AERO 213 – Science of Materials CHAPTER 7 – Mechanical Properties • Key mechanical design properties: stiffness, strength, hardness, durability, and toughness. • Stress on a material consists of the nature of the applied load and its duration, as well as

environmental conditions. The load can be tensile, compressive, or shear, and its magnitude may be constant with time or may fluctuate continuously.

• To utilize the correct material, one must understand the relationships between the microstructure (i.e. internal features) of materials and their mechanical properties.

7.2 – Concepts of Stress and Strain • One of the most common mechanical stress-strain tests is performed in tension.

o A specimen is deformed, usually to fracture, with a gradually increasing tensile load that is applied axially along the long axis of a specimen.

o Destructive – the test specimen is permanently deformed and usually fractured. o Recorded as load or force versus elongation.

o Engineering Stress – 0

FA

σ = (7.1)

where F is the instantaneous load applied perpendicular to the specimen cross section, and 0A is the cross-sectional area. Units of MPa.

o Engineering Strain – 0

0 0

il l ll l

ε − ∆= = (7.2)

where 0l is the original length before a load is applied and il is the instantaneous length. Strain is unitless, but meters per meters or inch per inch are often used. Sometimes expressed as a percentage in which the strain is multiplied by 100.

o Compression tests are conducted if the in-service forces are of this type. The same equations as tension tests are utilized, however the resulting values are negative. Tension tests are usually performed because they are easier to conduct, and for most

structural applications, this information is not needed. Compressive tests are used when a material’s behavior under large and permanent (i.e. plastic) strains is desired, or when the material is brittle in tension.

o Shear and Torsional Tests – 0

FA

τ = (7.3)

where F is the load or force imposed parallel to the upper and lower faces each of which has an area 0A Shear Strain – Defined as the tangent of the strain angle θ and represented by γ Torsion is a variation of pure shear in which a structural member is twisted. Torsional

forces produce a rotational motion about the longitudinal axis of one end of the member relative to the other end.

A shear stress τ is a function of the applied torque T , whereas shear strain γ is related to the angle of twist φ .

7.3 – Stress-Strain Behavior • The degree to which a structure deforms or strains depends on the magnitude of an imposed

stress. Stress and strain are proportional through this relationship for most metals that are stressed in tension at relatively low levels. o Hooke’s Law – Eσ = ∈ (7.5)

where E is the constant of proportionality (GPa or psi) and is known as the modulus of elasticity or Young’s modulus.

o Moduli of elasticity of groups of materials: polymers < metals < ceramics o Elastic Deformation – Deformation in which stress and strain are proportional. Results in a linear relationship.

o The modulus of elasticity, E , may be thought of as stiffness, or a material’s resistance to elastic deformation. The greater the modulus, the stiffer is the material, or the smaller is the elastic strain that results from the application of a given stress.

o Elastic deformation is non-permanent. o For a non-linear relationship the tangent or secant modulus is normally used.

• Shear stress and shear strain are proportional to each other…

Gτ γ= (7.7) where G is the shear modulus – the slope of the linear elastic region of the shear stress-strain curve.

7.5 – Elastic Properties of Materials • When a tensile stress is imposed on a metal specimen, an elastic elongation and

accompanying strain z∈ result in the direction of the applied stress (arbitrarily taken to be the z direction), as indicated in Figure 7.9. As a result of this elongation, there will be constrictions in the lateral (x and y) directions perpendicular to the applied stress; from these contractions, the compressive strains x∈ and y∈ may be determined. If the applied stress is uniaxial (only in the z direction) and the material is isotropic, then x y∈ =∈ . A parameter termed Poisson’s ratio v is defined as the ratio of the lateral and axial strains and is

represented by the expression yx

z z

v∈∈

= − = −∈ ∈

and 0

xyzd

d∆

∈ = (7.8)

• For isotropic materials, shear and elastic moduli are related to each other and to Poiosson’s ratio according to ( )2 1E G v= + (7.9) o In most metals, G is about 0.4E . Thus, if one modulus is know, the other may be

approximated. o Isotropic – Identical in all areas; no variation.

• Plastic Deformation – Occurs at the point where the stress is no longer proportional to the strain, and permanent, nonrecoverable changes occur. o Occurs very gradual for metals and increases more rapidly with rising stress.

7.6 – Tensile Properties • Yielding – Occurs when the gradual elastic-plastic transition initially departs from linearity

of the stress-strain curve. o Also known as the proportional limit.

• Yield Strength – The stress corresponding to the intersection of the elastic portion of the line and the stress-strain curve as it bends over in a plastic region. Units of (MPa or psi) o Yield-Point Phenomenon – Occurs when the elastic-plastic transition is very well defined

and occurs abruptly. • Tensile Strength – The stress at the maximum on the engineering stress-strain curve.

o All deformation up to this point is uniform throughout the region of the tensile specimen. • Necking – When the fracture ultimately occurs at the neck.

• Ductility – A measure of the degree of plastic deformation that has occurred at a fracture.

o Brittle – A metal that experiences very little or no plastic deformation upon fracture. o Expressed quantitatively as either percent elongation or percent reduction in area.

0

0

% 100fl lEL

l−

= ×

(7.11)

where fl is the fracture length, and 0l is the original gauge length.

o Percent Reduction in Area – 0

0

% 100fA ARA

A−

= ×

(7.12)

where 0A is the original cross-sectional area and fA is the cross-sectional area at the point of fracture.

o Ductility is important because it indicates to a designer the degree to which a structure will deform plastically before fracture, and it specifies the degree of allowable deformation during fabrication operations.

o Brittle materials are approximately considered to be those having a fracture strain of >5%. o For metals, yield and tensile strengths decrease with increasing temperatures, while

ductility increases with increasing temperatures. • Resilience – The capacity of a material to absorb energy when it is deformed elastically and

then, upon unloading, to have this energy recovered.

o Modulus of Resilience – 0

y

rU dσ∈

= ∈∫ (7.13a)

which is the strain energy per unit volume required to stress a material from an unloaded state up to the point of yielding.

Assuming a linear elastic region, 12r y yU σ= ∈ (7.13b)

where y∈ is the strain at yielding.

• Incorporating Hooke’s Law, 21 1

2 2 2y y

r y y yUE Eσ σ

σ σ

= ∈ = =

(7.14)

• Toughness o A property that is indicative of a material’s resistance to fracture when a crack (or other

stress-concentrating defect) is present. o The ability of a material to absorb energy and plastically deform before fracturing. o For the static (low strain rate) situation, a measure of toughness in metals (derived from

plastic deformation) may be ascertained from the results of a tensile stress-strain test. It is the area under the σ −∈ curve up to the point of fracture.

o For a metal to be tough, it must display both strength and ductility. 7.7 – True Stress and Strain • True Stress – Defined as the load F divided by the instantaneous cross-sectional area tA over

which the deformation is occurring (i.e. the neck, past the tensile point)

Tf

FA

σ = (7.15)

• True Strain – 0

ln tT

ll

∈ = (7.16)

• If no volume change occurs, then ( )1 and ln(1 )T Tσ σ= +∈ ∈ = +∈ (7.18a-b) o Valid only to the onset of necking; beyond this point true stress and strain should be

computed from actual load, cross-sectional area, and gauge length measurements. • True stress-true strain relationship in the plastic region of deformation (to the point of

necking), n

T TKσ = ∈ (7.19) where and K n are constants that vary from alloy to alloy. The parameter n is often termed the strain-hardening exponent and has a value less than unity.

7.8 – Elastic Recovery After Plastic Deformation • Upon release of the load during the course of a stress-strain test, some fraction of the total

deformation is recovered as elastic strain. There will also be an elastic strain recovery associated with the fracture.

7.9 – Compressive, Shear, and Torsional Deformation • Metals may experience plastic deformation under the influence of applied compressive, shear,

and torsional loads. The resulting stress-strain behavior into the plastic region will be similar to the tensile counterpart. However, for compression, there will be no maximum because necking does not occur; furthermore, the mode of fracture will be difference from that for tension.

CERAMICS 7.10 – Flexural Strength • The stress-strain behavior of ceramics is not usually ascertained by a tensile test because…

o It is difficult to prepare and test specimens having the required geometry. o It is difficult to grip brittle materials without fracturing them. o Ceramics fail after only about 0.1% strain, which necessitates that tensile specimens be

perfectly aligned to avoid the presence of bending stresses, which are not easily calculated. • Because of these difficulties, a bending test is more frequently employed.

o A rod with either a circular or rectangular cross-section is bent until fracture. o At the point of loading, the top surface of the specimen is placed in a state of compression,

whereas the bottom surface is in tension. o Stress is computed from the specimen thickness, the bending moment, and the moment of

inertia of the cross section. • The maximum tensile stress exists at the bottom specimen surface directly below the point of

load application. o The stress at fracture using this flexure test is known as the flexural strength, modulus of

rupture, fracture strength, or bend strength. o For a rectangular cross section, the flexural strength fsσ is given by…

2

32

ffs

F Lbd

σ = (7.20a)

where fF is the load at fracture, L is the distance between support points.

o For a circular cross section, the flexural strength fsσ is given by…

3f

fs

F LR

σπ

= (7.20b)

where R is the specimen radius. • Flexural strength values will vary depend on specimen size. 7.11 – Elastic Behavior • The moduli of elasticity for ceramic materials are slightly higher than for metals. POLYMERS 7.13 – Stress-Strain Behavior • Three types of stress-strain behavior for polymers…

• Modulus of elasticity and ductility in percent elongation are determined for polymers in the

same manner as for metals. • Polymers are have much less tensile strength than metals but have much more plasticity. • Polymers are also much more sensitive to temperature. 7.16 – Hardness • Hardness is a measure of a materials resistance to localized plastic deformation (e.g. a small

dent or a scratch) • Hardness tests are performed more frequently than any other mechanical test because…

o They are simple and inexpensive. o The test is nondestructive. o Other mechanical properties may be estimated from hardness data, like tensile strength.’

