ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation

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Fig. 8 Shear strain rate versus shear stress (logarithmic scale) for high purity Zn-22Al for various grain

sizes at 493 K. Source: Ref 14

The Relationship between Strain Contributed by Boundary Sliding and Total Strain

Grain-boundary sliding (GBS) is a process that occurs during the high-temperature creep of polycrystalline

materials and in which one grain slides over another grain under the action of a shear stress. Sliding is shown

schematically in Fig. 9. As a result of the sliding of the two grains offsets are produced at their common

boundary; pr is the sliding vector, u is the component of sliding resolved along the stress axis, ν is the

component measured perpendicular to both the stress axis and the specimen surface, and w is the component

measured perpendicular to the stress axis but in the plane of the surface. Also, as shown in Fig. 9, the

orientation of the grain boundary is defined by two angles θ and α: θ is the angle between the stress axis and the

trace of the boundary in the plane of the surface, and α is the internal angle made by the boundary trace on a

longitudinal section cut perpendicular to the surface.

Fig. 9 Schematic representation of grain-boundary sliding

In superplasticity experiments, the longitudinal offset, u, and/or the transverse offsets, w, are considered in

calculating the strain contribution from boundary sliding to total strain. As described elsewhere (Ref 24, 25),

the strain due to boundary sliding, ε

gbs

, is obtained from and through the following expressions:

gbs

= Ψ /

(Eq 5a)

gbs

= φ /

(Eq 5b)

where Ψ is a geometric constant equal to 0.8 (Ref 24) for the longitudinal offset, is the average offset, is the

average linear intercept grain size, φ is a geometric constant equal to 1.5 for the transverse offset, and is the

average transverse offset (Ref 24, 25). The method of calculating ε

gbs

from u is applicable only when the grains

are equiaxed, a condition that is satisfied during superplastic deformation. The contribution of boundary sliding

to the total strain, ξ, is calculated from the equation:

ξ = ε

gbs

/ ε

(Eq 6)

In taking measurements of GBS, several steps are carried out. First, prior to testing, one of the flat surfaces of

each specimen is polished to a mirror-like scratch-free surface, and very fine lines (marker lines) are placed on

the polished gage surface, either parallel or perpendicular to the specimen axis. One method of creating such

lines is by drawing a lens tissue containing 1 μm diamond paste across the surface only once in either the

longitudinal or the transverse direction. Longitudinal marker lines are used to measure the sliding offset

perpendicular to the stress axis, w. Transverse marker lines are used to measure the sliding offset parallel to the

stress axis, u. Figure 10 illustrates the two types of marker lines, longitudinal and transverse, and the two

classes of offset, w and u. It is also possible to take measurements of sliding using a printed-grid technique.

However, as reported elsewhere, the technique can be unsatisfactory due to the resolution difficulties in the

scanning electron microscope (SEM) (Ref 24). Second, tensile specimens are deformed at a constant

temperature on a testing machine operated at a constant crosshead speed. Tensile tests are conducted to a

predetermined strain in the range of 20 to 100% at various initial strain rates. Third, after testing and cleaning in

an ultrasonic cleaner, the specimens are examined in the SEM and a number of representative

photomicrographs are taken within the gage length. These micrographs are enlarged, and individual sliding

offsets are measured. In general, about 200 to 500 readings are taken on each one of the specimens.

Fig. 10 Schematic configuration of maker lines, longitudinal and transverse, and their offsets, w

(perpendicular to stress axis) and u (parallel to the stress axis), respectively

Superplastic alloys such as Zn-22Al and Pb-62Sn have two phases, and in this case, there are two

intercrystalline boundaries (e.g., Al-Al and Zn-Zn) and an interphase boundary (e.g., Al-Zn). To ensure that

sampling of measurement for GBS does not favor a certain type of boundary, the number of each type of

boundary counted experimentally should be selected according to an approach proposed by Gifkins (Ref 26).

Such an approach, which was adopted in examining sliding behavior in high-purity Pb-62Sn (Ref 26, 27) and

Zn-22Al (Ref 28), is extended here to describe the general case of a superplastic material consisting of two

phases.

