Finite element modeling of thermal expansion in polymer/zrw₂o₈ composites

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2011 Finite element modeling of thermal expansion in polymer/zrw₂o₈ composites Gregory J. Tilton The University of Toledo Follow this and additional works at: Recommended Citation Tilton, Gregory J., "Finite element modeling of thermal expansion in polymer/zrw₂o₈ composites" (2011). Theses and Dissertations This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Thesis entitled Finite Element Modeling of Thermal Expansion in Polymer/ZrW 2 O 8 Composites by Gregory J. Tilton Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Mechanical Engineering Dr. Lesley M. Berhan, Committee Chair Dr. Maria R. Coleman, Committee Member Dr. Yong X. Gan, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo December 2011

3 Copyright 2011, Gregory J. Tilton This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Finite Element Modeling of Thermal Expansion in Polymer/ZrW 2 O 8 Composites by Gregory J. Tilton Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Mechanical Engineering The University of Toledo December 2011 Composite materials are being more frequently used in a wide variety of industries. Their high strength to weight ratio makes them a desirable material in many applications. In some specific cases, polymer based composites can be subjected to large changes in temperature causing undesirable amounts of expansion. To reduce the composite s thermal expansion, materials that have negative coefficients of thermal expansion are used as a filler material. Zirconium tungstate (ZrW 2 O 8 ) is a metal oxide which exhibits thermal behaviors not seen in most other materials. When subjected to a positive temperature change, ZrW 2 O 8 will decrease in volume as opposed to most other materials which show an increase in volume. This makes ZrW 2 O 8 an ideal candidate to be used as filler material in these polymer composites to reduce their overall thermal expansion. While experimental research on ZrW 2 O 8 composites has previously been completed, this research looked at the finite element modeling of these composite materials and tried to gain a better understanding of their possibilities. Initial two-dimensional models were created using COMSOL Multiphysics with basic geometries for both the matrix and filler. The results from these tests showed that the filler geometry had little effect on the expansion results and volume fraction was the most important factor. To further test this, more complex models were created using three-dimensional iii

5 geometries with the same volume fractions. These results confirmed the findings of the two-dimensional tests by showing similar expansion. These results were then compared to published experimental data where it was found that all the models showed less expansion than the physical experiments of the same volume fraction. The difference between the finite element analysis (FEA) and experimental results was attributed to the interaction between the filler and matrix materials. In the models, the bond between the two was considered perfect, with no voids or separation, leading to the filler material having more effect on the overall properties of the composite. In real-world testing, this perfect bond would be nearly impossible to achieve. To build on this idea and gain a better understanding of how the experimental testing compared to the FEA, models with no bond between the filler and matrix were created. Using the results from these models, as well as the models with a perfect bond, an upper and lower bound of expansion were able to be created. All published experimental data looked at was contained within these FEA-created bounds. This showed that while some bond was likely made between the filler and matrix materials, there was room for improvement if less expansion was desired. iv

6 Table of Contents Abstract iii Table of Contents v List of Tables viii List of Figures ix List of Abbreviations xii List of Symbols xiii 1 Introduction General Overview Problem Statement Overview of Principles Composite Materials Thermal Expansion and Zirconium Tungstate Finite Element Analysis Organization of Thesis Literature Review Negative Thermal Expansion Materials v

7 2.2 Experimental Research of ZrW 2 O 8 Composites Mathematical Models for Predicting the CTE of Composites Finite Element Modeling of Particulate Composites Modeling Approach Materials Used and their Properties Setup for COMSOL Multiphysics Modeling to Compare Geometry Two-Dimensional Models Three-Dimensional Models Modeling to Examine Matrix/Filler Bond Analysis of Models Results and Discussion Two-Dimensional Models Three-Dimensional Models Comparisons to Experimental Data and Development of Upper and Lower Bounds Comparison to Mathematical Models Conclusions and Future Work Conclusions Future Work References 59 A MATLAB Codes 67 A.1 MATLAB Code for Creating Random Coordinates for Spherical Inclusions vi

8 A.2 MATLAB Codes for Creating a Mapped Mesh and Randomly Assigning Material Properties A.2.1 Code for Developing Random Filler Elements in Mapped Mesh 70 A.2.2 Code for Assigning CTE Properties A.2.3 Code for Assigning Elastic Modulus Properties A.2.4 Code for Assigning Poisson s Ratio A.2.5 Code for Assigning Density Properties vii

9 List of Tables 1.1 A list of materials and their coefficient of thermal expansion, α, at 20 C Materials used in FEA models and their required properties Numerical results from the two-dimensional modeling Numerical results from the three-dimensional modeling viii

10 List of Figures 1-1 Comparison between positive and negative CTE materials Example of different types of composites Structure of ZrW 2 O Diagram showing thermal contraction in oxides Chart showing the effect of ZrW 2 O 8 on a polymer matrix Figure displaying the opposing forces in a ZrW 2 O 8 composite Chart comparing different mathematical models for predicting the CTE of composites Starting COMSOL window Model builder tree in COMSOL An example of a two-dimensional mesh in COMSOL A screen-shot of the MBT showing changes made to the default solver Two-dimensional models with seven percent filler volume Two-dimensional models with nineteen percent filler volume Two-dimensional models with thirty-seven percent filler volume Three-dimensional model with seven percent filler volume made of one cylinder Three-dimensional model with seven percent filler volume made of nine cylinders ix

11 3-10 Three-dimensional model with seven percent filler volume made of twentyfive cylinders Three-dimensional model with seven percent filler volume made of ten spheres Three-dimensional model with seven percent filler volume made of fifty spheres Three-dimensional model with a 10 by 10 by 50 element mapped mesh COMSOL results settings for determining the change in volume of the models Total displacement plot for a two-dimensional, pure polymer model Total displacement plots for two-dimensional models with seven percent filler volume Total displacement plots for two-dimensional models with nineteen percent filler volume Total displacement plots for two-dimensional models with thirty-seven percent filler volume Bar graph comparing all of the two-dimensional models studied A graph showing volume fraction versus relative expansion from the twodimensional data Total displacement plot for 3-D model with one cylindrical filler particle Total displacement plot for 3-D model with nine cylindrical filler particles Total displacement plot for 3-D model with twenty-five cylindrical filler particles Total displacement plot for 3-D model with ten spherical filler particles Total displacement plot for 3-D model with fifty spherical filler particles Total displacement plot for 3-D model made with a mapped mesh of cubeshaped particles x

12 4-13 Bar graph comparing the three-dimensional models studied Bar graph comparing the FEA models to experimental data Graph showing the upper and lower bounds of expansion created by FEA models compared to Tani s data Graph showing the upper and lower bounds of expansion created by FEA models compared to Sharma s data Graph showing the upper and lower bounds of expansion created by FEA models compared to Tani s data Graph showing the upper and lower bounds of expansion created by FEA models compared to Sharma s data Geometry showing a possible way to model the interface between the matrix and filler xi

13 List of Abbreviations ASME American Society of Mechanical Engineers CTE Coefficient of Thermal Expansion (α) E Modulus of Elasticity FEA FEM Finite Element Analysis Finite Element Modeling MBT Model Builder Tree NTE Negative Thermal Expansion PDE Partial Differential Equation PI Polyimide PR Phenolic Resin RAM Random-access Memory ROM Rule of Mixtures ZrW 2 O Zirconium Tungstate xii

14 List of Symbols A c total area of composite A f area of filler material V c total volume of composite V f volume of filler material α coefficient of thermal expansion ν Poisson s ratio ρ density E Young s modulus K bulk modulus G shear modulus v f volume fraction of filler material T temperature T change in temperature K Kelvin, unit of temperature xiii

