Mechanical Properties of Engineered Materials 2008 Part 6 docx

Following a simple review of the physical basis for plasticity in different classes of monolithic materialsceramics, metals, intermetallics, and polymers, empirical plastic flow rulesare

on subsequent loading and unloading, and these can lead ultimately tofailure In some cases, the dimensional and shape changes associated withplasticity may lead to loss of tolerance(s) and premature retirement of astructure or component from service An understanding of plasticity is,therefore, important in the design and analysis of engineering structuresand components.

This chapter presents a basic introduction to the mechanisms andmechanics of plasticity in monolithic materials Following a simple review

of the physical basis for plasticity in different classes of monolithic materials(ceramics, metals, intermetallics, and polymers), empirical plastic flow rulesare introduced along with multiaxial yield criteria Constitutive equations ofplasticity are then presented in the final section of the chapter

Copyright © 2003 Marcel Dekker, Inc

5.2 PHYSICAL BASIS FOR PLASTICITY

5.2.1 Plasticity in Ceramics

Most ceramics only undergo only elastic deformation prior to the onset ofcatastrophic failure at room temperature Hence, most reports on themechanical properties of ceramics are often limited to elastic properties.Furthermore, most ceramists report flexural properties obtained underthree- or four-point bending Typical strength properties of selected ceramicmaterials are presented inTable 5.1 Note that ceramics are stronger (almost

15 times stronger) in compression than in tension Also, the flexuralstrengths are intermediate between the compressive and tensile strengthlevels Reasons for these load-dependent properties will be discussed insubsequent chapters For now, it is simply sufficient to state that the trendsare due largely to the effects of pre-existing defects such as cracks in theceramic structures

The limited capacity of ceramic materials for plastic deformation isdue largely to the limited mobility of dislocations in ceramic structures Thelatter may be attributed to their large Burgers (slip) vectors and unfavorable(for plastic deformation) ionically/covalently bonded crystal structures.Plastic deformation in ceramics is, therefore, limited to very small strains(typically < 0.1–1%), except at elevated temperatures where thermally acti-vated dislocation motion and grain boundary sliding are possible In fact,the extent of plasticity at elevated temperatures may be very significant in

FIGURE5.1 Schematic illustration of plastic strain after unloading

ceramics deformed at elevated temperature, and superplasticity (strain levels

up to 1000% plastic strain) has been shown to occur due to creep ena in some fine-grained ceramics produced

phenom-However, in most ceramics, the plastic strains to failure are relativelysmall (<1%), especially under tensile loading which tends to open up pre-existing cracks that are generally present after processing Also, since inci-pient cracks in ceramics tend to close up under compressive loading, thestrength levels and the total strain to failure in compression are often greaterthan those in tension Furthermore, very limited plasticity (permanentstrains on removal of applied stresses) may occur in some ceramics or cera-mic matrix composites by microcracking or stress-induced phase transfor-mations

Microcracking generally results in a reduction in Young’s modulus, E,which may be used as a global/scalar measure of damage (Fig 5.2).If weassume that the initial ‘‘undeformed’’ material has a damage state of zero,while the final state of damage at the point of catastrophic failure corre-sponds to a damage state of 1, we may estimate the state of damage usingsome simple damage rules For an initial Young’s modulus of E0 and anintermediate damage state, the damage variable, D, is given simply by

D ¼ 1  E=E0 Damage tensors may also be used to obtain more rigorousdescriptions of damage (Lemaitre, 1991)

Plasticity in ceramics may also occur by stress-induced phase mations This has been observed in partially stabilized zirconia (ZrO2alloyed with CaO, Y2O3,or CeO to stabilize the high-temperature tetragonal

transfor-TABLE5.1 Strength Properties of Selected Ceramic Materials

Material

Compressivestrength(MPa (ksi)]

Tensilestrength[MPa (ksi)]

Flexuralstrength[MPa (ksi)]

Modulus ofelasticity[GPa (106psi)]

phase down to room temperature) Under monotonic loading, the stable tetragonal phase can undergo stress-induced phase transformationsfrom the tetragonal to the monoclinic phase This stress-induced phasetransformation is associated with a volume increase of  4%, and cangive rise to a form of toughening known as transformation toughening,which will be discussed inCh 13.

