This document introduces concepts of stiffness and strength in fiber-reinforced composite materials. It discusses how the microscopic properties of the constituents - fibers and matrix - relate to the macroscopic engineering properties of the composite. Specifically:
- Composites are made of a fiber reinforcement within a matrix. The fibers carry most of the load while the matrix holds them together and protects them.
- The stiffness and strength of a unidirectional composite can be estimated using simple "rule of mixtures" formulas that are functions of the fiber and matrix properties and their volume fractions.
- The fiber properties have a dominant effect on the longitudinal stiffness and strength, while the transverse properties are more dependent on the matrix properties.
-
Optimization of Heavy Vehicle Suspension System Using CompositesIOSR Journals
A leaf spring is a simple form of spring, commonly used for the suspension in wheeled vehicles. Leaf
Springs are long and narrow plates attached to the frame of a trailer that rest above or below the trailer's axle.
There are mono leaf springs, or single-leaf springs, that consist of simply one plate of spring steel. These are
usually thick in the middle and taper out toward the end, and they don't typically offer too much strength and
suspension for towed vehicles. Drivers looking to tow heavier loads typically use multi leaf springs, which
consist of several leaf springs of varying length stacked on top of each other. The shorter the leaf spring, the
closer to the bottom it will be, giving it the same semielliptical shape a single leaf spring gets from being thicker
in the middle. The automobile industry has shown increased interest in the replacement of steel spring with
fiberglass composite leaf spring due to high strength to weight ratio. In this thesis a leaf spring used in a heavy
vehicle is designed. While designing leaf spring following four cases are considered: by changing the thickness,
changing no. of leaves, changing camber and changing span .Present used material for leaf spring is Mild Steel.
The objective of this thesis is to compare the load carrying capacity, stiffness and weight savings of composite
leaf spring with that of steel leaf spring. The design constraints are stresses and deflections. In this thesis, the
material is replaced with composites since they are less dense than steel and have good strength. The strength
validation is done using FEA software ANSYS by structural analysis. Modal analysis is also done to determine
the frequencies. Analysis is done by layer stacking method for composites by changing number of layers 3, 5, 11
and 23. The composites used are Aramid Fiber and Glass Fiber
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Composites are made by combination of two or more natural or artificial materials to maximize their useful properties and minimize their weaknesses.
Example: The oldest and best-known composites,
Natural: Wood combination of cellulose fibre provides strength and lignin is the "glue" that bonds and stabilizes. Bamboo is a very efficient wood composite structure.
o is a very efficient wood composite structure
Artificial: The glass-fibre reinforced plastic (GRP), combines glass fiber (which are strong but brittle) with plastic (which is flexible) to make a composite material that is tough but not brittle.
70 to 90% of load carried by fibers
Provide structural properties to the composite
Stiffness
Strength
Thermal stability
Provide electrical conductivity or insulation
Example: Glass, Carbon, Organic Boron, Ceramic, Metallic
Function of Fiber/Dispersion phase
LARGE PARTICLE REINFORCED COMPOSITE ,,,An Overview
Seminar done as a part of METALLURGY AND MATERIAL SCIENCE.
Good PPT to study,
included most of the points to study
HASEEB KM
S3 ME
MUTHOOT INSTITUTE OF TECHNOLOGY AND SCIENCE, COCHIN
REINFORCED POLYMERIC COMPOSITE (RPC) MATERIALSSameer Ahmad
INTRODUCTION
A microscopic mixture of two or more different materials. One typically being the continuous phase (matrix), and the other being the discontinuous phase (reinforcement).
Its properties are strongly dependent on the composite structure. Composites could be natural or synthetic.
Constituents of composite materials.-
Technology industries have a very important direct and indirect impact on Finnish business life now and also in the future. A remarkable potential lies particularly in the small- and medium-sized enterprises. To develop and maintain industrial competitiveness we develop technological solutions which can widely be exploited by the Finnish manufacturing industries.
This presentation gives an overview of the VTT’s experience and offering for industry and is focused to computational material development, robotics, additive manufacturing, and concept design.
Optimization of Heavy Vehicle Suspension System Using CompositesIOSR Journals
A leaf spring is a simple form of spring, commonly used for the suspension in wheeled vehicles. Leaf
Springs are long and narrow plates attached to the frame of a trailer that rest above or below the trailer's axle.
