EXPERIMENTAL CHARACTERIZATION OF THE VISCOELASTIC BEHAVIOR OF A CURING EPOXY MATRIX COMPOSITE FROM PRE-GELATION TO FULL CURE by RYAN J. THORPE B.A.Sc. (Integrated Engineering), University of British Columbia, 2010 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Materials Engineering) The University of British Columbia (Vancouver) April 2013 © Ryan J. Thorpe, 2013 Abstract Process models have been used to predict the flow and stress development during manufacturing of composite structures for over 20 years. To date, these models have been treated separately, and the required material property models (viscosity and modulus) have also been treated separately. The latest breakthrough in process modeling of composite structures is an integrated stress-flow model. However, a consistent viscosity and viscoelastic material model required by the integrated stress-flow model has not been developed. Presented in this thesis is a consistent material model that predicts the viscoelastic liquid and viscoelastic solid behavior of a commercially available thermoset polymer, namely MTM45-1 epoxy. The goal here is to show that a single material model can predict the viscosity and viscoelastic modulus for all temperature, degree of cure and time scales encountered in composite manufacturing. The model was generated by fitting a generalized Maxwell model to test results from both dynamic mechanical analysis and rheological tests. Both resin and prepreg samples were examined. Thermorheological complex behavior was captured by applying linear temperature dependence to the unrelaxed modulus. The effect of cure was accounted for by applying a degree of cure dependent shift function. The relaxed modulus was predicted using the cross-link concentration and the theory of rubber. Excellent agreement was found when comparing predictions from the model to experimental data ranging from temperatures of -50°C to 245°C, degree of cure of 0.01 to 1.0 and frequencies of 0.01Hz to 10Hz. ii Table of Contents Abstract .................................................................................................................................................. ii Table of Contents................................................................................................................................... iii List of Figures ......................................................................................................................................... vi List of Symbols ........................................................................................................................................ x Acknowledgements ............................................................................................................................. xiv 1 Introduction .................................................................................................................................... 1 1.1 Applications and Processing of Composites .............................................................................. 1 1.2 Process Modeling ..................................................................................................................... 4 1.2.1 1.3 Objectives ................................................................................................................................ 7 1.3.1 2 3 Current Needs .................................................................................................................. 6 Thesis Structure ................................................................................................................ 8 Literature Review ............................................................................................................................ 9 2.1 Viscoelastic Material Behavior.................................................................................................. 9 2.2 Time-Temperature Superposition........................................................................................... 13 2.3 Effect of Cure ......................................................................................................................... 16 2.4 Viscosity Material Characterization ........................................................................................ 17 Test Methods ................................................................................................................................ 21 3.1 Dynamic Mechanical and Thermal Analysis ............................................................................ 22 3.1.1 3.1.1.1 Homogeneous Beam Sample Preparation ................................................................... 26 3.1.1.2 Homogeneous Beam: Calculating Complex Modulus................................................... 30 3.1.1.3 Homogeneous Beam: Test Procedures........................................................................ 30 3.1.2 3.2 Homogeneous Beam Tests ............................................................................................. 26 Bi-Material Beam Tests ................................................................................................... 32 3.1.2.1 Bi-Material Beam Sample Preparation ........................................................................ 34 3.1.2.2 Bi-Material Beam: Calculating Modulus ...................................................................... 35 3.1.2.3 BMB: Test Procedure .................................................................................................. 40 Rheometer ............................................................................................................................. 40 3.2.1 Rheometer Sample Preparation...................................................................................... 41 iii 4 3.2.2 Rheometer: Calculating Complex Modulus ..................................................................... 42 3.2.3 Rheometer: Test Procedure ............................................................................................ 43 3.3 Differential Scanning Calorimeter........................................................................................... 44 3.4 Cure Kinetics .......................................................................................................................... 46 3.5 Thermo Gravitational Analysis................................................................................................ 46 3.6 Comparing Rheometer and DMTA Results .............................................................................. 47 Results .......................................................................................................................................... 50 4.1 4.1.1 DMTA ............................................................................................................................. 50 4.1.2 Rheometer ..................................................................................................................... 59 4.1.3 Overlay........................................................................................................................... 61 4.2 Trends .................................................................................................................................... 63 4.2.1 Continuity of DMTA and Rheometer ............................................................................... 63 4.2.2 Effect of Thermoplastic Constituent on Modulus ............................................................ 65 4.2.3 Effect of Temperature and DOC on Un-Relaxed Modulus ................................................ 67 4.2.4 Effect of Fiber ................................................................................................................. 70 4.3 4.2.4.1 Fiber Bed Interaction .................................................................................................. 70 4.2.4.2 Shift Function ............................................................................................................. 72 Generalized Maxwell Model Fit .............................................................................................. 74 4.3.1 Relaxed Modulus ............................................................................................................ 76 4.3.2 Un-Relaxed Modulus ...................................................................................................... 82 4.3.3 Shift Function ................................................................................................................. 84 4.3.4 Relaxation Times and Weight Factors ............................................................................. 89 4.4 5 Experimental Data.................................................................................................................. 50 Goodness of Fit ...................................................................................................................... 92 Discussion and Conclusion ............................................................................................................ 95 5.1 Discussion .............................................................................................................................. 95 5.1.1 5.2 Predicting Viscosity and Modulus During Cure ................................................................ 95 Future Work ........................................................................................................................... 99 5.2.1 Experiments on Resin ................................................................................................... 100 5.2.2 Micromechanics ........................................................................................................... 101 5.2.3 Relaxation Modulus ...................................................................................................... 101 5.2.4 Effect of Pressure ......................................................................................................... 102 iv 5.2.5 5.3 Use Multi-Variable Fitting Software .............................................................................. 102 Conclusion ........................................................................................................................... 103 References .......................................................................................................................................... 105 v List of Figures Figure 1.1: Schematic of inter-relationship of process sub-component models. ....................................... 5 Figure 1.2: Schematic of integrated stress-flow model. ............................................................................ 7 Figure 2.1: Stress-strain relationship of a viscoelastic material in dynamic loading. ................................ 11 Figure 2.2: Phasor representation of complex modulus. ........................................................................ 12 Figure 3.1: Flow chart illustrating the test classes and the inter relationship of each.............................. 21 Figure 3.2: Maximum strain in DMTA homogeneous beam experiment as a function of amplitude and sample thickness. .................................................................................................................................. 23 Figure 3.3: Storage modulus of steel and MTM45-1/GA045 as a function of DMTA amplitude. .............. 24 Figure 3.4: TA Instrument Q800 DMTA with three point bend clamp as used for all homogeneous beam experiments. ......................................................................................................................................... 26 Figure 3.5: Neat resin (left) and prepreg (right) homogeneous beam samples. ...................................... 27 Figure 3.6: Lay-up used to manufacture a prepreg laminate. ................................................................. 28 Figure 3.7: Side view of the mold used to manufacture a neat resin panel. ............................................ 29 Figure 3.8: Example of DMTA homogeneous beam experiment: iso-frequency temperature ramp. ....... 31 Figure 3.9: Example of DMTA homogeneous beam experiment: iso-thermal frequency sweep. ............. 32 Figure 3.10: Bi-material beam experimental setup schematic. ............................................................... 33 Figure 3.11: Bi-material beam sample loaded in DMTA three point bend clamp..................................... 34 Figure 3.12: Prepreg (left) and resin (right) bi-material beam samples. .................................................. 35 Figure 3.13: Sensitivity of calculated prepreg modulus from BMB test to the steel shim’s modulus. ...... 37 Figure 3.14: MTM45-1 prepreg BMB modulus scaled to match the homogeneous beam modulus. Note that the view is expanded to show the cool down only. ......................................................................... 39 Figure 3.15: Anton-Paar MCR502 rheometer with disposable 25mm parallel plate geometry with thermal control from peltier plate and hood. ...................................................................................................... 41 Figure 3.16: Neat resin (left) and prepreg (right) rheometer samples in disposable bottom dish. Disposable upper plates (middle) are also shown. ................................................................................. 42 Figure 3.17: Evaluation of glass transition temperature using DSC heat capacity results. ....................... 45 Figure 4.1: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1/GA045 prepreg. .............................................................................................................................................................. 51 vi Figure 4.2: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1/HTS5631 prepreg. ................................................................................................................................................ 51 Figure 4.3: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1 neat resin. ..... 52 Figure 4.4: E’ and E” results from iso-thermal frequency sweep DMTA test on 3900-2/T800H prepreg. . 52 Figure 4.5: E’ results from iso-frequency temperature ramp DMTA tests of MTM45-1/GA045 prepreg samples with varying DOC. Note that each test was on a different sample. ........................................... 53 Figure 4.6: E” results from iso-frequency temperature ramp DMTA tests of MTM45-1/GA045 prepreg samples with varying DOC. Note that each test was on a different sample. ........................................... 54 Figure 4.7: E’ results from iso-frequency temperature ramp DMTA tests of MTM45-1/HTS5631 prepreg samples with varying DOC. Note that each test was on the same sample with the exception of X= 0.83, 0.9 and 1.0. ........................................................................................................................................... 