FURTHER REFINEMENTS: TRANSIENT BEHAVIOUR; VERY HIGH AND VERY LOW STRAIN RATES; HIGH PRESSURE
17.1 Transient Behaviour and Transient Maps
17.2 High StrainRates
17.3 Very Low Stresses
17.4 The Effect of Pressure on Plastic Flow
The maps shown so far in this book are based on simplified rateequations which describe behaviour at steady structure or at steady state (Chapters 1 and 2), ignoring the effects of workhardening and of transient creep. They ignore, too, deformation mechanisms which become important when strain rates are very large (among them phonon drag and adiabatic heating) or very small (such as threshold effects associated with diffusional flow), and the influence of large hydrostatic pressures. This was done because the data for these mechanisms are so meagre that their rates, and even their placings on the maps, are often uncertain. But as a better understanding of them becomes available they can be included. In this chapter we discuss the present level of understanding, and show maps illustrating their characteristics.
When strains are small, as they are in service life of most engineering structures, the steadystate approximation is a poor one. At ambient temperatures, metals workharden, so that the flow stress (at a given strainrate) changes with strain. At higher temperatures most materials show primary or transient creep as well as steadystate flow; if small strains concern us we cannot neglect their contribution.
Flow at steady structure or steady state is described (Chapter 1) by an equation of the form:
_{} 
(17.1) 
The state variables S_{i} (dislocation density and so forth) do not appear as independent variables because they are either fixed, or uniquely determined by σ_{s} and T. But during nonsteady flow the state variables change with time or strain:
S_{i} = S_{i}(t or γ) 
(17.2) 
and a more elaborate constitutive law is needed, containing either time t or a strain γ as an additional variable:
_{} 
(17.3) 
If this law is integrated to give:
γ = F(σ_{s}, T, t) 
(17.4) 
we can construct maps, still using σ_{s}/μ and T/T_{M} as axes, but with contours showing the strain γ accumulated during the time t. Such maps can include workhardening, and both transient and steady state creep (Ashby and Frost, 1976) [1].
In doing this, we lose some of the generality of the steadystate maps. The strainrate (which is used as the dependent variable in the steadystate maps) is a differential quantity which depends only on the current structure (S_{i}) of the material. The strain (which is the variable we use in the transient maps) is an integral quantity: it depends not on the current structure, but on its entire history. As a result, the constitutive laws we use are largely empirical, and the maps refer to monotonic loading at constant temperature only.
We start by listing the equations used to construct transient maps for a stainless steel, and then show three examples of them. They are computed from the data listed in Table 17.1. An application of such maps is described in Chapter 19, Section 19.3.
A stress σ_{s} produces an elastic strain:
_{} 
(17.5) 
Since we are now concerned with strain (not strainrate), this elastic contribution must be added to the plastic strain to calculate the total strain.
Polycrystal stressstrain curves can, in general, be fitted to a workhardening law, which for tensile straining, takes the form:
_{} 
where ε_{p} is the plastic tensile strain. Inverting, and converting from tensile to equivalent shear stress and strain, gives:
where _{} 
(17.6) 
and σ_{0s} is the initial shear strength. Data are available for many metals and alloys. Those for Type 316 stainless steel are listed in Table 17.1.
TABLE 17.1 Data for Type 316 stainless steel
Crystallographic and thermal data 


Atomic volume, Ω (m^{3}) 
1.21 x 10^{29} 

Burger's vector, b (m) 
2.58 x 10^{10} 
(a,b) 
Melting temperature, T_{M}(K) 
1810 

Modulus 


Shear modulus at 300 K, 
8.1 x 10^{4} 
(a) 
Temperature dependence of 
– 0.85 

Lattice diffusion 


Preexponential, D_{0}_{υ} (m^{2}/s) 
3.7 x 10^{5} 
(a) 
Activation energy, Q_{υ} (kJ/mole) 
280 

Boundary diffusion 


Preexponential, δD_{0b} (m^{3}/s) 
2 x 10^{13} 
(a) 
Activation energy, Q_{b} (kJ/mole) 
167 