• The Rockwell tests constitute the most common method used to measure hardness because they are so simple to perform and require no special skills. o A hardness number is determined by the difference in depth of penetration resulting from

the application of an initial minor load followed by a larger major load. The use of a minor load enhances test accuracy.

o Inaccuracies will result if the hardness goes outside a specified range, and will also occur if the specimen is too thin, if an indentation is made too near a specimen edge, or if two indentations are made too close to one another.

o The specimen thickness should be at least 10 times the indentation depth, whereas allowance should be made for at least three indentation diameters between the center of one indentation and the specimen edge, or to the center of a second indentation.

• The Brinell tests consist of a hard, spherical indenter being forced into the surface of the metal in question, similar to the Rockwell tests.

• Hardness is not a well-defined material property, and therefore conversion between the hardness scales of the various tests is difficult.

• Tear strength is the energy required to tear apart a cut specimen of standard geometry. • Correlation Between Hardness and Tensile Strength

( ) 3.45TS MPa HB= × (7.25)

( ) 500TS psi HB= × (7.25)

• Computation of Average Value – 1

n

ii

xx

n==∑

(7.26)

• Computation of Standard Deviation – ( )

1/22

1

1

n

ii

x xs

n=

− =

−

∑ (7.27)

7.20 – Design/Safety Factors • Design Stress – 'd cNσ σ= (7.28) • Safe Stress – Also known as working stress, is based on the yield strength of the material and

is defined as the yield strength divided by a factor of safety, N.

yw N

σσ = (7.29)

o Usually preferred as it is based on the anticipated maximum applied stress instead of the yield strength of the material.

o Values for N usually range from 1.2 to 4.0. CHAPTER 2 – Atomic Structure and Interatomic Bonding • Force-Potential Energy Relationships for Two Atoms –

E F dr= ∫ (2.4)

• Bonding Energy – The energy that would be required to separate these two atoms to an infinite separation.

• Primary or Chemical Bonds o Ionic Ionic bonding is bonding that occurs in compounds that are composed of both metallic

and nonmetallic elements. Atoms of a metallic element easily give up their valence electrons to the nonmetallic. Coulombic – The attractive force between positive and negative ions by virtue of their

net electrical charge. Ionic materials have relatively large bonding energies, which results in high melting

temperatures. Ionic materials are characteristically hard and brittle, and, furthermore, electrically

and thermally insulative.

For two isolated ions, the attractive energy is…

AAEr

= − (2.8)

An analogous equation for repulsive energy is…

R n

BEr

= (2.9)

Ionic bonding is termed nondirectional; that is, the magnitude of the bond is equal in all directions around an ion.

o Covalent Covalent bonding occurs when stable electron configurations are assumed by the

sharing of electrons between adjacent atoms. Each atom will share an electron for the bond and it is considered to belong to both.

Covalent bonding is termed directional; that is, it is between specific atoms and may exist only in the direction between one atom and another that participates in the electron sharing.

Covalent bonds may either be very strong, or very weak. o Metallic Metallic bonding is found in metals and their alloys. Metallic materials have one, two,

or at most three valence electrons. These valence electrons are not bound to any particular atom in the solid and are more or less free to drift throughout the entire metal. Thus, these valence electrons form a “sea of electrons” or an “electron cloud”.

The remaining nonvalence electrons and atomic nuclei form ion cores. These ion cores possess a net positive charge equal in magnitude to the total valence electron charge per atom. The free roaming electrons act as the glue for the ion cores.

Metallic bonds may either be very strong, or very weak. Metals are good conductors of both electricity and heat. Most metals and their alloys fail in a ductile manner at room temperature.

• Secondary, van der Waals, or Physical Bonds o These bonds are weak in comparison to the primary or chemical ones. o Secondary bonding exists between virtually all atoms or molecules, but its presence may

be obscured if a type of primary bonding is present. o Secondary bonding forces arise from atomic or molecular dipoles. Exists whenever there is

some separation of positive and negative portions of an atom or molecule. The bonding results from the coulombic attraction between the positive end of one dipole, and the negative region of an adjacent one.

o Hydrogen bonding is found to exist between some molecules that have hydrogen as one of the constituents.

o Fluctuating Induced Dipole Bonds A dipole may be created or induced in an atom or molecule that is normally electrically

symmetric. All atoms experience constant vibrational motion that can cause instantaneous and short-lived distortions of this electrical symmetry for some of the atoms or molecules and the creation of small electric dipoles. One of these dipoles can in turn produce a displacement of the electron distribution of an adjacent molecule or atom, which induces the second one also to become a dipole that is then weakly attracted or bonded to the first.

May occur between large groups of atoms or molecules, which forces are temporary and fluctuate with time.

o Polar Molecule-Induced Dipole Bonds Polar Molecules – Molecules in which permanent dipole moments exist by virtue of an

asymmetrical arrangement of positively and negatively charged regions. Polar molecules can also induce dipoles in adjacent nonpolar molecules, and a bond will

form as a result of attractive forces between the two molecules. Furthermore, the magnitude of this bond will be greater than for fluctuating induced dipoles.

o Permanent Dipole Bonds Van der Waals forces will also exist between adjacent polar molecules, and the

associated bonding energies are significantly greater than for bonds involving induced dipoles. The strongest secondary bonding type, the hydrogen bond, is a special case of polar molecular bonding.

• Water experiences a 9% volume increase upon freezing. o This occurs because in solid ice, each water molecule participates in four hydrogen bonds,

whereas liquid water only participates in two. CHAPTER 3 – Structure of Metals • A crystalline materials is one in which the atoms are situated in a repeating or periodic array

over large atomic distances; that is, long range order exists, such that upon solidification, the atoms position themselves in a repetitive three-dimensional pattern in which each atom is bonded to its nearest-neighbor atoms.

• The crystal structure of the material is the manner in which atoms, ions, or molecules are spatially arranged.

• A lattice (which respect to crystal structures) is a three-dimensional array of points coinciding with atom positions (or sphere centers).

• Unit cells are small entities comprised of a repetitive form of atoms. Also described as the basic structural unit or building block of the crystal structure that defines the crystal structure by virtue of its geometry and the atom positions within.

• Face-Centered Cubic Crystal Structure o The crystal structure found for many metals has a unit cell of cubic geometry with atoms

located at each of the corners and the centers of all the cube faces. 2 2a R=

o The number of atoms per unit cell can be computed using the following…

2 8

where the number of interior atoms the number of face atoms the number of corner atoms

f ci

i

f

c

N NN N

NNN

= + +

==

=

o Coordination Number – The number of nearest-neighbor or touching atoms. o Atomic Packing Factor (APF) – The sum of the sphere volumes of all atoms within a unit

cell (assuming the atomic hard-sphere model) divided by the unit cell volume. volume of atoms in a unit cellAPF

total unit cell volumeS

C

VV

= =

• Body-Centered Cubic Crystal Structure

o A metallic crystal structure having a cubic unit cell with atoms located at all eight corners and a single atom at the cube center.

43Ra =

o The number of atoms per BCC unit cell is…

2 8f c

i

N NN N= + +

• Hexagonal Close-Packed Crystal Structure o The top and bottom faces of the unit cell consist of six atoms that form regular hexagons

and surround a single atom in the center. o Number of atoms per HCP unit cell…

2 6f c

i

N NN N= + +

• Density Computations – Metals o Calculation of theoretical density through knowledge of the crystal structure…

where number of atoms associated with each unit cell atomic weight volume of the unit cell = Avagadro's n

C A

c

A

nAV N

nAVN

ρ =

===

23umber (6.022 10 atoms/mole)

×

• Because ceramics are composed of at least two elements and often more, their crystal structures are generally more complex than those of metals. The atomic bonding in these materials ranges from purely ionic to totally covalent.

• Cations – Positively charged because they have given up their valence electrons to the nonmetallic ions.

• Anions – Receive the negative charge from the cations CARBON • Carbon is an element that exists in various polymorphic forms as well as in the amorphous

state. o Diamond is a metastable carbon polymorph at room temperature and atmospheric

pressure. Its crystal structure is a variant of the zinc blende structure in which carbon atoms occupy all positions. Each carbon bonds to four other carbons, and these bonds are completely covalent.

o Graphite has a different crystal structure than diamond and is also more stable than diamond at ambient temperature and pressure. The graphite structure is composed of layers of hexagonally arranged carbon atoms; within the layers, each carbon atom is bonded to three coplanar neighbor atoms by strong covalent bonds. The fourth bonding electron participates in a weak van der Waals type of bond between the layers.