15 Chapter 1 Introduction 1.1 General Overview Thermal expansion is the tendency of a material to change in volume when it undergoes a change in temperature. It is referred to as thermal expansion because most types of matter increase in volume when subjected to a positive change in temperature. This property can often create problems for engineers who must account for this change in volume in their designs. While a majority of solids do not undergo significant changes for their given operating temperatures, certain cases exist where a large increase in volume is observed (e.g. [1 4]). This often leads to less than ideal situations where problems with fit and contact need to be addressed. One way to combat the high thermal expansion in polymers is through the use of composite materials. As stated earlier, most materials undergo thermal expansion and expand given a temperature increase. Certain materials undergo thermal contraction, or, negative thermal expansion, and shrink when subjected to an increase in temperature (Fig. 1-1). While none of these materials contain the properties to completely replace polymers, they can be used to form a composite material that will exhibit lower thermal expansion when compared to the original material. 1

16 (a) Positive CTE (b) Negative CTE Figure 1-1: Comparison between the effects of a positive and a negative CTE when subjected to an increase in temperature. Extensive experimental research on these composites has been conducted and results have shown that it is an effective way to reduce thermal expansion (e.g. [5 8]). This research focuses on the finite element modeling of them in the hopes of being able to correctly model the properties of these composites as well as gain a better understanding of how the matrix and filler interact. With this understanding, new experimental procedures could be developed to obtain better results and further decrease the amount of thermal expansion in these composites. 1.2 Problem Statement The objective of this work was to determine an accurate way to model composite materials containing ZrW 2 O 8 as a filler material. These types of composites pose problems in finite element modeling (FEM) because they contain a filler material that exhibits negative thermal expansion while the matrix material exhibits positive thermal expansion. In physical experimentation, this creates opposing forces within the composite leading to imperfect bonds and separation between the two materials. It is also likely that voids are created between the two materials during the production 2

17 of these composites. In a typical finite element model, the bond between the two materials is modeled as a perfect connection with no separation and the voids are not taken into consideration. This leads to inaccurate results that over-predict the effects of the shrinking filler material. To conduct this study, the FEM software package COMSOL Multiphysics was used. Different approaches to modeling the composite s internal geometry were looked at and a better understanding of the interface region between the two materials was gained. Comparisons were made to published experimental results that looked at different volume fractions subjected to a large temperature change. The objective was not only to develop a way to closely match the experimental results, but also to do so using the simplest model. This was to ensure that the results of this study were not only beneficial to those running computers with multiple processors and large quantities of RAM. 1.3 Overview of Principles This thesis looks at the finite element analysis (FEA) of thermal expansion in composite materials containing zirconium tungstate as the filler material. In order for the reader to better understand this work, a basic overview of the principals used to conduct this research will be given. Topics covered will include composite materials, thermal expansion, and FEA Composite Materials A composite is a material which consists of two or more separate materials that are combined. Their use has become more prevalent over the past few decades due to the ability to obtain desirable properties that would not be achievable through the use of a single material. Composite s constituents are categorized into two components; 3

18 (a) Sandwich Composite (b) Particle Composite (c) Fiber Composite Figure 1-2: Three cross-sections showing different ways filler material can be included in a composite. Each has its own advantages and disadvantages. matrix and filler. The matrix is the material that holds the composite together. It can be made up of a polymer, metal, or ceramic material and is usually chosen for its ductility, toughness, or electrical insulation property (e.g. [9 11]). The filler is the material that is mixed into the matrix to form the composite. It is held in place by the matrix and used to improve certain properties of the matrix material in order to meet design criteria. One important characteristic of a composite is its volume fraction (v f ). A composite s volume fraction is the ratio of one constituent s volume divided by the entire composite s volume. For a two-part composite, the volume of the filler material is usually referenced. For instance, a composite consisting of two materials and a volume fraction of 25% would contain one-quarter filler material and three-quarters matrix material. Filler material in composite materials can be included in various forms. (Fig. 1-2) These can range from tiny, nanoscopic particles to long, fibrous strands. In many cases, the way the filler is included can affect the final properties of the composite. When fibrous filler is used and the fibers are specifically oriented, the composite will exhibit anisotropic behavior. This means the composite will behave differently de- 4

19 pending on the direction of the load relative to the orientation of the fibers. However, those same fibers could be randomly oriented within the matrix and the composite would exhibit isotropic behavior. This means the composite would exhibit the same properties regardless of the loading direction. This paper looks at randomly oriented, particulate filler in a polymer matrix, displaying isotropic behavior Thermal Expansion and Zirconium Tungstate Thermal expansion is the tendency for a material s volume to change in response to a change in temperature [12]. Most materials undergo an increase in volume when subjected to a positive change in temperature, hence the name thermal expansion. However, some materials will exhibit thermal contraction, or negative thermal expansion (NTE), and decrease in volume when subjected to a positive temperature change [6]. The materials that exhibit this behavior typically only do so over a small temperature range, rendering them difficult to use in real-world situations. However, zirconium tungstate is a material that is being heavily researched due to its negative thermal expansion over a large temperature range (e.g. [5, 7, 8, 13 15]). Table 1.1: A list of materials and their coefficient of thermal expansion, α, at 20 C. Material α 10 6 (K 1 ) Ref. Aluminum 23.9 [16] Stainless Steel 17.3 [16] Copper 17.6 [17] Concrete 9.9 [17] Polymide 64.2 [7] Phenolic Resin 45.6 [5] ZrW 2 O [6] Zirconium tungstate is a metal oxide that has a coefficient of thermal expansion (CTE), α, in its cubic phase of K 1 [6]. Table lists various materials 5

20 Figure 1-3: Structure of ZrW 2 O 8 [6]. along with their CTE for comparison. The NTE behavior of ZrW 2 O 8 is seen from 2 to 350K and can be attributed to its internal structure [6]. When ZrW 2 O 8 is subjected to a positive temperature change, its rigid internal particles rotate, bringing them closer together, decreasing the overall volume. Figure 1-3 shows the internal structure of ZrW 2 O 8 that allows this. The NTE behavior can also be seen in other materials with similar particle structure, however none have been found to occur over as wide of a temperature range (e.g. [6, 18 20]) Finite Element Analysis Finite element analysis (FEA) is a numerical method to solving partial differential equations (PDE). In engineering, it is often necessary to develop a mathematical model to describe the behavior of a system. More often than not, this model will contain differential equations that can be difficult to solve, making it challenging to find an exact solution to the system. In FEA, this system is broken down into small subdomains, or elements, that are connected to each other by nodes. An algebraic expression can be developed at each of these nodes, allowing for an approximate 6

21 solution to the system to be obtained. Today, FEA is a very common technique used among engineers. It allows for computational models to be developed to real world engineering problems. The solutions of these models are used to verify experimental results, which leads to fewer experiments needing to be run, which leads to saving time and money. In this research, COMSOL Multiphysics v4.0, a commercial FEA package, was used to numerically solve thermal expansion problems. The results from the FEA were then compared to published experimental data. 1.4 Organization of Thesis The remaining chapters of this thesis will provide a detailed overview of the simulation approach taken to model ZrW 2 O 8 composites as well as how the simulations compare to published experimental data. Chapter 2 consists of a literature review of previous research in this area. It covers experimental research that has been done on NTE composites, specifically those that deal with ZrW 2 O 8 /polymer composites. It also reviews previous work in the mathematical and finite element modeling of composite materials. Chapter 3 covers the approach taken to model ZrW 2 O 8 composites in this research. A detailed explanation of the 2-D and 3-D models are covered as well as the different ways the filler material s geometry was looked at. Details of the set-up and input parameters used in COMSOL Multiphysics are also described. Chapter 4 contains the numerical results to the models as well as discussion about the results. In this chapter, results from the FEA are compared to similar experimental results as well as analytical equations. Finally, Chapter 5 includes the conclusions drawn from this research. It summarizes the results from the modeling and published experimental data as well as 7

22 provides reasoning to the results. It also outlines possible future work, providing goals and theoretical background to move forward with the research. The appendices provided at the end of the document contain the computer codes used to develop some of the geometries in FEA. 8

