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Development of VB Code to Generate Randomly Distributed Short
Fiber Composites and
Estimation of Mechanical Properties using FEM
Thesis submitted in partial fulfillment of the Requirements for
the degree of
Master of Science in
Mechanical Systems Design By
(Signature) BIJU BL
(Reg. No.102520085)
Under the guidance of
(Signature) Dr. BADARI NARAYANA KANTHETI
AEROSTRUCTURES UTC AEROSPACE SYSTEMS
BANGALORE
MANIPAL UNIVERSITY, MANIPAL
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This document contains no EAR or ITAR technical data
Development of VB Code to Generate Randomly Distributed Short
Fiber
Composites and Estimation of Mechanical Properties using FEM
Thesis submitted in partial fulfillment of the Requirements for
the degree of
Master of Science in
Mechanical Systems Design By
BIJU BL (Reg. No.102520085)
Examiner 1 Examiner 2
Signature: Signature:
Name: Name:
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UTC Aerospace Systems Netra Tech Park, EPIP Industrial Area
Sy.No.28 , Bengaluru 560 066, INDIA City, State/Province, Postal
Code www.utcaerospacesystems.com
This document contains no EAR or ITAR technical data
CERTIFICATE
This is to certify that this thesis work titled
Development of VB Code to Generate Randomly Distributed Short
Fiber Composites and
Estimation of Mechanical Properties using FEM
Is a bonafide record of the work done by BIJU BL
102520085
In partial fulfillment of the requirements for the award of the
degree of Master of Science in Mechanical Systems Design under
Manipal University, Manipal and the same has not been submitted
elsewhere for the award for any other degree
(Signature) Dr. BADARI NARAYANA KANTHETI
AEROSTRUCTURES UTC AEROSPACE SYSTEMS
BANGALORE
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ACKNOWLEDGMENTS
Foremost, I would like to express my sincere gratitude to my
advisor Dr. Badari Narayana Kantheti for the support and the
immense knowledge. His support was instrumental from the choosing
of the thesis topic through getting the report
completed. I would also like to thank my manager Mr. Ananda
Kumar for allowing me to be off work at times for completing this
thesis. I would also like to thank Mr Pradip Kumar Pandey, SBU
Head-Aerostructures and Ravishankar Mysore, Vice
President-Engineering for their approval of this thesis.
A special thanks to my family, especially to my parents and my
wife Kamya for supporting me always. Without her support I could
not have gathered so much time at home to spend on my studies and
on this project. Also credits go to my daughter Samiha for cheering
me up whenever I was fatigued.
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ABSTRACT
Short fiber reinforced polymers were developed largely to fill
the property gap between continuous fiber laminates used as primary
structures by the aircraft and aerospace industry and
non-reinforced polymers used largely in non-load bearing
applications. In some respects the short fiber systems couple
advantages from each of these property bounding engineering
materials. If the fibers are sufficiently long, stiffness levels
approaching those for continuous fiber systems at the same fiber
loading can be achieved, while the ability of the non-reinforced
polymer to be molded into complex shapes is at least partially
retained in the short fiber systems. Thus, short fiber reinforced
polymers have found their way into lightly loaded secondary
structures, in which stiffness dominates the design, but in which
there must also be a notable increase in strength over the
non-reinforced polymer. The physical properties may be determined
by conducting suitable experiments as per industry standards.
However, a specific set of experiments can only inform us about a
specific Fiber/matrix system. Hence, to design a composite system
by tuning its volume fraction, or fiber/matrix combination, or
orientation, then a very large number of experiments may have to be
conducted. Such a process for material property determination is
extremely tedious, prohibitively expensive, and time consuming.
Still further, exact fiber/matrix combinations may not be always
available for testing. Hence, there is a need for developing
mathematical models, which can reliably predict different
mechanical properties of composite materials. Such approaches are
very useful for engineers since they provide significant savings in
time and cost. Existing solutions for determining physical
properties of aligned short fiber composites are studied and
methods are identified to extend it to randomly oriented short
fiber composite. These were compared with experimental results
available in Literature.
A close correlation was observed between the properties obtained
by various empirical methods and by FE modelling of random fiber.
The data obtained should be a good starting point first degree
accuracy which can be used for preliminary analysis of components
manufactured using short fiber composite with random
orientation.
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LIST OF TABLES
Table No
Table Title Page No
2.7.6.1 Values for KR used in Eq 2.7.6-8 for shear lag models 34
2.7.7.1 Correspondence between Halpin-Tsai Eq 2.7.7-1 and
generalized self-consistent predictions 37
2.7.7.2 Traditional Halpin-Tsai parameters for short-fiber
composites
39
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LIST OF FIGURES
Figure No
Figure Title Page No
1.1 1 Family of Composites (Courtesy http://nptel.ac.in/) 1 1.1
2 Typical aerospace application brackets (Courtesy
http://www.gtweed.com/) 2
2.2 1 Loads on Composite, Fibres, & Matrix in a
Unidirectional Lamina
6
2.3 1 A Slab Like Model for Predicting Transverse Properties of
Unidirectional Composites
10
2.6 1 Force Equilibrium of an Infinitesimal Portion of
Discontinuous Fiber which is aligned to External Load
15
2.6 2 is a plot of variation of fiber strength for three
different fiber lengths.
18
2.7 1 Eshelby's inclusion problem. 20
2.7 2 Eshelby's equivalent inclusion problem. 21
2.7 3 Idealized fiber and matrix geometry used in shear lag
models. 32
2.7 4 Fiber packing arrangements used to find R in shear lag
models. (a) Hexagonal (Cox, 1952). (b) Hexagonal (Rosen, 1964) (c)
Square (Robinson & Robinson, 1994).
35
3.1 1 Comparison of Empirical models with experimental results
54
3.5 1 Flow chart for Random fiber generation VBA code 60
3.5 2 VBA code for Random number Generation I 61
3.5 3 VBA code for Random number Generation II 62
3.5 4 A Sample random short fiber composite specimen generated
by the VBA code
63
3.5 5 A Sample random short fiber composite specimen generated
by the VBA code
64
3.5 6 A Sample random short fiber composite specimen generated
by the VBA code
64
3.6 1 Patran Session file for creating random fibers of volume
fraction 0.2
65
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3.6 2 (a) Geometric model (b) FE model of random fibers of
volume fraction 0.2
65
3.7 1 Patran Session file for creating random fibers of volume
fraction 0.15
66
3.7 2 (a) Geometric model (b) FE model of random fibers of
volume fraction 0.15
66
3.8 1 Patran Session file for creating random fibers of volume
fraction 0.1
67
3.8 2 (a) Geometric model (b) FE model of random fibers of
volume fraction 0.1
67
3.9 1 Load application in the FE model 68
3.10 1 Stress on model with same material applied for fiber and
matrix
69
3.10 2 Stress on cross sections with same material applied for
fiber and matrix
69
4.1 1 Comparison of Empirical model results for E with FE of
random fibers
72
4.2 1 Comparison of Empirical model results for G with FE of
random fibers
73
4.3 1 Comparison of Empirical model results for Strength for
random fibers
74
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List of Notations
A Area
a cross sectional dimension of fiber
A Strain concentration tensor C Compliance
d Fiber diameter
E Elastic modulus
Eshelbys Tensor
G Shear modulus
l length of fiber
lc Critical Fiber length
P Load
r Radius of fiber
S Stiffness
t Thickness
v Volume fraction
Strain
efficiency factor
Poissons ratio
Normal Stress
Shear stress
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Contents Page No
Acknowledgement i Abstract ii List Of Tables iii List Of Figures
iv
Chapter 1 INTRODUCTION 1
Introduction 1
Motivation 3
Organization of Report 4
Chapter 2 LITERATURE REVIEW 5
The Need for Predictive Models for Determining Composite
Properties
5
Predicting Longitudinal Modulus of Unidirectional Lamina 6
Predicting Transverse Modulus of Unidirectional Lamina 9
Shear Modulus and Poissons Ratio 12
Transverse Strength 12
About Short-Fibre Composites 13
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Modulus of Short-Fiber Composites 19
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In Eq 3.7.7-2 the underlined term is typically negligible, and
dropping it gives the familiar rule of mixtures for E11of a
continuous-fiber composite. However, dropping the underlined term
in Eq 3.7.7-3 and using a rule of mixtures for 12 is not
necessarily accurate if the fiber and matrix Poisson ratios differ.