Powerlaw creep 


Exponent, n 
7.9 

Dorn constant, A 
1.0 x 10^{10} 
(a) 
Obstaclecontrolled glide 


0 K flow stress, _{} 
6.5 x 10^{3} 

Preexponential, _{}(s^{1}) 
10^{6} 
(a) 
Activation energy, ∆F/μ_{0}b^{3} 
0.5 

Workhardening 


Initial yield stress, σ_{0s}/μ_{0} 
7.5 x 10^{4}–2.2 x 10^{7} T 

Hardening exponent, m 
0.31 + 6 x 10^{5} T 
(c) 
Hardening constant, K_{s}/μ_{0} 
2.5 x 10^{3}–5.7 x 10^{7} T 

Transient powerlaw creep 


Transient strain, γ_{t} 
0.087 
(d) 
Transient constant, C_{s} 
46.0 

(a) Except where noted, the data are the same as those given in Table 8.1. (b) All maps are normalized to 1810°C (T_{M} for pure iron). This choice is arbitrary; one could use 1680 K (the solidus for 316 stainless steel) thereby expanding the abscissa slightly. The choice influences the computation only via the normalized temperature dependence of the modulus, T_{M}/μ_{0} (dμ/dT) in evaluating this we used T_{M} = 1810 K for consistency. (c) Based on the data of Blackburn (1972). The temperature T is in degrees centigrade. (d) These data are based on an average of values given by Garofalo et al. (1963) and Blackburn (1972). A more precise description of the transient creep of 316 stainless steel requires two transient terms (Blackburn, 1972). 
When an intrinsically soft material such as a metal is loaded, dislocations are generated and the material usually workhardens. If the stress is now held constant these dislocations rearrange, finally attaining a steady structure, and the sample creeps at a steady state. During the rearrangement, the sample creeps faster than at steady state. This normal transient has been studied and modelled by Dorn and his coworkers (Amin et al., 1970 [2]; Bird et al., 1969 [3]; Webster et al., 1969 [4]).
On loading an intrinsically hard material (Si, Ge, ice, probably most oxides, silicates, etc.) it appears that too few dislocations are immediately available to permit steady flow. As they move they multiply; during this process the creeprate increases to that of the steady state. This inverse transient has been studied and modelled by Li (1963) [5], Alexander and Haasen (1968) [6], Gilman (1969) [7] and others. We shall restrict the discussion to that of normal transients. Many engineering texts and papers use a law:
_{} 
(17.7) 
when n and q are positive and greater than unity and σ and ε are the tensile stress and creep strain, and t is time. Differentiating and rearranging gives laws of the two forms:
_{} 
(17.8) 
Though analytically convenient, these laws are physically unsound. Both predict infinite creep rates at zero time (or strain) and no steady state; and, like all integral formulations, they cannot predict the effect of changes of stress (see the discussion of Finnie and Heller, 1959 [8]). Some of these difficulties are removed in the formulation of Dorn and his coworkers (Webster et al., 1969 [4]; Amin et al., 1970 [2]). They demonstrate remarkable agreement of creep data for Al, Mo, Ag, Fe, Cu, Ni, Nb and Pt with the creep law.
_{} 
(17.9) 
where _{} is the steadystate creeprate, ε_{t} is the total transient strain, and C is a constant. We shall use this equation to construct maps, though it, too, is incapable of describing transient behaviour due to change of stress during a test.
Converted to shear stress and strainrate, eqn. (17.9) becomes:
_{} 
(17.10) 
where _{} is the steadystate strainrate (and thus is identical with the rate used to construct the steady state maps, eqn. (2.21)), _{}and _{}. Data for Type 316 stainless steel are given in Table 17.1.
In a pure, onecomponent system, there is a small transient associated with diffusional flow. On applying a stress, grain boundary sliding generates an internal stress distribution which decays with time, ultimately reaching the steadystate level. The transient strain must be of the same order as the elastic strain σ_{s}/μ (since it is associated with the redistribution of internal stresses). The time constant is determined by the relaxation process involved; in this case, diffusion over distances comparable with the grain size, giving a relaxation time of about _{}where _{} is the steadystate strainrate by diffusional flow and thus is identical with the rate used to construct the steadystate maps, eqn. (2.29). The strain then becomes:
_{} 
(17.11) 
This transient involves no new data. In alloys, larger transients with larger relaxation times, associated with the redistribution of solute by diffusion, appear. We shall not consider them here.
Figs. 17.1, 17.2 and 17.3 show transient maps for Type 316 stainless steel with a grain size of 100 µm. The first shows the areas of dominance of each mechanism after a time of 10^{4} s (about 3 hours); the second after a time of 10^{8 }s (about 3 years), the third after 10^{9} s (30 years).