The approach by Gifkins (Ref 26) is based on the following assumptions: (a) the ratio of the volume fraction of

the two phases of a micrograin superplastic alloy is equivalent to the ratio in the linear transverse intercept, and

(b) grains make a single transverse array. If the ratio of the volume fraction of α and β phases in the alloy is f,

then this ratio, on the basis of the above assumptions, may be expressed as:

/ Y

(Eq 7a)

where X and Y are the number of α and β grains respectively, and

and

are the linear sizes of α and β

phases, respectively. This expression may be rewritten using the following consideration. If x (α-α) boundaries

and y (α-β) boundaries are counted, there would be x (α) grains forming (α-α) boundaries and y/2 (α) grains

forming α-β boundaries, that is, X = x + y/2. Then, the number of β grains, Y, and the number of (β-β)

boundaries, z, are given, respectively, by:

(Eq 7b)

and

z = Y - y/2

(Eq 7c)

The following example, based on the recent results reported for the superplastic Zn-22Al alloy (Ref 28),

illustrates the application of Eq 7b(b) and (c) to boundary-sliding measurements. For this alloy, α is the

aluminum-rich phase, β is the zinc-rich phase, and f = 0.75. Accordingly, Eq 7a(a) gives 100 zinc grains for

every 88 aluminum grains when

= 303 μm and

= 2.8 μm. In performing sliding measurements, the

number of Al-Al and Zn-Al boundaries sampled were close to 60 and 150, respectively. Using x = 60, y/2 = 75,

= 2.8 μm, and

= 3.3 μm, Eq 7b(b) and (c) result in 78 Zn-Zn boundaries, a number used in the analysis

reported in Ref 28.

Figures 11(a) and (b) show typical photographs of commercial purity Zn-22Al tested to an elongation of about

20% at 4 × 10

-5

-1

in the superplastic region (Ref 28). The photographs show several features associated with

the occurrence of sliding in the alloy: (a) grain displacements are often large and sharp, as illustrated, for

example, by Fig. 11(a); (b) striated bands (Fig. 11a) are usually observed at Zn-Al boundaries and Zn-Zn

boundaries; and (c) grain rotation is observed to occur in both phases, the aluminum-rich phase (dark) and the

zinc-rich phase (bright) (Fig. 11b).

Fig. 11 Typical area of the gage length for a Zn-22Al sample tested at 4 × 10

-5

-1

to an elongation of

20%. (a) A large offset at a Zn-Al boundary is shown at A and a striated band at a Zn-Zn boundary is

marked at B and at C. (b) Grain rotation in a zinc-rich phase is shown at D. Source: Ref 28

To characterize the offset measurements, two types of histograms are normally constructed (Ref 24, 27, 28, 29).

The first type is based on the relative frequency of various boundaries, whereas the second type is based on the

relative sliding contribution of these boundaries. These two types of histograms are illustrated using some of

the recent results reported for Zn-22Al (Ref 27). In Fig. 12(a), the histogram of the relative frequency of

boundaries is shown. In this case, the relative frequency of the number of each boundary (Al-Al, Zn-Al, and

Zn-Zn) is plotted against the offset w in increments of 0.1 μm at

= 4 × 10

-5

-1

, which corresponds to the

super-plastic region. As noted in the figure, the largest displacements occur at Zn-Al interphase boundaries, the

smallest displacements occur at Al-Al intercrystalline boundaries, and Zn-Zn intercrystalline boundaries exhibit

slightly lower values than those at Zn-Al boundaries. In addition, the histogram shows the following features:

(a) all displacements are in the range of 0 to 0.8 μm; (b) the maximum frequency occurs at the boundaries with

the offset interval of 0.1 to 0.2 μm; and (c) the largest sliding offsets at Al-Al intercrystalline boundaries are

found in the interval of 0.4 to 0.5 μm.

Fig. 12 Histograms for commercial purity Zn-22Al with a grain size of 5.4 μm for offset increments of

0.1 μm at

= 4 × 10

-5

-1

and 493 K. (a) Histogram showing the relative frequency of the number of

boundaries. (b) Histogram showing the relative sliding contribution to the total sliding for each type of

boundary

In Fig. 12(b), the histogram of relative sliding is presented. As shown in the figure, the percentage contributions

to the total sliding for each type of boundary in Zn-22Al are plotted for the offset w in increments of 0.1 μm

= 4 × 10

-5

-1

. This histogram is more practical than that based on the frequency of the number of each type of

boundary (Fig. 12a) because a small number of boundaries with a large amount of offsets can significantly

contribute to the total sliding. According to the histogram, maximum contribution occurs at boundaries with

displacement offsets in the interval of 0.2 to 0.3 μm.