POLYMORPHISM AND ALLOTROPY • Polymorphism – When a metal or nonmetal has more than one crystal structure. • Allotropy – Occurs when polymorphism is found in elemental solids. CRYSTAL SYSTEMS • Lattice Parameters – The unit cell geometry is completely defined in terms of six parameters:

the three edge lengths , , and a b c and three interfacial angles , , and α β γ . • Each of these parameters represents a distinct crystal system, which is composed of the

following seven crystal systems: cubic, tetragonal, hexagonal, orthorhombic, rhombohedral (trigonal), monoclinic, and triclinic.

CRYSTALLOGRAPHIC DIRECTIONS • A crystallographic direction is defined as a lone directed between two points, or a vector. The

following steps are used to determine the three directional indices: o A vector of convenient length is positioned such that it passes through the origin of the

coordinate system. Any vector may be translated throughout the crystal lattice without alteration if parallelism is maintained.

o The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a,b,and c.

o These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values

o The three indices, not separated by commas, are enclosed in square brackets, thus: [uvw]. The u, v, and w integers correspond to the reduced projections along the x, y, and z axes, respectively.

CRYSTALLOGRAPHIC PLANES • Miller indices are used to graphically represent crystallographic planes.

CLOSE-PACKED CRYSTAL STRUCTURES

SINGLE CRYSTALS • Single Crystal – The result of a crystalline solid when the periodic and repeated arrangement

of atoms is perfect or extends throughout the entirety of the specimen without interruption. POLYCRYSTALLINE MATERIALS • Grains – A collection of many small crystals. • Polycrystalline – A crystalline structure composed of grains, or small collections of crystals. • Grain Boundary – An atomic mismatch within the region where two grains meet. ANISOTROPY • Anisotropy is the directionality of properties of a single crystal. • Isotropic – Substances where measured properties are independent of direction NONCRYSTALLINE SOLIDS • Noncrystalline solids lack a systemic and regular arrangement of atoms over relatively large

atomic distances. • Polymers may be completely noncrystalline or sometimes semicrystalline consisting of

varying degrees of crystallinity. CHAPTER 5 – Imperfection in Solids • Crystalline Defect – A lattice irregularity having one or more of its dimensions on the order of

an atomic diameter. • Point Defect – Defects associated with one or two atomic positions. POINT DEFECTS IN METALS • Vacancy – The simplest of the point defects, also known as a vacant lattice site where an

atom normally occupies but from which it is missing. • ALL crystalline solids contain vacancies, and it is not possible to create such a material that

is free of these defects. The presence of vacancies increases the entropy of the crystal. • The equilibrium number of vacancies vN for a given quantity of material depends on and

increases with temperature…

exp vv

QN NkT

= −

where N is the total number of atomic sites, vQ is the energy required for the formation of a vacancy,T is the absolute temperature in kelvins, and k is the gas or Boltzmann’s constant.

23 51.38 10 J/atom K or 8.62 10 eV/atom Kk − −= × ⋅ × ⋅ Thus, the number of vacancies increases exponentially with temperature.

• Self-Interstitial – An atom from the crystal that is crowded into an interstitial site – a small void space that under ordinary circumstances is not occupied.

POINT DEFECTS IN CERAMICS • Defect Structure – Often used to designate the types and concentrations of atomic defects in

ceramic materials. • Electroneutrality – The state the exists when there are equal numbers of positive and

negative charges from the ions. • Frenkel Defect – A defect that involves a cation-vacancy and a cation-interstitial pair. • Schottky Defect – A defect found in AX materials consisting of a cation/anion-vacancy pair.

• Stoichiometry – Defined as a state for ionic compounds wherein there is the exact ration of

cations to anions predicted by the chemical formula. • A ceramic compound is nonstoichiometric if there is any deviation from this exact ratio. IMPURITIES IN SOLIDS • Alloys – Metals in which impurity atoms have been added intentionally to impart specific

characteristics to the material. o Alloying is used in metals to improve mechanical strength and corrosion resistance.

• Solid Solution – Formed as a result of the addition of impurity atoms to a metal. • Solvent – The element or compound that is present in the greatest amount. • Solute – Used to denote an element or compound present in a minor concentration. • Two types of impurity point defects are found in solid solutions…

o Substitutional – Solute or impurity atoms replace or substitute for the host atoms. Several factors determine the degree to which the impurities dissolve to the host.

• Atomic Size Factor • Crystal Structure • Electronegativity • Valences

o Interstitial – Impurity atoms fill the voids or interstices among the host atoms. For metallic materials that have relatively high atomic packing factors, these

interstitial positions are relatively small. Consequently, the atomic diameter of an interstitial impurity must be substantially smaller than that of the host atoms.

SPECIFICATION OF COMPOSITION • The composition of an alloy is expressed in terms of its constituent elements. This is most

commonly done in the two following ways… o Weight Percent – The weight of a particular element relative to the total allow weight.

11

1 2

100mCm m

= ×+

o Atom Percent – The number of moles of an element in relation to the total moles of the elements in the alloy.

' 11

1 2

100m

m m

nCn n

= ×+

'1

11

mmnA

=

• Composition Conversions

(1)(2) Conversion of weight percent to atom percent. (3)(4) Conversion of atom percent to weight percent. (5)(6) Conversion of weight percent to mass per unit volume. (6)(7) Computation of density. (8)(9) Computation of atomic weight. DISLOCATIONS – LINEAR DEFECTS • A dislocation is considered to be a linear or one-dimensional defect around which some of the

atoms are misaligned. • Edge Dislocation – A linear defect that centers on the line that is defined along the end of the

extra half-plane of atoms. • Dislocation Line – The line in the edge dislocation that is perpendicular to the plane. • Screw Dislocation – Formed by a shear stress that is applied. • Mixed Dislocations – Exhibit characteristics of all three dislocation types. INTERFACIAL DEFECTS • Interfacial defects are boundaries that have two dimensions and normally separate regions of

the materials that have different crystal structures and/or crystallographic orientations. o External surfaces are one of the most obvious boundaries along which the crystal

structure terminates. Surface atoms are not bonded to the maximum number of nearest neighbors and are therefore in a higher-energy state than the atoms at interior positions. The bonds of these surface atoms that are not satisfied give rise to a surface energy which the materials tend to reduce by minimizing the surface area.

o Grain boundaries are the boundaries separating two small grains or crystals having different crystallographic orientations in polycrystalline materials. Within the boundary region, which is probably just several atom distances wide, there is some atomic mismatch in a transition from the crystalline orientation of one grain to that of an adjacent one. When this orientation mismatch is slight, on the order of a few degrees, the term small- (or low-) angle grain boundary is used (edge dislocation). When the angle of misorientation is parallel to the boundary, a twist boundary results (screw dislocation).

o Phase boundaries exist in multiphase materials in which a different phase exists on each side of the boundary; furthermore, each of the constituent phases has its own distinctive physical and/or chemical characteristics.

o Twin boundaries are special types of grain boundaries across which there is a specific mirror lattice symmetry; that is, atoms on one side of the boundary are located in mirror-image positions to those of the atoms on the other side.

ATOMIC VIBRATIONS • Atomic vibrations may be thought of as imperfections that are a result of inconsistencies in

the vibrations of atoms about their lattice positions within the crystal. CHAPTER 8 – Deformation and Strengthening Mechanisms • Edge and screw are the two fundamental dislocation types. • In an edge dislocation, localized lattice distortion exists along the end of an extra half-plane of

atoms, which also defines the dislocation line. • An edge dislocation moves in response to a shear stress applied in a direction perpendicular

to its line. • Slip is the process by which plastic deformation is produced by dislocation motion. • The crystallographic plane along which the dislocation line traverses is the slip plane. • When metals are plastically deformed, some fraction of the deformation energy,

approximately 5%, is retained internally; the remainder is dissipated as heat. The major portion of this stored energy is as strain energy associated with dislocations.

• During plastic deformation, the number of dislocations increases dramatically. • The slip system depends on the crystal structure of the metal and is such that the atomic

distortion that accompanies the motion of a dislocation is a minimum. The slip plane is the plane that has the densest atomic packing.

• Direction of the slip plane corresponds to the direction in this plane that has the highest linear density of packed atoms.

• Resolved Shear Stress – Dependence on applied stress and orientation of stress direction relative to slip plane normal and slip direction. o One slip system is generally oriented most favorably, and is identified as the plane having

the largest resolved shear stress. cos cosRτ σ φ λ=

• Critical Resolved Shear Stress represents the minimum shear stress required to initiate slip and is a property of the material that determines when yielding occurs.

( )maxcos cos

crssy

τσφ λ

=

• The minimum stress necessary to induce yielding occurs when a single crystal is oriented such that 45φ λ= = ; at this point… 2y crssσ τ=

• Slip lines are the lines that appear on the surface of a specimen as a result of dislocations.

• For polycrystalline metals, because of the random crystallographic orientations of the numerous grains, the direction of slip varies from one grain to another.

• The shape of newly formed grains coincides slightly with its neighboring grains. • Polycrystalline metals are stronger than their single-crystal equivalents, which means that

greater stresses are required to initiate slip and the attendant yielding. This is due in large part to the geometrical constraints. If one crystal has the most favorable orientation, the crystal next to it may not which then inhibits slip until a higher stress level is applied.