There are mono leaf springs, or single-leaf springs, that consist of simply one plate of spring steel. These are
usually thick in the middle and taper out toward the end, and they don't typically offer too much strength and
suspension for towed vehicles. Drivers looking to tow heavier loads typically use multi leaf springs, which
consist of several leaf springs of varying length stacked on top of each other. The shorter the leaf spring, the
closer to the bottom it will be, giving it the same semielliptical shape a single leaf spring gets from being thicker
in the middle. The automobile industry has shown increased interest in the replacement of steel spring with
fiberglass composite leaf spring due to high strength to weight ratio. In this thesis a leaf spring used in a heavy
vehicle is designed. While designing leaf spring following four cases are considered: by changing the thickness,
changing no. of leaves, changing camber and changing span .Present used material for leaf spring is Mild Steel.
The objective of this thesis is to compare the load carrying capacity, stiffness and weight savings of composite
leaf spring with that of steel leaf spring. The design constraints are stresses and deflections. In this thesis, the
material is replaced with composites since they are less dense than steel and have good strength. The strength
validation is done using FEA software ANSYS by structural analysis. Modal analysis is also done to determine
the frequencies. Analysis is done by layer stacking method for composites by changing number of layers 3, 5, 11
and 23. The composites used are Aramid Fiber and Glass Fiber
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Composites are made by combination of two or more natural or artificial materials to maximize their useful properties and minimize their weaknesses.
Example: The oldest and best-known composites,
Natural: Wood combination of cellulose fibre provides strength and lignin is the "glue" that bonds and stabilizes. Bamboo is a very efficient wood composite structure.
o is a very efficient wood composite structure
Artificial: The glass-fibre reinforced plastic (GRP), combines glass fiber (which are strong but brittle) with plastic (which is flexible) to make a composite material that is tough but not brittle.
70 to 90% of load carried by fibers
Provide structural properties to the composite
Stiffness
Strength
Thermal stability
Provide electrical conductivity or insulation
Example: Glass, Carbon, Organic Boron, Ceramic, Metallic
Function of Fiber/Dispersion phase
LARGE PARTICLE REINFORCED COMPOSITE ,,,An Overview
Seminar done as a part of METALLURGY AND MATERIAL SCIENCE.
Good PPT to study,
included most of the points to study
HASEEB KM
S3 ME
MUTHOOT INSTITUTE OF TECHNOLOGY AND SCIENCE, COCHIN
REINFORCED POLYMERIC COMPOSITE (RPC) MATERIALSSameer Ahmad
INTRODUCTION
A microscopic mixture of two or more different materials. One typically being the continuous phase (matrix), and the other being the discontinuous phase (reinforcement).
Its properties are strongly dependent on the composite structure. Composites could be natural or synthetic.
Constituents of composite materials.-
Technology industries have a very important direct and indirect impact on Finnish business life now and also in the future. A remarkable potential lies particularly in the small- and medium-sized enterprises. To develop and maintain industrial competitiveness we develop technological solutions which can widely be exploited by the Finnish manufacturing industries.
This presentation gives an overview of the VTT’s experience and offering for industry and is focused to computational material development, robotics, additive manufacturing, and concept design.
The Venus Papers by Lydia Towsey SAMPLEBurning Eye
The title sequence of The Venus Papers takes as it's starting point Botticelli's 15th Century painting, 'Birth of Venus' where Venus is depicted arriving on a shell at a Cypriot beach. Transported to the 21st century UK; Lydia explores how Venus would respond to what she would find and what would she do next?
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...MSEJjournal1
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
1. INTRODUCTION TO COMPOSITE MATERIALS
David Roylance
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
March 24, 2000
Introduction
This module introduces basic concepts of stiffness and strength underlying the mechanics of
fiber-reinforced advanced composite materials. This aspect of composite materials technology
is sometimes terms “micromechanics,” because it deals with the relations between macroscopic
engineering properties and the microscopic distribution of the material’s constituents, namely
the volume fraction of fiber. This module will deal primarily with unidirectionally-reinforced
continuous-fiber composites, and with properties measured along and transverse to the fiber
direction.
Materials
The term composite could mean almost anything if taken at face value, since all materials are
composed of dissimilar subunits if examined at close enough detail. But in modern materials
engineering, the term usually refers to a “matrix” material that is reinforced with fibers. For in-
stance, the term “FRP” (for Fiber Reinforced Plastic) usually indicates a thermosetting polyester
matrix containing glass fibers, and this particular composite has the lion’s share of today’s
commercial market. Figure 1 shows a laminate fabricated by “crossplying” unidirectionally-
reinforced layers in a 0◦-90◦stacking sequence.