23 Chapter 2 Literature Review Research in the areas of negative thermal expansion (NTE) and composite materials has been on the rise over the last two decades. Researchers are interested in creating stronger, lighter, thermally stable materials to be used in a wide range of applications [5, 7, 8, 21, 22]. This chapter will review previous research involving NTE materials, specifically zirconium tungstate, as well as the mathematical and finite element modeling of composite materials. 2.1 Negative Thermal Expansion Materials As stated earlier, thermal expansion is the tendency for a material s volume to change in response to a change in temperature. It is a material property quantified by a coefficient of thermal expansion (CTE), α, where α V = V 1 ( V/ T ) (2.1) for volumetric calculations. In Equation 2.1, V represents volume, T represents temperature, and it is assumed that pressure is held constant. For most materials, the value of α is positive, representing expansion when temperature is increased. How- 9

24 Figure 2-1: Diagram showing thermal contraction in oxides. Circles represent oxygen and squares represent cations. As temperature is increased, the movement of the oxygen atoms causes the cations to become closer together. Therefore, the overall volume decreases [24]. ever, in some materials, the value of the CTE is negative in at least one direction and in a few of those materials, the CTE is isotropically negative [6]. The study of NTE materials has been ongoing for roughly twenty years. Early researchers found that there were several oxide systems that exhibited the behavior [23 30]. In this research, however, it was found that the materials only exhibited NTE behavior over a small, high temperature range. In addition, the contraction of the material was usually small and anisotropic. From there, researchers sought to determine the source of the NTE behavior in these materials. It was found that the excited oxygen atoms linked to other elements were the cause [24]. The linkages start linear, but then thermally bend when temperature is increased (Fig. 2-1). This thermal bending causes a decrease in length between atoms which leads to a decrease in volume in at least one direction. One particular material that gained the attention of researchers of negative thermal expansion was zirconium tungstate (ZrW 2 O 8 ). This material is a ternary oxide first discovered by Graham et al. in 1959 [31]. In 1967, the phase relations of the oxide were reported by Chang et al. [32]. This aided in the discovery of its NTE behavior 10

25 in 1996 by Sleight et al. [14]. This research shows that the NTE behavior exists from 0.3 K to its decomposition temperature of 1050 K. It also states that, because of its cubic symmetry, the NTE properties are isotropic over its entire stability range. Similar NTE behavior was also found by Sleight et al. for HfW 2 O 8, but the cost to produce it is much higher [21]. These findings have led to ZrW 2 O 8 become a highly researched material for its potential uses in composite materials. Production of ZrW 2 O 8 is somewhat challenging and has been carried out in different ways [8, 15, 33 36]. For use in composites, small particles of ZrW 2 O 8 of uniform size are desirable. This keeps particles from settling during the formation of the composite, leading to a more homogeneous mixture [13]. Nanoparticles of ZrW 2 O 8 have also been created to further improve homogeneity as well as mechanical strength [7]. 2.2 Experimental Research of ZrW 2 O 8 Composites The knowledge of zirconium tungstate s NTE behavior led to its use as filler in composite materials. The use of polymers as the matrix material allows for a wide range of applications with low cost and ease of production [37]. The addition of filler materials within the polymer allows for fine tuning of the material properties by combining the favorable properties of the polymer with favorable properties from a filler material [38 42]. Incorporating ZrW 2 O 8 into matrix materials has allowed researchers to reduce the overall CTE of the composite [5,8,13,21,22]. This is desirable due to polymers usually high CTE, reducing their usefulness in certain applications [43 45]. Figure 2-2 shows an example of this research conducted by Tani et al. where ZrW 2 O 8 was added to a phenolic resin matrix. As the volume fraction of ZrW 2 O 8 is increased, the overall expansion of the composite is reduced. Researchers are now beginning to look into the interface between the filler and the matrix [7, 46 49]. This is especially important in composites with ZrW 2 O 8 as a 11

26 Figure 2-2: Chart showing the effect of ZrW 2 O 8 on a phenolic resin matrix during a temperature increase. As the volume fraction of ZrW 2 O 8 increases, the overall expansion of the composite decreases [5]. filler because of the opposing forces created during a positive temperature change. Because of this, the interface between the two materials is subjected to high stress. When the stress becomes too high and the two materials separate, the effect of the filler material is reduced. Researchers are currently trying different surface treatments to the ZrW 2 O 8 to improve the bond between the filler and matrix [7]. Improvement to this interface region allows the ZrW 2 O 8 to further reduce the composite s CTE. 2.3 Mathematical Models for Predicting the CTE of Composites Several mathematical models have been developed to predict the overall CTE in composite materials [50 53]. These range from simple, linear relationships to 12

27 Figure 2-3: Figure displaying the opposing forces in a ZrW 2 O 8 composite. The gray area represents the ZrW 2 O 8 filler and the white area represents the matrix material. The dotted line is the interface between the two materials where high stresses would be seen when subjected to a temperature increase. complex equations that account for material stiffness and matrix/filler interaction. This interaction can greatly affect the resulting CTE of the composite which allows these models to give more accurate results [5]. A chart comparing results from each model can be seen in Figure 2-4. The simplest of the mathematical models is called the Rule of Mixtures (ROM) [53]. The ROM is a linear average of the matrix s and filler s CTE based on volume fraction. The first order equation is expressed as α c = v f α f + (1 v f )α m (2.2) where α c, α f, and α m are the coefficients of thermal expansion of the composite, filler, and matrix, respectfully and v f is the volume fraction of the filler. The ROM model does not account for any relationship between the filler and matrix and also does not account for the stiffness of either material. It assumes a uniform stress distribution throughout the composite [7]. Another mathematical model for predicting the CTE of a composite is called the 13

28 Turner model [50]. This model builds upon the ROM and incorporates each materials bulk modulus, which accounts for stiffness. The model is expressed as α c = v fk f α f + (1 v f )K m α m v f K f + (1 v f )K m (2.3) where K f and K m represent the filler and matrix bulk moduli, respectively. α c, α f, α m, and v f are the same as from the ROM. One will notice that if K f is equal to K m, then the equation simplifies to the ROM. The last model to be covered is called the Schapery model [51]. This model consists of two equations that develop an upper and lower bound for the effective CTE of composites. Schapery used energy principles to develop this model that works for isotropic and anisotropic composites and also is independent of filler geometry. The Schapery model also uses a model for elastic modulus developed by Hashin-Shtrikman (H-S) [54] within the CTE equations. The upper bound is expressed as and the lower bound is expressed as αc u = α m + K f (K m Kc)(α l f α m ) (2.4) Kc l K m K f α l c = α m + K f K u c (K m K u c )(α f α m ) K m K f (2.5) where the superscript u and l refer to the upper and lower bounds and K u c andk l c are calculated from the H-S model using the equations below. Kc u 1 v f = K f + 1 K m K f + 3v (2.6) f 3K f +4G f K l c = K m + v f (2.7) 1 K f K m + 3(1 v f ) 3K m+4g m In the Schapery and H-S equations, all variables are the same as in the previous 14

29 Figure 2-4: Chart comparing different mathematical models for predicting the CTE of composites. All calculations were done for a composite with a polymer matrix and ZrW 2 O 8 filler. two models and G m and G f represent the shear moduli of the matrix and filler, respectively. A chart comparing the results of the three models can be seen in Figure Finite Element Modeling of Particulate Composites Finite element analysis (FEA) is a numerical procedure to solve differential equations that are difficult or impossible to solve analytically [55]. The finite element 15