Halpin and Tsai argue for this latter approximation on the grounds
that laminate stiffnesses are insensitive to 12. In adapting their
approach to short-fiber composites, Halpin and Tsai noted that must
lie between 0 and . If =0 then Eq 3.7.7-1 reduces to the inverse
rule of mixtures ,
1 = +
while for = the Halpin-Tsai form becomes the rule of
mixtures,
= + Halpin and Tsai suggested that was correlated with the
geometry of the reinforcement and, when calculating E11, it should
vary from some small value to infinity as a function of the fiber
aspect ratio l/d. By comparing model predictions with available 2-D
finite element results, they found that =2(l/d) gave good
predictions for E11of short-fiber systems. Also, they suggested
that other engineering constants of short-fiber composites were
only weakly dependent on fiber aspect ratio, and could be
approximated using the continuous-fiber formulae. The resulting
equations are summarized in Table 3.7.7-2. The early references and
do not mention G23. When this property is needed the usual approach
is to use the value given in Table 3.7.7-1. While the Halpin-Tsai
equations have been widely used for isotropic fiber materials, the
underlying results of Hermans and Hill apply to transversely
isotropic fibers, so the Halpin-Tsai equations can also be used in
this case. The Halpin-Tsai equations are known to fit some data
very well at low volume fractions, but to under-predict some
stiffnesses at high volume fractions. This has prompted some
modifications to their model. proposed making a function of vf, and
by curve fitting found that
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Strength of Short fiber composites 47
Chapter 3 METHODOLOGY 48
Stiffness Estimation 49
Strength Estimation 55
Calculation of SFC stiffness for Low aspect ratio PEEK Carbon
Fibre composite
57
Generating Random oriented short fibre composite stiffness from
FE
59
VBA Code for random fiber generation 60
FE Creation and analysis for Fibre volume fraction of 0.2 65
FE Creation and analysis for Fibre volume fraction of 0.15
66
FE Creation and analysis for Fibre volume fraction of 0.1 67
Loads and Boundary Conditions 68
Validation of Stress continuity in FE model 69
Calculation of Elastic Constants from FE Results 70
Chapter 4 RESULT ANALYSIS 72
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Modulus of Elasticity 72
Modulus of Rigidity 73
Strength of Composite 74
Chapter 5 CONCLUSION AND FUTURE SCOPE 75
Work Conclusion 75
Future Scope of Work 75
REFERENCES 76 ANNEXURES (OPTIONAL) PROJECT DETAILS
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CHAPTER 1 2. INTRODUCTION
2.1.Introduction
High strength to weight ratio is always a premium for an
aircraft. The lower the
structural weight the higher payload it can carry with lesser
fuel consumption. This is the scenario in which Light weight alloys
and composites becomes crucial for aerospace industry. Composites
are the most important materials to be adapted for aviation since
the use of aluminium in the 1920s. Composites are multi-phase
materials that are combinations of two or more organic or inorganic
components. One material with continuous phase serves as a
"matrix," which is the material that holds everything together,
while the other material with dispersed phase serves as
reinforcement, in the form of fibres embedded in the matrix. Until
recently, the most common matrix materials were "thermosetting"
materials such as epoxy, bismaleimide, or polyimide. The
reinforcing materials can be glass fibre, boron fibre, carbon
fibre, or other more exotic mixtures. Classification of composites
is shown in Figure 2.1-1.
Figure 2.1-1 Family of Composites (Courtesy
http://nptel.ac.in/)
Even though modern aircrafts like Boeing 787, Airbus A380 and
A350 feature large composite structures, a gap still exists for
metal-replacement of smaller lightly loaded secondary structures
with complex-shaped parts such as structural brackets, fittings or
frames/intercostals , where injection moulding has insufficient
performance but use of traditional continuous fiber composite
materials is typically impractical due to complex component
geometry. Figure 2.1-2 shows some typical aerospace brackets
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which are traditionally made of metallic materials and which can
be replaced by chopped fiber composites.
Figure 2.1-2 Typical aerospace application brackets (Courtesy
http://www.gtweed.com/)
Technologies to produce complex shaped near-net moulded
components for a number of commercial aerospace applications using
chopped fiber composites is under development. It is thus important
to structurally validate the components as experimental data for
physical properties and strength for components made with such
composites are not widely available. For this accurate prediction
of physical properties like strength and stiffness is very
important. It is here that the objective of this project is trying
to bridge the gap.
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2.2.Motivation
Proven experimental data is not available for physical and
structural properties of chopped fiber composites. Empirical,
mathematical & numerical models exist but the application of
the same in aerospace industry is yet to be explored. This project
deals with the use of such models to predict the physical
properties of chopped fiber composites which can be used in FE
simulations to structurally evaluate components manufactured from
such composites.
Identify close form solution for strength and stiffness for
chopped fiber composite with aligned fibers
Identify close form solution for strength and stiffness for
non-aligned fibers and to understand what more is needed for random
orientation.
Generate a VBA code for forming a unit volume of composite with
randomly oriented short fibers
Validate the mechanical properties using FEM for single fiber or
certain fiber combinations with random orientations
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2.3. Organization of the report
This report is organised into 5 Chapters. Chapter 1-Introduction
throws light on the current situation in the industry on short
fibre composites and its importance. Followed by Chapter
2-Literature Survey in which a research on the existing empirical
methods for predicting mechanical properties is pursued. Chapter
3-Methodolgy details the calculations and process followed in the
current thesis work followed by Chapter 4-Results which presents
and analyses the results from the study. This report is closed by
providing the conclusion and the future scope of work in Chapter 5-
Conclusion and Future Scope of Work.
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CHAPTER 2 3. LITERATURE SURVEY
This Chapter discusses the existing theories pertaining to the
physical property prediction of aligned short fibre composite and
compare it with experimental results available in literature for
different combination of fiber and matrix.
3.1.The Need for Predictive Models for Determining Composite
Properties
Mechanical properties of a composite material depend on:
Properties of constituent materials
Orientations of each layer
Volume fractions of each constituent
Thickness of each layer
Nature of bonding between adjacent layers
These properties may be determined by conducting suitable
experiments as per industry standards. However, a specific set of
experiments can only inform us about a specific Fibre/matrix
system. Hence, to design a composite system by tuning its volume
fraction, or fiber/matrix combination, or orientation, then a very
large number of experiments may have to be conducted. Such a
process for material property determination is extremely tedious,
prohibitively expensive, and time consuming. Still further, exact
fibre/matrix combinations may not be always available for testing.
Hence, there is a need for developing mathematical models, which
can reliably predict different thermo-mechanical properties of
composite materials. Such approaches are very useful for engineers
since they provide significant savings in time and cost.
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3.2.Predicting Longitudinal Modulus of Unidirectional Lamina
Consider a unidirectional composite lamina with fibres which are
continuous and uniform in geometric and mechanical properties, and
mutually parallel throughout the length of the lamina. It is also
assumed that the bonding between fibre and matrix is perfect, and
thus strains experienced by fibre (f), matrix (f) and composite (c)
are same in longitudinal direction (1-direction). For such a
composite, when loaded in 1-direction, the total external load Pc
will be shared partly by fibres, Pf, and partly by matrix, Pm. This
is shown in Figure 3.2-1
Figure 3.2-1 Loads on Composite, Fibres, & Matrix in a
Unidirectional Lamina
It is further assumed that fibres and matrix behave elastically.
Thus, the expression for stress in fibres, and matrix can be
written in terms of their moduli (Ef, and Em) and strains as:
f = Ef f Eq 3.2-1
and
m = Em m Eq 3.2-2
Further, if total cross-sectional areas of fibres and matrix are
Af and Am, respectively, then:
Pf = Aff = AfEf f, Eq 3.2-3
and
Pm = Amm= AmEm Eq 3.2-4
Further, we know that load on composite, Pc, is sum of Pf and
Pm. Thus,
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Pc = Acc = Aff + Amm, Eq 3.2-5
or
c = (Af/Ac)f + (Am/Ac)m Eq 3.2-6
However, for a unidirectional composite, Af/Ac and Am/Ac are
volume fractions for fibre and matrix, respectively. Hence,
c = Vff + Vmm = (VfEf + VmEm) Eq 3.2-7
And if Eq 3.2-7 is differentiated with respect to strain (which
is same in fibre and matrix) then,
dc /d = Vf(df /d) + Vm(dm/d), or
Ec = VfEf + VmEm Eq 3.2-8
Equations Eq 3.2-7 and Eq 3.2-8 show that contributions of
fibres and matrix to average composite tensile modulus and stress
are proportionately dependent on their respective volume fractions.
In general, matrix material has a nonlinear stress-strain
response curve. For unidirectional composites having such
nonlinear matrix materials Eq 3.2-7 works well in terms of
predicting their stress-strain. However, the stress-strain response
curve in such materials may not show up as strongly nonlinear,
since
fibres, especially when their volume fractions are high,
dominate their stress-strain response. The higher the fibre volume
fraction, the closer is the stress-strain curve for a
unidirectional lamina to that for the fibre.
Experimental data pertaining to tensile test specimens of lamina
agree very well with Eq 3.2-7 and Eq 3.2-8. However, the results
for compressive tests are not all that agreeable. This is because
fibres under compression tend to buckle, and this tendency is
resisted by matrix material. This is analogous to a structure with
several columns on an elastic foundation. For a unidirectional
composite, the compressive response is strongly dependent on shear
stiffness of matrix material.
Further, Eq 3.2-7 shows us that load shared by fibres may be
increased either by increasing fibre stiffness or by increasing its
volume fraction. However, experimental
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data show that it becomes impractical to aim for fibre volume
fractions in excess of 80% due to issues of poor fibre wetting and
insufficient matrix impregnation between fibres. To predict
longitudinal strength of a unidirectional ply requires one to
understand the nature of deformation of such a ply as load
increases. In general, stress-the stress strain response of
unidirectional plies under tension undergoes four stages of
change.
In first stage, when stresses are small, fibre as well as matrix
materials exhibit elastic behaviour. Subsequently, matrix starts
becoming plastic, while most of the fibres continue to extend
elastically. In the third stage, both fibres and matrix deform
plastically. This may not happen in case of glass or graphite
fibres, as they are brittle in nature. Finally, the fibres fracture
leading to sudden rise in matrix stress, which in turn leads to
overall composite failure. A unidirectional lamina starts to fail
in tension, when its fibres are stretched to their ultimate
fracture strain. Here it is assumed that all of its fibre fails at
the same strain level. If at this stage, the volume fraction of
matrix is below a certain threshold, then it will not be able to
absorb extra stresses transferred to it due to breaking of fibres.
In such a scenario, the entire composite lamina will fail.
Thus, ultimate tensile strength of a unidirectional ply can be
calculated as:
uc = Vfuf + (1-Vf)m Eq 3.2-9
Where uc and uf are ultimate tensile strengths of ply and fibre,
respectively, and m is stress in matrix at a strain level equalling
fracture strain in fibre.