54 Figure 4.8: E” results from iso-frequency temperature ramp DMTA tests of MTM45-1/HTS5631 prepreg samples with varying DOC. Note that each test was on the same sample with the exception of X= 0.83, 0.9 and 1.0. ........................................................................................................................................... 55 Figure 4.9: MTM45-1 resin BMB test (ramp 1°C/min to 180C, cure 120 minutes). .................................. 56 Figure 4.10: MTM45-1 resin BMB test (ramp 2°C/min to 140C, cure 240 minutes). ................................ 56 Figure 4.11: MTM45-1 resin BMB test (ramp 1°C/min to 180C, cure 120 minutes). ................................ 57 Figure 4.12: MTM45-1 resin BMB test (ramp 2°C/min to 130C, cure 240 minutes, ramp 2°C/min to 180C, cure 120 minutes). ................................................................................................................................ 57 Figure 4.13: MTM45-1/GA045 prepreg BMB test (ramp 1°C/min to 180C, cure 120 minutes). ............... 58 Figure 4.14: Figure 4.15: MTM45-1/GA045 prepreg BMB test (ramp 1°C/min to 180C, cure 120 minutes). .............................................................................................................................................................. 58 Figure 4.16: G’ results from iso-frequency temperature ramp rheometer tests of MTM45-1 resin and MTM45-1/HTS5613 prepreg. ................................................................................................................. 59 Figure 4.17: G” results from iso-frequency temperature ramp rheometer tests of MTM45-1 resin and MTM45-1/HTS5631 prepreg. ................................................................................................................. 60 Figure 4.18: G’ results from iso-frequency iso-thermal rheometer tests of MTM45-1 resin. ................... 60 Figure 4.19: G” results from iso-frequency iso-thermal rheometer test of MTM45-1 resin. .................... 61 Figure 4.20: Overlay of DMTA and rheometer MTM45-1 resin tests that show G’ as a function of temperature and DOC ranging from 0.01 to 1.0. .................................................................................... 62 Figure 4.21: Overlay of DMTA and rheometer MTM45-1 resin tests that show G’ as a function of temperature and DOC ranging from 0.01 to 1.0. .................................................................................... 63 vii Figure 4.22: Overlay of results from rheometer, BMB and homogeneous beam experiments of MTM45-1 resin and MTM45-1/GA045 prepreg. ..................................................................................................... 64 Figure 4.23: The effect of a thermoplastic constituent (MTM45-1/GA045 & 3900-2/T800H) on the storage modulus compared to systems without a thermoplastic constituent (MTM45-1 & MTM451/HTS5631). All samples are fully cured. ................................................................................................ 65 Figure 4.24: Storage modulus of homogeneous beams partially cured to varying DOC of MTM451/GA045 prepreg showing a secondary transition from the thermoplastic constituent. ......................... 66 Figure 4.25: Glassy storage modulus of MTM45-1/HTS5631 prepreg and MTM45-1/GA045 prepreg as a function of temperature and frequency. ................................................................................................ 67 Figure 4.26: Results of homogeneous beam tests on MTM45-1/HTS5631 showing the temperature and frequency dependence of the storage modulus. .................................................................................... 68 Figure 4.27: Glassy storage modulus of MTM45-1/HTS5631 prepreg as a function of temperature for partially cured samples of varying DOC ranging from 0.295 to 1.0. ........................................................ 69 Figure 4.28: G’ results from rheometer temperature ramp tests (1, 2, 3, 4 & 5 °C/min) of MTM45-1 prepreg and resin showing the effect of the fiber bed stiffness.............................................................. 70 Figure 4.29: G’ prepreg vs. G’ resin showing the effect of the fiber bed interaction................................ 71 Figure 4.30: Storage modulus master curve of fully cured MTM45-1 resin, MTM45-1/GA045 prepreg and MTM45-1/HTS5631 prepreg. ................................................................................................................. 72 Figure 4.31: Temperature shift function of fully cured MTM45-1 resin, MTM45-1/GA045 prepreg and MTM45-1/HTS5631 prepreg. ................................................................................................................. 73 Figure 4.32: Schematic representation of a generalized Maxwell model. ............................................... 75 Figure 4.33: Iterative approach to fitting the generalized Maxwell model sub-components. .................. 76 Figure 4.34: Normalized relaxed modulus as a function DOC for functionalities of 3, 4 and 5. ................ 80 Figure 4.35: Determining the reference relaxed modulus of MTM45-1 resin using iso-frequency temperature ramp results of a fully cured sample. ................................................................................ 81 Figure 4.36: Fitting the un-relaxed modulus to experimental data. ........................................................ 83 Figure 4.37: The effect of temperature and DOC on the piecewise shift function. .................................. 85 Figure 4.38: Schematic showing the free volume at the glass transition temperature as a function of DOC. ...................................................................................................................................................... 86 Figure 4.39: G’ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin. ................................................................................................................................. 92 viii Figure 4.40: G“ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin. ................................................................................................................................. 93 Figure 4.41: G” comparison of rheometer and generalized Maxwell model predictions of MTM45-1 resin. .............................................................................................................................................................. 93 Figure 4.42: G’ comparison of rheometer and generalized Maxwell model predictions of MTM45-1 resin. .............................................................................................................................................................. 94 Figure 4.43: G’ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin for frequencies ranging from 0.01Hz to 10Hz. ........................................................... 94 Figure 5.1: Comparison of MTM45-1 resin G” results from generalized Maxwell model predictions to isothermal viscosity rheometer tests. ........................................................................................................ 96 Figure 5.2: Comparison of MTM45-1 resin G” results from generalized Maxwell model predictions to temperature ramp viscosity rheometer tests. ........................................................................................ 96 Figure 5.3: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a typical cure cycle (1°C/min ramp to 180°C, 120 minute cure). ................................................................ 97 Figure 5.4: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a low temperature cure cycle (2°C/min ramp to 140°C, 240 minute cure)................................................. 98 Figure 5.5: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a two stage cure cycle (2°C/min ramp to 130°C, 240 minute cure, 2°C/min ramp to 180°C, 120 minute cure)...................................................................................................................................................... 99 ix List of Symbols (t ) Time dependent strain o Constant stress D(t ) Creep compliance o Constant strain (t ) Time dependent stress E (t ) Relaxation modulus p Peak strain p Peak stress t Time Frequency Phase angle E* Complex flexure modulus E' Real flexure modulus E '' Imaginary flexure modulus Er Relaxed modulus Eu Un-relaxed modulus gi Weight factor of the ith Maxwell element i Relaxation time of the ith Maxwell element aT Temperature shift factor T Temperature * Complex viscosity x ' Real viscosity " Imaginary viscosity Shear rate Ymax Maximum deflection in three point bending Fmax Maximum force in three point bending I Moment of inertia max Maximum strain c Distance to neutral axis Radius of curvature M Bending moment Poisson ratio ts Sample thickness L Length between three point bending supports w Width of a bi-material beam sample tst Thickness of steel shim in bi-material beam sample tc Thickness of resin or prepreg in bi-material beam sample Est Modulus of steel in bi-material beam sample Ec Modulus of resin or prepreg in bi-material beam sample I st Moment of inertia of steel beam in bi-material beam sample ' E HB Homogeneous beam storage modulus ' E BMB Bi-material beam storage modulus G* Complex shear modulus xi G' Real shear modulus G" Imaginary shear modulus max Maximum shear stress max Maximum shear strain Tg Glass transition temperature Tg 0 Initial glass transition temperature Tg Final glass transition temperature Material constant used to fit DeBenedetto equation X Degree of cure G23 Shear modulus in the 2-3 direction E2 Extensional modulus in the 2 direction 23 Poisson ratio in the 2-3 direction E f ( transverse ) Transverse fiber modulus Vf Fiber volume fraction Tm Melt temperature aT , X Temperature and degree of cure shift factor G Xr 1 Reference relaxed modulus for fully cured material TGr Temperature where G Xr 1 is selected Effective cross link network concentration X 1 Reference effective cross link network concentration for fully cured material P Probability that a given arm is not connected to cross link network r Stoichiometric ratio limit X 1 xii X gel Degree of cure when gelation occurs GTu 273 Un-relaxed modulus at 273K fg Factor for the variation of free volume at the glass transition temperature C1 WLF constant C2 WLF constant fp Factor to scale the polynomial sub-domain of the shift function a' Fit parameter constant b' Fit parameter constant c' Fit parameter constant d' Fit parameter constant P1 Fit parameter constant P2 Fit parameter constant L1 Fit parameter constant L2 Fit parameter constant xiii Acknowledgements This work was only possible thanks to the help of many great individuals. Without a doubt, the most significant contribution came from my supervisor Dr. Anoush Poursartip. His support, encouragement, passion and direction were truly inspirational. Secondly, I would like to acknowledge the funding provided by the Natural Science and Engineering Research Council of Canada. In particular, I would like to thank Dr. Fernand Ellyin for the initial Undergraduate Research Assistantship that ultimately led to this thesis work. I would like to thank every member of the UBC Composite Group I have come to know and call friends. A special thanks to James Kay, Kamyar Gordnian, Sardar Malekmohammadi, Kevin Hsiao, Lelya Farhang, Nathan Slesinger, Kalle Keil, Casey Keulen and Christophe Moubuchon for such great discussions, ideas and knowledge. I am grateful to Convergent Manufacturing Technologies for the resources and technical support they provided throughout my research. In particular, Dr. Abdul Arafath shared with me many stimulating discussions and a wealth of knowledge. Additionally, I would like to thank Malcolm Lane, Dave Vanee and Dr. Anthony Floyd. Finally, I am so fortunate and grateful to my father, mother, brothers and friends. Their unconditional love, support and motivation truly made this work possible. xiv 1 Introduction 1.1 Applications and Processing of Composites Composite materials have been used for centuries. The first historical reference dates to the Egyptian period when clay/straw composites and plywood (by orientation of wood) were used in structural applications to improve mechanical properties (Jones, 1975). Modern composite materials and more specifically fiber reinforced polymers have been used for decades. In recent years, they have gained further attention in a wide variety of applications where weight and mechanical performance are important. The aerospace industry has committed to using composite materials for the primary structure of the new and future aircraft such as the Boeing 787 Dreamliner, Airbus A350 and Bell 407. The energy sector is utilizing composite materials for large structures such as wind turbine blades. The largest wind turbine blade to date is the Siemens B75 which is 75m long. The marine, automotive and sporting goods industries are also using composites in ever increasing quantities. In all, the global end use market value of goods made with composites was $55.6B in 2011 (Kazmierski, 2012). Fiber reinforced composite materials, referred to as composites from here on, are composed of a polymer matrix reinforced with fibers. High performance applications such as aerospace typically use carbon fiber as reinforcement, while low performance applications such as sporting goods and boat hulls typically use glass fiber as reinforcement. The matrix is typically a thermoset polymer such as epoxy or polyester. However, thermoplastics are showing promise for products that demand a high production rate such as in the automotive industry. One reason composite materials are in such high demand is their exceptional mechanical properties. Composites offer very high strength and stiffness at relatively low weights. Additionally, they are very tough and resistant to corrosion (Callister, 2007). Composites by nature lend themselves well to design 1 optimization. The anisotropic nature of composite materials allows designers to further reduce the weight of structures by optimizing load paths and this is achieved by placing fibers in specified directions and stacking order. Manufacturing of composites has many advantages over other material classes. Most composite manufacturing processes yield net shape parts, meaning that little or no machining is required. This can offer significant savings in manufacturing costs by reducing the steps required to produce a part. Another advantage is that the tool side of a part can have very high surface quality. In many applications, a gel coat finish results in parts that require no further cosmetic treatment such as sporting good products and boat hulls. With composites, complex geometries can be produced. For example, each sheet of aluminum used to make an aircraft component such as wing or fuselage can only have one plane of curvature. Composites on the other hand can be made with multiple planes of curvature since in the uncured state the material can easily form over complex geometries. Very large structures can be produced in a single processing step, eliminating the need to fasten and join many smaller components together. This can significantly reduce labour and weight of finished goods, allow for less design complexity and eliminate many sources of stress concentration. In a general sense, composite structures are produced by combining the reinforcement fibers and the polymer matrix on a mold, applying pressure and curing the polymer. Many processing techniques to achieve this exist including autoclave vacuum bagging, oven vacuum bagging, resin transfer molding, vacuum infusion processes, compression molding and spray-up. The highest performance of these processes is autoclave vacuum bagging of carbon fiber reinforced epoxy polymer composites. Since this class of material is studied in this thesis, the remaining discussion will be focused on carbon fiber reinforced epoxy composite material. 