8.9 – Strengthening by Grain Size Reduction • Typically, ductility is sacrificed when an alloy is strengthened. • The ability of a metal to deform plastically depends on the ability of dislocations to move. • Restricting or hindering dislocation motion renders a material harder and stronger. • The grain barrier acts as a barrier to dislocation motion for two reasons:

o Because the two grains are of different orientations, a dislocation passing into grain B will have to change its direction of motion; this becomes more difficult as the crystallographic misorientation increases.

o The atomic disorder within a grain boundary region will result in a discontinuity of slip planes from one grain into the other.

• A fine-grained material (one that has small grains) is harder and stronger than one that is coarse grained because the former has a greater total grain boundary area to impede dislocation motion.

• Hall-Petch Equation 1/2

0y yk dσ σ −= +

where d is the average grain diameter, and sigma and k are constants for a particular material. 8.10 – Solid-Solution Strengthening • Solid-Solution Strengthening – Strengthening of metals achieved by alloying with impurity

atoms that go into either substitutional or interstitial solid solution. • Alloys are stronger than pure metals because impurity atoms that go into solid solution

typically impose lattice strains on the surrounding host atoms. • The resistance to slip is greater when impurity atoms are present because the overall lattice

strain must increase if a dislocation is torn away from them. Furthermore, the same lattice strain interactions will exist between impurity atoms and dislocations that are in motion during plastic deformation. This, a greater applied stress is necessary to first initiate and the continue plastic deformation for solid-solution alloys, as opposed to pure metals; this is evidenced by the enhancement of strength and hardness.

8.11 – Strain Hardening • Strain hardening is the phenomenon by which a ductile metal becomes harder and stronger

as it is plastically deformed. o Also called work hardening, or, because the temperature at which deformation takes place

is “cold” relative to the absolute melting temperature of the metal, cold working.

0

0

%CW 100dA AA

−= ×

• Through this method, as the hardness and strength increase, the ductility decreases. • Strain hardening is often utilized commercially to enhance the mechanical properties of

metals during fabrication procedures. • In the mathematical expression relation true stress and strain, the parameter n is called the

strain-hardening component.

8.12 – Recovery • During recovery, some of the stored internal strain energy is relieved by virtue of dislocation

motion (in the absence of an externally applied stress), as a result of enhanced atomic diffusion at the elevated temperature.

8.13 – Recrystallization • Recrystallization is the formation of a new set of strain-free and equiaxed grains (i.e., having

approximately equal dimensions in all directions) that have low dislocation densities and are characteristic of the pre-cold-worked condition.

• The new grains form as very small nuclei and grow until they completely consume the parent material, processes that involve short-range diffusion.

• Also, during recrystallization, the mechanical properties that were changed as a result of cold working are restored to their pre-cold-worked values; that is, the metal becomes softer and weaker, yet more ductile.

• Recrystallization Temperature – The temperature at which recrystallization just reaches completion in 1 h.

• Recrystallization proceeds more rapidly in pure metals than in alloys. • Hot working takes place at temperatures above the recrystallization temperature. This allows

the material to remain relatively soft and ductile during deformation because it does not strain harden, and thus large deformations are possible.

8.14 – Grain Growth • Grain growth occurs after recrystallization is complete. The strain-free grains will continue to

grow if the metal specimen is left at the elevated temperature. • As grains increase in size, the total boundary area decreases, yielding an attendant reduction

in the total energy; this is the driving force for grain growth.

0n nd d Kt− =

For many polycrystalline materials, the grain diameter d varies with time t. 8.16 Noncrystalline Ceramics • For ceramics in which the bonding is highly covalent, slip is difficult, and they are brittle

because of the following reasons… o The covalent bonds are relatively strong. o There are limited numbers of slip systems. o Dislocation structures are complex.

• Viscosity – A measure of a noncrystalline material’s resistance to deformation. 8.18 – Semicrystalline Polymers • For many polymers, it has been observed that tensile strength increases with increasing

molecular weight.

n

ATS TSM∞= −

CHAPTER 4 – Polymer Structures 4.2 – Hydrocarbons • A hydrocarbon bond is composed of hydrogen and carbon.

o The bond is covalent. o Each carbon atom has four electrons that may participate in covalent bonding, whereas

every hydrogen atom has only one bonding electron. • Molecules that have double and triple covalent bonds are termed unsaturated.

o This means that each carbon atom is not bonded to the maximum of four other atoms. Therefore, it is possible for another atom or group of atoms to become attached to the original molecule.

• A saturated hydrocarbon is one such that all bonds are single, and therefore no new atoms may be joined without the removal of others that are already bonded.

• Parrafin family molecules have strong covalent bonds in each molecules, but only weak hydrogen and can der Waals bonds exist between the molecules, and thus these hydrocarbons have relatively low melting and boiling points.

4.3 – Polymer Molecules • Polymer molecules are gigantic in size compared to hydrocarbons.

o Often referred to as macromolecules. o Each molecule is bound together by covalent interatomic bonds.

• Repeat units are molecular units that are successively repeated along the chain. o Repeat units are generally enclosed in parentheses, and the subscript n indicates the

number of times it repeats.

o Monomer refers to the small molecules from which a polymer is synthesized. o NOT TO BE CONFUSED WITH EACHOTHER

4.4 – The Chemistry of Polymer Molecules • When all of the repeating units along a chain are of the same type, the resulting polymer is

called a homopolymer. • Chains may be composed of two or more different repeat units called copolymers. • A monomer that has an active bond that may react to form two covalent bonds with other

monomers forming a two-dimensional chainlike molecular structure is considered to be bifunctional in nature.

• Functionality refers to the number of bonds that a given monomer can form. • Trifunctional monomers have three active bonds from which a three-dimensional molecular

network structure results.

4.5 – Molecular Weight • Extremely large molecular weights are observed in polymers with very long chains. • During polymerization, not all of the chains will grow to the same length. This results in a

distribution of chain lengths or molecular weights. • The number-average molecular weight nM is obtained by dividing the chains into a series of

size ranges and then determining the number fraction of chains within each size range.

n i iM x M=∑

Where iM represents the mean (middle) molecular weight of size range i and ix is the fraction of the total number of chains within the corresponding size range.

• A weight-average molecular weight wM is based on the weight fraction of molecules within the various size ranges.

w i iM w M=∑

Where iM represents the mean molecular weight within a size range and iw denotes the weight fraction of molecules within the same size interval.

• The degree of polymerization is an alternative way of expressive average chain size of a

polymer. It represents the average number of repeat units in a chain. nMDP

m=

Where m is the repeat unit molecular weight. • Many polymer properties are affected by the length of the polymer chains.

o For example, the melting or softening temperature increases with increasing molecular weight.

o At room temperature, polymers with very short chains will generally exists as liquids, and as the molecular weight increases, so does the solidity of the material.

4.6 – Molecular Shape • Molecules can form random coils and molecular entanglements. • These random coils and molecular entanglements are responsible for a number of important

characteristics of polymers, including the large elastic extensions displayed by the rubbers. • Rotational flexibility is dependent on repeat unit structure and chemistry.

4.7 – Molecular Structure • The physical characteristics of a polymer depend not only on its molecular weight and shape,

but also on differences in the structure of the molecular chains. • Linear polymers are those in which the repeat units are joined together end to end in single

chains. o These long chains are flexible and may be thought of as a mass of spaghetti. o For linear polymers, there may be extensive van der Waals and hydrogen bonding between

the chains. • Branched polymers are polymers that are synthesized in which side-bar chains are connected

to the main ones. o May be a result of side reactions that occur during the synthesis of the polymer. o The chain packing efficiency is reduced with the formation of side branches, which results

in a lowering of the polymer density. • Cross-linked polymers occur when adjacent linear chains are joined to one another at various

positions by covalent bonds. o The process of crosslinking is achieved either during synthesis or by a nonreversible

chemical reaction. o This crosslinking is often accomplished by additive atoms or molecules that are covalently

bonded to the chains. o Many of the rubber elastic materials are cross-linked, in which this process is called

vulcanization. • Network polymers occur when multifunctional monomers forming three or more active

covalent bonds make three-dimensional networks. o A polymer that is highly cross-linked may also be classified as a network polymer. These

materials have distinctive mechanical and thermal properties; the epoxies, polyurethanes, and phenol-formaldehyde belong to this group.

• Polymers are not usually of only one distinctive structural type. A predominantly linear polymer hay have limited branching and crosslinking.

4.8 – Molecular Configurations • Isomerism is found in polymer molecules wherein different atomic configurations are possible

for the same composition. o Stereoisomerism denotes the situation in which atoms are linked together in the same

order (head to tail) but differ in their spatial arrangement. Isostatic configuration occurs when all the R groups are situated on the same side of

the chain.

Solid wedges represent bonds that project out of the plane of the page, and dashed

ones represent bonds that project into the page. Syndiotactic configuration the R groups alternate sides of the chain.

Atactic configuration describes random positioning of the R groups. o Geometrical isomerism are possible within repeat units having a double bond between

chain carbon atoms. A cis structure is one in which the side group bonded to the carbon atom exists on

one side of the chain.

A trans structure occurs when the bonds are on opposite sides of the chain.

