EFFECTIVE THERMAL CONDUCTIVITY OF EPOXY MATRIX COMPOSITES FILLED WITH ALUMINIUM POWDER

EFFECTIVE THERMAL CONDUCTIVITY OF EPOXY MATRIX COMPOSITES FILLED WITH ALUMINIUM POWDER
EFFECTIVE THERMAL CONDUCTIVITY OF EPOXY
MATRIX COMPOSITES FILLED WITH
ALUMINIUM POWDER
A Project Report Submitted in Partial Fulfillment of the Requirements for the Degree of
B. Tech.
(Mechanical Engineering)
By
LIZA DAS
Roll No. 107ME044
Department of Mechanical Engineering
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
MAY, 2011
EFFECTIVE THERMAL CONDUCTIVITY OF EPOXY
MATRIX COMPOSITES FILLED WITH
ALUMINIUM POWDER
A Project Report Submitted in Partial Fulfillment of the Requirements for the Degree of
B. Tech.
(Mechanical Engineering)
By
LIZA DAS
Roll No. 107ME044
Under the supervision of
Dr. Alok Satapathy
Associate Professor
Department of Mechanical Engineering, NIT, Rourkela
Department of Mechanical Engineering
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
MAY, 2011
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the work in this thesis entitled Effective Thermal
Conductivity of Epoxy Matrix Composites Filled With Aluminium Powder by
Liza Das, has been carried out under my supervision in partial fulfillment of the
requirements for the degree of Bachelor of Technology in Mechanical
Engineering during session 2010 - 2011 in the Department of Mechanical
Engineering, National Institute of Technology, Rourkela.
To the best of my knowledge, this work has not been submitted to any other
University/Institute for the award of any degree or diploma.
Dr. Alok Satapathy
(Supervisor)
Associate Professor
Dept. of Mechanical Engineering
National Institute of Technology
Rourkela - 769008
ACKNOWLEDGEMENT
I would like to express my deep sense of gratitude and respect to my supervisor
Prof.
Alok Satapathy for his excellent guidance, suggestions and support. I
consider myself extremely lucky to be able to work under the guidance of such a
dynamic personality. I am also thankful to Prof. R. K. Sahoo, H.O.D, Department
of Mechanical Engineering, N.I.T., Rourkela for his constant support and
encouragement.
Last but not the least, I extend my sincere thanks to all other faculty members and
my friends at the Department of Mechanical Engineering, NIT, Rourkela, for their
help and valuable advice in every stage for successful completion of this project
report.
Date:
LIZA DAS
N.I.T. ROURKELA
Roll No. 107ME044
Mechanical Engineering Department
C O N T E N TS
Page No.
ABSTRACT
i
LIST OF TABLES AND FIGURES
ii
Chapter 1
Introduction
1-6
Chapter 2
Literature Review
7-12
Chapter 3
Materials and Methods
13-21
Chapter 4
Results and Discussion
22-29
Chapter 5
Conclusions and Future Scope
30-32
REFERENCES
33-38
ABSTRACT
Guarded heat flow meter test method is used to measure the thermal conductivity
of Aluminium powder filled epoxy composites using an instrument UnithermTM
Model 2022 in accordance with ASTM-E1530. In the numerical study, the finiteelement package ANSYS is used to calculate the conductivity of the composites.
Three-dimensional spheres-in-cube lattice array models are used to simulate the
microstructure of composite materials for various filler concentrations. This study
reveals that the incorporation of aluminium particulates results in enhancement
of thermal conductivity of epoxy resin and thereby improves its heat transfer
capability. The experimentally measured conductivity values are compared with
the numerically calculated ones and it is found that the values obtained for various
composite models using finite element method (FEM) are in reasonable agreement
with the experimental values.
Key Words:
Polymer Composite, Metal Powder Reinforcement, Thermal
Conductivity, FEM
(i)
LIST OF FIGURES AND TABLES
Figures:
Figure 1.1:
Classification of composites based on reinforcement type
Figure 3.1:
Preparation of particle filled composites by hand-lay-up technique.
Figure 3.2:
Determination of Thermal Conductivity Using Unitherm™
Figure 4.1:
Boundary conditions
Figure4.2:
Three-dimensional sphere-in-cube model with various particle
concentration
Figure 4.3:
Temperature profiles for composite with particle concentration of
(a) 0.4 vol%
(b) 1.4 vol%
(c) 3.34 vol%
Figure 4.4:
Comparison between Experimental results and FEM Analysis
Tables:
Table 3.1:
List of composites fabricated by hand-lay-up technique
Table 4.1
Densities of the composites under this study
Table 4.2:
Thermal conductivity values for composites obtained from FEM
and Experiment.
