Solutions Manual
For
Composite Materials
Science & Engineering
Third Edition
Krishan K. Chawla
Springer, Inc., New York
2013
CHAPTER 1
1.1
Bone is a major structural-material that supports the human body. It is a composite material
consisting of a mineral matrix, hydroxyapatite containing organic fibers, collagen. The
organization of the two constituents varies according to the functional needs. Cortical (dense)
bone has a concentric ringlike structure extending along the shaft of the bone. Cancellous
(spongy) bone consists of a space filling network of small beams of bone resulting in a porous,
compliant material. Many long bones consist of hollow tubes of cortical bone with enlarged
extremities filled with cancellous bone fibers oriented along the principal stress directions
resulting from the major loading. As expected, cortical bone is stronger in the axial direction
than in the transverse direction.
Wood is another versatile natural composite. It consists of crystalline cellulose fibers embedded
in an amorphous matrix of lignin and hemicellulose. The matrix contributes to the stiffness of
the wood as well as serves to protect the crystalline fibrous cellulose from moisture. Generally,
the fiber volume fraction in wood is about 50% and it has a high strength to weight ratio and
toughness. It should be pointed out this fiber reinforced composite material forms the cellular
walls and this structure is essentially the same in all woods This composite is about as strong as
aluminum but less stiff. The overall structure of wood is that of a foam composite with the foam
cells and the cellulose fibers aligned predominantly long the wood grain. The different wood
types have different properties mainly owing to different shapes and sizes of their cells. The cell
wall material is a fibrous composite and is about the same in different woods.
1.2
Capacitors with enhanced permitivity can be made of BaTiO3, i.e., ferroelectric grains and in an
antiferroelectric matrix (e.g., NaNbO3). The resulting composite shows relatively little variation
in the dielectric constant with voltage and shows a higher capacitance than either component at
high bias field (see Fig. 1.2).
In a magnetoelectric composite consisting of BaTiO3 and cobalt ferrite, the interaction of
different properties in the two phases results in a third (product) property. When a magnetic field
is applied to the composite, the ferrite grains change shape because of magnetostriction. This
strain serves as input to the piezoelectric grains and causes electrical polarization.
Magnetoelectric effects a hundred times larger than those in Cr2O3 can be obtained (1, 2) .
A composite consisting of lithium flouride filled with alumina can be used as a humidity sensor
(3). The differential contraction between LiF and Al2O3 leads to microcracking in the composite.
Moisture penetrates into these microcracks and changes the surface resistivity. The surface
resistance is a very sensitive measure of moisture absorbed and it is this characteristic that is
exploited in the making of A12O3/LiF sensors.
References:
1.J. van Suchtelen, Philips Res. Report, 27 (1972) 28.
2.R.E. Newnham, Ann. Rev. Mater. Sci., 16 (1986) 47.
3.B.C. Tofield and D.E. Williams, Solid State Ionics, 83 (1983) 1299.
Fig 1.2
1.3
The Voyager became the first plane to fly around the world without refueling in December,
1986. The Voyager covered 4272 km (2670 miles) on less than 3200 kg (~ 5120 lbs) of fuel. The
plane had to be the lightest thing possible and the payload essentially consisted of fuel. The
plane was sandwich construction: two plies of carbon fiber/epoxy tapes and a Nomex honey
comb core. Epoxy was used as matrix and adhesive. Few fasteners were used. The wings and
fuselage contained about 8 kg (12.8 lbs) of metals. Its takeoff weight was 9700 lbs (6062 kg), of
which 7000 lbs (4375 kg) was fuel. All wing and fuselage structures were loaded with fuel.
CHAPTER 2
2.1
Nonwoven fibrous mats are cheaper to produce because the number of steps required, in going
from fiber to mat, is s all. Weaving, on the other hand, can be very complex and consequently
expensive.
Nonwoven mats are porous and have lower strength as well as flexibility compared to the woven
fabrics.
2.2
Here is one possible inorganic polymeric chain structure of glass fiber:
2.3
Optical glass fiber is a thin, flexible, and transparent guide through which light can be
transmitted. Absorption and scattering of light traveling through such fibers result in signal
attenuation. Intrinsic signal attenuation is a function of the wavelength and this component of
optical loss is the lowest in GeO2 - SiO2glasses. Extrinsic absorption losses occur from transition
metal and OH impurities. These are present to a considerable degree in the glass fiber produced
by direct melting. In the vapor phase deposited GeO2 - SiO2 glasses, transition metal ion
impurities can be reduced to < 1 ppb. OH absorption can be reduced by carefully preparing dry
glass fibers. Complete elimination of OH is difficult.