30 method takes a given geometry and breaks it down into smaller parts (elements) and uses a system of algebraic equations to describe the relationship at the points (nodes) connecting the elements [6]. This allows the original differential equations to be simplified into a system of algebraic equations that can be quickly solved using a computer to come up with an approximate solution to the problem. FEA allows researchers to test ideas and theories without having to purchase materials and produce samples. It can also be used to verify experimental results. In the study of thermal expansion in composites, FEA is a tool that can give an idea of the effect certain filler materials will have on the overall composite and what volume fractions researchers should test experimentally. Research in the area of FEA of particulate composites has been mostly limited to composites with positive CTEs for both the filler and matrix [56 61]. In these studies, the effect of particle size, shape, and distribution within the FEA model have been looked at and compared to mathematical models and experimental results. To date, little research on the FEA of composites with negative CTE fillers has been completed. One study in the area looked at the FEA of closely packed tetrahedra ZrW 2 O 8 at 60% volume loading in a copper matrix [62]. This study used FEA to calculate the thermal mismatch stresses within the composite and found that they were high enough to trigger a pressure-induced phase transformation. Another publication looked at how thermal and mechanical properties, rates of cooling/heating, geometry, and packing fraction influenced the overall expansion and thermal stress in composites with ZrW 2 O 8 filler and either copper or ziconium oxide matrix [6]. Here researchers concluded that the thermal stresses are larger when the amount of filler is larger and that it is advantageous to use matrix and filler materials with similar magnitudes of CTE. The research of this thesis built upon these works and studied the FEA of polymer/zrw 2 O 8 composites. 16

31 Chapter 3 Modeling Approach This research focused on the finite element modeling of thermal expansion in polymer based composites with zirconium tungstate as a filler material. The FEA was completed with the software package COMSOL Multiphysics version 4.0 using the structural mechanics module. This software allowed for two and three-dimensional models to be created with different numbers of filler particles and different filler geometries. This chapter will provide a detailed explanation of the approach to modeling including the materials used and their properties, the various geometries of the 2-D and 3-D models, the setup for COMSOL, and how the solutions to the simulations were analyzed. 3.1 Materials Used and their Properties The materials studied in this research were polymer based composites with the negative thermal expansion material ZrW 2 O 8 as a filler. Two different polymer matrix materials were looked at in order to compare the results from the FEA to experimental results published by Tani et al. [5] and Sharma et al. [7] The experiments conducted by Tani and Sharma were on ZrW 2 O 8 composites using phenolic resin (PR) and polyimide (PI) as the matrix materials, respectively. To run FEA of thermal expansion 17

32 on these composites, four material properties are required for each material. These are Young s modulus (E), Poisson s Ratio (ν), density (ρ), and coefficient of thermal expansion (α). Then, in order to calculate the CTE using the mathematical models discussed in section 2.3, the materials bulk moduli (K) and shear moduli (G) must also be known. All of the materials used are listed in table 3.1 with each of their required properties. Table 3.1: Materials used in FEA models and their required properties Material E(GP a) ν ρ( kg ) α( 10 6 ) K(GP a) G(GP a) m 3 K ZrW 2 O PI PR Setup for COMSOL Multiphysics This research was conducted using the finite element software COMSOL Multiphysics version 4.0. This software was used to create two and three-dimensional models, simulate a change in temperature applied to the models, and analyze the effects of the temperature change on the models dimensions. This section will overview the settings and initial setup of COMSOL for this research. When first opening COMSOL multiphysics, a window similar to the one in figure 3-1 is seen. When starting a new model, this is where options such as space dimension, physics, and type of study are selected. For this research, both 2-D and 3-D models were used, so one of those two options was chosen first. Next, COMSOL prompts users to choose which physics modules they would like to use. COMSOL is capable of running multiple physics modules on one model, however, for this research, only the solid mechanics module under the structural mechanics section was needed. Lastly, 18

33 Figure 3-1: Opening window seen when COMSOL is first launched. This research used both 2-D and 3-D modeling, the solid mechanics module under structural mechanics physics, and a stationary study. COMSOL needs to know what type of study will be run, which in this case was stationary. After inputing the initial settings, the model is ready to be created. At this point, the geometries of the matrix and filler are drawn. Details about the different geometries studied and how they were created can be seen in sections 3.3 and 3.4. Once the geometry of the model is made, the matrix and filler material properties need to be assigned. To do this, two materials were created for each model under the material branch of the model builder tree (MBT). (Fig. 3-2) Then, the properties listed in Table 3.1 were input to the appropriate matrix or filler material. The next step was to apply boundary conditions to the model. This is done under the solid mechanics branch of the MBT. For all models, roller supports were used as the only boundary conditions. The rollers were added to the x = 0, y = 0, and z = 0 19

34 Figure 3-2: An example of what the model builder tree in COMSOL looked like for the finite element models created for this research. surfaces (or just the x and y if the model was 2-D) to allow for thermal expansion in the positive directions similar to what is seen in Figure 1-1(a). After applying the boundary conditions, the loads were applied to the models. While there were no external forces applied in any of the simulations, the loading was applied through a change in temperature. To do this, the thermal expansion option was added to the model under the linear elastic material model branch of the MBT. Once added, this option allowed for two temperatures to be defined. The first was the strain reference temperature, T ref, which was the initial temperature of the model. Then, the user defined temperature, T, was the final temperature. Subtracting T ref from T allowed for the change in temperature, T, to be calculated. For all models, T was positive and T ref was kept at K. After the boundary conditions and loads were applied, the model was ready to be meshed. Meshing in FEA is where the geometry of the model is broken down into 20

35 Figure 3-3: An example of a two-dimensional, free triangular mesh in COM- SOL set at the finer size setting. smaller pieces, called elements, which are connected by nodes. Each element is then solved individually creating a simpler way to obtain an approximate solution. When meshing models, it is generally understood that a finer mesh (more elements) will give a more accurate solution. However, a finer mesh will also require more time for the computer to solve the study. Therefore, it is necessary to determine the appropriate balance between computation time and an acceptable level of accuracy. A mesh was created in COMSOL by right-clicking on the mesh branch of the MBT and selecting free triangular for 2-D models and free tetrahedral for 3-D models. Then, the size of the mesh was set to the finer setting. This gave an initial starting point for the mesh which could then be adjusted either finer or courser based on the time it took to solve the problem. An example of a two-dimensional mesh set to the finer setting can be seen in Figure 3-3. The last step in setting up COMSOL for this research was to adjust the default 21

36 Figure 3-4: A screen-shot of the MBT showing the changes made to the default solver. Changes were made under the Advanced, Iterative 1, and Multigrid 1 branches to achieve faster solution times with no effect to the accuracy of the results. solver settings. This allowed for the solutions to be found in a slightly faster time while having no negative effects on the accuracy of the solution. To do this, the study branch of the MBT was right-clicked on and show default solver was selected. Once all the drop-down menus are expanded, various options are available to change on the solver. For the models run in this research, the first change was made under the Advanced branch which was found under Stationary Solver 1. Here the matrix symmetry was changed from automatic to symmetric. The next change was made on the Iterative 1 branch. Here the solver was changed from GMRES to conjugate gradients. The last change was made under the Multigrid 1 branch where the solver setting was changed from a geometric to an algebraic multigrid. The branches these changes were made under can be seen in Figure

37 3.3 Modeling to Compare Geometry The first portion of this research looked to compare the FEA results of polymer/zrw 2 O 8 composites with the same volume fraction but different geometries. The goal was to determine if varying the geometry of the model had a significant effect on the results. Two-dimensional and three-dimensional models were compared as well as different filler geometries and numbers. Details of the geometries modeled and compared can be seen in the following two sections Two-Dimensional Models The initial modeling completed for this research was done using two-dimensional models. These models were created using the same materials and volume fractions and then subjected to the same temperature change. The goal was to compare these models to see the effect different filler geometries and number of inclusions has on the relative expansion of the composite after being subjected to a positive temperature change. The first model created was a completely isotropic polymer to use as a control. The geometry was a one-meter by one-meter square created by right-clicking on the Geometry branch of the MBT and selecting the Square option. The material properties of phenolic resin were applied to the square geometry and the model was subjected to a 70K temperature increase. The geometry was meshed and the study was run using the settings described in Section 3.2. After running the control model, filler material was added to the geometry. To model the composites in two dimensions, smaller geometries, or inclusions, were added to the polymer square. This portion of the research looked to compare the effects of changing the shape and number of these inclusions. To do this, three sets of models differing only by volume fraction were created. Each set consisted of eight models with different filler geometry that equated to the same amount of matrix 23