Table 4.3:
Percentage errors with respected to the experimental value
(ii)
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1
Chapter
Introduction
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Chapter 1
INTRODUCTION
Composite Materials:
Composites Materials are combinations of two phases in which one of the phases, called
the reinforcing phase, which is in the form of fiber sheets or particles and are embedded
in the other phase called the matrix phase. The primary functions of the matrix are to
transfer stresses between the reinforcing fibers or particles and to protect them from
mechanical and environmental damage whereas the presence of fibers or particles in a
composite improves its mechanical properties such as strength, stiffness etc. A composite
is therefore a synergistic combination of two or more micro-constituents which differ in
physical form and chemical composition and which are insoluble in each other. Our
objective is to take advantage of the superior properties of both materials without
compromising on the weakness of either. Composite materials have successfully
substituted the conventional materials in several applications like light weight and high
strength. The reasons why composites are selected for such applications are mainly due to
their high strength-to-weight ratio, high tensile strength at elevated temperatures, high
creep resistance and high toughness. Typically, the reinforcing materials are strong with
low densities while the matrix is usually a ductile or tough material. If the composite is
designed and fabricated correctly it combines the strength of the reinforcement with the
toughness of the matrix to achieve a combination of desirable properties not available in
any single traditional material. The strength of the composites depends primarily on the
amount, arrangement and type of fiber or particle reinforcement in the resin. [1]
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Types of Composite Materials:
Basically, composites can be categorized into three groups on the basis of matrix
material. They are:
a) Metal Matrix Composites (MMC)
b) Ceramic Matrix Composites (CMC)
c) Polymer Matrix Composites (PMC)
a)
Metal Matrix Composites:
These Composites have many advantages over monolithic metals like higher specific
strength, higher specific modulus, better properties at elevated temperatures and lower
coefficient of thermal expansion. Because of these attributes metal matrix composites are
under consideration for wide range of applications viz. combustion chamber nozzle (in
rocket, space shuttle), housings, tubing, cables, heat exchangers, structural members etc.
b)
Ceramic matrix Composites:
The main objectives in producing ceramic matrix composites is to increase the toughness.
Naturally it is hoped and indeed often found that there is a concomitant improvement in
strength and stiffness of ceramic matrix composites.
c)
Polymer Matrix Composites:
These are the most commonly used matrix materials.In general the mechanical properties
of polymers are inadequate for many structural purposes. In particular their strength and
stiffness are low compared to metals and ceramics. These difficulties are overcome by
reinforcing other materials with polymers.
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Secondly the processing of polymer matrix composites do not require high pressure and
high temperature. The equipments which are required for manufacturing polymer matrix
composites are simpler. For this reason polymer composites developed rapidly and soon
became popular for structural applications. Polymer composites are used because overall
properties of the composites are superior to those of the individual polymers. The elastic
modulus is greater than the neat polymer but are not as brittle as ceramics.
Classification of polymer composites:
Fig. 1.1:
Classification of composites based on reinforcement type
Polymer composites can be classified into three groups on the basis of
reinforcing material. They are:
(a) Fiber reinforced polymer ( FRP )
(b) Particle reinforced polymer ( PRP )
(c) Structural polymer composites (SPC)
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(a) Fiber reinforced polymer:
The fiber reinforced composites are composed of fibers and a matrix. Fibers are the
reinforcing elements and the main source of strength while matrix glues all the fibers
together in shape and transfers stresses between the reinforcing fibers. The fibers carry
the loads along their longitudinal directions. Sometimes, filler is added to smoothen the
manufacturing process and to impact special properties to the composites .These also
reduces the production cost. Most commonly used agents include asbestos, carbon/
graphite fibers, beryllium, beryllium carbide, beryllium oxide, molybdenum, aluminum
oxide, glass fibers, polyamide, natural fibers etc. Similarly common matrix materials
include epoxy, phenolic resin, polyester, polyurethane, vinyl ester etc. Among these
materials, resin and polyester are most widely used. Epoxy, which has higher adhesion
and less shrinkage than polyesters, comes in second for its high cost.