• SUMMARIZATION:

o Polymer moleculaes may be characterized in terms of their size, shape, and structure. o Molecular size is specified in terms of molecular weight or degree of polymerization. o Molecular shape relates to the degree of chain twisting, coiling, and bending. o Molecular structure depends on the manner in which structural units are joined together.

Linear, branched, cross-linked, and network structures are all possible, in addition to several isomeric configurations (isotactic, syndiotactic, atactic, cis, and trans).

4.9 – Thermoplastic and Thermosetting Polymers • Thermoplastic polymers soften when heated (and eventually liquefy) and harden when cooled.

o This process is totally reversible and may be repeated. o On a molecular level, as the temperature is raised, secondary bonding forces are

diminished (by increased molecular motion) so that the relative movement of adjacent chains is facilitated when a stress is applied.

o Most linear polymers and those having some branched structures with flexible chains will generally be thermoplastic. These materials are normally fabricated by the simultaneous application of heat and pressure.

• Thermosetting polymers are network polymers. o They become permanently hard during their formation and do not soften upon heating. o Network polymers have covalent crosslinks between adjacent molecular chains that,

during heating, anchor the chains together to resist the vibrational and rotational chain motions at high temperatures.

• Thermoset polymers are generally harder and stronger than thermoplastics and have better dimensional stability.

4.10 – Copolymers • A block copolymer is one in which identical repeat units are clustered in blocks along the

chain and a graft polymer is one in which homopolymer side branches of one type may be grafted to homopolymer main chains that are composed of a different repeat unit.

• Degree of polymerization for a copolymer...

i jm f m=∑

Where if and jm are, respectively, the mole fraction and molecular weight of repeat unit j in the polymer chain.

4.11 – Polymer Crystallinity • Polymer crystallinity can be thought of as the packing of molecular chains to produce an

ordered atomic array. • Molecular substances having small molecules (e.g., water and methane) are normally either

totally crystalline (as solids) or totally amorphous (as liquids). • As a consequence of their size and complexity, polymer molecules are often only partially

crystalline (or semicrystalline), having crystalline regions dispersed within the remaining amorphous material.

• Percent crystallinity depends on specimen density, and densities of totally crystalline and totally amorphous materials.

( )( )

% crystallinity = 100c s a

s c a

ρ ρ ρρ ρ ρ

−×

−

o The degree of crystallinity of a polymer depends on the rate of cooling during solidification as well as on the chain configuration.

o For linear polymers, crystallization is easily accomplished because there are few restrictions to prevent chain alignment. Any side branches interfere with crystallization, such that branched polymers never are highly crystalline; in fact, excessive branching may percent any crystallization whatsoever.

o Most network and cross-linked polymers are almost totally amorphous due to the crosslinking prevents alignment into a crystalline structure.

o For copolymers, as a general rule, the more irregular and random the repeat unit arrangements, the greater is the tendency for the development of noncrystallinity.

o Crystalline polymers are usually stronger and more resistant to dissolution and softening by heat.

4.12 – Polymer Crystals

CHAPTER 5 – Imperfections in Solids 5.1 – Introduction • ALL crystalline materials contain large numbers of various defects or imperfections. • A crystalline defect refers to a lattice irregularity having one or more of its dimensions on the

order of an atomic diameter. • Point defects are those that are associated with one or two atomic positions. 5.2 – Point Defects • A vacancy or vacant lattice site is the simplest of the point defects, and is a site that would

normally be occupied but from which an atom is missing. o ALL crystalline solids contain vacancies, and, in fact, it is not possible to create such a

material that is free of these defects. o The presence of the vacancies increases the entropy of the crystal.

• The equilibrium number of vacancies for a given quantity of material depends on and increases with temperature according to…

exp vv

QN NkT

= −

Where N is the total number of atomic sites, vQ is the energy required for the formation of a vacancy, T is the absolute temperature in kelvins, and k is the gas or Boltzmann’s constant. o Boltzmann’s constant has a value of 23 51.38 10 J/atom K or 8.62 10 eV/atom Kk − −= × ⋅ × ⋅ o Thus, the number of vacancies increases exponentially with temperature.

• A self-interstitial is an atom from the crystal that is crowded into an interstitial site – a small void space that under ordinary circumstances is not occupied. o In metals, a self-interstitial introduces relatively large distortions in the surrounding

lattice because the atom is substantially larger than the interstitial position in which it is situated. Therefore, the formation of this defect is not highly probable, and it exists in very small concentrations, which are significantly lower than for vacancies.

5.3 Point Defects in Ceramics • Point defects involving host atoms may exist in ceramic compounds, however, because

ceramic materials contain ions of at least two kinds, defects for each ion type may occur. • The term defect structure is often used to designate the types and concentrations of atomic

defects in ceramics. o Because the atoms exist as charged ions, when defect structures are considered, conditions

of electroneutrality must be maintained. o Electroneutrality is the state that exists when there are equal numbers of positive and

negative chargers from the ions. Consequently, defects in ceramics do not occur alone, but involve a cation-vacancy

and a cation-interstitial pair, known as a Frenkel defect. This defect can be thought of as being formed by a cation leaving its normal position

and moving into an interstitial site. A Schottky defect is found in AX materials and is a cation vacancy-anion vacancy.

• Stoichiometry may be defined as a state for ionic compounds wherein there is the exact ratio of cations to anions predicted by the chemical formula.

5.4 – Impurities in Solids • A pure metal consisting of only one type of atom just isn’t possible. • An alloy is a metal in which impurity atoms have been added intentionally to impart specific

characteristics to the material. • A solid solution or a new second phase will result depending on the kinds of impurities, their

concentrations, and the temperature of the alloy. • A solvent is the element or compound that is present in the greatest amount.

o Occasionally, solvent atoms are also called host atoms. • A solute is used to denote an element or compound present in a minor concentration. • A solid solution forms when, as the solute atoms are added to the host material, the crystal

structure is maintained and no new structures or formed. A solid solution is compositionally homogeneous, meaning the impurity atoms are randomly and uniformly mixed in the solid.

• Impurity point defects in solid solutions: o Substitutional occurs when solute/impurity atoms replace or substitute for host atoms.

The degree to which the solute dissolves in the solvent depends on… • Atomic size factor – Appreciable quantities of a solute may be accommodated

in this type of solid solution only when the difference in atomic radii between the two atom types is less than about 15%± . Otherwise the solute atoms will create substantial lattice distortions and a new phase will form.

• Crystal structure – For most appreciable solid solubility the crystal structures for metals of both atom types must be the same.

• Electronegativity – The more electropositive one element and the more electronegative the other, the greater is the likelihood that they will form an intermetallic compound instead of a substitutional solid solution.

• Valences – Other factors being equal, a metal will have a stronger tendency to dissolve another metal of higher valency than one of a lower valency.

5.5 – Point Defects in Polymers • The point defect concept is different in polymers than in metals because of the chainlike

macromolecules and the nature of the crystalline state for polymers.

5.6 – Specification of Composition • The two most common ways to express the composition of an alloy in terms of its constituent

elements is… o Weight percent is the weight of a particular element relative to the total alloy weight.

(wt%) 1

11 2

100mCm m

= ×+

Where 1m and 2m represent the weight (or mass) of elements 1 and 2. o Atom percent is the number of moles of an element in relation to the total moles of the

elements in the alloy. '1

11

mmnA

=

where '1m and 1A denote the mass (in grams) and atomic weight.

' 11

1 2

100m

m m

nCn n

= ×+

5.7 – Dislocations – Linear Defects • A dislocation is a linear or one-dimensional defect around which some of the atoms are

aligned in the incorrect manner, or misaligned. • An edge dislocation ( )⊥ is a dislocation in which an extra portion of a plane of atoms, or half-

plane, whose edge terminates within the crystal. It is a linear defect that centers on the line that is defined along the end of the extra half-plane of atoms. o The dislocation line is considered to be where the edge dislocation is perpendicular to the

plane of the page. o In the region around the dislocation lines, there is some localized lattice distortion. o The atoms above the dislocation line are squeezed together, and those below are pulled

apart from each other.

• A screw dislocation may be thought of as being formed by a shear stress that is applied to produce a distortion which causes the upper front region of the crystal to be shifted one atomic distance to the right relative to the bottom portion. o The atomic distortion associated with a screw dislocation is also linear and along a

dislocation line. o The screw dislocation derives its name from the spiral or helical path or ramp that is

traced around the dislocation line by the atomic planes of atoms.

• Most common are mixed dislocations that are neither pure edge nor pure screw, but exhibit

components of both types.

• The magnitude and direction of the lattice distortion associated with a dislocation are express in terms of a Burgers vector.

• The nature of a dislocation (i.e., edge, screw, or mixed) is defined by the relative orientations of the dislocation line and Burgers vector. o For an edge, they are perpendicular. o For a screw, they are parallel. o For a mixed dislocation, they are neither perpendicular nor parallel.