meta-Stress-induced phase transformations occur gradually in partially bilized zirconia, and they give rise to a gradual transition from linearity inthe elastic regime, to the nonlinear second stage of the stress–strain curveshown inFig 5.3.The second stage ends when the stress-induced transfor-mation spreads completely across the gauge section of the specimen This isfollowed by the final stage in which rapid hardening occurs until failure It isimportant to note that the total strain to failure is limited, even in partiallystabilized zirconia polycrystals Also, as in conventional plasticity, stress-induced transformation may be associated with increasing, level, or decreas-ing stress–strain behavior (Fig 5.4)

sta-5.2.2 Plasticity in Metals

In contrast to ceramics, plastic deformation in metals is typically associatedwith relatively large strains before final failure This is illustrated inFig 5.5using data obtained for an aluminum alloy In general, the total plasticstrains can vary between 5 and 100% in ductile metals deformed to failure

at room temperature However, the elastic portion of the stress–strain curve

is generally limited to strains below  0:1 to 1% Furthermore, metals andtheir alloys may exhibit stress–strain characteristics with rising, level, or

FIGURE5.2 Schematic showing the change in modulus due to damage duringloading and unloading sequences

decreasing stress, as shown in Fig 5.4 Materials in which the stress levelremains constant with increasing strain [Fig 5.4(b)] are known as elastic–perfectly plastic Materials in which the stress level decreases with increasingstrain are said to undergo strain softening [Fig 5.4(c)], while those in whichthe stress level increases with increasing strain are described as strain hard-ening materials, Fig 5.4(a).

FIGURE5.3 Schematic of the three stages of deformation in material going stress-induced phase transformation (After Evans et al., 1981.)

under-FIGURE5.4 Types of stress–strain response: (a) strain hardening; (b) elastic–perfectly plastic deformation; (c) strain softening

Strain hardening occurs as a result of dislocation interactions in thefully plastic regime These may involve interactions with point defects(vacancies, interstitials, or solutes), line defects (screw, edge, or mixed dis-locations), surface defects (grain boundaries, twin boundaries, or stackingfaults), and volume defects (porosity, entrapped gases, and inclusions) Thedislocation interactions may give rise to hardening when additional stressesmust be applied to overcome the influence of defects that restrict dislocationmotion This may result in rising stress–strain curves that are characteristic

of strain hardening behavior,Fig 5.4(a)

As discussed earlier, the stress–strain curves may also remain level[Fig 5.4(b)], or decrease or increase continuously with increasing strain,Fig 5.4(c) The reasons for such behavior are generally complex, and notfully understood at present However, there is some limited evidence thatsuggests that elastic–perfectly plastic behavior is associated with slip planar-ity, i.e., slip on a particular crystallographic plane, while strain softeningtends to occur in cases where slip localizes on a particular microstructuralfeature such as a precipitate The onset of macroscopic yielding, therefore,corresponds to the stress needed to shear the microstructural feature Oncethe initial resistance to shear is overcome, the material may offer decreasingresistance to increasing displacement, giving rise ultimately to strain soft-ening behavior, Fig 5.4(c)

Since the moving dislocations interact with solute clouds, serratedyielding phenomena may be observed in the stress–strain behavior [Fig5.6) Different types of serrated yielding phenomena have been reporteddue to the interactions of dislocations with internal defects such as solutesand interstitials The phenomenon is generally referred to as the Portevin–

FIGURE 5.5 Stress–strain behavior in an aluminum alloy (After Courtney,

1990 Reprinted with permission from McGraw-Hill.)

LeChatelier effect, in honor of the two Frenchmen who first reported it(Portevin and LeChatelier, 1923) The serrations are caused by the pinningand unpinning of groups of dislocations from solutes that diffuse towards it

as it moves through a lattice The mechanisms is particularly effective atparticular parametric ranges of strain-rate and temperature (Cottrell, 1958).Finally in this section, it is important to discuss the so-called anom-alous yield phenomena that has been reported in some plain carbon steels(Fig 5.7) The stress–strain plots for such materials have been observed toexhibit double yield points in some annealed conditions, as shown in Fig.5.7 The upper yield point (UYP) corresponds to the unpinning of disloca-tions from interstitial carbon clouds Upon unpinning, the load drops to alower yield point (LYP) Lu¨der’s bands (shear bands inclined at  458degrees to the loading axis) are then observed to propagate across the

FIGURE5.6 Types of serrated yielding phenomena: (a) Type A; (b) Type B; (c)Type C; (d) Type S (Types A–C After Brindley and Worthington, 1970; Type SAfter Pink, 1994 Reprinted with permission from Scripta Met.)