Particle reinforced polymer:
Particles which are used for reinforcing include ceramics and glasses such as small
mineral particles, metal particles such as aluminum and amorphous materials, including
polymers and carbon black. Particles are used to enhance the modulus and to decrease the
ductility of the matrix. Some of the useful properties of ceramics and glasses include high
melting temp., low density, high strength, stiffness; wear resistance, and corrosion
resistance etc. Many ceramics are good electrical and thermal insulators. Some ceramics
have special properties; some have magnetic properties; some are piezoelectric materials;
and a few special ceramics are even superconductors at very low temperatures. One
major drawback of ceramics and glass is their brittleness. An example of particle –
reinforced composites is an automobile tyre, which has carbon black particles in a matrix
of poly-isobutylene elastomeric polymer.
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Structural Polymer Composites:
These are laminar composites which are composed of layers of materials held together by
matrix. This category also includes sandwich structures. Over the past few decades, we
find that polymers have replaced many of the conventional materials in various
applications. The most important advantages of using polymers are the ease of
processing, productivity and cost reduction. The properties of polymers are modified
using fillers and fibers to suit the high strength and high modulus requirements. Fiberreinforced polymers offer advantages over other conventional materials when specific
properties are compared. That‟s the reason for these composites finding applications in
diverse fields from appliances to spacecrafts.
A lot of work has been carried out on various aspects of polymer composites, but a few
researchers have reported on the thermal conductivity modification of particulate filled
polymers. In view of this, the present work is undertaken to estimate and measure the
effective thermal conductivity of epoxy filled with metal powders.
******
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2
Chapter
Literature Review
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Chapter 2
LITERATURE REVIEW
The purpose of this literature review is to provide background information on the issues
to be considered in this thesis and to emphasize the relevance of the present study. This
treatise embraces some related aspects of polymer composites with special reference to
their thermal conductivity characteristics. The topics include brief review:
1) On Particulate Reinforced Polymer Composites
2) On Thermal Conductivity of Polymer Composites
3) On Thermal Conductivity Models
(1) On particulate filled polymer composites:
Hard particulate fillers consisting of ceramic or metal particles and fiber fillers made of
glass are being used to improve the mechanical properties such as wear resistance [2].
Various kinds of polymers and polymer matrix composites reinforced with metal
particles have a wide range of industrial applications such as heaters, electrodes [3],
composites with thermal durability at high temperature [4] etc. These engineering
composites are desired due to their low density, high corrosion resistance, ease of
fabrication and low cost [5-7]. The inclusion of inorganic fillers into polymers for
commercial applications is primarily aimed at the cost reduction and stiffness
improvement [8,9]. Along with fiber reinforced composites, the composites made with
particulate fillers have been found to perform well in many real operational conditions.
When silica particles are added into a polymer matrix to form a composite, they play an
important role in improving electrical, mechanical and thermal properties of the
composites [10,11]. Currently, particle size is being reduced rapidly and many studies
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have focused on how single-particle size affects mechanical properties [12-18]. The
shape, size, volume fraction, and specific surface area of such added particles have been
found to affect mechanical properties of the composites greatly. In this regard,
Yamamoto et al. [19] reported that the structure and shape of silica particle have
significant effects on the mechanical properties such as fatigue resistance, tensile and
fracture properties. Nakamura et al. [20-22] discussed the effects of size and shape of
silica particle on the strength and fracture toughness based on particle-matrix adhesion
and also found an increase of the flexural and tensile strength as specific surface area of
particles increased.
(2) On Thermal Conductivity of Polymer Composites
Considerable work has been reported on the subject of heat conductivity in polymers by
Hansen and Ho [23], Peng et. al [24], Choy and Young [25], Tavman [26] etc. It is well
known that thermal transport increases significantly in the direction of orientation and
decreases slightly in the direction perpendicular to the orientation. Reports are available
in the existing literature on experimental as well as numerical and analytical studies on
thermal conductivity of some filled polymer composites [27-39]. The fillers most
frequently used are aluminum particles, copper particles, brass particles, short carbon
fiber, carbon particles, graphite, aluminum nitrides and magnetite particles. Progelhof et.
al [40] was the first to present an exhaustive overview on models and methods for
predicting the thermal conductivity of composite systems. Procter and Solc [41] used
Nielsen model as a prediction to investigate the thermal conductivity of several types of
polymer composites filled with different fillers and confirmed its applicability. Nagai
[42] found that Bruggeman model for Al2O3/epoxy system and a modified form of
Bruggeman model for AlN/epoxy system are both good prediction theories for thermal
conductivity. Griesinger et. al [43] reported that thermal conductivity of low-density
poly-ethylene (LDPE) increased from 0.35 W/mK for an isotropic sample, to the value of
50 W/mK for a sample with an orientation ratio of 50. The thermal and mechanical
properties of copper powder filled poly-ethylene composites are found by Tavman [44]
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while Sofian et al. [45] investigated experimentally on thermal properties such as thermal
conductivity, thermal diffusivity and specific heat of metal (copper, zinc, iron, and
bronze) powder filled HDPE composites in the range of filler content 0–24% by volume.