5.8 – Interfacial Defects • Interfacial defects are boundaries that have two dimensions and normally separate regions of

the materials that have different crystal structures and/or crystallographic orientations and are comprised of… o External Surfaces

One of the most obvious boundaries is the external surface, along which the crystal structure terminates. Surface atoms are not bonded to the maximum number of nearest neighbors and are therefore in a higher-energy state than the atoms at interior positions. The bonds of these surface atoms that are not satisfied give rise to a surface energy. Thus, to reduce this energy, materials tend to minimize the total surface area if at all possible.

o Grain Boundaries The grain boundary is the boundary separating two small grains or crystals having

different crystallographic orientations in polycrystalline materials. Various degrees of crystallographic misalignment between adjacent grains are

possible and this misalignment is known as a tilt boundary. The angle of misorientation, θ , is known as a twist boundary which can be described

by an array of screw dislocations.

o Phase Boundaries

Phase boundaries exist in multiphase materials, in which a different phase exists on each side of the boundary; furthermore, each of the constituent phases has its own distinctive physical and/or chemical characteristics.

o Twin Boundaries A twin boundary is a special type of grain boundary across which there is a specific

mirror lattice symmetry; that is, atoms on one side of the boundary are located in mirror-image positions to those of the atoms on the other side.

The region of material between these boundaries is termed a twin. Twins results from atomic displacements that are produced from applied

mechanical shear forces (mechanical twins: BCC, HCP) and also during annealing heat treatments following deformation (annealing twins: FCC).

CHAPTER 6 – Diffusion • Diffusion is the phenomenon of material transport by atomic motion. • Interdiffusion or inpurity diffusion is the process by which atoms of one metal diffuse into the

atoms of another material. o Can be thought of as a net drift or transport or atoms from high- to low-concentration

regions of the material. • Diffusion also occurs for pure metals, but all atoms exchanging positions are of the same type.

This is known as self-diffusion. 6.2 – Diffusion Mechanisms • From an atomic perspective, diffusion is just the stepwise migration of atoms from lattice site

to lattice site. • For an atom to make such a move, it must meet two conditions…

o There must be an empty adjacent site. o The atom must have sufficient energy to break bonds with its neighbor atoms and then

cause some lattice distortion during the displacement. This energy is vibrational in nature, and at some specific temperature, a small

fraction of the total number of atoms are capable of diffusive motion by virtue of the magnitudes of their vibrational energies. However, this fraction increases with an increasing temperature.

• Two dominate proposals for metallic diffusion… o Vacancy Diffusion

This mechanism involves the interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy.

This process necessitates the presence of vacancies. And the extent to which vacancy diffusion can occur is a function of the number of these defects that are present; significant concentrations of vacancies may exist in metals when the temperatures are elevated.

Diffusing atoms and vacancies exchange positions, thus the diffusion of atoms in one direction corresponds to the motion of vacancies in the opposite direction. Therefore, both self-diffusion and interdiffusion occur by this mechanism.

o Interstitial Diffusion This mechanism involves atoms that migrate from an interstitial position to a

neighboring one that is empty. This is found for interdiffusion of impurities such as hydrogen, carbon, nitrogen,

and oxygen as they have atoms that are small enough to fit into the interstitial positions necessary.

In most metal alloys, interstitial diffusion occurs much more rapidly than diffusion by the vacancy mode because the interstitial atoms are smaller and therefore more mobile. Furthermore, there are more empty interstitial positions than vacancies; hence, the probability of interstitial atomic movement is greater than for diffusion through vacancies.

6.3 – Steady-State Diffusion • Diffusion is a time-dependent process meaning that in a macroscopic sense, the quantity of an

element that is transported within another is a function of time. • Diffusion flux ( )J is the rate of mass transfer and is defined as the mass, (or equivalently, the

number of atoms) M diffusing through and perpendicular to a unit cross-sectional area of solid per unit of time…

1M dMJAt A dt

= =

• If the diffusion flux foes not change with time, a steady-state condition exists. • Steady-state diffusion is the diffusion of atoms of a gas through a plate of metal for which the

concentrations (or pressures) of the diffusing species on both surfaces of the plate are held constant.

• The concentration profile is when the concentration C is plotted versus position/distance within the solid x .

• The slope at a particular point on this curve is called the concentration gradient.

concetration gradient = dCdx

• Steady-state diffusion in a single ( )x direction is defined as the flux being proportional to the concentration gradient. Sometimes called Fick’s first law...

dCJ Ddx

= −

• In the equation above, D is called the diffusion coefficient which is expressed in 2

ms

.

o The negative sign in the expression above indicates that the direction of diffusion is down the concentration gradient, from a high to a low concentration.

• The driving force is used in the context of what compels a reaction to occur. o In Fick’s first law, the driving force is the concentration gradient.

6.4 – Nonsteady-State Diffusion • Nonsteady-state diffusion is that which takes place when the diffusion flux and the

concentration gradient at some particular point in a solid vary with time, with a net accumulation or depletion of the diffusing species resulting.

• Fick’s second law… C CDt x x

∂ ∂ ∂ = ∂ ∂ ∂

Simplifies expression below if the diffusion coefficient is independent of composition. 2

2

C CDt x

∂ ∂=

∂ ∂

6.8 – Diffusion in Ionic and Polymeric Materials CHAPTER 17 – Thermal Properties • Thermal property refers to the response of a material to the application of heat. As a solid

absorbs energy in the form of heat, its temperature rises and its dimensions increase. • Heat Capacity – Heat capacity indicates a material’s ability to absorb heat from the external

surroundings; it represents the amount of energy required to produce a unit temperature rise. dQCdT

=

• In most solids, the principal mode of thermal energy assimilation is by the increase in vibrational energy of the atoms. o Atoms in solid materials are constantly vibrating at very high frequencies and with

relatively small amplitudes. o The vibrations of adjacent atoms are coupled by virtue of atomic bonding.

• The variation with temperature of the vibrational contribution to the heat capacity at constant volume for many relatively simple crystalline solids at low temperatures can be described by…

3vC AT=

• Other energy-absorbative mechanisms also exist that can add to the total heat capacity of a

solid, however, in most instances, these are minor relative to the magnitude of the vibrational contribution.

• Thermal Expansion – Most solid materials expand upon heating and contract when cooled The change in length with temperature for a solid material is described as…

00

0

( )ft f

l lT T

lα

−= −

or

0t

l Tl

α∆= ∆

• Volume change can be described as…

0v

V TV

α∆= ∆

• From an atomic perspective, thermal expansion is reflected by an increase in the average distance between atoms. Thermal expansion is due to the asymmetric curvature of this potential energy trough rather than the increased atomic vibrational amplitudes with rising temperature. For each class of materials (metals, ceramics, polymers), the greater the atomic bonding energy, the deeper and more narrow is this potential energy trough. As a result, the increase in interatomic separation with a given rise in temperature will be lower.

• Thermal conduction is the phenomenon by which heat is transported from high- to low-

temperature regions of a substance. The property that characterizes the ability of a material to transfer heat is the thermal conductivity…

dTq kdx

= −

• Heat is transported in solid materials by both lattice vibration waves (phonons) and free electrons. A thermal conductivity is associated with each of these mechanisms, and the total conductivity is the sum of the two contributions, or l ek k k= + .

• Thermal stresses are the stresses included in a body as a result of changes in temperature. Thermal stress is important because these stresses can lead to fracture or undesirable plastic deformation.

0( )t f tE T T E Tσ α α= − = ∆

CHAPTER 9 – Failures in Materials • The usual causes of failure in a structure is improper material selection and processing and

inadequate design of the component or misuse. 9.2 – Fundamentals of Fracture • Simple fracture is the separation of a body into two or more pieces in response to an imposed

stress that is static (i.e., constant or slowly changing with time) and at temperatures that are low relative to the melting temperature of the material. o Fatigue is another cause of failure and appears as a result of cyclically imposed stresses.

• For metals, there are two possible types of fracture: o Ductile fracture is characterized by extensive plastic deformation in the vicinity of an

advancing crack. Furthermore, the process proceeds relatively slowly as the crack length is extended.

o Brittle fractures have cracks that may spread extremely rapidly, with very little accompanying plastic deformation.

• Any fracture process involves two steps – crack formation and propagation – in response to an imposed stress.

• Ductile fracture is preferred to brittle fracture because (1) brittle fracture occurs suddenly and catastrophically without any warning, and (2) ductile fractures give the warning that failure is imminent due to the presence of plastic deformation. Also, more strain energy is require to induce ductile fracture since these materials are generally tougher.

9.3 – Ductile Fracture • Highly ductile materials will neck down to a point fracture, showing virtually 100% reduction

in area. • The ductile fracture process occurs in several stages…

o After necking begins, small cavities, or microvoids, form in the interior of the cross section. o As deformation continues, these microvoids enlarge, come together, and coalesce to form

an elliptical crack, which has its long axis perpendicular to the stress direction. o The crack continues to grow in a direction parallel to its major axis by this microvoid

coalescence process. o Finally, fracture ensues by the rapid propagation of a crack around the outer perimeter pf

the neck by shear deformation at an angle of about 45 degrees with the tensile axis – this is the angle at which the shear stress is a maximum.

9.4 – Brittle Fracture • Brittle fracture takes place without any appreciable deformation and by rapid crack

propagation. The direction of crack motion is very nearly perpendicular to the direction of the applied tensile stress and yields a relatively flat fracture structure.

• For brittle crystalline materials, crack propagation corresponds to the successive and repeated breaking of atomic bonds along specific crystallographic planes which is known as cleavage. o This fracture type is said to be transgranular because the fracture cracks pass through the

grains. • Some alloys have crack propagation along grain boundaries which is termed intergranular.