gauge sections of the tensile specimens, as the strain is increased further(Fig 5.7) Note that the stress remains relatively constant in the so-calledLu¨der’s strain regime, although serrations may be observed with sufficientlysensitive instrumentation The strain at the end of this constant stress regime

is known as the Lu¨ders strain This corresponds to the point at which theLu¨der’s bands have spread completely across the gauge section of the speci-men Beyond this point, the stress generally increases with increasing due tothe multiple interactions between dislocations, as discussed earlier for con-ventional metallic materials (Fig 5.5)

5.2.3 Plasticity in Intermetallics

As discussed inChap 1,intermetallics are compounds between metals andother metals Due to their generally ordered structures, and partially cova-lently or ionically bonded structures, intermetallics generally exhibit onlylimited plasticity at room-temperature Nevertheless, some ductility hasbeen reported for ordered gamma-based titanium aluminide intermetallics

FIGURE5.7 Anomalous yielding in 1018 plain carbon steel (After Courtney,

1990 Reprinted with permission from McGraw-Hill.)

with duplex
2þ  microstructures These two phase intermetallics haveroom temperature plastic elongations to failure of about 1–2% due to defor-mation by slip and twinning (Kim and Dimiduk, 1991) Their limited room-temperature ductility has been attributed to the soaking up of interstitialoxygen by the
2 phase This results in a reduction in interstitial oxygencontent in the gamma phase, and the increased dislocation mobility of dis-locations in the latter which gives rise to the improved ductility in two-phasegamma titanium aluminides (Vasudevan et al., 1989).

Niobium aluminide intermetallics with plastic elongations of 10–30%have also been developed in recent years (Hou et al., 1994; Ye et al., 1998).The ductility in these B2 (ordered body-centered cubic structures) interme-tallics has been attributed to the partial order in their structures Similarimprovements in room-temperature (10–50%) ductility have been reported

in Ni3Al intermetallics that are alloyed with boron (Aoki and Izumi, 1979;Liu et al., 1983), and Fe3Al intermetallics alloyed with boron (Liu andKumar, 1993)

The improvements in the room-temperature ductilities of the nickeland iron aluminide intermetallics have been attributed to the cleaning up ofthe grain boundaries by the boron additions However, the reasons for theimproved ductility in ordered or partially ordered intermetallics are still notfully understood, and are under investigation Similarly, anomalous yield-point phenomena (increasing yield stress with increasing temperature) andthe transition from brittle behavior at room temperature to ductile behavior

at elevated temperature are still under investigation

5.2.4 Plasticity in Polymers

Plasticity in polymers is not controlled by dislocations, although tions may also exist in polymeric structures Instead, plastic deformation inpolymers occurs largely by chain sliding, rotation, and unkinking (Figs 1.7

disloca-and1.8) Such chain sliding mechanisms do not occur so readily in dimensional (thermoset) polymers (Fig 1.8) However, chain sliding mayoccur relatively easily in linear (thermoplastic) polymers when the sliding ofpolymer chains is not hindered significantly by radical side groups or othersteric hindrances The plastic deformation of polymers is also associatedwith significant changes in entropy, which can alter the local driving forcefor deformation

three-Elasticity and plasticity [Fig 5.8(a)] in rubbery polymers may result instrain levels that are between 100 and 1000% at fracture Such large strainsare associated with chain sliding, unkinking, and uncoiling mechanisms.Furthermore, unloading does not result in a sudden load drop Instead,unloading follows a time-dependent path, as shown in Fig 5.8(b)

Elasticity and plasticity in rubbery polymers are, therefore, often timedependent, since time is often required for the polymer chains to flow toand from the deformed configurations Cyclic deformation may result inhysterisis loops since the strain generally lags the stress (Fig 5.9), andanomalous stress–strain behavior may also be associated with chain inter-actions with distributed side groups which are often referred to as sterichindrances.