Strength loss sources in such fibers are mainly surface damage due to contamination and the
presence of microcracks (e.g., bubbles, etc.). Polymer surface coatings are used to minimize the
damage. Fibers are also proof tested to breaking strains in the range of 0.5 - 1%.
Optical fibers are generally made from GeO2 - SiO2 glass system. A modified chemical vapor
deposition (MCVD) technique is used to obtain a fiber consisting of a GeO2 - SiO2core and pure
SiO2 cladding. First the cladding is deposited on the inside of a hot silica tube. When sufficient
cladding has been obtained, the reactants are changed to obtain the core glass. In the final stage,
the temperature is increased and the tube is collapsed to form a solid preform rod. This preform
is converted into a fine filament by a drawing process and protective polymeric layers are
applied. A clean atmosphere must be maintained throughout all these operations to avoid
introduction of impurities. 5-10 km long fibers (strain to fracture 2-3%) can generally be pulled.
Direct melt techniques are used under certain circumstances.
Reference
K.K. Chawla, Fibrous Materials, Cambridge University Press, Cambridge, England, 1998.
If we did the same exercise for Kevlar 29 fiber, we would find the rod diameter to be 2.6 mm.
2.5
Figure below shows the stress-strain curves of Kelvar 49 and Kevlar 29. The strain to fracture
values for Kevlar 49 and Kevlar 29 are 2.4% and 4.0%, respectively. Kevlar 49 shows an almost
linear stress-strain curve right up to fracture. Both the fibers are semi-crystalline. The straining
involves crystal lattice elongation through valence angle deformation and bond stretching of the
polymer chain. In the case of Kevlar 29, the deviation from linearity starts at 1% strain. This
corresponds to chain breakage and other irreversible processes such as crystallite rotation toward
the fiber axis.
2.6
Crack propagation is easy along the longitudinal axis since it only requires rupture of weak
hydrogen bonds. Even when the predominant fracture is transverse to the fiber axis, there is
always present fibrillation along the fiber axis.
2.7
Both Kevlar and Nomex are aramid fibers. The chemical stmcture of Kevlar has ph!'a phenylene
rings (PPDT or PPTA) while Nomex (MPD-l) has meta-linked rings. The two chemical
structures are shown below:
Kevlar: p-phenylenediamine (PPD-T) or p-poly (p-phenyleneterephthalamide) or PPTA
The aromatic linkages in the para position result in a linear configuration parallel to the
fiber axis. The result is a high degree of crystallinity because of ease of packing, high
strength, high modulus, but low to strain fracture.
Nomex: m-phenylenodiamine and iso-phthaloyl chloride (MPD-1)
The aromatic linkages in the meta position result in an irregular chain configuration,
consequently a lower degree of crystallinity, lower modulus and strength than K-49
fiber.. The bending and breaking of bonds at an angle to the fiber requires very low stress.
The essential difference bet w e en K e v l a r and Nomex is thus in the orientation of
the aromatic rings. This, o f course, res ult s i n a spectacular difference i n strain -tofracture values for the t wo: 28% for Nomex and 2-4% for Kevlar. Kevlar h a s also
slightly h i gher thermal stability.
2.8
Asbestos is the name of several naturally occurring minerals which are silicate-based and are
fibrous in form. They have a crystalline structure and are very resistant to heat, acids, alkalis,
other chemicals. Asbestos fibers have relatively low strength but they are not attacked by insects
or microorganisms as is the case with the vegetable fibers.
Its use is being curbed because it has been shown to cause lung cancer if inhaled. It is thought
that the generally large aspect ratio (length/ diameter) of asbestos fiber is related to causing the
tumor. Some respiratory diseases are also caused by inhalation of asbestos fibers.
2.9
(i) Gripping of extremely small whiskers is very difficult.
(ii) Proper axial alignment of whiskers is difficult.
(iii) Precise measurements of the length and diameter is difficult In particular, an error diameter
will be squared when the cross-sectional area is computed.
