEARLY SINGULAR THETA''*FVR*l«R,THETA ,4 *WR,WR*«a«» THETA" , FVR 3 *G«P , THETA *** NEARLY SINGULAR FVR**GtP,lfJR 2 *K3FP12 FVR«at , ,GFP12 FVR«GTP , FVR*GFP12 FVRMGnP ,FVR 2 *GFP12 FyR^^GW , FVR«GFP12«G»» , WR**GtP FVRwGMP , FVR*GFP12 FVR*G»> , FVR«GFP12 FVR«GfP , FVRJ«GFP12 FVR*GfF , FVR 2 «GFP12 FVR a «GMP , FVR*GFP12«GMP , JWR 2 «GrF R J 97.86 97.75 98.01 98.46 95.49 91.04 96.70 97.73 97.97 96.52 97.10 98.90 96.91 92.37 95.08 97.42 96.85 FVR 2 «VVR,VVR»WTO f FVRiKIP,THETA ,, «FVR»Kaf s 99.09 FVfi 2 «GHP,VVR*GM > l GFP12«GMP 99.54 TABLE XVIII- LONG. THERHQL EXPANSION fCTEll) 121 INPUT FVR Nl 0.3 N2 ©.4 N3 0.5 N4 0.6 K5 0.7 wi 0.3 U2 0.4 U3 0.5 w 0.6 U5 0.7 N6 0.3 H7 0.4 N8 0.5 N9 0.6 NIO 0.7 W6 0.3 W7 0.4 U8 0.5 W9 0.6 W10 0.7 CONi VARIES COM2 VARIES INTERACTION M3DEL TEBTE ACCfcyiU) THETA 2 *nVR,MWR 2 ,FVR*EFPU*fWR,E« ,2 *EFPl THETA 2 *WR,THETA M ,EFPl ,, ,EK' 2 *FVR,rWR 2 «EFPl THETA 2 *WR f JWR,EW' 2 *WR,EH> 2 *EFPl FVR*EFP1 , THETA»FVR*EFP1 ,EMP 2 *WR THETA,EfF*WR *** NEARLY SINGULAR THETA' l ,r«R 2 *EfP THETA",EfP 2 *lWR, THETA' , ,WR 2 *WR THETA M ,FVR 2 *IWR MVR 2 «EfF , EFPi 2 *EM> , FVR a 11VR 2 «EtP , FVR»»EFP 1*IWR , MVR 2 *WR IWR 2 *E^P,EFP1,FVR ,, MVR 2 «EM? , EFP1 2 *fNR MUR 2 *EMP , EFPi«MVR - MVR 2 »EfP , FVR*EFF1«*WR MVR 2 *ETF , FVR*EFP1*MVR MVR 2 *ECF , EFPi 2 *WR MVR 2 *EfF,EFT»l 2 *lWR MVR 2 *EFP1 ,EHP 2 *HVR THETA , MVR 3 , EFP 1 2 « WR , FVR* WR*EFP 1 MVR 2 *EfP,FVR*EFPl»lWR f FVR M ,FVR 2 «WR. . . 92.51 96.38 97.26 96.32 90.66 80.81 87.98 75.2© 82.97 99.29 99.17 98.94 98.94 99.33 98.35 98.55 98.56 99.00 98.20 96.82 99.84 TQHUE XIX- TOMB. THERMAL EXPANSION (CIE22) 122 inpitt Nl N2 N3 N4 N5 Wi U2 U3 W4 U5 N6 H7 N8 N9 NiO W6 W7 UB 119 W10 COK1 COM2 INTERACTION HDCEL FVR TERMS ACOaMfel) 0.3 THETA 2 *ttVR,WR 0.4 THETA x *M0R,MVR**FVR f FVR*EFPl*EM» 0.5 THCTA 2 *Mra,lWR**FVR,FVR" 0.6 THETA 2 »MVR,MVR 2 «FVR,EfF 2 »VVR,»VR 2 MWR 0.7 FVR 2 ,THETA,THETA 2 *FVR 0.3 THETA 2 MWi,Wm,EFPl 2 *WR 0.4 THETA 2 «ttVR,MyR,FVR*EFPl*MVR 0.5 TOETA 2 «MUR f WR 2 *FVR t EFPl 2 «VVR 0.6 •^^ETA 2 *WR,*^^ETA,•^^ETA ,, 0.7 *** NEARLY SINGULAR 0.3 FVR.WR 3 0.4 FVR.IWR* 1 0.5 FVR,1WR 2 0.6 FVR,EWP 2 *EFP1 0.7 FVR,1WR - 0.3 FVR f FVR*EFPl*EMP 0.4 FVR.MVR" 0.5 FVR 0.6 F^^MyR" 0.7 FVR,EMP 2 *FVR VARIES THETA 2 ««JR VARIES FVR, FVR 3 , MVR 2 *EM > 99. 6® 99.38 99.48 99.73 99.81 95.16 98.71 95.91 95.69 99.70 99.59 99.67 99.70 99.82 99.26 98.97 98.88 99.57 99.29 99.32 99.95 TABLE KK- POISSON RATIO; MAJOR fNUC12) 123 INPUT FVR INTERACTION MODEL TERTB ACCKFTfcl) Nl 0.3 *** NEARLY SINGULAR N2 0.4 THETA, EFP2*HVR N3 0.5 THETA.GFPl 12*MVR N4 0.6 THETA, EFP1*MVR N5 0.7 . THETA ,FVR*EFF2 Wl 0.3 *** NEARLY SINGULAR U2 0.4 THETA , THETA ,I *FVR*GFP1 2 W3 O.S THETA W4 0.6 THETA , WR*GFP 1 2 W5 0.7 THETA, FVR*MVR N6 0.3 FVR N7 0.4 FVR N8 0.5 FVR N9 0.6 FVR.FVRhMVR NIO 0.7 FVR U6 0.3 FVR U7 0.4 FVR,WR*EFP2 UB 0.5 FVR W9 0.6 FVR, EFP1*EFP2 , GFP12*MVR Wl© 0.7 FVR,FVR*EFP2 CON1 VARIES *** SINGULAR CON2 VARIES MVR , FVR* WR , EFP 1 *MVR 97.96 96.71 98.17 96.48 84.73 89.43 84.27 92.10 97.83 98.48 97.77 98.52 99.17 97.32 96.50 96.38 98.41 96.97 99.77 TABLE XXI- POISSON RATIO: IHNOR (NUC21) 124 INPUT FVR Nl 0.3 N2 0.4 N3 0.5 N4 0.6 N5 0.7 yi 0.3 U2 0.4 U3 0.5 W4 0.6 U5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 Nie 0.7 us 0.3 M7 0.4 UB 0.S W9 0.6 W10 0.7 CON1 VARIES CON2 VARIES INTERACTION MODEL TERMS ACCEPTED THETA,THETA ,, *FVR*EFP1 THETA, FVRHEFP 1 THETA,FVR*EFF1 ,EFP2*<FP12 THETA, THETA ,, hFVR*EFP1 , EFP2 THETA,THETA ,, hFVRhEFP1 THETA,FVR*GFP12 THETA, EFP2»MVR THETA, FVRhEFPI THETA THETA , FVRHEFP 1 , THETA ,| hFVRhWVR , EFP2*1WR FVRhEFP 1 , FVR*EFP2 FVRhEFPI , FVRHEFP2 FVRhEFPI ,FVR*EFP2,FVR*GFP12 FVRhEFP 1 , FVRHEFP2 FVRhEFPI , FVRHEFP2 FVRhEFPI ,GFP12*WR FVRHEFP 1 , FVR*EFP2 , FVRHMVR FVRhEFPI, EFP2 FVRhEFPI , FVR»EFP2 FVR»EFP 1 , FVR*EFP2 THETA,FRVH«T12,EFF2 f THETA ,, HFVR*WR, . . FVRHEFP1 , FVRHWtfR, EFP2 , WR»GFPi2 91.69 94.66 95.10 97.15 95.82 91.16 89.52 92.06 92.53 95.60 95.48 94.69 95.52 92.85 91.77 87.83 86.48 84.36 89.84 91.55 98. 7© 98.35 TPrag XXII- LONGITUDINAL TENSILE STRENGTH (SCXXT) 125 INPUT FVR Nl 0.3 K2 0.4 K3 0.5 N4 0.6 N5 0.7 Ul 0.3 U2 0.4 W3 0.5 W4 0.6 W5 0.7 N6 0.3 N7 0.4 N8 0.5 N9 0.6 Nie> 0.7 W6 0.3 W7 0.4 U8 0.5 U9 0.6 W10 0.7 C0H1 VARIES C0N2 VARIES INTERACTION MODEL TERMS ACCfcKim THETA a *FVR*SFPT THETA' 4 *FVR«SFPT f IWR" 1 THETA a *FVR*SFFT THETA ,, *FVR«SFFr , FVR 2 *EMP THETA g *FVR*SFPT FVR*SFPT, FVR*EFP 1*IWR FVR*SFPT,FVR 2 *MVR EFP 1*SFPT, EMP*MtfR EPP 2 *MVR , THETA 2 *SFFT» MVR THETA 2 *FVR*SFFT FVR*SFPT, FVR 2 *EMP FVR*SFPT , FVRwEFT* 1 FVR*SFPT, WR 2 *FVR FVR*SFPT FVR*SFPT, FVR*EMP FVR*SFFT , MVR 2 *EFP 1 FVR*SFPT FVR*SFPT FVR«SFFT FVR*SFPT 17.72 47.65 27.65 44.67 45.35 39.18 42.87 33.97 45.09 32.56 52.95 64.41 39.12 47.13 27.43 49.71 25.19 32.16 34.06 35.09 THETA u «FVRHSFPT f FVR»VVR*IWR,FVR»EFPl*lWR 81.20 FVR*SFPT,FVR*WR,IWR 2 *SFFr 84.79 TABLE XXIII- L0NGITt3DINAL COffRESSIVE STRENglH (SCXXC) 126 input FUR INTERACTION IBDEL TERIS ACCEPTED HI 0.3 SFPChSIFC N2 0.4 FVR«JNR K3 0.5 NONE H4 0.6 SFPOK3T N5 0.7 FVRNGFP12 Ul 0.3 WR" U2 0.4 THETA ,, U3 0.5 EfP*SW»C , THETA 2 *EMP U4 0.6 THETA"* U5 0.7 NONE N6 0.3 SFPC , CFPi2»Srf , C N7 0.4 NONE K8 0.5 NONE N9 0.6 NONE N10 0.7 GFP12*EMP W6 0.3 FVR*1WR U7 0.4 WR UB 0.5 WR W9 0.6 CTP12" W10 0.7 NONE CON1 VARIES FVR*WR CON2 VARIES FVR*WR,SFPC R 2 12.53 19.45 9.81 10.20 10.40 9.32 23.32 9.20 20.04 14.96 11.91 10.76 9.85 9.10 46.48 44.44 127 INTERACTION MDDEL INPUT 0.3 TERTE ACCEPTED Nl HVR^SMT N2 0.4 MUR 2 *STFT N3 0.5 EW^SIFT N4 0.6 FVR 2 *1WR N5 0.7 NONE Ul 0.3 FVR 2 *WR f SreT W2 0.4 SfFMWR • U3 0.5 WR 2 *FVR U4 0.6 FVR*WR*EFF2 , EfP U5 0.7 FVR*S1FT N6 0.3 FVR*1WR N7 0.4 WR 2 *FVR N8 0.5 FVR*EFP2*STFT N9 0.6 STFT 2 *!