2 The bulk material used to make high performance composite structures is composed of a sheet of carbon fibers that have been pre-impregnated with an un-cured epoxy resin. This material is known as prepreg and must be stored in a freezer to minimize cure progression prior to layup and curing. Manufacturing a prepreg part typically uses the following steps: Stack prepreg sheets onto mold. Place vacuum bag over prepreg laminate. Apply vacuum. Place mold and laminate in autoclave. Increase pressure and temperature inside the autoclave. Allow laminate to cure. Return pressure and temperature in the autoclave to ambient. Remove part from mold. Final machining, finishing and assembly of part. The manufacturing process such as the one described above, often leads to finished parts with unwanted issues such as residual stress and porosity. Residual stresses can lead to dimensional problems such as warping and deformation. Additionally, residual stresses can affect the strength of the material and therefore play a role in determining the allowable stresses in a structure. Porosity is another major issue and can be detrimental to the material’s final strength, stiffness, fatigue resistance and moisture resistance (Kaw, 2006). Compaction and resin flow can lead to variations in part thickness and fiber volume fraction. A significant amount of knowledge in composite manufacturing has been empirically gained through trial and error. This is very costly and not very time effective. A more effective strategy to manage these unwanted issues is the use of process models. Over recent years there has been a significant demand on 3 engineers to better understand the composites manufacturing processes and therefore improve predictions of the resulting finished parts. This pressure has led to an increased use and advancement of composite process modeling (Advani & Sozer, 2003). 1.2 Process Modeling Process models are mathematical representations of manufacturing processes and are used to predict the behavior and properties of the resulting structure. Some of the phenomena that are predicted by process models are heat transfer, resin cure kinetics, resin flow, gas extraction, laminate compaction and residual stress development. Process models are developed from the governing equations of the respective physical phenomena, and each requires its own set of boundary conditions, process conditions and material properties. The process models are coupled and each sub-component is continually updated during a simulation. The following describes the physical phenomena during typical manufacturing and the associated process model sub-components required to predict the physical phenomena: Vacuum is applied. Entrapped gases are extracted from the laminate. o Gas Transport Model Temperature and pressure increase. Resin begins to flows and compaction of the laminate occurs. Volatiles and gases continue to be extracted. o Thermal Model, Flow Model, Gas Transport Model Cure initiates and advances. During thermally activated crosslinking, exothermic heat is generated. o Cure Kinetics Model, Thermal Model 4 Material is strained due to cure shrinkage, thermal expansion/contraction and applied pressure resulting in stress. Above the material’s glass transition temperature, stress relaxation occurs almost instantaneously. Below the material’s glass transition temperature, stress relaxation is insignificant over process time scales. o Stress Model, Thermal Model, Cure Kinetics Model Because of the interconnection between sub-components, the inputs required by one sub-component often depend on the outputs from another. For example, during a process simulation both the stress and flow models receive updated degree of cure and temperature predictions from the cure kinetics and thermal models. Additionally, the stress model receives updated volume fraction prediction from the flow model. This example is illustrated in Figure 1.1. Figure 1.1: Schematic of inter-relationship of process sub-component models. The stress and flow models are currently treated as separate sub-components despite being dependent on some of the same process conditions (DOC and temperature). Lack of coupling between flow and stress models is arbitrary and results in loss of predictive fidelity for cases where resin flow changes local 5 volume fractions significantly, or where there are significant thermal and cure gradients in the part meaning that it is effectively impossible to run the flow and stress models sequentially. The flow model requires a viscosity material model which is generally empirically fit to experimental data. The stress model requires a modulus material model. The modulus model can vary in complexity depending on the required accuracy of the process model predictions. From simplest to the most complex, some approaches that have been used are elastic material model, CHILE (Cure Hardening Instantaneous Linear Elastic) material model, quasi viscoelastic material model and viscoelastic material model. 1.2.1 Current Needs Recently, an integrated stress-flow model has been developed (Haghshenas, Vaziri, & Poursartip, 2009) and is schematically shown in Figure 1.2. This model has the ability to significantly increase the efficiency of running process models. Further, this integrated framework utilizes consistent material property values and provides a more complete picture than current flow and stress sub-component models. For example it is possible that due to poor thermal management, a large part may have one end that is uncured where flow is dominant and the other end is cured where stress is dominant. The formulation developed requires a consistent material model that describes the viscosity and viscoelastic modulus properties. However, no such material model currently exists for curing thermoset polymers. The focus on this thesis is on the material property models required for flow, stress and integrated stress-flow models. 6 Figure 1.2: Schematic of integrated stress-flow model. 1.3 Objectives The objective of this work is to develop a material model that is capable of describing the viscoelastic liquid and viscoelastic solid behavior of a curing thermoset epoxy resin over the entire range of cure and temperature. This objective is pursued by the following: Experimentally investigate the viscoelastic behavior of epoxy resin and prepreg composite materials using dynamic mechanical analysis. Study the relationship between viscosity and viscoelastic modulus. Evaluate if the complex modulus (loss and storage modulus) is continuous between liquid (Rheometer) and viscoelastic solid (DMTA) states. Attempt to use a generalized Maxwell model to describe the complex modulus for all liquid and solid experimental data as a function of temperature, frequency and degree of cure. Achieving the aforementioned objective would provide the material model required for implementation of the integrated stress-flow model. 7 1.3.1 Thesis Structure The following structure is used to present the work in this thesis. A brief overview of each chapter is also provided in this section. Chapter 1 (Introduction) provides an introduction to composite materials and positions this work within the industry. The applications, manufacturing and process modeling of composites are discussed. Additionally, the objective of this thesis is summarized. Chapter 2 (Literature Review) includes a literature review of the current methods used to model viscosity and viscoelastic material properties of thermosetting polymers. Chapter 3 (Test Methods) is a summary of the experimental tests used to generate the presented data. Additionally, the assumptions and calculations used to generate material properties data from test measurements are discussed. Chapter 4 (Results) presents the experimental results along with trends that were observed in the material behavior. The methodology and fit to a generalized Maxwell model is described and the fit of the model is compared to experimental data. Chapter 5 (Conclusion) includes the results of using the generalized Maxwell model to predict material behavior during a typical cure cycles. Also included in this chapter are sections that discus future work and conclusion. 8 2 Literature Review 2.1 Viscoelastic Material Behavior Equation Chapter 2 Section 2 Viscoelastic material behavior is a time dependent deformation as a result of an applied stress or strain. Since the 1940’s, polymers have been documented as exhibiting viscoelastic behavior (Tobolsky & Eyring, 1943). The response of a viscoelastic material is somewhere between that of an idealized fluid (Newton’s Law) and an idealized linear elastic solid (Hooke’s Law) which can be represented as a dashpot and spring respectively (Ferry, 1980). A viscoelastic material can be represented as combination of springs and dashpots in series, parallel or combination of both (Crawford, 1998). Viscoelastic behavior is generally categorised into either creep or stress relaxation. Creep is defined as the time dependent strain when a constant stress is applied to a material. The creep compliance is mathematically represented as D (t ) (t ) 0 (2.1) where D(t ) is the creep compliance, 0 is the constant applied stress and (t ) is the time dependent strain. Stress relaxation is defined as the time dependent stress when a constant strain is applied to a material. The stress relaxation modulus is mathematically represented as E (t ) (t ) 0 (2.2) where E (t ) is the relaxation modulus, (t ) is the time dependent stress and 0 is the constant applied strain (Brinson & Brinson, 2008). 9 A common experimental method for determining viscoelastic material properties is dynamic loading (Ferry, 1980) (Lakes, 1999) (Brinson & Brinson, 2008). In such a test, sinusoidal displacements are applied to the material and the stress-strain relationship is observed. The mathematical representation is as follows (t ) p sin( t ) (2.3) (t ) p sin( t ) (2.4) where (t ) is the time dependent strain, p is the peak strain, is the angular frequency, (t ) is the time dependent stress, p is the peak stress and is the phase angle. The resulting modulus is mathematically represented as E* p p (2.5) where E * is the complex modulus. The time dependent stress and strain relationship are not in phase for viscoelastic materials as shown in Figure 2.1. 10 Figure 2.1: Stress-strain relationship of a viscoelastic material in dynamic loading. The phase angle is the shift between the peak strain and peak stress. A perfectly elastic material has a phase angle of 0, a perfectly viscous liquid has a phase angle of phase angle between 0 and 2 2 and a viscoelastic material has a . The resulting complex modulus is usually represented using phasor notation as the in phase and out of phase components as shown in Figure 2.2. The out of phase component of the complex modulus is also referred to as the loss or imaginary modulus and the in phase component of the complex modulus is also referred to as the storage or real modulus. Mathematically it is represented as E * E ' iE " (2.6) E * E '2 E "2 (2.7) where E ' is the storage modulus and E " is the loss modulus. 11 E‘’ E‘ Figure 2.2: Phasor representation of complex modulus. A generalized Maxwell model (Prony series) has been used to describe the stress relaxation, storage and loss modulus of thermoset epoxy polymers (Vreugd, Jansen, Ernst, & Bohm, 2010) (Ruiz & Trochu, 2005) (O'Brien, Mather, & White, 2001) (Simon, McKenna, & Sindt, 2000) (Prassatya, McKenna, & Simon, 2001) (White & Kim, 1998) (Kim & White, 1996) (Brinson, 1968). The general form of the generalized Maxwell model is n E '( ) E r E u E r gi i 1 n E ''( ) E u E r gi i 1 E (t ) E E E r u 2 i2 1 2 i2 i 1 2 i2 n r g e i t i (2.8) (2.9) (2.10) i 1 r u where E is the relaxed modulus, E is the un-relaxed modulus, t is time, is the frequency, i is the relaxation time of the ith Maxwell element and g i is the weight factor of the ith Maxwell element (Ferry, 1980). 12 2.2 Time-Temperature Superposition The observation that temperature and time have an equivalent effect on the mechanical properties of viscoelastic materials was first proposed by Leaderman (Leaderman, 1943). This equivalency is known as Time-Temperature Superposition (TTS) and has become the standard methodology for describing the effect of temperature on viscoelastic behavior. Both frequency domain and time domain material properties can be expressed by TTS which applies a horizontal shift to the material property curve to a shorter or longer time scales. In general t aT E (t ) T 1 E T 2 E ( ) T 1 E aT T 2 (2.11) (2.12) where aT is the shift factor and T1 and T2 are arbitrary reference temperatures. If a material behaves according to TTS, it is called a thermo-rheologically simple material (Lakes, 1999). Most of the prominent work on viscoelastic material characterization of thermoset polymers assumes thermo-rheological simple behavior (Kim & White, 1996) (Prassatya, McKenna, & Simon, 2001) (Goertzen & Kessler, 2006) (Ruiz & Trochu, 2005) (Simon, McKenna, & Sindt, 2000) (White & Kim, 1998) (Crowson & Arridge, 1979). In contrast, thermo-rheologically complex materials cannot be described by a purely horizontal shift as was found by other researchers (Adolf, Cambers, & Caruthers, 2004) (Scholtens & Booij, 1980). An empirical relationship for the shift function, well known as the WLF equation, was first introduced by Williams, Landel and Ferry in the following form: log aT C1 (T TREF ) C2 (T TREF ) 13 (2.13) It was observed that for many polymers, when the glass transition temperature is used as the reference temperature, C1 and C2 were found to be 17.44 and 51.6 respectively (Williams, Landel, & Ferry, 1955). Above Tg the WLF equation shows strong agreement with experimental data provided that different values of C1 and C2 be used for different classes of materials (Brinson & Brinson, 2008) (Lakes, 1999) (Adolf, Cambers, & Caruthers, 2004) (Ferry, 1980). Although Williams, Landel and Ferry state that TTS and the WLF equation is not valid below Tg (Williams, Landel, & Ferry, 1955), other researchers have contested this and proposed alternate shift functions below Tg in some cases and above and below Tg in others: Rusch proposed a modified form of the WLF equation in which temperature is replaced by an effective temperature the material would have if the total free volume was equal to the equilibrium free volume. This was done in order to predict relaxation behavior in the glassy region (Rusch, 1968). log( aT ) C1 (Te Tg ) C 2 (Te Tg ) T; T Tg w f (T ) Te T ; T Tg f (2.14) (2.15) where Te is the effective temperature, w f (T ) is the non-equilibrium portion of the total free volume and f is the difference in coefficient of thermal volumetric expansion between the equilibrium liquid and equilibrium solid. The resulting shift function is composed of WLF above Tg , an inflection point at Tg and linear below Tg. 14 The Arrhenius activation energy has been proposed to describe the shift function below the materials glass transition temperature (Goertzen & Kessler, 2006) (Brinson & Brinson, 2008) (Gilbert, Ashby, & Beaumont, 1986). log(aT ) Ea 1 1 2.303 R T Tg (2.16) where Ea is the activation energy and R is the gas constant. Others have simply used a linear shift function below the glass transition temperature (Brinson, 1968) or for the entire shift function above and below the glass transition temperature (Kim & White, 1996) (Ruiz & Trochu, 2005). log(aT ) ao a1 T (2.17) where ao and a1 are constants. The Vogel dependence has also been used to estimate the shift function (Simon, McKenna, & Sindt, 2000) (Prassatya, McKenna, & Simon, 2001). log(aT , X ) C C T T Tg T (2.18) where C is a constant. This form was stated to be capable of describing the WLF when T (Tg 51.6) , however in their work it was assumed that (Tg 51.6) 0 which effectively reduces equation (2.18) to the Arrhenius Activation energy equation. A common theme in all of the works examined is that WLF provides a good estimate of the shift function above Tg, an inflection point occurs at Tg and that a linear fit below Tg provides a reasonable estimate of the shift function. 