(iv) Any nonuniformity in the cross-sectional area along the length will also make things
difficult.
In general, precise measurement of load and elongation is difficult.
CHAPTER 3
3.1
In metals, the outermost electrons of each atom form a cloud of electrons. This cloud of electrons
is shared among the atoms of a metal, which results in a nondirectional cohesion or bonding in
metals. The stress at a crack tip or at a dislocation is easy to relieve in metals because it requires
only a small shear stress to make the dislocation move. In a ceramic or polymer, there exist
directional and very local chemical bonds which lead to a highly directional and localized
electron sharing. Consequently, the motion of a defect, such as a dislocation, requires the
breaking and reestablishment of such bonds. Such an activity requires, frequently, a force to
move a defect greater than that to cause fracture. That is why a tiny cracklike defect can cause
catastrophic failure under stress in ceramics and polymers.
3.2
Elastic constants are decreased by the presence of porosity. Pores are like a second phase with
a zero modulus. Various expressions are used. According to Ishai and Cohen (1)
where E and Eo are the Young’s moduli at porosity volume fraction p and 0, respectively.
From an expression due to Mackenzie (2) for relative shear moduli, we get the following
expression for Young’s moduli:
where v and
Coble and Kingery(3) assumed that E = 0 at p= 1 and found C = -0.91 with v0 = 0.3.
Assuming that v does not vary with porosity, i.e., v = Vo for all p, the Mackenzie equation
becomes
Phani and M u k e r j e e (4) d e r i v e d semi-empirically the following equation
where b is a pore distribution geometry factor and n depends on pore geometry.
For spherical pores, b has a value between 1 and 1.91 and 2.
(1) 0. Ishai and L.J. Cohen, Inst. J. Mech. Sci., 9 (1967) 539.
(2) J.K. Mackenzie, Proc. Phys. Soc. (London) 63B (1950) 2.
(3) R.L. Coble and W.D. Kingery1 J. Amer. Ceram. Soc. 39 (1956) 377.
(4) K.K. Phani and R.N. Mukerjee, J. Materials Sci., 22 (1987) 3453.
3.3
(i) Polymers show much more pronounced mechanical relaxation processes than do metals, e.g.,
stress relaxation, creep, mechanical hysteresis, etc. These processes make the mechanical
properties of polymers much more time and temperature dependent than metals.
(ii) Elastomeric polymers can undergo large amounts of , nonlinear, elastic, and reversible
strains than can metals.
3.4
3.5
In general, the fatigue resistance of a polymer improves with the degree of crystallinity. Semicrystalline polymers seem to have a high resistance to fatigue crack propagation than amorphous
polymers mainly because of their ability to undergo plastic deformation at the crack tip.
Crystalline polymers also dissipate energy more efficiently when crystallites are deformed. It is
also thought (see, for example, R.W. Hertzberg and J.A. Manson, Fatigue of Engineering
Plastics, Academic Press, New York, 1980, p. 130) that the fatigue process modifies the
polymer substructure, some kind of cold drawing, which makes the polymer exceedingly
strong. A fourfold decrease in fatigue crack propagation rate in high-density polyethylene was
observed when the crystallinity increased from 47 to 55 percent (1).
Reference
1. F.X. de Charentenay, F. Laghouati, and J. Dewas, Deformation, Yield and Fracture of
Polymers, Plastics and Rubber Inst., 1979, p. 61.
3.6
In thermal effects on fatigue in polymers, the most important parameter is the cycling frequency.
Thus, if the cycling frequency is so high that isothermal conditions do not prevail, then the
hysteretic heating effect in each cycle will cause the elastic modulus of the polymer to
decrease, resulting in a premature failure. In this regard, it is worth noting that carbon fibers
being better thermal conductors than most other nonmetallic fibers, carbon fiber reinforced
polymers would be expected to show lower hysteretic heating and thus better fatigue resistance.
3.7
Processing is done at high temperatures and in a glassy state, say, between 1000 – 1400° C. The
ease of glass flow is exploited to form intricate shapes. The nucleation of crystals may take place
between 450 – 700 °C while the growth of these crystals may occur between 600 – 900°C.
CHAPTER 4
4.2
Diffusion and reaction kinetics, in general, increase in a nonlinear way with temperature.