References cited in this section

7. W.A. Backofen, I.R. Turner, and D.H. Avery, Superplasticity in an Al-Zn Alloy, Trans. ASM, Vol 57,

1964, p 980

8. W.B. Morrison, Superplasticity of Low-Alloy Steels, Trans. ASM, Vol 61, 1968, p 423

9. J. Hedworth and M.J. Stowell, The Measurement of Strain-Rate Sensitivity in Superplastic Alloys, J.

Mater. Sci., Vol 6, 1971, p 1061

10. G.B. Gibbs, Creep and Stress Relaxation Studies with Polycrystalline Magnesium, Philos. Mag., Vol

13, 1966, p 317

11. A. Arieli and A. Rosen, Measurement of Strain-Rate Sensitivity Coefficient in Superplastic Ti-6A1-4V

Alloy, Scr. Metall., Vol 10, 1976, p 471

12. E.W. Hart, A Phenomenological Theory for Plastic Deformation of Polycrystalline Metals, Acta.

Metall., Vol 18, 1970, p 599

13. P.K. Chaudhury and F.A. Mohamed, Effect of Impurity Content on Superplastic Flow in the Zn-22% Al

Alloy, Acta. Metall., Vol 36, 1988, p 1099

14. P.K. Chaudhury, V. Sivaramakrishnan, and F.A. Mohamed, Superplastic Deformation Behavior in

Commercial and High Purity Zn-22 pct Al, Metall. Trans. A, Vol 19, 1988, p 2741

15. S. Yan, J.C. Earthman, and F.A. Mohamed, Effect of Cd on Superplastic Flow in the Pb-62 wt% Sn

Eutectic, Philos. Mag. A, Vol 69, 1994, p 1017

16. B.Y. Chirouze, D.M. Schwartz, and J.E. Dorn, The Mechanism of Basal Creep in Mg-12 at.% Li from

500 to 885 K, Trans. ASM, Vol 51, 1967, p 51

17. K.L. Murty, F.A. Mohamed, and J.E. Dorn, Viscous Glide, Dislocation Climb and Newtonian Viscous

Deformation Mechanisms of High Temperature Creep in Al-3Mg, Acta Metall., Vol 20, 1972, p 1009

18. M.S. Soliman, T.J. Ginter, and F.A. Mohamed, Philos. Mag. A, Vol 48, 1993, p 63

19. F.A. Mohamed and T.G. Langdon, A Comparison of Constant Strain Rate and Creep Testing

Procedures in Superplasticity, J. Mater. Sci., Vol 17, 1982, p 1925

20. H.I. Huang, O.D. Sherby, and J.E. Dorn, Activation Energy for High Temperature Creep of High Purity

Aluminum, Trans. MS/AIME, Vol 23, 1956, p 117

21. F.A. Mohamed and T.G. Langdon, The Determination of the Activation Energy for Superplastic Flow,

Phys. Status Solidi (a), Vol 33, 1976, p 375

22. J.E. Bird, A.K. Mukerjee, and J.E. Dorn, Correlations between High-Temperature Creep Behavior and

Structure, Proc. of a Symp. on Quantitative Relation Between Properties and Microstructure, D.G.

Brandon and A. Rosen, Ed., Israel University Press, 1969, p 22

23. F.A. Mohamed and T.G. Langdon, Creep at Low Stress Levels in the Superplastic Zn-22% Al

Eutectoid, Acta Metall., Vol 23, 1975, p 117

24. P. Shariat, R.B. Vastava, and T.G. Langdon, An Evaluation of the Roles of Intercrystalline and

Interphase Boundary Sliding in Two-Phase Supersonic Alloys, Acta Metall., Vol 30, 1982, p 285