*** N10 0.7 FVR*EfF U6 0.3 JTO 2 *E«P W7 0.4 WVR 2 *WR U8 0.5 SIFT 2 *WR U9 0.6 FVR 2 *EFP2 W10 0.7 wr 2 «smt C0H1 VARIES THETA* , «SWT*WVR, FVR*EFF2*WR , *S VF\ C0N2 VARIES SIf»T 2 *11WR, FVR*WR*IWR 31.60 37.23 9.61 47.59 25.39 43.94 16.32 24.10 30.29 10.47 8.94 13.54 9.40 9.13 35.13 19.34 12.89 29.27 36.77 73.42 76.40 128 - rows- »W- TRANSVERSE COffRESSIVE STRENGTH (SCYYC) INTERACTION KJCEL INPUT FVR TESTE ACCEPTED Nl 0.3 SlfC*WR N2 0.4 FVR 2 *EM» N3 0.5 NOME N4 0.6 FVR**I»R N5 0.7 NONE Ul 0.3 FVR**WR W2 0.4 FVR*»EMP U3 0.5 SMPC***WR U4 0.6 FVR«WR*EFP2 , EM 1 US 0.7 SMPOIWR N6 0.3 sit>c 2 «wm N7 0.4 EFF2»IWR N8 0.5 FVR*EFP2 N9 0.6 NONE N10 0.7 S»»C 2 *1WR, FVR 2 *1WR U6 0.3 INR^EfF W7 0.4 EFP2*1WR UB 0.5 NONE U9 0.6 MWl 2 «SJf»C Ul® 0.7 IWR - COK1 VARIES THETA a «SM»C»«WR, FVR M C0N2 VARIES WR 11 , FVR 2 *WR, IWR 2 «SB 33.39 32.99 42.31 26.24 43.86 21.13 25.75 18.63 11.57 9.03 9.87 19.07 32.50 14.58 32.85 35.79 76.43 75.59 129 TAHLE XXVI- IN PLANE SHEAR STENGTH (SCXV5) FVR 0.3 INTERACTION MODEL INPOT TERMS ACCEPTED Nl FVR*GFP1 2*G»» , THETA" N2 0.4 FVR*GFP12*Erf> N3 0.5 THETA N4 0.6 THETA a «0 : P12 , SWSJtfNR N5 0.7 NOME Ul 0.3 THETA , FVR* WR*EMP , THETA^SHPS , F U2 0.4 THETA ,, *FVR , THETA ,, «<jFP 1 2 U3 0.5 THETA' 1 m 0.6 THETA, FVR* WR U5 0.7 THETA a *FVR N6 0.3 NONE N7 0.4 SMPS«IWR N8 0.5 NONE N9 0.6 SPPS f Sl*»S a N10 0.7 FVR 2 «IWR U6 0.3 SnPS*tNR,GnP ,, u? 0.4 FVR«a : P12*J1VR U8 0.5 NONE 119 0.6 FVR*GFP12*MVR Wl© 0.7 SHPS C0N1 VARIES THETA^FVR , FVR 2 *SMPS CON2 VARIES FVR*WR, 1WR" , FVR**WR 27.64 13.51 14.97 30.84 52.20 26.58 12.89 22.33 10.72 11.24 16.14 11.40 28.58 8.28 19.20 17.73 36.74 61.46 CHAPTER IV DISCUSS ION A. Overview ■Hie numerical simulations conducted show that certain assumptions about the statistical distribution of local nonuniformities in fiber composites lead directly to quantifiable variations in material properties. The advantages inherent in the stochastic characterization are numerous. The development of quality control and reliabilty measures for composites is crucial to their acceptance in aircraft designs. The reduction in needed experimental data achievable through judicious simulation of the wide variety of available composite material systems could significantly lower the costs of material selection and acceptance testing. In the results of this study, the confidence intervals calculated can be interpreted as the product of an experimental program, specifically designed as an analog of the physical processes which occur in real materials. 130 131 B. Histograms and Distributions Data cases 1, 2, and 3 demonstrate the differences between a deterministic base case and random cases with narrow and wide dispersion of input data about the base case. In Fig. 30, it is apparent that the deterministic case 1 value of 1575© ksi. for longitudinal modulus falls near the mean of the case 2 data. However, the case 3 sample appears to have a mean slightly lower (approximately 15C00 ksi.). It should be noted that the size of the interval over which the sample occurs is noticeably larger in the widely distributed case 3 run. Transverse modulus, (Fig. 31) demonstrates a higher mean value for the wide distribution than for the narrow, which is greater than the deterministic value of 1065 ksi. reported in Table II. The increased transverse modulus is related to the added stiffness available in fibers with high misalignment relative to longitudinal direction. Shear modulus, (Fig. 32) is measurably changed by nonuniformities. The deterministic value of 516 ksi is exceeded by the case 2 value of approximately 62« ksi, which is further exceeded by the case 3 value near 900 ksi. Fiber misaligment has a significant effect in shear modulus variation. Poisson's ratios (Fig. 33, 34) show similar trends in location of sample means and relative dispersion of the sample for the data studied. Poisson's ratios generally increase with fiber misaligment and volume fraction changes. 132 The coefficients of thermal expansion (Figs. 35, 36) for the sample studied reflect the longitudinal contraction of graphite fibers when heated. The longitudinal coefficient of thermal expansion for flS-graphite fiber is -0.550 x 10~V F, while the transverse coefficient is 0.560 x 10~ 5 / F. The offset orientation of crystal lattice planes in graphite fibers can explain this behavior. These values, the fiber misalignnent, and fiber volume ratio near 0.5 all contribute to the occurrence of a negative longitudinal coefficient of thermal expansion for the composite. At higher fiber volume ratios, the values calculated would be less than in the present case, because of the contolling fiber behavior for high fiber volume ratio. The longitudinal strengths (Fig. 38, 39) are significantly reduced when nonuniform! ties are present. The deterministic case 1 value of 203 ksi. for tensile strength is compared to a mean near 160 ksi for case 2 and a mean near 130 for case 3. In compression, the deterministic value of 165 ksi. compares to means near 100 ksi. and 80 ksi. for the narrow and wide distributions, respectively. The failure mode in compression varies in the random samples. Transverse strengths (Fig. 40, 41) show sensitivity to the variations assumed. Misalignments, volume fraction nonuniformities, and constituent strength variations all contibute to reduction in the strength values. Sub-ply shear failures occur, which undermine the already low transverse composite strengths. In plane shear strength (Fig. 42) values decline from 10.01 ksi. for case 1 to a mean near 8.0 ksi. for case 2. However, case 3 shows a 133 value of a mean near 8.© also. It appears that the added shear strength due to fiber misalignment is balanced by the reduced strength due to variable fiber volume fraction. C. Confidence Curves The effects of various shape parameters of fiber strength are shown in Figs. 43 and 44. The higher weibull distribution shape parameter of 20 produces a narrow distribution of fiber strength values. The composite that has few weaker fibers is expected to be .stronger, and Fig. 43 denonstrates this for lonitudinal tensile strength. However, compressive failure (Fig. 44) is a nore complex phenomenon. In the region of low fiber volume ratio, the 'rule of mixtures' failure criteria for a subply can control the failure node. At higher fiber volune ratio, however, compressive failure can be initiated by delamination, or by a shear failure in a sub-ply. The mixture of failure nodes in compressive failure is not well understood, but can explain the seeming inconsistency of the intersection of the curves in Fig. 44. At a fiber volune of 0.7, the weakest fibers (a = 10) are in the strongest composite, when strength is normalized with respect to fiber compressive strength. The effects of various shape parameters for matrix strengths are studied in Figs. 45, 46, and 47. Transverse tensile and compressive strengths show expected reductions for lower matrix strengths. In-plane shear strength shows lower dispersion at a large fiber volume of 0.7, and also declines in general for higher fiber volume. 