Within a field, one mechanism is dominant: it has contributed more strain to the total than any other. Superimposed on the fields are contours of constant shear strain: they show the total strain accumulated in the time to which the map refers: 10^{4}, or 10^{8} or 10^{9} s in these examples.
The maps show an elastic field; within it, the elastic strain exceeds the total plastic strain (steady plus transient) due to all mechanisms. Above it lies the field of lowtemperature plasticity; the spacing of the strain contours reflects workhardening. The powerlaw creep and diffusional flow fields occupy their usual relative positions, but the boundaries separating them from each other and from the other mechanisms move as strain accumulates with time, because the various transients have different time constants.
An example of the use of these maps is given in Chapter 19, Section 19.3.
Fig. 17.1. A transient map for Type 316 stainless of grain size 100 µm, for a time of 10^{4} s (about 3 hours).
Fig. 17.2. As Fig. 17.1, but for a time of 10^{8} s (about 3 years).
Fig. 17.3. As Fig. 17.1, but for a time of 10^{9} s (about 30 years).
Under impact conditions, and in many metalworking operations (Chapter 19, Section 19.3), strainrates are high. They lie in the range 1/s to 10^{6}/s, well above that covered by the maps shown so far. In this range, phonon and electron drags and relativistic effects can limit dislocation velocities at low temperatures; and at high, the power law which describes creep breaks down completely. Further, if the material is deformed so fast that the heat generated by the deformation is unable to diffuse away, then it may lead to a localization of slip known as adiabatic shear.
Phonon and electron drags, and powerlaw breakdown, are easily incorporated into deformation maps by using the rate equations given in Chapter 2. The main problem is that of data: there are very few reliable measurements from which the drag coefficient B, and the powerlaw breakdown coefficient α', can be determined.
A moving dislocation interacts with, and scatters, phonons and electrons. If no other obstacles limit its velocity, a force σ_{s}b per unit length causes it to move at a velocity:
_{} 
(17.12) 
As the temperature increases, the phonon density rises, and the drag coefficient, B, increases. Experimental data (for review, see Klahn et al., 1970 [9] and Kocks et al., 1975 [10]) show much scatter, but are generally consistent with a drag coefficient which increases linearly with temperature:
_{} 
(17.13) 
where B_{e} is the electron drag coefficient, and B_{p} is the phonondrag coefficient at 300 K.
B can be measured by direct observation of dislocation motion during a stress pulse, and can be inferred from measurements of internal friction, and from tensile or compression tests at very high strain rates. The three techniques, properly applied, show broad agreement (Klahn et al., 1970) [11]. For the metals and ionic crystals for which measurements exist, B increases from about 10^{5} Ns/m^{2} at 4.2 K to about 10^{4} Ns/m^{2} at room temperature. Using the Orowan equation (eqn. (2.2)), we find:
_{} 
The high strainrate experiments of Kumar et al. (1968) [12], Kumar and Kumble (1969) [13] and of Wulf (1979) [14] allow the difficult term ρb^{2}μ/B_{p} to be evaluated; in all three sets of experiments the result is close to 5 x 10^{6 }/s at room temperature. Combining these results gives an approximate rateequation for phonon plus electron drag:
_{} 
(17.14) 
where _{}is measured in units of s^{l}. This equation has been used in constructing the maps described below.
As the dislocation velocity approaches that of sound, the stress required to move it increases more rapidly. This is in part due to the relativistic constriction of the strain field which causes the elastic energy to rise steeply, imposing a limiting velocity, roughly that of shear waves, on the moving dislocation. There is evidence (Kumar et al., 1968) [12] that the mobile dislocation density, too, rises towards a limiting value, so that, from eqn. (2.2) an upper limiting strainrate, _{}exists which we take to be 10^{6} s^{l}. Then the approach to this limit is described by the relativistic correction to the drag equation:
_{} 
(17.15) 
These equations must be regarded as little more than first approximations, and they are fitted to minimal data. But they serve to show, roughly, the regimes on deformation maps in which the mechanisms have significant influence.
The transition from pure powerlaw creep to glidecontrolled plasticity was described in Chapter 2, Section 2.4. An adequate empirical description is given by eqn.(2.26), which reduces, at low stresses, to the simple powerlaw of eqn.(2.21). The important new parameter is α', the reciprocal of the normalized stress at which breakdown occurs. Table 17.2 lists approximate values of α' derived from the data plots of previous chapters.
TABLE 17.2 The powerlaw breakdown parameter