Crystalline polymers (Fig 1.9) may also exhibit interesting stress–strain behavior The minimum in the stress–strain curve is due to colddrawing and the competition between the breakdown of the initial crys-talline structure, and the reorganization into a highly oriented chainstructure

5.3 ELASTIC–PLASTIC BEHAVIOR

A generic plot of stress versus strain is presented inFig 5.10 This shows atransition from a linear ‘‘elastic’’ regime to a nonlinear ‘‘plastic regime.’’The linear elastic regime persists up to the proportional limit, at which thedeviation from linear elastic behavior occurs However, the onset of non-linear stress–strain behavior is generally difficult to determine experimen-tally An engineering offset yield strength is, therefore, defined by drawing aline parallel to the original linear elastic line, but offset by a given strain(usually an engineering strain level of 0.002 or 0.2%)

FIGURE5.8 Elastic–plastic deformation in rubbery polymers (a) Rubber randdeformed at room temperature (After Argon and McClintock, 1990) (b)Viscoelasticity in a rubbery polymer (After Hertzberg, 1996 Reprinted withpermission from John Wiley.)

The arbitrary offset strain level of 0.002 is recommended by the ASTME-8 code for tensile testing for the characterization of stresses required forbulk yielding However, it is important to remember that the offset strainlevel is simply an arbitrary number selected by a group of experts with aconsiderable amount of combined experience in the area of tensile testing.Above the offset yield strength, A, the stress may continue to increasewith increasing applied strain The slope of the stress–strain curve in the

FIGURE5.9 Hysterisis loop in a cyclically deformed polymer

FIGURE 5.10 Schematic of stress–strain behavior in the elastic and plasticregimes

plastic regime depends largely on the underlying dislocation interactions.The resulting shape changes in the gauge sections of tensile specimens areillustrated in Fig 5.11 Stretching in the vertical direction is accompanied byPoisson contraction in the elastic regime However, the contraction in thehorizontal direction is countered by hardening during the initial stages ofplastic deformation in which the gauge section deforms in a relatively uni-form manner, Fig 5.12(a) The rate of rate of hardening is, therefore,greater than the rate of horizontal contraction, and the total volume ofdeformed material remains constant, Fig 5.12(a) This inequality persistsuntil the ultimate tensile strength, M, is reached in Fig 5.11 At this stresslevel, the rate of hardening is equal to the rate contraction of the gauge area,

as shown in Fig 5.12(b)

Beyond the point M, in the stress–strain plot, geometrical instabilities(internal microvoids and microcracks within the gauge section) dominatethe plastic response, and the rate of horizontal contraction is greater thanthe rate of hardening, Fig 5.12(c) The deformation is thus concentratedwithin regions with the highest crack/microvoid density, and a phenomenonknown as ‘‘necking’’ [Figs 5.11 and 5.12(c)] occurs beyond the ultimate

FIGURE 5.11 Schematic illustration of gauge deformation in the elastic andplastic regimes

tensile stress This involves the gradual reduction in the cross-sectional area

in the regime of concentrated deformation This reduction occurs because ofthe rate of horizontal contraction is now greater than the rate of hardening,Fig 5.12(c) Necking may continue until the geometrical instabilities coa-lesce In any case, catastrophic failure occurs when a critical condition isreached

It is important to note here that the onset of necking may be delayed

by the application of hydrostatic stresses to the gauge section of a tensilespecimen This was first shown by Bridgman (1948) who demonstrated thatthe ductility of metals could be increased significantly with increasing hydro-static stress This is because the hydrostatic stresses tend to close up poresand voids that lead ultimately to necking and fracture

The geometrical instabilities are, therefore, artifacts of the test tions and specimen geometries that are used in tensile tests (Fig 5.13) Notethat the tensile specimen geometries (usually dog-bone shapes) are typicallydesigned to minimize stress concentrations in the region of transition fromthe grip to the gauge sections This is done to avoid fracture outside the

condi-FIGURE5.12 Hardening versus geometrical instability: (a) rate of hardening >rate of geometrical instability formation; (b) rate of hardening ¼ rate of geo-metrical instability formation (onset of necking); (c) rate of hardening < rate ofgeometrical instability formation (necking down to failure) (After Courtney,

1990 Reprinted with permission from McGraw-Hill.)

gauge section Also, the engineering definitions of stress and strain may not

be applicable to situations in which the cross-sectional area changes cantly during incremental plastic deformation to failure (Figs 5.11and5.12).True stress and true strain levels must, therefore, be defined, especially

signifi-in the plastic regime The true engsignifi-ineersignifi-ing stress, T, is given by the ratio ofapplied load, P, to the actual cross-sectional area, A This gives

FIGURE 5.13 Types of tensile specimen geometries: (a) cylindrical sections, (b) dog-bone specimen (wedge grips); (c) dog-bone specimen (pinloaded)

FIGURE 5.13 Types of tensile specimen geometries: (a) cylindrical