Thus, accelerated tests done at high temperatures cannot be translated to low temperatures
unless the variation of diffusion and reaction kinetics data with temperature is known. Besides,
it is quite possible that at high temperatures reaction products may form which are
thermodynamically not predicted at low temperatures.
4.3
The short beam shear test for measuring interlaminar shear stress exploits the fact that in
a three-point bend test the ratio of the shear stress, τ in the beam interior and the tensile
stress, σ in the outermost layer of the beam is given by
where h is the beam height and S is the span of the bend specimen. Thus, by making the beam
small enough, we can make r so large that the composite will fail by interlaminar shear.
Among the problems of this test are:
•
•
•
Difficult to avoid damage under the loading points.
If tensile failure of fibers precedes the shear failure or if a combination of tensile and
shear failures occur, then the test is invalid. Thus, any transverse tensile or
compressive stresses present will complicate the situation.
In laminated composites, the results will depend on the ply stacking sequence. The
maximum shear stress will not necessarily occur at the center of the beam.
The short beam test, in summary, works for a composite beam which can be treated as a
homogeneous material.
CHAPTER 5
5.1
A prepreg is a thin lamina of unidirectional (or sometimes woven) fiber/polymer composite
protected on both sides with easily removable separators. Prepregs have the following
advantages:
•
handling ease
•
prefixed volume fractions of components
•
no mixing of resin, hardener, and catalyst required
•
shelf life at room temperature of a few weeks
•
Deep freeze shelf can be many months
•
good control of polymer viscosity → easy processing, low porosity laminates
•
quality control of fiber/polymer composite performance before making the actual
component
5.2
Injection molding techniques suffer from tremendous flow variations during mold filling, which
result in a heterogeneous distribution of fibers (see Fig. in the text). Other limitations:
i.
rather low fiber volume fractions.
ii.
difficult to incorporate continuous or very long fibers.
5.3
In a thermally cured PMC, the fiber surface treatments have been established over the years for
certain systems. For example, carbon fibers meant for use in an epoxy matrix are given an
oxidizing treatment while silanes are put on glass fibers for use in an epoxy matrix. Electron
beam curing may not be effective in developing an appropriate interface. In the conventional
thermal curing, initially there is a decrease in viscosity of the matrix (exothermic curing reaction)
followed by an increase in viscosity as the curing proceeds. This initial viscosity decrease allows
the polymeric matrix to wet the fibers. This important stage showing a decrease in viscosity is
missing in electron beam curing.
5.5
Some polymers, when heated to a certain temperature, decompose and form flammable gaseous
compounds. Following are the important aspects in imparting fire resistance to polymers and
PMCs.
Modify the polymer by impregnation by or adding flame retardants that release scavenging
agents that remove free radicals normally involved in flame initiation and propagation. This can
be accomplished by adding halogenated compounds such as chlorinated paraffins, alicyclic
compounds, and bromo-aromatic additive. The flame retardant and/or its decomposition products
volatilize simultaneously with the gases generated by the polymer and thus inhibit the vapor
phase combustion of the fuel gases.
Flame retardants such as antimony oxides, and some bromide and chloride compounds, tend to
reduce the decomposition products of some polymers.
Heat stabilizers (for example some metal carboxylates and organic compounds) react with
polymers such that they interrupt the degradative chain reaction.
If the fuel used during combustion is a condensed phase, then flame retardancy can be achieved
by modifying the decomposition products. The flame retardant alters the pathway of thermal
degradation by providing a low energy process such carbonization rather than generate
combustible gases. In some cases, the flame retardant forms a protective coating that insulates
the polymer.
Addition of antioxidants reduces the amount of free radicals. Hydroperoxides play a key role in
the oxidation of hydrocarbons via a degradative free-radical chain reaction.
CHAPTER 6
6.1
The greatest advantage of casting methods is that they give a near-net shape of the product, i.e.,
it requires little or no further maching or finishing. Hence, the casting processes are generally
cheaper than other processes.
The greatest disadvantage of casting methods is the presence of porosity. Porosity can result
from either normal shrinkage during the liquid to solid transformation or gas evolution. Some of
the excessive porosity can be eliminated by good casting practice, e.g., avoiding turbulence in
the melt, solidifying under a small pressure, etc. The latter technique is used to a great advantage
in squeeze casting. Nonmetallic inclusions are another problem in the casting route. Filteration of
the metal, through ceramic foam filters or steel mesh pads, is one way out.