25. T.G. Langdon, The Effect of Surface Configuration on Grain Boundary Sliding, Metall. Trans., Vol 3,

1972, p 797

26. R.C. Gifkins, Factors Influencing Deformation of Superplastic Alloys, Mater. Forum, Vol 15, 1991, p

27. K.T. Park, S. Yan, and F.A. Mohamed, Boundary Sliding Behavior in High-Purity Pb-62% Sn, Philos.

Mag. A, Vol 72, 1995, p 891

28. K. Duong and F.A. Mohamed, Effect of Impurity Content on Boundary Sliding Behavior in the

Superplastic Zn-22% Al Alloy, Acta Mater., Vol 46, 1998, p 4571

29. R.B. Vastava and T.G. Langdon, An Investigation of Intercrystalline and Interphase Boundary Sliding

in the Superplastic Pb-62% Sn Eutectic, Acta Metall., Vol 27, 1979, p 251

Superplastic Deformation at Elevated Temperatures

Farghalli A. Mohamed, University of California, Irvine

Deformation Regions and Their Characteristics

Over the past three decades, the mechanical behavior of micrograin superplastic alloys has been extensively

studied. As a result of these investigations, two findings are well documented. First, micrograin superplasticity

is a diffusion-controlled process that can be represented by the following dimensionless equation (Ref 22):

(Eq 8a)

with

(Eq 8b)

where is the shear creep, k is Boltzmann's constant, T is the absolute temperature, D is the diffusion

coefficient that characterizes the creep process, G is the shear modulus, b is the Burgers vector, A is a

dimensionless constant, d is the grain size, s is the grain-size sensitivity, τ is the applied shear stress, n is the

stress exponent, Q is the activation energy for the diffusion process that controls the creep behavior, and D

the frequency factor for diffusion.

Second, the relationship between stress (τ) and strain rate ( ) is often sigmoidal (Ref 23, 30, 31, 32). Under

creep-testing conditions, this sigmoidal relationship is manifested by the presence of three regions (Fig. 6):

region I (low-stress region), region II (intermediate-stress region), and region III (high-stress region). The

characterization of the sigmoidal trend in terms of these regions is mainly based on the value of the stress

exponent, n (n = (∂ ln /∂ ln τ)

T,d

) or strain rate sensitivity, m [m = (∂ ln τ/∂ ln )

T,d

] at constant temperature

and grain size. Of these three regions, regions I and II have been the subject of many studies that have aimed at

not only establishing the deformation characteristics of both regions but also providing interpretations of these

characteristics in terms of deformation mechanisms. The characteristics of these three regions are briefly

described below.

Region III. In this region (the high-stress region), the stress exponent is higher than 3, the apparent activation

energy is higher than that for grain boundary diffusion, and the creep rates are sensitive to changes in grain size.

Region II (the intermediate-stress region) covers several orders of magnitude of strain rate and is characterized

by a stress exponent, n, of 1.5 to 2.5, an apparent activation energy, Q

, that is close to that for boundary

diffusion, Q

, and a grain size sensitivity, s, of about 2. Contribution of GBS to the total strain is high and

maximum ductility occurs. Because of these characteristics, region II is often referred to as the superplastic

region. Experimental data show that superplastic behavior in this region is not sensitive to the presence of

impurities. This finding is demonstrated in Fig. 13, where the experimental data of (a) Pb-62Sn and (b) Zn-22Al

are plotted in terms of 8aEq 8b, as normalized creep rate versus normalized shear stress, τ/G, on a logarithmic

scale. Normalized creep rate is given by ( kT / D

Gb)(d/b)

, where D

= D

exp(-Q

/RT). In plotting the data

on Pb-62Sn, the values of D

, Q

, b, and s were taken as 1 cm

/s, 61 kJ/mol, 3.2 × 10

-8

cm, and 2.3,

respectively (Ref 17). In plotting the data on Zn-22Al, the values of D

, Q

, b, and s were taken as 1 cm

/s, 78

kJ/mol, 2.7 × 10

-8

cm, and 2.4, respectively (Ref 33). In addition to demonstrating that the creep rates of two

grades of Pb-62Sn, having different impurity content, are identical in this region, Fig. 13(a) indicates that the

same creep data cluster about a single straight line that extends over four orders of magnitude of strain rate.

This result, which is similar to that shown for Zn-22Al in Fig. 13(b), confirms not only that region II behavior

is insensitive to impurity atoms, but also that region II (the superplastic region) does not represent a transition

region between region I (low-stress region) and region III (high-stress region), that is, region II is a

consequence of the operation of a distinct deformation process.