They observed a moderate increase in thermal conductivity up to 16% of metal powder
filler content. Mamunya et. al [46] also reported the improvement in electrical and
thermal conductivity of polymers filled with metal powders. In a recent research
Weidenfeller et al. [47] studied the effect of the interconnectivity of the filler particles
and its important role in the thermal conductivity of the composites. They prepared PP
samples with different commercially available fillers by extrusion and injection molding
using various volume fractions of filler content to systematically vary density and thermal
transport properties of these composites. Surprisingly, they measured that the thermal
conductivity of the PP has increased from 0.27 up to 2.5W/mK with 30 vol% talc in the
PP matrix, while the same matrix material containing the same volume fraction of copper
particles had a thermal conductivity of only 1.25W/m-K despite the fact that copper
particles have a thermal conductivity approximately 40 times greater than that of talc
particles. Tekce et. al [48] noticed the strong influence of the shape factor of fillers on
thermal conductivity of the composite. While Kumlutas and Tavman [49] carried out a
numerical and experimental study on thermal conductivity of particle filled polymer
composites, Patnaik et. al reported the existence of a possible correlation between thermal
conductivity and wear resistance of particulate filled composites [50].
(3) On Thermal Conductivity Models
Many theoretical and empirical models have been proposed to predict the effective
thermal conductivity of two-phase mixtures. Comprehensive review articles have
discussed the applicability of many of these models [27, 51]. For a two-component
composite, the simplest alternatives would be with the materials arranged in either
parallel or series with respect to heat flow, which gives the upper or lower bounds of
effective thermal conductivity.
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For the parallel conduction model:
(2.1)
where, kc, km, kf are the thermal conductivities of the composite, the matrix and the filler
respectively and Ø is the volume fraction of filler.
For the series conduction model:
(2.2)
The correlations presented by equations (2.1) and (2.2) are derived on the basis of the
Rules of Mixture (ROM). Tsao [52] derived an equation relating the two-phase solid
mixture thermal conductivity to the conductivity of the individual components and to two
parameters which describe the spatial distribution of the two phases. By assuming a
parabolic distribution of the discontinuous phase in the continuous phase, Cheng and
Vachon [53] obtained a solution to Tsao‟s [52] model that did not require knowledge of
additional parameters. Agari and Uno [54] propose a new model for filled polymers,
which takes into account parallel and series conduction mechanisms.
According to this model, the expression that governs the thermal conductivity of the
composite is:
)
(2.3)
where, C1, C2 are experimentally determined constants of order unity. C1 is a measure of
the effect of the particles on the secondary structure of the polymer, like crystallinity and
the crystal size of the polymer. C2 measures the ease of the particles to form conductive
chains. The more easily particles are gathered to form conductive chains, the more
thermal conductivity of the particles contributes to change in thermal conductivity of the
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composite and C2 becomes closer to 1. Later, they modified the model to take into
account the shape of the particles [55]. Generally, this semi-empirical model seems to fit
the experimental data well. However, adequate experimental data is needed for each type
of composite in order to determine the necessary constants. For an infinitely dilute
composite of spherical particles, the exact expression for the effective thermal
conductivity is given as:
(2.4)
where K, Kc and Kd are thermal conductivities of composite, continuous-phase (matrix),
and dispersed-phase (filler), respectively, and Ø is the volume fraction of the dispersed
phase. Equation (2.4) is the well-known Maxwell equation [56] for dilute composites.
Objective of the present Investigation
The objectives of this work are outlined as follows:
1. Fabrication of a new class of low cost composites using micro-sized
aluminium powder as the reinforcing filler with an objective to improve the
heat conducting properties of neat epoxy.
2. Measurement of thermal conductivity (k) of these particulate filled polymer
composite experimentally.
3. Estimation of equivalent thermal conductivity of this particulate-polymer
composite system using Finite Element Method (FEM).
4. Validation of the FEM analysis by measuring the thermal conductivity values
experimentally.
