Solid Solution Strengthening of Face-Centered Cubic Alloys

Solid Solution Strengthening of Face-Centered Cubic Alloys K. R. EVANS Shell Development Company Houston, Texas

I. Introduction II. T h e Structure of Solid Solution Alloys A. Dislocation Density B. Stacking Fault Energy C. Dislocation Arrangements D . Solute A t o m Distribution III. General Plastic Properties of Solid Solution Alloys A. The Stress-Strain Curve B. Yield Point P h e n o m e n a C. Deformation Twinning D . The Portevin-LeChatelier Effect IV. Observations of Solid Solution Strengthening A . Temperature D e p e n d e n c e of the Yield Stress B. Composition D e p e n d e n c e of the Yield Stress V. Interpretations of Solid Solution Strengthening A. Dislocation Locking Mechanisms B. Dislocation Friction Mechanisms C. D i s l o c a t i o n Structure Mechanisms References

113 115 115 116 119 124 126 127 132 133 135 136 136 140 144 144 152 160 168

I. Introduction

Solid solution strengthening is commonly defined as the increase in yield strength of a material resulting from the presence of solute atoms located in substitutional or interstitial lattice sites. This definition applies to the majority of solid solution strengthened alloy systems; however, it is somewhat restrictive, for in the broadest sense it should make provision for strengthening 113

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species such as vacancy-impurity atom combinations, defect complexes formed during irradiation of materials and others. The latter types of strengthening species do not influence the properties of materials encountered in normal service and will not be included in this discussion. In fact, rather than attempt an all-inclusive discussion of solid solution strenghtening, the discussion to follow will be limited to occurrence of the phenomenon in substitutional metallic alloys. These alloys are by far the most technologically significant and comprehensively investigated solid solution strengthened systems. However, it should be recognized that solid solution strenghtening is neither restricted to metallic systems or characteristic only of substitutional strengthening agents, for in addition to the more unusual aspects of the phenomenon mentioned above it is well documented in alkali halide systems (Metag, 1932; Barrett and Wallace, 1956) and especially in interstitial metallic solid solutions (Wert, 1950; Evans, 1962; Ravi and Gibala, 1970). Solid solution strengthening of metals has been known and taken advantage of for technological purposes since early recorded history. For example, bronze tools have been found in ancient Cretan ruins which date back to around 3500 B . C . However, it is only within recent years and development in the mid-1930s of the concept of dislocation structures in crystalline materials that a foundation has existed for actually understanding the physical mechanisms responsible for solid solution strengthening. In these terms, the problem of understanding solid solution strenghtening is one of defining the manner in which solute atoms dictate the need for a higher stress to mobilize dislocations in an alloy than in the pure metal. It would be somewhat presumptuous to claim that solid solution strengthening is well understood mechanistically at this time. One need only to examine the shift in emphasis of reviews (Parker and Hazlett, 1954; Hibbard, 1958; Fleischer and Hibbard, 1963; Haasen, 1968) of the subject with time to recognize the dynamic nature of its understanding. However, it does appear that the basic mechanisms which lead to strengthening have been discussed at some level of understanding even though the uniqueness of one or more may be subject to debate. These mechanisms are conveniently categorized according to which of the following concepts forms its basis: (1) the immobilization of dislocations by solute atoms, (2) the impedance of mobile dislocations by solute atoms, or (3) the indirect influence solute atoms have on establishing dislocation structures which provide discrete obstacles to dislocation motion. It is the intention of this review to outline the concepts of each of these mechanisms and discuss their compatibility with experimental evidence. The discussion will be preceded by definition of the structure of solid solutions in terms of their defect character and solute distribution, and a summarization of experimental evidence.

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

115

II. The Structure of Solid Solution Alloys The addition of solid solution solute atoms to a metal can influence its physical, chemical, and mechanical properties both indirectly and directly. Indirect influences on properties occur as the result of effects solute atoms may have on the defect structure of a metal, i.e., primarily the density and distribution of dislocations and their stacking fault energy. Direct influences on properties result from the very presence of the solute atoms themselves (for they possess physical, chemical, and mechanical properties which differ from the solvent atoms) and the manner in which they are distributed in the metal. Both the indirect and direct influences of solute atoms have been suggested as explanations for solid solution strengthening in face-centered cubic alloys. Accordingly, it is important to understand the nature of defect structures in solid solutions and factors influencing solute distribution in such alloys. A. Dislocation

Density

Dislocations are considered to originate in metals as they solidify from the molten state by one or more of the following mechanisms (Elbaum, 1959; Evans and Flanagan, 1966): (1) shear strains resulting from abrupt compositional fluctuations and the resulting localized changes in lattice parameter, (2) thermal strains arising from constraints during cooling, (3) collapsing of vacancy disks, and/or (4) mechanical stresses. The first listed mechanism for dislocation formation involving solute composition is of particular interest in this dicussion because of its potential relationship to solid solution strengthening. Goss et al (1956), Jackson (1962), and Tiller (1958, 1965) have considered details of the role of solute concentration in influencing dislocation formation. Their models consider that during the cooling process from the molten state microscopic regions of solute segregation are produced. Differences in lattice parameter between the segregated region and matrix will result in creation of a stress, a, given by (Goss et aL, 1956; Tiller, 1965) a = Ac(da/a)E

(1)

where Ac is the atom fraction difference in solute concentration across the segregated region, a is the lattice parameter of the solvent metal, da is the lattice parameter difference between the solvent and solute, and E is Young's modulus. If the magnitude of this localized stress is greater than the yield stress of the matrix metal near its melting point, dislocations can be introduced into the metal just ahead of the solid-liquid interface. The density, p, of

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dislocations formed in this manner can be expressed as (Goss et al, 1956; Tiller, 1965) (2)

p^Acida/d^isby

1

where s is the spacing between segregated regions and b is the Burgers vector of the solvent metal. Dislocation densities as high as 1 0 / c m have been rationalized according to this mechanism. Direct evidence for existence of a dislocation formation mechanism of the above type has been obtained by Goss et al. (1956) for germanium-silicon crystals. In this study, dislocation etch pits were observed in alloy crystals adjacent to regions of high silicon content which were revealed upon etching because of a variation in chemical reactivity with composition. Hendrickson and Fine (1961) examined the dislocation structures of a series of silveraluminum alloys containing up to 6 at. % aluminum. These observations showed that after extensive annealing the mean cell diameter defined by a dislocation boundary array decreased markedly with increasing aluminum content, resulting in an increase in dislocation density with solute. Akhtar and Teghtsoonian (1969) have reported the dislocation density on the basal plane of magnesium-aluminum single crystals increases with the square root of the solute concentration. H a m m a r et al. conducted recent etch pit density experiments on a series of very dilute ( < 0 . 2 0 at. %) silver-tin and silverindium single crystals. Their results showed that dislocation density did not increase systematically in silver with the dilute tin and indium solute additions. The larger size difference between silver and tin than silver and indium had no appreciable influence on dislocation density or on dislocation arrangements in terms of developing dislocation cell walls and a well-defined substructure network. 10

2

An interesting observation by H a m m a r et al. (1967) was an increase in dislocation density by as much as a factor of five for silver and silver alloy crystals within 0.01 in. from their surface. The observed increase was more pronounced for the alloy crystals than for pure silver. This is an interesting observation considering arguments that surface dislocation sources dictate the plastic properties of copper single crystals (Fourie, 1967). B. Stacking Fault Energy While the influence of solute atom concentration on dislocation density may be open to question, it is well established that solutes in solid solution have a pronounced effect on the stacking fault energies of close-packed metals. Stacking faults in face-centered cubic alloys can be considered as strips of material with a hexagonal close-packed atom stacking sequence whose interfaces with the matrix consist of partial dislocations. Intrinsic and extrinsic

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SOLID SOLUTION STRENGTHENING OF F C C ALLOYS

{111}

•

(a)

(b)

Fig, 1. Schematic illustration o f (a) intrinsic and (b) extrinsic stacking faults in a fee metal from viewing position normal and parallel to their plane.

type faults are illustrated in Fig. l a and b, respectively. Intrinsic faults form when the normal face-centered cubic stacking sequence ABCABC is interrupted by removal of a {111} plane while extrinsic faults form upon the insertion of an extra {111} plane. The equilibrium separation distance, d , between partial dislocations is given by 0

d oc Gb b \y 0

x

(3)

2

where G is the shear modulus in the slip plane, b and b are Burgers vectors of the two partials, and y is the stacking fault energy. Stacking fault energy can dictate the stress required for deformation processes to occur if the process is dependent upon constriction of the partial dislocations by the applied stress (Seeger, 1957). Stacking fault energy is a measurable parameter which is used to characterize the nature of dislocations in an alloy, y has commonly been determined experimentally by one of the following methods (Christian and Swann, 1965): (1) T method, y is related to the stress required for constriction of the stacking fault so that cross-slip and the initiation of Stage III deformation can be initiated in single crystals of the alloy. (2) Twin boundary method, y is twice the twin boundary energy which can be measured for certain alloys in terms of experimentally determinable grain boundary energies. (3) Node method, y is directly related to the radii of curvature of partial dislocations at the extended nodes of lightly deformed crystals as detected by transmission electron microscopy. x

I H

2

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Experimental difficulties with each method make it difficult to obtain precise values of y as can be appreciated from comparison of values listed in Table I which have been determined by each method for various facecentered cubic metals. It should be noted that there is some degree of qualitative consistency between each method. Theoretical calculations indicate that the equilibrium separation of partial dislocations with stacking fault energies of 200, 40, and 20 erg/cm are around 1.6, 12, and 50 interatomic distances for edge dislocations and 1, 5, and 7 interatomic distances for screw dislocations (Christian and Swann, 1956). On this basis, dislocations in aluminum and nickel are not dissociated to any great extent, while in copper, silver, and gold extension is expected to be significant. Stacking fault energies of dislocations in body-centered cubic metals are considered to be in excess of 200 erg/cm and, thus, are essentially undissociated. 2

2

TABLE

I

STACKING FAULT ENERGIES FOR METALS"

(erg/cm ) 2

Ag Cu Au Ni Al a

65, 15 170, 50 30 300 230

2y(twin) (erg/cm ) 2

— 24

y (node) (erg/cm ) 2

21 70

—

—

60 142

180

—

Christian and Swann, 1965.

The addition of solid solution solute atoms to pure metals results in rather dramatic reductions in y as is indicated in Fig. 2a and b for copper-base and silver-base alloys, respectively. These figures illustrate the point that as the solute concentration of an alloy is increased, dislocations become more widely dissociated. Solutes of high valency in monovalent solvent metals have a more dramatic effect in reducing y for a given concentration than do lower valence solutes. Values of y in Fig. 2a and b do not show the scatter indicated in Table I for pure metals because of the experimentally consistent procedures employed by a single imvestigator. The dramatic reductions in stacking fault energy indicated in Fig. 2a and b are consistent with direct transmission electron microscopy observations. Such significant reductions in y have been shown to have a significant influence on the mechanical properties of solid solutions, particularly their creep behavior; however, a direct correlation between y and solid solution strengthening is not observed (Hendrickson and Fine, 1961).