38 and filler materials. The three volume fractions modeled were 7%, 19%, and 37%. For each volume fraction, eight different models were simulated. These consisted of four different filler arrangements with two different filler geometries. The two filler geometries used were square and circular and the four different filler arrangements consisted of models with one-fourth of an inclusion (using symmetry), one inclusion, nine inclusions, and twenty-five inclusions. All the variations of the filler created a matrix of twenty-four models to be simulated for this section. Figures 3-5, 3-6, and 3-7 show all of the models simulated for the two-dimensional portion of the study. To create the models in COMSOL, the geometry of the filler was added to the control model of pure polymer. This was done through the use of the Square or Circle option under the Geometry branch of the MBT. The size of the square or circular inclusions depended on the volume fraction and number of filler particles. To quickly determine these sizes, equations 3.1 and 3.2 were used for square and circular inclusions, respectively. v f = n(a f) = n[(l f)(w f )] A c (l c )(w c ) = n[(x sq)(x sq )] (1)(1) = nx2 sq 1 x sq = vf n (3.1) v f = n(a f) A c = πd2 cir n( ) 4 (l c )(w c ) = n(πr2 cir) (1)(1) = nπr2 cir 1 r cir = vf nπ (3.2) In Equation 3.1, v f is the volume fraction and n is the number of square inclusions within the composite. A f is the area of one inclusion, which is equal to the length times the width of the square (l c and w c ) which are both equal and called x sq. A c is the area of the entire composite, which is equal to the length times the width of the model (l c and w c ). Both of these values are equal to one meter making the area one square meter. Equation 3.2 is for circular inclusions and all the variables are the same as in equation 3.1 except for r cir which is the radius of one inclusion. Once the size of the filler particles was determined, the Array tool under Trans- 24

39 Figure 3-5: Geometry of two-dimensional models with 7% percent filler volume. Each grid division represents 0.5 m and all models have an initial total volume of 1 m 2. Figure 3-6: Geometry of two-dimensional models with 19% percent filler volume. Each grid division represents 0.5 m and all models have an initial total volume of 1 m 2. 25

40 Figure 3-7: Geometry of two-dimensional models with 37% percent filler volume. Each grid division represents 0.5 m and all models have an initial total volume of 1 m 2. forms was used on models with more than one inclusion to create a symmetric pattern centered within the polymer square. After the full geometry of the filler was created, the Difference tool under Boolean Operations was used to subtract the filler geometry from the matrix. When doing this, the Keep input objects box was checked so that the filler s geometry was not deleted after the operation. However, because that option was selected, the original matrix square needed to be removed using the Delete Entities tool. This left two domains within the model. One being the matrix with voids, and the other being the filler inclusions. Material properties were then added to both domains with the matrix and filler consisting of phenolic resin and ZrW 2 O 8, respectively. All models were subjected to a 70 K temperature increase with boundary conditions, mesh settings, and solver settings as described in section 3.2. Results from these models can be seen in Chapter 4. 26

41 3.3.2 Three-Dimensional Models Three-dimensional models were also tested and compared to the results of the two-dimensional models. A similar matrix of tests was created that varied the shape of the filler as well as the number of inclusions. The only volume fraction tested was 7% due to the computation time required to run a simulation. While only one volume fraction was tested, results were still conclusive and able to be compared to the two-dimensional models. The exterior dimensions of the three-dimensional models were based off of the samples tested by Tani et al. [5] with a length of 20 mm, a width of 4 mm, and a height of 4 mm. Filler was added to the block in the shape of either cylinders, spheres, or cubes. Cylindrical filler models consisted of either one, nine, or twenty-five, 20 mm tall cylinders with radii depending on the number of inclusions. Equation 3.3 was used to determine these radii and is similar to the equations used in two-dimensional modeling. The geometry of these models can be seen in Figures 3-8, 3-9, and v f = n(v f) = n( πd 2 cyl h cyl ) 4 V c (l c )(w c )(h c ) = n(πr2 cyl (20)) (4)(4)(20) = nπr2 cyl 16 r cyl = 16vf nπ (3.3) For three-dimensional models containing spherical filler particles, Equation 3.4 was used to determine the radius of the spheres based on the volume fraction and number of particles. Models were made using ten and fifty spherical inclusions located randomly within the matrix. To generate the random coordinates, a MATLAB code was used. This code ensured that the correct number of inclusions were used and that the spheres did not overlap or break the surface of the matrix. This code can be seen in Appendix A. Because the location of the spherical inclusions was random within the matrix, the results of the simulation differed slightly with each run. To account for this, five different simulations were run and the results were averaged to 27

42 Figure 3-8: Three-dimensional model with seven percent filler volume made of one cylinder. The initial volume of the model is 320 mm 2. Figure 3-9: Three-dimensional model with seven percent filler volume made of nine cylinders. The initial volume of the model is 320 mm 2. 28

43 Figure 3-10: Three-dimensional model with seven percent filler volume made of twenty-five cylinders. The initial volume of the model is 320 mm 2. obtain the data point for each specific case. The geometry of these models can be seen in Figures 3-11 and v f = n(v f) = n( 4 3 πr3 sph ) V c (l c )(w c )(h c ) = n( 4 3 πr3 sph ) (4)(4)(20) = nπr3 sph 240 r sph = 3 240vf nπ (3.4) The last style of three-dimensional model studied was quite different than the previous two. This model used MATLAB and COMSOL together to create a model with a specific mesh and then randomly assigned either matrix or filler material properties to individual elements until the desired volume fraction was achieved. To do this, five MATLAB files were created, all of which can be seen in Appendix A. The first of these codes defined the size of the mapped mesh (a grid pattern mesh consisting of a certain number of blocks) as well as the volume fraction and then randomly assigned each block as either matrix or filler material. This code output 29

44 Figure 3-11: Three-dimensional model with seven percent filler volume made of ten spheres. The initial volume of the model is 320 mm 2. Figure 3-12: Three-dimensional model with seven percent filler volume made of fifty spheres. The initial volume of the model is 320 mm 2. 30

45 the coordinates of the elements to become filler material which was then used by the other four codes to assign material properties. Each required material property (E, nu, rho, and alpha) has a MATLAB code that looked at the coordinates generated by the first code and then assigned the correct values for that specific material property to each particular element. Once the codes for each material property were written, the m-files needed to be placed in the COMSOL root folder so they could be found and used when the models were run. To use these MATLAB files within COMSOL, a few settings needed to be changed from previous models. Firstly, the codes needed to be defined within COMSOL so that they could be found when needed. To do this, a MATLAB function was added under the Global Definitions branch of the MBT. Here, each MATLAB file that defined a material property was added to the list and the variables it used were defined. Next, the geometry of the models was created. This was, simply, a solid, 20 mm by 4 mm by 4 mm bar with roller supports on three sides. The only load applied to this bar was, again, a temperature increase of 90 K. To define the material properties of this bar, no values were input under the Material branch of the MBT. Instead, one material was added and the names of MATLAB files defined above were input into the value fields, telling COMSOL to look to the results of the MATLAB code to find the material property. The last step to create these models was to create the mesh. For these tests, a mapped, grid-pattern, mesh was used. It was important that the mesh defined within COMSOL matched the grid that was input in the MATLAB codes. For this research, a mapped mesh of ten blocks, by ten blocks, by fifty blocks was used. To create this within COMSOL, the first step was to add a mapped mesh under the Mesh 1 branch of the MBT. Then, the x = 0, y = 0, and z = 0 faces were added to the selection box. The next step was to define the size of the mesh by adding the Size option under the Mapped 1 branch of the MBT. Here, the custom option was selected and the maximum element size was set to m to 31

46 produce the desired 10 by 10 by 50 element mesh. The last step in the creation of the mesh was to add the Swept option under Mesh 1 to convert the face mesh to a three-dimensional mesh of the whole block. Once all the proper options were applied to the mesh, the build button was pressed and COMSOL output that 5, 000 elements were created. Figure 3-13 shows what this mesh looked like within COMSOL. Figure 3-13: Three-dimensional model with a 10 by 10 by 50 element mapped mesh. The initial volume of the model is 320 mm 2. After creating the mesh the models were ready to be run. The solver was setup the same way as all previous models which is shown in Figure 3-4. As with the spherical three-dimensional models, five separate models were run and the results were averaged. These results can be seen in Chapter Modeling to Examine Matrix/Filler Bond The models created in the previous section were based off the experiments of Tani et at. [5] where expansion data was collected for phenolic resin/zrw 2 O 8 composites. After running the simulations it was found that the FEA models tended to 32

47 over-predict the negative CTE effect of the ZrW 2 O 8 filler material. More detailed results can be seen in the next chapter, but this section will describe the modeling techniques that were used to gain a better understanding of the interface between the two materials. It was assumed that the difference between the experimental data and the FEA models could be attributed to the bond between the matrix and filler materials. In the FEA models, this was created as a perfect bond with no voids or separation. In experimental procedures, this perfect bond is nearly impossible to achieve. Therefore, the goal of this modeling was to establish the upper and lower bounds of the data by creating a model with a perfect bond between the filler and matrix as well as a model with no bond between the filler and matrix. The model used for the perfect bond case was already created for the previous section and the results could be carried over. It was decided to use the geometry of the model that showed the most expansion of all the two or three-dimensional models. This way, when the bond between the matrix and filler was removed, it would show the most expansion and better represent the worst-case scenario. However, choosing a different geometry would make little difference as all the models showed similar results. The next step was to create a model with no bond between the filler and matrix. To do this, the filler particles of the model were completely removed, leaving empty voids within the matrix. By doing this, the effect of the filler was removed from the model, which gave the same effect as if there was no bond between the two materials. Therefore, these models would show much more expansion than the models with a perfect bond and represent the worst-case scenario in experimentation. Using the modeling techniques just described, FEA was completed to compare to the experiments of Tani et al. and Sharma et al. To compare to the Tani experiments, models contained the properties of phenolic resin and zirconium tungstate at volume 33

48 fractions of 7%, 19%, and 37%. To compare to the Sharma experiments, models contained the properties of polyimide and zirconium tungstate at volume fractions of 5%, 10%, and 15%. The results of these models as well as how they compare to the published experimental results can be seen in Section Analysis of Models The final step to the modeling conducted in this research was to evaluate the models. This consisted of simply determining the final total volume after the 70 K temperature change. To do this, post-processing was conducted within COMSOL after the models had been run. After a model finished its study, COMSOL displays a surface plot of total displacement and puts the user in the Results section of the MBT. The first step was to adjust the plot to show the deformation of the model. To do this, under 2D Plot Group 1, Surface 1 is right-clicked on and Deformation is selected. The plot should now show a scaled, deformation plot that extents past the black lines of the original shape. The next step was to have COMSOL calculate the change in volume of the model. To do this, the Derived Values section was right-clicked on and either Surface Integration or Volume Integration was selected depending on if the model tested was two-dimensional or three-dimensional, respectively. In the settings of the integration, all domains were selected and the expression was set to solid.evol to integrate over the volumetric strain. With those settings chosen, the equals button was selected and COMSOL then output a value for volumetric strain in the results table. This value represented the change in volume of the model. Therefore, knowing the initial volume of the model (1 m 2 for 2-D, and 320 mm 2 for 3-D), the final volume of the model was calculated by adding the value of volumetric strain to the initial volume. From there, the relative expansion was calculated using Equation

49 to allow for direct comparison between the 2-D and 3-D models. Comparisons were also made between these results and the mathematical models discussed earlier. The results and discussions about them can be seen in Chapter 4. RelativeExpansion = V final V initial 1 (3.5) Figure 3-14: COMSOL results settings for determining the change in volume of the models. 35

50 Chapter 4 Results and Discussion The models described in Chapter 3 were run and evaluated as described in Section 3.5. The results of these models will be given and discussed in this chapter. First, the two-dimensional models will be looked at, followed by the three-dimensional ones. Then, the models will be compared to published experimental results and modifications will be made to create an upper and lower bound for expansion. Finally, the results will be compared to the mathematical models described in Section 2.3 to determine which model the data most closely followed and how the upper and lower bounds compare. 4.1 Two-Dimensional Models The two-dimensional models detailed in Section were run and evaluated as described in Section 3.5. The results of these tests will be discussed in this section. The first model run was a one-meter by one-meter square made up of only phenolic resin; the matrix material. This model was created to determine the baseline of expansion for the pure polymer and to better understand the effects of adding the filler material. The total displacement plot for this model can be seen in Figure 4-1 and the relative expansion can be seen in Table

51 Figure 4-1: Total displacement plot for a two-dimensional, pure polymer model. This model represents the baseline of all future models and has an even, linear growth in the positive x and y directions. It was subjected to a temperature change of 70 K and the scale factor of the displacement is 200. When looking at the displacement plot, take note of its linear shape and even displacement in both the positive x and y directions. It should be noted that the plot has a scale factor on the displacement of 200. This allows the changes in shape to be more easily seen and will be used on all 2-D plots. After using COMSOL to determine the change in area, Equation 3.5 was used to calculate the relative expansion using the initial and final areas in place of the volumes. This was found to be 0.292%. Next, the two-dimensional models containing 7% filler material were run. These models were all created in the same COMSOL file and run at the same time, but analysed individually. The total displacement plot for these models can be seen in Figure 4-2 and the expansion data can been seen graphically in Figure 4-5 and numerically in Table 4.1. When looking at the total displacement plot, the effects of the ZrW 2 O 8 can be seen, but only slightly, and the displacement of all eight models 37

52 appears to be similar. This was confirmed by calculating the relative expansion for each model and plotting the results. The graph in Figure 4-5 shows that there was little deviation in the expansion of the eight models suggesting that filler shape and arrangement have little effect on the overall expansion. The average value of relative expansion for the 7% models was 0.252% with a standard deviation of %. This equates to a 0.070% reduction in relative expansion when compared to the pure polymer model. Figure 4-2: Total displacement plots for two-dimensional models with seven percent filler volume. The average relative expansion of the models was 0.252% meaning an average reduction in expansion of 0.070% was seen when compared to the pure polymer model. Next, the models with 19% filler were run. Like the 7% models, these were all created in the same COMSOL file and run and the same time, but analysed separately. The total displacement plot for these models can be seen in Figure 4-3. In these plots, the filler material s effect is more noticeable, especially in the models with only one particle where the expansion is not as evenly distributed as the models with multiple inclusions. Despite these differences, the relative expansion of the models were all 38

53 similar with an average value of 0.187% and a standard deviation of %. This gives an average reduction of expansion of 0.135% when compared to the pure polymer model. This data can be seen graphically in figure 4-5 and numerically in Table 4.1. Figure 4-3: Total displacement plots for two-dimensional models with nineteen percent filler volume. The effects of the ZrW 2 O 8 filler can be seen in the shape of the displaced plots and an average reduction in expansion of 0.135% was calculated when compared to the pure polymer model. The last set of two-dimensional models run were those containing 37% filler. These models were run the same way as those with 7% and 19% and the total displacement plot can be seen in Figure 4-4. The negative effect of the filler material is very noticeable in these plots with the expanded shape appearing only slightly larger than the original geometry, even with the scale factor of 200. These models also gave the largest variation in relative expansion with an average value of 0.096% and a standard deviation of %. This can be seen graphically in Figure 4-5 and numerically in Table 4.1. The 37% filler reduced the expansion of the polymer by 0.226%. To compare the results of the 2-D models, the graph seen in Figure 4-5 was 39

54 Figure 4-4: Total displacement plots for 2-D models with 37% percent filler volume. These models had the largest variance in expansion with an average value of 0.096% and a standard deviation of %. The 37% filler reduced the expansion of the polymer by 0.226%. created. This graph shows that the addition of the ZrW 2 O 8 reduced the expansion of all models when compared to the pure polymer model, represented by the horizontal, orange line. It also compares the different arrangements of filler particles tested for each volume fraction. It was found that at each volume fraction, the arrangement or shape of the filler particles was not an important factor in the overall expansion. This can be seen by each volume fraction producing very similar expansion data for all models tested. Knowing this, the expansion data was averaged and plotted versus volume fraction, as shown in Figure 4-6. A linear best-fit line was added to these data points and fit very well with an R 2 value of This means that as more filler material is added to a composite, the reduction in expansion should be linear. If this prediction is projected past the data of the models, a composite exhibiting no expansion over the tested temperature range would require a volume fraction of approximately 52%. 40

55 Table 4.1: Numerical results from the two-dimensional modeling. This data shows that the filler geometry of 2-D models has little effect on the expansion results with low standard deviations calculated for each volume fraction. This is represented graphically in Figure 4-5. Volume Filler Num. of Final A Relative Avg Std Fraction Shape Inclusions (m 2 ) Exp. Exp. Dev. 0% NA % 0.322% NA 7% Sq % 7% Sq % 7% Sq % 7% Sq % 0.252% % 7% Cir % 7% Cir % 7% Cir % 7% Cir % 19% Sq % 19% Sq % 19% Sq % 19% Sq % 0.187% % 19% Cir % 19% Cir % 19% Cir % 19% Cir % 37% Sq % 37% Sq % 37% Sq % 37% Sq % 0.096% % 37% Cir % 37% Cir % 37% Cir % 37% Cir % 41

56 Figure 4-5: Bar graph comparing all of the two-dimensional models studied. The orange line at the top represents the expansion of the pure polymer model. It can be seen that all models reduced the expansion of the polymer and that each volume fraction gave similar results for all filler arrangements. Figure 4-6: This is a plot of volume fraction versus relative expansion created from the averaged data of the two-dimensional models. A best-fit line was added to show the linearity of the data and also projects that a volume fraction of approximately 52% will produce a composite with no expansion. 42

57 4.2 Three-Dimensional Models After running and evaluating the two-dimensional models, the three-dimensional models were looked at. The goal was to see if modeling in three-dimensions affected the outcome of the results in a significant manner. To do this, various models were created at a volume fraction of 7% and compared to each other, as well as the results from the two-dimensional testing. The models were based off the experiments of Tani et al. [5] where composite bars of 4 mm 4 mm 20 mm were subjected to a 70 K temperature increase. The first model run had a filler geometry of one cylinder running through the center of the bar. This model s total displacement plot can be seen in Figure 4-7. When looking at the plot, it is apparent that there was an even displacement throughout the bar, mostly in the y and z directions. The long, cylindrical filler prevented the bar from expanding in the x direction. The relative expansion of this model was calculated to be 0.255%. Figure 4-7: Total displacement plot for 3-D model with one cylindrical filler particle. Expansion is seen mainly in the y and z directions with little growth along the x axis. 43

58 For the next model, an array of nine cylinders was used for the filler geometry. Like the model created with one cylinder, this one also showed even growth in the y and z directions with the cylinders preventing much expansion in the x direction. This can be seen in Figure 4-8. The relative expansion was calculated to be 0.256%. The model shown in Figure 4-9 with twenty-five cylinders, again, showed similar behavior. Its relative expansion was found to be 0.262%. Figure 4-8: Total displacement plot for 3-D model with nine cylindrical filler particles. Expansion is similar to the model with one cylindrical particle. For the next 3-D models, the filler was made up of spherical particles. One model was made with ten inclusions and another with fifty. The total displacement plots of these models can be seen in Figures 4-10 and 4-11, respectively. When looking at these plots, the first thing noticed is that they are drastically different than the ones of the models with cylindrical filler geometry (the displacement scale is kept the same for all models). The plots for the spherical models show large displacement in the x direction while the plots for the cylindrical models showed very little change in that direction. This would lead one to believe that the expansion of the spherical 44

59 Figure 4-9: Total displacement plot for 3-D model with twenty-five cylindrical filler particles. Little difference was seen between the three models with cylindrical-shaped filler. models is significantly greater than the expansion of the cylindrical models, however, this is not the case. The relative expansion of the ten sphere model was found to be 0.265% and the fifty sphere model was 0.258% (both values obtained by averaging five models results) showing that they exhibited a similar change in volume as the cylindrical models. This meant that the spherical particles allowed the models to expand primarily in the x direction while the cylindrical models prevented this growth, which then was then made up for in the y and z directions. The last three-dimensional model created used a mapped mesh to create cubeshaped elements. Matlab was then used to randomly select enough elements to give the composite the desired 7% filler material and the rest were given matrix material properties. The total displacement plot of this model can be seen in Figure 4-12 where it can be seen that it expanded similarly to the spherical models with large displacement in the x direction. The relative expansion, found by averaging five runs, was found to be 0.262%. 45

60 Figure 4-10: Total displacement plot for 3-D model with ten spherical filler particles. The shape of the filler allows this model to expand primarily in the x direction, which is different from the models with cylindrical filler. However, the final volumes did not differ significantly. Figure 4-11: Total displacement plot for 3-D model with fifty spherical filler particles. This model behaved similarly to the model with ten spheres and produced a similar final volume. 46

61 Figure 4-12: Total displacement plot for 3-D model made with a mapped mesh of cube-shaped particles. This model used Matlab to randomly assign material properties to the mesh and contained the smallest particles tested. The results were most similar to the spherical models. In Table 4.2, all of the numerical data from the three-dimensional models can be seen. Looking at the relative expansion and standard deviation for all the models shows that little changes were seen when the filler geometry was changed, which agrees with the results from the two-dimensional modeling. This trend can be seen graphically in Figure 4-13 where the average two-dimensional, 7% volume fraction results were added as the last column. This shows that the three-dimensional models showed similar expansion to the two-dimensional ones. All of the three-dimensional models exhibited slightly more growth than the two-dimensional models, but not enough to be deemed significant. The standard deviation of the three-dimensional models was also higher than the two-dimensional ones, but no pattern to explain this was seen. 47

62 Table 4.2: Numerical results from the three-dimensional modeling. This data shows that the filler geometry of 3-D models has little effect on the overall expansion of the model. The standard deviation of the data is low and follows the trend seen in the 2-D modeling. This data is represented graphically in Figure Filler Initial Final Relative Standard Geometry V (mm 3 ) V (mm 3 ) Expansion Deviation 1 Cylinder % 9 Cylinders % 25 Cylinders % 10 Spheres % % 50 Spheres % Mapped Mesh % Figure 4-13: Bar graph comparing the three-dimensional models studied. The orange line at the top represents the expansion of the pure polymer model. It can be seen that all models reduced the expansion of the polymer and that each filler configuration showed similar expansion behavior. Results from the 2-D models are also included and shown to be similar. 48

63 4.3 Comparisons to Experimental Data and Development of Upper and Lower Bounds When looking at the results of the two and three-dimensional models evaluated for the previous sections, it can be seen that all models of the same volume fraction produced similar expansion results. While the consistency of the results is a plus, how the models compare to real-world experimental data is also important. To do this, the results of the experiments of Tani et al. [5] were looked at as the same materials were used. Figure 4-14 compares the results of the 7% models with the experimental data of Tani. The two models chosen for the graph were the 7% models that showed the highest and lowest expansion. The model that showed the least expansion was the two-dimensional model with nine square-shaped filler particles. The model that showed the most expansion was the three-dimensional model with twenty-five cylindrical filler particles. Looking at the plot shows that even the model with the most expansion is considerably less than the experimental data published by Tani. This means that all the FEA models over-predicted the effect of the ZrW 2 O 8 filler material and reduced the expansion more than what is seen in physical experiments. While exactly matching experimental results was not a goal of this research, the comparison in Figure 4-14 does give a better understanding of the FEA models. In the models, the filler and matrix materials are considered to be perfectly bonded together with no voids or separation. In real-world experiments, this is nearly impossible to achieve. When a composite material is made, small air bubbles can get trapped between the two materials which leads to imperfect bonds. In composites with ZrW 2 O 8 filler material, this has larger consequences as the opposing forces created during a temperature increase only accentuate these imperfections. This explains why the all the FEA models showed less expansion than the experimental data. The perfect bond between the filler and matrix gave the negative thermal expansion filler more effect, 49

64 Figure 4-14: Bar graph comparing the FEA models to experimental data. The two models chosen represent the highest and lowest expansion of all the 2-D and 3-D, 7% models. This shows that all modeling produced results that showed less expansion that the experiments of Tani et al. decreasing the final volume. Knowing that the bond between the filler and matrix is perfect in the FEA models, it is considered to be the lower bound of possible expansion for the composite. This means that in a physical experiment, a test specimen will not show less expansion than this lower bound provided they are tested at the same volume fraction. The obvious next step is to then determine the upper bound for the expansion. A specimen would undergo the most expansion if the ZrW 2 O 8 filler was not connected to the matrix material at all, or, in other words, the complete opposite of the previous FEA models. To do this in FEA, models were created similarly to the previous models, but the filler particles were deleted leaving voids within the matrix material. To reduce the amount of modeling needed, the results from the previous two sections were used to guide this section. Geometry for the models was chosen based on the previous 50

65 data where the model showing the most expansion was used. This ensured that the upper bound represented the worst case scenario and the lower bound maintained consistency. The model showing the most expansion was the three-dimensional model with twenty-five cylindrical filler particles and the expansion data from it was used for the lower bound. It was then modified to represent no bond between the matrix and filler by deleting the twenty-five cylindrical filler particles, leaving twenty-five cylindrical voids within the matrix. Volume fractions of 7%, 19%, and 37% were tested and the results can be seen in Figure Figure 4-15: Graph showing the upper and lower bounds of expansion created by FEA models compared to Tani s data. The data obtained by Tani et al. falls between the upper and lower bounds in blue and green, respectively, for all volume fractions. In Figure 4-15, it can be seen that the experimental data obtained by Tani et al. falls between the upper and lower bounds created by the finite element models for all volume fractions. The experimental data also tends to be closer to the upper bound, meaning that there was likely many voids and separation between the filler 51

66 and matrix within the specimens. To further validate the FEA upper and lower bounds, the experiments of Sharma et al. [7] were also compared to the models. In these experiments, ZrW 2 O 8 filler was added to a polyimide (PI) matrix material and the effects on the CTE were looked at. Surface treatments were also done to the ZrW 2 O 8 to improve the bond between the filler and matrix. The FEA models were adjusted to account for the different matrix material properties and different volume fractions of 5%, 10%, and 15%. The results can be seen in Figure Figure 4-16: Graph showing the upper and lower bounds of expansion created by FEA models compared to Sharma s data. Again, the data falls within the bounds and follows a similar trend as the Tani data. Looking at the graph in Figure 4-16, it can be seen that the upper and lower bounds created by the FEA models encased the results of Sharma s experiments. In these experiments, Sharma et al. tested samples of PI/ZrW 2 O 8 composites with and without surface treatments to the ZrW 2 O 8 in attempt to improve the bond between the two materials. In the graph, the unmodified tests are represented by 52

67 the red bar and labelled Sharma Exp. and the samples with surface modifications are represented by the orange bar and labelled Sharma Exp. Mod. While little difference was seen between the two experiments, the modified tests did show a slight reduction in expansion for the 5% and 10% samples but exhibited an increase in expansion in the 15% samples. Despite these differences, the upper and lower bounds still remained above and below all the results and the data followed a similar trend to Tani s, tending to be closer to the upper FEA bound. This means that there is room to improve the bond between the filler and matrix if less overall expansion is desired. For a better understand of Sharma s data and the effect of the surface treatments on the expansion of the composites, please see item [7] in the references section. 4.4 Comparison to Mathematical Models The mathematical models discussed in Section 2.3 were developed to predict the CTE of composite materials. These included the rule of mixtures, the Turner model, and the Schapery upper and lower bounds and can be seen graphically in Figure 2-4. Figures 4-17 and 4-18 show the same plot with the bounds created by the FEA for Tani s and Sharma s data added. It can be seen that the line for the lower bound falls between the Schapery limits. This makes sense as both of the experimental results closely followed the Shapery upper limit and the FEA models showed that a lower CTE was possible. The upper bound predicts the CTE to be higher than any of the other models, climbing above the rule of mixtures and showing the greatest difference between the lower bound at a 50% filler loading. This makes sense because with a high filler volume fraction, there is more surface area that could or could not be bonded to the matrix material leading to a better chance for variance in the results. After 50%, the difference starts to become smaller again as the filler becomes the dominant material and less bond area is seen because of the reduction in matrix material. 53

68 Figure 4-17: Graph showing the upper and lower bounds of expansion created by FEA models compared to Tani s data. Figure 4-18: Graph showing the upper and lower bounds of expansion created by FEA models compared to Sharma s data. 54

69 Chapter 5 Conclusions and Future Work 5.1 Conclusions The main goal of this research was to determine the effect of filler geometry on the finite element modeling of polymer/zrw 2 O 8 composites. Through the research it was determined that the filler geometry had little effect on the expansion results of the FEA. All two and three-dimensional models of the same volume fraction showed similar expansion over the 70 K temperature change despite drastic differences in their geometry. It should be noted, however, that all of the three-dimensional models showed slightly more expansion than all of the two-dimensional models. This means that the two-dimensional models gave the filler material more effect, slightly reducing the amount of expansion witnessed when compared to the three-dimensional models. These results can be seen in Sections 4.1 and 4.2. After determining the effect of filler geometry on the outcome of the models, the results were compared to the experiments of Tani et al. where it was found that all the models showed less expansion than the experimental data of the same volume fraction. While the FEA results were close enough to rule out any errors in modeling, the difference was significant and consistent enough that an explanation was required. It 55

70 was decided that the discrepancy was most likely caused by the interface between the filler and matrix materials. In the FEA models, the bond between the two materials was considered perfect, with no voids or separation. In real world experimentation, a perfect bond is nearly impossible to achieve. During the creation of particulate composites, voids can be created and separation between the two materials can occur, all of which lead to the filler material having less effect on the overall properties. This meant that the modeling completed represented the best possible case for reducing the coefficient of thermal expansion in the composite. To gain a better understanding of the experimental results, the opposite case needed to be modeled where there was no bond between the filler and matrix material. Together, the results of the models with a perfect bond and the results of the models with no bond created a lower and upper bound for the possible expansion of the polymer/zrw 2 O 8 composite. These bounds were verified using the experimental results of Tani and Sharma et al. where their results fell within the bounds at all volume fractions. These results can be seen in Section Future Work Research in the field of negative thermal expansion materials is headed in the right direction. The experiments of Sharma et al. that look to improve the bond between a polymer matrix and ZrW 2 O 8 filler are areas that need to be addressed if researchers wish to create materials with lower coefficients of thermal expansion. While, currently, the polymer/zrw 2 O 8 composites created are able to reduce the CTE of the polymer, this research has shown that further reduction could be seen with a more efficient bond between the two materials. The work conducted for this paper also leaves an opportunity for future research. To start, the findings of this research could be verified through the recreation of the 56

71 models alongside real world experiments. This would ensure full confidence in the research having gathered both the experimental data as well as completing the finite element analysis. While completing the modeling, more volume fractions could be modeled to add more data points to the upper and lower bounds created as in Section 4.3. The data used to create the bounds in Figure 4-17 were based on only three data points and a best-fit line. These could be better represented if more volume fractions were modeled. Figure 5-1: Geometry showing a possible way to model the interface between the matrix and filler. The effect of the ZrW 2 O 8 could be reduced by adjusting the size and properties of the red layer surrounding the filler. Another possible area of future research would be to attempt to closely match experimental results using FEA. The unmodified results of Sharma and Tani both showed similar differences to the bounds created by the finite element models. This means that if one could determine how to properly model the interface between the two materials when standard composite production techniques are used, accurate 57