5. Recommendation of these composites for suitable applications.
*******
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3
Chapter
Materials and Methods
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Chapter 3
MATERIALS AND METHODS
This chapter describes the materials and methods used for the processing of the
composites under this investigation. It presents the details of the characterization and
thermal conductivity tests which the composite samples are subjected to. The numerical
methodology related to the determination of thermal conductivity based on finite element
method is also presented in this chapter of the thesis.
MATERIALS
Matrix Material:
Epoxy LY 556 resin, chemically belonging to the „epoxide‟ family is used as the matrix
material. Its common name is Bisphenol-A-Diglycidyl-Ether. The low temperature curing
epoxy resin (Araldite LY 556) and the corresponding hardener (HY 951) are mixed in a
ratio of 10:1 by weight as recommended. The epoxy resin and the hardener are supplied
by Ciba Geigy India Ltd. Epoxy is chosen primarily because it happens to the most
commonly used polymer and because of its insulating nature (low value of thermal
conductivity, about 0.363 W/m-K).
Filler Material (Aluminium Powder):
It is a silvery white metal. Aluminium is remarkable for the metal's low density and for
its ability to resist corrosion . Its thermal conductivity and density values are 250 W/m.K
and 2.7 g/cc.Structural components made from aluminium and its alloys are vital to
the aerospace industry and are very important in other areas of transportation and
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building. Its reactive nature makes it useful as a catalyst or additive in chemical mixtures,
including ammonium nitrate explosives, to enhance blast power.
Composite Fabrication:
The low temperature curing epoxy resin (LY 556) and corresponding hardener (HY951)
are mixed in a ratio of 10:1 by weight as recommended. Aluminium powder with
average size 100-200 μm are reinforced in epoxy resin (density 1.1 gm/cc) to prepare the
composites. The dough (epoxy filled with Al powder) is then slowly decanted into the
glass tubes, coated beforehand with wax and uniform thin film of silicone-releasing
agent. The composites are cast by conventional hand-lay-up technique in glass tubes so as
to get cylindrical specimens (dia. 9 mm, length 120 mm). Composites of four different
compositions (with 0.4, 1.4, 3.34 and 6.5 vol % of Al respectively) are made. The
castings are left to cure at room temperature for about 24 hours after which the tubes are
broken and samples are released. Specimens of suitable dimension are cut using a
diamond cutter for further physical characterization and thermal conductivity test.
Samples
Composition
1
Epoxy + 0 vol% (0 wt%) Filler
2
Epoxy + 0.4 vol% (1.01 wt%) Filler
3
Epoxy + 1.4vol% (3.4 wt%) Filler
4
Epoxy + 3.34 vol% (7.8wt%) Filler
5
Epoxy + 6.5 vol % (14.7wt%) Filler
Table 3.1: List of particulate filled composites fabricated by hand-lay-up technique
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Fig. 3.1 Preparation of particulate filled composites by hand-lay-up technique
THERMAL CONDUCTIVITY CHARACTERIZATION
Experimental Determination of Thermal Conductivity:
Unitherm™ Model 2022 is used to measure thermal conductivity of a variety of
materials. These include polymers, ceramics, composites, glasses, rubbers, some metals
and other materials of low to medium thermal conductivity. Only a relatively small test
sample is required. Non-solids, such as pastes or liquids can be tested using special
containers. Thin films can also be tested accurately using a multi-layer technique.
The tests are in accordance with ASTM E-1530 Standard.
Operating principle of UnithermTM 2022:
A sample of the material is held under a uniform compressive load between two polished
surfaces, each controlled at a different temperature. The lower surface is part of a
calibrated heat flow transducer. The heat flows from the upper surface, through the
sample, to the lower surface, establishing an axial temperature gradient in the stack. After
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reaching thermal equilibrium, the temperature difference across the sample is measured
along with the output from the heat flow transducer. These values and the sample
thickness are then used to calculate the thermal conductivity. The temperature drop
through the sample is measured with temperature sensors in the highly conductive metal
surface layers on either side of the sample.
Fig. 3.2 Determination of Thermal Conductivity Using Unitherm™ Model 2022
By definition thermal conductivity means “The material property that describes the rate at
which heat flows with in a body for a given temperature change.” For one-dimensional
heat conduction the formula can be given as equation 3.1:
(3.1)
Where Q is the heat flux (W), K is the thermal conductivity (W/m-K), A is the cross
sectional area (m2) T1-T2 is the difference in temperature (K), x is the thickness of the
sample (m). The thermal resistance of a sample can be given as:
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(3.2)
Where, R is the resistance of the sample between hot and cold surfaces (m2-K/W). From
Equations 3.1 and 3.2 we can derive that
(3.3)
In Unitherm 2022, the heat flux transducer measures the Q value and the temperature
difference can be obtained between the upper plate and lower plate. Thus the thermal
resistance can be calculated between the upper and lower surfaces. Giving the input value
of thickness and taking the known cross sectional area, the thermal conductivity of the
samples can be calculated using Equation 3.3.