119

2

f [erg/cm ]

SOLID SOLUTION STRENGTHENING OF F C C ALLOYS

2

STACKING FAULT ENERGY (erg/cm )

(a)

(b)

ELECTRON CONCENTRATION

Fig. 2. Effect of solute concentration o n the stacking fault energies of (a) copper alloys, and (b) silver alloys (Christian and Swann, 1965).

C. Dislocation

Arrangements

The distribution of dislocations has been studied extensively in pure copper (Evans and Flanagan, 1966; Basinski and Basinski, 1964) and silver (Levinstein and Robinson, 1962) by etch pit methods; however, comparable observations in alloy systems are very limited. Hendrickson and Fine (1961) showed that dislocations in silver-aluminum crystals tended to align themselves in sub-boundary arrays which defined a cell size that decreased with increasing aluminum concentration. H a m m a r et al. (1967) showed that for very dilute (less than 0.20 at. % solute) silver-indium and silver-tin crystals only a

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slight tendency existed to form sub-boundaries. As mentioned earlier, subboundaries had a preference for forming near the crystal surface and were somewhat more pronounced in the silver-tin than the silver-indium alloys. Transmission electron microscopy observations have been extensively used to establish the detailed nature of dislocation arrangements in alloys (Swann, 1962; Swann and Nutting, 1962; Hirsch, 1963). Such studies have show that dislocation distribution is strongly dependent upon solute concentration. Dislocations in pure metals occur in isolated patches 5-10 ixm long. The patches of dislocations contain a high density of edge dislocation dipoles. The distance between dislocation patches is rather significant, leaving large areas dislocation free. The general effect of increasing the solute content of an alloy and thereby decreasing its stacking fault energy appears to be t o : (1) promote a more uniform distribution of dislocations rather than the " p a t c h " distribution observed in pure metals and dilute alloys, and (2) confine dislocation movement to the primary slip system and inhibit secondary slip. These tendencies become particularly apparent upon observing the dislocation distributions in deformed alloys. Strained high stacking fault energy metals exhibit sharp, well defined cell walls of high dislocation density. The cell diameter decreases with increasing solute concentration and the structure of the cell walls becomes increasingly diffuse. A solute concentration is eventually reached where a cell structure is no longer apparent and dislocations are distributed in planar arrays on the primary slip system in a screw orientation. At very high solute concentrations the density of stacking faults, which are very wide due to small values of y, is quite high and the structure is defined as being heavily faulted. Swann and Nutting (1962) have made a series of transmission electron microscopy observations on prestrained copper alloys and defined the variation in dislocation arrangement with solute concentration in terms of stacking fault energy as follows: (1) Cellular structure (y > 8 erg/cm , corresponding to < 20 at.% Zn, < 3.6 at.% Al, < 2 at.% Si). The dislocation structure consists of cells whose walls consist of dislocation tangles formed by the cross-slip of screw dislocations as illustrated in Fig. 3a. Increasing solute content tends to reduce the cell size and increase the dislocation density within the cells. (2) Planar structure (y < 8 erg/cm > 1 erg/cm , corresponding to 30-37 at.% Zn, 4.5-8.0 at.% Al). Dislocations are arranged on easily recognizable slip planes (see Fig. 3b) with many long dislocation pile-ups at grain boundaries. (3) Faulted structure (y < 1 erg/cm , corresponding to > 10 at.% Ge, > 4 at.% Si). Structure consists of heavily faulted bands as shown in Fig. 3c. 2

2

2

2

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

121

Fig, 3A, Dislocation distributions in slightly strained copper alloys: C o p p e r - 2 a t . % aluminum, x 14,000 (Swann and Nutting, 1962).

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Fig. 3B. Dislocation distributions in slightly strained copper alloys: C o p p e r - 8 a t . % germanium strained 5%, x 14,000 (Swann and Nutting, 1962).

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

123

Fig. SC. Dislocation distributions in slightly strained copper alloys: C o p p e r - 1 0 a t . % germanium strained 5%, x 7,000 (Swann and Nutting, 1962).

124 D. Solute Atom

K. R. EVANS

Distribution

The distribution of solute atoms in solid solution alloys is generally not random. Solute atoms may preferentially segregate about dislocations as the result of one or more of three basic dislocation-solute atom interactions: elastic interactions, chemical interactions, and electrical interactions. Furthermore, most solid solutions are not ideal and a preference does exist for a specific atom to have either like or unlike neighbors. Those concepts are significant for they have provided the basis for several solid solution strengthening theories which will be discussed later. 1. ELASTIC DISLOCATION-SOLUTE A T O M INTERACTION

A substitutional solute atom is generally not of the same size as the solvent atom it replaces. The size difference causes a localized dilation or contraction of the lattice, resulting in a characteristic stress field about the solute atom. The stress field for a substitutional solute atom in a face-centered cubic matrix will be hydrostatic. Dislocations are also characterized by their stress fields. Hydrostatic and shear components exist about edge dislocations while only a shear component is associated with screw dislocations. The strain energy of the whole lattice can be minimized by an interaction between the stress fields of the dislocations and solute atoms in the manner illustrated for an edge dislocation in Fig. 4. Both the strain energy of the dislocation and dilation field about the solute atom will be reduced if solute atoms larger than the solvent atoms migrate to A type lattice sites and those smaller migrate to B type sites. Solute atoms are generally not the same size as solvent atoms, so such interactions are expected to be rather common. The maximum dislocation-solute elastic interaction occurs at the dislocation core where the stress field is a maximum. The stress field decreases rapidly with reciprocal distance from the dislocation core; therefore, the interaction can be significantly influenced by thermal fluctuations. Accordingly,

o

o

o

o

o

o

o

o

o

o

o

o

o

o

q

o

o

o

o

i

B

o

o

o

#

o

o

o

o

o

o

o

o

A

o

o

o

o

o

o

o

Fig. 4. Illustration o f strain energy minimization by solute segregation t o an edge dislocation. A site replaced by large solute a t o m and B site replaced by small solute a t o m .

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

125

the interaction is a maximum at 0 ° K and decreases rapidly with increasing temperature. The theoretical aspects of the elastic interaction have been treated in some detail (Bilby, 1950; Cottrell, 1963). 2. CHEMICAL DISLOCATION-SOLUTE A T O M INTERACTION

Dislocations in close-packed lattices tend to separate into two partial dislocations separated by a stacking fault as illustrated in Fig. 1. Although the affected region is only on the order of two atoms thick, the interatomic forces in the fault region are significantly different from those in the matrix. Accordingly, when an alloy is in thermodynamic equilibrium, the concentration of solute atoms in the fault region will not necessarily be the same as that of the matrix, i.e., a chemical driving force will cause solute atoms to migrate to the fault. Suzuki (1952) has calculated details of the chemical interaction. It is found that the chemical dislocation-solute atom interaction is on the order of one-tenth the maximum elastic dislocation-solute atom interaction; however, whereas the elastic interaction is strongly temperature dependent, the chemical interaction is not. 3. ELECTRICAL DISLOCATION-SOLUTE A T O M INTERACTION

The dilation existing around the core of a dislocation causes localized changes in the energy level of the Fermi surface (Cottrell etal, 1953). Conduction electrons in the lattice will tend to redistribute themselves (from the compressed side to the dilated side of an edge dislocation) in an effort to maintain a uniform Fermi surface level everywhere in the metal. Solute atoms having a higher valence than atoms of the base metal are a natural source of the conduction electrons necessary to smooth out the Fermi surface level. Accordingly, a tendency will exist for these solutes to segregate to dislocations resulting in the formation of an electrical dipole at the dislocation. The nature of the interaction has been estimated to be very small compared to the elastic dislocation-solute atom interaction (Cottrell et al, 1953). For copper alloys the electrical interaction is only expected to be one-third to one-seventh that of the elastic interaction. As a result, electrical interactions are not expected to have a rate-controlling influence on the mobility of dislocations. The electrical interaction is dependent upon the localized dilation about a dislocation in the same manner as the elastic interaction and, accordingly, will exhibit the same temperature dependence. 4.

LOCAL ORDERING

Disregarding their defect structures, the distribution of solute atoms in solid solution alloys is generally not random. A random distribution of solutes will only occur when the interaction energies between like and unlike neighboring atoms is exactly the same (Flinn, 1962). In general, some preferential

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interaction will exist so that preponderance of like or unlike neighbors will exist. This is best illustrated in Fig. 5 where the nearest neighbor local order parameter, a is used to characterize the randomness of the solid solution. a is defined as l5

1

«i =

1

()

~ (^ABK)

4

where p is the probability of finding a particular A atom as nearest neighbor to a B atom in a solid solution of A and B atoms, and m is the mole fraction of A atoms which is the corresponding probability in a random solid solution. AB

A

R a n d o m Solution Average Number of Unlike Neighbors =3 a =0 (a) /

Short R a n g e Order Average Number of Unlike Neighbors =4 (B)

'

(

Clustering

Average Number of I Unlike Neighbors = 2 (0 Fig. 5. Two-dimensional illustration of (a) random solution, (b) short-range order, and (c) clustering (Flinn, 1962).

Figure 5 illustrates the two-dimensional distribution of solutes for a, = 0 (a completely random solid solution), oc = — £ (the case for short-range order, i.e., a preponderance of unlike neighbors compared to the random distribution), and a, = -f y (the case for clustering, i.e., a preponderance for like neighbors compared to the random distribution). The presence of such nonrandom solute distributions can influence the stress required to move dislocations in solid solutions as will be discussed later. i

III. General Plastic Properties of Solid Solution Alloys The presence of solute atoms has some rather dramatic effects on the plastic properties of face-centered cubic metals. This section discusses these influences with respect to observations for pure metals.

127

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

A. The Stress-Strain

Curve

SHEAR STRESS

Early investigators used hardness measurements to establish the occurence of solid solution strengthening in lead (Goebel, 1922), copper (Norbury, 1 9 2 3 ; Brick et al, 1943) and silver (Frye and Hume-Rothery, 1942) alloys; however, the tensile or compressive testing of single crystal and polycrystalline specimens provides a more accurate measurement of yielding, and at an early date became a standard testing procedure for studying strengthening (von Goler and Sachs, 1 9 2 9 ; Sachs and Weerts, 1 9 3 0 ; Osswald, 1933). Figures 6a and b

SHEAR STRAIN

STRAIN

(a)

(b)

Fig. 6. Schematic stress-strain curves for fee (a) single crystals and (b) polycrystalline specimens illustrating methods for determining the critical resolved shear stress and yield stress, respectively.

schematically illustrate stress-strain curves typical for face-centered cubic single crystal and polycrystalline specimens, respectively. The critical resolved shear stress (CRSS), T , defining the yielding of a single crystal is determined by simple extrapolation of the flow stress from the Stage I hardening region to zero plastic strain. The stresses T„ and T define discontinuities in the stress-strain curve indicating the onset of Stage II and Stage III work-hardening processes. Complete descriptions of single crystal deformation behavior are provided in reviews by Clarebrough and Hargreaves (1959) and Mitchell (1964). The yield stress,
i h

0

0

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Yield stress measurements for defining solid solution strengthening behavior can be obtained from either polycrystalline or single crystal specimens. The influence work hardening rate may have on a values measured for polycrystalline alloys can be minimized by using single crystal specimens. The yield strength of polycrystalline specimens is also strongly influenced by their grain size. This can make analysis of strengthening behavior difficult, particularly in comparing data from various investigators. For this reason, it is more convenient and probably necessary to study solid solution strengthening mechanisms in terms of single crystal data where structural variations are minimized, although not altogether eliminated. This is not to say that available polycrystalline data should be ignored in considering strengthening, for as will become evident in the discussion to follow, they have contributed significantly to the current understanding of strengthening. The most obvious influence on the mechanical properties of a metal by solid solution alloying is an increase in yield stress as illustrated in Fig. 7 for

T r u e S t r e s s ( c r ) i n 1000 psi

0

0.01

0.02 0.03 0.04 € - True S t r a i n

0.05

0.06

Fig. 7. True stress-true strain curves for polycrystalline nickel-iron (Parker and Hazlett, 1954).