15 2.3 Effect of Cure There have been a limited number of works investigating the effect of cure on the viscoelastic properties of thermoset polymers. The most significant challenge is that experiments above the material’s glass transition temperature will cause curing of partially cured samples. The following notable works study the effect of cure on the shift function: Kim and White applied a DOC dependent linear shift function. The slope of the shift function was assumed to be exponentially dependent on the DOC (Kim & White, 1996). log(aT , X ) ao ( X ) a1 ( X ) T (2.19) 1 ao ( X ) 1.4 e1 X 0.712 (2.20) a1 ( X ) 30 ao ( X ) (2.21) Simon et. al. proposed that a horizontal shift based on the degree of cure could be applied in a similar fashion to Time-Temperature Superposition. The shift function used replaced the reference temperature with the material’s DOC dependent glass transition temperature (Simon, McKenna, & Sindt, 2000). log (aT , X ) C C T Tg ( X ) (2.22) Ruiz and Trochu found that the difference between shift functions from various DOC was small enough that it could be ignored and a linear fit was used (Ruiz & Trochu, 2005). O’Brien et. al. applied a cure dependent un-relaxed modulus to the prony series (O'Brien, Mather, & White, 2001). 16 log( Eiu ) Ai Bi X (2.23) where Ai and Bi are two constants that represent the un-relaxed modulus of the ith Maxwell element. 2.4 Viscosity Material Characterization There has been extensive work in modeling the viscosity of thermoset polymers. In these systems, the viscosity is very sensitive to cross linking and temperature therefore, great accuracy is required for successful implementation in process modeling (Halley & MacKay, 1996). Much of the work assumes Newtonian fluid behavior and therefore neglects the shear-rate effect on viscosity. The majority of the models found in the literature are empirically fit and use time and temperature as state variables. In doing so, the cure kinetics and viscosity models are essentially combined into the same model. In terms of implementation with the sub-component modeling structure used in composite process modeling, this is very undesirable. Since process models utilize a cure kinetics sub-component, the viscosity model used should be in terms of DOC and temperature. Noteworthy viscosity models that use DOC and temperature as state variables are as follows: Simple Empirical Model: Uses a two term relationship in which one term predicts viscosity as a function of temperature and the second predicts the viscosity as a function of DOC (Lane, Seferis, & Bachmann, 1996). This method is employed in current state of the art process modeling software such as RAVEN 3 (CMT, 2009). Ea T e RT f ( X ) 17 (2.24) Where, T is the viscosity at infinitely high temperature and f ( X ) is the reaction term. Note that different forms of f ( X ) can be used such as the molecular weight or gel model below. Molecular Weight Model: Uses molecular weight and temperature to predict viscosity (Mijovic & Lee, 1989) (Lipshitz & Macosko, 1976). Ae B R T M w Mw o C D RT (2.25) where, M w is the average molecular weight, M wo is the average initial molecular weight and A, B, C & D are constants. Note that this model requires that the relationship between molecular weight and degree of cure is known. Gel Model: Uses a ratio of the initial viscosity based on the difference between the DOC and the gelation DOC (Castro & Macosko, 1980). X gel o X gel X A (2.26) where, o is initial viscosity, X gel is the degree of cure at gelation and A is a constant. Modified WLF: Modification of WLF and inclusion of molecular weight (Hale, Garcia, Macosko, & Manzoine, 1989). o e 1 A B (T Tg ) 18 Mw (2.27) Modification of WLF by making C1 and C2 a function of DOC (Chiou & Letton, 1992). C1 (T Tg ) ref e C2 ( X ) ( T Tg ) C2 ( X ) e A B X (2.28) (2.29) where, C1 is a constant and ref is the reference viscosity. Since it has been observed that epoxy resins may exhibit shear thinning, some researchers have proposed viscosity models that include shear rate, DOC and temperature. Although the general form differs (Modified WLF, Arrhenius, Power, or combination) all of the works use an initial viscosity term that is a function of shear rate or frequency (Halley & MacKay, 1996). The dynamic viscosity from small strains oscillatory shear tests can be related to the complex modulus by G* * (2.30) ' G" (2.31) " G' (2.32) where * is the complex viscosity, ' is the real viscosity and '' is the imaginary viscosity (Morrison, 2001). Further, the Cox-Merz relationship is an empirical rule that relates rotational and oscillatory viscosity in the following form 19 ( ) * ( ) (2.33) where is the rotational shear rate (Morrison, 2001). 2.5 Combined Viscoelastic and Viscosity Material Model There have been very few attempts in the literature to model the viscoelastic and viscosity behavior in a single model. This may be due to a lack of demand for such a model prior to the development of an integrated stress-flow model. Another reason may be due to the difficulty in generating consistent experimental data for both the flow and glassy regimes. Experiments for viscous and viscoelastic solid materials utilize very different equipment. A further challenge with thermoset polymers is the cure advancement that occurs when the temperature is above the material’s glass transition temperature. One work was found that presents a model that predicts the storage and loss modulus of polybutadiene from the flow to glassy regime (Palade, Verney, & Attane, 1996). This work used fractional calculus to describe the viscoelastic material response but did not provide great fit to experimental data in the glassy regime. 20 3 Test Methods This chapter provides the test methods used to generate all experimental results presented in this thesis. Dynamic Mechanical and Thermal Analysis (DMTA) and rheometry were used to determine the complex modulus of prepreg and epoxy resin samples. Although both of these machines function fundamentally in the same manner, the DMTA measures solid samples and the rheometer measures samples that are in a liquid or gelled state. Differential Scanning Calorimetry (DSC) was used to experimentally determine the Degree of Cure (DOC) of samples prior to mechanical testing. A previously developed cure kinetics model was used to predict the DOC during mechanical testing (CMT, 2009). Figure 3.1 illustrates the relationship between the different types of tests. Additionally, Thermo Gravitational Analysis (TGA) tests were performed to determine the temperature at which material degradation occurs. Figure 3.1: Flow chart illustrating the test classes and the inter relationship of each. Equation Section 3 21 3.1 Dynamic Mechanical and Thermal Analysis The dynamic mechanical behavior of both resin and composite samples was determined using a TA Instrument Q800 DMTA. The machine applies a sinusoidal loading to a sample. Based on the peak force, peak displacement and phase angle (shift between peak force and peak displacement) the complex modulus of the sample was calculated. All the DMTA tests used three point bending. Although other clamps are available for use with the DMTA such as cantilever, tension and shear, all of these were found to give poor absolute modulus values when testing materials of known properties. A number of general considerations were evaluated before beginning the DMTA study: Amplitude If inappropriate amplitude is used, four problems arise. First, the TA Instrument Q800 can only apply a limited force of 18N. Therefore, the combination of sample geometry and amplitude must be such that less than 18N is required. These can be determined using Ymax Fmax L3 48 E I (3.1) where Ymax is the maximum amplitude, Fmax is the maximum force applied, L is the support span, E is the modulus and I is the beam’s moment of inertia (Hibbeler, 2008). Second, the material may be damaged if the strains applied are too large. In the case of epoxy and carbon/epoxy composites, strains of less than 0.1% are considered low enough that matrix cracks or damage will not occur (Jones, 1975). The maximum strain ( max ) can be determined using 22 c (3.2) EI M (3.3) max where c is the distance to the neutral axis, is the radius of curvature and M is the applied moment (Hibbeler, 2008). Figure 3.2 shows the maximum strain as a function of amplitude and sample thickness. Figure 3.2: Maximum strain in DMTA homogeneous beam experiment as a function of amplitude and sample thickness. Third, the material must remain within its linear regime. This was accomplished by applying increasing amplitudes to a sample and determining the amplitude at which non-linear behavior is observed. 23 Fourth, due to the low end resolution of the DMTA, amplitudes that are too small will not provide meaningful results. The amplitude and resulting force must be large enough that results are reproducible and good in terms of absolute values. Determining the minimum amplitude required was accomplished by testing materials with known properties. Figure 3.3 shows such tests for steel and MTM45-1/GA045. At very low amplitudes, the measured modulus is affected by amplitude. Above 75 microns, the DMTA measurements are no longer affected by the amplitude and modulus values match the reference modulus. 8E+09 7E+09 E' (Steel) [Pa] 2E+11 6E+09 5E+09 1.5E+11 4E+09 1E+11 Steel MTM45-1/GA045 3E+09 2E+09 5E+10 1E+09 0 0 0 100 200 300 400 500 E' Prepreg (MTM45-1/GA045) 2.5E+11 600 Amplitude [um] Figure 3.3: Storage modulus of steel and MTM45-1/GA045 as a function of DMTA amplitude. Thermal Lag: Thermal lag refers to the difference between the actual temperature of the sample and the thermocouple inside the DMTA furnace. The thermal lag also refers to the temperature profile that will develop through the sample. Higher temperature ramp rates will create a larger thermal lag. Using low 24 temperature ramp rates or using isothermal tests can minimize or eliminate thermal lags. However, tests take longer and spending more time at temperatures above the materials glass transition temperature can cause cure advancement of the sample. This effect is discussed below and its trade-off must be evaluated. A second consideration relating to thermal lag during temperature ramp tests is the frequency used. Tests at low frequency can have significant temperature variation over a single loading cycle. For example, the temperature will change by 3.33C over one loading cycle for a frequency of 0.01Hz and a temperature ramp of 2cpm. Lower frequencies and higher ramp rates will magnify this effect. Cure Advancement: During a DMTA test the DOC will advance if sufficiently high temperatures are reached. The temperature that will cause this effect depends on the glass transition temperature of the material. Unless the material is completely cured, cure advancement will initiate when the glass transition temperature is approached. Note that manufacturer’s recommended cure cycles do not result in materials that are completely cured and are typically only cured 95% or less. A cure kinetics model implemented in RAVEN3 (see section 3.4) was a valuable tool for managing cure advancement. 25 3.1.1 Homogeneous Beam Tests Homogeneous beam tests consist of a simply supported beam loading in three point bending as shown in Figure 3.4. The samples are beams made of homogeneous resin or unidirectional prepreg. Figure 3.4: TA Instrument Q800 DMTA with three point bend clamp as used for all homogeneous beam experiments. 3.1.1.1 Homogeneous Beam Sample Preparation Homogeneous beam samples as shown in Figure 3.5, were made from the following neat resin and prepreg systems: Advanced Composite Group (ACG): MTM45-1 (Neat Resin) ACG: MTM45-1/GA045 (Prepreg) ACG: MTM45-1/HTS5631 (Prepreg) 26 Toray: 3900-2/T800H (Prepreg) Figure 3.5: Neat resin (left) and prepreg (right) homogeneous beam samples. Preparation of prepreg samples followed all recommendations from the manufacturer. The only exception was modifying the cure temperature and cure time to produce samples of varying DOC. Using RAVEN3 software, the NCAMP MTM45-1 model (CMT, 2009) was used to design cure cycles to achieve desired DOC ranging from 0.14 to 1.0. The following steps were used to produce a prepreg homogeneous beam sample: Lay-up prepreg sheets into uni-directional laminate 8 plies thick (De-bulk for 5 minutes every 4plies). Bag laminate as shown in Figure 3.6. 27 Apply vacuum to laminate. Cure laminate. Cut laminate into beams (TYP 10 x 60 mm) using a Buehler IsoMet4000 linear precision diamond saw. Perform DSC to determine the DOC (see section 3.3). Figure 3.6: Lay-up used to manufacture a prepreg laminate. Preparation of neat epoxy resin samples was more challenging than for the prepreg samples and no directions or recommendations from the manufacturer exist. Since compaction and volume fraction are not relevant, pressure was not required during cure. In order to produce a panel that had low porosity, de-gasing the resin before cure was required. During the cure cycle, the mold must be very level 28 otherwise the thickness of the sample will not be uniform. The following steps were used to produce a homogeneous resin beam sample: Place a sufficient volume of resin film to fill the mold coated with FreKote release wax (Figure 3.7). Heat mold to 80C. Place mold in desiccator and pull vacuum for 5 minutes. Release vacuum and remove mold from desiccator. Place mold in oven. Cure material according to MRCC. Cut laminate into beams (TYP 10 x 60 mm) using a Buehler IsoMet4000 linear precision diamond saw. Figure 3.7: Side view of the mold used to manufacture a neat resin panel. 