Materials and class 
α' 

f.c.c. metals (Cu, Al, Ni) 
10^{3} 

b.c.c. metals (W) 
2 x 10^{3} 

h.c.p. metals (Ti) 
5 x 10^{2} →^{ }10^{3} 

Alkali halides (NaCI) 
2 x 10^{3} 

Oxides (MgO, UO_{2}, Al_{2}O_{3}) 
10^{3} → 2 x 10^{3} 

Ice 
2 x 10^{3} 
Analyses of the onset of adiabatic shear vary in generality and complexity, but almost all are based on the same physical idea: that if the loss of strength due to heating exceeds the gain in strength due to the combined effects of strain hardening and of strainrate hardening (which are locally higher if deformation becomes localized), then adiabatic shear will occur (Zener and Hollomon, 1944 [15]; Baron, 1956 [16]; Backofen, 1964 [17]; Culver, 1973 [18]; Argon, 1973 [19]; Staker, 1981 [20]).
Deformation generates heat, causing the flow strength σ_{y} to fall. Workhardening, or an increase in strain rate, raises σ_{y}. Treatments of diffuse necking (Considčre, 1885 [21], for example) assume that instability starts when the rate of softening first exceeds the rate of hardening. If the current flow strength is σ_{y} and all work is converted into heat, then the heat input per unit volume per second is:
_{} 
(17.16) 
The flow strength σ_{y} depends on strain, strainrate and temperature:
_{} 
Instability starts when dσ_{y} = 0, that is, when:
_{} 
(17.17) 
This equation is the starting point of most treatments of adiabatic localization (see, for instance, Baron, 1956 [16]; Culver, 1973 [18] or Staker, 1981 [20]).
Consider first the case when no heat is lost. (For this truly adiabatic approximation to hold, the strainrate must be higher than the value _{}, calculated below.) At low temperatures we can assume (as Staker, 1981 [20], does) that _{}, so that the instability condition simplifies to:
_{} 
(17.18) 
or, in words: workhardening is just offset by the fall in strength caused by heating. If heating is uniform:
dq = C_{p}dT= σ_{y}dε 
or
_{} 
(17.19) 
If workhardening is described by a powerlaw:
σ_{y} = Kε^{m} 
(17.20) 
we obtain the critical strain for localization under truly adiabatic conditions:
_{} 
(17.21) 
where _{} 
The quantity ψ is a dimensionless material property. Typically it lies in the range –0.5 to –6. The smaller number is appropriate if the yield stress varies with temperature only as the modulus does; the larger number is typical of a material with a strongly temperaturedependent yield strength, such as the b.c.c. metals below 0.1 T_{M}. For many engineering metals at room temperature, its value is about –3. Then the critical strain depends mainly on the current strength σ_{y}, the workhardening exponent m, the specific heat C_{p} and the melting point, T_{M}.
Eqn. (17.21) defines a sufficient condition for the onset of adiabatic shear provided no heat is lost from the sample. It is the basis of the approach used by Culver (1973) [18] and Bai (1981) [22], and by Staker (1981) [20] who supports it with data on explosively deformed _{} AISI 4340 steel, heattreated to give various combinations of σ_{y} and m. But the assumption of no heat loss holds only when the rate of deformation is sufficiently large. So a second condition must also be met: that the strainrate exceeds a critical value which we now calculate approximately.
Consider a uniform deformation (and thus heat input) but with heat loss to the surroundings at a rate (Carslaw and Jaeger, 1959 [23], or Geiger and Poirier, 1973 [24]):
_{} 
(17.22) 