129

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

the true stress-true strain curves for a series of polycrystalline nickel-iron alloys. However, testing of single crystal specimens provides a more sensitive indication of specific details solute atoms have on the stress-strain curve. This is illustrated in Fig. 8 for a series of copper-zinc solid solution crystals where a number of generalizations other than the strength increase can be made as follows: (1) Stage I, the easy glide region, increases with solute content (Rosi, 1954; Garstone 1956; Mitchell and Thornton, 1963; Van der Planken and Deruyttere, 1969), (2) the rate of work hardening in Stage I decreases somewhat with increasing solute content (Mitchell, 1964), (3) the rate of work 14 12

e

^ 8 CO

oc U

4 2 0

0

0.5 6

1.0

-

-

Fig. 8. Stress-strain curves of copper and copper-zinc single crystals at 295°K. The numbers on the curves represent the zinc content of the alloys (Mitchell, 1964).

hardening in Stage II decreases slightly with increasing solute content (Rosi, 1954; Mitchell and Thornton, 1963; Van der Planken and Deruyttere, 1969), and (4) the stress, T , , required to initiate Stage III increases with solute content (Mitchell and Thornton, 1963). Some of these generalizations have been reported to be inconsistent with observations in other face-centered cubic alloys, notably aluminum alloys; however, the very restricted solubility of alloying elements in aluminum makes it very difficult to avoid second phase precipitation during specimen preparation. It is this difficulty that is most likely to be responsible for reported inconsistencies with alloys exhibiting significant solubility. For this reason, solid solution strengthening data reported for aluminum alloys is not emphasized in the observations presented and discussed below. Direct observation of dislocation structures in deformed pure metals and alloys by transmission electron microscopy has provided some indication of h

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the specific influence solute atoms have on dislocation movements and arrangements. Some of these features were discussed above. Etch pit observations and slip-band studies of deformed single crystal specimens are also capable of providing information about the yielding and deformation processes occurring in metal crystals. Etch pit studies have been fairly well confined to pure metals such as copper (Basinski and Basinski, 1964; Young, 1961, 1962b) and silver (Levinstein and Robinson, 1962). The limited studies reported for alloy crystals (Meakin and Wilsdorf, 1960) have not provided sufficient results for comparison with results obtained for pure metals. An extensive number of slip-band studies have not been conducted for alloy crystals; however, the results of those available provide the basis for several generalizations. Slip bands in alloy crystals exhibiting easy glide are long (on the order of 600-1000 jum) and straight, located on the primary slip system as observed in pure metals (Mader, 1957). The length of the slip bands and their mean spacing is independent of strain in easy glide; however, for a given strain the mean spacing between slip bands increases with increasing solute content both in easy glide and Stage II deformation. The height of slip steps observed in copper is on the order of 20 Burgers vectors (Fourie, 1960). In Stage II deformation for both pure metals and alloys the mean spacing between slip bands decreases with increasing strain (Swann and Nutting, 1962). These results indicate that in easy glide a limited number of dislocation sources are activated in both pure metals and alloys which operate during all of easy glide. Solute concentration appears to have the effect of decreasing the number of sources which can be activated and concentrating the strain produced in the planes of these sources; hence, the larger mean spacing between slip bands and larger slip-band steps observed in alloys. Studies in pure metals indicate easy glide terminates when the applied stress becomes large enough to activate a significant number of dislocation sources on secondary slip systems. The longer extent of easy glide in alloys may be related to solute strengthening of the secondary slip systems; however, it is more likely to be related to stress fields associated with dislocation configurations formed on the primary slip system during easy glide (Mitchell, 1964). It should be mentioned that Stage I observed in the deformation of alloy crystals does not necessarily occur by easy glide. Easy glide appears to take place in dilute copper (Garstone and Honeycombe, 1957; Haessner and Schreiber, 1957; Brindley et al, 1962) and nickel-cobalt (Meissner, 1959) solid solutions; however, in many copper alloys Stage I does not occur by the activation of a uniformly distributed number of dislocation sources along the specimen gage length but rather by propagation of a Luders front along the gage length (Brindley et al, 1962; Piercy et al, 1955; Koppenaal and Fine, 1961; Haasen and King, 1960). Brindley et al. (1962) found for a series

SOLID SOLUTION STRENGTHENING OF FCC

131

ALLOYS

of copper-zinc solid solutions that Stage I occurred by easy glide for dilute zinc concentrations (1 at. %), Luders front propagation for high zinc concentrations (20 at. %), and initially Luders front propagation followed by easy glide for intermediate zinc concentrations (5 and 10 at.%). The dependence of Luders strain upon solute content has been interpreted to be the result of the locking of dislocations by solute atoms. Dislocation movement will be initiated at a weakly locked source and strain will be concentrated in this region until sufficient stress is concentrated at the Luders front to initiate adjacent locked sources. The applied stress required to propagate the Luders front may not be sufficient to activate secondary slip systems for reasons to be mentioned in the following paragraph. Accordingly, once the Luders front has propagated the length of the specimen, easy glide may proceed until a sufficiently large stress is available for activation of secondary slip systems and the initiation of Stage II hardening. Overshooting" is a term describing the phenomenon of a single crystal continuing to deform on its primary slip system toward the [Oil] primary direction even when the tensile axis has rotated or " overshot" the [001 ]—[Tl 1 ] symmetry boundary rather than moved toward the [112] direction by a double-slip process where the conjugate slip system becomes activated and is more highly stressed than the primary system. Overshooting can occur in pure face-centered cubic metals but is much more pronounced in alloys, increasing with solute content and decreasing temperature. Overshooting has been interpreted in terms of solute locking on the conjugate slip system; however, it would appear the phenomenon is more likely to be related to the existence of widely extended dislocations on the primary slip system and/or a higher dislocation density on the primary slip system (Mitchell, 1964; Piercy et al, 1955). 44

A corresponding increase in the critical stress, T , required to initiate Stage III deformation with solute content is related to the stacking fault energy. Stage III deformation is initiated when a shear stress of sufficient magnitude is available to produce cross-slip (Seeger, 1957). Cross-slip is a thermally activated process consisting of the constriction of extended dislocations and their subsequent separation and movement on a close-packed plane other than the primary slip plane. The energy for constriction is therefore dependent upon solute content through the stacking fault energy; hence, the observed relationship between T and solute concentration. T for a given alloy decreases with increasing test temperature because more thermal energy is available to participate in the thermally activated process. Solute atoms can also produce several other rather dramatic changes in the shape of stress-strain curves for both polycrystalline and single crystal specimens, such as the occurrence of yield point phenomena upon initial yielding, the Portevin-LeChatelier effect (serrated yielding), and deformation twinning. h i

i h

h i

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Although each of these phenomena in itself is outside the scope of this presentation, they do reflect the influence of dislocation-solute atom interactions and will be described briefly. B. Yield Point Phenomena

Pure copper

1% Zn

5% Zn

10% Zn

30% Zn

Resolved shear stress

2

(kg/mm )

Yield points occur during the testing of a crystalline material when a larger stress is required to activate dislocation sources than to keep them operative. During tensile testing this phenomena is revealed by a decrease in stress from the level required to initiate plastic deformation (the upper yield stress) to that required to maintain it (the lower yield stress).

Shear strain % Fig. 9. Yield point formation in copper-zinc alloys deformed at r o o m temperature (Ardley and Cottrell, 1953).

Yield points are particularly well defined in alloy single crystals as illustrated in Fig. 9 for a series of copper-zinc alloys (Ardley and Cottrell, 1953). Polycrystalline alloys also exhibit yield point phenomena upon initial yielding (Hutchison and Honeycombe, 1967; Soler-Gomez and M c G . Tegart, 1970); however, the yield drops are not as pronounced as those observed in single crystals. In general, the magnitude of the yield point increases with decreasing test temperature and increasing solute content (see Fig. 9), although exceptions to the first generalization have been reported for some polycrystalline alloys (Hutchison and Honeycombe, 1967; Soler-Gomez and M c G . Tegart, 1970). The magnitude of the yield point in face-centered cubic alloys has also been shown to be dependent upon exposure time at sufficiently high temperatures, i.e., these alloys are subject to strain aging. Rather extensive strain-aging studies have been carried out for silver-aluminum (Hendrickson and Fine, 1961), copper-aluminum (Koppenaal and Fine, 1961), and copperzinc (Boiling, 1959) alloys. These observations have pointed out a correlation

133

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

between the magnitude of the yield point and diffusivity of solute atoms. Accordingly, the role of solute atoms in producing yield point phenomena and strain aging is their ability to migrate to dislocations, and undergo dislocation-solute atom interactions of the types described earlier. Before dislocations which have experienced a solute atom interaction can be mobilized to participate in the yielding process, an increment of stress must be provided to overcome the interaction energy. The chemical interaction has been considered responsible for yield points occurring in silver-aluminum (Hendrickson and Fine, 1961) and copper-aluminum (Koppenaal and Fine, 1961) alloys. The immobilization of dislocations by dislocation-solute atom interactions is consistent with the concept of dynamical yielding proposed by Johnston (1962) and Hahn (1962). The constant strain rate, s, imposed upon a test specimen can be expressed as e = bp v

(5)

m

where p is the mobile dislocation density and v is the mean velocity of dislocations. The stress dependence of the dislocation velocity may be expressed as m

(6)

*=[T*/TS]'

where T* is the effective shear stress doing work on a dislocation, i* is the effective shear stress required for unit velocity, and w* is a constant for a given material and testing conditions. Equation (5) illustrates that for a constant strain rate test dislocations can move at relatively low velocities when the mobile density is large, but their velocity must be high when the mobile dislocation density is low. In the latter case, Eq. (6) shows that a relatively high stress must be applied to satisfy dynamic conditions. The yield point in face-centered cubic alloys appears to be adequately interpreted in terms of dislocation-solute atom interactions which decrease p to values requiring higher than characteristic dislocation velocities. The high stresses required to produce these velocities are also capable of activating additional dislocation sources, thereby permitting a reduction in dislocation velocity and applied stress to levels determined by the solid solution strengthened lattice. Either the density of immobilized sources or the degree of immobilization (or both) are influenced by the amount of solute permitted to diffuse to extended dislocations, thereby accounting for the strain aging observations. m

C. Deformation

Twinning

Plastic deformation in crystals can take place by twinning as well as the more frequently observed translational slip mechanisms. Deformation twinning usually occurs by the initiation of twin movement in one part of the

134

K. R. EVANS

2

Nominal stress (kg/mm )

specimen which then propagates discontinuously throughout the specimen length. The initiation of twinning results in a sudden drop in stress followed by a nearly constant stress level which is required to propagate twins the length of the specimen. When this has been achieved, normal work hardening may take place. A typical example of such behavior is shown in Fig. 10 for copper-10 at. % indium crystals tested at 77° and 293°K (Honeycombe, 1968).