29 3.1.1.2 Homogeneous Beam: Calculating Complex Modulus The complex modulus ( E*) for each data point was calculated using the values of maximum displacement (Ymax ) and maximum applied load ( Fmax ) as measured by the DMTA. The complex modulus was calculated using F E * max Ymax 2 6 L ts 1 1 (1 ) 6 I 10 L 2 (3.4) where is the Possion ratio, L is the support span and ts is the sample thickness (TA Instruments, 2011). The phase angle ( ) is the lag between Ymax and Fmax which is measured by the DMTA. The phase angle was used to calculate Storage ( E ') and Loss ( E ") moduli as follows: E ' E * cos( ) (3.5) E '' E * sin( ) (3.6) 3.1.1.3 Homogeneous Beam: Test Procedures Two classes of Homogeneous Beam tests were performed; iso-frequency temperature ramps and isothermal frequency sweeps. Iso-frequency temperature ramp tests were used to determine the effect of temperature for a given frequency. Figure 3.8 shows a typical iso-frequency temperature ramp. The experimental setup for this class of tests was as follows: 30 Amplitude: 150µm Frequency: 1 Hz Temperature Ramp Rate: 1°C /min, 2°C /min Temperature Range: -30°C to 250°C 0.01N 9.E+09 0.45 8.E+09 0.4 7.E+09 0.35 6.E+09 0.3 5.E+09 0.25 E' 4.E+09 0.2 E" 3.E+09 0.15 Tan Delta 2.E+09 Tan Delta E' & E" [Pa] Pre-Load Force: 0.1 1.E+09 0.05 0.E+00 0 0 50 100 150 Temperature [C] 200 250 Figure 3.8: Example of DMTA homogeneous beam experiment: iso-frequency temperature ramp. The iso-thermal frequency sweep tests investigate the effect that frequency has on the modulus for a given temperature. Figure 3.9 shows results from typical tests for differing temperatures, however, most temperatures have been omitted for clarity. The experimental setup and procedure for iso-thermal frequency tests is as follows: Amplitude: 150µm Frequency: 0.01Hz to 10 Hz Iso-Thermal Range: 0°C to 250°C 31 Pre-Load Force: 0.01N Procedure: 1. 2. 3. 4. 5. Equilibrate at 0°C Iso-Thermal hold for 5 minutes Perform frequency sweep Increase temperature by 5°C Repeat step 1-4 until 250°C Figure 3.9: Example of DMTA homogeneous beam experiment: iso-thermal frequency sweep. 3.1.2 Bi-Material Beam Tests The bi-material beam (BMB) experiment consists of a simply supported beam in 3 point bending. A BMB sample consists of two materials bonded together. In the presented work, two types of BMB are 32 examined; steel - resin film BMB and steel - prepreg BMB. In both cases the epoxy constituent evolves from a viscous liquid to a visco-elastic solid during the experiment. Figure 3.10 illustrates the BMB experimental setup and Figure 3.11 shows a BMB sample in the three point bend clamp. The steel shim serves to support the resin film or prepreg when the epoxy has insufficient modulus to support itself. The load is applied directly to the steel shim via a steel ball placed in the middle of the BMB. A small hole in the resin or composite layer enables the steel ball to rest on the steel shim. This is to ensure the displacement measurements are not subject to creep as would be the case if the load were applied directly to the resin film or prepreg. Figure 3.10: Bi-material beam experimental setup schematic. 33 Figure 3.11: Bi-material beam sample loaded in DMTA three point bend clamp. 3.1.2.1 Bi-Material Beam Sample Preparation BMB samples consist of uncured prepreg or resin film that is laminated to a steel shim as shown in Figure 3.12. Prior to curing, when the resin is at a low degree of cure, the lamination is dependent on the ‘tack’ of the resin. As the cure progresses, the strength of adhesion to the steel shim increases. The following steps were used to produce a BMB sample: 1. Roughen steel shim with sand paper. 2. Clean steel shim with acetone. 3. Stack resin film or prepreg sheets (unidirectional layup) on steel shim to a total thickness of 1mm. 4. De-bulk laminate for 5 minutes. 5. Trim excess resin or prepreg to match the dimension of the steel shim using a utility knife. 6. Cut a small hole in the resin or prepreg in the center of the beam. 7. De-bulk sample for 5 minutes. 8. Place a ball bearing in the hole created in step 6. 34 Figure 3.12: Prepreg (left) and resin (right) bi-material beam samples. 3.1.2.2 Bi-Material Beam: Calculating Modulus Since the DMTA measures the stiffness of the BMB system based on the force and amplitude, the raw data from the DMTA must be processed in order to extract the resin film or prepreg modulus. The following methodology used for BMB data extraction is based on the work of (Curiel & Fernlund, 2008) and (Twigg & Poursartip, 2003). The procedure described in this section was implemented in Microsoft Excel and utilized raw measured data from the DMTA. The effective stiffness ( E I ) Eff of an elastic beam in three point bending is calculated using ( E I ) Eff 35 F L3 48 Y (3.7) where F , L and Y are applied force, length of the support span and deflection of the beam at mid span respectively. This equation assumes the beam is made of homogenous material, that perfect bonding occurs and that the materials behave elastically. When equation (3.7) is applied to the BMB sample, the effective stiffness of the entire BMB system is calculated. Since the loading is dynamic, the BMB effective stiffness is modified to account only for the storage component using ( E ' I ) Eff ( E * I ) Eff cos Fmax L3 cos 48 Ymax (3.8) where the force ( Fmax ) , displacement (Ymax ) and phase angle ( ) are measured by DMTA. Calculating the effective stiffness of a bi-material uses ( E I ) Eff 2 tc w t st3 tc Est Ec tc E 4 6 4 c 12 tst Est tc Ec tst t st Est 3 tc Est tst tst Ec tc (3.9) where w is the beam width, tc is the composite (resin or prepreg) thickness, tst is the steel shim thickness, Ec is the composite modulus and Est is the steel shim modulus. The only unknown in equation (3.9) is the prepreg or resin modulus. Solving for Ec is a matter of adjusting this value until equations (3.8) and (3.9) converge. A macro was written that utilizes the ‘solver’ function to solve the value of Ec for each data point. The modulus of the steel shim used has an important effect on the calculated modulus of the resin or composite. When examining the DMTA’s reproducibility of the steel shim modulus, some variability results. Over 7 identical trials, the mean value and standard deviation of the steel shim’s modulus were 207.15 GPa and 5.79 GPa respectively. This implies that for a 3 sigma confidence interval, the measured 36 range of Young’s modulus for the steel shim can range from 189.63 to 224.37 GPa. Figure 3.13 shows that when the prepreg or resin modulus is below 100MPa, it is very sensitive to the steel modulus. Figure 3.13: Sensitivity of calculated prepreg modulus from BMB test to the steel shim’s modulus. As a result of this measurement variability, calculated composite modulus values below 100MPa have large uncertainty and therefore not used for further analysis. In an effort to reduce this effect, the modulus used for the steel in the data reduction was calculated from the minimum stiffness of the BMB when the resin modulus was at a minimum. It is assumed that the stiffness contribution to the BMB at this point is entirely from the shim. This is based on the 9 orders of magnitude difference between the resin modulus (≈20-100 Pa) and the steel shim modulus (≈200 GPa). This calculation ensures that the modulus used for the steel shim results in a minimum resin modulus of zero. This is accomplished by finding the data point that has the lowest drive force ( Fmin ) throughout the experiment and its associated displacement ( y ) . The measured steel shim modulus for a given experiment was calculated using 37 Fmin L3 ESt 48 y I St (3.10) where the moment of inertia ( I st ) of the steel shim is calculated using I St 1 w h3 12 (3.11) where b and h are the width and thickness of the steel shim respectively. In all cases, the calculated steel shim modulus used for the data reduction, were within the 3 sigma range found when testing the steel shim. Two additional sources of error that affect the calculated resin or prepreg modulus have been identified. They are the thickness variation in the prepreg samples and variation in the cross sectional shape in the resin sample. In both cases the practical solution was to scale the BMB modulus to match that of the fully cured homogeneous beams as shown in Figure 3.14. To determine how much scaling is required, a reference temperature must first be selected. The scaling factor applied to every BMB data point is ' EHB (Tref ) ' EBMB (Tref ) ' (3.12) where EHB (Tref ) is the storage modulus of the homogeneous beam at the reference temperature and ' EBMB (Tref ) is the storage modulus of the bi-material beam at the reference temperature. 38 9.E+09 200 8.E+09 E' [Pa] 6.E+09 BMB 100 Homogeneous Beam 5.E+09 BMB: Scaled 4.E+09 Temperature 50 3.E+09 2.E+09 Temperature [C] 150 7.E+09 0 300 350 400 450 500 Time [Min] Figure 3.14: MTM45-1 prepreg BMB modulus scaled to match the homogeneous beam modulus. Note that the view is expanded to show the cool down only. The error associated with the prepreg BMB samples was speculated to be a result of poor compaction since no pressure was applied during the cure. This leads to a sample that is thicker than if it had been cured with pressure as is typical in processing. Comparing a 4-ply prepreg lay-up cured with and without vacuum yields a typical material thickness of 0.54mm and 0.75mm respectively. Since the modulus is a function of the thickness cubed, the calculated modulus is very sensitive to thickness. Although the resin BMB samples did not suffer from lack of compaction like the prepreg BMB samples, they were subject to error as a result of their non-rectangular cross-sections. This was due to the surface tension of the resin while in a liquid state. Since equation (3.9) applies to rectangular cross section, error is introduced. To solve this, the calculated resin BMB modulus was scaled to match the fully cured homogeneous beams in the same manner as the prepreg BMB. 39 3.1.2.3 BMB: Test Procedure The BMB tests were intended to give insight into the modulus development during a cure cycle; therefore, the thermal cycles used for BMB tests were based on typical cure cycles. All tests were performed with a constant frequency. The following experimental setup and procedure was used for the BMB tests: Amplitude: 150µm Frequency: 0.1Hz, 1 Hz Temperature Ramp Rates: 1°C /min, 2°C /min Cure Temperature: 180°C, 130°C Pre-Load Force: 0.01N Procedure: 1. 2. 3. 4. 5. Equilibrate at -20°C Iso-Thermal hold for 5 minutes Ramp to cure temperature Iso-Thermal for cure time Ramp temperature to ambient 3.2 Rheometer Rheological data presented in this work was provided by Dr. Pascal Hubert of McGill University and by Convergent Manufacturing Technologies (CMT, 2011). All experiments were conducted in a TA Instruments AR2000 rheometer by applying oscillatory shear at a frequency of 1Hz to a sample using a parallel plate configuration as shown in Figure 3.15. 40 Figure 3.15: Anton-Paar MCR502 rheometer with disposable 25mm parallel plate geometry with thermal control from peltier plate and hood. The rheometer works in a very similar manner to the DMTA in the sense that an oscillatory strain is applied to the material and force required is measured. The difference with the DMTA is that the rheometer applies shear strains to the sample. 3.2.1 Rheometer Sample Preparation Neat epoxy resin and composite prepreg rheometer samples are shown in Figure 3.16. Preparing a sample for a rheometer test was completed using the following steps: Stack resin film or prepreg sheets (unidirectional layup) to achieve a thickness of approximately 1mm. De-bulk for 5 minutes. Trim excess resin or prepreg using a utility knife and the upper plate as a template. Load the sample in the rheometer. 41 Figure 3.16: Neat resin (left) and prepreg (right) rheometer samples in disposable bottom dish. Disposable upper plates (middle) are also shown. 3.2.2 Rheometer: Calculating Complex Modulus During a test, the force, displacement and tan delta are measured by the rheometer. The complex modulus was calculated using G* G ' iG " max max (3.13) G ' G * cos (3.14) G '' G * sin (3.15) where max and max are calculated based on the geometry by the rheometer’s built in software. 42 3.2.3 Rheometer: Test Procedure Two types of rheometer test results were available; temperature ramp tests and iso-thermal tests. All samples were initially uncured and for all tests a constant frequency of 1Hz was used. The following procedures were used for the rheometer tests: Temperature Ramp Rheometry Test Frequency: 1 Hz Temperature Ramp Rates: 1°C /min, 2°C /min, 3°C /min, 4°C /min, 5°C /min Temperature Range: 30°C to 200°C Procedure: 1. Ramp temperature until the required force reaches the machine’s limit. Iso-Thermal Rheometry Test Frequency: 1 Hz Temperature Ramp Rates: 2°C /min Iso-Thermals: 125°C, 130°C, 135°C, 140°C, 145°C Procedure: 1. Ramp to Iso-Thermal temperature. 2. Hold temperature until the required force reaches the machine’s limit. 43 3.3 Differential Scanning Calorimeter A Differential Scanning Calorimeter (DSC) was used to quantify the DOC of partially cured DMTA samples by identifying the materials glass transition temperature (Tg). The DSC used was a TA Instruments Discovery DSC. In all tests modulated temperature ramps were used and enabled calculation of reversible heat capacity. Heating ramps of 4°C/min were used for all tests and was recommended by TA Instruments. Calibration of the DSC also used the same 4°C/min temperature ramp. Very small amounts (in the order of 1mg) of material were required for each DSC test. DSC samples were collected by trimming off DMTA samples of interest. The DSC sample weight was calculated as the difference between the empty pan and the pan with the sample. A Swiss Made EP125SM analytical scale was used for all mass measurements. When examining the heat capacity as a function of temperature of a polymer, a transition occurs at the glass transition temperature. The temperature where this inflection point occurs was determined using the analysis software ‘Trios’ that is built into the DSC system. Figure 3.17 shows how the inflection point is used for determining the Tg of a partially cured sample. 