134 The fiber misalignment effects are studied in Figs. 48-57. Longitudinal modulus (Fig. 48) shows narrow intervals and slight reductions for greater misalignment. Transverse modulus (Fig. 49) and in plane shear modulus (Fig. 50) are enhanced by fiber misalignment. Longitudinal tensile and compressive strengths are degraded by misalignment (Figs. 51, 52). Transverse tensile and compressive strengths are enhanced (Figs. 53, 54). In-plane shear strength shows total separation of confidence intervals between curves with different degrees of misalignment. Poisson's ratios (Figs. 56, 57) increase for high fiber misalignment values. The fiber stiffness effects (Figs. 58-67) are very small for the distribution parameters studied. D. Examination of Regression ftodels The regression models for thernoelastic properties denonstrate resonably high predictive capability in the simple models assumed. Ifarginal improvements are achieved in expanding the models to include higher order interaction terms. Further improvement is gained by using sample data from the wide range of volume percent values. The higher multiple correlation coefficients of these models may be due to the additional information available in the sample size of 1TO that was used. The nearly singular predictor matrices which occur in the higher order models indicate that terms must by selectively removed to eliminate linearity between assumed predictor terms. The regression results support the use of the simple models for thernoelastic 135 properties, because improvements in predictive capability in the higher order node Is for the same data are small. Strengths are not node led well by the sinple or the interaction node Is. The predictors chosen are average properties, whereas the strengths are based on the weakest points in the material. Even the unidirectional cases (N6-N10, W6-W10) present data that the interaction node Is have considerable difficulty in accomodating. Somewhat greater predictive value is gained by using the expanded data for strength model prediction. Using fourth order algebraic functions, values of the multiple correlation coefficient square approach 857. for longitudinal tensile strength. The other strengths generally have poorer results. CHAPTER V CONCLUSIONS A tractable, constituent based, probabilistic analysis procedure for fiber conposites has been developed using the ICAN program as a basis. Within the limitations of the mechanics of material model, properties and strengths of a variety of composite material configurations can be simulated. This study quantifies the thermoelast ic and strength properties of a graphite/epoxy ply subject to assumed uncertainties for fiber misalignment, constituent volume fractions, and constituent properties. The results show several advantages of probabilistic characterization of this material. These include the identification of unforseen variations in composite material properties, and the mechanical effects of local nonuniformities. The relative importance of the various fabrication and material variables on composite properties is identified, and the resulting behavior quantified. The advantages of a probabilistic formulation of composite material 136 137 properties over a deterministic one are numerous. Conparison of the results of this study with test data oould reveal sone souroes of previously unaccounted scatter in the data. Expected value ranges oould be predicted for experimental results. Since the simulations provide data that is analagous to experimental data at lover cost, laboratory classification, material selection, and acceptance testing of composites can be guided by the information made available by these methods. Although the method presented provides results for only the basic ply, extension of the simulation to include lamination angle variations in a general layup is feasible. Since finite element material property cards are generated, structural analysis of components with randomly varied properties defined at a number of points in the body can supply a more realistic description of the random nature of structural response due to material inhomogeneity. The stochastic formulation of material properties is generally recognized as one requirement of failure theories for materials. Although the failure criteria in the models used in this study are conservative, progressive failure of fiber composites could be modeled by incorporating load redistribution and material property recalculation in the vicinity of failed material. REFERENCES 1. Harter, H. Leon: A Survey on the Literature on the Size Effect on Material Strength. AFFDL-TR-77-11, Ik-ight Patterson AFB, April, 1977. 2. Griffith, A. A.: The Phenonena of Rupture and Flow in Solids. Philosophical Transactions of the Royal Society of London A, Vol. 221, pp. 163-198. 3. Murthy, P. L. N. and Chamis, C. C. : Integrated Composites Analyzer (ICAN) User's and Progr aimer ' s Maunual. NASA TP 2515, March 1986. 4. Flaggs, D. L. : ADVLAM- An Advanced Conposite Laminate Analysis Code. Lockheed Missies and Space Company, Inc., 1983. 5. Chamis, C. C. and Sinclair, J. H. : Micromechanics of Intraply Hybrid Composites: Elastic and Thermal Properties. NASA TM 79253. 6. Kural, M. H. and Min, B. K.: The Effects of Matrix Plasticity on the Thermal Deformation of Continuous Fiber Graphite/fetal Composites. J. Coup. Mater., v. 18, Nov. 1984, pp. 519-535. 7. Bolotin, V. V. : Statistical Methods in Structural Mechanics (trans. S. Aroni). San Francisco, Holden Day, Inc. 1969 8. Sobol, I. M. (USSR Academy of Sciences): The Monte Carlo Method. Chicago, The University of Chicago, 1974.' 