If the fibre volume fraction does not exceed a certain threshold
(Vmin), then even if all the fibres break, the matrix will take the
total load on the composite. In such a condition, the ultimate
tensile strength of composite may be written as:
uc = Vmum= (1-Vf)um Eq 3.2-10
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The relation for Vmin can be developed by equating Eq 3.2-9 And
Eq 3.2-10, replacing Vf by Vmin, and solving for the latter. This
is shown in Eq 3.2-11.
Vmin = (um - m )/(uf + um m) Eq 3.2-11
Further, a well-designed unidirectional lamina requires that its
ultimate tensile strength should exceed that of matrix. This can
happen only when; uc = Vfuf + (1-Vf)m um , Where, um is ultimate
tensile strength of matrix.
This equation is satisfied only if fibre volume fraction exceeds
a certain critical value, which is defined as:
Vcrit = (um - m )/(uf - m ) Eq 3.2-12
Thus, if:
Vf < Vmin, then failure of matrix will coincide with failure
of composite, while fibres will fail prior to failure of
matrix.
Vf = Vmin, then failure of matrix, fibre and composite will
happen at the same time. Vf > Vmin, then failure of fibre, will
immediately lead to failure of matrix as well as of the
composite.
Vf > Vcrit, then failure of fibre will immediately lead to
failure of matrix and also the composite. In such a case the
strength of unidirectional composite will exceed that of
matrix.
3.3.Predicting Transverse Modulus of Unidirectional Lamina
Figure 17.1 shows a simple model for predicting transverse
modulus of unidirectional lamina. Here, the model constitutes of
two slabs of materials, fibre and matrix, of thicknesses tf and tm,
respectively. The overall thickness of composite slab is tc, which
is sum of tf and tm. It may be noted here that these thicknesses of
fibre and matrix are directly proportional to their respective
volume fractions.
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Figure 3.3-1 A Slab Like Model for Predicting Transverse
Properties of Unidirectional Composites
In such a system, externally imposed stress on the composite c)
is assumed to be same as that seen by fibre (f) and also by matrix
(m). This is in contrast to the model developed for predicting
longitudinal modulus, where we had assumed that strains, and not
stresses, in composite, fibre and matrix are equal. Further, in
such a model, which is akin to springs in series, the overall
displacement in composite c) in transverse direction due to
external load is a sum of displacement in fibre (f) and
displacement in matrix (m).
c = f + m Eq 3.3-1
Further, recognizing the relation between strains in each
constituent, and their thicknesses, above equation can be rewritten
as:
c tc = m tm + f tf Eq 3.3-2
Dividing above equation by thickness of composite (tc), and
realizing that tf/tc, and tm/tc equal Vf and Vm, respectively, we
get:
c = m Vm + f Vf
Eq 3.3-3
In linear-elastic range, strain is a ratio of stress and the
modulus. Hence, above equation can be further re-written as:
(c/Ec)= (m/Em)Vm + (f/Ef)Vf Eq 3.3-4
However, we had earlier assumed that externally applied stress
on the composite (c) is same as that seen by fibre (f) and also by
matrix (m). Thus, previous equation can be rewritten as:
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1/Ec= Vm/Em + Vf/Ef Eq 3.3-5
Or alternatively,
Ec = ( EfEm)/([(1-Vf)Ef + VfEm] Eq 3.3-6
Ec = ( EfEm)/([(1-Vf)Ef + VfEm] Eq 3.3-7
Eq 3.3-5 and Eq 3.3-6 give us an estimate for transverse modulus
of unidirectional lamina. The relation shows that a significant
increase in fibre volume fraction is required to raise overall
transverse modulus in moderate amounts. This is in stark contrast
with longitudinal modulus, which is linearly dependent on fibre
volume fraction.
Eq 3.3-5and Eq 3.3-6, even though based on a simple model, is
not borne out well be experimental data. To address this
inconsistency, several alternative models have been developed.
However, in we will use simple and generalized expressions for
transverse modulus developed Halpin and Tsai. These are relatively
simple relations, and hence easy to use in design practice. The
results from Halpin and Tsai are also quite accurate especially if
fibre volume fraction is not too close to unity. As per Halpin and
Tsai, transverse modulus (ET) can be written as:
ET/Em = (1 + Vf)/(1 - Vf) Eq 3.3-8
Where,
= [(Ef/Em) - 1] / [(Ef/Em) + ] Here, is a parameter that
accounts for packing and fibre geometry, and loading condition. Its
values are given below for different fibre geometries. = 2 for
fibres with square and round cross-sections. = 2a/b for fibres with
rectangular cross-section. Here a is the cross-sectional dimension
of fibre in direction of loading, while b is the other dimension of
fibres cross-section.
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3.4.Shear Modulus and Poissons Ratio
A perfectly isotropic material has two fundamental elastic
constants, E and . Its shear modulus and bulk modulus can be
expressed in terms of these two elastic constants. Likewise, a
transversely isotropic composite ply has four elastic constants.
These are:
EL, i.e. elastic modulus in longitudinal direction. ET i.e.
elastic modulus in transverse direction. GLT i.e. longitudinal
shear modulus. LT i.e. Poissons ratio
A detailed discussion on the mathematical logic underlying
existence of these four constants will be conducted in a subsequent
lecture. Till so far, we have developed relations for EL, and ET.
Now we will learn about similar relationships for GLT and LT.
Halpin and Tsai have developed relations similar to Eq 3.3-7 which
can be used predict longitudinal shear modulus, GLT. This is shown
below.
GLT/Gm = (1 + Vf)/(1 - Vf) Eq 3.4-1
Where,
= [(Gf/Gm) - 1] / [(Gf/Gm) + 1] For predicting Poissons ratio
LT, we exploit the fact that a longitudinal tensile strain in fibre
direction, will generate Poisson contraction in transverse
direction in both, matrix and fibre materials. In this context, we
also use the fact that relative strain values for such a
contraction will be proportional to each constituent materials
volume fraction. Thus, overall Poissons ratio LT for the composite
can be written as:
LT = fVf + fVm Eq 3.4-2
3.5. Transverse Strength
We have seen that a unidirectional ply, when put to tension in
fibre direction tends to break at stress values which exceed matrix
tensile strength. This is particularly true when fibre volume
fraction exceeds Vcrit. Similarly, fibres play a central role in
significantly enhancing the stiffness of the ply in fibre
direction, and the overall stiffness of the system tends to far
surpass that of pure matrix. This occurs because
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fibres, which are stronger and stiffer vis--vis matrix, carry a
major portion of external load, thereby enhancing composites
stiffness and strength. However, the same may not be said for a
unidirectional ply loaded in tension in the transverse direction.
This is because load-sharing between fibre and matrix in a
transversely loaded ply is very less. In contrast, the extent of
load sharing between fibre and matrix in a longitudinally loaded
ply is very significant. When a unidirectional load is subjected to
transverse tension, fibres which are far stiffer vis--vis matrix,
act to constrain matrix deformation. Such a constraint on matrix
deformation tends to increase plys transverse modulus, though only
marginally (unless fibre volume fraction is high). The deformation
constraints imposed on matrix by fibres tend to generate strain and
stress concentrations in matrix material. These stress and strain
concentrations cause the matrix to fail at much lesser values of
stress and strain, than a sample of matrix material which has no
fibres at all. Thus, unlike longitudinal strength, transverse
strength tends to get reduced for composites due to presence of
fibres. This reduction in transverse strength of a unidirectional
ply is characterized by a factor, S, the strength-reduction-factor.
The exact value of this factor can be calculated by using a
combination of advanced elasticity formulations and numerical
solution techniques. The strength of unidirectional ply in
transverse direction, uT, can be written as:
uT = uf /S Eq 3.5-1
3.6.About Short-Fibre Composites
It was seen earlier that unidirectional composites tend to be
very stiff and strong in fibre direction, but very weak in the
transverse direction. Their weakness in transverse direction is
attributable to presence of significant stress concentration at the
interface of matrix and fibre. Given these attributes,
unidirectional composites are very useful in applications where
state of stress is well known. In such applications, lamination
sequence of composite can be tailor-made to bear external loads
optimally. However,
if externally applied loads are Omni-directional, or if their
direction can vary in time,
then such laminates fabricated by stacking up unidirectional
laminas may not necessarily meet our design needs.
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We may still be able design a laminate for such cases (that is
when loading is uni-
directional) which is equally strong in all directions, but even
in such a design, the top and bottom layers will be weak in
transverse directions, and failure could get initiated from here.
Hence, in such applications, it is useful to have laminas which
have in
plane isotropy. One way to produce such lamina is by using short
fibres which are randomly oriented. Such composites, in general are
significantly less expensive than unidirectional composites. The
fibre lengths in these are from 1 to 8 cm. Such composites are used
extensively in general purpose applications, such as car body
panels, boats, household goods, etc. In most of such applications,
glass fibre is used as the reinforcing material for matrix.
In composite materials, fibres are invariably surrounded by
matrix material. Hence, external load is directly applied to
matrix, and from here, it gets transferred to fibres. A part of
this load gets transferred to fibres at their ends, while remaining
portion of this load gets transferred to fibres through their
external cylindrical surfaces. For unidirectional composites with
continuous fibres, transfer of load at fibre ends may be very small
vis
-
-
vis load transfer through fibres external surface. This is
because fibres are very long, and hence their cylindrical surfaces,
across which load gets transferred through shear
-
mechanism, are sufficiently long. In such fibres, the effect of
load transfer through fibre ends may not significantly affect
overall mechanics of load transfer. However, in short
-
fibre composites the same may not be necessarily true. In such
composites, the length of the fibre is not sufficiently long such
that much of load transfer happens across cylindrical surfaces of
fibres. Thus, in such fibres, both the ends, as well as external
cylindrical surfaces of fibres play a significant role in
matrix
-
to-
fibre load transfer. Hence, it is important to understand role
of end-
effects in
context of load transfer to fibres. Without this understanding,
our understanding of reinforcing effects in short
-
fibre composites will be inaccurate and flawed.
