msd final project_v1 11-1

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

(Signature) BIJU BL

(Reg. No.102520085)

Under the guidance of

(Signature) Dr. BADARI NARAYANA KANTHETI

AEROSTRUCTURES UTC AEROSPACE SYSTEMS

BANGALORE

MANIPAL UNIVERSITY, MANIPAL


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:

UTC Aerospace Systems Netra Tech Park, EPIP Industrial Area Sy.No.28 , Bengaluru 560 066, INDIA City, State/Province, Postal Code www.utcaerospacesystems.com


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

i This document contains no EAR or ITAR technical data

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.

ii This document contains no EAR or ITAR technical data

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.

iii This document contains no EAR or ITAR technical data

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

iv This document contains no EAR or ITAR technical data

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


64


64

3.6 1 Patran Session file for creating random fibers of volume fraction 0.2

65

v


3.6 2 (a) Geometric model (b) FE model of random fibers of volume fraction 0.2

65


66


66


67


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

vi This document contains no EAR or ITAR technical data

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

vii This document contains no EAR or ITAR technical data

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

viii This document contains no EAR or ITAR technical data

Modulus of Short-Fiber Composites 19

ix This document contains no EAR or ITAR technical data

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

x


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



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

xi This document contains no EAR or ITAR technical data

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

1 This document contains no EAR or ITAR technical data

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


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.


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


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.


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.


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,


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


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


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.


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:


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.


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


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.


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


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


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


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.


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.


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


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


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 inhomogeneity (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


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


[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.


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


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


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,


- = $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


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


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


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


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)


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:


++ = [>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).


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


++ = + 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.


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,


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


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


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.


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%.


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.


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)


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)


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


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.


4.1.Stiffness Estimation


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


4.2.Strength Estimation


4.3.Calculation of SFC stiffness for Low aspect ratio PEEK Carbon Fibre composite

58 This docume

msd final project_v1 11-1

Documents

continuous fiber systems

mechanical systems design

fiber loading

degree signature

fem thesis

continuous fiber laminates

signature biju bl

degree of master