O

i

i

25

50

•

i 75

• 100

Elongation (%) Fig. 10. Deformation twinning in c o p p e r - 1 0 a t . % indium single crystals ( H o n e y c o m b e , 1968).

Deformation twinning has frequently been observed in body-centered cubic and hexagonal close-packed metals but is a relatively recent observation in face-centered cubic metals. The phenomenon was initially observed in pure copper (Blewitt et al., 1957; Thornton and Mitchell, 1962) at 4°K, and has subsequently been found to occur in silver (Suzuki and Barrett, 1958), gold (Suzuki and Barrett, 1958), and nickel (Haasen, 1958) at low temperatures, generally well below 77°K. Twinning is observed at low temperatures in pure metals for it is only in this temperature region that the applied stress levels approach those necessary to activate twinning systems. Subsequent studies have shown deformation twinning to occur in the following alloys: silver-gold (Suzuki and Barrett, 1958), copper-zinc (Mitchell and Thornton, 1963; Haasen and King, 1960), and copper-aluminum, copper-germanium, coppergallium, and copper-indium (Haasen and King, 1960). The influence of solute

135

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

2

Twin Stress (kg/mm )

15

f

• CU+AI A CU + GE

5

0

20 40 60 Stacking Fault Energy, y (erg/cm ) 2

Fig. 11. Twin stress as a function of stacking fault energy for copper alloys (Venables, 1964).

content is to lower the stress required to initiate dislocation movement on twin systems. The reduction in twinning stress, T , with increased solute concentration appears related to the stacking fault energy as suggested by the dependence of T on y as shown in Fig. 11. Mechanisms for deformation twinning in face-centered cubic alloys have been presented by Suzuki and Barrett (1958) and Venables (1964). t

t

D. The Portevin-LeChatelier

Effect

The Portevin-LeChatelier effect (also known as serrated yielding or jerky flow) has been observed in both body-centered cubic and face-centered cubic alloys for some time. It occurs in both single crystal (Garstone et al, 1956; Radelaar et al, 1970; MacEwan and Ramaswami, 1970) and polycrystalline (Brindley and Worthington, 1969; McCormick, 1971; Wijler et al, 1972) specimens, and for a constant strain-rate test consists of abrupt oscillations in stress on a stress-strain curve as shown in Fig. 12. The effect is rather unique in that oscillations in stress are sensitively dependent upon both the temperature of test and the imposed strain rate (see Fig. 12). Interpretations of the phenomenon have been in terms of dynamic strain aging (Cottrell, 1953; McCormick, 1972; Brindley and Worthington, 1972), i.e., the conditions of test define a critical condition in terms of solute atom mobility with respect to dislocation velocity. The test condition occurs when solute atoms have sufficient mobility to migrate to moving dislocations, interact with them during the deformation process, thereby requiring an increase in applied stress to free dislocations from their newly acquired atmosphere in order to

136

K . R . EVANS

1000 ,

LOAD ( k g )

800

600

400

SOLUTION TREATED AT 4 9 0 °C TEST TEMPERATURE 23°C GAGE LENGTH 5 4 mm CROSS SECTION AREA 2 0 mm

2

~0

1

2

3

4

5

6 7 8 ELONGATION (mm)

9

F/^r. / 2 . Load-elongation curves at various strain rates for an magnesium (type 2024) alloy ( R o s e n and Bodner, 1969).

10

11

12

aluminum-copper-

maintain the imposed strain rate. The sequence of interaction and liberation continues, resulting in continual yield point formation which can be interpreted in the manner described earlier. Details of the effect are discussed more fully in the references cited above.

IV. Observations of Solid Solution Strengthening The number of references to solid solution strengthening in the literature since the 1920s is rather impressive; however, more careful observation will show that very little systematic data is available for body-centered cubic and hexagonal close-packed alloys, and that the preponderance of data for facecentered cubic alloys is limited to copper and silver solid solutions. However, sufficient data for face-centered cubic alloys is available upon which to formulate and evaluate mechanisms for solid solution strengthening. A. Temperature Dependence of the Yield Stress The yield stress and critical resolved shear stress for both pure metals and solid solution alloys are very temperature dependent. Figure 13 illustrates schematically the temperature dependence of yielding for a pure metal or alloy. Three temperature regions can be defined as follows: (1) Region 1 is defined by a significant increase in yield stress with decreasing temperature. The stress in this region is considered to be the sum of

13

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

Y I E L D S T R E S S OR C R S S

137

TEMPERATURE Fig. 13. Schematic illustration for the temperature dependence o f the yield stress and critical resolved shear stress ( C R S S ) .

two components—a temperature independent stress, r , defined for Region II and a temperature dependent stress, T * , required to assist thermal fluctuations overcome short-range obstacles to yielding by a thermally activated process. Figure 13 defines the magnitude of T * at temperature 7\ as T * T * becomes zero at a critical temperature, T where the magnitude of thermal fluctuations is large enough to provide sufficient thermal energy to overcome the short-range obstacles without the assistance of an externally applied stress. The energy of short-range obstacles in many face-centered cubic alloys is such that T is on the order of 350°-500°K. (2) Region 11 is defined by a temperature interval where the yield stress of an alloy is dependent upon temperature only through the temperature dependence of its elastic modulus. The yield stress, r , in this region is commonly referred to as the " p l a t e a u stress." r is the stress commonly used to provide a measure of solid solution strengthening. (3) Region 111 is defined by a decrease in yield stress with increasing temperature. The yield stress in this temperature region is influenced by diffusional processes whose consideration is beyond the scope of this discussion. The critical temperature, T , where this region starts is dependent upon the m o bility of solute atoms present. In general, 7 , occurs around two-thirds of the melting point of an alloy. p

R

C9

c

p

p

D

D

The temperature dependent component of the yield stress-temperature curve is very much dependent upon both the amount of solute present and the nature of the solute. Figure 14 shows that the rate of increase in stress with

138

K. R. EVANS

2

( UJUI/6>I) sSldD TEMPERATURE (°K) Fig. 14. Critical resolved shear stress versus temperature for a series of copper-aluminum alloy single crystals (Koppenaal and Fine, 1962).

decreasing temperature increases significantly with the aluminum content of copper single crystals. This type of behavior is well documented for both single crystal (Hendrickson and Fine, 1961; Garstone and Honeycombe, 1957; Thornton and Mitchell, 1962; Evans and Flanagan, 1968; Suzuki and Kuramoto, 1968) and polycrystalline (Hutchison and Honeycombe, 1967; Hutchison and Pascoe, 1972) test speciemns. Figure 15 illustrates polycrystalline behavior with results for copper-indium alloys. Figure 16 demonstrates the dramatic influence different solutes of the same concentration level (6 at. % have on the yield stress-temperature curve for polycrystalline silver. Gold, for example, does not appreciably alter the yield stress of silver or its temperature dependence, while antimony exhibits a very substantial influence increasing rj by a factor of two at 375°K and a factor of four at 77°K. Cottrell and Stokes (1955) conducted an experiment where a specimen deformed to a specific strain at a temperature T was unloaded and stressed to yield at a lower temperature T . The experimental method established a dislocation structure at 7\ corresponding to a stress of T ( r ) , while at T a stress of T(T ) was required to initiate plastic deformation in the structure developed at 7\. Cottrell and Stokes found the ratio T ( r ) / t ( r ) to be independent of strain for aluminum single crystals, an observation now known as the Cottrell-Stokes law. Constancy of the flow stress ratio has also been 0

x

2

x

2

2

2

l

139

1% PROOF

2

STRESS ( N / m m )

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

0

100

200 300 400 TEMPERATURE (°K)

500

600

2

STRESS ( K G / M M )

Fig. 15. Temperature dependence of the yield stress for a series of copper-indium polycrystalline specimens (grain size = 50 ± 5 /xm (Hutchison and Pascoe, 1972).

TEMPERATURE

<°K)

Fig. 16. Temperature dependence o f the yield stress o f polycrystalline silver alloys containing 6 a t . % solute (Hutchison and H o n e y c o m b e , 1967).

140

K. R. EVANS 1.5 Cu-5at.% Si \

—

77°k/ 194°K

O

C117A

A

C119B

t

•

C129B

V

C134A

O

C128A

O

C119C

T

T/ 194°K

1.4

t

T

V

\

a

-

" 194°K^^^ r

D

1.1 — 1.0

^ — 6 *

PURE Cu

I

03

T

,

77°k/

1 2

T

194°K

,

1

.

4 SHEAR STRESS, T

1

.

1

A

fi

1

1

in

(kg/mm ) 2

] 9 4 0 | <

fi,gr. 17, Temperature sensitivity of the flow stress with increasing state of deformation for copper-5 a t . % silicon crystals (Evans and Flanagan, 1968a).

reported for copper, (Makin, 1958), silver (Basinski, 1959) and nickel (Haasen, 1958) single crystals. Detailed studies (Diehl and Berner, 1960) on copper crystals have shown deviations from a constant ratio to exist in easy glide. A limited number of observations in alloy crystals show that the flow stress ratio is constant in the easy glide region (Hendrickson and Fine, 1961; Koppenaal and Fine, 1962; Evans and Flanegan, 1968a) but decreases very markedly with strain in Stage II deformation as illustrated in Fig. 17 for a copper-silicon alloy. B. Composition Dependence of the Yield Stress The plateau stress, T , is a proper index of strengthening and should be used to define for an alloy system its strengthening rate with composition, dzjdc. This criterion should be adhered to for the simple reason that it establishes an experimental condition all investigators can reproduce in defining the strengthening ability of various solute atoms in a given metal and the ability of various metals to be strengthened. Comparison of strengthening data from various sources is then feasible. Care has not been taken in many studies, p

141

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

especially early ones, to insure that the yield stresses or critical resolved shear stresses were measured in the athermal temperature region. Convenience led many investigators to make their measurements at room temperature. For face-centered cubic alloys this is frequently not a serious problem for the critical temperature, T defining the lower boundary of the athermal region is usually close enough to room temperature that little error is associated with room temperature data. The early studies of von Goler and Sachs (1929) found that the critical resolved shear stress of copper-zinc single crystals increased rapidly and linearly with zinc concentration in fairly dilute amounts. This observation has since been reported for both single crystal (Linde et al, 1950; Linde and Edwardson, 1954) and polycrystalline (French and Hibbard, 1950) test specimens. However, it is apparent from the early work of Sachs and Weerts (1930) and Osswald (1933) on alloy systems exhibiting complete solubility (silvergold and copper-nickel systems, respectively) that the linear relationship between strengthening and solute content is only an approximation in the dilute range, and that the more general relationship demonstrated by these systems is parabolic with a maximum near 50 a t . % solute. Figure 18 shows results obtained for silver-gold alloys. c9

0.6

~ 0.4 E E •* 0.2

0 0

20

40

60

80

100

Atomic % Au Fig. 18. C o m p o s i t i o n dependence of the critical resolved shear stress for silver-gold single crystals (Sachs and Weerts, 1930). A

g

A

u

Systematic single crystal studies have only been conducted for copper, silver, gold, and lead alloys. The composition dependence of the plateau stress for these alloy systems is shown in Figs. 19-22 and, in general, exhibits parabolic behavior. These data show that the ability to strengthen a specific base metal varies greatly from solute to solute. This is further illustrated in Fig. 23 where the stress-strain curves for polycrystalline silver-6 at. % solute alloys are significantly dependent upon the type of solute. The composition dependence of the yield stress for polycrystalline alloys should be the same as for single crystals unless care is not taken to maintain a constant specimen grain size.