Here k is the thermal conductivity and R a characteristic dimension of the sample (the radius of a cylindrical sample for example); α is a constant of order 2; T is the temperature of the sample and T_{s} is that of the heat sink.
The heat balance equation now becomes:
_{} 
(17.23) 
where V is the volume of the sample and A its surface area (Estrin and Kubin, 1980 [25], for example, base their analysis on this equation). Taking A/V = 2/R we find:
_{} 
(17.24) 
Now the factor (C_{p}R^{2})/(2αk) = τ is the characteristic time (in seconds) for thermal diffusion to occur and is almost independent of temperature except near 0 K (it depends only on the temperature dependencies of k and C_{p}). Eqn. (17.24) now becomes:
_{} 
(17.25) 
If the critical strain for adiabatic shear is ε_{c}, we may write:
_{} 
Heat loss to the surroundings is significant only if the second term on the lefthand side of eqn. (17.25) becomes comparable to, or larger than, the first; adiabatic conditions therefore apply when:
_{} 
Using eqn. (17.21), we find the minimum strainrate for adiabatic conditions to be, approximately:
_{} 
(17.26) 
At an approximate level, then, adiabatic shear is expected when two conditions are met simultaneously: the strain must exceed the critical strain given by eqn. (17.21) and the strainrate must exceed the critical strainrate given by eqn. (17.26). In reality, shear localization can occur even when there is heat loss. Analyses which include it are possible (Estrin and Kubin, 1980) [25] but are complicated, and, at the level of accuracy aimed at here, unnecessary.
Using the equations and data developed above, the influence of drag (eqn. (17.14)), or of relativistic effects (eqn. (17.15)) and of adiabatic heating (eqn. (17.26)) can be incorporated into any one of the four types of map shown in Chapter 1. The first two are straightforward; the last requires further explanation.
The parameters R and α which enter eqn. (17.26) are poorly known. But if in some standard state (say, room temperature) it is found that adiabatic localization occurs at a given strain rate _{}, then in some other state (say 4.2 K) it will occur at the strainrate _{}where:
_{} 
(17.27) 
where the superscript ^{0} refers to the standard state and the unsuperscripted parameters are the values in the other state. We have used the fact that commercially pure titanium at room temperature shows adiabatic localization at strainrates above 10^{2 }/s (Winter, 1975 [26]; Wulf, 1979 [14]; and Timothy, 1982 [27]) to construct maps (Fig. 17.4) which show the field in which it will occur. The most useful is that with axes of strainrate and temperature (Fig. 17.4); it displays most effectively the region in which high strainrate effects are unimportant. The same information can, of course, be crossplotted onto the others.
The maps are based on data described in Chapter 6, and on those listed in Table 17.3. In addition to the usual fields, they show a regime of dragcontrolled plasticity (eqn. (17.14)), the relativistic limit (eqn. (17.15)) and the regime in which adiabatic heating can cause localization (eqn. (17.26)). Adiabatic localization, of course, can occur in compression or torsion, as well as in tension; but in tension the simple necking instability may obscure the adiabatic localization because it occurs first. Further details and examples are given by Sargent and Ashby (1983) [28].
TABLE 17.3 Further material data for commercialpurity titanium