9.5 – Principles of Fracture Mechanics • Fracture mechanics allows the quantification of the relationships between material

properties, stress level, the presence of crack-producing flaws, and crack propagation mechanisms.

• Measured fracture strengths for most brittle materials are significantly lower than those predicted by theoretical calculations based on atomic bonding energies because of the presence of microscopic flaws or cracks that always exist under normal conditions at the surface and within the interior of a body of material. o These flaws are sometimes called stress raisers.

• Assuming a crack is similar to an elliptical hole through a plate and is oriented perpendicular to the applied stress, the maximum stress occurs at the crack tip and may be approximated by…

1/2

02mt

aσ σρ

=

Where 0σ is the magnitude of the nominal applied tensile stress, tρ is the radius of curvature of the crack tip, and a represents the length of a surface crack, or half the length of an internal

crack depending on the need. • The effects of a stress raiser is more significant in brittle than in ductile materials because in

a ductile metal, plastic deformation ensues when the maximum stress exceeds the yield strength. This leads to a more uniform distribution of stress in the vicinity of the stress raiser and to the development of a maximum stress concentration factor less than the theoretical value determined.

• The critical stress required for crack propagation in a brittle material is described by the expression…

1/22 sc

Eaγσ

π =

Where cσ is the critical stress, E is the modulus of elasticity, sγ is the specific surface energy, and a is one-half the length of an internal crack.

• When the magnitude of a tensile stress at the tip of a flaw exceeds the value of this critical stress, a crack forms and then propagates, which results in fracture.

• Fracture toughness is a result of the relationship between the critical stress for crack propagation and crack length…

c cK Y aσ π= where cK is the fracture toughness, a property that is a measure of a material’s resistance to

brittle fracture when a crack is present. • Plane strain occurs when a load operates on a crack where there is no strain component

perpendicular to the front and back faces. • Plane fracture toughness is approximated by…

IcK Y aσ π= where IcK is the plane strain fracture toughness for mode I crack displacement.

• Brittle materials, for which appreciable plastic deformation is not possible in front of an advancing crack, have low IcK values and are vulnerable to catastrophic failure. On the other hand, IcK values are relatively large for ductile materials.

• The plane strain fracture toughness is a fundamental material property that depends primarily on the temperature, strain rate, and microstructure.

• The design critical stress is defined by…

Icc

KY a

σπ

=

• The maximum allowable flaw size is defined by… 21 Ic

cKa

Yπ σ =

9.6 – Brittle Fracture of Ceramics • Both crystalline and noncrystalline ceramics almost always fracture before any plastic

deformation can occur in response to an applied tensile load. • Stress raisers in brittle ceramics may be minute surface or interior cracks (microcracks),

internal pores, and grain corners, which are virtually impossible to eliminate or control. 9.7 – Fracture of Polymers • The fracture strengths of polymeric materials are low relative to those of metals and

ceramics. As a general rule, the mode of fracture in thermosetting polymers (heavily cross-linked networks) is brittle. o For thermoplastic polymers, both ductile and brittle modes are possible, and many of these

materials are capable of experiencing a ductile-to-brittle transition. 9.9 – Fatigue and Cyclic Stresses • Fatigue is a form of failure that occurs in structures subjected to dynamic and fluctuating

stresses. The term fatigue is used because this type of failure normally occurs after a lengthy period of repeated stress or strain cycling.

• Fatigue failure is brittlelike in nature even in normally ductile metals, in that there is very little, if any, gross plastic deformation associated with failure. The process occurs by the initiation and propagation of cracks, and typically the fracture surface is perpendicular to the direction of an applied tensile stress.

• Applied stress may be axial (tension-compression), flexural (bending), or torsional (twisting) in nature.

• Three different fluctuating stress-time modes are possible o One is represented schematically by a regular and sinusoidal time dependence where the

amplitude is symmetrical about a mean zero stress level, for example alternating from a maximum tensile stress to a minimum compresses stress of equal magnitude. This is referred to as a reversed stress cycle.

o Another type, termed repeated stress cycle, occurs when the maxima and minima are asymmetrical relative to the zero stress level.

o Lastly, there can be a random variation in amplitude and frequency.

9.10 – The S-N Curve

• The figure above is a schematic of a rotating-bending test apparatus commonly used for

fatigue testing. The compression and tensile stresses are imposed on the specimen as it is simultaneously bent and rotated.

• Fatigue limit is the point below which fatigue failure will not occur. The fatigue limit represents the largest value of fluctuating stress that will not cause failure for essentially an infinite number of cycles.

• MOST NONFERROUR ALLOYS (E.G., ALUMINUM, COPPER, MAGNESIUM) DO NOT

HAVE A FATIGUE LIMIT, IN THAT THE S-N CURVE CONTINUES ITS DOWNWARD TREND AT INCREASINGLY GREATER N VALUES. THUS, FATIGUE WILL ULTIMATELY OCCUR REGARDLESS OF THE MAGNITUDE OF THE STRESS.

• For these materials, one fatigue response is specified as fatigue strength which is defined as the stress level at which failure will occur for some specified number of cycles.

• Another important parameter that characterizes a material’s fatigue behavior is fatigue life. It is the number of cycles to cause failure at a specified stress level, as taken from a S-N plot.

• Fatigue data is often scattered, so it is usually graphed to a best fit curve.

9.11 – Fatigue in Polymeric Materials • Polymers may experience fatigue under conditions of cyclic loading. As with metals, fatigue

occurs at stress levels that are relative to the low yield strength. 9.12 – Crack Initiation and Propagation • The process of fatigue failure is characterized by three distinct steps:

o Crack initiation, in which a small crack forms at some point of high stress concentration. o Crack propagation, during which this crack advances incrementally with each stress cycle. o Final failure, which occurs very rapidly once the advancing crack has reached critical size.

• Cracks associated with fatigue failure almost always initiate (or nucleate) on the surface of a component at some point of stress concentration.

• The region of a fracture surface that formed during the crack propagation step may be characterized by two types of markings termed… o Benchmarks are of macroscopic dimensions and may be observed with the unaided eye.

These markings are found for components that experienced interruptions during the crack propagation stage. Each benchmark band represents a period of time over which crack growth occurred.

o Striations are microscopic in size and subject to observation with the electron microscope. Each striation is thought to represent the advance distance of a crack front during a single load cycle. Striation width depends on, and increases with, increasing stress range.

• Benchmarks and striations are fatigue fracture surface features having similar appearances, but are nevertheless different in both origin and size. There may be thousands of striations within a single benchmark.

• The presence of benchmarks and striations upon examination of the failure indicate that it was fatigue induced, however, the absence of these features does not exclude fatigue from the cause of the failure.

9.13 – Factors That Affect Fatigue Life • The primary factors of fatigue life include mean stress level, geometrical design, surface

effects, and metallurgical variables, as well as the environment. o Mean stress is related to S-N plots because it is taken from the stress amplitude that a

material is subject to. Increasing the mean stress level leads to a decrease in fatigue life. o For many loading situations, the maximum stress within a component or structure occurs

at its surface. Consequently, most cracks leading to fatigue failure originate at surface positions, specifically at stress amplification sites. Surface effects are dependent on… Design factors can have a significant influence on a components fatigue characteristics.

Any notch or geometrical discontinuity can act as a stress raiser and fatigue crack initiation site. The sharper the discontinuity (i.e., the smaller the radius of curvature) the more severe is the stress concentration. Avoid these as much as possible.

Surface treatments can be applied in a variety of ways. The most effective methods of increase fatigue performance is by imposing residual compressive stresses within a thin outer surface layer. Thus, a surface tensile stress of external origin will be partially nullified and reduced in magnitude by the residual compressive stress.

9.14 – Environmental Effects • Thermal fatigue is normally induced at elevated temperatures by fluctuating thermal

stresses; mechanical stresses from an external source need not be present. • The magnitude of thermal stress developed by a temperature change depends of the

coefficient of thermal expansion and the modulus of elasticity according to… l E Tσ α= ∆

• Thermal stresses will not arise if mechanical restraint is absent.• Failure that occurs by the simultaneous action of a cyclic stress and chemical attack is termed

corrosion fatigue.o Small pits may form as a result of chemical reactions between the environment of the

material, which may serve as points of stress concentration and therefore as cracknucleation sites.

CHAPTER 15 – Composites • A composite is considered to be any multiphase material that exhibits a significant proportion

of the properties of both constituent phases such that better combination of properties form. • According to this principle of combined action, better property combinations are fashioned by

the judicious combination of two or more distinct materials. • Many composite materials are composed of just two phases: one is termed the matrix, which

is continuous and surrounds the other phase, often called the dispersed phase.

15.5 – Influence of Fiber Orientation and Concentration • The arrangement or orientation of the fibers relative to one another, the fiber concentration,

and the distribution all have a significant influence on the strength and other properties of fiber-reinforced composites.

• With respect to orientation, two extremes are possible: (1) a parallel alignment of the longitudinal axis of the fibers in a single direction, and (2) a totally random alignment.

• Continuous fibers are normally aligned, whereas discontinuous fibers may be aligned, randomly oriented, or partially oriented. Better overall composite properties are realized when the fiber distribution is uniform.