Numerical Analysis: Concept of Finite Element Method (FEM) and ANSYS:
The Finite Element Method (FEM), originally introduced by Turner et al. [58] in 1956, is
a powerful computational technique for approximate solutions to a variety of "real-world"
engineering problems having complex domains subjected to general boundary conditions.
FEM has become an essential step in the design or modeling of a physical phenomenon
in various engineering disciplines. A physical phenomenon usually occurs in a continuum
of matter (solid, liquid, or gas) involving several field variables. The field variables vary
from point to point, thus possessing an infinite number of solutions in the domain.
The basis of FEM relies on the decomposition of the domain into a finite number of
subdomains (elements) for which the systematic approximate solution is constructed by
applying the variational or weighted residual methods. In effect, FEM reduces the
problem to that of a finite number of unknowns by dividing the domain into elements and
by expressing the unknown field variable in terms of the assumed approximating
functions within each element. These functions (also called interpolation functions) are
defined in terms of the values of the field variables at specific points, referred to as nodes.
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Nodes are usually located along the element boundaries and they connect adjacent
elements. The ability to discretize the irregular domains with finite elements makes the
method a valuable and practical analysis tool for the solution of boundary, initial and
eigen value problems arising in various engineering disciplines.
The FEM is thus a numerical procedure that can be used to obtain solutions to a large
class of engineering problems involving stress analysis, heat transfer, fluid flow etc.
ANSYS is general-purpose finite-element modeling package for numerically solving a
wide variety of mechanical problems that include static/dynamic, structural analysis (both
linear and nonlinear), heat transfer, and fluid problems, as well as acoustic and
electromagnetic problems.
Basic Steps in FEM:
The finite element method involves the following steps.
First, the governing differential equation of the problem is converted into an integral
form. There are two techniques to achieve this:
(i) Variational Technique
(ii) Weighted Residual Technique.
In variational technique, the calculus of variation is used to obtain the integral form
corresponding to the given differential equation. This integral needs to be minimized to
obtain the solution of the problem. For structural mechanics problems, the integral form
turns out to be the expression for the total potential energy of the structure. In weighted
residual technique, the integral form is constructed as a weighted integral of the
governing differential equation where the weight functions are known and arbitrary
except that they satisfy certain boundary conditions. To reduce the continuity requirement
of the solution, this integral form is often modified using the divergence theorem. This
integral form is set to zero to obtain the solution of the problem. For structural mechanics
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problems, if the weight function is considered as the virtual displacement, then the
integral form becomes the expression of the virtual work of the structure.
In the second step, the domain of the problem is divided into a number of parts, called as
elements. For one-dimensional (1-D) problems, the elements are nothing but line
segments having only length and no shape. For problems of higher dimensions, the
elements have both the shape and size. For two-dimensional (2D) or axi-symmetric
problems, the elements used are triangles, rectangles and quadrilateral having straight or
curved boundaries. Curved sided elements are good choice when the domain boundary is
curved. For three-dimensional (3-D) problems, the shapes used are tetrahedron and
parallelepiped having straight or curved surfaces. Division of the domain into elements is
called a mesh. In this step, over a typical element, a suitable approximation is chosen for
the primary variable of the problem using interpolation functions (also called as shape
functions) and the unknown values of the primary variable at some pre-selected points of
the element, called as the nodes. Usually polynomials are chosen as the shape functions.
For 1-D elements, there are at least 2 nodes placed at the endpoints. Additional nodes are
placed in the interior of the element. For 2-D and 3-D elements, the nodes are placed at
the vertices (minimum 3 nodes for triangles, minimum 4 nodes for rectangles,
quadrilaterals and tetrahedral and minimum 8 nodes for parallelepiped shaped elements).
Additional nodes are placed either on the boundaries or in the interior. The values of the
primary variable at the nodes are called as the degrees of freedom.