6.2
6.3
An unconstrained annealed metal, i.e., unreinforced matrix metal, will show a characteristically
large amount of plastic deformation and a low work hardening rate. When a composite is made
by introducing fibers into a metal, the generally strong bond between the metal and the fiber will
make the insitu deformation of the matrix quite constrained. The constraint comes from the fact
that the metal is not free to contract laterally. The difference in the Poisson’s ratio of the fiber
and the matrix () results in transverse stresses even when a uniaxial stress is applied. The
constrained matrix under a triaxial stress will show a stress-strain curve higher than that of the
unconstrained metal. (see Fig. below) The important point is that the insitu stress-strain behavior
of matrix is different from that of the same metal when in an unreinforced condition.
6.4
Fiber reinforced composites in general and eutectic composites in particular are characterized by
an extraordinarily large amount of interfacial surface area. A volume of 1 mm3 of a lamellar
eutectic can easily have more 400 mm2 of interfacial area. This would lead to an extensive
tendency for spheroidization of lamellar eutectic microstructure when exposed to high
temperatures due to interdiffusion of the two phases. When these interfaces are of a low energy
variety, the eutectic microstructure may exhibit an unusually high thermal stability. If the
interface has a high energy, structural changes will occur first at the sites of large mismatch. For
example, pits may form at lamellar faults. Another possibility is that of interphase boundary
sliding at high temperatures. Thermal stresses due to the expansion coefficient mismatch
represent another serious problem in any kind of composite. Interlamellar sliding, due to thermal
cycling, has been observed in Al-CuAl2. Coarsening or degeneration of the reinforcing phase can
result on thermal cycling, especially under conditions of large temperature gradients. For
example, it was observed (1) that under a temperature gradient of 5 K mm-1, perpendicular to the
fiber axis, in Al3Ni/Al composite, the fibrous phase (Al3Ni) coarsened 5 times more rapidly than
under isothermal conditions. It would appear that systems having sluggish interfacial kinetics
will do well.
References
(1) D.R.H. Jones and G.J. May, Acta Met., 23 (1975) 29
6.5
Some of the advantages of pressure casting are:
•
Independent of wettability of reinforcement by the liquid metal.
•
Can use a wide range of alloys as a matrix.
•
Near net shape capability, can produce complex shapes with good details.
•
Superior properties in the as cast state because of solidification under pressure because
the process results in a high solidification rate and low porosity, which in turn give the as
cast alloy matrix properties equal to those of a wrought alloy matrix.
•
Relatively simple and cost effective process.
6.6
Following are some of the advantages of metal matrix composites over monolithic metals.
•
Weight savings over monolithic metals
•
Better dimensional stability
•
Higher strength and stiffness than conventional metals
•
Higher temperature capability than conventional metals
•
Improved cyclic fatigue properties
6.7
Some of the advantages of metal matrix composites vis à vis polymer matrix composites are.
•
Higher operating temperature
•
Higher thermal conductivity
•
Higher electrical conductivity
•
No problems of grounding, space charging
•
Better properties in the transverse direction
•
Better resistance to radiation (laser, UV, nuclear, etc.)
•
Little or no outgassing
•
Little or no moisture absorption
CHAPTER 7
7.1
Mechanical damage of fibers can occur at excessively high pressures. At very high processing
temperatures one must guard against grain growth or softening in the reinforcement as well as
any adverse chemical reaction between the fiber and the matrix. For example, oxidation of
carbon fibers at high temperature is highly undesirable.
7.2
i.
ii.
iii.
iv.
A greater control of the composition as well as the degree of homogeneity is attainable
Possibility of forming unique multiphase matrices
The fluid starting materials have a relative ease of penetrating a fibrous perform.
Lower processing temperatures.
7.3
Carbon fibers have a negative axial coefficient of thermal expansion (CTE). Thus, appropriately
combining them with a glass or glass-ceramic matrix can result in a composite with an almost
zero in-plane CTE over a range of temperature. Figure below shows the in-plane CTE as a
function of fiber content for 0/90 cross-ply carbon fiber/glass (1).