Consider a short-
fibre of length l embedded in matrix which is shown in Figure
3.6-1. The figure also shows the details of an infinitesimal
portion of fibre of length dz, which experiences normal stress in
length direction, and shear stress, , along its cylindrical
surface. Please note that while normal stress at one end of
infinitesimal fibre is f, it is f +df at its other end. This
variation in normal stress along the length
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of infinitesimally long fibre is because some of the load gets
transferred from matrix to fiber due to application of shear stress
on its cylindrical surface.
Figure 3.6-1 Force Equilibrium of an Infinitesimal Portion of
Discontinuous Fiber which is aligned to External Load
From principles of static equilibrium, equation of force
equilibrium for this infinitesimally sized portion of fiber. r2f +
(2 r dz)f = r2(f + df ) Cancelling out term r2f from both sides,
and rearranging remaining terms : df /dz = 2/r Integrating above
equation yields, f = fo+ (2/r) dz, where the integral limits are 0
to z. Quite often, fiber separates from the matrix due to presence
of large stress concentration. In other cases, matrix yields at the
fiber end. The implication of either case is that the integration
constant for above equation, fo, is zero. Thus, above equation can
be rewritten as: f = (2/r) dz The integral equation shown earlier
can be evaluated if variation of shear stress, , with respect to
coordinate z, is known. At this point, an assumption is made that
the shear stress at the interface of fiber and matrix is constant
along fiber length, and equals matrix yield shear, i.e. y. Such an
assumption may be made for a system
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where matrix material transmits maximum possible stress to
fiber, which would be y. For such a case, the integral equation can
be simplified as:
f = 2y z/r Eq 3.6-1
For short fibers, maximum fiber stress is expected to occur at
mid-
length, i.e. z = l/2, while it will be zero at its extremities
for reasons explained earlier. Hence, the equation written above
will hold good only for values of z = 0 to l/2, and for the region
z = l/2 to l, the equation will have to have a negative slope.
Further, the
maximum value of fiber stress will be, as per above
equation:
f_max = y l/r, corresponding to z = l/2 Eq 3.6-2
Eq 3.6-1 and Eq 3.6-2 place no limit on the upper bound for
fiber stress, and can approach very large values if l is made very
large. However, in reality there will indeed be a limit, which will
correspond to the stress borne by continuous and infinitely long
fibers in unidirectional plies. This stress, as calculated earlier
is Ef/Emc. Equating this value to maximum fiber stress in
short-fiber (as per Eq 3.6-2) gives us a load
-
transfer length, lt, which is required to achieve maximum
possible stress in fiber. This is shown below.
f_max = y lt/r = Ef/Emc Eq 3.6-3
f_max = y lt/r = Ef/Emc Eq 3.6-4
Or,
lt/r = f_max /y = (Ef/Emc) / y Eq 3.6-5
Thus, a fiber which is at least lt long develops maximum fiber
stress (Ef/Emc) as defined earlier, when the externally applied
stress is c.
Hence, if we increase external stress c, we will have to
increase lt to ensure
maximum load in fiber, as f_max, which equals Ef/Emc, will also
increase. But, there
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is a limit beyond which external stress c cannot be increased.
This limit corresponds to a point when the stress in fiber equals
its ultimate strength (uf), At this limit, any further stress in
external stress will lead to failure of fiber Thus, the condition
for maximum possible stress in fiber is:
f_max = uf = Ef/Emc Eq 3.6-6
For such a limiting stress, there is a corresponding minimum
fiber length which is required to support such a level of stress.
Mathematically, the value of minimum fiber length can be calculated
from Eq 3.6-5 and is given below.
lmin/r = uf/y Eq 3.6-7
Thus, any design of a short
fiber composite should ensure that its fiber is at least lmin
long, because in such a system the overall composite strength will
be maximized. If fibers are shorter than this critical length, then
composite strength would not be at its maximum value, thereby
adding weight and cost to the structure. Finally, if l is very
large compared to lmin, then composite increasingly behaves as one
with continuous fibers.
Till so far, It was assumed that the matrix material in
fiber-matrix interface region is perfectly plastic. This is not
entirely true. In reality, most matrix materials exhibit
elasto-plastic behaviour. Developing analytical solutions for such
systems is not easy. Hence, numerical methods may be used to solve
such problems to get better understanding of load transfer
mechanisms in short-fiber composites. Several such studies have
shown that load transfer at fiber ends is not significant, and
hence our earlier assumption of fo being zero, stands validated,
though in an approximate sense.
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Figure 3.6-2 is a plot of variation of fiber strength for three
different fiber lengths.
Following observations can be made from Figure 3.6-2. If fiber
length is less than lt, then the normal stress in fiber is zero at
either ends of fiber, and it reaches a peak value at mid-fiber
length. In such a case, the longer the fiber, the higher is the
value of peak normal stress which occurs at its mid-length. If
fiber length equals lt, then normal stress in fiber gets maximized.
However, the shape of stress plot still remains triangular.
Finally, if fiber length exceeds lt, then normal stress in fiber:
Rises from zero to a maximum value over part of the fiber length.
Remains constant once it has maximized. Falls back to zero, over
remaining part of fiber length. Utilization of fiber strength is
maximized in the third configuration.
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3.7.Modulus of Short-Fiber Composites
There are many models to predict stiffness of uniaxial short
fiber composites. In selecting models for consideration, we impose
the general requirements that each model must include the effects
of fiber and matrix properties and the fiber volume fraction,
include the effect of fiber aspect ratio, and predict a complete
set of elastic constants for the composite. Any model not meeting
these criteria was excluded from consideration. All of the models
use the same basic assumptions:
The fibers and the matrix are linearly elastic, the matrix is
isotropic, and the fibers are either isotropic or transversely
isotropic. .
The fibers are axisymmetric, identical in shape and size, and
can be characterized by an aspect ratio l/d
The fibers and matrix are well bonded at their interface, and
remain that way during deformation. Thus, we do not consider
interfacial slip, fiber/matrix debonding or matrix
micro-cracking.
3.7.1. Eshelby's equivalent inclusion
A fundamental result used in several different models is
Eshelby's equivalent inclusion (Eshelby, 1959) & (Eshelby,
1961). Eshelby solved for the elastic stress field in and around an
ellipsoidal particle in an infinite matrix. By letting the particle
be a prolate ellipsoid of revolution, one can use Eshelby's result
to model the stress and strain fields around a cylindrical fiber.
Eshelby first posed and solved a different problem, that of a
homogeneous inclusion Figure 3.7-1. Consider an in finite solid
body with stiffness Cm that is initially stress-free. All
subsequent strains will be measured from this state. A particular
small region of the body will be called the inclusion, and the rest
of the body will be called the matrix. Suppose that the inclusion
undergoes
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Figure 3.7-1 Eshelby's inclusion problem.
Starting from the stress-free state (a), the inclusion undergoes
a stress-free transformation strain T(b). Fitting the inclusion and
matrix back together (c) produces the strain state C(x) in both the
inclusion and the matrix.
Some type of transformation such that, if it were a separate
body, it would acquire a uniform strain T with no surface traction
or stress. T is called the transformation strain, or the eigen
strain. This strain might be acquired through a phase
transformation, or by a combination of a temperature change and a
different thermal expansion coefficient in the inclusion. In fact
the inclusion is bonded to the matrix, so when the transformation
occurs the whole body develops some complicated strain field C(x)
relative to its shape before the transformation. Within the matrix
the stress is simply the stiffness times this strain,
m(x)= Cm C(x) Eq 3.7.1-1
But within the inclusion the transformation strain does not
contribute to the stress, so the inclusion stress is
I =Cm (C-T) Eq 3.7.1-2
The key result of Eshelby was to show that within an ellipsoidal
inclusion the strain C is uniform, and is related to the
transformation strain by
C= ET Eq 3.7.1-3
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E is called Eshelby's tensor, and it depends only on the
inclusion aspect ratio and the matrix elastic constants. Mura gave
a detailed derivation and applications (Mura, 1982) , and
analytical expressions for Eshelby's tensor for an ellipsoid of
revolution in an isotropic matrix appear in many papers. The strain
field C(x) in the matrix is highly non-uniform (Eshelby, 1959), but
this more complicated part of the solution can often be ignored.
The second step in Eshelby's approach is to demonstrate equivalence
between the homogeneous inclusion problem and an inhomogeneous
inclusion of the same shape. Consider two infinite bodies of
matrix, as shown in Fig. 2. One has a homogeneous inclusion with
some transformation strain T. The other has an inclusion with a
different stiffness Cf , but no transformation strain. Subject both
bodies to a uniform applied strain A at infinity. We wish to find
the transformation strain T that gives the two problems the same
stress and strain distributions.
Figure 3.7-2 Eshelby's equivalent inclusion problem.