Property 
Value 
Reference 

α' 
5 x 10^{2} 
Doner and Conrad (1973) [29] 

m 
0.11–8.6 x 10^{5}T 
Harding (1975)[30] 

k (Wm^{1} K^{1}) 
5.8 (at 4.2 K); 33 (at 80 K); 20 (at 273 K) 
Am. Ins. Phys. (1972) 


0.94–(4.7 x 10^{4}T) for T < 468 K 


_{} 
2.4–(3.6 x 10^{3}T)for 468 < T< 664 K 0.0 for T > 664 K 
Tanaka et al. (1978) [31] 
TABLE 17.4 Apparent threshold stresses for creep in pure metals
Material 
Grain size
(a) 
Temp (K) 
τ_{tr} (MN/m^{2}) (b) 
τ_{tr} /μ (c) 
References 
Cd 
80 → 300 
300 
0.2 
7.5 x 10^{6} 
Crossland (1974) [32] 
Mg 
25 → 170 
425 → 596 
0.88 → 0.09 
6 x 10^{5} → 6 x 10^{6} 
Crossland and Jones (1977)[33] 
Ag 
40 → 220 
473 → 623 
1.0 → 0.3 
4 x 10^{5} → 1.3 x 10^{5} 
Crossland (1975) [34] 
Cu 
35 
523 → 573 
0.6 → 0.4 
1.5 x 10^{5 } → 1 x 10^{5} 
Crossland(1975 [34] 
Ni 
130 
1023 
0.2 
3.5 x 10^{6} 
Towle(1975) [35] 
A1 
160 → 500 
913 
0.08 
5 x 10^{6} 
Burton (1972) [36] 
αFe 
53 → 89 
758 → 1073 
0.3 → 0.05 
6 x 10^{6} → 1 x 10^{6} 
Towle and Jones(1976) [37] 
βCo 
35 → 206 
773 → 1113 
1.4 → 0.6 
9 x 10^{6} → 1.7 x 10^{5} 
Sritharan and Jones (1979) [38] 
(a) Grain size = 1.65 x mean linear intercept.
(b) Threshold stress in shear (_{} when the tensile threshold σ_{0} is given).
(c) Shear moduli at the test temperature calculated from data listed in Table 4.1, 5.1 and 6.1.
Fig. 17.4. A strainrate/temperature map for titanium, showing the fields of dragcontrolled plasticity and adiabatic shear. The relativistic limit is coincident with the top of the diagram.
At very low stresses there is evidence that the simple rate equations for both powerlaw creep (eqn. (2.21)) and for diffusional flow (eqn. (2.29)) cease to be a good description of experiments. For pure metals the discrepancies are small, but for metallic alloys, and for some ceramics, they can be large. Most commonly, the strainrate decreases steeply with stress at low stresses, suggesting the existence of a "threshold stress" below which creep ceases, or, more accurately, a stress below which the creeprate falls beneath the limit of resolution of the experiment (typically 10^{9 }/s).
Observed "threshold stresses", τ_{tr}, for pure metals, alloys and one ceramic are listed in Tables 17.4 and 17.5. In pure metals, τ_{tr} increases with decreasing grain size and with decreasing temperature, and is of general order 5 x 10^{6} μ. In alloys containing a stable dispersion (of ThO_{2}, or of Y_{2}O_{3}, for example) it can be 10 to 100 times larger.
In single crystals or largegrained polycrystals, powerlaw creep dominates for all interesting stresses. In pure metals the power law is wellbehaved at low stresses. A fine dispersion of second phase introduces an apparent threshold stress, below which creep is too slow to measure with ordinary equipment. The most complete studies are those of Dorey (1968), Humphries et al. (1970) and Shewlelt and Brown (1974, 1977) [39, 40] who introduced up to 9% of SiO_{2}, Al_{2}O_{3} and BeO into single crystals of copper. They found that, on introducing the dispersion, the power n rose from the value characteristic of pure copper (about 5) to a much larger value (10 or more) at low stresses, giving the appearance of a threshold on a log_{ } vs. 1og σ diagram. The data in these tests extends down to a strain rate of 10^{–6}/s, at which apparent thresholds of order 10^{4} μ were observed.
At low temperatures, plastic flow in a dispersionhardened crystal requires a stress sufficient to bow dislocations between the dispersed particles; if their spacing is l, this Orowan stress is roughly:
_{} 
(17.28) 
(Chapter 2 and Table 2.1). Shewfelt and Brown demonstrate that if dislocations can climb round particles instead of bowing between them, then flow is possible at a stress as low as onethird of this, so that a creep threshold of _{} will appear (where μ, of course, is the modulus at the test temperature). Their data, and those of Dorey (1968) and of Humphries et al. (1970) are consistent with this prediction.