9. Nance, R. E. and Overstreet, C. : A Bibliography of Random Number Generation. Computing Review, Oct. 1972, pp. 495-508. 10. Lehner, D. H. : Mathematical Methods in Large-Scale Computing Units, Proceedings of the 2nd Symposium on Large-Scale Digital Computing Machinery, Cambridge, Harvard University Press, 1951 pp. 141-146. 11. Haimersly, J. M. and Handscomb, D. C. : Monte Carlo Methods. London, Menthuen I Co. Ltd., 1964, pp. 28-31. 12. Oh, Kong P.: A Monte Carlo Study of the Strength of Unidirectional Fiber-Reinforced Conposite Materials. Journal of Composite Materials, Vol. 13, p. 311. 13. Box, G. E. P., and Mueller, M. E. : A Note on the Generation of Random Normal Deviates. Annals of Mathematical Statistics, Vol. 29(2), 1958, pp. 610-611. 138 139 14. Howell, L. W. and Rheinfurth, M. H. : Generation of Pseudo-Random Numbers. NASA TP 2105, 1982, pp. 7-8. 13. Welbull, W.: Statistical Tneory of Strength of mterlals. (In English) Ingenoirs Vetenskaps Akademien Handlinger, Vol. 151, p. 16. Chamis, C. C: Simplified Composite Micronechanics Equations for Strength, Fracture Toughness, Impact Resistance, and Environmental Effects. NASA Tfl 83696, 1984. 17. Ryan, Thonas A., Jr.: mNITAB. The Pennsylvania State University Computation Center, Jan 15, 1981. 18. Draper, N.R. and Smith, H. : Applied Regression Analysis. New York, John Wiley * Sons, Inc., 1981. 19. Ginty, C. A.: unpublished data. APPENDIX A 140 „ „„,,„ ,»»»»»»»•» «■ » c c :::::::::::::::::::.: ""rT^T^ - C PROBABILISTIC INTEGRATED COMPOSITES AMLYZERC PI C»H> J J £ > rnnPUTERCODE TOR AHALYSI5 OF PROBABILISTIC VARIATIONS IN »"• iaaSF Si f S j s E r RO s E si: P ;? D o^!;iiST5oH S H ?o Y oB?«H C CoSJSsiTE T pHOKllTuI »HS"0E5METR?V WHICH ARE THEH INPUT TO 1CJIN. » C ^O^J^'EUSFSSSk* OF ICJHUHICH ARE NMIW » g :::::::::::...».:.--«. c E . tut* T<t I tUSTER PROGRAM FOR "ICHH" WHICH ALLOCATES C»- DVHM1XCRLW "™riCKNT STOAAOE FOR THE ARRAY VARIABLES — C— IH "ICRH" HMD "PIC AN" CODES. cSn /PS"'/ nAXlEN.NUM) mxLEH • moo CALL SPIHIT STOP END C «M» 8 IHPUT T MT«lET N tO KWIM If PAOAAAILISTIC ANALTSIS IS DESIRED COmON /IUII4/ INHWI.OUTr.INr.IHPr.IHDS.IDBK loSlC*t lSTiT. ) »HOWV.WMTV.riMTV ^ C ^».S8 U HS..UN N O.OUTr DATA PIN/51/ SS IKh:!J!J! fife.™ ir <.MOT. B5TAT) 00 TO JM C KM IPIH.llll) RUNS C SR UP POINTERS TOR WASTER ARRAT lII! - iiil ♦ Nuns us) > u») ♦ Hurts uo • ui) ♦ Nuns us) ■ u*> * Nuns Ut> • us) ♦ NUtlS 1(71 > Lit) * HI L(«l ■ t(7) ♦ ML II • Ml) 11 ■ LI2) LS • US) L« " LIO IS • US) It ■ U«) 17 ■ ID) C WOP*'RUNs' , TI«ES THROUGH DATA CREATION AND ICAN ROUTINE M cUt"uP0»T^!MW?.MLJ).»tL»).MlS).»<U..»(t7).»(L.).Nl.NU f lS) •CUIND IDBK CALL ICAHHN ENOriU OUTf 111 CONTINUE RENIN* OUTP CO TO lilt SI* CONTINUE CALL COPT ^ CALL ICANHN , i!,, &S! i.H:!S!. x .».».u) ins roimm*) ^ »••• continue 5 AETUAN END i- fMTraER ML.MLC.HHS.WT.I».IHPr mean tu.inm.wM* CMMCTKMt ►"";»"« C0W10M /SEtO/ ISEED is» MP'"®*' l «» IH UNirOMI MNMM HOME. OEHERRTOR SEED c remind xsuor S»D(ISEEDr.*> ISEEO C REWIND IMPr HRITEIIHW.?) (OECK(I). !■».») C inura.ll IDEHTUl.ia.NlC.NnS iriNi.n.MM) m to SI WRITE (Wr.MI STOP c si MiXTE fmrr.iii imiTiii.iit.Mic.wB C mO(MH.U) C0«S»T.»M0lEV.nMTV.VOI.m.COMW KlTE "iHPF.in CMM ICftD tPIM.ll> ■ 1Dt URITE (lHPr.U) BIDE REM IPIN.lU MHDV MinuNrr.iiinNM C •5K B l Jp".l"VMT(»).TO.TCU.Pt(72.1).PL(».» Sf <I0tH?(»>.EQ.PW) GO TO II SSS iSS* 1HWICII STOP «• IF CRNOIEVI 00 TO I* DO it IR ■ I.Ht THET»«IR> ■ THtlU •2 COHTINUE CO TO III u , &t Ki"«.THnU.TH5IC.t, THETMH) • » 1«S CONTINUE c ir criMT»» oo to n« DO 111 X* " '• HMS .. wrrcim • vrwu lit CONTINUE 00 TO 1M , „,„ 114 DO its I* • i«hhs 119 CIU UMHDIXI CALL USMMHXll „_„„, u - MTa yi "W.2t: x : JWtS 8 ft" ir a .5. ••»•> co to in .vrrait ■ v 12S continue c it* xr ivoimi «o TO !«• DO 111 X* " I*""* vvr<i«> • *»'"" IM 8W8. wwpam - vvrciM'iM. Itl CONTINUE c "•.ssHiEiSin^ii^i-.w^; 1 i "- mil "- 21 S CONTINUE C «E»0 W»DINO CONDITIONS UHITEHNPr.m iMJJTlViSSii IB I HBSCi.ilO.tlBStJ.lM SttTM»!i , .rBaTi l .wai? i i!i..!i*. , ..*i c c c c c JM CONTINUE REM OUTPUT OPTIONS IEMXPIH.2I) IDENTtil.IOUT URITE UNPr.Jl) IDEMNSl.IOUT INCREMENT »N0 PEEILE SEED rOR FUTURE RUNS ISEED " ISEED ♦ II* REMIND ISEEDr WRITE t ISEEDr. »> ISEEO 1 4 1 • I roantT ux.iiHiDEHTm •.»•> rORIMT II*» , foEuT {"THERE IS » MX UP IN THE UYER PROPERTIES CARD') , FORIUT UI.JIII i« roRtuT m.m> u rORtUT <u) mmAt !! SSSS iK:Sri8J.p 4 i».«*«.r».>».«.».*.M.i« u foriut (M.aM.ux.M«>srt-si 17 roRn»T m.m.ar*. 2. *»«.»?••*' it roRnkT ui.ira.o it FORtUT CM.7FB.O 21 FORIUT (M.II) 21 FORMAT (M.Itl 22 format <«» .. 21 rOMMT (*E1I.J» mmmM 2 J FORMAT C INPUT EBROR. RETURN END NttS MUST BE SET EQUAl TO NL.') U1 SUBROUTINE URRNDIEJ «H*anM«« •• »»»*»»»» DkU Z/l/ INTEGER *. X C0W10H /SEED/ I5EED ir IX .EQ. •» 00 TO I X • • n ■ !"■»• rn • n X ■ ISEED . A ■ >""!• ♦ S I t ■ IWBtMX. MI rx • x z ■ rx/rn RETURN END 51»5Kl!;j55.2S5!J!il!ii55;»i"S"!iliJ C OMirORH MNDOtl VARIABLES Ji.^S^-Bi-oxx"" »•««•»■••« ° REAL M.ltU.SIOItt.Xl.Xl.t ?ViSIc!iAMi"2-uoo«xin-'«.»)-<cos(2»pi»xz))..nu RRURH END •b •o c . Ha R8S88SSSS.nK!HS!8;!&SS """m:^^ C SUBROUTINE TOR OCNCRkTIHO OMf« VRRIATES WITH PARAMETERS C RLMIDk AND K. H|(|M ,„ M , H „o»HMMMH „»*»*»**»»»»*» DIMENSION U(IH) oincNSioH pum> COntlON /SEED/ I3E» DO »• X • I.K _ 31 CAU UMHO(U(I)» pen ■ od) DO It* I • 2>K in pti> • uu> « pcx-n . .. I ■ <-l.l/AtAt1DA» » MM(»CK» RETURN END 00 C THIS ROUTIHE GENERATES THE DESIRED UEIBULL MSTMBUTtD MHDOH C VARIABLE DISCAIPTIOHS C ALPHA ■ SHAPE PARAMETER S K l\ I UHirOWt?*DIST5lSUTED RAHDOH VARIABLE OH 11.11 C 1 • UEIBULI. DISTRIBUTED RANDOM VARIABLE C USt IS HADE Or THE HEIBULL DISTRIBUTION rUHCTIOH C NX) ■ 1 - EXM - <X/BET»> •» ALPHA) TOR X .OE. ZERO c ^IEtI « ( l -»UKJCOHXD) "« Cl/ALPHA) RETURN EHD SUBROUTINE COPT ,***»»»»•»»*•»*•••• • I "^"^m ;»««»« IHFUT DRTR INTO THE FILE TO RE R«D PinCHSION IDEMT(S) C ^!£rL E S:H.?^!.I» S .«HPV.HOHUOr IHTEOER NL.NLC.HNS.INT.IR INTEGER MM. POUT DATA PIN/41/. POUT/J/ RERL TU.TCU.NIS.niS DM* PtT/' P"'' c k>d(pin.4) (decmii.im.m) WRITE (POUT. I) (DECK(I).I-l.tR) c K»D(HH.t» iBtHUli;?lf; H i!t , K!2 nhs WRITE (POUT.H) IOCHTIII.Hl.HLC.MIB C ■£»»( PXM.lt I COHMT HRITC (POUT. ID CWSRT itiotPiM.in cs»h; MUTE (POUT.U) CSKNt RERO (PXN.lt) IIDE WRITE (POUT. 11) MOE RERO (PIN.lt) RXNDV HRITE(P0UT.1S)UNDV RERD(PIM.tt) NONUOr _ WRITE (POUT.ll) NONUOr C HERD tRTER DRTR 11. mO (PIN.l*) IDEKT(t).INP(»).IPnR».TU.TCU.PL(«.IR).THET»(IR). - , ir , (i0EHT(t).NE.PW)00TOH5 CO TO 1*4 m% 1M WRITE «POWT.«) „_._,,, URITE (POUT.t) XOEMT(J) m MRITE (POUT.U) IOEHT(t).INP(III)."«I»>•■"'• tCU • P, • < "•" , • ITHETRUM.PtCT.IR) 5r (Xt.M.NL) 00 TO 1M IR«IR«l CO TO lit C RERD MRTERIRL DRTR Jt C 5 II* IR'« "'S:i:w!S!:K:A!:K»M""" • URITE CPOUT.IM IDENT(4».(CODES(l.J.IR).J«l.2).VrP(IR>.VVP(IRI. ir IM.Efl.HHSI CO TO 121 oo to m C RERD tOkDINO CONDITIONS lit !»■• 1S ' ■rln'lPTN 111 IDEHT(J>.HBS(1.IRJ.HBS(2.IR>.NBS<J.IR>.THCS SSlTEPOut I ) SDWT?J).HBs}ia ? ).NBS^ ■MD (PIN. It ) IDENTI J) .HBS( I . IK) .n«S< 2 . IR ) .I1BS( S . IB ) URITC (POUT.lt) I0ENT(J).nBS(l.IBI.nBS(2.W).t1BS(J.I«) mD»PIN.IB I0ENT(1».(0BS(I.IB).IM ; * WRITE (POUT.M) IDENT(J>.(DBS(I.IR).W.O IP UR.EQ.IOC> 00 TO HI 00 TO 111 141 CONTINUE C RERO OUTPUT OPTIONS C RERD(PIH.tt) IDEMT(J).IOUT URITC (P0UT.2I) IDENT(5).I0UT 2 rORMT (IX,1IHIDEHT(2) •.»•> t rORtUT (2IR4) i fS5S*T ("THERE IS R MX UP IN THE LRTER PROPERTIES CRRDM « format us. mi) 11 rORHRT (ftl.SIl) 12 rORH»T (Li) IS FORtUT (I*) is roRtuT (ki.2ii.sra.)) i» roRn»T (M.2ii.sra.]> It rORWRT (M.2»*.2r«.2.2R«.Jrt.2» 17 rORtUT (RB.2fc4.2rB.2.2k4.jri.2) IB FORtlRT Ul.7ri.4l It rORHRT (Rl.7rt.4l 21 rORIHT (Rl.lt) 21 rORMRT (Rt.IB) 22 rORHRT (15) 2) rORtIRT (4E1I.S) RETURN END U» subroutine vrrconi prp . prs . mr . wis . codes . wis i C "sUiioOTIHETO SOPPU W»M»TIOMS IM CONSTITUENT PROPERTIES g M .S5.555S!5!.!X.I!!5.1l55!.21I.S!:!:ilI.!:i:2^5?2!:ii!l5i INTEGER PIN DRTR PIN/4?/ DMENSioSi ount( iti .pfmn . 1 1 .pps( si . n . pupc u . n .pns( it . i » . C VAM McFpROPMT* MHICH COMESPOMDS TO A BOOLEAN UITM VALUE 'TRUE' C DO 9t J • l.MMS C OENERRTE rilER PROPERTIES C RERBI PIN. 11*1 > BOOL.SHERN.STDEV IM. NOT. ROOt I 60 TO S CRLL URAND(Xl) CALL UR»ND(X2) CILL NOMKXI.Xt.SNERN.STDEV.CrPtn PFPU.J) ■ EPPI1 » RER»(PIN.t«tl> BOOL.SnEM.STDEV XPC.NOT. tOOLI 00 TO « CRLL URANDIXll CRLL SSom?XI.Xt.SnE»N.STDEV.ErPI2) prp(«,J> • Err« 4 READ! PIN. I MM BOOL.StlERN.STDEV in. NOT. ROOD 00 TO 1 CRLL URRND(X1> CALL URRHDIX2) CRLL N0RmXl.X2.SHERN.STDEV.crPI?) rrru.Jt • omt 7 READ! PIN. 1(»1> B00L.3NCAN.STDEV tN. HOT. BOOL) 00 TO • CRLL URRHDIXI) CRLL URRNO(XII CR LL NORM X I . X* . SHERN , STDEV . Or P2 J ) pppir.ji * orris • ■ERDIPIN.IRII) ROOL.BETR.RLPHR irt.NOT. ROOt) CO TO * CRLL URRND(XI) CRLL UEIR(XI.RLPHR.BETR.SrPT) _ prp(i«.J) » srpT „ » RERD<PIM.IIII> ROOL.BETR.RLPHR ir(.NOT. BOOL I CO TO II CRLL URRNO(XI) CRLL HEIRIXI.RLPHR.BETR.SrPO prp(i».J) • srpc c II CONTINUE C GENERATE HATAIX PROPERTIES ° 2* UtD(l>IN,lMI) BOOl.StlEAN.STDEV IM.HOT. BOOL) GO TO 21 CALL UAANDtXl) ^tNoSmil'xi.SnEAN.STDEV.EMNPl PdP(S.J) « ENNP ° 21 REAO(PIN.lMl) BOOL.BETA. ALPHA ir(.HOT. IOOL) GO TO 22 CALL UMHD(Xl) __ . CALL WEIBU1. ALPHA. BETA. SMTP) PHP(*.J) ■ SNTP ° 22 IEAD<PIN.1MI> IOOL. BETA. ALPHA in. HOT. IOOL) GO TO 21 S& Kxil.LPHA.BETA.SnCP, pnp<n.j) - shcp C 2S REAOJPIN.1M1I BOOL. RET A. ALPHA in.HOT. BOOL) GO TO 24 CALL URAHD(Xl) CALL WEIRCX1. ALPHA. BETA.SI13P) ptipcii.J) • snsp 24 CONTINUE REWIND PIN St CONTINUE lilt rORHATCl«X.LI.2E2(.tl) RETURN END C C 1 APPENDIX B 154 155 This appendix outlines the theories and equations in the ICAN program that are used in this project. In the first section on composite micronechanics , the elastic and thermal properties of a composite ply are defined with respect to its principal material axes. The next section, demoted to laminate theory, contains the transformations and summations of ply properties used to arrive at laminate properties. The last section contains a brief discussion of the failure criteria. 1. Composite micromechanics The theory for calculation of the properties of a unidirectional fiber composite ply based on the properties, volume fractions, and orientation of its constituents is known as composite micronechanics . In this section, the subscripts f , m, v, and / represent fiber, matrix, void, and laminate, respectively. The symbolic notation and the equations used are summarized below. Volume fractions: k„ + k ♦ k = 1 £ m v Longitudinal Ifedulus: =711 = Wil + k m E m Transverse Modulus: m E /22 = E /33 i-^i* - vw Shear Moduli: 156 7 12 1 - ^ (1 - G n /G £12 ) 723 i - JT £ (i - Q m n m ) Poisson's Ratios: W /12 = W /13 - W m * M W £12 ~ *J U /23 - k f U f23 + K "712 m E /ii **2 Coefficients of therm 1 expansion /ll n L * m m ill' *11 J a, 711 1 + k (E /E. f , - 1) m* m /ll ' a /22 = a .« x ' *r > 1 + VAu I E /il + k A - E flO 1 + W/ a 33 = a /22 157 2. Laminate Theory This section describes the nethods which are used to calculate the elastic properties of laminates from the properties, orientation, and distribution of individual laminae. The elastic properties are then used to predict the response of the laminate to external loads. The methods used to predict stresses in the laminae under application of external loads are also described. Failure loads can be predicted by using these methods; as described in a following section. a. Generalized Hooke's Law The stresses acting at a point in a solid can be represented by the stresses acting on the planes normal to the coordinate directions, or equivalents, on the surfaces of an infinitesimal cube as shown in Fig. B-l. The stresses (a. .) on each face are resolved into three components: one normal stress and two shearing stresses. The first subscript refers to the direction normal to the plane in which the stress acts and the second subscript to the direction in which the stress acts. The stress components shown on the faces of the cube are taken as positive and can be taken as the forces (per unit area) exerted by the material outside the cube upon the material inside. A stress component is positive if it acts in the positive direction on a positive face of the cube. Thus normal tensile stresses are positive, and normal compressive stresses are negative. Mine stress components must be used to define the state of stress at a point, nam-ly «r n , o^ C33, " 23 . n and a . There are nine corresponding strain *31» a 12' °32' a 13' *^ 21* 1S8 components, following the sane subscript convention. For bodies in which each strain conponent is a linear function of all six stress conponent s, the generalized Hooke's Law can be expressed a ij = E ijkl *kl where E is a fourth order tensor of elastic constants. For nine ijkl stress components and nine strain conponent s, there mist be 81 elastic constants defining E i ^ Certain reductions in the number of independent constants for an anisotropic body are due to symmetry properties of the tensor E. .... By considering nonent equilibrium about the center of the cube, it can be shown that at any point a^ = a^, a 31 = a 13 , and a i2 = a^. Thus, E. jkl is symmetric with respect to the first two indices. Second, because the strains are symmetric (that is, * = e ), E. .. . must be symmetric with respect to the second two ij ji 7 ijkl indices. This reduces the number of elastic constants to 36. Further reduction to the final 21 elastic constants for a general anisotropic material is accomplished by assuming the existence of a strain energy density function, such that with the property J 32 - - •-■ d *ij 1J From the generalized Hooke's Law, -22_ - E 5 de. . ijkl kl Partial differentiation with respect to * fcl yields 159 as. [a* J = E ijki c kl l *"iJ Since the order of partial differentiation is imnaterial, as kl [ m 1 a au df kl and the subscripts can be interchanged to yield a 3s, [as J = ^lij kl l_ "ij so that E ijkl = ^lij Thus the first pair of subscripts in E. jfcl can be interchanged with the second pair without any change in the values. The nunter of elastic constants is thus reduced to 21. b. Lamina Constitutive Relation Several sinplif ications to the generalized Hooke's Law can be node for the special case of a thin orthotropic material, which approximates a unidirectional fiber composite lamina. By considering the invariance of elastic properties under coordinate transf orimt ion for planes of symretry, the tensor E. jfcl can be reduced to the following nine constants: E U11 E 1122 E 1133 E 1122 E 2222 ^233 E 1133 E 2233 E 3333 'ijkl It is now convenient to make the following notation changes: 160 *11 " °l '22 '33 a 23 = T 23 = a 4 a 13 = T 13 = a 5 a 12 - T 12 * °6 e li " e l ! 22" *2 e 33 * e 3 ^23 = Y 23 * U 2e 13 = V 13 " *5 2e l2 " ¥ 12 " *6 The generalized form of Hooke's Law can now be written 6 a . = T C . £ . for i ,J = 1 , . . . , 6 x >1 '" The matrix C. . is known as the stiffness matrix, and s. are the engineering strain components. In matrix form Hooke's Law is witten a l °2 a 3 _ T 23 T 31 T 12 C ll C 12 C 13 * C 12 C 22 C 23 C 13 C 23 C 33 C 44 0- C 55 C 66 '23 '31 r 12 vfcere the coordinate axes coincide with the symmetry axis of the material. For laminae that are assumed sufficiently thin, the through the thickness stresses are zero. Thus ° 3 = a 4 = ^5 = ®» for P lane stress. It is apparent that e^ = fg= The stress strain relations for a thin unidirectional lamina are written r 12' '11 '12 *12 < 2 2 2Q '66* 161 '2 l TV 12 J using the tensorial strain T * 12 instead of the engineering strain Y^. The Q terns are known as reduced stiffnesses, i.e. Ej *11 = C ll - *12 - C 12 ^22 ' C 22 1 - Wi2«21 V 12 E 2 1 - V 12 U 21 E 2 1 - u 12 u 21 u 2 ,E, 1 - wia«2i *66 l = T < C 11 " C 12) = G 12 where E lf E 2 , v l2 , v 21 , and G 12 are the ply elastic constants, neasured with respect to the natural material system. It may be noted that only four of these constants are independent. The stress- strain relation above shows that there is no coupling between tensile and shear strains, as long as the applied stresses are coincident with the principal material directions. However, coupling appears when a lamina is tested at arbitrary angles with respect to the principal material directions. The general form of the stress-strain relation for any angular orientation of a lamina is considered next. c. Stiffness matrix transformations A lamina is loaded along a coordinate system x-y oriented at soma 162 angle * with respect to the principal material directions as shown in Fig. B-2. Since stress and strain are second order tensors, they are transforned by = [T] 12 xy and '2 l 12 - [TJ . T* xy where [T] is the transformation matrix for plane stress and plane strain transformed by clockwise rotation about the (3,z) axes, given by cos 2 * sin 2 * 2 sin© cos* sin 2 * cos 2 * -2 sin* cos* -sin* cos* sin* cos* cos 2 * - sin 2 * [T] = Inversion and substitution yields xy J = [T] _1 [Q][T] y i xy which is the stress strain relation for a lamina referred to arbitrary axes. For simplicity, the notation [ Q ] is introduced [Q] = [T] _1 [Q][T] where [Q] is called the transforned reduced stiffness matrix. Using the approach outlined above, it is possible to obtain 163 expressions for the elastic properties referred to the x-y coordinate system. d. Elastic properties of laminates A number of assumptions are made in laminate theory to obtain theoretical predictions. These are: 1. the lamina are perfectly bonded and do not slip relative to each other 2. the bond between the laminae is inf initesimally thin 3. the laminate has the properties of a thin sheet These assumptions allow the laminate to be treated as a thin elastic plate. The classical hypothesis of Kirchhoff is applied to derive the strain distribution throughout the plate under external forces. Because the laminate is composed of laminae oriented in different directions with respect to each other, the stress-strain equation for each layer (k) is defined as *11 *12 *16 xy *12 Q 22 Q 26 I *16 ?26 *66 l l lT * xy Thus for a given strain distribution, the stress in each layer can be determined. The strain at any point in a laminate undergoing deformation must be related to the displacements and curvatures of its midplane. The discussion which follows assumes that the laminate is thin. Kirchhoff plate theory is used in this formulation. The deformation of an arbitrary section of a laminate is showi in Fig. B-3. It is assuned that lines straight and perpendicular to the 164 midplane before defornation remain so after deformation. This is equivalent to neglecting transverse shearing deformations. Comparing Fig. B-4(b) with Fig. B-4(a), in which the normals to the midplane remain perpendicular after deforrsBticn, it is seen that the upper and lower surfaces of the plate must not shift their relative positions. It is obvious that the resistance of a thin plate to such deformation is large, much larger than its resistance to deformations perpendicular to the midplane. It is assumed that the point B at the midplane undergoes displacements u 0f v„, and w along the x, y, and 2 axes, respectively. The displacement u in the x direction of a point C located on the normal ABCD at a distance z from the midplane is given by u = u - za where a is the slope of the midplane in the x direction, dw The last two equations can be used to obtain the displacement u of an arbitrary point at a distance 2 from the midplane as dw Similarly, dw v - v ° " z "aT Since the strains normal to the midplane are neglected (plane strain) , the displacement w at any point is taken equal to the displacement w at the midplane. The strains in terms of displacement u and v are 165 du du d 2 Wo £ x ~ Ox = 0x z 0x a 3v dv 3 2 w *y " dy = dy~ ~ Z 3y z 3u dv du dv d 2 *to xy dy dx dy 3x axdy In terns of midplane strains and plate curvatures, the strains in a laminate wary linearly through the thickness, £ X f «° 1 X k X £ y = £° y + z k y y 1 xy J y° 1 xy J k 1 xy J where midplane strains are given by x "y xy du ax - dv ay" du ay~ dv ( aiT and the plate curvatures by a 2 u k X ax 2 a 2 w k y = •*■ ay 1 " k . *yJ a*w dxdy The stresses in any (k) lamina can be obtained by substituting the previous equation into the stress strain equation a X a y = 1 xy J k $11 ^12 Q 16 Q 12 °22 Q 26 $16 $26 $66 1 x y xy J + z *y 166 e. Laminate Stiffness ffetrix Classical laminate theory provides a method of oaloulating the resultant forces and moments per unit length acting on the laminate by integrating the stresses acting in each lamina through the thickness (h) of the laminate. Resultant forces are obtained by The moment resultants are obtained by integration through the thickness of the corresponding moments of stresses about the mi dp lane: f ' X J- r ■ xy J ., h/2 azdz h/2 X h/2 a z dz y -h/2 h/2 t z dz h/2 ** The units of N , N , N are force per unit length and n^ H y , n^ are moment per unit length. The sign conventions are showi in Fig. B-5. Using the resultant force and moment relations, a system is defined that is statically equivalent to the laminate stress system, but applied 167 at the midplane. Tnus, the external loading has been reduced to a system that does not contain the laminate thickness or z coordinate explicitly. For a laminate consisting of n laminae (Fig. B-fi). the resultant force-romsnt system acting at the midplane can be obtained by adding integrals representing the contribution of each layer by N X * h/2 a X N y = -h/2 a y N 1 xy T 1 xy ' n r dz = I k=l J Vi xy dz r m l X * h/2 a X n y = -h/2 a y 1 xy ' T 1 xy J z dz = n I k=l a X \ a Vi y T 1 xy J z dz Using the expressions for the stresses in the k-th lamina derived earlier, and noting that the midplane strains and plate curvatures are constant not only within the lamina, but for all laminae, it is apparent that they can be taken outside the integral sign. TTie stiffness matrix [Q] is constant within a lamina so it also can be taken outside the integration to give [ H 1 X n N y = I k=l N 1 xy J , *11 «12 °16 *12 ^22 *26 1^16 Q 26 Q 66 I Vj dz x y xy n 1 k=l *1I *i2 Q 16 «12 *22 ^26 «16 *26 *66 z dz xy 168 11 X n y = 1 M 1 xy J k=l *ii Q 12 ^16 Q 12 Q 22 <? 26 *16 *26 *66 ik I z dz x y xy k=l *il 9 12 ^IG *12 *22 *26 *16 *26 *66 'k T^ 2 „ I z dz J Vi xy Three new matrices, A. ., B , and D ., are defined, where k=l k=i i n k=i These new matrices, A, B, and D, simplify the resultant force and nonent relations, and are know as the extensional, coupling, and bending stiffness matrices, respectively. The total plate constitutive equation is then M M = A B B D k It may be recalled that in an orthotropic lamina with arbitrary orientation the shear stress is coupled with the normal strain and the normal stresses are coupled with the shear strain. In general, a resultant shearing force on a laminated plate produces midplane normal strains in addition to the expected shearing strain. Similarly, a 169 resultant nornal force will induce shear strains in addition to midplane normal strains. The nonzero coupling natrix B in the plate constitutive equation explains the coupling between bending and extension of the laminated plate. Thus, normal and shear forces at the midplane induce not only midplane deformations, (and hence, midplane strains) but also twisting and bending, producing plate curvatures. Similarly, resultant bending and twisting nonents induce midplane strains, f. Lamina stresses and strains The aim of the analysis of a laminated composite is to determine the stresses and strains in each of the laminae forming the laminate. These stresses and strains are used with failure criteria to predict the loads for failure initiation for a laminate. The failure criteria are discussed in the section devoted specifically to that purpose. The strains in a lamina caused by external loading are a function of laminate midplane strains and plate curvatures, as previously discussed. Once the lamina strains are known, lamina stresses can be found using the lamina stress-strain law. Thus, the starting point for calculating lamina stresses is the determination of laminate midplane strains and plate curvatures in terms of the applied loading. The plate constitutive equation given previously can be inverted to give the midplane strains and plate curvatures explicitly in terms of the resultant external forces and nonents. ThB result of the inversion process is 17© f° A' B' N A' B' M k = C D* n = B» D« II where A', B' , and D* are simplified farms of the inversion process results, and are functions of the A, B, and D matrices of the original form of the plate constitutive equation. It is now apparent that with these equations, an analysis of a laminate subjected to external forces and moments can be conducted: 1. calculate midplane strains and plate curvatures k A' B' B« D' N M 2. calculate lamina stresses in global (x-y) system xy <>il Q 12 Q 16 Q 12 Q 22 Q 26 *16 ^26 %G k X s* y 1 xy J + z k 1 X k y k l Xy J J 3. calculate lamina stresses in natural (longitudinal and transverse to fiber) system. - [T] '12 «y The strain variations in a lamina are calculated in an analagous manner. Tne stress-strain variation is compared with the allowable stresses and strains in each lamina. Thus the load at which failure is initiated in one of the lamina can be calculated, as long as a strength criteria exists in terms of the lamina natural axis system. The formulation of lamina failure criteria is discussed in the next section. 171 3. Strength TTieories It is assured that the strength of a laminate must be related to the strengths of the individual laminae. A simple failure criteria consists of evaluating the lamina strengths in their principal material directions subject to induced stresses or strains at the boundaries of the lamina. In this context, it is assumed that the lamina and its constituents behave in a linear elastic manner to failure. The strength analysis described here assumes that the behavior of each lamina in an arbitrary laminate is the sane as the behavior observed in the natural axis system **en the lamina is part of any other laminate under the sane stresses or strains. In other words, it is assured that the strength criteria for a lamina in plane stress is valid for any orientation of the lamina in a laminate. In the ICAN program, the lamina strengths are calculated using the expressions given below. Longitudinal tension s /iit " s rr < k f + WW Longitudinal compression: Tne longitudinal compressive strength mist be computed on the basis of three different criteria: a. rule of mixtures S /iiC = S fC < k f + WW b. de lamination S /11C - < 13 S /12 + S -C> 172 c. fiber microbuckling S r 2°. /11C - 1 - k £ (l - G B /G fl2 ) Transverse tension S /22T - S mT (FACT/DEl«H) Transverse conpression S /22C " S „C ' DBBM Transverse shear [( F i- 1+ V G n 2 ) F 2 G /i2 S ^ c = — FACT where F. and F 2 are given by F -1 * 1 1 " 4 <Ol £ r a .x- y 4k V irk The variable EEHOM is introduced for convenience: EEHOM = [1 - w*J(l - E m /E f22 )] >l 1 ♦ T{T - 1) ♦ */,(* - 1) ; tfiere 9 is given by F. - E HI 1 " E [1 - *,(1 - E^/E^Jj V22 9 = F l-' 173 The variable FACT is used to correlate the strengths of KB and Kevlar fiber conposites with the experimentally observed values. Since neither of these fibers is used in this work, FACT takes the value unity. 174 a 33 t / a 32 / / a 3l/ / °23 °13j ^ >a22 J V a 12 °2lJ Fig. B.l- Components of Stress acting on elemental unit cube. Fig. B.2- Rotation of coordinates from 1-2 to x-y. 175 r Zc 1_J ■♦X •> INITIAL CHOSS-SCCTION »> DEFOaMCD C«OSS-SECTlOM Fig. B.3- Bending geometry in the x-z plane. a. D«A«cf«rf bar witkmvt aSaor b. OtlbcM bar wtffc chaor aV- •Wrenaafiant farwafioiM Fig. B.A-Shearing force deformations on straight cross section. -uojiusAuoo uoi^eiou xapuj ajeuiuiei -9*9 *6ij s^ueainssj juauioui pue ssajjs sjeu -S'g 'Btj 9£T REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports. 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188). Washington, DC 20503. 1. AGENCY USE ONLY (Leave blank) REPORT DATE January 1996 3. REPORT TYPE AND DATES COVERED Final Contractor Report 4. TITLE AND SUBTITLE Probabilistic Fiber Composite Micromechanics 6. AUTHOR(S) Thomas A. Stock 5. FUNDING NUMBERS WU-505-63-5B G-NAG3-550 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSES) Cleveland State University Fenn Tower 1010 1983 E. 24th St. Cleveland, Ohio 44115 8. PERFORMING ORGANIZATION REPORT NUMBER E-10082 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESSES) National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135-3191 10. SPONSORING/MONITORING AGENCY REPORT NUMBER NASA CR-198443 11. SUPPLEMENTARY NOTES This report was submitted as a thesis in partial fulfillment of the requirements for the degree Master of Science in Civil Engineering to Cleveland State University, Cleveland, Ohio 441 15. Project Manager, Christos C. Chamis, Structures Division, NASA Lewis Research Center, organization code 5200, (216) 433-3252. 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified -Unlimited Subject Category 05 This publication is available from the NASA Center for Aerospace Information, (301)621-0390. 12b. DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) Probabilistic composite micromechanics methods are developed that simulate expected uncertainties in unidirectional fiber composite properties. These methods are in the form of computational procedures using Monte Carlo simulation. The variables in which uncertainties are accounted for include constituent and void volume ratios, constituent elastic properties and strengths, and fiber misalignment. A graphite/epoxy unidirectional composite (ply) is studied to demon- strate fiber composite material property variations induced by random changes expected at the material micro level. Regression results are presented to show the relative correlation between predictor and response variables in the study. These computational procedures make possible a formal description of anticipated random processes at the intraply level, and the related effects of these on composite properties. 14. SUBJECT TERMS Computational simulation; Uncertainties; Probabilistic distribution; Monte Carlo; Fiber content; Void content; Misalignment; Graphite fibers; Epoxy matrix; Micro level; Unidirectional: Random processes 15. NUMBER OF PAGES 192 16. PRICE CODE A09 17. SECURITY CLASSIFICATION OF REPORT Unclassified 18. SECURITY CLASSIFICATION OF THIS PAGE Unclassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified 20. LIMITATION OF ABSTRACT NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89} Prescribed by ANSI Sid. Z39-18 298-102 national Aeronautics and bpace Administration Lewis Research Center 2 000 Brookpark Rd Cleveland, OH 44135-3191 Official Business Penalty tor Private use $300 P0S ™ ASTE * «*—«.- Do N o, B . lum