44 Heat Capacity (Normalized) 1.1 1.05 Midpoint Tg: 167.5C 1 0.95 0.9 120 140 160 Temperature [C] 180 200 Figure 3.17: Evaluation of glass transition temperature using DSC heat capacity results. Using the material’s Tg, the DOC of the sample was determined by referencing the DeBenedetto equation which relates Tg and DOC. The DeBenedetto equation is an empirical relationship in the form Tg Tg 0 X (Tg Tg 0 ) 1 (1 ) X (3.16) where X is the DOC, Tg 0 is the uncured glass transition temperature, Tg is the fully cured glass transition temperature and is a material constant. The constants for MTM45-1 are as follows (CMT, 2009): Tg 0 : 3[C] Tg : 220 [C] : 0.7 45 3.4 Cure Kinetics Determining the DOC during rheometry and BMB tests was accomplished using a cure kinetics model. This model required the material’s thermal history (time and temperature) as inputs. When the time and temperature dataset of an experiment was used, the result was a prediction of the DOC for each data point. The commercially available RAVEN3 was used to implement the NCAMP ACG MTM45-1 cure kinetics model (CMT, NCAMP ACG MTM45-1 Material Model, 2009). RAVEN3 was also used to design cure cycles that would produce DMTA samples that were partially cured to a specified DOC. Using RAVEN3 saved significant time and material since different cure cycles could be run through the software to predict the final DOC. 3.5 Thermo Gravitational Analysis Thermo Gravitational Analysis (TGA) was used to determine the temperature at which material degradation occurs. The TGA measures the mass of a sample while heating it. The degradation temperature is taken as the temperature when material begins to lose mass. The following steps were used to determine the degradation temperature: Load sample into TGA. Ramp temperature at 1cpm. Note the temperature when mass loss initiates. 46 3.6 Comparing Rheometer and DMTA Results The results from rheology and DMTA can only be compared if they are in a consistent form. This includes consistency of the modulus (shear, extensional, bulk) as well as consistency of the sample form (resin or prepreg). A schematic of the DMTA 3-point bend and rheometer parallel plate set up is shown in Figure 3.18: Applied load and fiber orientation in DMTA 3-point bend setup.Figure 3.18 and Figure 3.19 respectively, to illustrate the applied loading relative the fiber orientation. Figure 3.18: Applied load and fiber orientation in DMTA 3-point bend setup. 47 Figure 3.19: Applied load and fiber orientation for a rheometer parallel plate setup. To estimate the shear modulus from the extensional modulus, Hooke’s law was used. For an isotropic material such as neat resin Gr Er 2 (1 r ) (3.17) where Gr is the shear modulus of the resin, Er is the extensional modulus of the resin and r is the Poisson ratio of the resin. Relating the shear and extensional modulus for prepreg samples required the 3 dimensional constitutive equations of generalized Hooke’s law. For a transversely isotropic material G23 E2 2 (1 23 ) 48 (3.18) where G23 is the 2-3 direction (or plane of isotropy) shear modulus of the prepreg, E2 is the 2 direction extensional modulus of the prepreg and 23 is the 2-3 Poisson ratio of the prepreg (Herakovich, 1998). As the resin state varies between liquid and solid, the Poisson ratio also varies. In work by Bogetti and Gilespie, the conclusion was that the macroscopic composite properties of the composite were not significantly sensitive to the Poisson ratio (Bogetti & Gillespie, 1992). In this work, constant Poisson ratio of 0.59 for 23 and 0.36 for resin were assumed (Herakovich, 1998). Extracting the resin properties from the results of tests completed with prepreg samples required the use of micromechanics. In the presented work, the simple iso-stress rule of mixtures was used and is of the following form: E2 where Em E f ( transverse ) E f ( transverse ) (1 V f ) Em V f (3.19) Em is the matrix modulus, E f ( transverse ) is the transverse fiber modulus and V f is the fiber volume fraction. The value of E f ( transverse ) used was 20 GPa (Hughes, 1991) and the value of V f used was 0.6 (ACG, 2008). 49 4 Results This chapter contains the findings and results from the experiments as described in Chapter 3. The experimental data generated from DMTA and rheometry is presented along with general trends and observations of the experimental data. A generalized Maxwell model is presented and the procedure used to fit it to the experimental results is explained. Finally the predictions of the model are compared to the experimental results. 4.1 Experimental Data All DMTA and rheometer experimental data used in this thesis is presented here. Further an overlay of the two classes of tests provides a map of the complex modulus behavior for all ranges of cure and temperature typical to composite processing. 4.1.1 DMTA DMTA Homogeneous Beam: Fully Cured Iso-Thermal Frequency Sweep Tests The results from fully cured homogeneous beam are shown from Iso-Thermal Frequency Sweep tests. The presented results illustrate the effect of temperature and frequency on the complex modulus. The raw outputs of E’ and E” are shown in Figure 4.1, Figure 4.2, Figure 4.3 and Figure 4.4 for MTM451/GA045, MTM45-1/HTS5631, MTM45-1 neat resin and 3900-2/T800H respectively. 50 Figure 4.1: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1/GA045 prepreg. Figure 4.2: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1/HTS5631 prepreg. 51 Figure 4.3: E’ and E” results from iso-thermal frequency sweep DMTA test on MTM45-1 neat resin. . Figure 4.4: E’ and E” results from iso-thermal frequency sweep DMTA test on 3900-2/T800H prepreg. 52 DMTA Homogeneous Beam: Partially Cured Iso-Frequency Temperature Ramp Tests A series of Homogeneous Beam tests was completed on MTM45-1/HTS5831 and MTM45-1/GA045. Each test was composed of numerous iso-frequency temperature ramp tests each with a different DOC. The resulting plots illustrate the effect of temperature and DOC on the storage and loss modulus. Figure 4.5 and Figure 4.6 show the results for MTM45-1/GA045 which used a different sample for each curve. Figure 4.7 and Figure 4.8 shows the results for MTM45-1/HTS5631 which used the same sample for all but X = 0.83, 0.9 and 1.0. Using the same sample helped to reduce the error introduced by sample to sample variability. Figure 4.5: E’ results from iso-frequency temperature ramp DMTA tests of MTM45-1/GA045 prepreg samples with varying DOC. Note that each test was on a different sample. 53 Figure 4.6: E” results from iso-frequency temperature ramp DMTA tests of MTM45-1/GA045 prepreg samples with varying DOC. Note that each test was on a different sample. Figure 4.7: E’ results from iso-frequency temperature ramp DMTA tests of MTM45-1/HTS5631 prepreg samples with varying DOC. Note that each test was on the same sample with the exception of X= 0.83, 0.9 and 1.0. 54 Figure 4.8: E” results from iso-frequency temperature ramp DMTA tests of MTM45-1/HTS5631 prepreg samples with varying DOC. Note that each test was on the same sample with the exception of X= 0.83, 0.9 and 1.0. Bi-Material Beam Results from bi-material beam tests are presented in this section. Tests were completed using MTM45-1 resin and MTM45-1/GA045 prepreg. These results show the development of the storage modulus during cure cycles typical to composite manufacturing. 55 Figure 4.9: MTM45-1 resin BMB test (ramp 1°C/min to 180C, cure 120 minutes). Figure 4.10: MTM45-1 resin BMB test (ramp 2°C/min to 140C, cure 240 minutes). 56 Figure 4.11: MTM45-1 resin BMB test (ramp 1°C/min to 180C, cure 120 minutes). Figure 4.12: MTM45-1 resin BMB test (ramp 2°C/min to 130C, cure 240 minutes, ramp 2°C/min to 180C, cure 120 minutes). 57 Figure 4.13: MTM45-1/GA045 prepreg BMB test (ramp 1°C/min to 180C, cure 120 minutes). Figure 4.14: Figure 4.15: MTM45-1/GA045 prepreg BMB test (ramp 1°C/min to 180C, cure 120 minutes). 58 4.1.2 Rheometer The following section presents the results of G’ and G” from rheometer iso-thermal and temperature ramp tests. The experimental data of MTM45-1 resin and prepreg were provided in raw form by Convergent Manufacturing Technologies (CMT, 2011). Figure 4.16: G’ results from iso-frequency temperature ramp rheometer tests of MTM45-1 resin and MTM45-1/HTS5613 prepreg. 59 Figure 4.17: G” results from iso-frequency temperature ramp rheometer tests of MTM45-1 resin and MTM45-1/HTS5631 prepreg. Figure 4.18: G’ results from iso-frequency iso-thermal rheometer tests of MTM45-1 resin. 60 Figure 4.19: G” results from iso-frequency iso-thermal rheometer test of MTM45-1 resin. 4.1.3 Overlay Overlaying results from sections 4.1.1 and 4.1.2 shows the material response over the range of the DMTA and the rheometer from the uncured to the fully cured state. Figure 4.20 and Figure 4.19 show G’ and G” respectively. These provide a visual picture of how temperature and DOC affect the modulus. Additionally, these two figures provide the experimental data that the generalized Maxwell model is fit to in section 4.3. 61 Figure 4.20: Overlay of DMTA and rheometer MTM45-1 resin tests that show G’ as a function of temperature and DOC ranging from 0.01 to 1.0. 62 Figure 4.21: Overlay of DMTA and rheometer MTM45-1 resin tests that show G’ as a function of temperature and DOC ranging from 0.01 to 1.0. 4.2 Trends 4.2.1 Continuity of DMTA and Rheometer In order to show the continuous modulus development from an uncured to a cured state, it is necessary to look at both DMTA and rheometry results. Overlaying the storage modulus from a rheometer test (1°C/min ramp), BMB test (1°C/min to 180°C Isothermal) and homogeneous beam test show the modulus development during the manufacturer’s recommended cure cycle. All data presented in this section was collected under identical frequency (1 Hz) and thermal histories (time and temperature). 63 Figure 4.22 shows continuous development of the storage modulus for both neat resin (MTM45-1) and prepreg (MTM54-1/GA045). The DOC as predicted by the cure kinetics model is also shown. Figure 4.22: Overlay of results from rheometer, BMB and homogeneous beam experiments of MTM45-1 resin and MTM451/GA045 prepreg. . There is a gap between the BMB and rheometer data which is due to the limitation of each test. Despite this, continuity is observed in the modulus development spanning 8 orders of magnitude. The loss modulus was omitted since the BMB tests do not accurately capture this. The continuity of modulus is an important result since a single model will be used to predict the behavior across DMTA and rheometry ranges in section 4.3. 64 4.2.2 Effect of Thermoplastic Constituent on Modulus Classically, composites are thought of as containing two phases; the fiber reinforcement and the matrix. However, numerous high performance systems contain a third thermoplastic constituent. Two such examples are Toray 3900-2/T800H which advertises a ‘thermoplastic toughening’ constituent (Toray, 2001) and ACG MTM45-1/GA045 which contain thermoplastic fibers intended to hold the prepreg together during layup (ACG, 2008). Results of DMTA homogeneous beam tests are shown in Figure 4.23 which demonstrates an effect intrinsic to composites containing a thermo-plastic constituent. Both systems containing a thermoplastic phase (MTM45-1/GA045 and Toray 3900-2/T800H) show a secondary transition indicated by the decrease in modulus. The two systems that do not contain a thermoplastic phase (MTM45-1/HTS5631 and MTM45-1 neat resin) do not show a secondary transition. The same experimental procedure and operating parameters were used for all four materials presented. Figure 4.23: The effect of a thermoplastic constituent (MTM45-1/GA045 & 3900-2/T800H) on the storage modulus compared to systems without a thermoplastic constituent (MTM45-1 & MTM45-1/HTS5631). All samples are fully cured. 65 This secondary transition is believed to be associated with the thermoplastic constituent and occurs at the respective melt temperature (Tm). This is confirmed for MTM45-1/GA045 in Figure 4.24 which shows the effect of temperature and degree of cure on the storage modulus. Each curve is the result of a homogeneous beam test of different samples each with different DOC ranging from 0.14 to 1.0. Figure 4.24: Storage modulus of homogeneous beams partially cured to varying DOC of MTM45-1/GA045 prepreg showing a secondary transition from the thermoplastic constituent. When Tg of the epoxy is greater than that Tm of the thermoplastic constituent, the secondary transition is clearly observed in the storage modulus. Further, the Tm of the thermoplastic does not change despite the cure advancement of the epoxy. 66 4.2.3 Effect of Temperature and DOC on Un-Relaxed Modulus The un-relaxed modulus is the limit that the glassy modulus approaches for infinitely high frequencies. In examining numerous tests, it is clear that the glassy modulus is dependent on temperature. This observed behavior holds true for all temperature ranges explored (as low as -30C). With the exception of secondary transitions (see section 4.2.2), this temperature dependence of the glassy storage modulus appears to be generally linear. Figure 4.25 shows the glassy storage modulus dependence on temperature and frequency for fully cured MTM45-1/HTS5631 and MTM45-1/GA045. Figure 4.25: Glassy storage modulus of MTM45-1/HTS5631 prepreg and MTM45-1/GA045 prepreg as a function of temperature and frequency. 67 To better understand the effect frequency has on the storage modulus, MTM45-1/HTS5631 results are shown in Figure 4.26 as storage modulus vs frequency. Due to the limitations of the DMTA it is not possible to go to higher frequencies. As a result, the expected limit that the modulus approaches at infinitely high frequencies cannot be observed. Figure 4.26: Results of homogeneous beam tests on MTM45-1/HTS5631 showing the temperature and frequency dependence of the storage modulus. The glassy storage modulus of partially cured beams also shows linear temperature dependence. The storage modulus of MTM45-1/HTS5631 for different DOC ranging from 0.295 to 0.99 is shown in Figure 4.27. 68 Figure 4.27: Glassy storage modulus of MTM45-1/HTS5631 prepreg as a function of temperature for partially cured samples of varying DOC ranging from 0.295 to 1.0. There are two significant implications. First is that a simple horizontal shift is not sufficient to describe the material behavior for varying DOC as is assumed for thermo-rheologically simple TTS. Second is that the glassy modulus is shown to be independent of DOC. 69 4.2.4 Effect of Fiber 4.2.4.1 Fiber Bed Interaction Figure 4.28 below shows the results from numerous rheometer temperature ramp tests of MTM45-1 resin and prepreg. The storage modulus of the prepreg samples reaches a plateau despite the continued decrease of modulus of the resin samples. This is due to the interaction of the fiber bed (Hubert & Poursartip, 1998). Due to the waviness of the fibers, the fiber bed is able to resist shear deformation in proportion to the shear modulus of the fiber bed. This effect was found to be significant for high fiber volume fraction composites and has since been the subject of model development (Gutowski, Cai, Bauer, & Boucher, 1987) (Cai & Gutowski, 1992). Figure 4.28: G’ results from rheometer temperature ramp tests (1, 2, 3, 4 & 5 °C/min) of MTM45-1 prepreg and resin showing the effect of the fiber bed stiffness. 