The powerlaw behaviour of coarsegrained polycrystals is more complicated (Lund and Nix, 1976 [41]; Lin and Sherby, 1981 [42]). As with single crystals, a stable dispersion introduces an apparent threshold. In some instances it has the characteristics described above: the creeprate becomes too small to measure below about _{}. But in others, perhaps because of the stressconcentrating effect of grain boundary sliding or the influence of a substructure created by previous working, the threshold is not quantitatively explained.
In finegrained polycrystals there is a further aspect. Creep does not stop when powerlaw creep is suppressed because it is replaced by diffusional flow. This mechanism, too, is retarded by alloying, but to a lesser degree than powerlaw creep. For this reason, diffusional flow at low stresses is of particular interest to us here.
Diffusional flow at low stresses
When a grain or phase boundary acts as a sink or source for a diffusive flux of atoms, the flux has its sources and sinks in boundary dislocations which move in a nonconservative way in the boundary plane. Electron microscopy reveals dislocations of the appropriate kind (Gleiter, 1969 [43]; Schober and Balluffi, 1970 [44]) which multiply by the action of sources, so that their density increases with the stress up to about 10^{7} m/m^{2}. Both theory and experiment show that their Burger's vectors b_{b} are not lattice vectors (they are smaller) and therefore that they are constrained to remain in the boundary plane when they move.
The model for diffusional flow must now be modified in two ways. First, one must ask how its rate is changed by the presence of discrete sinks and sources. The answer (Arzt et al., 1982) [45] is that the change is negligible unless the density of boundary dislocation is exceedingly low; we shall ignore it here. But second, one must ask how the mobility of defects influences the creeprate. The answer is that in pure metals of normal grain size, the mobility is high; and the creeprate is unaffected; that is why much data for pure metals follow the simple diffusioncontrolled rateequation (2.29). Only at very low stresses does the selfstress of the boundary dislocations limit their mobility, giving a threshold. But in alloys or compounds, particularly those containing a fine, stable dispersion of a second phase, the mobility of boundary dislocations is much reduced. Then anomalously slow creep, and large apparent threshold stresses, are found. We now develop these ideas a little further.
A boundary dislocation of the kind of interest here cannot end within a solid. It must either be continuous, or link, at nodes (at boundary triplejunctions, for instance) to one or more other dislocations such that the sum of the Burger's vectors flowing into each node is zero. If this line now tracks across the undulating surface making up the boundary between grains, its length fluctuates, and it may be pinned by the nodes. If the selfenergy of the line is:
_{} 
(17.28) 
then the length fluctuation, or the pinning, will introduce a threshold for dislocation motion. Both are easily modelled (e.g. Burton, 1972 [36]; Arzt et al., 1982 [45]) and lead in an obvious way to the result:
_{} 
(17.29) 
where d is the grain size and C is a constant close to 1. Taking b_{b} to be 10^{10} m (about onethird of a lattice Burger's vector), we predict threshold stresses between 2 ´ 10^{7} μ (for a grain size of 500 µm) and 10^{5} μ (for a grain size of 10 µm). They are a little smaller than those reported in Table 7.4, which probably reflect a combination of this with impurity drag, discussed next.
The influence of this threshold on the map for nickel is shown in Fig. 17.5. It was computed by replacing σ_{s} in the diffusional flow equation (eqn. (2.29)) by (σ_{s} — τ_{tr}). The map has been extended downwards by a decade in stress to allow the threshold to be seen. Over a wide range of stress, creep follows eqn. (2.29). Only near τ_{tr} is any change visible.
Fig. 17.5. A map for pure nickel with a threshold τ_{tr} of 8 ´ 10^{7} μ. The selfenergy of the boundary dislocations will introduce a threshold of general magnitude _{}.