The inclusion (a) with transformation strain T has the same
stress T and strain as the in- homogeneity (b) when both bodies are
subject to a far-field strain A:
For the first problem the inclusion stress is just Eq 3.7.1-2
with the applied strain added,
I = Cm (A + C - T ) Eq 3.7.1-4
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While the second problem has no T but a different stiffness,
giving a stress of
I = Cf (A + C ) Eq 3.7.1-5
Equating these two expressions gives the transformation strain
that makes the two
problems equivalent. Using Eq 3.7.1-3 and some rearrangement,
the result is
-[ Cm + (Cf - Cm ) E] T = (Cf - Cm ) A Eq 3.7.1-6
Note that T is proportional to A, which makes the stress in the
equivalent
inhomogeneity proportional to the applied strain.
3.7.2. Dilute Eshelby model
One can use Eshelby's result to find the stiffness of a
composite with ellipsoidal fibers at dilute concentrations. To find
the stiffness one only has to find the strain-concentration tensor
A. To do this, first note that for a dilute composite the average
strain is identical to the applied strain,
^ = A Eq 3.7.2-1
Since this is the strain at infinity. Also, from Eshelby, the
fiber strain is uniform, and is given by
^f = A + C Eq 3.7.2-2
Where, the right-hand side is evaluated within the fiber. Now
write the equivalence between the stresses in the homogeneous and
the inhomogeneous inclusions, Eq 3.7.1-4 and Eq 3.7.1-5,
Cf (A + C) = Cm (A + C - T ) Eq 3.7.2-3
Then use Eq 3.7.1-3, Eq 3.7.2-1 and Eq 3.7.2-2to eliminate T, A
and C from this equation, giving
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[I + ESm (Cf - Cm)] ^f = ^ Eq 3.7.2-4
Comparing this to
f = A Eq 3.7.2-5
Shows that the strain-concentration tensor for Eshelby's
equivalent inclusion is
AEshelby = [I + ESm (Cf - Cm)]-1 Eq 3.7.2-6
This can be used in
C= Cm + vf (Cf- Cm )A Eq 3.7.2-7
To predict the moduli of aligned- fiber composites, a result
(Russel, 1973) was developed. Calculations using this model to
explore the effects of particle aspect ratio on stiffness (Chow,
1977) were also presented. While Eshelby's solution treats only
ellipsoidal fibers, the fibers in most short- fiber composites are
much better approximated as right circular cylinders. In their
paper the relationship between ellipsoidal and cylindrical
particles (Steif & Hoysan, 1987) was considered , who developed
a very accurate finite element technique for determining the
stiffening effect of a single fiber of given shape. For very short
particles, l/d = 4, they found reasonable agreement for E11 by
letting the cylinder and the ellipsoid have the same l/d. The
ellipsoidal particle gave a slightly stiffer composite, with the
difference between the two results increasing as the modulus ratio
Ef = Em increased. Henceforth we will use the cylinder aspect ratio
in place of the ellipsoid aspect ratio in Eshelby-type models.
Because Eshelby's solution only applies to a single particle
surrounded by an infinite matrix, AEshelby is independent of fiber
volume fraction and the stiffness predicted by this model increases
linearly with fiber volume fraction. Modulus predictions based on
Eq 3.7.2-5 and Eq 3.7.2-7 should be accurate only at low volume
fractions, say up to vf of 1%. The more difficult problem is to
find some way to include interactions between fibers in the model,
and so produce accurate results at higher volume fractions. We next
consider approaches for doing that.
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3.7.3. Mori-Tanaka Model
A family of models for non-dilute composite materials (Mori
& Tanaka, 1973) had evolved. Later a particularly simple and
clear explanation (Benveniste, 1987) of the Mori-Tanaka approach
was given to introduce the approach. Suppose that a composite is to
be made of a certain type of reinforcing particle, and that, for a
single particle in an infinite matrix, we know the dilute
strain-concentration tensor AEshelby
f=AEshelby^ Eq 3.7.3-1
The Mori-Tanaka assumption is that, when many identical
particles are introduced in the composite, the average fiber strain
is given by
f =AEshelby^m Eq 3.7.3-2
That is, within a concentrated composite each particle sees a
far-field strain equal to the average strain in the matrix. Using
the alternate strain concentrator defined in eqn
f = Am Eq 3.7.3-3
The Mori-Tanaka assumption can be re-stated as
A^MT=AEshelby Eq 3.7.3-4
Then
A=A^ [(1-vf) I + vf A^]-1 Eq 3.7.3-5
Gives the Mori-Tanaka strain concentrator as
AMT=AEshelby[(1-vf) I + vf A]-1 Eq 3.7.3-6
This is the basic equation for implementing a Mori-Tanaka model.
The Mori-Tanaka approach for modelling composites was first
introduced by Wakashima et. al (Wakashima, Umekawa, & Otsuka,
1974) for modelling thermal expansions of composites with aligned
ellipsoidal inclusions. Mori and Tanaka treat only the homogeneous
inclusion problem (Mori & Tanaka, 1973), and say nothing
about
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composites. Mori-Tanaka predictions for the longitudinal modulus
of a short-fiber composite were again developed to also include the
effects of cracks and of a second type of reinforcement by (Taya
& Mura, 1981) and (Taya & Chou, 1981). Their method was
generalised (Weng, 1984), lead to the usage of the Mori-Tanaka
approach to develop equations for the complete set of elastic
constants of a short-fiber composite (Tandon & Weng, 1984).
Tandon andWengs equations for the plane-strain bulk modulus k23and
the major Poisson ratio 12 must be solved iteratively. The usual
development of the Mori-Tanaka model differs somewhat from
Benvenistes explanation. For an average applied stress , the
reference strain 0 is defined as the strain in a homogeneous body
of matrix at this stress,
= Cm 0 Eq 3.7.3-7
Within the composite the average matrix strain differs from the
reference strain by some perturbation
= 0+ Eq 3.7.3-8
A fiber in the composite will have an additional strain
perturbation ~ f , such that
= + + Eq 3.7.3-9
While the equivalent inclusion will have this strain plus the
transformation strain T.
The stress equivalence between the inclusion and the fiber then
becomes
( + + ) = ( + + ) Eq 3.7.3-10
Compare this to the dilute version, Eq 3.7.2-3, noting that A in
the dilute problem is
equivalent to (0 + ) here. The development is completed by
assuming that the extra fiber perturbation is related to the
transformation strain by Eshelbys tensor,
= Eq 3.7.3-11
Combining this with Eq 3.7.3-8 and Eq 3.7.3-9 reveals that Eq
3.7.3-11 contains the essential Mori-Tanaka assumption: the fiber
in a concentrated composite sees the
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average strain of the matrix. Some other micromechanics models
are equivalent to the Mori-Tanaka approach, though this equivalence
has not always been recognized. Chow considered Eshelbys inclusion
problem and conjectured that in a concentrated composite the
inclusion strain would be the sum of two terms: the dilute result
given by Eshelby and the average strain in the matrix (Chow,
1978).
() = + () Eq 3.7.3-12
This can be combined with the definition of the average strain
from eqn (7) to relate the inclusion strain ( C)f to the
transformation strain T
() = (1 ) Eq 3.7.3-13
Chow then extended this result to an inhomogeneity following the
usual arguments, Eq 3.7.1-4 to Eq 3.7.2-6. This produces a
strain-concentration tensor
! = [# + $1 %& ' (]*+ Eq 3.7.3-14
Which is equivalent to the Mori-Tanaka result Eq 3.7.3-6. Chow
was apparently unaware of the connection between his approach and
the Mori-Tanaka scheme, but he seems to have been the first to
apply the Mori-Tanaka approach to predict the stiffness of
short-fiber composites. A more recent development is the equivalent
poly-inclusion model (Ferrari, 1994). Rather than use the
strain-concentration tensor A, Ferrari used an effective Eshelby
tensor ,, defined as the tensor that relates inclusion strain to
transformation strain at finite volume fraction:
() = , Eq 3.7.3-15
Once ,has been defined, it is straightforward to derive a
strain-concentration tensor A and a composite modulus. Ferrari
considered admissible forms for ,, given the requirements that
,must (a) produce a symmetric stiffness tensor C, (b) approach
Eshelbys tensor E as volume fraction approaches zero, and (c) give
a composite stiffness that is independent of the matrix stiffness
as volume fraction approaches unity. He proposed a simple form that
satisfies these criteria,
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- = $1 % Eq 3.7.3-16
The combination of Eq 3.7.3-15 and Eq 3.7.3-16 is identical to
Chows assumption Eq 3.7.3-15 and, for aligned fibers of uniform
length, Ferraris equivalent poly-inclusion model, Chows model, and
the Mori- Tanaka model are identical. Important differences between
the equivalent poly-inclusion model and the Mori-Tanaka model arise
when the fibers are not oriented or have different lengths.
3.7.4. Self-Consistent Models A second approach to account for
finite fiber volume fraction is the self-consistent method (Hill,
1965) and (Budiansky, 1965). The original work focused on spherical
particles and continuous, aligned fibers. The application to
short-fiber composites was developed (Laws & McLaughlin, 1979)
and (Chou, Nomura, & Taya, 1980). In the self-consistent scheme
one finds the properties of a composite in which a single particle
is embedded in an infinite matrix that has the average properties
of the composite. For this reason, self-consistent models are also
called embedding models. Again building on Eshelbys result for a
ellipsoidal particle, we can create a self-consistent version of Eq
3.7.2-6 by replacing the matrix stiffness and compliance tensors by
the corresponding properties of the composite. This gives the
self-consistent strain-concentration tensor as
. = [# + &$ %]*+ Eq 3.7.4-1
Of course the properties C and S of the embedding matrix are
initially unknown. When the reinforcing particle is a sphere or an
infinite cylinder, the equations can be manipulated algebraically
to find explicit expressions for the overall properties. For short
fibers this has not proved possible, but numerical solutions are
easily obtained by an iterative scheme. One starts with an initial
guess at the composite properties, evaluates E and then ASC from Eq
3.7.4-1, and substitutes the result into Eq 3.7.2-7 to get an
improved value for the composite stiffness. The procedure is
repeated using this new value, and the iterations continue until
the results for C converge. An additional, but less obvious, change
is that Eshelbys tensor E depends on the matrix properties, which
are now transversely isotropic. Expressions for Eshelbys tensor
for
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an ellipsoid of revolution in a transversely isotropic matrix
were given in (Chou, Nomura, & Taya, 1980) and (Lin & Mura,
1973). With these expressions in hand one can use Eq 3.7.4-1
together with Eq 3.7.2-7 to find the stiffness of the composite.
This is the self-consistent approach used for short-fiber
composites. A closely-related approach, called the generalized
self-consistent model, also uses an embedding approach. However, in
these models the embedded object comprises both fiber and matrix
material. When the composite has spherical reinforcing particles,
the
embedded object is a sphere of the reinforcement encased in a
concentric spherical shell of matrix; this is in turn surrounded by
an infinite body with the average composite properties. The
generalized self-consistent model is sometimes referred to as a
double embedding approach. For continuous fibers the embedded
object is a cylindrical fiber surrounded by a cylindrical shell of
matrix. The first generalized self-consistent models were developed
for spherical particles (Kerner, 1956), and for cylindrical fibers
(Hermans, 1967). Both of these papers contain an error, which was
discussed and corrected later (Christensen & Lo, 1979). While
the generalized self-consistent model is widely regarded as
superior to the original self-consistent approach, no such model
has been developed for short fibers.
3.7.5. Bounding Models
A rather different approach to modelling stiffness is based on
finding upper and lower bounds for the composite moduli. All
bounding methods are based on assuming an approximate field for
either the stress or the strain in the composite. The unknown field
is then found through a variational principle, by minimizing or
maximizing some functional of the stress and strain. The resulting
composite stiffness is not exact, but it can be guaranteed to be
either greater than or less than the actual stiffness, depending on
the variational principle. This rigorous bounding property is the
attraction of bounding methods. Historically, the Voigt and Reuss
averages were the first models to be recognized as providing
rigorous upper and lower bounds. To derive the Voigt model,
/ 012 = + $ % = + Eq 3.7.5-1
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One assumes that the fiber and matrix have the same uniform
strain, and then minimizes the potential energy. Since the
potential energy will have an absolute minimum when the entire
composite is in equilibrium, the potential energy under the uniform
strain assumption must be greater than or equal to the exact
result, and the calculated stiffness will be an upper bound on the
actual stiffness. The Reuss model,
&34566 =& + $& &% = & + & Eq 3.7.5-2
is derived by assuming that the fiber and matrix have the same
uniform stress, and then maximizing the complementary energy. Since
the complementary energy must be a maximum at equilibrium, the
model provides a lower bound on the composite stiffness. Detailed
derivations of these bounds are provided in (Wu C & McCullough,
1977). The Voigt and Reuss bounds provide isotropic results
(provided the fiber and matrix are themselves isotropic), when in
fact we expect aligned-fiber composites to be highly anisotropic.
More importantly, when the fiber and matrix have substantially
different stiffness then the Voigt and Reuss bounds are quite far
apart, and provide little useful information about the actual
composite stiffness. This latter point motivated Hashin and
Shtrikman to develop a way to construct tighter bounds. Hashin and
Shtrikman developed an alternate vibrational principle for
heterogeneous materials (Hashin & Shtrikman, 1963). Their
method introduces a reference material, and bases the subsequent
development on the differences between this reference material and
the actual composite. Rather than requiring two variation
principles, like the Voigt and Reuss bounds, their single variation
principle gives both the upper and lower bounds by making
appropriate choices of the reference material. For an upper bound
the reference material must be as stiff or stiffer than any phase
in the composite (fiber or matrix), and for a lower bound the
reference material must have a stiffness less than or equal to any
phase. In most composites the fiber is stiffer than the matrix, so
choosing the fiber as the reference material gives an upper bound
and choosing the matrix as the reference material gives a lower
bound. If the matrix is stiffer than the fiber, the bounds are
reversed. The resulting bounds are tighter than the Voigt and Reuss
bounds, which can be obtained from the Hashin-Shtrikman theory by
giving the reference material infinite or zero stiffness,
respectively. Hashin and Shtrikmans original bounds apply to
isotropic composites with isotropic constituents. Frequently the
bounds are regarded as applying to composites with spherical
particles, orientation
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must also obey the bounds. Walpole re-derived the
Hashin-Shtrikman bounds using classical energy principles (Walpole,
1966), and extended them to anisotropic materials (Walpole, 1966).
Walpole also derived results for infinitely long fibers and
infinitely thin disks in both aligned and 3-D random orientations
(Walpole, 1969). The Hashin-Shtrikman-Walpole bounds were extended
to short-fiber composites in (Willis, 1977) and (Wu C &
McCullough, 1977). These workers introduced a two-point correlation
function into the bounding scheme, allowing aligned ellipsoidal
particles to be treated. Based on these extensions, explicit
formulae for aligned ellipsoids were developed in (Weng, 1992) and
(Eduljee, McCullough, & Gillespie, 1994). The general bounding
formula, shown here in the format developed by Weng, gives the
composite stiffness C as
= 78 + 89[8 + 8]*+ Eq 3.7.5-3
Where the tensors Qf and Qm are defined as
8 = [# + 0&0 ' 0(]1:;N ('Y>N ( ]
Eq 3.7.6-3
With
ZN = STD++N Eq 3.7.6-4
It is convenient to rewrite this as an expression for the
average fiber strain,
++ = [>++ Eq 3.7.6-5
where l is a length-dependent efficiency factor,
[> = [1 W:; 'Y>N ('Y>N ( ] Eq 3.7.6-6
Note that l is a scalar analog of the strain-concentration
tensor A defined in Eq 3.7.2-5, and (1/) is a characteristic length
for stress transfer between the fiber and the matrix.
Table 3.7.6-1Values for KR used in Eq 3.7.6-8 for shear lag
models
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Fiber Packing KR
Cox 2/3=3.628 Composite Cylinders 1 Hexagonal
/23=0.907 Square /4=0.785
It was found that the coefficient H by solving a second
idealized problem (Cox, 1952). The concentric cylinder geometry is
maintained, but the outer cylindrical surface of the matrix is held
stationary and the inner cylinder, which is now rigid, is subjected
to a uniform axial displacement. An elasticity solution for the
matrix layer then gives
S = 2T^@;(3?K)
Eq 3.7.6-7
This part of the problem was simplified by assuming that the
matrix shell was thin compared to the fiber radius, (R- rf)
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1964), and later (Carman & Reifsnider, 1992) so that the
concentric cylinder model in Figure 3.7-3 would have the same fiber
volume fraction as the composite. This is the same R as the
composite cylinders model explained in (Hashin & Rosen, 1964).
More recently, a square array of fibers was assumed (Robinson &
Robinson, 1994), and chose R as half the distance between centres
of nearest neighbours (Figure 3.7-4 c). Each of these choices gives
a somewhat different dependence of l on fiber volume fraction, with
larger values of KR producing lower values of E11. Shear lag models
are usually completed by combining the average fiber stress in Eq
3.7.6-3 with an average matrix stress to produce a modified rule of
mixtures for the axial modulus:
++ = [>c + $1 c% Eq 3.7.6-10
Figure 3.7-4 Fiber packing arrangements used to find R in shear
lag models. (a) Hexagonal (Cox, 1952). (b) Hexagonal (Rosen, 1964)
(c) Square (Robinson & Robinson, 1994).
However, the matrix stress in this formula is not consistent
with the basic concepts of average stress and average strain. Note
that
= c + c Eq 3.7.6-11 must hold for 11, as for any other component
of strain. Combining this with Eq 3.7.6-5 to find the average
matrix strain, and following through to find the composite
stiffness (with Poisson effects neglected), gives a result that is
consistent with both the assumptions of shear lag theory and the
basic concepts of average stress and strain:
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++ = [>c + $1 [>c% = + c( )[>
Eq 3.7.6-12
This equation is an exact scalar analog of the general tensorial
stiffness formula, Eq
3.7.2-7. For the cases in this paper, the difference between Eq
3.7.6-10 and Eq 3.7.6-12 is small. A model by (Fukuda & Kawata,
1974) for the axial stiffness of aligned short-fiber composites is
closely related to shear lag theory. They begin with a 2-D
elasticity solution for the shear stress around a single slender
fiber in an infinite matrix. The usual shear lag relation, Eq
3.7.6-1, is used to transform this into an equation for the fiber
stress distribution, which is then approximated by a Fourier
series. The coefficients of a truncated series are evaluated
analytically using Galerkins method. This is a dilute theory, in
which modulus varies linearly with fiber volume fraction.
Like any shear lag theory, Fukuda and Kawatas theory predicts
that E11 approaches the rule of mixtures result as the fiber aspect
ratio approaches infinity. But for short fibers Fukuda and Kawatas
theory gives much lower E11 values than shear lag theory. In Fukuda
and Kawatas theory, the ratio of fiber strain to matrix strain is
governed by the parameter (l/d)(Em/Ef). In contrast, for shear lag
theory, Eq 3.7.6-6, the governing parameter is l/2, which is
proportional to (l/d)(Em/Ef). Thus, for high modulus ratio and low
aspect ratio, Fukuda and Kawatas theory tends to under predict
E11.
3.7.7. Halpin-Tsai Equations
The Halpin-Tsai equations (Ashton, Halpin, & Petit, 1969)
have long been popular for predicting the properties of short-fiber
composites. A detailed review and derivation is provided by (Halpin
& Kardos, The Halpin-Tsai Equations: A Review, 1976), from
which the main points are summarized. The Halpin-Tsai equations
were originally developed with continuous-fiber composites in mind,
and were derived from the work of (Hermans, 1967) and (Hill, 1964).
Hermans developed the first generalized self-consistent model for a
composite with continuous aligned fibers (see Section 3.7.4).
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Halpin and Tsai found that three of Hermans equations for
stiffness could be expressed in a common form:
= 1 + d[1 [ BeW[ ='fKfg( 1'fKfg( + 1
Eq 3.7.7-1
Here P represents any one of the composite moduli listed in
Table 1, and Pf and Pm are the corresponding moduli of the fibers
and matrix, while is a parameter that depends on the matrix Poisson
ratio and on the particular elastic property being considered.
Hermans derived expressions for the plane-strain bulk modulus k23,
and for the longitudinal and transverse shear moduli G12and G23.
The parameters for these properties are given in Table 1. Note that
for an isotropic matrix
Table 3.7.7-1 Correspondence between Halpin-Tsai Eq 3.7.7-1 and
generalized self-consistent predictions of (Hermans, 1967) and
(Kerner, 1956). After (Halpin & Kardos, The Halpin-Tsai
Equations: A Review, 1976)
P Pf Pm Comments
k23 kf km 1 2N1 + Plane strain bulk modulus, aligned fibers G23
Gf Gm 1 + 3 4N Transverse shear modulus, aligned fibers G12 Gf Gm 1
Longitudinal shear modulus, aligned fibers
K Kf Km 2(1 2)1 + Bulk modulus, particulates G Gf Gm 7 5)8 10
Shear modulus, particulates
(Hill, 1964) showed that for a continuous, aligned-fiber
composite the remaining stiffness parameters are given by
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++ = + 4 l +mK +mg nNo 1pNq p pr Eq 3.7.7-2
+N = + + l +mK +mg n o1pNq p pr Eq 3.7.7-3
This completes Hermans model for aligned-fiber composites; note
that one must know k23 to find E11and 12. We now know that Hermans
result for G23 is incorrect, in that it does not satisfy all of the
fiber/matrix continuity conditions (Hashin, 1983). It is, however,
identical to a lower bound on G23 derived by (Hashin, 1965).
Hermans remaining results are identical to Hashin and Rosens
composite cylinders assemblage model (Hashin & Rosen, 1964), so
Hermans k23, and thus his E11 and 12, are identical to the
self-consistent results of (Hill, 1965).
The Halpin-Tsai form can also be used to express equations for
particulate composites derived by (Kerner, 1956), who also used a
generalized self-consistent model. Table 3.7.7-1 gives the details.
Kerners result for shear modulus G is also known to be incorrect,
but reproduces the Hashin- Shtrikman-Walpole lower bound for
isotropic composites, while Kerners result for bulk modulus K is
identical to Hashins composite spheres assemblage model (Hashin,
1962). See (Christensen & Lo, 1979) and (Hashin, 1983) for
further discussion of Kerners and Hermans results. To transform
these results into convenient forms for continuous-fiber
composites, Halpin and Tsai made three additional ad hoc
approximations:
Eq 3.7.7-1 can be used directly to calculate selected
engineering constants, with E11or E22 replacing P.
The parameters in Table 3.7.7-1 are insensitive to m, and can be
approximated by constant values.
The underlined terms in Eq 3.7.7-2 and Eq 3.7.7-3 can be
neglected.
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Table 3.7.7-2 Traditional Halpin-Tsai parameters for short-fiber
composites, used in Eq 3.7.7-1. For G23 see Table 3.7.7-1.
P Pf Pm Comments
E11 Ef Em 2(l/d) Longitudinal modulus
E22 Ef Em 2 Transverse modulus
G12 Gf Gm 1 Longitudinal shear modulus
12 Poisons Ratio = f f + m m
In Eq 3.7.7-2 the underlined term is typically negligible, and
dropping it gives the familiar rule of mixtures for E11of a
continuous-fiber composite. However, dropping the underlined term
in Eq 3.7.7-3 and using a rule of mixtures for 12 is not
necessarily accurate if the fiber and matrix Poisson ratios differ.
Halpin and Tsai argue for this latter approximation on the grounds
that laminate stiffnesses are insensitive to 12. In adapting their
approach to short-fiber composites, Halpin and Tsai noted that must
lie between 0 and . If =0 then Eq 3.7.7-1 reduces to the inverse
rule of mixtures (Halpin & Kardos, The Halpin-Tsai Equations: A
Review, 1976), 1 = + Eq 3.7.7-4
while for = the Halpin-Tsai form becomes the rule of
mixtures,
= + Eq 3.7.7-5 Halpin and Tsai suggested that was correlated
with the geometry of the reinforcement and, when calculating E11,
it should vary from some small value to infinity as a function of
the fiber aspect ratio l/d. By comparing model predictions with
available 2-D finite element results, they found that =2(l/d) gave
good predictions for E11of short-fiber systems. Also, they
suggested that other engineering constants of short-fiber
composites were only weakly dependent on fiber aspect ratio, and
could be approximated using the continuous-fiber formulae (Halpin,
1969). The resulting equations are summarized in Table 3.7.7-2. The
early references (Ashton,
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Halpin, & Petit, 1969) and (Halpin, 1969) do not mention
G23. When this property is needed the usual approach is to use the
value given in Table 3.7.7-1. While the Halpin-Tsai equations have
been widely used for isotropic fiber materials, the underlying
results of Hermans and Hill apply to transversely isotropic fibers,
so the Halpin-Tsai equations can also be used in this case. The
Halpin-Tsai equations are known to fit some data very well at low
volume fractions, but to under-predict some stiffnesses at high
volume fractions. This has prompted some modifications to their
model. (Hewitt & Malherbe, 1970) proposed making a function of
vf, and by curve fitting found that
d = 1 + 40+ Eq 3.7.7-6 This gave good agreement with 2-D finite
element results for G12 of continuous fiber composites. (Lewis
& Nielsen, 1970) & (Nielsen, 1970) focused on the analogy
between the stiffness G of a composite and the viscosity of a
suspension of rigid particles in a Newtonian fluid, noting that one
should find / m = G / Gm when the reinforcement is rigid (Gf/Gm)
and the matrix is incompressible. They developed an equation in
which the stiffness not only matches dilute theory at low volume
fractions, but also displays G/Gm) as vf approaches a packing limit
vfmax. This leads to a modified Halpin-Tsai form
= 1 + d[1 s()d[ Eq 3.7.7-7
with retaining its definition from Eq 3.7.7-1. Here the function
(vf) contains the maximum volume fraction vfmax as a parameter. is
chosen to give the proper
behaviour at the upper and lower volume fraction limits, which
leads to forms such as
s$% = 1 + o1 tutuN r Eq 3.7.7-8
s$% = 1 v1 Cwx o 1 (/tu)rz Eq 3.7.7-9
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The Nielsen and Lewis model improves on the Halpin-Tsai
predictions, compared to experimental data for G of
particle-reinforced polymers (Lewis & Nielsen, 1970) and to
finite element calculations for G12 of continuous-fiber composites
(Nielsen, 1970), using vfmax values from 0.40 to 0.85.
Recently (Ingber & Papathanasiou, 1997) tested the
Halpin-Tsai equation and its modifications against boundary element
calculations of E11 for aligned short fibers. They found the
Nielsen modification to be better than the original Halpin-Tsai
form. Hewitt and deMalherbes form could be adjusted to fit data for
any single l/d, but was not useful for predictions over a range of
aspect ratios.
3.7.8. Fiber Efficiency Factor Approach (Blumentritt, VU, &
Cooper, 1975) proposed a method to calculate the ultimate
strength and the Youngs modulus of the composite in the plane of
the fibers. Their results are summarized below
5{ = b|5} + (1 }) Eq 3.7.8-1 { = b~} + (1 }) Eq 3.7.8-2
where, uc is the ultimate strength of the composite, K is the
fiber efficiency factor for strength, uf is the ultimate strength
of the fiber, Vf is the fiber volume fraction, m is the matrix
stress at the fracture strain of the composite, Ec is the
modulus of the composite, KE is the fiber efficiency factor for
modulus, Ef is the modulus of the fiber and Em is the matrix
modulus.
(Blumentritt, VU, & Cooper, 1975) measured the mechanical
properties of discontinuous fiber reinforced thermoplastics
fabricated using six types of reinforcement fiber and five types of
thermoplastics matrix resin. The fibers used were Dupont type 702
nylon 6/6, Dupont type 73 poly (ethylene terephthalate), Kuralon
poly (vinyl alcohol), Owens-Cornings type 801 E-glass, Dupont
Kevlar-49, and Union Carbide Thornel 300 graphite. The poly (vinyl
alcohol) fibers were 5 mm in length and the glass fibers were 6.3
mm in length. The other fibers were all 9.5 mm in length. The five
thermoplastics used were Dupont Surlyn 1558 type 30 ionomer, Dupont
Alathon 7140 high-density polyethylene, Huels grade L-1901 nylon
12, General Electric Lexan 105-111 polycarbonate and Dupont Lucite
47 poly (methyl
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methacrylate) (PMMA). All together thirty different combinations
of fiber/resin were tried.
The specimens in their study were made by a hand lay-up process
and then compression moulded. The moulded panels had a thickness of
about 1 mm. Tensile tests were conducted at an elongation rate of
5.1 mm/min to determine KE and K.