Fig. In plane coefficient of thermal expansion (-20º to +80ºC) as a function of fiber content for
0/90 cross-ply HM-carbon-fiber-reinforced borosilicate glass
Reference 1. K.M. Prewo and E.J. Minford, “Thermal Stable Composites – Graphite Reinforced
Glass,” Proc. Of SPIE – Intl. Soc. For Opt. Eng., 505 (Aug. 1984) 188-191.
CHAPTER 8
8.1
This is a very serious problem. Cutting and trimming of prepregs results in a considerable
quantity of scrap. Recovery and recycling of carbon fibers is an economical proposition because
of the high cost of carbon fibers. Carbon fibers in a thermoset matrix are the biggest problem.
Use as landfill is perhaps the least desirable method. There are two methods that can be used to
recover the carbon fibers:
i. Epoxy in the prepreg is only partially cured and thus is soluble in common organic
solvents such as acetone or methyl ethyl ketone. This process can be used to remove resin
and the fiber sizing. It is important that the carbon fiber surface should not suffer any
damage because that will affect its subsequent use as a short fiber reinforcement.
ii.
Thermal degradation. Essentially, this method is to burn the resin matrix and recover the
carbon fibers. Carbon fibers get oxidized in air at about 400ºC when heated in air.
Therefore, one should remove the resin matrix at temperatures less than 400ºC in air or at
higher temperatures in an inert atmosphere.
There is no economical way as yet to remove a thermoset resin such as an epoxy from fully
cured laminates. That is where thermoplastic matrix composites come in. They have the
advantage that they can be repeatedly melted and reprocessed. But, it should be pointed out that
the resin properties degrade with each heat exposure. Thus, it is likely that the aerospace scrap or
a used aircraft part consisting of a thermoplastic matrix containing carbon fibers will be recycled
for use in some sector that does not have very rigorous and high specifications.
8.2
Epoxy and less frequently polyester resins are typical examples of thermosetting resins.
Polyimide resins can have a use-temperature between 225º and 300 ºC. Chemically,
condensation-type polymides are thermoplastic. Examples are LARC TPI and Avimid N.
Addition-type polymides are thermosets. Examples include bismaleimide resins such as PMR15, Thermid MC-600, and IP-600.
A variety of thermoplastic resins is available. These have a linear molecular structure and are
repeatedly meltable, i.e., unlike thermosets, thermoplastics can be reprocessed. Some of the
commercial, so-called high temperature thermoplastics are: polyetheretherketone (PEEK),
Polysulfone (Udel P-1700), Polyphenylene sulfide (Ryton), etc.
8.3
Interface is the essentially bidimensional region between any two phases. If another phase is
introduced deliberately or if it forms due to a reaction between the matrix and reinforcement,
then this new phase will be called an interphase. Note that the presence of an interphase will
create two interfaces: reinforcement/interphase and interphase/matrix.
8.6
The failure of Columbia space shuttle was initiated by the impact of the thermal protection
system (TPS) by a piece of insulating foam from an external tank. This foam impact caused a
hole in the TPS made of C/C panel. This resulted in a breach of the thermal protection system on
the leading edge of the left wing. During reentry of the vehicle, hot gases entered the shuttle
through the hole, melted the aluminum structure and led to the tragedy.
CHAPTER 9
9.1
Nb3Al, Nb3Ga, and Nb3Ge cannot be prepared directly by reacting solid Nb with an appropriate
liquid or by a solid state diffusional reaction between Nb and an appropriate bronze. In each
case, one or more solute-rich compounds form which are more stable.
In Nb-Sn and V-Ga system, the Al5 compounds are the only compounds formed by solid state
diffusion.
9.2
Jc of Nb3Sn at high magnetic fields is inversely proportional to its grain size until the grain
diameter becomes less than 30-50 nm. Below this grain size, Jc decreases. If no special efforts
are made, the grain size in Nb3Sn and V3Ga tends to be several hundreds of nm. The bronze
route allows the use of low temperatures suited to minimizing the grain size and maximizing the
grain boundary area and current densities.
9.3
Any left over, unreacted bronze will dilute the intrinsic superconducting critical current density
of the Al5 compound. Similar is the effect of any non-superconducting metallic reinforcing
elements and the pure copper which is used for electrical stability.
9.4
Nb3Sn, the Al5 compound, becomes unstable below ~ 775ºC. Thus, either the phase diagram is
wrong or the presence of copper in the bronze extends the stability range of the Al5 compound to
lower temperatures.
9.5
Yes, because a very large scale use of superconductors is in fusion reactors and high energy
accelerators (e.g., the superconducting supercollider) where irradiation is likely to introduce
defects which, in turn, will affect the Tc, Jc, and Hc2 values.
9.8
The main sources of mechanical loading in superconducting composites in large magnets are:
i.
Fabrication induced stresses such as the bending stress as the superconductor wire is
wound into a coil and the uniaxial tension due to pretensioning of wire.
ii.
Thermal expansion and contraction differences between the superconductor and any of
the support structure. Temperature gradients may result during cooling or heating. In
particular, when a portion of the superconductors reverts (quenches) to the normal high
temperature state, the remainder of the superconductor will be put in tension.
Temperature gradients exacerbate the already complex situation due to the differential in
the expansion of different components.
iii.
Electromagnetic forces of very high magnitude can develop in large solenoids due to the
Lorentz forces. These are also called magnetic stresses and they come into being when
the superconducting winding is energized. The magnetic hoop stress in each wire in a
solenoid is given by
σ = J.B.R
where J is the current density, B is the magnetic field strength, and R is the radius of
winding. Clearly, Lorentz force increases linearly with the size of the magnet (radius). In small
research magnets, this stress will generally be less than 0.1 Gpa. But in a large magnet having a
bore several meters in diameter, the magnetic hoop stress can be greater than 1 Gpa, i.e., much
more than the fracture strength of the superconductor or any of the support structure materials.
This is generally taken care of by providing a rigid clamping system.
CHAPTER 10
10.1
10.2
Halpin-Tsai equaitons
10.3
10.4
10.5
For fiber failure to occur, i.e., to avoid interface failure, we have the following condition
Taking ℓ/d (=1000) of the fiber equal to (ℓ/d)c, we get the minimum interfacial strength
required to avoid interface failure:
10.6
10.7
10.8
10.9
10.10
Note, we have taken Ef2 to be 162 GPa, slightly less than 05 Ef1.
10.11
10.12
10.13
E11 = Vf Ef1 + Em Vm
10.14
(a) For tensile loading of the compsoite wire, we have
The component that has a lower yeild strain will yield plastically first. The yield strain of a
component can be found by using Hooke’s law: ey = σy/E. Thus,
Therefore, copper will yield first.
(b)
Using the rule-of-mixtures, we have the composite yield strength as the strength corresponding
to a strain of 6.6 10 –4 (see part (a) above)
(c)
CHAPTER 11
11.1
11.2
11.3
11.5
11.6
11.7
CHAPTER 12
12.1
12.2
12.3
12.4
CHAPTER 13
13.1
Fatigue crack initiation sites:
•
Voids
•
Inclusions
•
Interface between the reinforcement and the matrix
•
Interface between laminae
•
Free edges of a laminate with a ply sequence that results in out of plane stresses at the
free edge
•
Extremities of short fiber or whiskers
13.2
Hysteretic heating can be a problem in PMCs. Polymers, being poor conductors of heat,
do not dissipate heat easily. It is not uncommon to generate temperature differences
between the surface and the interior of a PMC under conditions of cyclic fatigue. All other
things being equal, the hysteretic heating will increase as a function of frequency. Such
heating can lead to a decrease in the fatigue life of the PMC.
Similar effects can be expected in CMC but the softening due to heating may not be as
critical as in PMCs.
CHAPTER 14
14.2
Angle
(deg)
0
5
10
15
20
25
30
35
45
90
Tensile
(GPa)
1.5
0.7
0.37
0.24
0.18
0.13
0.1
0.085
0.065
0.04
Compression
(GPa)
-1.5
-1.05
-0.7
-0.5
-0.35
-0.25
-0.22
-0.21
-0.2
-0.25
The plot shows the variation of stresses with angle.
2.0
Tensile
1.5
Compression
Stress [ GPa ]
1.0
0.5
0.0
0
10
20
30
40
50
-0.5
-1.0
-1.5
-2.0
Angle [ o ]
60
70
80
90
http://www.springer.com/978-0-387-74364-6