70 To show the dependence of the prepreg modulus on the resin modulus, G’ results from the prepreg tests were plotted against the resin tests in Figure 4.29. Both results from rheometer and bi-material beam tests are shown. Additionally, the Iso-Stress prediction from equation (3.18) is shown. Figure 4.29: G’ prepreg vs. G’ resin showing the effect of the fiber bed interaction. These results imply that classical micromechanics do not work when the fiber bed modulus is dominant (Malekmohammadi, Thorpe, & Poursartip, 2011). Below a critical value of resin modulus, the prepreg modulus depends on the fiber bed modulus only. For the purpose of the work presented in this thesis, this finding requires that the material model fitting (Section 4.3) be completed on resin rheometry data in order to avoid complications arising from the fiber bed interaction. 71 4.2.4.2 Shift Function Time-Temperature-Superposition (simple horizontal shift only) was used to construct master curves from Iso-Thermal Frequency DMTA tests of three different materials. One of the samples was neat resin (MTM45-1) and two of the samples were prepreg (MTM45-1/GA045 & MTM45-1/HTS5631). All samples were fully cured prior to the DMTA tests. Each test was conducted using the same experimental setup and the TTS for all three datasets used 220°C (the fully cured glass transition temperature) as the reference temperature. Figure 4.30 and Figure 4.31 show the master curves and shift functions for the three materials. Figure 4.30: Storage modulus master curve of fully cured MTM45-1 resin, MTM45-1/GA045 prepreg and MTM45-1/HTS5631 prepreg. 72 Figure 4.31: Temperature shift function of fully cured MTM45-1 resin, MTM45-1/GA045 prepreg and MTM45-1/HTS5631 prepreg. The shift functions for the three materials show strong agreement. This confirms experimentally the assumption that the matrix shift function is equal to the composite shift function (Zobeiry, 1997). Near the transition temperature associated with the thermoplastic constituent (MTM5-1/GA045 sample) the shift function deviates from the 2 systems that do not contain the thermoplastic phase. Note that data for the resin sample was not collected above the glass transition temperature due to creep of the sample. Equation Section 4 73 4.3 Generalized Maxwell Model Fit The generalized Maxwell model used is taken from (Simon, McKenna, & Sindt, 2000). The only alteration here is that the un-relaxed modulus (Gu) was made to be a funtion of temperature. The following equations were used to describe the storage, loss and relaxation modulus1: G '(, T , X ) G r ( X , T ) [Gu (T ) G r ( X , T )] gi G "(, T , X ) [Gu (T ) G r ( X , T )] gi 2 aT2, X i2 1 2 aT2, X i2 aT , X i 1 2 aT2 , X i2 G(t, T , X ) G r ( X , T ) [Gu (T ) G r ( X , T )] gie ( t / aT , X i ) (4.1) (4.2) (4.3) Where, G’ – Storage Modulus G” – Loss Modulus G – Relaxation Modulus Variables ω – Frequency T – Temperature X – Degree of Cure (Chemical Conversion) t - Time 1 Note that the relaxation modulus is not evaluated against experimental data in this thesis and is included here only to show the Maxwell model’s ability to describe relaxation behavior in addition to the storage and loss modulus. 74 Functions of ω, t, T and X Gr – Relaxed Modulus [See Equation (4.4)] Gu – Un-relaxed Modulus [See Equation(4.12)] aT,X – Shift Function [See Equation (4.13)] Constants (Fit Parameters) τi – Relaxation Time of the ith Maxwell Element gi – Weight Factor of the ith Maxwell Element The general form of the Maxwell model is shown schematically in Figure 4.32 which is composed of n Maxwell elements (spring and dashpot in series) in parallel with a single spring. Figure 4.32: Schematic representation of a generalized Maxwell model. The approach used to fit the Maxwell model to experimental data is described for each sub-component (Relaxed Modulus, Un-Relaxed Modulus, Shift Function and Relaxation Time/Weight Factors) of the model. Each sub-component has an effect on each of the other sub-components; therefore an iterative approach was required for fitting as shown in Figure 4.33. 75 Figure 4.33: Iterative approach to fitting the generalized Maxwell model sub-components. 4.3.1 Relaxed Modulus The relaxed modulus depends on the DOC of the material and whether the material has reached gelation or not. The relaxed modulus is in the following form: 0; X X gel Gr ( X ,T ) r G ( X , T ); X X gel 76 (4.4) Where, Gr : Relaxed Modulus X : Degree of Cure X gel : Degree of Cure at Gelation The DOC at which gelation occurs can be predicted based on the functionality of the thermosetting system. The exact solution to Miller and Macosko’s approach was solved by (Gordnian, 2012) for functionalities of 3, 4 and 5. The DOC at which gelation occurs is 0.707, 0.577 and 0.5 for functionalities of 3, 4 and 5 respectively. Pre-gelation, it is assumed that the material has a relaxed modulus of zero. In this regime an infinite cross-link network has not yet formed and the material is a visco-elastic liquid which by definition has a relaxed modulus of zero (Morrison, 2001). Post gelation, the material is a visco-elastic solid and the relaxed modulus is assumed to behave based on rubber theory as developed by (Miller & Macosko, 1976). The relaxed modulus is a function of the effective cross-link concentration (determined by DOC) and temperature as follows: G Xr 1 T G ( X ,T ) TG r X 1 r X 1 Where, 77 (4.5) G Xr 1 : Reference Relaxed Modulus [Pa] T : Temperature [K] TGr : Reference Temperature [K] X 1 : Effective Cross-link Concentration X 1 : Reference Effective Cross-link Concentration The effective cross-link network concentration drives the behavior of the relaxed modulus post gelation as DOC increases. Gordnian’s solutions to the Miller and Macosko equation are shown in Figure 4.34 for functionalities of 3, 4 and 5 based on the following equations: Functionality: 3 1 3 f 3 1 P 2 Pf 3 (4.6) 1 1 rX 2 (4.7) Functionality: 4 3 4 f 4 2 P 1 P 1 P Pf 4 1 4 1 3 2 2 rX 2 Functionality: 5 78 (4.8) (4.9) 3 4 f 5 5 P 2 1 P 5 P 1 P 1 Pf 5 1 108 120 81 80 2 12 48 2 2 4 6 rX rX r X 3 1 P 5 2 3 (4.10) 4 1 108 120 81 3 80 2 12 48 2 2 4 rX rX r X 3 1 3 (4.11) Where, Pf : Probability that a given arm is not connected (Subscript denotes the functionality) r : Stoichiometric ratio limit (r 1) 79 Figure 4.34: Normalized relaxed modulus as a function DOC for functionalities of 3, 4 and 5. The constants used in equation (4.5) are determined from an iso-frequency temperature ramp (homogeneous beam) test. Figure 4.35 shows such a test and the constants used for the relaxed modulus of MTM45-1. 80 GXr 1 40 MPa X 1 1 TGr 518.15 K X 1 Figure 4.35: Determining the reference relaxed modulus of MTM45-1 resin using iso-frequency temperature ramp results of a fully cured sample. The following constants were fit to the relaxed modulus for MTM45-1 resin: G Xr 1 : TG' 40 MPa : 518.15 K X 1 X 1 : 1 functionality : 4 X gel : 0.577 r : 1 81 4.3.2 Un-Relaxed Modulus The un-relaxed modulus is the limit that the storage modulus approaches at infinitely high frequencies. As was shown in section 4.2.3, the glassy modulus is linearly dependent on temperature and independent of DOC. The un-relaxed modulus used is in the following form: u u T 273 G (T ) G GT' 2 GT' 1 T T2 T1 (4.12) Where, T : Temperature [K] GTu 273 : Un-relaxed Modulus at T = 273K = 0°C T1 & T2 : Reference Temperatures [K] GT' 1 & GT' 2 : Storage Modulus at T1 & T2 [Pa] The values of T1 & T2 and GT' 1 & GT' 2 used to fit equation (4.12) are found using the storage modulus determined from a homogeneous beam (iso-frequency temperature ramp) test. Any two data points may be used provided that both are in the glassy regime. Since the DMTA cannot test beams at infinitely high frequencies, GTu 273 is selected by trial and error until the glassy storage modulus predicted by the Maxwell model matches experimental data for the frequencies tested. Figure 4.36 illustrates the fit of equation (4.12) to experimental data. 82 Figure 4.36: Fitting the un-relaxed modulus to experimental data. The following constants were fit to the un-relaxed modulus for MTM45-1 resin: GTu 273 : 2.25 GPa GT' 2 GT' 1 : T2 T1 -3.6 MPa/K 83 4.3.3 Shift Function The shift function depends on the difference between temperature and the material’s glass transition temperature (T Tg ) . Further, the function used is a piecewise function. The general form of each subfunction is based on experimental data presented in literature and on section 4.2.4.2. Above the glass transition temperature the WLF equation is applied. Between 0K and 50K below the glass transition temperature, a 2nd order polynomial is used. The constants are selected such that the polynomial is concave down which captures the inflection point generally observed when T Tg 0 . At temperatures more than 50K below the glass transition temperature a linear function is applied. The shift function is of the following form: log aT , X C 1 T Tg fg ; T Tg 0 C2 f g T Tg 2 f p p1 T Tg p2 T Tg ; 0 T Tg 50 L1 L2 T Tg ; 50 T Tg (4.13) Details of each sub-function are described below in the following sections. Figure 4.37 shows the effect of temperature and DOC of the shift function. 84 Figure 4.37: The effect of temperature and DOC on the piecewise shift function. WLF Sub-Function Fit The WLF equation has been well documented and cited as describing the shift function for polymers above their glass transition temperature. The method used extends the free volume derivation of WLF by (Gilbert, Ashby, & Beaumont, 1986) log aT C1 T Tg fg C2 f g T Tg f g a ' b ' X (4.14) (4.15) where C1 and C2 are the WLF constants, a ' and b ' are constants and f g describes the change of the material’s free volume at the glass transition temperature as a function of DOC. Since the glass 85 transition temperature increases with DOC, the free volume at the glass transition temperature also increases with DOC. This is shown schematically in Figure 4.38. Figure 4.38: Schematic showing the free volume at the glass transition temperature as a function of DOC. The following algorithm was used to fit the WLF Shift Sub-Function: 1. Select C1 & C2 by fitting equation (4.14) to experimentally determined shift function from TTS of Iso-Thermal Frequency Sweep experiment of a fully cured sample (note that f g , X 1 1 ). 2. Select f g , X 0 1 . " " 3. Select prony series weight factors to fit G Maxwell to G Rheometer for X 0 . 4. Calculate a ' and b ' such that f g increases linearly between f g , X 0 and f g , X 1 . " " 5. Check fit of G Maxwell compared to G Rheometer for X 0 0.6 . 6. Repeat steps 2 – 5 until fit in step 5 is satisfied. 86 The following constants were fit to the WFL sub-function for MTM45-1 resin: C1 : 12.5 C2 : 45 a' : 0.691 b' : 0.309 2nd Order Polynomial Sub-Function Fit The polynomial shift function fit is of the following form log aT p1 T Tg p2 T Tg 2 (4.16) where p1 & p2 are fit constants. To account for the DOC, equation (4.16) is scaled by f p c ' d ' X where c '& d ' are fit constants. 87 (4.17) The following algorithm was used to fit the Polynomial Sub-Function: 1. Select p1 & p2 to fit equation (4.16) to experimentally determined shift function from TTS of Iso-Thermal Frequency Sweep experiment of a fully cured sample. ' ' '' '' 2. Select prony series weight factors fit G Maxwell to G DMTA and G Maxwell to G DMTA for X 1 . ' ' 3. Check fit of G Maxwell to G DMTA for each partial cure Iso-Frequency Temperature Ramp DMTA test. ' ' 4. For each DOC in step 3, determine scalar ( f p ) required for best fit of G Maxwell to G DMTA . 5. Plot ( f p ) vs X from step 4. 6. Select c '& d ' of equation (4.17) to fit step 5. The following constants were fit to the Polynomial Sub-Function for MTM45-1 resin: p1 : -0.299 p2 : -0.003 c' : 1.0 d' : 0 88 Linear Sub-Function Fit The linear shift function is in the following form L1 L2 T Tg (4.18) where L1 & L2 are constants. L1 is selected such that equations (4.16) and (4.18) have the same value at T Tg 50 . Based on the assumption that the glassy modulus is independent of DOC as described in section 4.2.3, L2 0 . 4.3.4 Relaxation Times and Weight Factors Each Maxwell element requires a relaxation time and weight factor. The choice of relaxation times affects the weight factors required to fit the model to the experiments. Therefore the relaxation times are selected before fitting the weight factors. Once selected, the relaxation times remain constant and it is the weight factors that are used to fit the model to experimental data. Both the range of relaxation times and the number of relaxation times were selected before any fitting. The range determines the extent to which the model can be tuned. The number of relaxation times (equal to the number of Maxwell elements) determines the smoothness of the model. Insufficient elements results in oscillations of the model. An excessive number of elements is redundant which 89 makes fitting more difficult and implementation of the model less computationally efficient. The fewest number of elements should be used that provides a sufficiently smooth model. The sum of the weight factors must equal one. This ensures that the un-relaxed modulus retains a physical meaning. The weight factors determine how much each element contributes at the corresponding relaxation time. Therefore the weight factors are used to fine tune the model fit to experiments. The weight factors are fit to different types of experiments depending on the associated relaxation times. Relaxation Time (Log) Weight Factor Fit To 10 to 15 G” – Rheometer 9 to 7 No Experimental Data – Smooth fit between DMTA and Rheometer 6 to -15 G’ – DMTA, G” - DMTA The following constants were fit to the relaxation times and weight factors for MTM45-1 resin: Log Time [s] Weight Factor -15 0.009561 -14 0.009561 -13 0.009561 -12 0.009561 -11 0.019121 -10 0.019121 -9 0.019121 -8 0.019121 90 -7 0.019121 -6 0.019121 -5 0.019121 -4 0.038243 -3 0.057362 -2 0.076485 -1 0.114728 0 0.133849 1 0.15297 2 0.133849 3 0.095606 4 0.019121 5 0.003824 6 0.001338 7 0.000382 8 9.56E-5 9 3.82E-5 10 1.15E-5 11 2.87E-6 12 7.66E-7 13 9.56E-8 14 3.82E-8 15 1.91E-10 91 4.4 Goodness of Fit The following figures compare the predictions of G’ and G” from the generalized Maxwell model fit (Section 4.3) to the experimental results. Figure 4.39: G’ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin. 92 Figure 4.40: G“ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin. Figure 4.41: G” comparison of rheometer and generalized Maxwell model predictions of MTM45-1 resin. 93 Figure 4.42: G’ comparison of rheometer and generalized Maxwell model predictions of MTM45-1 resin. Figure 4.43: G’ comparison of DMTA partially cured beams and generalized Maxwell model predictions of MTM45-1 resin for frequencies ranging from 0.01Hz to 10Hz. 94 5 Discussion and Conclusion 5.1 Discussion 5.1.1 Predicting Viscosity and Modulus During Cure The objective of the work presented in this thesis was to develop a consistent material model to predict the viscoelastic properties of a curing thermosetting polymer. The value of this work is ultimately dependent on its ability to predict the viscosity and modulus during a typical manufacturing cure cycle. In this section the generalized Maxwell model fit in section 4.4 is used to predict the curing behavior of MTM45-1 resin. It is critical that the model accurately predicts both the viscosity during low degree of cure when flow predictions dominate and the modulus for higher degree of cure when stress predictions dominate. These predictions are shown to describe the curing behavior very well when compared to experimental data. In order to compare the rheometer viscosity tests and the model predictions, the model was used to predict the loss modulus for each data point in a given rheometer viscosity test. The temperature and frequency input into the model are raw data from the experiment. The DOC input into the model is from the cure kinetics model which predicts the DOC for each data point using the experimental time and temperature raw data. The resulting G” predictions from the generalized Maxwell model and the experimental G” results are shown in Figure 5.1 and Figure 5.2 for the iso-thermal and temperature ramp tests respectively. It is very clearly shown that the experimental viscosities are well predicted by the generalized Maxwell model. 95 Figure 5.1: Comparison of MTM45-1 resin G” results from generalized Maxwell model predictions to iso-thermal viscosity rheometer tests. Figure 5.2: Comparison of MTM45-1 resin G” results from generalized Maxwell model predictions to temperature ramp viscosity rheometer tests. 96 Predicting the storage modulus to compare with BMB tests was done in a very similar manner to the viscosity predictions above. For each data point of a given BMB test, the generalized Maxwell model was used to predict the storage modulus. The temperature and frequency input were raw data from the BMB experiment and the DOC was determined using the cure kinetics model. Figure 5.3 below shows the G’ results of a BMB MTM45-1 resin test as well as the predictions from the generalized Maxwell model. The cure cycle used one recommended by the manufacturer which ramps at 1°C/min ramp to 180°C and cures for 120 minutes. Very good agreement between the model and experiment is observed. On the cool down sudden small drops of the BMB modulus result from minor delamination at the resin and steel interface. This was common in neat resin BMB samples. Figure 5.4 shows similar results a low temperature cure cycle of 2°C/min ramp to 140°C and cures for 240 minutes. Again, very good agreement between the model and experiment is observed. Figure 5.3: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a typical cure cycle (1°C/min ramp to 180°C, 120 minute cure). 97 Figure 5.4: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a low temperature cure cycle (2°C/min ramp to 140°C, 240 minute cure). Figure 5.5 below shows a two stage cure cycle of 2°C/min ramp to 130°C, 240 minute cure, 2°C/min ramp to 180°C, 120 minute cure. The match between the model and BMB experiment do not fit as well as the previous two BMB tests. However, the maximum error is only approximately 10%. This error during 130°C hold is likely a result of the hold temperature being very close to the material’s glass transition temperature. At and near the material’s glass transition temperature, the modulus is most sensitive to changes in temperature, degree of cure and frequency. Therefore a small error in any of these experimental variables is not captured the model. 98 Figure 5.5: Generalized Maxwell model predictions of G’ compared to MTM45-1 resin BMB test using a two stage cure cycle (2°C/min ramp to 130°C, 240 minute cure, 2°C/min ramp to 180°C, 120 minute cure). 5.2 Future Work The greatest success of this work is showing that a generalized Maxwell model can be used to predict both viscosity and modulus of a curing thermoset epoxy for all DOC and realistic temperature ranges during typical cure cycles. The data set that the model was fit to was the result of numerous types of experiments. Furthermore, producing a consistent data set of resin shear modulus required assumptions and calculations. This was because it was not possible to perform all tests with neat resin samples. Therefore the absolute values predicted by the model can likely be improved by using all resin samples. Alternatively, better micromechanics could be employed using CCA (Composite Cylinder Assemblage) and a non-constant Poisson’s ratio. This does not change the findings of this work and would only result in a fine tune adjustment of the fitting parameters. Most likely only the un-relaxed modulus fit would 99 need to be adjusted to better represent the absolute resin shear modulus. Future research should be directed to improving the absolute values. Additionally, future work should evaluate the generalized Maxwell model’s ability to predict the relaxation modulus, determine the effect of pressure and utilize fit optimization using multi-variable fitting software. 5.2.1 Experiments on Resin Performing all tests on neat resin samples would eliminate the calculations and assumptions made in section 3.6 which were used to convert DMTA prepreg data into resin shear modulus. In this work, there was limited success performing DMTA tests on neat resin beams due to excessive sample creep near the material’s glass transition temperature. An effort was made to use a cantilever clamp instead of the three point bend clamp. The cantilever clamp did reduce the sample creep, however very poor absolute values resulted for glassy samples. This is consistent with the recommendation from TA Instruments which state that stiff, glassy polymers should be tested using the three point bend clamp and low stiffness rubbery polymers should be tested using the cantilever clamp. One method that could be employed is to perform cantilever tests on neat resin beams and then scale the results to match the absolute modulus values determined from a three point bend test. A second method that could be used to test neat resin samples is to use a rheometer equipped with a torsion clamp. Such a clamp is used to apply torsion to a beam sample and essentially converts a rheometer into a DMTA. A significant issue that would require investigation is the accuracy of the resulting absolute modulus values. If poor absolute values result, there is no advantage over using a DMTA cantilever clamp and scaling to match the modulus from three point bend tests. If good absolute 100 values result, this method would be ideal for generating complex resin shear modulus since it would be consistent with complex modulus generated from a parallel plate rheometer test and no calculations would be required. 5.2.2 Micromechanics If in future work the prepreg DMTA results are converted to resin shear modulus, it is possible to improve the accuracy of these calculations. The first and most important item would be to replace the iso-stress micromechanics with CCA micromechanics. Iso-stress is nothing more than a rule of mixtures and is only a first order approximation. Although there is error introduced by the use of iso-stress, it does not detract from this work which was aimed to show that a generalized Maxwell could in fact be used to predict the viscosity and modulus of a curing thermoset epoxy. Adjusting the model such that the glassy modulus fits the CCA prediction as opposed to the iso-stress predictions would be a matter of fine tuning the fit parameters of the un-relaxed modulus and the appropriate relaxation times. A second assumption that should be addressed is the Poisson’s ratio used. In this work, a constant was assumed however in reality the Poisson’s ratio is not constant as was discussed in section 4.2.4.1. 5.2.3 Relaxation Modulus The focus of the presented work has been on the complex modulus. The relaxation modulus can be predicted using the generalized Maxwell model and the same fit parameters as shown by (Simon, McKenna, & Sindt, 2000). Since no relaxation modulus data was collected in this research, it is recommended that further study investigate the accuracy of the relaxation modulus predicted by the generalized Maxwell model compared to experimental data. 101 5.2.4 Effect of Thermoplastic Constituent The presented material model in this thesis does not attempt to predict the effect of a thermoplastic constituent on the composite modulus. It was however shown that the modulus of a composite that contains a thermoplastic constituent will be affected near and above the melt temperature of the thermoplastic. Adapting the generalized Maxwell model to account for a composite that contains a thermoplastic constituent would provide a further improvement to the work presented. This could likely be achieved by using a second prony series and fitting it to the thermoplastic constituent. An inverse rule of mixtures could then be used to predict the response of a system with various fractions of the thermoplastic constituent. This would be a very valuable predictive tool since it is possible to produce composites with varying volume fractions of thermoplastic as a result of different processing conditions. 5.2.5 Use Multi-Variable Fitting Software Optimization and automation of the fitting would provide a better fit to the experimental data and save significant time. The fitting method used in this work was completed in Excel and utilized the solver function with the objective of minimizing the error between the model and the experiment. However, as the complexity and number of variables increased, the solver function became inadequate and an iterative trial and error approach was used. As shown in the results and discussion sections, good fit was achieved however this can likely be improved. Using advanced multivariable fitting software and coding the fit algorithm of section 4.3 would certainly improve the model fit. A second benefit of such an implementation would be the time saved since the trial and error method used was very labour intensive. 102 5.3 Conclusion The goal of this work was to show that a single material model can describe both the viscosity and viscoelastic behavior of a curing thermoset epoxy polymer. This goal was achieved by using a generalized Maxwell model and cure dependent temperature shift function. The following conclusions have been drawn: A generalized Maxwell model can describe both the viscosity and viscoelastic behavior of MTM45-1 thermoset epoxy polymer. The model was shown to be accurate for ranges of degree of cure from 0.01 to 1.0, temperature from -50C to 245C and frequencies from 0.01 to 10Hz. In the context of process modeling, these ranges represent most degree of cure, temperature and time scales that would be encountered in typical manufacturing process. The complex modulus is continuous and consistent between rheometer tests (viscoelastic liquid material behavior) and DMTA tests (viscoelastic solid material behavior) for dynamic oscillatory experiments. Overlaying rheometer and DMTA tests showed continuity between G’ and G”. Further, the successful implementation of the generalized Maxwell model shows that the required continuity and consistency of complex modulus is present. Thermo-rheologically complex behavior was observed in all experiments. The un-relaxed modulus was shown to be a function of temperature and therefore a simple horizontal shift does not describe the material behavior. Using a temperature dependent un-relaxed modulus and a temperature and DOC dependent relaxed modulus was shown to be suitable to predict the thermo-rheologically complex behavior. The glassy modulus of epoxy resin is independent of the degree of cure. Well below the material’s glass transition temperature, the modulus depends only on temperature and 103 frequency. Further, approaching infinitely high frequencies, the un-relaxed modulus depends only on temperature. Classical micromechanics break down when the resin modulus is low and the fiber bed stiffness dominates the composite modulus. Below a critical resin modulus, the composite modulus was shown to be independent of resin modulus and depends only on the fiber bed properties. Thermoplastic constituents were shown to produce a secondary transition associated with the respective melt temperature. The significance is that a drop in storage modulus occurs at the melt temperature and is not affected by the DOC of the thermoset constituent. The current state of viscosity and viscoelastic material models used in composite process modeling is to treat the two phenomena entirely separate. 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Experimental characterization of the viscoelastic behavior of a curing epoxy matrix composite from pre-gelation… Thorpe, Ryan J. 2013
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Title | Experimental characterization of the viscoelastic behavior of a curing epoxy matrix composite from pre-gelation to full cure |
Creator |
Thorpe, Ryan J. |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | Process models have been used to predict the flow and stress development during manufacturing of composite structures for over 20 years. To date, these models have been treated separately, and the required material property models (viscosity and modulus) have also been treated separately. The latest breakthrough in process modeling of composite structures is an integrated stress-flow model. However, a consistent viscosity and viscoelastic material model required by the integrated stress-flow model has not been developed. Presented in this thesis is a consistent material model that predicts the viscoelastic liquid and viscoelastic solid behavior of a commercially available thermoset polymer, namely MTM45-1 epoxy. The goal here is to show that a single material model can predict the viscosity and viscoelastic modulus for all temperature, degree of cure and time scales encountered in composite manufacturing. The model was generated by fitting a generalized Maxwell model to test results from both dynamic mechanical analysis and rheological tests. Both resin and prepreg samples were examined. Thermo-rheological complex behavior was captured by applying linear temperature dependence to the un-relaxed modulus. The effect of cure was accounted for by applying a degree of cure dependent shift function. The relaxed modulus was predicted using the cross-link concentration and the theory of rubber. Excellent agreement was found when comparing predictions from the model to experimental data ranging from temperatures of -50°C to 245°C, degree of cure of 0.01 to 1.0 and frequencies of 0.01Hz to 10Hz. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-04-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0073820 |
URI | http://hdl.handle.net/2429/44271 |
Degree |
Master of Applied Science - MASc |
Program |
Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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