Composite materials science and engineering 3rd edition pdf

Composite Midterm

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 between Kevlar and Nomex is thus in the orientation of the aromatic rings. This, o f course, results i n a spectacular difference in strain -to- fracture values for the two: 28% for Nomex and 2-4% for Kevlar. Kevlar h a s also slightly higher 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 Muker jee (4) de r ived 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. Semi- crystalline 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.

(1) D.R.H. Jones and G.J. May, Acta Met., 23 (1975) 29

References

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. A greater control of the composition as well as the degree of homogeneity is attainable ii. Possibility of forming unique multiphase matrices

iii. The fluid starting materials have a relative ease of penetrating a fibrous perform. iv. 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 PMR- 15, 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)

Tensile (GPa)

Compression (GPa)

0 1.5 -1.5 5 0.7 -1.05

10 0.37 -0.7 15 0.24 -0.5 20 0.18 -0.35 25 0.13 -0.25 30 0.1 -0.22 35 0.085 -0.21 45 0.065 -0.2 90 0.04 -0.25

The plot shows the variation of stresses with angle.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 10 20 30 40 50 60 70 80 90

St re

ss [

G Pa

]

Angle [ o ]

Tensile

Compression