Many composites used today are at the leading edge of materials technology, with perfor-
mance and costs appropriate to ultrademanding applications such as spacecraft. But heteroge-
neous materials combining the best aspects of dissimilar constituents have been used by nature
for millions of years. Ancient society, imitating nature, used this approach as well: the Book of
Exodus speaks of using straw to reinforce mud in brickmaking, without which the bricks would
have almost no strength.
As seen in Table 11, the fibers used in modern composites have strengths and stiffnesses
far above those of traditional bulk materials. The high strengths of the glass fibers are due to
processing that avoids the internal or surface flaws which normally weaken glass, and the strength
and stiffness of the polymeric aramid fiber is a consequence of the nearly perfect alignment of
the molecular chains with the fiber axis.
1
F.P. Gerstle, “Composites,” Encyclopedia of Polymer Science and Engineering, Wiley, New York, 1991. Here
E is Young’s modulus, σb is breaking stress, b is breaking strain, and ρ is density.
1
2. Figure 1: A crossplied FRP laminate, showing nonuniform fiber packing and microcracking
(from Harris, 1986).
Table 1: Properties of Composite Reinforcing Fibers.
Material E σb b ρ E/ρ σb/ρ cost
(GPa) (GPa) (%) (Mg/m3
) (MJ/kg) (MJ/kg) ($/kg)
E-glass 72.4 2.4 2.6 2.54 28.5 0.95 1.1
S-glass 85.5 4.5 2.0 2.49 34.3 1.8 22–33
aramid 124 3.6 2.3 1.45 86 2.5 22–33
boron 400 3.5 1.0 2.45 163 1.43 330–440
HS graphite 253 4.5 1.1 1.80 140 2.5 66–110
HM graphite 520 2.4 0.6 1.85 281 1.3 220–660
Of course, these materials are not generally usable as fibers alone, and typically they are
impregnated by a matrix material that acts to transfer loads to the fibers, and also to pro-
tect the fibers from abrasion and environmental attack. The matrix dilutes the properties to
some degree, but even so very high specific (weight-adjusted) properties are available from these
materials. Metal and glass are available as matrix materials, but these are currently very ex-
pensive and largely restricted to R&D laboratories. Polymers are much more commonly used,
with unsaturated styrene-hardened polyesters having the majority of low-to-medium perfor-
mance applications and epoxy or more sophisticated thermosets having the higher end of the
market. Thermoplastic matrix composites are increasingly attractive materials, with processing
difficulties being perhaps their principal limitation.
Stiffness
The fibers may be oriented randomly within the material, but it is also possible to arrange for
them to be oriented preferentially in the direction expected to have the highest stresses. Such
a material is said to be anisotropic (different properties in different directions), and control of
the anisotropy is an important means of optimizing the material for specific applications. At
a microscopic level, the properties of these composites are determined by the orientation and
2
3. distribution of the fibers, as well as by the properties of the fiber and matrix materials. The
topic known as composite micromechanics is concerned with developing estimates of the overall
material properties from these parameters.
Figure 2: Loading parallel to the fibers.
Consider a typical region of material of unit dimensions, containing a volume fraction Vf of
fibers all oriented in a single direction. The matrix volume fraction is then Vm = 1 − Vf . This
region can be idealized as shown in Fig. 2 by gathering all the fibers together, leaving the matrix
to occupy the remaining volume — this is sometimes called the “slab model.” If a stress σ1 is
applied along the fiber direction, the fiber and matrix phases act in parallel to support the load.
In these parallel connections the strains in each phase must be the same, so the strain 1 in the
fiber direction can be written as:
f = m = 1
The forces in each phase must add to balance the total load on the material. Since the forces in
each phase are the phase stresses times the area (here numerically equal to the volume fraction),
we have
σ1 = σf Vf + σmVm = Ef 1Vf + Em 1Vm
The stiffness in the fiber direction is found by dividing by the strain:
E1 =
σ1
1
= Vf Ef + Vm Em (1)
This relation is known as a rule of mixtures prediction of the overall modulus in terms of the
moduli of the constituent phases and their volume fractions.
If the stress is applied in the direction transverse to the fibers as depicted in Fig. 3, the slab
model can be applied with the fiber and matrix materials acting in series. In this case the stress
in the fiber and matrix are equal (an idealization), but the deflections add to give the overall
transverse deflection. In this case it can be shown (see Prob. 5)
1
E2
=
Vf
Ef
+
Vm
Em
(2)
Figure 4 shows the functional form of the parallel (Eqn. 1) and series (Eqn. 2) predictions for
the fiber- and transverse-direction moduli.