Fig. 13 Normalized creep rate versus normalized shear stress (logarithmic scale). (a) For Pb-62Sn doped

with cadmium at intermediate stresses (region II) and for high-purity Pb-62Sn at both intermediate and

low stresses. Source: Ref 15. (b) For commercial-purity Zn-22Al at intermediate stresses (region II) and

for high-purity Zn-22Al at both intermediate and low stresses. Source: Ref 14

Region I (the low-stress region) is characterized by a stress exponent of 3 to 5 (Ref 23, 30, 31, 32), an apparent

activation energy higher than that for grain-boundary diffusion, and a decrease in both ductility and

contribution of GBS to total strain. However, the creep behavior in this region exhibits essentially the same

grain-size sensitivity noted in region II. Recent experimental evidence (Ref 13, 14, 15, 33) has revealed that

steady-state creep in region I, unlike that in region II, is influenced by impurities. This evidence has been

demonstrated by two main experimental observations: (a) Zn-22Al (Ref 13, 14) and Pb-62Sn (Ref 15) do not

exhibit region I when the impurity level in the alloy is reduced to about 6 ppm; and (b) the stress exponent, n,

and the apparent activation energy for creep, Q

, in region I, unlike those in region II, are sensitive to impurity

content (Ref 13, 33). Observation (a) is demonstrated by Fig. 14(a) and (b), which show the absence of region I

in high-purity Zn-22Al (Ref 13, 14) and high-purity Pb-62Sn (Ref 15), respectively. As indicated by Fig. 6,

region I is present in commercial purity Zn-22Al that contains 100 ppm of impurities (Ref 14). Observation (b)

is demonstrated by Fig. 15, in which the values of the apparent activation energy in regions I and II are plotted

as a function of applied stress for three grades of Zn-22Al containing different impurity levels; grades 1, 2, and

3 contain 180, 100, and 6 ppm of impurities. As shown by the figure, Q

in grade 3 (high-purity grade) is equal

to 85 kJ/mol (~Q

), even at stresses typical of those used in investigating region I in grades 1 and 2; under the

condition of constant stress, Q

in region I for grade 2 (100 ppm of impurities) is smaller than that for grade 1

(180 ppm of impurities).

Fig. 14 Shear strain rate versus shear stress (logarithmic scale). (a) For high-purity Zn-22Al having a

grain size of 3.5 μm at various temperatures from 433 to 493 K. Source: Ref 14. (b) For high-purity Pb-

62Sn having a grain size of 8 μm at various temperatures from 392 to 422 K. Source: Ref 15

Fig. 15 The dependence of the average apparent activation energy on stress for grades 1, 2, and 3 of Zn-

22Al containing 180, 100, and 6 ppm of impurities, respectively (grain size = 2.5 μm). Source: Ref 13

References cited in this section

13. P.K. Chaudhury and F.A. Mohamed, Effect of Impurity Content on Superplastic Flow in the Zn-22% Al

Alloy, Acta. Metall., Vol 36, 1988, p 1099

14. P.K. Chaudhury, V. Sivaramakrishnan, and F.A. Mohamed, Superplastic Deformation Behavior in

Commercial and High Purity Zn-22 pct Al, Metall. Trans. A, Vol 19, 1988, p 2741

15. S. Yan, J.C. Earthman, and F.A. Mohamed, Effect of Cd on Superplastic Flow in the Pb-62 wt% Sn

Eutectic, Philos. Mag. A, Vol 69, 1994, p 1017

17. K.L. Murty, F.A. Mohamed, and J.E. Dorn, Viscous Glide, Dislocation Climb and Newtonian Viscous

Deformation Mechanisms of High Temperature Creep in Al-3Mg, Acta Metall., Vol 20, 1972, p 1009

22. J.E. Bird, A.K. Mukerjee, and J.E. Dorn, Correlations between High-Temperature Creep Behavior and

Structure, Proc. of a Symp. on Quantitative Relation Between Properties and Microstructure, D.G.