T

p

2

(kg/mm )

3.5.-

SOLUTE

C O N T E N T (%)

Fig. 19. C o m p o s i t i o n dependence of the plateau stress for copper-germanium (Peissker, 1965), copper-silicon (Evans and Flanagan, 1968b), c o p p e r - a l u m i n u m (Koppenaal and Fine, 1962), copper-zinc (Mitchell and T h o r n t o n , 1963), and copper-nickel ( M e n o n and Flanagan, 1973) alloy single crystals. r -

T

p

2

(kg/mm )

0.7

SOLUTE CONTENT (%) Fig. 20. C o m p o s i t i o n dependence of the plateau stress for silver-aluminum (Henrickson and Fine, 1961), silver-indium (Haasen, 1965), silver-cadmium ( K a n and H a a s e n , 1 9 6 9 1970), and silver-gold ( K l o s k e and Fine, 1969) alloy single crystals.

143

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

2

( ujuj/6>J) j.

d

SOLUTE

CONTENT (%)

T

p

2

(kg/mm )

Fig. 21. C o m p o s i t i o n dependence o f the plateau stress for gold-gallium (Jax et al, 1970), g o l d - i n d i u m (Jax et al, 1970), g o l d - p a l l a d i u m (Kratochvil, 1970), g o l d - z i n c (Jax et al, 1970) and g o l d - c a d m i u m (Jax et al, 1970) alloy single crystals.

" 0

0.5

1.0

1.5 2.0 2.5 3.0 3.5 SOLUTE CONTENT (%)

4.0

4.5

Fig. 22. C o m p o s i t i o n dependence of the plateau stress for l e a d - c a d m i u m , lead-tin, lead-bismuth, and lead-tellurium alloy single crystals ( K o s t o r z and Mihailovich, 1970).

144

K.

R.

EVANS

N A T U R A L STRAIN

Fig. 23. Stress-strain curves for silver and silver alloys containing 6 at. % solute (Hutchinson and H o n e y c o m b e , 1967).

V. Interpretations of Solid Solution Strengthening Mechanisms proposed for solid solution strengthening can be classified in one of three categories: ( 1 ) locking mechanisms, dislocations at rest are locked in place by mobile solute a t o m s ; (2) friction mechanisms, mobile dislocation movement is restricted by internal stress fields associated with the presence of relatively immobile solute atoms; and (3) structure mechanisms, the indirect influence solute atoms have on the dislocation structure in an alloy dictates the magnitude of the stress required for dislocation mobility. The concepts of each mechanism have been reasonably well developed; however, it is not yet possible unambiguously to relate strengthening to a single mechanism or the combined influence of more than one. Therefore, the approach in the discussion to follow will be to outline the concepts of proposed mechanisms and then examine experimental evidence for consistency with their predictions. A. Dislocation Locking

Mechanisms

N o n r a n d o m solute distributions described earlier for alloys in thermodynamic equilibrium can have important consequences in determining the stress levels required to initiate dislocation motion. As discussed earlier, yield point

145

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

phenomena in alloys is well accounted for by such mechanisms. It has also been common for investigators to relate solid solution strengthening to the stress required to free dislocations from their solute environments. In terms of this type of mechanism, once dislocations have been freed the additional stress required for their liberation should no longer be necessary and the stress level for continued plastic deformation would be expected to be independent of solute content. The strengthening data in Figs. 19-22 shows that this is not the case. This inconsistency is the major argument against interpreting solid solution strengthening in terms of a locking mechanism. The discussion below outlines the nature of the dislocation-solute atom interactions and is followed by a description of the evidence available for each interaction. 1. INTERACTION ENTHALPIES

The stress required to reverse or overcome solute atom interactions is proportional to the binding free energy responsible for the interaction. A number of treatments have been made to estimate the binding enthalpies (entropy effects are neglected) of the elastic (Cottrell, 1953), electrical (Cottrell et al, 1953), chemical (Suzuki, 1952; Flinn, 1962) and local order (Fischer, 1954; Flinn, 1962) interactions. They can be expressed as (Fiore and Bauer, 1967) "elastic

where solute to the of the

(7)

=4Gbe r\sm6/R) h

G is the shear modulus, b is the Burgers vector, r is the radius of the atom, (sin 6/R) are the polar coordinates of a solute atom with respect dislocation core, and s is the size misfit parameter expressed in terms lattice parameter, a, as e = (\/a)(da/dc). h

b

"electrical

= 0.0175 b(N

y

- l)(sill 0/R)

(8)

where N is the number of valence electrons associated with the solute atom. v

"chemical

=(Q/A)(0y/3c)

(9)

where Q is the solute atomic volume, h is the stacking fault thickness, y is the stacking fault energy, and c is the concentration of solute. "order

=

(10)

CZ(j)

where z is the lattice coordination number and cj) is an interaction parameter expressed as =("AA + " B B - 2 / /

a

b

)

(11)

where H is the solvent-solvent bond enthalpy, H is the solute-solute bond enthalpy, and H is the solvent-solute bond enthalpy. AA

BB

AB

146

K. R. EVANS

These derived expressions must be considered only as reasonable approximations, for they are based on a number of assumptions. The elastic interaction, for example, is computed using linear elasticity theory which is not applicable in the dislocation core, and an approximation for position of the solute atom in the core must be made. Considered in a qualitative manner, Eqs. (7)-(10) should yield reasonable estimates of the strengthening contribution from each mechanism. In these terms, H is estimated to be much larger than the other enthalpies. / / i t r i c a i is considered to be on the order of 0-15 / / and / / about 0.10 / / . One means of discriminating between the rate-controlling influence of each locking mechanism is to examine its temperature dependence. The (sin 9/R) term of Eqs. (7) and (8) defines the location of a solute atom with respect to the dislocation core. As the distance from the core, R, becomes large, H and / / become very small. Because of this (\/R) dependence, thermal fluctuations in the alloy lattice strongly influence the magnitude of H and / / e l e c t r i c y maximum values at 0°K (thermal fluctuations are minimal) and approach zero at a critical temperature where thermal fluctuations are sufficiently large that the probability for their overcoming the binding force is large. If solid solution strengthening is dictated by either //elastic / / e l e c t r i c this critical temperature would correspond to the temperature T in Fig. 13 where the yield stress becomes athermal, and the temperature dependence of yielding could then be interpreted in terms of the temperature dependence of their binding enthalpies. elastic

e

e l a s t i c

c h e m i c a l

e c

e l a s t i c

elastic

e l e c t r i c

elastic

s u c n

o

t n a t

t n e

n

a

v

e

r

c

In contrast, / / m i c a i * / / o r d e r do not exhibit a temperature dependence for they are determined by physical quantities (stacking faults and a relatively infinite order dimension) of such dimensions that thermal fluctuations do not have any influence. a

n

a

c h e

2. ELASTIC DISLOCATION-SOLUTE A T O M INTERACTION

The elastic dislocation-solute atom interaction (commonly referred to as Cottrell locking) has been used to account for Region I solid solution strenthening because of both its temperature dependence and theoretical magnitude. Suzuki (1957) examined the low temperature strengthening of copper-nickel, copper-zinc, and gold-silver single crystals with respect to his quantitative prediction for the elastic interaction. Satisfactory correlation was found for the copper-nickel alloys, while the prediction could only account for one-half the low temperature strength of the copper-zinc alloys and very little of that observed for gold-silver alloys. A later investigation by Suzuki and K u r a m o t o (1968) concluded the elastic interaction could account well for the Region I strengthening of high solute concentration copper-aluminum alloys.

147

SOLID SOLUTION STRENGTHENING OF F C C ALLOYS

TABLE

II

M I S M A T C H PARAMETERS U S E D BY FLEISCHER ( 1 9 6 3 ) TO A N A L Y Z E THE S T R E N G T H E N I N G OF C O P P E R A L L O Y S

Solute element Al As Ga Ge In Ni Pd Pt Si Sn Zn

e=

\ e ' — 3e |

Percentage o f effect due to size

^_ — x 10

e' G

e„

-0.61

+0.064

0.80

24

25

-1.07

+0.122

1.44

26

73

-0.64

+0.078

0.87

27

36

-0.84

+0.089

1.11

24

53

-1.17

+0.262

1.96

40

111

+0.48

-0.031

0.57

16

20

-0.27

+0.089

0.54

50

22

-0.38

+0.114

0.72

47

31

-0.76

+0.020

0.82

7

-1.18

+0.282

2.03

42

120-130

-0.38

+0.056

0.55

13

17.5

s

b

G

d

C

35

Utilization of Eq. (7) employing misfit parameters from Table II for nickel, zinc, and silicon in copper leads to the expectation that the low temperature strength of copper-zinc alloys should be nearly twice that of copper-nickel alloys and much larger than copper-silicon alloys. In fact, the low temperature strength of these alloys for a given concentration is comparable (see Fig. 24). In these same terms, the temperature dependence of silver-aluminum alloys should be negligible; however, it is found to be comparable to that illustrated in Fig. 14 for copper-aluminum single crystals. Suzuki (1957) has suggested that some of these discrepancies can be accounted for by comparing low temperature strengthening to the sum of the temperature dependent enthalpies (H + / / i e c t r i c a i ) - On this basis, the temperature dependent strength of copper-aluminum alloys should be twice that of copper-silicon alloys; however, as indicated in Fig. 24 they are comparable (Evans and Flanagan, 1968b). elastic

e

A significant observation is that T (defined in Fig. 13) is nearly the same for silicon, aluminum, nickel, and zinc in copper (Fig. 24). Considering that T is the temperature at which the probability is large for thermal fluctuations to overcome the temperature dependent energy resisting yielding, it would appear that this energy is nearly the same in each of the alloy systems. This is not consistent with the predictions of Eqs. (7) and (8). The above discussion adds further support to the argument expressed at the beginning of this section against the role of locking mechanisms influencing solid solution strengthening. c

c

148

K. R. EVANS

Z

( WW/6>i) SS381S HV3HS Q3A10S3H lVDIildD TEMPERATURE ( K)

Fig. 24. Temperature dependence of the critical resolved shear stress for copper alloy single crystals in terms o f their electron per a t o m (e/a) ratio (Evans and Flanagan, 1968b). O C u - A g (Koppenaal and Fine, 1962); # C u - S i (Evans, unpublished); • C u - N i (Suzuki, 1957); • C u - Z n (Suzuki, 1957).