The prediction of transverse modulus given by the series slab model (Eqn. 2) is considered
unreliable, in spite of its occasional agreement with experiment. Among other deficiencies the
3
4. Figure 3: Loading perpendicular to the fibers.
assumption of uniform matrix strain being untenable; both analytical and experimental studies
have shown substantial nonuniformity in the matirx strain. Figure 5 shows the photoelastic
fringes in the matrix caused by the perturbing effect of the stiffer fibers. (A more complete
description of these phtoelasticity can be found in the Module on Experimental Strain Analysis,
but this figure can be interpreted simply by noting that closely-spaced photoelastic fringes are
indicative of large strain gradients.
In more complicated composites, for instance those with fibers in more than one direction
or those having particulate or other nonfibrous reinforcements, Eqn. 1 provides an upper bound
to the composite modulus, while Eqn. 2 is a lower bound (see Fig. 4). Most practical cases
will be somewhere between these two values, and the search for reasonable models for these
intermediate cases has occupied considerable attention in the composites research community.
Perhaps the most popular model is an empirical one known as the Halpin-Tsai equation2, which
can be written in the form:
E =
Em[Ef + ξ(Vf Ef + VmEm)]
Vf Em + VmEf + ξEm
(3)
Here ξ is an adjustable parameter that results in series coupling for ξ = 0 and parallel averaging
for very large ξ.
Strength
Rule of mixtures estimates for strength proceed along lines similar to those for stiffness. For
instance, consider a unidirectionally reinforced composite that is strained up to the value at
which the fibers begin to break. Denoting this value fb, the stress transmitted by the composite
is given by multiplying the stiffness (Eqn. 1):
σb = fb E1 = Vf σfb + (1 − Vf )σ∗
The stress σ∗ is the stress in the matrix, which is given by fbEm. This relation is linear in Vf ,
rising from σ∗ to the fiber breaking strength σfb = Ef fb. However, this relation is not realistic
at low fiber concentration, since the breaking strain of the matrix mb is usually substantially
greater than fb. If the matrix had no fibers in it, it would fail at a stress σmb = Em mb. If the
fibers were considered to carry no load at all, having broken at = fb and leaving the matrix
2
c.f. J.C.. Halpin and J.L. Kardos, Polymer Engineering and Science, Vol. 16, May 1976, pp. 344–352.
4
5. Figure 4: Rule-of-mixtures predictions for longitudinal (E1) and transverse (E2) modulus, for
glass-polyester composite (Ef = 73.7 MPa, Em = 4 GPa). Experimental data taken from Hull
(1996).
to carry the remaining load, the strength of the composite would fall off with fiber fraction
according to
σb = (1 − Vf )σmb
Since the breaking strength actually observed in the composite is the greater of these two
expressions, there will be a range of fiber fraction in which the composite is weakened by the
addition of fibers. These relations are depicted in Fig. 6.
References
1. Ashton, J.E., J.C. Halpin and P.H. Petit, Primer on Composite Materials: Analysis,Technomic
Press, Westport, CT, 1969.
2. , Harris, B., Engineering Composite Materials, The Institute of Metals, London, 1986.
3. Hull, D. and T.W. Clyne, An Introduction to Composites Materials, Cambridge University
Press, 1996.
4. Jones, R.M., Mechanics of Composite Materials, McGraw-Hill, New York, 1975.
5. Powell, P.C, Engineering with Polymers, Chapman and Hall, London, 1983.
6. Roylance, D., Mechanics of Materials, Wiley & Sons, New York, 1996.
5
6. Figure 5: Photoelastic (isochromatic) fringes in a composite model subjected to transverse
tension (from Hull, 1996).
Figure 6: Strength of unidirectional composite in fiber direction.
Problems
1. Compute the longitudinal and transverse stiffness (E1, E2) of an S-glass epoxy lamina for
a fiber volume fraction Vf = 0.7, using the fiber properties from Table 1, and matrix
properties from the Module on Materials Properties.
2. Plot the longitudinal stiffness E1 of an E-glass/nylon unidirectionally-reinforced composite,
as a function of the volume fraction Vf .
3. Plot the longitudinal tensile strength of a E-glass/epoxy unidirectionally-reinforced com-
posite, as a function of the volume fraction Vf .
4. What is the maximum fiber volume fraction Vf that could be obtained in a unidirectionally
reinforced with optimal fiber packing?
5. Using the slab model and assuming uniform strain in the matrix, show the transverse
modulus of a unidirectionally-reinforced composite to be
6