To get the exact solution, the expression for the primary variable must contain a complete
set of polynomials (i.e., infinite terms) or if it contains only the finite number of terms,
then the number of elements must be infinite. In either case, it results into an infinite set
of algebraic equations. To make the problem tractable, only a finite number of elements
and an expression with only finite number of terms are used. Then, we get only an
approximate solution. (Therefore, the expression for the primary variable chosen to
obtain an approximate solution is called an approximation). The accuracy of the
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approximate solution, however, can be improved either by increasing the number of
terms in the approximation or the number of elements.
In the fourth step, the approximation for the primary variable is substituted into the
integral form. If the integral form is of variational type, it is minimized to get the
algebraic equations for the unknown nodal values of the primary variable. If the integral
form is of the weighted residual type, it is set to zero to obtain the algebraic equations. In
each case, the algebraic equations are obtained element wise first (called as the element
equations) and then they are assembled over all the elements to obtain the algebraic
equations for the whole domain (called as the global equations). In this step, the algebraic
equations are modified to take care of the boundary conditions on the primary variable.
The modified algebraic equations are solved to find the nodal values of the primary
variable.
In the last step, the post-processing of the solution is done. That is, first the secondary
variables of the problem are calculated from the solution. Then, the nodal values of the
primary and secondary variables are used to construct their graphical variation over the
domain either in the form of graphs (for 1-D problems) or 2-D/3-D contours as the case
may be.
Advantages of the finite element method over other numerical methods are as follows:
 The method can be used for any irregular-shaped domain and all types of boundary
conditions.
 Domains consisting of more than one material can be easily analyzed.
 Accuracy of the solution can be improved either by proper refinement of the mesh or
by choosing approximation of higher degree polynomials.
 The algebraic equations can be easily generated and solved on a computer. In fact, a
general purpose code can be developed for the analysis of a large class of problems.
*******
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2011
4
Chapter
Results and Discussion
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2011
Chapter 4
RESULTS AND DISCUSSION
PHYSICAL CHARACTERIZATION
Density and void fraction:
The theoretical density of composite materials in terms of weight fraction can easily be
obtained as for the following equations given by Agarwal and Broutman [56].
(4.1)
Where, W and ρ represent the weight fraction and density respectively. The suffix f, m
and ct stand for the fiber, matrix and the composite materials respectively.
In case of hybrid composites, consisting of three components namely matrix, fiber and
particulate filler, the modified form of the expression for the density of the composite can
be written as:
(4.2)
Where, the suffix „p’ indicates the particulate filler materials.
Mechanical Engineering Department, N.I.T. Rourkela
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B. Tech. Project Report
Samples
2011
Density of composite
(gm/cc)
1
Composition
(for Al filled epoxy)
Epoxy + 0 vol% (0 wt %) Filler
2
Epoxy + 0.4 vol% (1.01 wt %) Filler
1.1066
3
4
Epoxy + 1.4vol% (3.4 wt %) Filler
Epoxy + 3.34vol% (7.8 wt %) Filler
5
Epoxy + 6.5vol % (14.7 wt %) Filler
1.12253
1.1534
1.2046
1.1
Table 4.1 Density of the composites under this study
THERMAL CONDUCTIVITY CHARACTERIZATION
Description of the problem
The determination of effective properties of composite materials is of paramount
importance for functional design and application of composite materials. One of the
important factors that influence the effective properties and can be controlled to an
appreciable extent is the microstructure of the composite. Here, microstructure means the
shape, size distribution, spatial distribution and orientation distribution of the reinforcing
inclusion in the matrix. Although most composite possess inclusion of random
distributions, great insight of the effect of microstructure on the effective properties can
be gained from the investigation of composites with periodic structure. System with
periodic structures can be more easily analyzed because of the high degree of symmetry
embedded in the system.
Using the finite-element program ANSYS, thermal analysis is carried out for the
conductive heat transfer through the composite body. In order to make a thermal analysis,
three-dimensional physical models with spheres-in-a-cube lattice array have been used to
simulate the microstructure of composite materials for four different filler concentrations.
Furthermore, the effective thermal conductivities of these epoxy composites filled with
Mechanical Engineering Department, N.I.T. Rourkela
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2011
Aluminium particle up to about 6.5% by volume is numerically determined using
ANSYS.
Assumptions
In the analysis of the ideal case it will be assumed that
1. The composites are macroscopically homogeneous
2. Locally both the matrix and filler are homogeneous and isotropic
3. The thermal contact resistance between the filler and the matrix is negligible.
4. The composite lamina is free of voids
5. The problem is based on 3D physical model
6. The filler are arranged in a square periodic array/uniformly distributed in matrix.
Numerical Analysis
In the numerical analysis of the heat conduction problem, the temperatures at the nodes
along the surfaces ABCD is prescribed as T1 (=1000C) and the convective heat transfer
coefficient of ambient is assumed to be 2.5 W/m2-K at ambient temperature of 27°C. The
heat flow direction and the boundary conditions are shown in Fig. 4.1. The other surfaces
parallel to the direction of the heat flow are all assumed adiabatic. The temperatures at
the nodes in the interior region and on the adiabatic boundaries are unknown. These
temperatures are obtained with the help of finite-element program package ANSYS.
Fig.4. 1 Boundary conditions
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2011
(a)
(b)
(c)
Fig. 4.2 Typical 3-D spheres-in-cube models with particle concentration of aluminum
powder (a) 0.4 vol% (b) 1.4 vol% and (c) 3.34 respectively.
Mechanical Engineering Department, N.I.T. Rourkela
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B. Tech. Project Report
2011
(a)
(b)
( c)
Fig. 4.3 Temperature profiles for aluminium filled epoxy composites with particle
concentration of (a) 0.4 vol% (b) 1.4 vol% and (c) 3.34 vol% respectively
Mechanical Engineering Department, N.I.T. Rourkela
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B. Tech. Project Report
Sample
1
2
3
4
2011
Al
Content
(vol %)
0
0.4
1.4.
3.34
Effective thermal conductivity of composites
Keff (W/m-K)
FEM
(Spheres-in-cube Model )
0.369
0.382
0.408
Experimental
Value
0.363
0.364
0.369
0.385
Table 4.2 Thermal conductivity for composites obtained from FEM and Experiment
Composite
Sample
Al
Content
(Vol. %)
Percentage errors associated with FEM
results w.r.t. the experimental value (%)
1
0.4
1.3
2
3
1.4
3.34
3.5
5.9
Table 4.3 Percentage errors with respect to the experimental value
GRAPH
Keff vs vol. %
0.42
0.41
0.4
Keff
0.39
0.38
0.37
Keff(FEM)
0.36
Keff(Exp.)
0.35
0.34
0
0.4188
1.413
3.34
vol %
Figure 4.4
Comparison between Experimental results and FEM Analysis
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2011
Thermal conductivities of epoxy composites filled with aluminium particles to 6.5% by
volume are numerically estimated by using the spheres-in-cube model and the numerical
results are compared with the experimental results and also with some of the existing
theoretical and empirical models The temperature profiles obtained from FEM analysis
for the composites with particulate concentrations of 0.4, 1.4, and 3.34 vol % are
presented in Figures 4.3a - 4.3c respectively.
This study shows that finite element method can be gainfully employed to determine
effective thermal conductivity of these composite with different amount of filler content.
The value of equivalent thermal conductivity obtained for various composite models
using FEM are in reasonable agreement with the experimental values for a wide range of
filler contents from about 0.4 vol.% to 3.34 vol.%. Incorporation of Al results in
enhancement of thermal conductivity of epoxy resin. With addition of 1.413 % and 3.34
vol. % of Al, the thermal conductivity improves by about 3.4 % and 6.3 % respectively
with respect to neat epoxy resin.
******
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5
Chapter
Conclusions
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2011
Chapter 5
CONCLUSIONS AND
SCOPE FOR FUTURE WORK
Conclusions
This numerical and experimental investigation on thermal conductivity of aluminium
filled epoxy composites have led to the following specific conclusions:
Successful fabrication of epoxy based composites filled with micro-sized Al by
hand-lay-up technique is possible.
Finite element method can be gainfully employed to determine effective
thermal conductivity of these composite with different amount of filler content.
The value of equivalent thermal conductivity obtained for various composite
models using FEM are in reasonable agreement with the experimental values
for a wide range of filler contents from about 0.4 vol.% to 3.34 vol.%.
Incorporation of Al results in enhancement of thermal conductivity of epoxy
resin. With addition of 3.34 vol. % of Al, the thermal conductivity improves by
about 6.3 % with respect to neat epoxy resin.
These new class of Al filled epoxy composites can be used for applications
such as electronic packages, encapsulations, die (chip) attach, thermal grease,
thermal interface material and electrical cable insulation.
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Scope for future work
This work leaves a wide scope for future investigators to explore many other aspects of
thermal behavior of particulate filled composites. Some recommendations for future
research include:
Effect of filler shape and size on thermal conductivity of the composites
Exploration of new fillers for development of thermal insulation materials
******
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2011
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******
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