3. ELECTRICAL DISLOCATION-SOLUTE A T O M INTERACTION

The theoretical weakness of the electrical interaction does not offer much support for it having an influence on solid solution strengthening, even considering it as a supplement to the elastic interaction. However, a well-defined correlation exists between solid solution strengthening and the electron per atom ratio (e/a) of alloys (Hutchison and Honeycombe, 1967; Evans and Flanagan, 1968b; Hutchison and Pascoe, 1972). This is demonstrated in Figs. 24 and 25 for both the temperature and composition dependence of yielding of copper and silver alloys, respectively. The critical resolved shear stress for the alloys appears to be reasonably well defined by a single parabolic

149

i

1

1

1

r

1.1

1.2

1.3

1.4

1.5

C R I T I C A L RESOLVED SHEAR STRESS

2

(kg/mm )

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

0

1.6

ELECTRON/ATOM RATIO Fig. 25. Plateau stress of copper and silver alloy single crystals as a function of their electron per a t o m (e/a) ratio (Evans and Flanagan, 1968b). # C u - A l (Koppenaal and Fine, 1962); O C u - Z n (Suzuki, 1957); • C u - A g (Garstone and H o n e y c o m b e , 1957); • C u - G e (Garstone and H o n e y c o m b e , 1957); A A g - A l (Hendrickson and Fine, 1961); A C u - S i (Evans and Flanagan, 1968); + A g - I n (Haasen, 1964).

function in terms of (e/a) ratio (Fig. 25), and both the magnitude of the critical resolved shear stress extrapolated to 0°K and its rate of increase with decreasing temperature also appear to correlate well with the (e/a) ratio. Both the plateau stress and temperature dependent component of the yield stress for polycrystalline alloys are reported to correlate well with (e/a) ratio. This has been demonstrated for both silver (Hutchison and Honeycombe, 1967) and copper alloys (Hutchison and Pascoe, 1972) although in the latter system silicon and germanium solute atoms (which cause a very rapid increase in yield stress) provide anamolous behavior. Figure 26 demonstrates the correlation observed for copper-base alloys. Van der Planken and Deruyttere (1969) tested a series of lead alloy single crystals in liquid air and found that (dx/dc) correlated well with a term consisting of the sum of a size factor and electronic factor. The electronic factor used was the sum of the electronegativities of the solute and solvent atoms. The ability to correlate strengthening data with (e/a) ratio must be considered to be well established; however, the significance of the correlation is yet open to debate. Flinn did point out that the original calculations by Cottrell et al. (1953) leading to the weak interaction predicted by Eq. (8) were based upon the simplified assumption of a spherical Fermi surface. It has not been determined whether a more sophisticated calculation would predict

150

CT

0

2

(N/mm )

K. R. EVANS

Fig. 26. Yield stress of polycrystalline copper alloys as a function of their electron per a t o m (e/a) ratio (Hutchison and Pascoe, 1972).

a significantly stronger electrical interaction. The significance of the (e/a) ratio on strengthening may be indirect. Differences in electronic structure between solute and solvent atoms may have the effect of locally altering the modulus of the alloy. This consideration will be explored later. 4. CHEMICAL DISLOCATION-SOLUTE A T O M INTERACTION

The temperature independence of the chemical interaction between dislocations and solutes (commonly referred to as the Suzuki interaction) has frequently suggested it as a source of r . The possibility of more than one mechanism contributing to r cannot be discounted; however, there are specific reasons for concluding that the chemical interaction is not a predominant strengthening mechanism. For example, the chemical interaction is expected to be more significant the more rapidly solute additions decrease the stacking fault energy of the alloy [Eq. (9)]. Because the stacking fault energy is expected to approach zero at the solubility limit silver-aluminum alloys with relatively small solubilities should be stronger than copper-aluminum and copper-zinc alloys whose solubilities are substantial; however, the opposite is observed experimentally (Hendrickson and Fine, 1961d). Observed strengthening of copper by nickel should be related to a substantial change in stacking fault energy (Menon and Flanagan, 1973); however, experiments suggest the stacking fault energy of copper is not significantly altered by nickel additions (Nakajima and Numakura, 1965). p

p

151

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS 5. S H O R T - R A N G E O R D E R

Short-range order or clustering (commonly known as Fischer interactions) contributions to athermal strengthening have been predicted by Fischer (1954) and Flinn (1958). Flinn (1958) found the yield stress of silver-gold, coppergold, and copper-zinc alloys to be in agreement with predicted values. Suzuki (1957) concluded that short-range order made a contribution to T of gold-silver and copper-zinc alloys. Svitak and Asimow (1969) took advantage of the fact that the short-range order contribution to strengthening should be heat-treatment sensitive (as is the chemical interaction) to study strengthening in quenched and slowly cooled silver-gold single crystals. The extreme heat treatments did not influence the critical resolved shear stress of the alloys but the magnitude of an initial yield point was strongly influenced. Cohen and Fine (1962) had earlier concluded that the influence of short-range order is only to cause the occurrence of yield points. This conclusion is consistent with the concept of deformation in an ordered lattice illustrated in Fig. 27. Slip of one Burgers vector along an ordered plane is shown to reduce the number of unlike neighbors to the random number, while further slip results in no additional change. Thus, it is fairly conclusive that ordering or clustering does not contribute to T . p

P

r—Slip Plone

Initial

Structure

II Unlike Bonds Across Slip Plane

Slip Plane Slip of b 9 Unlike Bonds Across Slip Plane

Slip Plane Slip of 2b 9 Unlike Bonds Across Slip Plane

Fig. 27. Destruction of local order along a slip plane by a unit of slip (Flinn, 1962).

152

K. R. EVANS

B. Dislocation Friction

Mechanisms

Friction mechanisms for solid solution strengthening propose that strengthening arises as the result of the resistance to dislocation motion exerted by the presence of randomly distributed solute atoms in the alloy. A frictional resistance of this type does not account for yield point formation. Locking mechanisms are consistent in this regard; however, the concept of a frictional resistance to dislocation motion is, in principle, consistent with an increase in lower yield stress with solute content. 1. M O T T - N A B A R R O THEORY

Mott and Nabarro ( 1 9 4 8 ) proposed a theory for solid solution strengthening based upon the difference in size between solute and solvent atoms. Their model considered solute atoms to be randomly distributed throughout the alloy matrix and to act as spherical inclusions defined by an atomic radius, r . Elasticity theory shows that a shear strain, e , exists a distance R' from such an inclusion given by 0

s = y

e rl/R

(12)

b

where s is the lattice misfit parameter providing an index of the size difference between the solvent and solute atoms. A change in solute concentration will not alter the magnitude of the shear strain induced by a solute atom, but will influence the distribution or "wavelength," A, of such strain centres. In a solid solution alloy A will be on the order of b/c . Mott and Nabarro recognized that the internal stresses associated with localized shear strains about solute atoms provided a resistance to moving dislocations. In order for dislocations to move through the alloy lattice the externally applied stress must be large enough to overcome the internal stress field and, therefore, be equal to at least some average of the internal stress, o An average of the arithmetic magnitude of a led Mott and Nabarro to show that the yield strength, a , of an alloy should be b

1/3

v

t

0

G = 2Ge c 0

(13)

h

Algebraic summation of the internal stresses is zero, leading to the conclusion that they cannot influence strength. However, Mott and Nabarro recognized that the flexibility of dislocations must be considered in treating the problem, and for this case, algebraic summation of the internal stresses is not zero. This consideration leads to a predicted value for the yield strength of (Mott, 1 9 5 2 ; Cottrell, 1 9 5 3 ) 2.5 G

s c l /3 h

(14)

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

753

AL

dc I0 PSI

0

2

da /dc(kN/nnm )

3

(b)

|E | b

Fig. 28. Strengthening rate o f copper alloys as a function o f the size misfit parameter according to (a) Fleischer (1963), and (b) Hutchison and Pascoe (1972).

154

K. R. EVANS

Investigators have frequently interpreted strengthening results in terms of the size difference between solvent and solute atoms. The results of early investigations on copper alloys suggested a linear relationship between the strengthening rate and size difference between solvent and solute atoms for both polycrystalline (French and Hibbard, 1950) and single crystal (Cottrell, 1953; Garstone and Honeycombe, 1957; Honeycombe, 1961; Friedel, 1964) specimens. Issue with this conclusion was taken when polycrystalline copper alloys having the same lattice parameters (but different solute concentrations) did not yield similar values for the yield strengths (Hibbard, 1958; Ainslie et al, 1959). Controversy has developed about the behavior of copper alloys with Fleischer (1963) reporting that the strengthening rate of a large number of copper alloys was not a linear function of the atomic size misfit parameter (see Fig. 28a), while Hutchison and Pascoe (1972) reported rather satisfactory correlation except for silicon, nickel, and zinc (see Fig. 28b). Correlation between strengthening rate and the misfit parameter was not good for either silver (Hutchison and Honeycombe, 1967) or lead (Van der Planken and Deruyttere, 1969) alloys. While extensive experimental data are not available from a wide variety of alloy systems it is apparent that the size factor alone cannot account for solid solution strengthening. Hutchison and Honeycombe (1967) found good correlation of their data with (e/a) ratio and allude to the presence of an electrical dislocation-solute atom interaction to account for their results. Van der Planken and Deruyttere (1969) added an electronic term to the misfit parameter to account for their observations in lead alloy crystals. Fleischer (1961, 1963) considered the influence the modulus difference between solute and solvent atoms may have on dislocation motion in addition to the size mismatch effect. This discussion follows. 2. FLEISCHER ANALYSIS

Fleischer (1961, 1963) maintained the basic elements of the M o t t - N a b a r r o model for strengthening but, in addition, postulated that the localized shear modulus difference between the randomly distributed solute and solvent atoms could also provide resistance to dislocation motion, i.e., contribute to strengthening. Incorporation of the modulus difference is significant considering the electron per atom correlations with strengthening established in preceding sections, for it reflects differences in the electronic structure of solute and solvent atoms. Assuming that immobile solute atoms are deformed by a moving dislocation to the same strains as if they were matrix atoms, simple elasticity theory was used to calculate the size misfit and modulus mismatch interaction forces between both edge and screw dislocations and substitutional solute atoms. Linear elasticity theory predicts there is no inter-

755

SOLID SOLUTION STRENGTHENING OF F C C ALLOYS

action of a screw dislocation with the size difference associated with a symmetrical substitutional atom; however, a second order effect produces a volume expansion around a screw dislocation which can interact with the dilation field of a substitutional atom. Fleischer (1961, 1963) summed these interaction forces and predicted that athermal strengthening should be linearly proportional to a weighted parameter s given by (15)

oce

h

where a is a constant which may be no less than 16 for edge dislocations (s = £ ) and is less than 16 for screw dislocations ( e = e ) , and s' is the modulus interaction parameter defined by s

e

G

(16)

£ ' = G

where (17) Fleischer (1963) expressed the hardening rate for eleven polycrystalline copper solid solution alloys in terms of both the size mismatch parameter e and the modulus mismatch parameter e '. A considerable amount of scatter existed when using each of these parameters alone, leading Fleischer to use the weighted function given by Eq. (15). Good correlation was achieved by choosing a = 3, characteristic of screw dislocation movement, as shown in Fig. 29 where dxjdc is proportional to e . This correlation was interpreted b

G

3/2

ii dc

10

3

PSI

Fig. 29. Strengthening rate of copper alloys as a function of the mismatch parameter e (Fleischer, 1963). s

156

K. R. EVANS

as indicating that yielding in copper solid solutions occurs by the movement of screw dislocations which are resisted by internal stresses resulting from both the size and modulus mismatch between solvent and solute atoms. Table II lists the mismatch parameters Fleischer used for his calculations and indicates the amount of strengthening associated with each contribution. In general, the modulus effect was found to predominate, contributing to 7 5 % or more of the observed strengthening in a number of alloy systems. On the basis of this approach, Fleischer (1964) was able to express the plateau stress as C'^

T = T + Gs '

(18)

1 2

3 2

P

0

where T is the critical resolved shear stress of the pure metal and z is a constant on the order of 760. Labusch (1970) subsequently employed a statistical averaging method for the interaction forces between dislocations and 0

x

'

I

05

-

W

L_

2.0 3.0

(a)

dtp

deb

Cu-alloys

[tig/mm^]

6.0 5.0 4.0

•E -hl1*3|6l

3.0

oe ' |t(|.16|6|

2.0

0

0 s

w

£

2.0

(b) Fig. 30. Strengthening rate (a) drjdc , parameter £ ' (Jax et al, 1970). 212

0

3.0 and (b) drjdc '

1 2

in terms of the mismatch

157

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

solute atoms to obtain the following expression for T T = T + Gs P

0

4 / 3

P

(19)

c ^/z 2

2

where z is a constant on the order of 550. Jax et al. (1970) examined data for copper, gold and silver alloy single crystals in terms of the concepts outlined above. In agreement with Fleischer, it was concluded that a weighted interaction parameter was needed to adequately describe available data; however, in contrast to Fleischer's conclusion that the onset of yielding was controlled by the motion of screw dislocations, i.e., a of Eq. (15) equals 3, these authors concluded the weighted parameter £ ' provided a better data fit, where 2

0

to = e ' + a | e j

( ) 2 0

c

and a = 16. The strengthening data analyzed by these authors is summarized in Table III. The data fit is illustrated in Fig. 30a for copper alloy crystals and implies strengthening is related to the difficulty in moving edge dislocations (a = 16) rather than screw dislocations. It was found that the strengthening rate expressed as either dtjdc or dxjdc was a monotonic function 111

213

TABLE III M I S M A T C H PARAMETERS U S E D BY J A X et al.

(1970) TO A N A L Y Z E THE

STRENGTHENING OF SILVER, C O P P E R , A N D G O L D A L L O Y S

Alloy

£

b

for [101] screw dislocation

for 60° dislocation

drjdc (kg/mm ) 112

2

drjdc (kg/mm ) 213

2

Ag-Zn Ag-Pd Ag-In Ag-Sn Ag-Au Ag-Cd

-0.046 -0.055 +0.085 +0.110 -0.005 +0.049

-0.52 +0.52 -0.52 -0.69 +0.57 -0.34

-0.38 +0.49 -0.49 -0.54 +0.48 -0.33

2.0 2.1 2.4 4.0 0.5 2.1

3.0 4.0 4.2 9.2 0.9 3.1

Cu-Ge Cu-Ga Cu-Zn Cu-Al Cu-Au

+0.093 +0.080 +0.060 +0.068 +0.153

-0.98 -0.80 -0.64 -0.50 -0.34

-0.92 -0.81 -0.56 -0.47

4.1 2.9 2.8 2.9 5.4

6.3 4.6 4.2 4.0 10.0

Au-Ga Au-Cd Au-Zn Au-In Au-Ag Au-Pd

-0.013 +0.043 -0.048 +0.075 -0.004 -0.052

-1.29 +0.17 -0.19 -0.60 +0.35 +0.67

— —

1.3 1.8 2.0 3.6 0.5 1.5

7.6 2.6 3.1 5.4 0.8 4.0

—

— —

— —

158

K. R. EVANS

of e ' , consistent with Eqs. (18) and (19), respectively (compare Figs. 30a and b). Jax et al. (1970) concluded that their experimental data was more consistent with Eq. (19) because T values as a function of c extrapolated to the critical resolved shear stress of the pure metal, whereas in terms of c , the extrapolated values were frequently negative. This is illustrated in Fig. 31 for gold-gallium alloys. A c concentration dependence for strengthening has also been reported by other investigators (Hammar et al, 1967; Riddhagni and Asimow, 1968). 0

2 / 3

p

1 / 2

2 / 3

0.10

005

015

Fig. 31. C o m p o s i t i o n dependence of the plateau stress as a function of c (Jax et al, 1970).

1 / 2

and

c

2/3

Kostorz and Mihailovich (1970] analyzed the strengthening of various lead alloy single crystals according to Fleischer's concepts. A concentration dependence of T proportional to either c or c appeared reasonable at concentration levels below 2 a t . % ; however, at higher concentrations anomalous behavior for tin, cadmium, and bismuth alloys became apparent. The strengthening rate expressed as either dx /dc or dx /dc was a monotonic function of both e and e , as indicated in Fig. 32. Cadmium solute data are observed to be inconsistent with the general trend and the results are relatively insensitive to the functions e and e . Mismatch parameters used for the calculations are listed in Table IV. The Fleischer-Labusch analysis has provided reasonably good correlation between athermal strengthening and selected interaction parameters with a 2 / 3

1/2

p

1/2

2/3

p

e

p

s

e

s

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

p

2/3

2

dr /dc (gm/mm )

159

Fig. 32. Strengthening rate o f lead alloy crystals as a function o f the mismatch parameters e and e (Kostorz and Mihailovich, 1970). e

s

limited number of exceptions such as lead-cadmium (Kostorz and Mihailovich, 1970) and polycrystalline silver alloys (Hutchison and Honeycombe, 1967). Validity of the analysis has been questioned on the basis that other interaction parameters provide equally good correlation (Van der Planken and Deruyttere, 1968; Hutchison and Honeycombe, 1967; Hutchison a n d Pascoe, 1972), that the observed correlation is dependent upon the manner in which modulus and size mismatch parameters are selected (Hutchison and Honeycombe, 1967; Riddhagni and Asimow, 1968), and that these bulk parameters d o n o t reflect conditions governing the actual, localized dislocationsolute atom interaction (Hutchison and Honeycombe, 1967; Fleischer, 1961). Regardless of the merit of these criticisms, the analysis must be recognized for its ability t o quantitatively correlate a large amount of strengthening data in terms of a reasonable physical model. This is a task other proposed mechanisms find difficult to d o .

TABLE IV PARAMETERS U S E D BY K O S T O R Z A N D M I H A I L O V I C H (1970) TO A N A L Y Z E S T R E N G T H E N I N G OF L E A D A L L O Y S

dr /dc (gm/mm )

dr /dc (gm/mm )

122 156 246 315

295 349 535 1016

l/2

p

e

Alloy

100

Pb-Tl Pb-Bi Pb-Sn Pb-Cd

-1.53 +2.29 -2.83 -4.70

2

b

-0.46 -0.4 -0.75 -1.49

2/3

p

2

160

K. R. EVANS

3. STATISTICAL DISLOCATION L I N E FLEXIBILITY M O D E L S

A number of investigators have adapted the M o t t - N a b a r r o concept of dislocation line flexibility m overcoming a strain center distribution to calculate strengthening behavior. The approaches followed are all very similar. A statistical computer calculation is made for a model assuming (1) a specific strain center distribution (usually a random array), (2) the nature of the dislocation-strain center interaction energy, and (3) the manner in which a dislocation deviates from a straight line as it encounters and overcomes a strain center. Results of the calculations differ according to the assumptions made, and basically serve for qualitative analysis with little opportunity for detailed comparison with experimental data. Accordingly, the results are not emphasized in this presentation. Strengthening has been predicted to be proportional to both c (Friedel, 1962; Foreman and Makin, 1966; Stefansky and Dorn, 1969; Suzuki, 1970) and c (Riddhagni and Asimow, 1968). Generally, the models predict strengthening to be a continually decreasing function with increasing temperature. Suzuki's (1970) calculation provides a notable exception where qualitative agreement with the schematic stress-temperature diagram of Fig. 13 was obtained. 1/2

2/3

C. Dislocation Structure

Mechanisms

Interpretation of solid solution strengthening in terms of the dislocation structure of an alloy is attractive because the concept provides continuity with theories of the work-hardened state of metals. While specific details of the mechanisms responsible for work hardening are subject to debate, the theories do agree that the magnitude of the flow stress of an alloy is dictated by its dislocation density (Seeger, 1957; Hirsch, 1962, 1963; Thornton et al, 1962; Kuhlmann-Wilsdorf, 1962). The suggestion that the magnitude of the critical resolved shear stress and yield stress are also dictated by dislocation density originated from Seeger's (1959) analysis of work hardening which defined the relationship between shear stress, T , and dislocation density as T = a'Gbp

(21)

112

where a' is a constant on the order of 0.5. This relationship led Seeger to postulate that predictions suggesting the grown-in dislocation density of an alloy to increase with solute content had a natural consequence in terms of dictating the yield stress. On the basis of Eq. (21) an increase in dislocation density by a factor of 25 could well account for the increases in r shown in Figs. 19-22. Dislocation density increases by this amount are totally reasonable within the framework of Tiller's predictions (1958, 1965) for the origin of dislocations by an impurity mechanism; however, it is not resolved whether p

161

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

this mechanism commonly occurs. Evidence exists both for and against its influence as discussed earlier. As a result, arguments for interpretation of solid solution strengthening in terms of dislocation structure are based on indirect evidence and ability of the theory to account for various observed yielding phenomena. Comparison of activation parameters characteristic of rate-controlling yielding and deformation processes in pure metals and alloys is frequently used to justify an interpretation of strengthening in terms of dislocation structure. Therefore, a brief discussion of the significance of activation parameters follows prior to examination of experimental observations. 1. ACTIVATION PARAMETERS

Work-hardening theories partition the flow stress below the temperature r ( s e e Fig. 13)(Seeger, 1957; Hirsch, 1962; Thornton etai, 1962; KuhlmannWilsdorf, 1962; Mitra and Dorn, 1962, 1963) as follows: D

T =

TQ +

(22a)

T*

where T is an athermal stress equivalent to x but not restricted to initial yielding. For the case of solid solution strengthening, Eq. (22a) can be expressed as g

p

T

0

=

T

P

+

(22b)

T*

The rate at which the activation of dislocations past obstacles occurs is dependent upon the rate at which mechanical and thermal energy are supplied to mobilize dislocations. Thus, the distinction expressed in Eqs. (22a) and (22b) merely arises from the ability of thermal energy to assist the applied stress in translating dislocations past obstacles to their motion. The retarding force profile of an obstacle to dislocation motion is illustrated in Fig. 33. An applied force, F , is considered to do work on a dislocation so that it can be translated past the obstacle by a thermally activated process. F is the sum of its rate insensitive and rate sensitive components, F and F*, respectively. Assuming a model of discrete obstacles to dislocation motion characterized by their mean spacing, L ', F can be expressed as appUed

applied

p

1

a p p l i e d

Applied = ^

P

+

^*

=

T Zi> = T Lb + T*Lb 0

p

(23)

Figure 33 shows that at the temperature of test, T , F is needed to push a dislocation to a distance X from the effective obstacle core where the activation enthalpy, H, is able to assist the externally supplied mechanical energy, x

applUd

1

t The case for when the obstacles are either dislocations or low concentrations of solute atoms.