From the measurements, KE had a range of 0.44 to 0.06, while K
had a range of 0.25 to 0. The average fiber efficiency factors were
0.19 for modulus and 0.11 for strength.
Average values of 0.43 and 0.25 for KE and K, respectively were
reported for similar composites with unidirectional fiber
orientation (Blumentritt, VU, & Cooper, 1974) They concluded
that for similar materials, the fiber efficiency factor of
unidirectional fiber composites was approximately twice that of
random-in-plane composites.
3.7.9. Christensen and Waals Model
(Christensen & Waals, 1972) examined the behaviour of a
composite system with a three-dimensional random fiber orientation.
Both fiber orientation and fiber-matrix interaction effects were
considered. For low fiber volume fractions, the modulus of the 3-D
composite was estimated to be
q* 6 + [1 + (1 + )] Eq 3.7.9-1
where, c < 0.2 and m is the Poissons ratio of the matrix. c
is the volume concentration of the fiber phase, which is equivalent
to the fiber volume fraction.
For a state of plane stress, the modulus is given as
3 + [1 + ] Eq 3.7.9-2
where, c < 0.2.
Comparisons were made between the predictions given by Eq
3.7.9-2 and data reported by (Lee, 1969). It was found that at low
fiber volume fractions, predictions from Eq 3.7.9-2 were within a
range of 0 ~ +15% higher than the test data. The difference between
the prediction and test data was attributed to partially
ineffective bonds and/or end effects for the chopped fibers.
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3.7.10. Approximation Model by Manera
(Manera, 1977) proposed approximate equations to predict the
elastic properties of randomly oriented short fiber-glass
composites. The invariant properties of composites defined by (Tsai
& Pagano, 1968) were used along with Pucks micromechanics
formulation (Manera, 1977). Manera made a few assumptions and
simplified Puck invariants equations. The assumptions included high
fiber aspect ratio (>300), two-dimensional random distribution
of fibers and treatment of randomly oriented discontinuous fiber
composites as laminates with an infinite number layers oriented in
all directions. The approximate equations can be expressed as
= } _1645 + 2` + 89 Eq 3.7.10-1 ^ = } _ 215 + 34` + 13 Eq
3.7.10-2 = 13 Eq 3.7.10-3
where, Vf is the fiber volume fraction, m is the Poissons ratio
of the matrix, Em is the modulus of the matrix, Ef is the modulus
of the fiber, ^E and ^G are the tensile (flexural) and shear moduli
of the composite, respectively and ^ is Poissons ratio of the
composite.
It can be seen from Eq.(2.22)-Eq.(2.24) that ^E, ^, and ^G
satisfy the relationship
^ = 2(1 + ) Eq 3.7.10-4 In order to get adequate precision in
the results, Manera chose Vf to be within
the range 0.1 Vf 0.4 and Em within the range 2Gpa Em 4Gpa.
Predictions of composite modulus by Eq 3.7.10-1 were compared with
test
data (Manera, 1977). The constituent properties of the
composites in the tests were 5 cm chopped glass fiber with
Ef=73Gpa, f =0.25 and polyester resin with Em=2.25Gpa and m=0.40.
The differences between the predictions and test data were less
than 5%.
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3.8.Theories for Random Fiber Composites Based on the Calculated
Properties of Unidirectional Fiber-Reinforced Composites
3.8.1. Tsai and Pagano
The in-plane modulus of a random fiber composite was proposed by
(Tsai & Pagano, 1968) to be
= 38++ + 58NN
Eq 3.8.1-1
Where E11 and E22 are longitudinal and transverse modulus of
unidirectional composite obtained from Halpin Tsai equation
(Section 3.7.7) Although Eq 3.8.1-1 have very simple form, the
predictions are only good at very low fiber volume fractions. At
high fiber volume fractions, the predicted modulus is much higher
than measured. (Blumentritt, VU, & Cooper, 1975) explained that
this was caused by the increase in concentration of defects within
the composite as the fiber content increases.
3.8.2. Lavengood and Goettler
(Lavengood & Goettler, 1987) established a general procedure
for predicting the average Young's modulus for randomly oriented
short fiber composites. When the fibers are two dimensionally
oriented, they derived the Reuss-type expression as: = 24++NN/(7NN
+ 17++) Eq 3.8.2-1 Where, ++ = + }( ) Eq 3.8.2-2
NN = [2}(U 1) + (U + 2)}(1 U)+ (U + 2)}] Eq 3.8.2-3
In which Em and Ef are Young's moduli of the matrix and fiber,
respectively. Vf is the volume fraction of the fiber; R is the
ratio of transverse fiber modulus to matrix modulus.
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3.8.3. Piggot
(Piggott, 1980) suggested the modulus for composites having
fibers which are random in three dimensions as
{ = '+(} + } Eq 3.8.3-1
3.8.4. Voigt and Reuss
The simplest models are those which make use of the rule of
mixtures (combining rules). Voigt assumed that each component was
subject to the same strain (isostrain), giving, { = } + } Eq
3.8.4-1
Alternately, Reuss assumed that each phase was subject to the
same stress (isostress), giving, { = /(} + }) Eq 3.8.4-2 where E
denotes modulus and V volume fraction, and the subscripts c, m and
f represent composite, matrix resin and fiber, respectively.
3.8.5. Cox
(Cox, 1952) who used a shear lag formulation to model the
longitudinal elastic modulus showed that the modulus of short-fiber
composites can be expressed as :
{ = (15)++ + (45)NN Eq 3.8.5-1
where Ell and E22 are defined as E11=Ec from Voigt model ( Eq
3.8.4-1) and E22=Ec from Reuss model (Eq 3.8.4-2)
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3.8.6. Hori & Onogi
(Hori & Onogi, 1951) proposed the following: { = (++NN)+/N
Eq 3.8.6-1
where Ell and E22 are defined as E11=Ec from Voigt model ( Eq
3.8.4-1) and E22=Ec from Reuss model (Eq 3.8.4-2)
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3.9.Strength of Short fiber composites
3.9.1. Kelly and Tysons Model
(Kelly & Tyson, 1965) developed a theory to predict the
strength of short fiber composites. Basically, it is an extension
of the rule of mixtures by taking into account the effects of both
the fiber length and fiber orientation. It was based on the
assumption that plastic flow will occur during stress transfer
between matrix and fibers, giving
{ = } _1 @{2@` + } Eq 3.9.1-1
where c is ultimate tensile strength of composite f , m ,
strengths of fiber and matrix, respectively and l, 1c, fiber length
and critical length of
One problem associated with Kelly and Tysons theory is that the
estimates of strength are higher than measured (Peijs, Garkhail,
Heijenrath, Oever, & Bos, 1998)
3.9.2. Piggot Model
(Piggott, 1980) accounted for both plastic and elastic effects
in the matrix in his fiber theory. Piggot's composite strength
model is expressed by lengthy equations that will not be presented
here. For composites having fibers which are random in three
dimensions, he also suggested an upper strength bound critical
length of fibers.
{ = _15`} + } Eq 3.9.2-1
3.9.3. Rileys model
(Riley, 1968) considered interaction between fibers by taking
into consideration of the stress transfer between fibers in a
rationalized fiber array such as a hexagonal arrangement, and
derived a strength equation as
{ = _67` }1 + '>> ( + (1 }) Eq 3.9.3-1
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CHAPTER 3 4. METHODOLOGY
This project is carried out in following steps.
i. All existing empirical solutions need to be compared with
experimental results from Literature for PEEK Carbon Fiber short
fibre composite
ii. The closest Empirical relation is chosen for further
studies.
iii. Using the chosen empirical solution the Elastic moduli for
PEEK Carbon fiber composite is calculated for low aspect ratio
fiber.
iv. A VBA code is created to generate block of composite with
randomly oriented fibers for fiber volume fraction up to 0.2.
v. Discretization of the unit volume of SFC is executed in
Patran and analysed to pull and torsion in Msc Nastran
vi. The results from empirical solution and FE is compared for
Elastic modulus
vii. If a close match is obtained then other elastic constants
are also determined using the same FE model.
viii. As empirical relationships does not exist for randomly
oriented short fiber composite, comparison is done with results for
aligned fiber composites
Comparison of Empirical relationships with Test results
This section details how the mechanical properties of random
oriented shot fiber composites can be determined and also compares
the results between Tsai and Pagano approximation and results from
Finite Element Method.
Out of all the available empirical solutions a comparison is
made with experimental results to choose the best solution. PEEK as
matrix and Carbon fiber combination is chosen for the study. Test
results are available for high aspect ratio shot fiber composite.
But the study in this report is about short fiber composite with
low aspect ratio.
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4.1.Stiffness Estimation
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Figure 4.1-1Comparison of Empirical models with experimental
results
Out of all the above methods most of them closely match with the
experimental results as the fiber volume fraction considered in
this study is on the lower side less than 0.2. Still Tsai and
Pagano model is the best suited for comparison with FE modelling
and subsequent stiffness estimation. This is for the reason that
this model considers aspect ratio also as a parameter. The FE
modelling and analysis method pursued in this study considers
fibers of very low aspect ratio as well.
7.00E+09
9.00E+09
1.10E+10
1.30E+10
1.50E+10
1.70E+10
1.90E+10
0.1 0.12 0.14 0.16 0.18 0.2
Mo
du
lus
of
ela
stic
ity
Fiber Volume Fraction
Experimental vs Empirical (E)
EXP
HO
COX
Piggot
L&G
T&P
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4.2.Strength Estimation
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4.3.Calculation of SFC stiffness for Low aspect ratio PEEK
Carbon Fibre composite
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