• The onset of composite failure begins as the fibers start to fracture. Composite failure is not

catastrophic because not all fibers fracture at the same time because there will always be considerable variations in the fracture strength of brittle fiber materials. Also, even after fiber failure, the matrix is still intact. Thus, the fractured fibers, which are shorter than the original ones, are still embedded within the matrix and consequently are capable of sustaining a diminished load as the matrix continues to plastically deform.

• The modulus of elasticity of a continuous and aligned fibrous composite in the direction of alignment (or longitudinal direction) is…

cl m m f fE E V E V= +

(1 )cl m f f fE E V E V= − +

• For longitudinal loading, the ratio of the load carried by the fibers to that carried by the matrix is…

f f f

m m m

F E VF E V

=

• A continuous and oriented giber composite may be loaded in the transverse direction, where the load is applied perpendicular to the direction of fiber alignment…

1 fm

ct m f

VVE E E

= +

(1 )m f m f

ctm f f m f f f m

E E E EE

V E V E V E V E= =

+ − +

• Longitudinal tensile strength is defined as… * ' *(1 )cl m f f fV Vσ σ σ= − +

where 'mσ is the stress in the matrix at fiber failure, and *

fσ is the fiber tensile strength. • For a discontinuous and aligned fiber composite having a uniform distribution of fibers and in

which cl l< the longitudinal strength is given by…

* * '1 (1 )2

ccd f f m f

lV Vl

σ σ σ = − + −

where * ' and f mσ σ represent the fracture strength of the fiber and the stress in the matrix when the composite fails.

• However, if the fiber length is less than critical, cl l< , the longitudinal strength is given by…

'* ' (1 )c

f m fcd

l V Vdτσ σ= + −

• For discontinuous and randomly oriented fiber composites, the “rule of mixtures” expression for the elastic modulus is…

cd f f m mE KE V E V+ +

where K is a fiber efficiency parameter that depends on fV and the /f mE E ratio. o The modulus increases with increasing volume fraction of fiber.

• Aligned fibrous composites are inherently anisotropic in that the maximum strength and reinforcement are achieved along the alignment (longitudinal) direction.

• In the transverse direction, fiber reinforcement is virtually nonexistent, which means that fracture usually occurs at relatively low tensile stresses.

15.6 – The Fiber Phase • An important characteristic of most materials, especially brittle ones, is that a small diameter

fiber is much stronger than the bulk material. This is a result of the probability of the presence of a critical surface flaw that can lead to fracture decreases with decreasing specimen volume, and this feature is used to advantage in fiber-reinforced composites.

• Fibers are either polycrystalline or amorphous with small diameters; fibrous materials are generally either polymers or ceramics.

• Whiskers are very thin single crystals that have extremely large length-to-diameter ratios. As a consequence of their small size, they have a high degree of crystalline perfection and are virtually flaw-free, which accounts for their exceptionally high strengths. They are among the strongest known materials.

15.7 – The Matrix Phase • The matrix phase of fibrous composites may be a metal, polymer, or ceramic. • Generally, metals and polymers are used as matrix materials because some ductility is

desirable. • The matrix phase serves several different functions…

o It binds the fibers together and acts as the medium by which an externally applied stress is transmitted and distributed to the fibers; only a small proportion of an applied load is sustained by the matrix phase. The matrix material should be ductile and the elastic modulus of the fiber should be

much higher than that of the matrix. o It protects the individual fibers from surface damage as a result of mechanical abrasion or

chemical reactions with the environment. Such interactions may introduce surface flaws capable of forming cracks, which may lead to failure at low tensile stress levels.

o It separates the fibers and, by virtue of its relative softness and plasticity, it prevents the propagation of brittle cracks from fiber to fiber. Essentially, the matrix phase serves as a barrier to crack propagation.

• Adhesive bonding forces between fiber and matrix must be high to prevent fiber pullout. The ultimate strength of the composite depends largely on the magnitude of this bond; adequate bonding is essential to maximize the stress transmittance from the weak matrix to the strong fibers of the composite.

15.8 – Polymer-Matrix Composites • Polymeric-matrix composites (PMCs) consists of a polymer resin as the matrix and the fibers

as the reinforcement medium. • Fiberglass is a composite consisting of glass fibers, either continuous or discontinuous,

contained within a polymer matrix. Referred to sometimes as E-glass, the fiber diameters normally range between 3 and 20 micrometers.

• Glass is a popular reinforcement material because… o It is easily drawn into high-strength fibers from the molten state. o It is readily available and may be fabricated into a glass-reinforced plastic economically

using a wide variety of composite-manufacturing techniques. o As a fiber it is relatively strong, and when embedded in a plastic matrix, it produces a

composite having a very high specific strength. o When coupled with various plastics, it possesses a chemical inertness that renders the

composite useful in a variety of corrosive environments. • Because the surface characteristics are so important, newly drawn fibers are normally coated

during drawing with a size, a thin layer of substance that protects the fiber surface from damage and undesirable environmental interactions.

• Despite having a high strength, fiberglass is not very stiff and are limited to service temperatures below 200 degrees Celsius.

• Carbon Fiber-Reinforced Polymer (CFRP) Composites are more high performance. o Carbon fiber is the most commonly used reinforcement in advanced polymer-matrix

composites because… Carbon fibers have high specific moduli and specific strengths. They retain their high tensile modulus and high strength at elevated temperatures,

but are susceptible to high-temperature oxidation. Carbon fibers are not affected by moisture or a wide variety of solvents, acids, and

bases at room temperature. These fibers exhibit a diversity of physical and mechanical characteristics, allow

composites incorporating these fibers to have specific engineering properties. Fiber and composite-manufacturing processes have been developed that are relatively

inexpensive and cost effective. • Hugh-temperature thermoplastic resins offer the potential to be used in future aerospace

applications; such materials include polyetheretherketone (PEEK), poly(phenylene sulfide) (PPS), and polyetherimide (PEI).

15.11 – Carbon-Carbon Composites • One of the most advanced and promising of engineering materials is the carbon-fiber

reinforced carbon-matrix composite, often term a carbon-carbon composite. • Their desirable properties include high tensile moduli and tensile strengths, which are

retained to temperatures in excess of 2000 degrees Celsius, resistance to creep, and relatively large fracture toughness values. They also have low coefficients of thermal expansion, and relatively high thermal conductivities which allows them to have a low susceptibility to thermal shock.

• However, their drawback is a propensity to high-temperature oxidation. 15.13 – Processing of Fiber-Reinforced Composites • Prepreg is the composite industry’s term for continuous-fiber reinforcement preimpregnated

with a polymer resin that is only partially cured. This material is delivered in tape form to the manufacturer, which then directly molds and fully cures the product without having to add any resin.

• The most widely used form of composite material for structural applications.

• At room temperature, the thermoset matrix undergoes curing reaction; therefore, the prepreg is stored at or below freezing.

• Actual fabrication begins with the lay-up – laying of the prepreg tape onto a tooled surface. • Final curing is accomplished by the simulataneous application of heat and pressure. • Filament winding is a process by which continuous reinforcing fibers are accurately

positioned in a predetermined pattern to form a hollow (usually cylindrical) shape. o The fibers, either as individual strands or as tows, are first fed through a resin bath and

then are continuously wound onto a mandrel, usually using automated winding equipment. After the appropriate number of layers have been applied, curing is carried out either in an oven or at room temperature after which the mandrel is removed.

15.4 – Laminar Composites • A structural composite is normally composed of both homogeneous and composite materials,

the properties of which depend not only on the properties of the constituent materials, but also on the geometrical design of the various structural elements.

• A laminar composite is composed of tw-dimensional sheets or panels that have preferred high-strength direction, such as is found in word and continusous and aligned fiber-reinforced plastics. o The layers are stacked and subsequently cemented together such that the orientation of

the high-strength direction varies with each successive layer. o Laminations may also be constructed using a fibric material such as cotton, paper, or

woven glass gibers embedded in a plastic matrix. Therefore, a laminar composite has a relatively high strength in a number of directions in the two-dimensional plane; however, the strength in any given direction is lower than it would be if all the fibers were oriented in that direction.

15.15 – Sandwich Panels • Sandwich panels, considered to be a class of structural composites, are designed to be

lightweight beams or panels having relatively high stiffnesses and strengths. o A panel consists of two outer sheets, or faces, that are separated by and adhesively bonded

to a thicker core.

• Construction: o The outer sheets ar emade of a relatively stiff and strong material, typically aluminum

alloys, fiber-reinforced plastics, titanium, steel, etc. o These outer sheets impart high stiffness and strength to the structure and must be thick

enough to withstand tensile and compressive stresses that result from loading. o The core materials typically consist of either rigid polymeric foams, wood, and honeycomes

that are lightweight and normally have a low modulus of elasticity. o The core provides continuous support for the faces. Additionally, it must have sufficient

shear strength to withstand transverse shear stresses and also be thick enough to provide high shear stiffness (to resist buckling of the panel).

• Another popular core consists of a “honeycomb” structure – thin foils that have been formed into interlocking hexagonal cells with aces oriented perpendicular to the face planes.

o The honeycomb material is normally either an aluminum alloy or an aramid polymer. o Strength and stuffness of honeycomb structures depend on cell size, cell wall thickness,

and the material from which the honeycomb was made.