Brandon and A. Rosen, Ed., Israel University Press, 1969, p 22

23. F.A. Mohamed and T.G. Langdon, Creep at Low Stress Levels in the Superplastic Zn-22% Al

Eutectoid, Acta Metall., Vol 23, 1975, p 117

30. F.A. Mohamed, S. Shen-Ann, and T.G. Langdon, The Activation Energies Associated with Superplastic

Flow, Acta Metall., Vol 23, 1975, p 1443

31. F.A. Mohamed and T.G. Langdon, Creep Behavior in the Superplastic Pb-62% Sn Eutectic, Philos.

Mag. A, Vol 32, 1975, p 697

32. S.H. Vale, D.J. Eastgate, and P.M. Hazzledine, The Low Strain Rate Behavior of Superplastic Zn-Al

Eutectoid Alloy, Scr. Metall., Vol 13, 1979, p 1157

33. P.K. Chaudhury, K.T. Park, and F.A. Mohamed, Effect of Fe on the Superplastic Deformation of Zn-22

pct Al, Metall. Mater. Trans. A, Vol 25, 1994, p 2391

Superplastic Deformation at Elevated Temperatures

Farghalli A. Mohamed, University of California, Irvine

Deformation Mechanisms

Region III Deformation Mechanisms. The deformation mechanism responsible for this region is not well

established. Early observations suggest that region III in superplastic alloys represents normal power-law creep,

which controls the behavior of large-grained metals at elevated temperatures (Ref 22). These observations (Ref

3) include (a) measurements of high stress exponents in several superplastic alloys, (b) the presence of

extensive dislocation activity in the interiors of grains, (c) the occurrence of changes in grain shape, (d)

increases in the texture after deformation, and (e) the close correspondence between the transition stresses from

region II (the superplastic region) to region III and those predicted from the equation that describes the

dependence of the average subgrain size, λ, formed during the creep of metals on the applied stress (Ref 22,

34). This correspondence implies that region III occurs at higher stresses where a stable subgrain structure

begins to develop (Ref 35). However, the above suggestion that the creep behavior of superplastic alloys in

region III is controlled by the same type of dislocation process that is dominant in metals at high temperatures

is not entirely satisfactory for two primary reasons. First, experimental results in a superplastic copper alloy

have revealed an inverse dependence of creep rate on grain size in region III (Ref 36); this behavior contrasts

with that of pure metals at high stresses where creep rates are essentially insensitive to changes in grain size.

Second, experimental evidence indicates that at high stresses no well-developed subgrains are formed in the

interiors of grains; only dislocation tangles are present (Ref 3). On the basis of these findings, it seems most

likely that region III is the result of the operation of some form of an intragranular dislocation process, which is

influenced by the presence of grain boundaries. It is worth mentioning that there are difficulties in establishing

the mechanical characteristics of region III due to the fast creep rates associated with this region.

Region II Deformation Mechanisms. As a result of considerable efforts over the past three decades, a number of

deformation mechanisms were proposed or speculated to explain the origin of superplastic flow, especially in

region II. Of these mechanisms, the most commonly considered ones are based on grain-boundary sliding

(GBS) (Ref 37, 38, 39, 40, 41, and 42 43). Depending on the nature of the accommodation process that is

necessary to relieve stress concentration, the GBS models may be divided into two types: diffusional

accommodation and dislocation accommodation.

GBS accommodated by diffusion flow is the basis of the model by Ashby and Verrall (Ref 37). This model,

which involves a grain-switching event, predicts the presence of a sigmoidal relationship between stress and

strain rate, the retention of an equiaxed grain structure, and the absence of a significant dislocation activity.

While these predictions are in harmony with experimental evidence, there are several problems associated with

the model. Of these problems, which are stated elsewhere in detail (Ref 14, 23, 30, and 31), the following two

are worth mentioning. First, the nature of the sigmoidal relationship between stress and strain rate, as predicted

by the model, is not consistent with experimental findings. According to the model, the stress exponent for

creep in region II is not constant, but varies continuously with stress (Ref 37). This trend contrasts the results

reported for superplastic alloys, in which region II exhibits a constant stress exponent over several orders of

magnitude of strain rate. Second, the model predicts that the apparent activation energies in region I and II are

the same. This prediction is in conflict with the finding that region I is associated with a higher apparent

activation energy.

Models based on GBS accommodated by dislocation motion were developed by several investigators (Ref 38,

39, 40, 41, 42, and 43). While these models, in general, involve GBS and some form of dislocation motion,

their details and assumptions are different (Table 2). However, despite their different details and assumptions,

all of the models mentioned can be described by a rate-controlling equation: