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University Microfilms Internationa! 300 N. Zeeb Road Ann Arbor, Ml 48106 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8402312 Lee, Woo II MICROWAVE CURING OF COM POSITES .' T h e U n iv ers ity o f M i c h i g a n University Microfilms International Ph.D. 1983 300 N. Zeeb Road, Ann Arbor, Ml 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MICROWAVE CURING OF COMPOSITES by Woo II Lee A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in The University of Michigan 1983 Doctoral Committee: Professor Professor Associate Professor Professor George S. Springer, Chairman William P. Graebel Professor Robert B. Keller Andrew F. Nagy Gene E. Smith R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RULES REGARDING THE USE OF MICROFILMED DISSERTATIONS Microfilmed or bound copies of doctoral dissertations submitted to The University of Michigan and made available through University Micro films International or The University of Michigan are open for inspection, but they are to be used only with due regard for the rights of the author. Extensive copying of the dissertation or publication of material in excess of standard copyright limits, whether or not the dissertation has been copy righted, must have been approved by the author as well as by the Dean of the Graduate School. Proper credit must be given to the author if any material from the dissertation is used in subsequent written or published work. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. to Eun Hee, Sun Goo, and Kyung Goo ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNO W LEDGM ENTS I would like to express deepest thanks to Professor George S. Springer course of this for his support and guidance during the research and his help in the preparation of this thesis. I also would like to thank my thesis committee of Professors William P. Graebel, F. Nagy, Robert B. Keller, Andrew and Gene E. Smith for their willingness to serve on my doctoral committee, and thank Professor Valdis V. Liepa in the Department of Electrical and Computer Engineering for his many valuable advice and help in the experiment. In addition, I would like to thank Messrs. and A. R. Allen for their assistance experimental apparatus, typing, in constructing Miss K. Bublitz and Mr. T. Kearns J. Wulster the for her help in for his advice on the experiment. This work was supported by the U. S. Air Force Systems Command, Materials Laboratory, Base, Dayton, Ohio with Dr. engineer. financial support The Wright-Patterson Air Force S. W. Tsai acting as a project received from this project is gratefully acknowledged. Finally, I would like to thank my family and friends for their support and encouragement. iii R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS D E D I C A T I O N ....................................................ii ACKNOWLEDGMENTS ............................................ iii LIST OF F I G U R E S .............................................. vi LIST OF T A B L E S .............................................. xi LIST OF A P P E N D I C E S ........................................ xii N O M E N C L A T U R E ............................................... xi i i SECTION I. INTRODUCTION 11 . MODEL .......................................... 1 • . ................................................ 3 2.1 Problem Statement .............................. 2.2 Electromagnetic Model 2.2.1 .......................... 3 5 Electric Field, Linearly Polarized TEM W a v e .................................... 6 2.2.2 Reflectance and Transmittance . . . . 2.2.3 Overall Reflection and Transmission Coefficients ............................ 17 19 2.2.4 Absorbed E n e r g y ........................... 22 2.2.5 Incident Isotropic Electromagnetic Wave ......................................25 2.3 Thermochemical Model ......................... 28 2.4 Resin Flow M o d e l ................................. 35 2.5 Input and Output Parameters 2.5.1 Composite Properties III. ................. 35 ................... 40 N U M E R I C A L ........................................... 44 3.1 Numerical Solution, Electromagnetic M o d e l ............................................ 44 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Numerical Solution, Thermochemical M o d e l .................................... 4 7 3.2.1 G r i d ................................ 47 3.2.2 Finite Difference Equations ......... 3.3 Numerical Solution, Resin Flow Model 3.4 Material Properties 50 . . . . 54 ......................... 54 3.5 Computer C o d e ........................... 54 IV. E X P E R I M E N T A L ...................................58 4.1 W a v e g u i d e ................................ 58 4.2 Microwave O v e n ........................... 61 4.3 Measurement of Electromagnetic P r o p e r t i e s ................................ 64 4.3.1 Dielectric Constants .................. 4.3.2 Measurement of the Reflectance . . . . 4.3.3 Measurement of the Transmittance 4.4 Microwave Curing V. .............................. EXPERIMENTAL VALIDATION OF THE MODELS 5.1 Electromagnetic Model NUMERICAL RESULTS 69 70 70 .......... 73 ....................... 73 5.2 Thermochemical and Resin Flow Models VI. ... 65 . . . . ..................... 74 85 6.1 Electromagnetic Wave-Composite Material Interactions .................................. 85 6.2 Microwave Curing-General Considerations 94 . . 6.3 Microwave Curing-Selection of Cure Cycles . . . . . .............................. 96 .VII. SUMMARY AND C O N C L U S I O N S .....................116 A P P E N D I C E S ................................................. 119 R E F E R E N C E S ................................................. 145 v R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF F IG U R E S Figure 1. Geometry of the Problem, Polarized TEM Wave. Incident Linearly 2. Designations of the Plies and the Interfaces. 3. Illustration of the Incident, Reflected, and Transmitted Electric Field Vectors at an Interface. Reflected Portion of the Right Traveling wave e T . (E I)r is the Reflected Portion of the Left Traveling Wave e T. 4. Illustration of the Off-axis axis (p-q) Coordinates. (1-2) and On'1 5. Illustration of the Electric Field Vectors in the m-th Ply (Linearly Polarized TEM W a v e ). 6. Illustration of the Electric Field Vectors at the Front (m=1) and Back (m=M+1) Interfaces (Linearly Polarized TEM Wave). 7. Illustration of the Overall Incident, Reflected, and Transmitted Energies (Linearly Polarized TEM Wave). 8. 18 Illustration of the Electric Field Vectors in the m-th Ply (Linearly Polarized TEM W a v e ). 24 Illustration of the Incident Isotropic Wave and the "Equivalent" Linearly Polarized TEM Waves. 29 10. Schematic of a Typical Cure Assembly. 11. Description of the Cure Assembly Used Thermochemical Model. 12. Arrangement of the Grid Points. 48 13. Illustration of the Control Volume about the 8-th Grid Point. 49 Schematic of the Waveguide Set-up. 60 14. in the vi with permission of the copyright owner. Further reproduction prohibited without permission. 31 15. 15. 17. 18. 19. 20. 21. Schematic of the Press Used during Microwave Curing. Illustration of the Waveguide Arrangement during the Measurements of the Dielectric Constants. Reflectances of Fiberite S2/9134B Glass Epoxy Uncured Unidirectional Composite as Functions of the Number of Plies. Comparisons between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization A n g l e , 6). 63 66 75 Reflectances and Transmittances of Hercules AS/3501-6 Graphite Epoxy Uncured Unidirectional Composite as Functions of the Number of Plies. Comparisons between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, o). 76 Reflectance of Fiberite S2/9134B Glass Epoxy Uncured Cross-ply Composite as a Function of the Number of Plies. Comparison between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, 6). 77 Reflectance as a Function of Polarization Angle 6 for Hercules AS/3501-6 Graphite Epoxy Uncured Single Ply Composite. Comparison between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves. 78 The Change in Reflectance with the Orientation of the Second Ply, 0, for a Twoply Hercules AS/3501-6 Graphite Epoxy Uncured Composite. Comparison between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, 6). 79 vi i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22. 23. 24. Temperature as Function of Time during Microwave Curing of 32-ply Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6 Graphite Epoxy Composites. Comparisons between the Data and Models. Cure Assembly Shown in Figures 14 and 23. Power Inputs to the Composites were as Indicated. 3! Mass Losses Normal (top) and Parallel (center) to the Tool Plate, and the Total Mass Loss (bottom) as Functions of Time during Microwave Curing of 32-ply Hercules AS/3501-6 Unidirectional Composites. Comparisons between the Data and Results Computed by the Models. Cure Assembly is Shown in Figures 14 and 23. The- Power Input W in' Cure Pressure, P q , and Bleeder Pressure, P., are as indicated. The Initial Resin Content was 42%. 82 Components of the Cure Assembly Used in Modelling the Temperature Distribution and the Resin Flow during Microwave Cure. Complete Cure Assembly is Shown in Figure 15. 84 25. 26. 27. The Variation in Reflectance and Transmittance with Polarization Angle, 5, for Single Plies of Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6 Graphite Epoxy Composites Exposed to Linearly Polarized TEM Waves. Results of the Model. Material Properties Listed in Appendix G. 87 Reflectances of a Two-ply Hercules AS/3501-6 Graphite Epoxy Composite Exposed to a Linearly Polarized TEM Wave (Polarization Angle, 6 = 90°) and to an Isotropic Wave. Results of the Model. Material Properties Listed in Appendix G. 89 Reflectances and Transmittances of Unidirectional, Cross-ply and Q u a s i isotropic Fiberite S2/9134B Glass Epoxy Composites Exposed to Linearly Polarized TEM Waves (Polarization Angle 0° ^ 6 ^ 90°) and to Isotropic Waves. Results of the Model. Material Properties Listed in Appendix G. 90 vi i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28. 29. 30. 31. 32. 33. 34. 35. Reflectances and Transmittances of Hercules Unidirectional, Cross-ply and Q u a s i isotropic AS/3501-6 Graphite Epoxy Composites Exposed to Linearly Polarized TEM ‘ Waves and to Isotropic Waves. Results of the Model. Material Properties Listed in Appendix G. 91 The Absorbed Energies in the m-th Ply of Fiberite S2/9134B Glass Epoxy and Hercules A S/3501-6 Graphite Epoxy Unidirectional Composites Exposed to Linearly Polarized TEM Waves. Results of the Model. Material Properties Listed in Appendix G. 93 Reflectances of Fully Cured 64 ply Fiberite S2/9134B Glass Epoxy (d=0.96cm) and Hercules AS/3501-6 Graphite Epoxy (d=0.77cm) Unidirectional Composites Coated with a Homogeneous Isotropic Material on the Front (Left), Back (Middle), and Both Front and Back (Right). Linearly Polarized TEM Wave Incident on the Front Surface. Results of the Model. Properties of the Composites Given in Appendix G. The Dielectric Constant of the Coating is e'. The Dissipation Factor of the Coating is e'^ = 0 . 95 Illustration of the Cure Cycle Used in the Parametric Study of Microwave Curing. 99 Manufacturer's Recommended Cure Cycle Hercules AS/3501-6 Prepreg [27]. for 101 Temperature Distribution Across the Composite as a Function of Time for Different Levels of Microwave Power Input. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 102 Temperature Distribution Across the Composite as a Function of Time for Different Power Cycles. Results Obtained by the Models. ^2 Temperature Distribution Across the Composite as a Function of Time for a Composite Thermally Insulated (Left) and for a Composite without Thermal Insulation (Right). Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 105 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36. Viscosity Distribution at Different Times Inside the Composite. Results obtained by the Models for the Cure Cycle Shown in Figure 31. 37. The Maximum Viscosity umax Inside the Composite as a Function or Time. Gel is Assumed to Occur When Viscosity Reaches 7 Pa»s. Results Obtained by the Model for the Cure Cycle Shown in Figure 31 at Power Inputs of 100 W and 200 W. 38. Gel Time as a Function of Power Input. Result of the Models .for the Cure Cycle Shown in Figure 31. 39. Number of Compacted Plies as a Function of Time for Different Microwave Power Inputs and Different Cure Pressures. The Results Obtained by the Models for the Cure Cycle Shown in Figure 31. The Result Shown for the Autoclave Cure Cycle is from Reference [13]. 40. Degree of Cure Distribution Across the Composite as a Function of Time. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 41. Minimum Degree of Cure as a Function of Time for Two Different Power Inputs. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 42. The Time Required to Reach the Gel point Using Microwave Curing with 100 W and 200 W Power Inputs and Using the Autoclave Cure Cycle Rerommended by the Prepreg Manufacturer. The Results for Microwave Curing were Obtained by the Models for the Cure Cycle Shown in Figure 31. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table 1. Input Parameters Required for the Electromagnetic Model. 2. Input Parameters Required for the Thermochemical Model. 3. Input Parameters Required for the Resin Flow Model. 4. Output Parameters Given by the Models. 5. The Reflectance, ^ r a n s m i t t a n c e , and Total Absorbed Energy < Calculated by the Closed Form Solutions (Eqs. 96-106) , and by the Computer Code xi R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF A P P E N D IC E S Appendix A. The Reflection Coefficient at In t e r f a c e . the m-th 120 B. The Attenuation Tensor for the m-th Ply. 123 C. Rate of Energy Absorption per Unit Volume by the m-th Ply. 126 D. Normal Incident Energy for an Isotropic Wave. 128 E. F. G. H. Relationship between the Platen Force and Air Bag Pressure. 131 Electric Field Strength Distribution Inside the Microwave Oven. 134 Properties of the Composites and Surrounding Materials. Reflectance for the Front Face of a Single Ply Composite. xi i R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 142 NOMENCLATURE A^j attenuation tensor ap amplitude of the incoming wave, V/m Bm constant defined in Eq. C (80) c specific heat of the composite, kJ/(kg*K) Q speed of light (=3.0x10 m/s) d total thickness of the composite, m dm thickness of m-th ply of the composite, E^ electric E? resultant electric field vector, m V/m field vector, V/m f frequency, Hz H amount of heat evolved by the chemical reactions, kJ/kg Hr ultimate heat of reaction during cure, H rate of heat generation by the chemical reactions, kJ/(kg*s) h^ thickness of the i-th layer, m h total thickness of the I total number of grid points j , K m = /“ thermal conductivity perpendicular to the plane of the composite, W/(m*K) M total number of plies Mf fiber mass, kg M re resin mass, kg m cure assembly, kJ/kg $ measured mass loss of the composite, Nj| j tensor defined by Eq. ng number of compacted plies percent (A.7) XI 1 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. pressure in the bleeder, kPa P1 electric V/m Pg applied cure pressure, R reflectance R reflectance at the front composite r reflection coefficient r^j reflection coefficient tensor $ r^j overall reflection coefficient T temperature, T ^ n ^t initial temperature, Tr transmittance t transmission coefficient t^j overall transmission coefficient tensor U^j unit matrix u number of layers below the VSWR voltage standing wave ratio v number of layers above the W^n microwave power w width of the waveguide, AX* parameter defined by Eq. x vertical coordinate normal to the tool Ax grid spacing x 1,X2 ,Xg field vector defined by Eq. (8) and (9), kPa face of a single ply tensor C C composite (Figure 11) composite (Figure 11) input, W m (109) coordinates defined in Figure Z impedance, z parameter defined by Eq. plate 1 ohm (110) xiv R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. GREEK SYMBOLS a degree of cure a •„ min minimum degree of cure a specified- degree of cure y propagation constant as defined by Eq. 6 polarization angle of linearly polarized T EM wave, degree ^ (4) — £ energy absorbed rate per unit volume, <5^nc incident energy flux, W/m^ £ absorbed energy per unit area per unit time the m-th ply, W/m m 3 W/m in r e f l e c t e d e n e r g y oe r u ni t a r e a D er u n i t t im e , W/m /_ total absorbed energy per unit area per time, W/m <; . c transmitted energy per unit area per unit time, w/m e complex dielectric constant e' dielectric constant e" dissipation n a tensor defined by Eq. 0 angle of ply orientation, X wavelength, m Xq wavelength in free space, m y resin viscosity, Umax maximum viscosity unit factor (39) degree Pa*s inside the composite, Pa*s complex permeability Vf fiber volume fraction of the composite £ a tensor defined by Eq. (38) xv R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. p density, 0 3 electrical conductivity of the composite, time, t t T kg/m g s cure time, c e i co mho/m gel time, s s angular velocity, rad/s SUBSCRIPTS A null position with the sample, B null position without the sample, f fiber 1 components of inc incident electromagnetic wave j components of a vector in 1, k components of a vector L lower surface of the cure assembly 1 layers of cure assembly m m-th ply or interface p direction parallel to the fibers q direction perpendicular to the fibers r reflected electromagnetic waves re resin T total t transmitted electromagnetic waves U upper wg waveguide 0 free space surface a vector m m in 1, 2, and 3 directions 2 and 3directions in 1, 2 ,and 3 directions of the cure assembly xvi R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1,2,3 1,2,3-directions (Figure g g-th grid or grid spacing 6 polarization angle 6 1) SUPERSCRIPTS c cured composite i isotropic wave q time step u uncured composite + electromagnetic wave travelling right (Figure 5) electromagnetic wave travelling left (Figure 5) from left to from right to xvi i R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S E C T IO N I INTRODUCTION Parts and structures constructed from fiber reinforced matrix composites are manufactured by arranging the uncured fiber-resin mixture into the desired shape and then curing the material at elevated temperatures and pressures. Generally, part the curing process in an autoclave, is accomplished by placing the with temperature and pressure inside the autoclave being maintained at prescribed levels. Autoclave curing parts made of thin, is suitable for individually cured uniform laminates. less suitable when the parts are large, uneven dimensions, simultaneously. gradients Autoclave curing thick, or have or when several different parts are cured In these cases, the unavoidable temperature inside the autoclave and the thermal the autoclave make is it difficult inertia of to ensure that the parts are cured uniformly and completely. Microwave curing offers the possibility of uniform, complete, the part. and economical cure regardless of the geometry of In order to utilize the microwave curing, full potential of the incident electromagnetic radiation and the a pplied pressure must be related to the thermal, chemical, and physical processes occurring in the composite 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 during cure. However, to date, microwave curing has only been studied by experimental methods these empirical studies do not shed [1-7], The results of light on the importance of the various parameters which affect the cure. The shortcoming of the empirical approach could be overcome by use of analytical models; unfortunately, no analytical model exists that could describe the behavior of composite materials during microwave curing. Therefore, the first and major objective of this study was to develop models which characterize the response of continuous reinforced thermosetting matrix composites and relates fiber to electromagnetic radiation, electromagnetic radiation to the relevant cure process variables such as temperature, viscosity, resin content, the incident degree of cure, resin and cure time. The second objective was to generate can be used to validate the models. test data which The third and final objective was to determine the response of composite materials to electromagnetic waves and to demonstrate usefulness, as well as the limitations, The emphasis the of microwave cure. in this investigation was on microwave curing of composites. Therefore, experimental results presented are the analytical and for linearly polarized, and for isotropic electromagnetic waves at the microwave frequency of 2.45 GHz. Nevertheless, the models and the numerical procedures developed here are general, and are applicable over the entire frequency spectrum. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S E C T IO N I I MODEL 2.1 Problem Statement Consider a composite laminate constructed of M plies, each ply consisting of unidirectional organic matrix. x 1 axis fibers embedded The fiber orientation with respect to the is denoted by the angle 6 (Figure 1). The composite is exposed on one side to a known electromagnetic The incoming electromagnetic plane field. field may be constrained to a (linearly polarized transverse electromagnetic TEM wave) or may be isotropic. polarized TEM wave, the direction field and the x 1 axis polarization angle) wave, In case of a linearly perpendicular to the composite. electric in an of the propagation must be The angle between the (referred to as the is denoted by the symbol 6. It is desired fo find the following parameters: a) the ratio of the reflected to the incident energy (r e f l e c t a n c e ) ; b) the ratio of the transmitted to the incident energy (transmittance); c ) the total absorbed energy; d) the absorbed energy as a function of position; e ) the temperature T as a function of time and 3 with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COMPOSITE FIBERS INCIDENT LINEARLY POLARIZED TEM WAVE Figure 1 Geometry of the Problem, Incident Linearly Polarized TEM Wave. 5 position; and f) the degree of cure of the resin as a function of time and position. In addition, for incoming linearly polarized TEM waves, is desired to find the overall it reflection a n d transmission coefficients. A model suitable for calculating is developed below in two parts. model, referred overall to first part of the as the electromagnetic model, reflection coefficient, transmission coefficient, absorbed energy. The the above parameters the reflectance, the transmittance, The second part, yields the the overall and the referred to as a thermochemical model, gives the temperature and the degree of cure. In addition, it is discussed how the information generated by these models can be used to calculate the viscosity and the resin flow. In order to emphasize the concepts and the solution methods, the model is presented for a composite in which the properties vary only across the thickness problem, (one-dimensional flat plate geometry). 2.2 Electromagnetic Model The electromagnetic model is developed linearly polarized TEM waves and for separately for isotropic waves. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 2.2.1 Electric Field, Linearly Polarized TEM Waves. A linearly polarized T EM wave of known amplitude ag and known wavelength X impinges on a composite material. The direction of propagation is perpendicular to the composite, i.e., the wave propagates in the x^ direction The x 1 a nd x2 components of the vector this electric field can be expressed as (£,) [8] e*P (jM,r ~ yXi) where ag is the amplitude, 1). representing = Q 0 cos<£ exp C j w T - 7X3) (E*) = CU the time, x^ (Figure (1) <2> 5 is the polarization angle, t is is the coordinate perpendicular to the plane of the composite with the origin the laminate, and j (Xg=0) at the front surface of has the common meaning, j=/-T. The parameter u is the angular velocity 60 and y = zirc/ Z (3) is the propagation constant c is the speed of light, e and u are the complex dielectric constant and the complex permeability of the medium through which the wave propagates. materials, ^ * s=1. For non-magnetic The parameter e* may be expressed as R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 where e' is dielectric constant and e" is the dissipation fa ctor; It is desired to find the electric inside of the material. field distribution A procedure suitable for performing the calculations is described below. To analyze the problem, we focus our attention on the m-th ply of the laminate (Figure 2). Electromagnetic waves.arrive at each interface from the right and from the left directions (Figure 3). Right and left traveling waves are represented by the vectors, e T, respectively. The subscript E^ and i has the values of i=1 or 2. At the interface, a portion of the arriving wave transmitted through the interface is reflected and a portion of it T he reflected portion of the right traveling wave is denoted by (E { ) r / and the reflected portion of the left traveling wave tangential is components [9]. (ET)r . The of the vectors representing the electromagnetic waves entering must be equal is denoted by and leaving This condition theinterface requires that the following equalities be satisfied at the interface E f = (El)± - ( E J r <6> E c = ( £ - ) t - ( E L )r n) We define now the following two parameters R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. B A C K 8 2 • tn <D 0 (0 4-t M <D +J C H 0) X • -P • nd c (0 • + tn Q) •H f-H Cu E E d) XX +> • • 0 • tn c m 0 •H +j (0 c CP •H tn <u Q CM OJ >_l CL — H 2 O UJ o g h2 q: Q) P 3 Cn •H u. LU cc 9 IxJ Ll I I— 2 o > 2: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTERFACE Ef (Et), pf Pf < (ET)r E7 (E D . IJ Figure 3 Illustration of the Incident, Reflected, and Transmitted Electric Field Vectors at an Interface. (Ei)r is the reflected portion of the right travelling v/ave . (Ei)r is the reflected portion of the left travelling w a v e E f . vo P: = (t;;r T The reflected and incident electric field vectors are related by the expressions ( E ^ = (A jK E -) <1°> CEL)r - - (rcj)(bj ) 'where r^j (1 1 ) is the reflection coefficient tensor at the interface. The subscripts components (i,j=1,2). i and j represent the Here, and in all subsequent analyses, the Einstein summation convention subscripts is used, i.e., repeated imply summations. For the interface between the m and m-1 ply the reflection coefficient tensor r^j H j where N^j and is (Figure 3), (Appendix A) - d 2) is defined as CoS0 - S i n 6 \ , / i - f j '■5iVi© coS&A O \/fc S 0 ( ] 3) l°S&) o t The subscripts p and q refer to the directions parallel and perpendicular to the fibers, p and q and and 0 is the angle between the 1 and 2 coordinate axes By substituting Eqs. (8)-( 11) (Figure 4). into Eqs. (6) and (7), we obtain R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 q Figure 4 Illustration of the Off-axis On-axis (p-q) Coordinates. (1-2 ) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 P-+ = Pr = U^j ( U;j + r y ) ( E j ) - ( Tcj) ( E j ) (P 5)(eJ) + (14) (Ucj -r£j- ) CEJ) os) represents the unit matrix. The electric field traveling through the material attenuated. Thus, the electric is field arriving at an interface is related to the electric field vector leaving the adjacent interface by the expressions (see Figure 5 and Appendix B) (E;+)m = (Aij)„<Pi)m (16> (Ei)m 1 A^. is the attenuation coefficient tensor in the m-th ply (Appendix B ) . , a > s e - s i n 6 s , e rJm 0 ~ The parameter y ' Sin0 ooSB coefficient tensor tensor S'Vife-1 Q >iiy\rSind ' 0 was defined in Eq. thickness of the m-th ply. \,M 8 (4), and dm is the The components of the reflection (r^j) and the attenuation coefficient (A^j) depend on the dielectric properties of the composite, wavelength. the fiber orientation Thus, in each ply, and the for a given material, incoming electric field, laminate layup, and the components of the r^j and A^j tensors are known. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •r fl UJ f I < ii PLY i— +LlT -TH CL 'oT 3 +q T <* n it UJ i UJ 1 i •UJ Figure 5 £ I ~1 Illustration of the Electric Field Vectors Ply (Linearly Polarized TEM W a v e ) . in the m-th 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 By utilizing Eqs. Eqs. (14) and (15), (16) and (17), together with the following expressions are obtained ,19) (P; )m_," (Pj)m ^Ajk)m.|( )rn-i + [ U £ 3 - < ^ ) B3(AjK)w (TS')nl The subscripts m and m-1 respectively. coefficient m-th ply. refer to the m-th and The parameter <20) (m-1)-th ply, (fjj)m is the reflection tensor at the interface between the The subscript k has the values of (m-1)-th and 1 and 2 (k=1,2). At the front material, we have interface (m=1), where the wave enters the (Figure 6) ( Ec* )0 " (E c )( = The components of the and Eqs. (21> ( A t j ), (22) ), (22) incident electric (E2 )0 , are given by Eqs. (21) and ( fj into Eqs. (1) and (14) and field vector, (2). (15), (E^q By substituting for m=1, we obtain R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Illustration of the Electric Field Vectors at the Front (m=l) and Back (rr.=M+l) Interfaces (Linearly Polarized TEM M+1 15 16 (E-)0 ( F f ; , = C Ucj + - ( rcj)I (.& j k ) ,( P* )0 (p a = ( r a a ) At the back interface material, (m=M+1), where no wave enters the the following equations hold (Figure 6) (E*)M = (Acj)M (Pj+JM <25> f E-' ) (26) = By substituting Eqs. for m=M+1, 0 (25) and (26) into Eqs. F’c )|v, t Uij + c 'pj (15), i (Ajk)„( (24), <27) - (28) For a laminate consisting of M plies, (23), (14) and we obtain < ' P a . , = ( (23) (27), and (28) represent Eqs. (19), 2M+2 equations (20), for the 2M+2 unknowns which are the pT and PT vectors. Note that the equations and unknowns are in vector Therefore, form. solutions must be obtained by expressing the equations component in form, and by solving for the x 1 and x ^ components of P* and P^. A procedure for determining the unknowns outlined in Section 3.1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. is 17 2.2.2 Reflectance and Transmittance The reflectance is the ratio of the reflected to the incident energy flux, and the transmittance is the ratio of the transmitted to the incident energy flux [10]. incident energy flux is (Figure 7) The [8] (29) ’o where Z q is the impedance 120ir ohms [11]. in free space and has the value of The reflected energy flux, (5r is [8] \(P: ) J (30) 0 Thus the reflectance is given by Z <fr KRD0I The transmitted energy f l u x , £ t is (31 ) [8] (32) 0 Accordingly, the transmittance is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m=1 m= M + 1 INCIDENT ENERGY inc 2Z TRANSMITTED ENERGY 2 LAMINATE . K pD m 2Z h I REFLECTED ENERGY . c - K flk J 1 ' 2Zn ' /I FRONT Figure 7 BACK Illustration of the Overall Incident, Reflected, and Transmitted Energies (Linearly Polarized TEM W a v e ) . 19 ^ T_ <ft — = = - \- (- P £lnc - w f 1 , 4 (33) I C E D J 2 In the above equations, (P^)g anc^ are e ^e c t r ^c field vectors entering and leaving the front interface, respectively the back (Figure 6). The electric interface is (Pj^M+l field vector leaving (Figure 6). These parameters can be calculated using the analysis developed in the previous 2.2.3 section. Overall Reflection and-Transmission Coefficients $ The overall reflection coefficient tensor r^j and the overall transmission coefficient tensor t^j are defined by the expressions [10,29] ( P t" ) a = (r j(e p 4 (34) 3 (t;p (E p 0 The electric field vectors (P- )n and 1 U (3 5 ) (P.+ )u ,. are to be 1 M+ I calculated by the method described previously in Section 2.2.1. is However, a knowledge of these vectors insufficient to readily calculate the r^j and t^j tensors. A procedure suitable for determining these tensors is described below. At the back interface equations apply (m=M+1), the following two (Section 2.2.1) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 ( F£Ih, = ( ^C. ” C U j t c r y )M+il C A j O m '(Pk+ )M £ ^ ) M+j (Ajk)|V| ( By rearranging these equations, (27) | (28) we obtain (36) ( P : \ = {<AcpM ru j k H f jk ) J ] ( -11 . r^+, < PT) m = ( < rCj;M/ u jk+ ( rjk )Mti] j ( Pk ; Mw w . These equations are now represented symbolically as . M'tl +■ <38) (P i)M = The tensors tensors (39) <39) and (see Eqs. into Eqs. contain only the known r^j and A^j 36 and 37). (19) and (20), vectors at the M - 1 interface Symbolically, j. By substituting Eqs. we obtain the electric (38) and field in terms of the result can be expressed as ^M r**s ,40) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 where the tensors and tensors A^j and r^j. The procedure is repeated step-by-step until the m=2 interface (P-), ( (23) is reached - Pi ^1 ~ ( 7ij As the final step, Eqs. again contain only the known Eqs. ^ +l (42) and (43) are substituted = the is obtained Jm+ i C,?Cj) ( P p M +i (44) (45> (44) can be inverted to yield ( 5:j) where into and (24), and after algebraic manipulations, following expression Equation (43) ((i|j) CEj)c 1 is the inverse matrix of (46) Equations (45) and (46) give (PDo It - is emphasized again «7 ( Sj'k) ' ( Efe )o that the and n]j tensors depend R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 only on the known reflection coefficient and attenuation coefficient tensors, expressions for r^'s and A^j's. However, and njj cannot be derived. closed form These tensors must be evaluated by a numerical method. Comparisons of Eqs. (46) and (47) with Eqs. (34) and (35) yield the overall reflection and transmission coefficients r c* = (48) <Sy>" " 2.2.4 Absorbed Energy The rate of absorbed energy per unit volume in each ply is taken to be uniform and constant across the ply. this approximation, volume the rate of absorbed energy per unit in the m-th ply = ^ With is (Appendix C) | ( E + ^ e m + c E , V ;" M ( E * ) c o j 6 m - ( E 4 )M S r n 0 m | (50) L where Op and Og are the electrical conductivities in the % e directions parallel and perpendicular to the fibers. E° and E^ are the x 1 and x2 components of the resultant electric R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 field vector E? passing through the ply (Figure 8). The vector E? is approximated by the expression CEL s)m = j (51) The energy absorbed by the m-th ply per unit area per unit time is (52) ~ where dm (Em ^ rw is the thickness of the m - t h ply. absorbed by the entire laminate unit area per unit time) The energy (total absorbed energy per is M & T1 Alternately, = Z (£ „) (53) ro-i the total absorbed energy per unit area per unit time may be calculated by the law of conservation of energy. This law requires that the following equality be satisfied /lncident\ /ReflectedX / Transmitted\ / AbsorbedX 1 Energy j = Energy I + [ Energy + f Energy \ Flux / \ Flux / \ Flux \ Flux J Equations (30) and er - w h e r e i s (32) together with Eq. <5 inc • ( I - R given by Eq. J (54) (54) yield - T r ) (29). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (55) 24 m - 1 K m U , " m ► . (p i+) m M EI U (AijUPPm -«■ (PD m I i[(P D m +(Aij )m(P7)m] Figure 8 Illustration of the Electric Field Vectors in the m-th Ply (Linearly Polarized TEM W a v e ) . R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 2.2.5 Incident Isotropic Electromagnetic Wave When an isotropic electromagnetic wave impinges on the surface, the wavelength X, and the total energy content are known. waves. The superscript i is used to denote The total energy flux normal one half of the total energy flux 6 j = £ l/ isotropic to the surface is (Appendix D) z <5S) In analyzing the problem, we replace the isotropic wave with a single unpolarized plane wave h a ving a total energy content (S^f and traveling in the di r e c t i o n perpendicular to the surface plane wave (x^ direction, Figure 9). This unpolarized is considered to be made up of plane waves, the vectors representing these waves being evenly distributed in a plane parallel plane). linearly polarized to the surface Each linearly polarized wave has the same wavelength X and the same energy content Since linearly polarized waves are evenly distributed, incident energy flux at any polarization angle (fjnJj = = £Y For each linearly polarized TEM wave reflectance (R)g, the transmittance absorbed energy per unit volume each ply 4 6 is ir (Tr)^, (5 7 ) the the rate of the energy absorbed by (Sm )g, and the total absorbed energy can be calculated by the method described in Sections 2.2.2 and 2.2.4. the the in the 6 plane, (£)g, ( x 1~ x 2 Once these parameters are known for each incident R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LINEARLY POLARIZED TEM WAVE WAVELENGTH: X NORMAL INCIDENT ENERGY FLUX OF ONE WAVE : WAVELENGTH: X TOTAL INCIDENT ]/ / / / / = £ / 4 7T / / / / / / v / / ISOTROPIC WAVE Figure 9 DIRECTION OF PROROGATION OF UNPOLARIZED TEM WAVE (SUM OF LINEARLY POLARIZED TEM WAVES) LINEARLY POLARIZED TEM WAVE Illustration of the Incident Isotropic Wave and the "Equivalent" Linearly Polarized TEM Waves. / / / / / / / / / / / z' Xi to CTv 27 linearly polarized TEM waves, for the incident they can then be calculated isotropic wave by the procedure described below. The isotropic reflectance R ‘ 3 is defined as (see Eq.31) / ei tr (58) <£*, the total reflected energy flux normal to the surface, is the sum of the reflected normal energy fluxes of the linearly polarized TEM waves £=$'(£,)*** The reflectance for a single polarized TE M wave (R)s By combining Eqs. expression = (57)-(60), reflectance 2T ~ I 2.7T o An expression for the I = — The result is ar f (TrLJS The rate of absorbed energy per unit volume is taken (6,) isotropic transmittance can be in a completely analogous manner. -JV (60) we obtain the following for the isotropic R‘ = is ( £ f)s/ ( & • * ) £ I developed ,59) (62) in the m-th ply to be the sum of the rates of absorbed energy per Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 unit volume due to each linearly polarized TEM wave 2t (63) The absorbed energy by the m-th ply per unit area per unit time is (64) The total energy absorbed by the laminate per unit area per unit time is (65) m-i Equations (1)— (65) complete the electromagnetic model. The input parameters required for the solution and the output parameters are specified in Section 2.5. The numerical method used to generate solutions is described in Section 3. 2.3 Thermochemical Model During microwave curing, the material to be cured is generally surrounded by various layers. Typically, composite will be placed on a solid plate. composite is a bleeder; on top of Above the bleeder, an air breather and another solid plate. the the there may be Sheets of porous or non-porous teflon release clothes are placed between the various layers (Figure 10). A microwave of known energy content and wavelength R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MICROWAVE \ i / PRESSURE I T E MP: T , | | TOOL 1 1 1 1 PLATE AIR BREATHER NON- POROUS ^ /T E F L O N BLEEDER ^ /P O R O U S TEFLON COMPOSITE NON-POROUS /TEFLON TOOL PLATE TEMP: T l t t t t t t t PRESSURE /t\ MICROWAVE Figure 10 Schematic of a Typical Cure Assembly. to VO 30 incident on both the top and bottom surfaces of this assembly. In addition, bottom surfaces, the temperatures at the top and the and calculate the temperature, , are specified. It is desired to the degree of cure, and the viscosity of the composite as functions of position and time. In order to generalize the problem, cure assembly illustrated in Figure 11. we consider the This assembly consists of an arbitrary number of layers above and below the composite. Each of these layers are taken to be homogeneous and isotropic with specified thicknesses and known material properties. Microwave energy absorbed by the layers surrounding the composite is taken to be negligible. By considering energy transfer only in the direction perpendicular to the plane of the composite (x direction), and by assuming perfect thermal contact between each layer, the temperature distribution at any position inside the assembly can be calculated by the following form of the energy equation t is the time, [12] x is the coordinate normal to the plane of the assembly with its origin on the bottom surface of the lowest layer. T is the temperature, and specific heat, conductivity respectively. p and C are the density K is the thermal in the x direction, (- .and H are the rates of heat generated by the absorbed microwave energy and by the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 INCIDENT MICROWAVE Tu LAYER: i =u+v+1 u+v+1 NUMBER OF LAYERS: v LAYER: I = u + 2 COMPOSITE (Layer: 1 = u + l} h u+ 2 u+1 /• Layer: 1 = u NUMBER OF LAYERS:u < Layer: I = 2 Layer: 1 T l / t\ NCIDENT MICROWAVE Figure 11 Description of the Cure Assembly Used in- the Thermochemical Model. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 chemical reactions, the composite, respectively. In the layers surrounding both £ and H are zero. For the composite, <£ as a function of position x may be found by the procedure described in Section 2.2.4. H is defined by the expression [13] clod • H -= Hr h r is the total or ultimate heat of reaction during cure, and a is the degree of cure defined as [13] H(r) A (6 8 ) - HR where H(t) is the heat evolved from the beginning of the reaction to some intermediate time, material a=0,and approaches unity. neglected, t . For an uncured for a completely cured material a If diffusion of chemical species is the degree of cure at each point inside the material can be calculated once the cure rate is known in the following way d o/ , ^ {gg) - - f 0 < •& ) In order to complete the model, the dependence of the cure rate on the composite and on the degree of cure must be known. This dependency may be expressed symbolically as R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 (70) The functional relationship in Eq. (70), along with the value of the heat of reaction Hp for the prepreg material under consideration, can be determined experimentally by the procedures described in Reference Solutions to Eqs. (66) and [13]. (67)— (70) can be obtained once the initial and boundary conditions are specified. The initial conditions require that the temperature inside the assembly and the degree of cure given before the start of cure inside the composite be (time t < 0). The boundary conditions require that the temperatures on the top and bottom surfaces of the assembly and the incident microwave energy be known as functions of time during cure Accordingly, (time the initial and boundary conditions corresponding to Eqs. (66) and (70) are Initial conditions: t <0 T init temperature in the assembly. Boundary conditions: R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. t ^O). 34 where and TL are the temperatures on the top and bottom surfaces of the assembly, respectively addition to the temperatures waves and T^, 10). In the electromagnetic incident on the top and bottom surfaces must be specified. aQ, (Figure For linearly polarized TEM waves, the amplitude the w a velength X, and the polarization angle 6 must be given. For isotropic waves, the wavelength X and the total energy content of the wave £.1 must be specified. Solutions to Eqs. the degree of cure (66)— (72) provide the temperature and in the composite as functions of position and time. Once these parameters are known, can be calculated, the resin viscosity provided a suitable expression relating resin viscosity to its temperature and degree of cure available. If the resin viscosity is is assumed to be independent of shear rate, then the relationship between viscosity, temperature, and degree of cure can be represented in the form: jji The manner 2 < T , ° < ) ' (73) in which the relationship between viscosity, temperature, described = and degree of cure can be established in Reference Equations is [13], (66)— (73) complete the thermochemical model. A numerical procedure suitable for generating solutions presented input parameters required for in Section 3.2. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is 35 the solutions are discussed subsequently (Section 2.5). 2.4 Resin Flow Model During the cure, a pressure is applied to the cure assembly (Figure composite, 10) to squeeze the excess resin out of the to consolidate the plies, void content. and to minimize the It is desired to estimate the resin flow out of the composite and the amount of resin functions of position and time. in the composite as These parameters can be calculated by the method proposed by Loos and Springer The detailed steps of the method are not given here, be found in Reference [14]. but can [14]. The equations proposed by Loos and Springer were incorporated into the numerical solutions (Section 3) to enable us to calculate the resin flow during microwave curing of composites. 2.5 Input and Output Parameters Solutions to the electromagnetic(Eqs. 1-65) thermochemical(Eqs. 66-73), and to the resin flow models (Eqs. 10-33 in Reference parameters be specified. [14]) require that the input The parameters needed for the solutions are summarized in Tables 1-3. Solutions to the electromagnetic, thermochemical, resin flow models provide the information listed and in Table 4. The methods used in obtaining solutions to the appropriate R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 Table 1 Input Parameters Required for the Electromagnetic Model A. Geometry 1) Thickness of each ply 2) Number of plies 3) Ply orientation B. Composite Material Properties 4) Dielectric constants parallel and perpendicular the fibers for the uncured composite to 5) Dielectric constants parallel and perpendicular the fibers for the cured composite to C. Incident Wave Properties^ 6) Wavelength 7) Amplitude 8) Polarization angle 9) Incident Energy' Flux 1. For incident linearly polarized T E M waves, only items 6, 7, and 8 are needed. For incident isotropic waves, only items 6 and 9 are needed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Table 2 Input Parameters Required for the Thermochemical Model These parameters are in addition to those required in the electromagnetic model (Table 1). A. Geometry 1) Length of the composite 2) Width of the composite 3) Number of layers above and below the composite 4) Thickness of each layer above and below the composite 5) Temperature at the upper surface of the cure assembly as a function of time 6) Temperature at the lower surface of the cure assembly as a function of time B. Composite Properties 7) Initial thickness of one ply 8) Initial resin mass fraction of one ply 9) Resin content of one compacted ply C. Resin Properties 10) Density 1 1) Specific heat 12) Thermal conductivity 13) Heat of reaction 14) Relationship between the cure rate, degree of cure temperature, D. Fiber Properties 15) Density 16) Specific heat 17) Thermal conductivity R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 38 Table 2. (cont.) E. Layer Properties 18) Density of each layer 19) Specific heat of each layer 20) Thermal conductivity of each layer F. Initial and Boundary Conditions 21) Initial temperature distribution in the composite and in the layers above and below the compos ite 22) Initial degree of cure of the resin in the composite R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Table 3 Input Parameters Required for the Resin Flow Model These parameters are in addition to those required in the electromagnetic and thermochemical models (Tables 1 and 2) A. Composite Material Properties 1) Apparent permeability of the prepreg plane of the composite 2) Flow coefficient fibers of the prepreg normal parallel to the to the B. Resin Properties 3) Relationship between the viscosity, degree of cure temperature, C. Bleeder Properties 4) Apparent permeability 5) Porosity D. Initial and Boundary Conditions 6) Cure pressure as a function of time 7) Pressure in the bleeder R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 40 equations are described in Section 3. 2.5.1 Composite Properties Solutions to the electromagnetic and thermochemical models described in Sections 2.2 - 2.4 require that the complex dielectric constant e , density p, specific heat C, heat of reaction H, and thermal conductivity normal to the fibers K of the composite, be known. These properties depend on the local resin and fiber contents and on the degree of cure of each ply. By assuming that the complex dielectric constant is directly proportional to the resin content of the composite, the changes in the complex dielectric constants in the m-th ply during cure may be approximated by the expressions c jy* f =W * 'JC C 'Jt U Mfio “ Mft. M ^ - M re yi C +l€f} 1 ™ {Ui (75) Mr. - M r ’ * The subscripts p and q represent the direction parallel and perpendicular to the fibers, respectively. u and c denote uncured and cured materials, re and re composites, are the resin masses The superscripts respectively. in the uncured and cured and M rg is the resin mass at any intermediate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4'1 Table 4 Output Parameters Given by the Models Electromagnetic Model^ 1) Overall reflection coefficient 2) Reflectance 3) Overall transmission coefficient 4) Transmittance 5) Absorbed energy in each ply 6) Total absorbed energy 7) Isotropic reflectance 8) Isotropic transmittance Thermochemical Model 9) Temperature distribution as a function of time and position 10) Degree of cure of resin as a function of time and position 11) Viscosity of the resin as a function of time and position Resin Flow Model 12) Number of compacted plies as a function of time 13) Amount of resin flow normal to theplane of the composite as a function of time 14) Amount of resin flow parallel to the plane of the composite as a function of time 15) Total time required for the cure 1. For incident linearly polarized TEM waves, only items 1 through 6 are calculated. For incident isotropic waves, only items 5 through 8 are calculated. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 time t . The parameters (ep)U f ^eq ^ U ' ^£p^C ' and ^eq ^ C are determined from the measurements of the dielectric constants (see Section 4.3.1). The parameter M re can be calculated by the resin flow model. The variations in p, C, H, and K with the degree of cure are generally unknown. Therefore, the change parameters with the degree of cure are neglected in these in this study, and only changes due to the resin content are taken into account. Accordingly, p, C, and H are calculated by the expressions developed in Reference [14] (76) (77) (78) where subscripts re and f refer to resin and fiber, respectively. M T is the total mass of the m-th ply, and M f are the masses of the resin and the fiber. the heat of reaction per unit mass for the resin thermal conductivity normal to the fibers and M rg (H R ^ re is only. (x direction, R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 43 Figure 11) K may be estimated from the expression Km Krc^'-i fn> [16] (79) + B/ Z, / where B = 2 - i ) (80) K c Kj and K re are the thermal conductivities of the fiber and the resin, ply (vf )m respectively. is given by The fiber volume fraction of m-th [16] (81 ) w/re. As noted previously, M re w ^t^1 position and time the variation in the resin mass is given by the numerical solution described in Section 3. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. SECTION III NUMERICAL Solutions to the electromagnetic, resin The thermochemical, and flow models must be obtained by numerical methods. numerical procedures used to obtain solutions and the associated computer code are described below. 3.1 Numerical Solution, Electromagnetic Model The starting point of the solution (39) is Eqs. (38) and in Section 2.2.3 M+i • + (38) (39) The components of the tensors Then, as was discussed an^ in Section 2.2.3, are known. the calculations proceed from ply to ply toward the front (40) (41) 0 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 j CPc^m At each step, calculated. reached, ^ ( F f ) ' M+! U Thus, once the front (82) , the components of tensors tensors composite (44), = / V 1J and n^j are interface (m=1) and n^j are known throughout the from m=1 to m=M+1. Then, using Eqs. the overall transmission coefficient t ^ j parameter is (35) and and the (p i)M + i are calculated ,* , ■tj: = ( 1 s-l fcj ) <49» (P-+) = (' - £ * )' ( y E J^ )<0 *■ 'm+I ' r C From the known values of of tensors and r^jf calculated at every (82) and (83). vectors, e T and (p T ) + , and the known values the parameters pt and PT are interface using Eqs. Once P^ and P^ are known, e (35> ‘J 7, at each (38) - (41), the electric (45), field interface are calculated using E q s .(16) and (17). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 The reflectance, transmittance, are calculated from Eqs. In case the wave composite (as, (31), (33), and (50) -(53). impinges on the two sides of the for example, absorbed energy and absorbed energies in a microwave oven) the is taken to be the sum of the absorbed energies of the waves entering from both sides. For isotropic waves, calculation of the isotropic reflectance, transmittance, intergration of (R)gf angle 5 (see Section 2.2.5, integration and absorbed energy requires an an<^ ^ ’V 5 over polarization E q s . 61-65). the interval was divided Here, in tt/6 segments and the integration was performed numerically by taking the parameters (R)g> ^T r ^5 radian segments. calculated Thus, ^5 t0 be constant over the isotropic n/6 reflectance was by the expression I R‘ = i f j2?r 2ir ° The isotropic volume were anc^ ^ I 11 ( R )sdS - C -Z(R ; W ^ rk (84: ^ j transmittance and absorbed energy rate per unit calculated in similar manners. It is noted that the foregoing calculation procedures R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 may also be applied to problems outside the composite. in which layers are placed These layers may then be treated as "plies" with known thicknesses and with known dielectric properties. 3.2 Numerical Solution, Thermochemical Model 3.2.1 Grid In order to calculate the temperature distribution inside the composite and in the layers surrounding the composite, the cure assembly one dimensional grids (Figure (Figure 11). 10) was divided into Inside the composite, grid points were located at the interface of each adjacent ply, and on the lower and upper surfaces of the composite. The layers surrounding the composite were also divided into grids. The distance between any two grid points as Ax„ (Figure p composite, 12). is denoted In the layers surrounding the the grid spacing Ax„ is constant. In the p composite, the grid spacings vary with time depending on the resin content of the ply. Each grid point is indicated by subscript g and time is designated by the superscript q. A control volume is placed around each grid point, shown in Figure 13. The material properties are the same between any two grid points between as (i.e., between g-1 and g, and g and g+1), but are different across the control volume. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 t ♦ 0-1 LAVER COMPOSITE PLIES — - — ■ LAYER a X/3 ♦ r LAYER t ^ =2 Kj3*\ // / / / ////// \ /////; s ; s ; y / r Figure 12 Arrangement of the Grid Points R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 I oa. oa. X X <J <u x t +j •P 3 O XI (d iH O > pH 0 V) 44 + G O U dJ -C OQ. --- 4-1 44 4-i O G •H c O o cu •H +J XI <d •H p >4 4-> a ui 3 -G r-H 4-1 H I H CQ n aj _J o IxJ 0c V4 3 Cd •H Pm I- oo $ R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 3.2.2 Finite Difference Equations Equation (66) together with Eqs. following expression (67) — <70) give the for the energy equation In order to proceed with the solution, we focus our attention to the control volume surrounding the (3-th grid point (Figure 13). control volume, By integrating Eq. (87) across the, we obtain S^<fCT)Jx -- K g ] + (8e> a. Solution to the above equation finite difference method in a form suitable [17]. is obtained by an Equation (88) for numerical calculations. the left-hand side of the Eq. (88) implicit is expressed The term on is approximated by where A t is the time step equal to the time between T1^. Here, on T refers and in the subsequent analyses, to the (3-th grid point; [18] p + 1 and the subscript (3 on all other parameters, it refers to the region between the 8-1 and 8“th grid Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 points.. The term on the right-hand side of Eq. (88) is approximated by a " (3 u (5-i where Axf is the distance between the grid points 8 and 8+1 p at time and A x ^ _ 1 is the distance between the grid points 8 and 8_1 at time t^. right-hand side « The second integral on the is expressed as r fljn>H‘ * £i By combining Eqs. (89) - (91), /z we obtain the following algebraic expression r ( K ]f + ( A f Ui'p L W ?-| + 1 (£94$ J? z _ /J< \l J-H jK Mxjj-/ P'1 1 °^ A* JL + z J ) /> Is*' | ( /£ V s > W £ -rf-" Z + ^ -yb f (f HriK 1- £ 7 ^ , -2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (92) 52 Equation (92) represents 1-1 linear equations corresponding to the 1-1 "interior" grid points (Figure equations, called a tridiagonal system, 12). This set of is solved for the temperature at each grid point at the new time T ^ +1 using p the Gaussian elimination method. An algorithm for the solution of the above tridiagonal system is given by Carnahan et al. [19]. The temperatures at the lower (T^+ 1 ) and upper (T^+j) boundary grid points are specified by the boundary conditions. The boundary temperatures may vary with time in an arbitrary manner. In the calculations, the boundary temperatures were .assumed to be constant. It is noted that the numerical procedure outlined previously is an implicit method of solution. The implicit method does not impose a limit on the size of the time step (At ), which can be used with a desired grid spacing ensure stability of the numerical solution. stability criterion procedure, is not required (Ax) Therefore, to a for the above numerical and the size of time step can be chosen ar b i t r a r i l y . Once the temperature (Tf+ 1 ) is known at each grid p point, the resin degree of cure can be determined at each grid point inside the composite. relationship between cure rate, cure, The functional temperature, and degree of is given by = -f C T , e O (70) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 By approximating the cure rate by the difference equation d°( ^ <?\s“ ^ dz )n A ” and substituting Eq. (93) into Eq. (70), we obtain the following expression for the degree of cure at the new time T * +1 £ n The viscosity of the grid point and at time V + -FCTp t^+ ' % A (54: resin can be determined at each 1 once the temperature (Tf+ 1 ) and p the degree of cure (a?+ 1 ) are known. p between viscosity, temperature, The exact relationship and degree of cure, depends on the resin system being studied (Eq. 73). A symbolic relationship between the viscosity at each grid point (u?+ ^), the temperature (T^+ 1 ), and the degree of cure p p (a?+ 1 ), can be expressed as p (95) The rate of microwave energy absorption per unit volume inside the composite is calculated from the solution of the electromagnetic model described in Section 3.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 3.3 Numerical Solution, Resin Flow Model A Numerical method of solution was developed by Loos and Springer described in detail in Reference for the resin flow model [14]. This method, [14], was adopted in this investi g a t i o n . 3.4 Material Properties Solution of the finite difference equation requires the complex dielectric constant (Eq.92) $ e , density p, specific heat C, heat of reaction H, and thermal conductivity normal to the fibers K be known at each grid point inside the composite. Once the mass of the resin and the total volume of the ply surrounding the grid point are determined, from Eqs. the aforementioned properties can be calculated (74) - (81). The resin mass and the ply volume are obtained from the resin flow model. 3.5 Computer Code A computer code (designated as "EMWAVE") was developed to implement the foregoing numerical procedures. To test the accuracy of the computer code pertaining to the electromagnetic model, results were obtained to problems for which known analytical solution exists. The parts of the code pertaining to the thermochemical and resin were not tested because flow models the accuracies of these portions of the code were evaluated by Loos and Springer [15]. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 In order pertaining to assess the accuracy of the computer code to the electromagnetic model, the transmittance, the reflectance, and the total absorbed energy were c alculated with the computer code for unidirectional 32 ply graphite epoxy and glass epoxy composites exposed to a po l arized TEM wave (6=90°). AS/3501-6 graphite epoxy and Fiberite S2/9134B glass epoxy composites were used are listed and (28), closed The properties of Hercules in the calculations. in Appendix G. By using Eqs. form solutions transmittance can be obtained. are These properties (23), (24), (27) for the reflectance and The resulting expressions [20] R = E u>sho( +■F sinho( - Q-Cosfl +• H sinfi (96) (rtf it) Tr - (97) E £<>5h o( + P S i n h o t ~ O ' + H ? where 4'fTkd (98) P 4-iTnd and R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. OQ ) 56 k + 1) - 4 n A = c B = ( n z+ k * - 1 ) + 4 k E = CnVkV// t F = 4-n ( n h k V i ) ( n l+fc1-/ A 4 k Q H = 4 k ( n + k L- i ) (100) ( 101) ( 102) (103) (104) ~ (105) where n and -k are the real and imaginary parts of respectively. The total energy absorbed may be calculated for the known value of R and Tr (see Eqs. - R 1 “ 54 and 55) - Tr doe) ' C ? . ^ (Y i < The reflectance, the transmittance, and total absorbed energy calculated by the computer code and by the closed form expressions results (Eqs. 96-106) are given in Table 5. The in this table show good agreement between the computer and the analytical solutions. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 TABLE 5 The Reflectance R, Transmittance Tr, and Total Absorbed Energy Calculated by Closed Form Solutions (Eqs. 96 - 106) and by the Computer Code S 2 / 9 13 4 B ( c u r e d , (0]3 2 , 6=90°) Closed Form Solution Tr Computer Code 0.181 0.181 0.750 0.750 0.069 0.069 AS/350 1-6(c u r e d , [0]3 2 , 6 = 90°) Closed Form Solution Computer Code 0.735 0.181 Tr 0.014 0.014 (** O rp 0.251 0.251 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. SECTION IV EXPERIMENTAL In this chapter, the experimental apparatus and procedures are described which were used to measure a) the dielectric properties of the material, b) the reflection and the transmission of linearly polarized T E M waves by the material, c) the temperature distribution inside the material during microwave heating, and d) the resin flow of the composite during microwave curing. Two types of apparatus were used waveguide and a microwave oven. in the tests: a A waveguide was used for measuring the dielectric constants and the reflected and transmitted energies. A microwave oven was utilized temperature and resin flow measurements. apparatus is described below. description test Each of in the the two In addition, a brief is given about the methods used to construct the specimens. 4.1 Waveguide In the experiments employing a waveguide, 2.45 GHz microwaves were generated by a microwave oscillator (General 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Radio Type 1360-B). The microwave oscillator was connected by coaxial cables to a waveguide adaptor S281A) via a frequency meter isolator (Hewlett-Packard (Hewlett-Packard 536A) and an (Huggins Lab HC-7082) (Figure 14). The adaptor was clamped to the waveguide. The waveguide section consited of two parts: a "slotted" (Hewlett-Packard S810A) and a "solid" section. material to be tested was placed between solid sections. clamps. The the slotted and the The two sections were fastened together by The waveguide was for S-band microwaves. Accordingly, dimensions the cross-section was rectangular, of 7.214 cm x 3.607 cm [21]. having The length of the slotted section was 32.4 cm. Three different types of solid sections were used: for measuring dielectric constants, reflectances, one one for measuring and one for measuring transmittances of the material. For measuring the dielectric constants, the solid section was a waveguide with an adjustable short Packard S920A). inside the The postion of the short (Hewlett- waveguide could be adjusted by a screw mechanism. measuring the reflectances, termination a resistor the solid section was a (Hewlett-Packard S910A). This termination had inside to minimize back reflection. transmittance measurements, an attenuator In For the the solid section consisted of (Polytechnic Research Development Type with an adaptor (Hewlett-Packard S281A) 171) attached to the far R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. STANDING WAVE INDICATOR MOVABLE PROBE o 0*1 MICROWAVE OSCILLATOR ADAPTOR FREQUENCY METER ISOLATOR SLOTTED SECTION WAVEGUIDE- Figure 14 Schematic of the Waveguide Set-up. SOLID SECTION 61 side. A probe measuring the electric into the slotted section. standing wave position indicator of the probe field strength was built The probe was connected to a (Hewlett-Packard 415B). The in the slotted section could be varied along the slot. During the transmittance measurements, the electric field strength was m e a s u r e d with a fixed probe located the adaptor attached to the solid section. attached to a standing wave indicator in The probe was (Hewlett-Packard 415B). The procedures used to determine the dielectric constants and the reflectances and transmittances are described in Section 4.3. 4.2 Microwave Oven A commercially available Model 700 watt, 2.45 GHz (Litton 1290) microwave oven was used in the curing tests. There were two controls built into this oven: one was a timer which controlled the length of the periods during which the power was on and off; which controlled the total the other one was a timer length of time during which the oven was operating. The temperatures inside the microwave oven were measured either by type T (c o p p e r - c o n s t a n t a n ) or by type J (iron-co n s t a n t a n ) thermocouples. shielded with either These thermocouples were stainless steel or inconel overbraid, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 respectively. The thermocouples were fed through a hole on the back of the oven, and the shield was grounded [22]. The output of each thermocouple was measured by a digital voltmeter. During cure, the pressure to the composite was applied, using the fixture shown in Figure constructed of polypropylene dissipation teflon 15. (dielectric constant factor e " = 0.0009 [ 2 3 ] ) (e'=2.3 , e"=0.002 The fixture was [23]). e'=2.2, and glass reinforced These materials were used to prevent significant energy absorption and corresponding temperature The rise of the fixture materials. fixture consisted of three 25.4 cm x 25.4 cm square polypropylene plates. The plates were separated by four 2.54 cm diameter ploypropylene rods. thick) a nd the top plate The bottom plate (5.08 cm (2.54 cm thick) were attached to the rods and were prevented from moving by polypropylene bolts. The middle plate up and down (2.45 cm thick) could freely slide the rods. A pancake shaped rubber bag (25.4 cm diameter) enclosed in glassfiber cloth was placed in between the top and the middle plates. The bag was pressurized by compressed air. A 20.3-cm x 20.3 cm (2.54 cm thick) glass reinforced teflon plate was placed on the bottom plate. to be cured was placed on this teflon plate. x 15 cm, Dams (2.5 cm 1.5 cm thick) made of glass reinforced teflon were placed around the composite. thick) The composite glass A 10.2 .cm x 10.2 cm (3.2 cm reinforced teflon plate was placed on the top Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. POLYPROPYLENE PLATE AIR BAG SUPPORT RODS GLASS REINFORCED TEFLON GLASS REINFORCED TEFLON DAM COMPOSITE Ch GLASS REINFORCED TEFLON POLYPROPYLENE PLATE Figure 15 Schematic of the Press Used during Microwave Curing. co 64 of the composite. By inflating the air bag, pressure could be exerted on the composite placed between the middle and the bottom plates. The relationship between the air pressure and the pressure applied to the composite was determined by replacing the composite with a load cell, and by measuring the load at different bag pressures. procedure is described further This calibration in Appendix E. The uniformity of the power distribution inside the oven was evaluated by the method proposed in Reference [24]. Nine 50 ml glass cups of water were placed inside the oven at different locations. The temperature rise of the water was measured by thermocouples during a 60 second time interval. These results are given maximum variation percent. The in the power across the entire oven was 46 This maximum variation of electric field occured across a distance of test specimen was variation in Appendix F. 13.5 cm. 10.1 cm. in electric Maximum dimension of the Across this distance, the field strength should be less than about 35 percent. 4.3. Measurement of the Electromagnetic Properties In measuring the electromagnetic properties, the composite material was placed between the slotted and the solid sections of the waveguide. vectors of The electric incoming waves were always direction(Figure 16). field in the x^ Tests were performed with fibers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 oriented in different directions with respect to the electric field. 4.3.1 Dielectric Constants In order to measure the dielectric constants of the material, the composite of thickness d was placed between the slotted and the solid section of the waveguide 16). The position of the adjustable short adjusted so that it was at a distance of back surface of the material. (Figure (plunger) was 1/4 X, „ from the wg The parameter Xwg„ is given J by (107) where Xg is the wavelength in free space, of the waveguide. manner, and w is the width By positioning the plunger in this it was assured that the electromagnetic wave had a maximum amplitude at the back surface of the material (x.j=d). This is referred to as the "open position." The wave generator was then turned on. The probe in the slotted section was moved until the location of the first minimum electric field strength (null position, *A ), and the location where the electric field had the maximum amplitude were found. section waveguide, From a scale mounted on the slotted the null position xA was recorded. the standing wave indicator, From the strength of the electric R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ELECTRIC FIELD FIRST NULL POSITION COMPOSITE MATERIAL PLUNGER MAXIMUM AMPLITUDE Figure 16 Illustration of the Waveguide Arrangement during the Measurements of the Dielectric Constants. 67 field at the maximum and min i m u m (E max , E • ). min locations were From the latter measurements, standing wave ratio (VSWR) was c a lculated recorded the voltage = [21] E f/)ax VSWR = <108> Emin The composite material was then taken out of the w a v eguide and was replaced by an a l u m i n u m plate. of the slotted section was again parameter AX found The null position (position Xg). A was calculated to? = Knowing the value of AX and VSWR, found from the Smith chart [21]. impedance at the front surface of calculated by the expression the impedance Once z was ratio z was known, the material Z(0) the was [21] Z(0) Z = (110) where 7 ) { L "fo Under rests 12.0 If ~.T ... . : = / , - ^ (111) f the "open p o s ition" condition employed (see above), the impedance material can be exDressed as at the front face in the of the [23] R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 (1 1 2 ) where /20V (113) ( 1 14) ( 115) From Eqs. (110) the dissipation - (115), the dielectric constant e' and factor e" could readily be calculated by a trial and error procedure. The dielectric constants thus measured are the values parallel to the direction of the incident electric fibers parallel field vector. Thus, by aligning the or perpendicular to the incoming wave, the dielectric constants of the composite normal or parallel to the fiber direction could be determined. The measured values of the dielectric constants are listed in Appendix G for Hercules AS/3501-6 graphite epoxy unidirectional composites and for Fiberit(e S2/9134B glass epoxy unidirectional composites. Graphite is a good electrical conductor. Therefore, the values of e' and e" measured for graphite epoxy composites along the fiber direction must be used with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 caution. For example, for Hercules AS/3501-6 graphite epoxy composites, measurements gave e T= 1 and e"=:25000 (Table G.2 in Appendix G). The high e"/e' absorption rate, and consequently a high heat-up rate by this material. However, because the material ratio implies a high energy high absorption rates do not occur is also a good reflector by the high value of e"), (as manifested and hence only a small the electromagnetic wave penetrates fraction of into the material. 4.3.2 Measurement of the Reflectances The reflectances were measured by placing the material between the slotted section and the solid section of the waveguide. With the wave generator turned on, VSWR was determined in the manner described in the previous section. The ratio of the incoming and the reflected electric fields (reflection coefficient) expression was calculated by the [25] VSWR - I V5 WR + / The ratio of the incoming energy energy flux (reflectance) ( 116) flux and the reflected is [10] 2 r ( 117) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 4.3.3 Measurement of the Transmittances The material was placed between the slotted and the solid section of the waveguide. turned on, the electric With the wave generator field intensity was measured with a probe positioned in the solid' section. The signal from the probe was displayed on the standing wave indicator decibels (dB). material in the waveguide. The measurement was repeated without the the incoming electric coefficient) in The ratio of the transmitted to field strength (transmission was calculated by the expression , t - [25] in-J-o(dZi-d e <) 10 (118) where dB^ and d B 2 are the dB readings with and without the material in the waveguide, respectively. The ratio of the transmitted and the incoming energy flux (transmittance) T = is given by [10] t (119) 4 .4 Microwave Curing Two types of experiments were performed with the microwave oven. One was to measure the temperature distributions during the cure of graphite epoxy and glass epoxy laminates; the other one was to measure the resin flow from laminates made of unidirectional graphite epoxy prepreg R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 tapes. During the temperature measurements, constantan) type T (copper- or type J (ir o n - constantan) thermocouples were imbedded into the specimens at different locations. thermocouple wires were shielded with either or inconel overbraids. The stainless steel During the temprature measurements, pressure was applied to the graphite epoxy samples, pressure was applied to the glass epoxy samples. while no The microwave oven was turned on and the temperatures as functions of time were recorded. The amounts of resin flow were measured using unidirectional graphite epoxy specimens. Aluminum foils extending approximately 5 mm from the edges were wrapped around each edge of the graphite epoxy specimens. The purpose of the foil was to minimize the buildup of the electric field concentrated around the edges. foils were perforated to permit resin foils. A thermocouple was also Before each test, The aluminum flow through the imbedded in the specimen. the composite layups and the bleeders were weighed on a Mettler analytical balance. The composite specimen was then placed in the press. Bleeders were placed on the top of the specimen and along the edges perpendicular to the fibers. Cork dams were placed along the edges parallel to the fibers. The air bag was then pressurized to the desired level and the press was inserted into the microwave oven. Next to the press, a small cup of water was placed in the oven to prevent possible buildup of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 the electromagnetic field. After the timer was set, During cure, the composite shrinks in height; calibration tests showed that changes the oven was turned on. in the force however, there were no appreciable (pressure) applied to the composite, due to the motion of the middle plate caused by the shrinkage. After the desired cure time was reached, removed from the oven, the press was the pressure was released, and the composite-bleeder assembly was cooled to room temperature. The weights of the composite and the bleeder were then measured. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SECTION V EXPERIMENTAL VALIDATION OF THE MODELS Experiments were performed to generate data which can be used to evaluate the accuracies of the electromagnetic, thermochemical, and resin flow models. Fiberite S2/9134B glass epoxy and Hercules AS/3501-6 graphite epoxy prepregs were used in the tests. The material properties used in the numerical calculations are listed in Appendix G. 5.1 Electromagnetic Model The reflectances of uncured glass epoxy and graphite epoxy composites were measured using a' waveguide. M e a s urements were performed with composites consisting of different and number of plies having different ply orientations, for different values of the polarization angle, ranging from 0° to 90°. In addition, functions of thicknesses (number of plies) 6, the transmittances as were measured for graphite epoxy composites. The data are given the in Figures 17-21. In these figures, results of the electromagnetic model are also included and are represented by solid lines. In the calculations, the values of the complex dielectric constants e* specified for free space and for the materials (Appendix G) had to be 73 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 m odified to account for the fact that the waves waveguide travel in a confined space. dielectric constant in the The complex in the waveguide eW g is related to the complex dielectric constant in free space e* by the expression * * = 6 (e*g = 0.28), (120’ in free space (Xg = 12.24 cm for and W is the width of the waveguide = 7.214 cm for S band waveguide were calculated 2o \ " f e i where Xg is the wavelength 2.45 GHz microwave), / [21]). (w The values of e*„ wg for air in front of and behind the material for the Fiberite S2/9134B glass epoxy prepreg in the direction parallel r (e„„ = 4.7 - j0.42) and wg J *$• perpendicular to the fibers (eWg = 4.5 - j0.40), and for the Hercules AS/3501-6 graphite epoxy prepreg in the directions parallel and perpendicular to the fibers and' 32.3 - j53.3). (e*g = 0.3 - j25000 The results of the model in Figures 17 - • • 21 were obtained with these e,,„ values. wg As can be seen from the Figures 17 - 21, the agreements between the data and the results of the model are excellent. This creates confidence in the accuracy of the electromagnetic model. 5,2 Thermochemical and Resin Flow Models The thermochemical and resin flow models had already Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 0.2 S 2 / 9 I 3 4 B (UNCURED) REFLECTANCE, oc O DATA -M O D E L 8=90 0 Figure 17 2 4 NUMBER OF PLIES, M Reflectances of Fiberite S2/9134B Glass Epoxy Uncured Unidirectional Composite as Functions of the Number of Plies. Com parisons between the Data and Results Com puted by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, o). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 1.0 8 =0 0.8 0.8 UJ LlJ H 06 8=90 o DATA MODEL 0.2 O Figure 18 A S /3 5 0 1 -6 (UNCURED) [0 ]M 2 8 10 4 6 NUMBER OF PLIES, M 12 0.6 o 2 £ h- 0.4 cn 2 < oc 0.2 14 Reflectances and Transmittances of Hercules AS/3501-6 Graphite Epoxy Uncured Unidirectional Composite as Functions of the Number of Plies. Comparisons between the Data and Results Computed by the Model. Data were Generated in a Wave guide with Incident Linearly Polarized TEM Waves (Polarization A n g l e , ,<5) . ar\ 0.2 S 2 /9 1 3 4 B (UNCURED) [0 /9 0 ] REFLECTANCE, 8=90 DATA MODEL 0.1 - 6 8 10 NUMBER OF PLIES, M Figure 19 Reflectance of Fiberite S2/9134B Glass Epoxy Uncured Cross-ply Composite as a Function of the Number of Plies. Comparison between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, 5). R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 78 A S / 3 5 0 1 -6 (UNCURED) 0.8 [0] REFLECTANCE a: 0.6 0.4 O DATA — MODEL 0.2 30 60 POLARIZATION ANGLE , 8 (degree) Figure 20 90 Reflectance as a Function of Polarization Angle 6 for Hercules AS/3501-6 Graphite Epoxy Uncured Single Ply Composite. Comparison between the Data and Results Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A S /3 5 0 1 -6 (UNCURED) 8=90° o 0.6 [ 0 /0 ] <-> 0.4 o Data — Model * 0.2 0 VO 30 60 ORIENTATION OF SECOND P L Y , Figure 21 90 6 { deg re e ) The Change in Reflectance with the Orientation of the Second Ply, o, for a Two-ply Hercules AS/3501-6 Graphite Epoxy Uncured Composite. Comparison between the Data and Resulus Computed by the Model. Data were Generated in a Waveguide with Incident Linearly Polarized TEM Waves (Polarization Angle, 6). 80 been tested by Loos and Springer were performed here [15]. Additional tests to evaluate these models when coupled with the electromagnetic model and when applied to microwave curing. The tests were performed in a microwave oven. unidirectional composites, Using temperature distributions and resin flows were measured parallel and perpendicular to the tool plate. In the experiments with glass epoxy composites, pressure was not applied and the resin flow was not measured. In the tests with graphite epoxy composites, pressure was applied at the beginning of the cure and was kept constant at 446 k P a ( a b s )(64.7 psia) during the cure. The pressure pressure in the bleeder was m a intained at ambient (101 kPa or 14.7 psia). The temperatures as functions of time measured at two different locations inside 32 ply composites are shown in Figure 22. The results of the resin flow measurements are presented in Figure on the abscissa. 23. In this figure, time t is plotted The ordinates represent either the total mass loss of the composite or the mass losses due to resin flow in the directions normal and parallel to the tool plate in time t . The mass losses shown in Figure 23 represent the mass loss with respect to the initial mass of the composite ( 121 ) initial noass R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 150 AS/3501-6 [0]32 x/d =0.5 x/d=0.875 50 105 TIME (min) 150 S2/9I34B 32 o x/d = 0.5 Ui (T 100 D x/d =0.875 £ tr UJ a. 2 tu 5 0 I- o Data 230 — Model TIME (min) Figure 22 Temperature as Function- of Time during Microwave Curing of 32-ply Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6 Graphite Epoxy Composites. Compar isons between the Data and Models. Cure Assembly Shown in Figures 15 and 24. Power Inputs to the Composites were as Indicated. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 10 0 Q. A S /3 5 0 1 -6 £ 32 20 o — DATA MODEL 105 TIME (min) Figure 23 Mass Losses Normal (top) and Parallel (center) to the Tool Plate, and the Total Mass Loss (bottom) as Functions of Time during Microwave Curing of 3 2-ply Hercules AS/3501-6 Unidirectional Composites. Comparisons be tween the Data and Results Computed by the Models. Cure Assembly is Shown in Figures 15 and 24. The Power Input W i n , Cure Pressure, P Q , and the' Bleeder Pressure, Pjj, are as Indicated. The initial Resin Content was 42%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 The temperature in the composite and the resin flow were calculated by the models. The microwave power transmitted into the material was less than the power output of the oven and was unknown. The power transmitted into the composite was determined by matching the results of the model to the first three temperature data points. This power level was then used in all subsequent calculations. In calculating the temperature a) inside the composite, the outside surfaces of the top and bottom teflon plates were taken to be at ambient temperature (Figure 24), b) the isotropic microwaves were taken to impinge equally on both sides of the cure assembly, c) and microwave energy absorption by all the layers surrounding the composite was neglected. In addition, for glass epoxy composites, energy release by chemical reactions was assumed to be negligible. The results of the models are represented by solid lines in Figures 22-23. . The calculated and measured temperature distributions and resin flows agree well. These agreements tend to confirm the validities of the combined electromagnetic-thermochemical-resin flow models and to demonstrate the usefulness of these models in simulating microwave curing of composites. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AMBIENT TEMPERATURE * PRESSURE, P0 I W I I I 3 2 mm GLASS REINFORCED TEFLON PLATE BLEEDER COMPOSITE GLASS REINFORCED TEFLON PLATE 2 5 mm AMBIENT Figure 24 TEMPERATURE Components of the Cure Assembly Used in Modelling the Temperature Distribution and the Resin Flow during Microwave Cure. Complete Cure Assembly is Shown in Figure 15. 00 S E C T IO N VI NUMERICAL RESULTS The models were used to generate illustrate the major composite material results which features of electromagnetic wave- interactions and of microwave fiber-reinforced organic matrix composites. curing of Calculations were performed for Fiberite S2/9134B glass epoxy and Hercules AS/3501-6 graphite epoxy composites. The material properties listed in Appendix G were used in the calculations. All the calculations were performed for electromagnetic waves having a frequency of 2.45 GHz. 6. 1 Electromagnetic Wave-Composite Material When an electromagnetic wave material, a fraction of the wave absorbed by the material, through the material. Interactions impinges on a composite is reflected, and the remainder The reflectance, a fraction is is transmitted transmittance, and amount of absorbed energy depend on the complex dielectric constants of the material, orientations, the number of plies, the ply and the characteristics of incident electromagnetic wave. The reflectance, the transmittance, and the amount of absorbed energy must be calculated by the model (Sections 2 and 3). Results, obtained for sample 85 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 problems, are presented subsequently. First, reflectance of a single ply is examined. however, The results the for a single ply aid in understanding the phenomena which occur when electromagnetic waves impinge on multilayered composites. The reflectance at the front face of a single ply composite exposed to a linearly polarized T EM wave is (Appendix H) (122) As indicated by the above equation, the reflectance R g depends on the complex dielectric constants parallel and perpendicular to the fibers, e $ Cr $ and e , and on the M polarization angle 5. When the complex dielectric constants nearly the same, $ $ £p and Eg are the reflectance becomes insensitive to the polarization angle 6. This is the case, for example, Fiberite S2/9134B glass epoxy composites(Figure for 25). When the complex dielectric constants differ considerably the fibers, in the directions parallel and perpendicular the reflectance becomes a strong function of the polarization angle. This of is the case for the Hercules AS/3501-6 graphite epoxy composite, one hand, to considered here. for polarization angle 6 = 90°, low and a large fraction of the wave On the the reflectance is transmitted across R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. is 67 . 1.0 1.0 S 2 /9 1 3 4 B cl: 2.45 GHz Ll I u 0.6 0.6 [0 ] 2 2 y 0.8 (CURED) 0.4 0.4 ul LU K 0.2 - d a .3 i— i■ 0.2 0 0.8 0.8- cr 0.6 0 0.6 2 < 0.4 0.4 01 0.2 0.2 uj _i u. UJ 30 60 TRANSMITTANCE, Tr 0.8 TRANSMITTANCE . Tr U 90 POLARIZATION ANGLE, 8 (degree) Figure 25 The Variation in Reflectance and Transmittance with Polarization Angle, 6, for Single Plies of Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6 Graphite Epoxy Composites Exposed to Linearly Polarized TEM Waves. Results of the Model. Material Properties Listed in Appendix G. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 the interface (Figure 25); on the other hand, the reflectance is nearly unity, wave is reflected. Thus, for 6 = 0°, and practically the entire significant amounts of electromagnetic energy can be transmitted into unidirectional graphite epoxy composites only when polarization angle 8 electromagnetic wave is 90°. Most, or all, is reflected the of the from graphite epoxy composites when the polarization angle is different than 90° for any given ply. Therefore, most of the electromagnetic wave is reflected when the composite is composed of multidirectional plies, as illustrated in Figure 26 and 28. The reflectance and transmittance of multilayer glass epoxy laminates are shown in Figure 27. Neither the ply orientation nor the polarization angle significantly affects either the reflectance or transmittance. above, the reason for this As discussed is that the values of the complex dielectric constants parallel and perpendicular fibers are nearly the same. electromagnetic waves, Thus, to the from the point of view of the material behaves in a q u a s i isotropic manner. As shown in Figure 27, for glass epoxy composites, the reflectance' increases and the transmittance decreases with thickness (number of plies). These trends are valid for laminates consisting of less than 80 plies. laminates consisting of more than 80 plies, reversal in these trends; transmittance increases there For is a the reflectance decreases and the slightly with the thickness. This Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 0.8 ISOTROPIC REFLECTANCE cc 0.6 8 =9 0 0.4 A S /3 5 0 1 -6 (CURED) 2 .4 5 G Hz 0.2 [ 0 /0 ] 30 60 ORIENTATION OF SECOND PLY, 0 Figure 26 9 90 (degree) Reflectances of a Two-ply Hercules AS/3501-6 Graphite Epoxy Composite Exposed to a Linearly Polarized TEM Wave (Polarization Angle, 6 = 90°) and to an Isotropic Wave. Results of the Model. Material Properties Listed in Appendix G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 _S2/9!34B (CURED) 2.45 GHz CO1 o 0 °< S < 9 0 ° AND ISOTROPIC UJ~0.5 (j 2 < P- \- 2 1.0 cn 2 < Lu 0.5 {£ 0.5 [ 0 /9 0 ] AND [ 0 / ± 4 5 / 90) 0 Figure 27 50 100 NUMBER OF PLIES, M 0 50 100 NUMBER OF PLIES, M Reflectances and Transmittances of Unidirectional, Cross-ply and Quasi-isotropic Fiberite S2/9134B Glass Epoxy Composites Exposed to Linearly Polarized TEM Waves (Polarization Angle 0° <_ 6 <_ 90°) and to Isotropic Waves. Results of the Model. Material Properties Listed in Appendix G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 ^ I 0.8— ^ 0.6 _ / \ y cr f ixJ 0.4 o ~ / 0.8 ><00 4 5 ° AND ISOTROPIC / ? 0.2 o 8=90° r1 0 ! i_ UJ uj 1.0 cr 0.8 I i A S /3501-6 (CURED) 2 .4 5 GHz [0 ] — . 1 1 0.4 AND l [o /9 Q ] 4 5 ° AND ISOTROPIC t 0.2 co k 0 °£ 8 < 90 ° AND ISOTROPIC I----- 1 0.6 8 = 90 ------------------------------ — Figure 28 £ 0.2 ID 0 °< 8 < 9 0 ° AND ISOTROPIC 0.1 [ 0 /± 4 5 / 9 0 ] I I 20 40 NUMBER OF PLIES, M I 60 0 40 NUMBER OF P LIES , M 20 60 Reflectances and Transmittances of Hercules Unidirectional, Cross—ply, and Quasi-isotropic AS/3501-6 Graphite Epoxy Composites Exposed to Linearly Polarized TEM Waves and to Isotropic Waves. Results of the Model. Material Properties Listed in Appendix G. 92 reversal in the trends transmittance reversal in the reflectance and in the increases slightly with the thickness. in the trends This in the reflectance and in the transmittance are caused by the interactions between the reflected and transmitted waves inside the material. For multilayered graphite epoxy composites, reflectance is very high (R = 1), except the for unidirectional composites exposed to linearly polarized TEM waves having polarization angle, laminates, 6=90° (Figure 28). Even for such the reflectance reaches 0.7 for laminates consisting of more than 32 plies. graphite epoxy composite both reflectance The transmittance of the is low because, for this material, (Figure 28) and absorption (Figure 29) are high. The absorbed energies as function of position across 16, 32, and 64 ply unidirectional uncured glass epoxy and graphite epoxy composites are shown in Figure interest It is of to note that the amount of energy absorbed per unit area per unit time may increase or may decrease across the thickness from the front towards the back. energy 29. The absorbed increases across the laminate when absorption across the laminate is low, so that a significant wave reaches the back surface. A portion of this wave reflected back into the material. contributes fraction of the is This reflected wave to the amount of absorbed energy, causing the higher absorption rate near the back surface. The absorbed energy decreases across the laminate when absorption is so high that most of the wave is absorbed R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [0 ] S2/9I34 B (UNCURED) 0.01 o tr LxJ 32 Ld AS/3501-6 (UNCURED) 2.45 GHz 8 = 90° 64 S 0.02 CD c r o if) WAVE GO < m= 1 m = M 0.01 30 0 10 PLY NUMBER , m Figure 29 20 30 40 50 60 The Absorbed Energies in the m-th Ply of Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6 Graphite Epoxy Unidirectional Composites Exposed to Linearly Polarized TEM Waves. Results of the Model. Material Properties Listed in Appendix G. 94 before it reaches the back surface. reflection from the back In this case, surface is insignificant, and the absorption is mainly due to the wave traveling from the front towards the back. absorption Under this condition, the decreases nearly exponentially with thickness. The results discussed in the foregoing and illustrated in Figures 25-29, must be born are of importance in microwave curing, and in mind when applying microwave to the curing of composites. Calculations were also performed to illustrate the effects of coatings (niade of homogeneous, dielectric materials) on the sides of the composite. covered by coatings are isotropic front, on the back, or on both The reflectances of composites illustrated in Figure 30. The reflectances change significantly with the values of the dielectric constant e^, and the thickness of the coating H. With appropriate choices of and H, the reflectance may be reduced to practically zero or may be increased to a value which is higher than the reflectance of the composite without coating. course, An increase in reflectance results, of in less energy being transmitted into and absorbed by the material. Reduced reflectance results in more energy being transmitted into and absorbed by the material. 6.2 Microwave Curing - General Considerations On the basis of the results presented Section in the previous (Section 6.1), the following major conclusions can R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .S 2 /9 I3 4 B 2.45 GHz 8 =90° WAVE WAVE Hd H :H , WAVE .A S /3 5 0 1 -6 1.0 - 1°]64 UD <J1 -€> 0 0.8 ^ UJ =4 o z 0.6 o y 0.4 Ll UJ 0.2 i. 0 Figure 30 0.5 1.0 1.5 2.0 0 .5 2.0 0 0.5 COATING THICKNESS. H (cm) 1.0 1.5 2.0 Reflectances of fully cured 64 ply Fiberite S2/9134B Glass Epoxy (d=0.96 cm) Hercules AS/3501-6 Graphite Epoxy (d=0.77 cm) Unidirectional Composites Covered with a Homogeneous Isotropic Material on the Front (Left), Back (Middle), and Both Front and Back (Right). Linearly Polarized TEM Wave Incident on the Front Surface. Results of the Model. Properties of the Composites Given in Appendix G. The Dielectric Constant of the The Dissipation Factor of the Cover is e " = 0. Cover is 96 be made regarding microwave a) Glass curing of composites: fiber reinforced epoxy matrix composites may be cured effectively by microwaves regardless of ply orientation and polarization angle. b) Unidirectional graphite epoxy composites may also be cured by microwaves. c) The curing efficiency depends strongly on the polarization angle 6.Most effective curing is achieved with 6 = 90°. Graphite epoxy composites consisting of multidirectional laminate cannot be cured effectively by microwaves. d) The energy absorbed the material and Correspondingly, across the laminate depends on on the thickness. the induced temperature distribution across the laminate may be non-uniform during microwave cure, and may either increase or decrease along the thickness. e) The amount of microwave energy absorbed by the material can be increased or decreased considerably by coatings. 6.3 Microwave Curinq-Selection When curing a composite, selected in such a way that of the Cure Cycle the cure cycle should be the following requirements are satisfied: a) The temperature is nearly uniform across the material and does not exceed a prescribed limit at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 any position during the cure. b) The magnitude of cure pressure is sufficiently high so that all of the excess resin is squeezed out from every ply of the composite. c) The resin is cured uniformly, and the degree of cure is above a specified limit throughout the composite at the end of the cure. d) The composite is cured in the shortest time. The models and the corresponding computer code were used to generate results which illustrate the effect ofthe microwave power level on the curing process for a given material. The calculations were performed for Hercules AS/3501-6 unidirectional 32 ply graphite epoxy composites bounded by teflon tool plates on both sides (Figure 24). The material properties used in the calculations are given in Appendix G. Resin flow only in the direction normal to the tool plate was considered. The purposes of the numerical calculations were to show the main features of microwave curing, and to demonstrate the procedures that can be used to select the proper cure cycle. The procedure used to select an appropriate cure cycle during autoclave cure was described by Loos and Springer [14], The same general steps may be used in selecting the microwave cure cycle. selection indicated, Therefore, is not given in detail; each step of cure cycle only the major steps are with emphasis on those features which are unique R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 to the microwave curing process. The microwave impinging on the composite was taken to be isotropic with the frequency of 2.45 GHz. is used most commonly This frequency in microwave heating. The total amount of microwave power absorbed by the composite depends on the geometries of the oven and the composite, and on the properties of the composite. The fraction of the total energy absorbed by the composite can be determined experimentally, as was discussed In the subsequent calculations, absorbed by the composite assumed to be known. in Section 5. the total amount of power (designated as "power Unless noted otherwise, input") was the calculations were performed with microwave power set at a constant level until the temperature inside the material reached the prescribed m a x i m u m value T * then cycled on and off in a manner , . max The power was r such as to keep the temperature constant across the laminate at the prescribed Tmax v a ^ue (Figure 31). If the power were to be kept at a constant level after Tm „„ was max reached, the temperature r inside the composite w o u l d 'increase beyond the allowed maximum. In the present calculations, at 177C. The outer the value of T was set surfaces of the lower teflon plate plate) and the upper teflon plate were taken ambient temperature of 22C to be at the (Figure 24). The cure and the bleeder pressures used calculations are shown (tool in Figure 31. in the The cure pressure P q R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 Tmax = 177 C IS REACHED TIME (m in) a Q_ CURE PRESSURE: P0 2C UJ ir D CO BLEEDER PRESSURE Pb= 16.7k Pa ( 5 in. H g) ir a. TIME (min) Figure 31 Illustration of the Cure Cycle Used in the Parametric Study of Microwave Curing. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 was applied at the beginning of cure process remained constant (x=0) and for the duration of the cure. magnitude of cure pressure is indicated showing the results. The pressure taken to be constant at 16.7 kPa. The in each figure, in the bleeder This value was is typical of the bleeder pressure used in vacuum bagging procedures. Calculations were also performed using the cure cycle recommended by manufacturer for Hercules AS/3501-6 graphite epoxy prepregs. is shown in Figure 32. This cycle Temperature Inside the Composite. During the cure, it is required that the temperature distribution across the composite be reasonably uniform, and that the temperature does not exceed the prescribed maximum value at any point. This condition can be satisfied by selecting the appropriate power input level and the appropriate power cycling after Tmax *s reac^e<^* As input levels result distributions. result illustrated in Figure 33, higher power in less uniform temperature However, higher power input levels also in a faster temperature rise and, hence, cure time. Thus, the power input level as to ensure a reasonably uniform in a shorter should be chosen so temperature distribution and a reasonably short cure time. It is interesting to note that the temperature distribution and the cure time depend mainly on the microwave energy input level input, (Watt) and are insensitive to the power (Figure 34). Thus, nearly the same temperature distributions are achieved by employing a lower R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 400 A S /3 5 0 1 -6 300 200 50 100 TEMPERATURE (F ) TEMPERATURE(C) 200 mmm 800 400 PRESSURE 600 (psi) 100 — 50 ABSOLUTE ABSOLUTE PRESSURE(kPa) CURE PRESSURE 200 BLEEDER PRESSURE 0 ■---------- 1---------- L 0 50 100 Figure 32 . I ..... 1.......... 1 150 200 150 TIME (min) 300 Manufacturer's Recommended Cure Cycle for Hercules AS/3501-6 Prepreg [27]. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A S /3 5 0 1 -6 [ 0 ] Tmax= <77C 200 ~Win=50W P0 = 1479 kPa Pb= 16.7 k Pa "Win = <00 W 20 60 O I— UJ 150 cr 30 ID <C cr 20 UJ CL 100 r = 1 min T= 10 min T= 2 0.5 Figure 33 1.0 0 0.5 POSITION, x /d 1.0 0 0.5 Temperature Distribution Across the Composite as a Function of Time for Different Levels of Microwave Power Input. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 1.0 200 W *t ♦♦♦ 200l \dTm m - 1001 s 20 P0 = 1479 kPa W Pb = 16.7 kPa 400T W lo l* 44 s 20 2 6s 20 o 10 \- 10 10 T = 2 min T = 2 min r 150 tr z> LU <i oc 5 LlI Q. UJ 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A S /3 5 0 1 -6 Tmax= (77 C 100 T = 2 min I i i i 0 i 1 i 0.5 i i i J 1.0 0 I L J I 0.5 I J L 1.0 0 I I L J 0.5 I I L 1.0 POSITION , x /d Figure 34 Temperature Distribution Across the Composite as a Function of Time for Different Power Cycles. Results Obtained by the Models. 104 power input maintained at a constant level, or higher power input, cycled on and off. Heat generated by the absorbed microwave energy is transferred out of the material. The amount of heat transferred can be reduced considerably by placing thermal insulators on both sides of the composite. Thermal insulation results in a more rapid temperature (Figure 35). increase In turn, a rapid temperature rise results in a reduced cure time. Gel Po i n t . Excess resin must be squeezed out of every ply before the gel point of the resin any point inside the composite. is reached at The computer code can be used to generate the viscosity distribution composite (Figure 36). inside the From this information, viscosity at any point inside the composite, can be determined, the maximum at any time, and a plot of maximum viscosity ym lil_v dA versus time can be constructed as illustrated in Figure 37. The gel point of the resin is assumed to occur when the viscosity of the resin reaches a certain value. Thus, knowing the viscosity corresponding to the gel point, by the time when gel occurs can be determined from the maximum viscosity umax versus time curve, as shown in Figure 37. For the present calculations, gel was taken to occur when the degree of cure a reaches 0.5 [13]. this degree of cure is about 7 Pa*s at viscosity, The viscosity at 177C [13]. For this Figure 37 indicates gel times of 21 and 13 minutes for 100 and 200 Watt power inputs, respectively. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 Thermally Insulated 20 o H 150 L u l tr 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission Not Insulated P0 =1479 kPa Pu =16.7 kPa A S /3 5 0 1 -6 T"= 5 min CL § 100 0 Figure 35 T = 1min 0.5 POSITION, x /d 1.0 0 0.5 POSITION, x /d 1.0 Temperature Distribution Across the Composite as a Function of Time for a Composite Thermally Insulated (Left) and for a Composite without Thermal Insulation (Right). Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 106 Gel times were calculated for different power inputs, the results of which are given in Figure 38. Resin Flow. Excess resin must be squeezed out from every ply of the composite before resin This resin flow results in any ply gels. in the compaction of plies [14], The number of compacted plies n g as a function of time is shown in Figure 39. In this figure, the number of compacted plies which results from the autoclave cure cycle recommended by the prepreg manufacturer is also indicated. As can be s e e n ,'microwave curing results in much faster compaction than autoclave curing with the manufacturer's cure cycle. Furthermore, the manufacturer the cure pressure recommended by (Figure 32) is insufficient to compact every ply in a 32 ply composite. kPa (160 psia) is required to compact all before gel time is reached Degree of Cure. process, the 32 plies (Figure 39). At the completion of the curing and the degree of cure should exceed a prescribed throughout the composite. The degree of cure will be uniform as long as the temperature distribution the composite is uniform Conversely, inside (Figures 33 and 40). if the temperature distribution composite becomes nonuniform, expected to be nonuniform. result 1100 the resin in the composite should be cured uniformly, value a A minimum pressure of inside the the degree of cure can also be Therefore, cure cycles which in uniform temperature distributions also result composites that are cured uniformly. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. in i 07 A S /3 5 0 1 - 6 20 V) 8. io ° - 5. - *» > hcn oo T =10 min to > P0 = 1479 kPa Pu = 16.7 kPa 0 .2 0 .4 0 .6 0.8 POSITION, x/d Figure 36 Viscosity Distribution at Different Times Inside the Composite. Result Obtained by the Models for the Cure Cycle Shown in Figure 31. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 A S /3 5 0 1 -6 J 32 s) max = 177 C MAXIMUM VISCOSITY, U ma)( (Pa W- = 100 W _________________ |_______ 0 10 u_ 20 TIME, T(m in) Figure 37 The Maximum Viscosity Inside the Composite as a Function of Time. Gel is Assumed to Occur When Viscosity Reaches 7 Pa*s. Results Obtained by the Model for the Cure Cycle Shown in Figure 31 at Power Inputs of 100 W and 200 W. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. [O]- c E ® 20 o» l±J 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A S /3 5 0 1 -6 30 _1 LU CD 10 t 0 7max = 177 C P0 = 1479 kPa Pb = 16.7 kPa 1 100 200 POWER INPUT, Wjn (W) Figure 38 Gel Time as a Function of Power Input. Result of the Models for the Cure Cycle Shown in Figure 31. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A S /3 5 0 1 -6 32 « 30 Tgei Win = 100W (P0 = 1100 k Pa) Tgel _i a_ Wjn=100W (P0 = 687kPa) 20 o MANUFACTURER'S CURE CYCLE (AUTOCLAVE CURE:P0 = 687 kPa =85psig) (P0 = 687 kPa) Tmax= 177 C P K = 16.7 kPa LlJ 00 0 50 100 49 150 TIME , T (min) Figure 39 Number of Compacted Plies as a Function of Time for Different Microwave Power Inputs and Different Cure Pressures. The Results Obtained by the Models for the Cure Cycle Shown in Figure 31. The Result Shown for the Autoclave Cure Cycle is from Reference [13]. 111 0.8 AS/3501-6 P0 =1479 kPa Pb =16.7 kPa ’ win =100 W | l i t 0.6 20 W ac o 0.4 15 LU LU CC o T=10 min y 0-2 — — i — i — i i 1 _L_ . 1 0.5 1 1 l-.O POSITION, x/d Figure 40 Degree of Cure Distribution Across the Composite as a Function of Time. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 The computer code can be used to generate results such as those shown in Figure 40. From these results, of cure as a function of position be determined. the degree inside the composite can Once this information is known, it can be determined if the composite was cured uniformly, and if the degree of cure exceeds the prescribed value of a throughout the composite. Cure Time. When curing a composite, that the curing process be completed of time. it is desirable in the shortest amount The cure is considered complete when the degree of • cure reaches a specified value a composite. at every point , m the The time required to reach this value can be established by first plotting the degree of cure as a function of position and time, as shown in Figure 41. this curve, From the lowest value of the degree of cure am -n in the composite at each time is determined, and a plot of the lowest value of the degree of cure as a function of time is constructed. The cure is considered complete when a reaches the specified value a shown in Figure 41, process . From a plot such as that the time required to complete the curing can be determined. For example, for a 32 ply unidirectional AS/3501-6 graphite epoxy composite, is reached in in 21 minutes with a power 13 minutes with a power input of input of 200 Watt After the gel time is reached, a = 0.9 100 Watt, (Figure 42). the composite usually is removed from the oven, and is postcured without pressure being applied. and It is interesting to compare the time R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 Tr = 4 7 min a =0.9 c E a 0.8 T r = 52 min LU tr z> 0.6 o Win = 100W Win = 2 00 W Ll O LU IU tr o 0.4 A S /3 5 0 1 - 6 IU 32 o z> 2 0.2 Tmax = 177 C P0 = 1479 kPa PK= 16.7 kPa 2 0 20 40 60 TIME, T ( min) Figure 41 Minimum Degree of Cure as a Function of Time for Two Different Power Inputs. Results Obtained by the Models for the Cure Cycle Shown in Figure 31. 114 150 AS/3501-6 [°]32 P0 = 6 87 k Pa ( 85 p s ig ) Pb = 16.7 kPa Tmax= 177 C c 100 i a> o» M AN U FAC TU R ER / RECOMMENDED AUTOCLAVE CURE CYCLE UJ 2 h- 50 UJ o Win = 100 W Win = 200 W [7 A Figure 42 The Time Required to Reach the Gel point Using Microwave Curing with 100 W and 200 W Power Inputs and Using the Autoclave Cure Cycle Recommended by the Prepreg Manufacturer The Results for Microwave Curing were Obtained by the Models for the Cure Cycle Shown in Figure 31. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. required to reach the gel point by microwave curing and by autoclave curing, using the cure cycle recommended by the prepreg manufacturer. A nearly 10-fold decrease can be achieved by microwave curing. This in gel time suggests that considerable reductions in cure time are offered by microwave curing. Finally, it is noted that, in addition to promoting internal heat generation, microwaves structure resin matrix. of the polymer considered in this may also affect the This effect was not investigation. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S E C T IO N V II SUMMARY AND CONCLUSIONS The following major tasks were completed during the course of this investigation: 1) Models were developed to describe the interaction of linearly polarized transverse electromagnetic waves and isotropic electromagnetic waves with continuous fiber reinforced organic matrix composites. model provides the reflectance, absorbed energy, The transmittance, total and absorbed energy distribution inside the composite. ■2) Models were developed which simulate microwave curing of fiber reinforced thermosetting resin matrix composites. power The models relate the microwave input and the cure pressure to the thermochemical and physical processes occurring in the composite during cure. Specifically, provide the following information the models for flat-plate composites cured by a specified cure cycle: a) the temperature T inside the composite as a function of position and time? b) the degree of cure of the resin a as a function of position and time; 116 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 c) the resin viscosity -\i as a function of position and time; d) the number of compacted plies n g as a function of time; e) the amount of resin in the bleeder as a function of time; and f) the thickness and the mass of the composite as functions of time. 3) On the basis of the models, a computer code was developed which can be used to generate results. 4) Experiments were performed in a waveguide and in a microwave oven with Fiberite S2/9134B glass epoxy and Hercules AS/3501-6 graphite epoxy composites. a) Using a waveguide, the reflectances and transmittances were measured for composites having different ply orientations and different thicknesses exposed to linearly polarized TEM waves with different polarization angles. b) Using a microwave oven, temperature distribution across glass epoxy and graphite epoxy composites, and resin flow out of graphite epoxy composites were measured during microwave cure. 5) Calculations were performed with the computer code for the conditions employed in the experiments. calculated results-were compared with the experimental data. These comparisons showed that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 118 the models adequately describe the reflectance and transmittance during exposure to linearly polarized TEM waves, and the temperature distributions the resin flow out of flat plate, in and unidirectional composites during microwave cure. 6) A parametric study was performed to illustrate: a) how the reflectance, transmittance, and absorbed energy depend on the characteristics of incident electromagnetic waves, geometry and the (thickness and ply orientation of the composite, thickness of coating), and on the material properties of the composite; and b) how the models and the associated computer code can be used to determine the appropriate power level during microwave curing which results a composite that in is cured uniformly in the shortest amount of time. In this investigation, results were generated only for electromagnetic waves at 2.45 GHz. the computer code are general, entire However, the model and and can be used over the frequency range. In the present form, the computer code can be applied to flat plate geometry only. It could be readily extended to composites made in different shapes. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IC E S 119 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX A THE REFLECTION COEFFICIENT AT THE M-TH INTERFACE The reflection coefficient tensor expression (Eq. r^- is defined by the 10 in Section 2.2.1) (Ec)r " C y H E j) u.i> where the notation is same as in Section 2.2.1. magnetic materials $ (-p, =1), the electric For non- field vectors directions parallel and perpendicular to the fibers directions, E? = hr and e (p and q Figure 4) are related to the components of the magnetic field vectors by the expressions e in the [11] HP (A.2) ^ (A.3) are the complex dielectric constants in the p and q directions. Coordinate transformation gives the relationship between the vector components directions and in the /Ep\ _ / 1 and 2 directions in the p and q [29] (Figure 4) / E, “ * & ) \ E ZJ <A' 4) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 h U p H|/ Equations e s i " 6 \ / H ^ \-5l'n0 60S $ / \ (A.2) - (A.5) yield /E|W^6 - S i n 0 \ / > / ^ Ez* ISiV»d 0 \/C0S$ SM0\/Hi (A.6) £°*d/\ Hz By defining the complex refraction coefficient tensor _ \Si'r\d The electric 0 \/OoSd sind\ 0 yejjy(-£/>)0 coS0/ field vector may be expressed as E-l = Nij Hj For a nonmagnetic electric (A.7) (u =1) homogeneous (A.8) isotropic material, field vector is related to magnetic the expression the field vector by [11] = N Ht- (a.9) where (\j - y ^ R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.10) 122 At the interface of two different homogeneous materials (material m-1 and material m), isotropic the reflection coefficient is [9] -I r By analogy = ( Nm-I +- s. Nm) ( Nlm-i" N m ) (see Eqs. A . 9 - A . 11), at the interface separating two anisotropic materials, the reflection coefficient tensor is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A. 11) 123 A P P E N D IX B THE ATTENUATION TENSOR FOR THE M-TH PLY In the following, tensor A^j is derived. an expression for the attenuation For the m-th ply of the composite, the attenuation tensor A^j is defined as (see Eq. 16, Section 2.2.1) Bi (B. 1 ) = ( A ijK f p The notations are the same as used in the text (Section 2 .2 .1 ). Inside a homogeneous isotropic material, the attenuated electric field vector E^ can be expressed in terms of the electric field vector P^ entering the material E. - Pc exp ( -ydm) where dm is the thickness of the ply, main text composite, (Eq. 4). [28] By analogy, (B.2) y was defined in the for the m-th ply of the the components of attenuated electric vector parallel and perpendicular to the fibers field (Figure 4) can be expressed as (B .3 ) E? = P|. e * p Jm) (B .4 ) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 Pp and Pg are related to P^, the coordinate transformation / Pr \ _ \pj Similarly, Pj components of this vector by [29] (Figure 4) I^ s,'nA / P-'l l-sfnacoseA fj <B-5) Ep and Eg can be expressed in terms of the components of attenuated electric field vector in the 1 and 2 directions BF\ / c o s e I £ s.V>0\/E.\ (B6) / I We introduce now the following two parameters AP s = Equations (b.7) C*p(-y<j.d m ) (B-8) (B.3) - (B.8) E|\ to ' give CoS0 Si r)d\j Pi ,S\r\b tosfy/l o A^A'Sfnfi \ "Sin6\/Ap A comparison of Eq. expression e>p(-ypdn) (B.1) 0 \j with Eq. (B. 9) (B.9) yield the desired for the attenuation coefficient tensor R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 CoSQ -Sind A l j = Sfnft LoSft Af> o \ / *'n8 (B. 10) 0 Af-'\-Sin() 9 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 A P P E N D IX C RATE OF ENERGY ABSORPTTON PER UNIT VOLUME BY THE M-TH PLY The rate of energy absorption per unit volume by a homogeneous where E^ and a isotropic material is the electric is [11] field vector is the electrical conductivity o' - ( f is the frequency, material. inside the material, [11] z'irf to) £ (c.2 ) and e" is the dissipation factor of the Eg is a constant £o = Analogously, * I0‘Z F a r a d / m (c.3 ) we express the rate of energy absorption per unit volume by the m-th ply as (C .4 ) where ap and are the electrical conductivities parallel and perpendicular to the fibers, respectively. are the components of resultant electric E^ and E^ field vector Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the 127 p and q directions. (E®) and (E®) can be related to the tr resultant electric H field vector components in the directions by the coordinate transformation - E, 0 “Ej The +• (Figure 4) Si/10 (C • 5) l~2. 1 and 2 components of the electric were found in Section 2.2.4 [29] 1 and 2 (c.6) field vector (Eq. 51) E,s= -j ( P,+ +■Aij Rj) + j-( E* = (E®) \ ( ?l + Azi p/ ^ + J ( ) (c-7> + Az3 ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c- 8) 128 A P P E N D IX D NORMAL INCIDENT ENERGY FOR AN ISOTROPIC WAVE [30] When an isotropic electromagnetic wave impinges on a surface, the total £ incident energy flux ~'1 is ‘ - <d . i > where dft is the solid angle shown in Figure D.1, and is the intensity, i.e., angle dfi. isotropic wave, For an distributed, and (<S-1)^ the incident energy flux in solid the energy is uniformly independent of the direction of incidence. A ccord ingly, ( C L) - (D. 2) 1 0 'a Z r The energy flux incident normal to the surface in solid angle dft is d£i J The total energy = ( £ L) CoS0 dJ2 Si (D.3) incident normal to the surface is r Et = JI Z ir 3 cose dfi (D-«) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cos0 Figure D.l The Geometry Used in Calculating the Normal Incident Energy Flux for Isotropic Electro magnetic Waves. 130 The solid angle d& is (D.5) where angles Eqs. (D.4) 0 and <{> are shown and in Figure D.1. By combining (D.5), we obtain 2* % 0 Integration of Eq. r^L S' J0 ZTf CoS $ st"n & ( D - 6) (D.6) yields <f‘ (D.7) z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 A P P E N D IX E RELATIONSHIP BETWEEN PLATEN FORCE AND AIR BAG PRESSURE The relationship between the force generated by the press platens and the indicated air bag pressure was determined, using a calibrated load cell, . as described in Section 4.2. The load cell was calibrated by connecting a Baldwin (Model SR-4) 8.9 kN Ellis Associates (2000 l b f ) strain gage load cell to an (Model B A M - 1) bridge amplifier and meter. The output of the Ellis bridge amplifier was Fluke digital voltmeter. within ± 0.25% over recorded on a Since the load cell its operating range is linear (0-2000 to lbf), only two points are necessary to construct a calibration curve [31]. The first calibration point was obtained by balancing the amplifier and bridge circuits of the Ellis bridge amplifier at the no load cell). point (no force applied to the These circuits were adjusted so that load the output voltage of the Ellis bridge amplifier was zero at the no load point. The second point o'" une r-’libration curve was obtained by loading the cell universal testing machine. to the cell, „o 2000 lbf on ^n Xnjtron With a force of 2000 lbf applied the gain of the Ellis bridge amplifier circuit was adjusted to give a measured output voltage of 2.0 V. Thus, when the applied force on the load cell ranged from 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 lbf to 2000 lbf, amplifier the output voltage of the Ellis ranged from 0 V to 2.0 V. Therefore, the load cell could be determined directly bridge the force on from the output voltage measurements of the Ellis bridge amplifier. The relationship between the platen force and the air bag pressure was determined for three different distance between the top and middle plates ha ). The results are given microwave curing experiment, mm. After the cure, height (air bag height, in Figure E.l. as the plies compacted, The results In each the initial height was h_ = 21 increased by about 2 mm, at the most composites. values of in Figure E.1 the air bag , for 32 ply suggest that a 2 mm change in air bag height would not cause a significant change in the platen force. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 AIR BAG PRESSURE ( psig) 0 10 20 30 40 1000 (lbf) 6 FORCE -1500 ■nrr 4 500 2 0 0 100 200 PLATEN PLATEN FORCE (kN) 8 300 AIR BAG PRESSURE (kPag) Figure E.l Force Generated by the Air Bag on the Press Platens as a Function of Air Bag Pressure for Different Air Bag Heights (ha ) . R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 A P P E N D IX F ELECTRIC FIELD STRENGTH DISTRIBUTION INSIDE THE MICROWAVE OVEN The distribution of the electric field strength inside the microwave oven was determined by placing nine glass cups in the oven, each cup being filled with 50 ml of water. temperature of the water in each cup were measured after 60 seconds of heating at full power (700 Watt). uniformities in the temperatures are uniformities in the electric Non indicative of the non field strength. temperature rise in each cup after 60 seconds The is shown below. K:-------------- 40.6 c m -------------- IT 41.3 cm The 28 ,0C 19. 1C 22. 5C 19.6C 18.6C 16.4C 18.9C 20.3C 17.7C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 A P P E N D IX G PROPERTIES OF THE COMPOSITE AND SURROUNDING MATERIALS This appendix contains the properties of Fiberite S2/9134B glass epoxy and Hercules AS/3501-6 graphite epoxy cured and uncured prepregs. The properties of Mochburg CW1850 bleeder cloth and glass reinforced teflon plate are also given. The properties are tabulated in Tables G.1 G.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 136 Table G.1 Properties of Fiberite S2/9134B Glass Epoxy Prepreg Dielectric constant (uncured and c u r e d ) 1 parallel to the fibers 5.46 perpendicular to the fibers 5.21 Dissipation factor (uncured and c u r e d ) 1 parallel to the fibers 0.42 perpendicular to the fibers 0.40 Initial (uncured) fraction prepreg resin mass Initial (uncured) prepreg thickness of the Resin density 0.30 1 .50x10-4 m 3 1 .2 6 x 103 k g / m 3 Specific heat of the resin 3 1.26 kJ/(kg*K) 3 Thermal conductivity of the resin 4 Fiber density Specific heat of the fiber 0.17 W/(m«K) 2.7 x 103 k g / m 3 4 0.83 kJ/(kg*K) Thermal conductivity of the fiber4 0.76 W / ( m * K ) 1. Measured using uncured prepreg. The dielectric constants and dissipation factors for the cured and uncured resins are nearly the same [2], 2. Specified by the manufacturer prepreg. 3. From Reference [14]. 4. From Reference [32], for each shipment of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Table G.2 Properties of Hercules AS/3501-1 Graphite Epoxy P r e p r e g 1 2 Dielectric constant (uncured) 1 parallel to the fibers perpendicular to the fibers 2 Dissipation factor (uncured) parallel to the fibers 33.0 ~ perpendicular to the fibers Dielectric constant (cured) 25,000 53.3 2 parallel to the fibers 1 perpendicular to the fibers 2 Dissipation factor (cured) parallel to the fibers perpendicular to the fibers Initial (uncured) fraction 14.5 ~ 25,000 75.8 prepreg resin mass 0.42 Initial uncured thickness of the prepreg 1.651x10 ^ m Thickness of one compacted ply 1.19 4 x 10-4 m Apparent permeability of the prepreg normal to the plane of the composite 5.8x10-16 m 2 Flow coefficient to the fibers of the prepreg parallel 170 Resin density 1. 26x 103 k g / m 3 Specific heat of the resin 1.26 kJ/(kg«K) Thermal conductivity of the resin Heat of reaction of the resin Fiber density Specific heat of 0.167 W/(m*K) 473.6 kJ/kg 1 .79x 103 k g / m 3 the fiber i .712 kJ/(kg *K) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 Table G.2 (cont.) Thermal conductivity of the fiber 26.0 W/(m«K) Relationship between the cure rate, temperature, and degree of cure see Table G.3 Relationship between viscosity, temperature, and degree of cure see Table G.3 1. Unless otherwise stated, [14]. all values are from Reference 2. Measured using a waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table G.3 Degree of Cure and Viscosity of Hercules 3501-6 Epoxy Resin [13] The relationship between the cure rate, temperature, and degree of cure is aX (K + K 2*)(I-c<)(B-°<) = K3(i-tf) <0. cX><3.3 where K 1 = A lex p ( - A E 1/RT) K 2 = A 2exp(-AE2/RT) K 3 = A 3exp(-AE3/RT) and A 1 = 2. 101x10^ min 1 A 2 = -2.014x10^ min 1 A 1 = 1.9 6 0 x 10 5 m i n -1 A E 1 = 8.07x104 J/mol A E 2 = 7.78x104 J/mol A E 3 = 5.6 6 x 104 J/mol R = 8.31434 J/mol B = 0.47 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 Table G.3 (cont.) The relationship between the viscosity, temperature, and degree of cure is ju - exp ( U / R T + Ko<) where 1^* 00 = 7 .93x 10“ 14 Pa *s U = 9.08 x 104 J/mol K = 14.1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 Table G.4 Properties of Mochburg CW1850 Bleeder Cloth and Glass Reinforced Teflon Mochburg Bleeder Cloth Apparent p e r m eability1 5 . 6 x 1 0 " 11 m 2 Porosity1 Density 0.57 2 0. 3 8 x 10 3 kg/m3 Specific heat 2 0.83 kJ/kg*K 2 Thermal conductivity 5.0 6 x 10 ~ 2 W/m-K Glass Reinforced Teflon Dielectric constant Dissipation Density factor 3 2.35 3 0.0035 4 2. 2 x 10 3 kg/ m 3 4 Specific heat 1.01 kJ/(kg*m) 5 Thermal conductivity 1. From Reference 0.40 W/(m*K) [14]. 2. Estimated using the rule of mixture with the properties of polypropylene [33], and the porosity value given above. 3. From Reference [23] for the frequency of 3.0 GHz. 4. Estimated using the rule of mixture with the properties of glass [32] and teflon [33], and the glass mass fraction of 20 %. 5. From Reference [34]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 A P P E N D IX H REFLECTANCE FOR THE FRONT FACE OF A SINGLE PLY COMPOSITE The reflectance R is defined by the expression (Eq. 31 in Section 2.2.2) KRj,0 R = The notation of the and is (H. 1 ) I(E D the same as in Section 2.2. incident electric The components field vectors are given by Eqs.(1) (2) (E*)0 - a0cosS expijwr-y*3) = a t SinS e/p (jwt ~y*i) The components of the incident electric («.2) (H.3) field vectors parallel and perpendicular to the fibers are given by the coordinate transformations <EfV [29] CoS0 -S inQ Si' nd C.°sd +\ -> (E R0 t (H.4) For a single ply of unidirectional composite with the fibers aligned in the 1 direction (0 = 0°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 (Ep) = ( E ^ o = a D ccs£ c * p ( j u > r - y t j ) ih.si (E?) =(El) = a0 sm$ c*p (ju>r'I*?,) (H. Similarly, the following equations hold for the components of the reflected electric (p ;)„ e = field vectors tensor (H.7) (H. 8) ( P i) 1 (free space value) coefficient (6 = 0°) ip;\ - (p,i ■ For 6) (Eq. and for 0 8 = 0 , the reflection A . 12) becomes 0 r.. ' ‘J (K.9) /€% 0 1 4 Also, from Appendix A (Eq. A.1), + we have ( P J, - ( Oj ) ( Ej ) By combining Eqs. < pr>; = (H.5) +■ to (H.10), (H.10) we obtain (epI irr7a (*s<k ( r T'i^ ief + c (H.11) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 (P ^ From Eq. (H.12), ^> (H.1), the 7 ^ -y^' 7 together with Eqs. following expression (H.5). (H.6), .1 2 ) (H.11)^ and is obtained .13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES 14 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 REFERENCES 1. Williams, N. H., "Curing Epoxy Resin Impregnated Pipe at 2450 MHz," Journal of Microwave Power, 2, 123-127 (1967). 2. Wilson, L. K. and Salerno, J. P., "Microwave Curing of Epoxy Resins," AVRADCOM Report N o . 78-46 (1978). 3. Gourdenne, A., Massarani, A.-H., Monchaux, P., Aussudre, S., and Thourel, L., "Cross-linking of Thermosetting Resins by Microwave Heating: Quantitative Approach," American Chemical Society, Division of Polymer Chemistry, Polymer P r e p r i n t s , 20, 4 71-474 (1979). 4. Strand, N. S., "Microwave Curing of Thermoset Resins," Society of Manufacturing Engineers Technical Paper EM79-368 (1979). 5. Gourdenne, A., Monchaux, P., Le Van, Q . , Aussudre, S., Thourel, L., Audo, N., and Nedelec, J., "Utilisation du Chauffage Dielectrique Micro-Ondes Pour La Preparation de Composites Epoxy-Verre et Epoxy-Graphite: Aspects Theoretiques et Practiques," in Proceedings of the Third International Congress of Composite M a t e r i a l s , Volume 2~| 1514-1520, Paris, France (1980). 6. Gourdenne, A. and Le Van, Q., "Intimate Study of the Microwave Curing," American Chemical Society, Division of Polymer Chemistry, Polymer P r e p r i n t s , 22, 125-127 <1981) 7. Haven, R. E., "New Approach to the High Frequency Electric Field Curing of Polymeric Composites," Ph.D. Thesis, Department of Mechanical Engineering, (1980) . MIT 8. Hayt Jr., W. H., Engineering Electr o m a g n e t i c s , McGrawHill (1981). 9. Bitter, F., Currents, (1956). Fields, and Particles, MIT Press 10. Liddel, H. M . , Computer-Aided Techniques for the Design of Multi-layer F i l t e r s , Adam Hilger Ltd. (198 1). 11. von Hippel, A. R . , Dielectrics and W a v e s , John Wiley Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 and Sons ( 1954 ). 12. Arpaci, V. S., (1966) . 13. Lee, W. I., Loos, A. C . , and Springer, G. S., "Heat of Reaction, Degree of Cure, and Viscosity of Hercules 3501-6 Resin," Journal of Composite M a t e r i a l s , 16, 510-520 (1982). 14. Loos, A. C. and Springer, G. S., "Curing of Epoxy Matrix Composites," Journal of Composite Materials, 135-169 (1983). Conduction Heat Transfer, Addison-Wesley 17, 15. Loos, A. C. and Springer, G. S., "Curing of Graphite/ Epoxy Composites," AFWAL-TR-83-4040 (1983). 16. Springer, G. S. and Tsai, S. W . , "Thermal Conductivities of Unidirectional Materials," Journal of Composite M a t e r i a l s , 166-173 ( 1967). 17. Forsythe, G. E. and Wasow, W. R . , Finite Difference Methods for Partial Differential E q u a t i o n s , John Wiley and Sons ( 1960 ) . 18. Rohsenow, W. M. and Hartnett, J. P.“, Handbook of Heat T r a n s f e r , McGraw-Hill ( 1973). 19. Carnahan, B., Luther, H. A., and Wilkes, J. 0., Applied Numerical M e t h o d s , John Wiley and Sons (1969). 20 . Heavens, 0. S., "Measurement of Optical Constants of Thin Films" in Physics of Thin F i l m s , Volume 2 , Academic Press (1964). 21 . 22 . Laverghetta, T. S., Microwave Measurements and T e c h n i q u e s , Artech House (1976). Chakraborty, Dev. P. and Brezovich, I. A., "Error Sources Affecting Thermocouple Thermometry in RF Electromagnetic Fields," Journal of Microwave Power, V7, 17-28 (1982). 23. von Hippel, A. R., Dielectric Materials and A ppl i c a t i o n s , MIT Press (1966). 24. Ishii, T. K., Yen, Y. H., and Kipp, R. J., "Improvement of Microwave Power Distribution by the Use of the First Order Principle of Geometrical Optics for S c i e n t i M c > Microwave Oven Cavitv," Journal of MicrowaVe Power*, *14,* 201-208 (1979). 25. Laverghetta, T. S., Handbook of Microwave T e s t i n g , Artech House ( 1981) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 27. Hercules Incorporated, Aerospace Division, "Magnamite Graphite Fiber Presentation," Magna, Utah (recieved 1981). 28. McCleod, H. A., Thin Film Optical F i l t e r s , American Elsevier Publishing Company ( 1969) . 29. Cornbleet, 30. Sparrow, E. M. and Cess, R. D., Radiation Heat T r a n s f e r , Brook/Cole Publishing Company (1970). 31 . Baldwin Lima Hamilton Corp., Electronics Division, "Instructions SR-4 Load Cells," Bulletin 4360, Waltham, M a s s a c h u s e t t e s , March, 1961. 32. Baumeister, T., Avallone, E. A., and Baumeister III, T ., Mark's Standard Handbook for Mechanical E n g i n e e r s , McGraw-Hill (1978). 33. Harper, C. A., Handbook of Plastics and Elastomers, McGraw-Hill ( 1 9 7 5 T 34. Manufacturing Handbook and Buyer's Guide Plastic T e c h n o l o g y , 29, Mid-June, 1983. S., Microwave O p t i c s , Academic Press (1976). 1983/1984, R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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