162

FORCE

K. R. EVANS

DISTANCE Fig. 33. Force-distance curve for an obstacle to dislocation m o t i o n .

T*Ab, produce the activation event. The product Ab is termed the activation volume where A is the effective area equal to X L over which work is being expended to produce the activation event. If the test were being conducted at a temperature T (T < Ti), additional mechanical work would need to be supplied to move the dislocation to X , for the thermal energy available at T is less than at T (kT < kT ). The temperature dependence of yielding can then be expressed in terms of the forces acting against the obstacle as X

2

2

2

2

1

2

x

(24) Methods for experimental determination of the parameters X, L, H , H, A, F , and F* have been described (Mitra and Dorn, 1962, 1963; Evans and Flanagan, 1968a). Such an analysis becomes significant in considering interpretations of solid solution strengthening in terms of dislocation structure if (1) the total obstacle energy, H = H + T*Ab, for alloys at yielding is the same as that for their pure base metal component at a stage of deformation where direct evidence for a rate-controlling dislocation mechanism is available, (2) the obstacle energies at yielding are independent of solute content, and (3) predictions for t (T )/ O(TI) calculated from retarding force parameters according to Eq. (24) are consistent with experimental observations. 0

p

0

0

T

2

163

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS 2. EXPERIMENTAL CORRELATION

The dependence of flow stress (i.e., the stress corresponding to a plastic strain) upon dislocation density according to Eq. (21) is well established for both pure metals (Bailey and Hirsch, 1960; Livingston, 1962; Young, 1962; Hordon, 1962; Bailey, 1963) and alloys (Venables, 1962; Mader et al, 1963). Experimental results for specimens deformed into easy glide and well into Stage II deformation are summarized in Figs. 34a and b.

•

O

P

T/G

o-:

• •

•

P

•

0

0-1

0-2

0-3

04

0-5

0-6

x10-

pb V2

3

(a)

t

3 a •

1 4

•

D

m°

0

1

2

3

4

5

6

7

flxlO

-3

(b) Fig, 34, F l o w stress dependence upon (dislocation d e n s i t y ) for (a) copper crystals employing etch pit counts, # Livingstone, x Y o u n g , O H o r d o n ; and (b) various metals and alloys employing electronmicroscopy observations to measure density, % copper, x c o p p e r - 0 . 8 % aluminum, A c o p p e r - 2 . 2 % aluminum, • c o p p e r - 4 . 5 % aluminum, • nickel, O n i c k e l - 4 0 % cobalt, + n i c k e l - 6 0 % cobalt, 3 silver, A copper (Mitchell, 1964). 112

164

K. R. EVANS

Shear

2

stress ( kg /mm )

It is also well established that the yield stress and critical resolved shear stress can be dependent upon dislocation structure. Experiments have shown the yield stress of polycrystalline metals to be directly related to their dislocation substructure (Ball, 1957). Of more direct interest have been experiments on pure metal and alloy single crystals showing their critical resolved shear stress to be dependent upon the density of forest dislocations. Washburn and Gollapudi (1968) prestrained large copper single crystals along an axis selected to keep a (111) slip plane inactive during the prestrain. Small tensile specimens oriented for single slip on the previously inactive plane were cut from the prestrained crystals and tested. Their results, shown in Fig. 35, demonstrate

Shear

strain

Fig. 35. Single slip stress-strain curves showing effect of increasing initial dislocation density by prestrain in multiple slip (Washburn and Gollapudi, 1968). N o prestrain, • 0.73 k g / m m , A 2.54 k g / m m , 6.7 k g / m m , # prestrained by shock loading (10 kbar pressure). 2

2

2

the significant influence initial dislocation density can have on pure metals. Suzuki (1970) conducted equivalent experiments in more detail for coppernickel and copper-aluminum single crystals. The results for a copper-0.25 % nickel alloy are shown in Fig. 36, where the yield stress is shown to be directly related to the forest dislocation density above some critical density ( < 1 0 / c m ) in a manner consistent with Eq. (21). Suzuki considered direct dislocationsolute interactions to dictate the yield stress below this critical density. While the above experiments are conclusive in demonstrating that the density of forest dislocations is able to determine the magnitude of the yield 6

2

165

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

io io*

Nt

io*

4

1

T fT—i

•o

7

1

r-

T

Cu--Nl 0.25 ot% 400

2

(gm/mm )

•/ •K/

<7

y

77

Stress

' /

/ro<>m T«mp. /

/

Yield

'

/

/

/

//

0

u

kL. 0

1 1 1 I 2 3xio» Square Root of Forest Dislocation Density

F/^r. 36. Critical resolved shear stress dependence u p o n (dislocation d e n s i t y ) copper-0.25 a t . % nickel single crystals (Suzuki, 1970).

172

for

stress, they do not preclude the possibility of other mechanisms controlling the yield stress, for a correlation between solute content and dislocation density remains to be established. H a m m a r et al. (1967) have conducted the only detailed study of dislocation densities formed in alloy crystals and found that no appreciable increases in density could be observed with solute additions up to 0.20 at. % even though the critical resolved shear stresses increased by a factor of 3-5. While very dilute solute additions were used in these experiments, they do represent the concentration region where strength increases occur at a high rate and should be reflected by corresponding increases in dislocation density if a structure mechanism dictates strength. Additional experiments of this type are needed before general conclusions can be established. Indirect evidence is available to suggest that dislocations formed during solidification do control strength. A consequence of Tiller's theory (1958, 1965) for the origin of dislocations according to an impurity mechanism is that the dislocation density should be a function of the rate the alloy is solidified from its liquid state. Hendrickson and Fine (1961) reported r to increase p

166

K. R. EVANS

significantly for silver-6 a t . % aluminum single crystals when their growth rate was increased from 0.37 to 0.9 in./hr. The temperature dependence of the critical resolved shear stress also appears to have increased somewhat with the corresponding increase in growth rate. These observations have been verified (Asimow, 1969; Asimow and Samal, 1970) for silver-2.9 a t . % indium alloy single crystals solidified at various rates, and suggest that increases in obstacle density occur with growth rate. Determination of the activation parameters for a number of alloys indicates that the obstacle energies for yielding are independent of the type of solute and its concentration. Argent et al. (1964-1965), Evans and Flanagan (1968a), Menon and Flanagan (1973), and Suzuki and K u r a m o t o (1968) report the obstacle energies for polycrystalline copper-zinc alloys, and copper-silicon, copper-nickel, and copper-aluminum single crystals, respectively, to be independent of solute concentration and the same as values determined for pure copper in Stage II deformation (Thornton and Hirsch, 1958; Sastry et al., 1971). These results are summarized in Table V. Haasen (1965) reports that energies calculated for a series of silver alloy solid solutions are independent of the type of solute added. These results are consistent with the well-defined experimental observation that T (defined in Fig. 13) is not significantly altered by the amount or kind of solute in an alloy and is relatively independent of the base metal (compare Figs. 14, 15, 16, and 24). Consideration of obstacle energies in the above manner supports the conclusion that the ratec

TABLE V R E P O R T E D OBSTACLE ENERGIES FOR C O P P E R A N D ITS S O L I D SOLUTION A L L O Y S

Obstacle energy, Ho (eV) Cu (commercial grade) Cu (electrolytic grade) Cu C u - 5 at. % Si C u - 2 . 7 at. % N i C u - 5 . 4 at. % N i C u - 0 . 7 at. % Al C u - 2 . 0 at. % Al C u - 3 . 0 at. % Al C u - 5 . 0 at. % Al Cu-lO.Oat. % Al C u - 1 5 . 0 a t . % Al

Reference

0.6

(Sastry et aL, 1971)

0.6

(Sastry et aL, 1971)

0.7 0.67 0.77 0.73 0.74 0.73 0.69 0.71 0.64 0.54

(Thornton and Hirsch, 1958) (Evans and Flanagan, 1968a) ( M e n o n and Flanagan, 1973) ( M e n o n and Flanagan, 1973) (Suzuki and K u r a m o t o , 1968) (Suzuki and K u r a m o t o , 1968) (Suzuki and K u r a m o t o , 1968) (Suzuki and K u r a m o t o , 1968) (Suzuki and K u r a m o t o , 1968) (Suzuki and K u r a m o t o , 1968)

167

SOLID SOLUTION STRENGTHENING OF FCC ALLOYS

controlling obstacles to dislocation motion in Stage II of pure face-centered cubic metals also dictate the yield stress of their alloys. This argument provides the most significant support for interpreting strengthening in terms of dislocation structure. Work-hardening theories for face-centered cubic metals consider the ratecontrolling obstacles to dislocation motion at temperatures below T to have a definite relationship to those between T and T (Seeger, 1957; Hirseh, 1962; Thornton et al, 1962; Kuhlmann-Wilsdorf, 1962; Mitra and Dorn, 1962, 1963). Experimental observations on silver (Hutchison and Honeycombe, 1967) and copper alloys (Hutchison and Pascoe, 1972) have led to the same conclusion. This has provided justification for utilization of low temperature stress (Menon and Flanagan, 1973) or strain-rate sensitivity (Mitra and Dorn, 1962, 1963; Evans and Flanagan, 1968a) measurements near 0°K to estimate the mean spacing between obstacles, L, so that the retarding force parameters F * and F , can be determined as a function of solute content and state of deformation. These determinations have been made for a number of pure metals (Mitra and Dorn, 1962, 1963), and silver-aluminum, (Evans and Flanagan, 1968b), copper-silicon (Evans and Flanagan, 1968b), and coppernickel (Menon and Flanagan, 1973) single crystals. These investigations agree that (1) F * at 0°K is independent of solute content and state of deformation and (2) F decreases with increasing solute content and increases with state of deformation in Stage II. The results for alloys are consistent with those for pure metals and again suggest that similar processes control yielding. Furthermore, utilization of F* and F in Eq. (24) has shown (Evans and Flanagan, 1968b) that the temperature dependence of the yield stress will be (1) smaller for pure metals and low solute content alloys compared to higher solute content alloys, (2) smaller for polycrystalline metals and alloys compared to their single crystal equivalents, and (3) independent of strain in easy glide for single crystals and decreasing with increasing state of deformation in Stage II, thereby accounting for deviations from the Cottrell-Stokes law (Evans and Flanagan, 1967). c

c

D

p

p

p

Interpretation of strengthening in terms of a dislocation structure mechanism is attractive on the basis of the indirect evidence outlined above. Critical experiments similar to those conducted by H a m m a r et al. (1967) in which the activation parameters for the yielding process are also determined are required in order to firmly establish the contribution of structure to solid solution strengthening in face-centered cubic alloys.

ACKNOWLEDGMENT

Appreciation is expressed t o Professor W. F . Flanagan of Vanderbilt University for helpful discussions of the subject matter and his review of the manuscript.

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EVANS

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