Mobilitycontrolled diffusional flow in alloys and compounds 
A solid solution, or dissolved impurities, can segregate to a boundary dislocation; then, when the dislocation moves, the segregant may diffuse along with it, exerting a viscous drag which can limit its mobility. Discrete particles of a second phase, too, interact with a boundary dislocation, pinning it, and introducing a threshold stress in much the same way that particles pin lattice dislocations and inhibit powerlaw creep. But because the average Burger's vector of the boundary dislocation is smaller (by a factor of perhaps 3) than that of a lattice dislocation, the extent of the segregation is less, and the pinning force is smaller. For this reason, strengthening mechanisms suppress powerlaw creep more effectively than diffusional flow, causing the latter to become dominant.
When the boundarydislocation motion is impeded, part of the applied stress (or of the chemicalpotential gradient it generates) is required to make the dislocations move, and only the remaining part is available to drive diffusion. The creeprate is then slower than that given by eqn. (2.29), to an extent which we now calculate.
The velocity υ of a boundary dislocation is related to the force F per unit length acting on it through a mobility equation (cf. eqn. (2.4)):
_{} 
(17.30) 
Here _{} is the part of the applied stress required to move the boundary dislocations. (We assume that, in pure shear, the boundaries are subjected to normal tractions of ± σ_{s}.) The strainrate is related to b_{b}, υ and the density of boundary dislocations, ρ_{b} (a length per unit area) by:
_{} 
(cf. eqn. (2.2)), where the factor 2 appears because the motion produces a normal, not a shear strain. Together these equations become:
_{} 
(17.31) 
This strainrate must match that produced by the transport of matter across the grain, by diffusion, driven by the remaining part of the stress, _{}. .From eqn. (2.29), this is:
_{} 
(17.32) 
Eliminating _{}, and solving for _{} gives:
_{} 
(17.33) 
This is the basic equation for diffusional flow when boundary dislocation mobility is limited (Ashby, 1969,1972) [46, 47]. Note that when M is large, the equation reduces to the classical diffusional flow law (eqn. (2.29)); but when M is small, it reduces to eqn. (17.31) with _{}. In particular, if boundary dislocations are pinned (M = 0), creep stops. To progress further we require explicit equations for ρ_{b} and M.
The most reasonable assumption for the density of boundary dislocations, ρ_{b}, is that it increases linearly with stress (Burton, 1972 [36]; Ashby and Verrall, 1973 [48]):
_{} 
(17.34) 
where α is a constant of order unity. This is simply the twodimensional analog of eqn. (2.3) for the dislocation density in a crystal.
When a solute or impurity drag restricts dislocation motion, the drag will obviously increase as the atom fraction C_{0}, of solute or impurities, increases. Arzt et al. (1982) [45]show that:
_{} 
(17.35) 
where D_{s} is the diffusion coefficient for the solute or impurity, and β is a constant (for a given impurity) of between 1 and 20. As an example of the influence of impurity drag, we set:
_{} 
(i.e. we set D_{s} = D_{b} and _{}) and compute maps for nickel using eqn. (17.33), and eqn.(17.34) for diffusional flow. The results, for two grain sizes, are shown in Figs. 17.6 to 17.9. The figures illustrate how the field of diffusioncontrolled flow, for which _{}, is replaced at low stresses by one of mobilitycontrolled flow ("interface reaction control") for which _{}. The size of this new field increases as the grain size decreases, and for a sufficiently large concentration C_{0}, or a sufficiently small diffusion coefficient D_{s}, it can completely replace diffusioncontrolled flow.
This field rarely appears in pure metals. The small grain size necessary to observe it will not survive the tendency to grain growth at high temperatures; and the mobility of boundary dislocations is too high. Impurities lower the mobility and suppress grain growth; larger alloying additions, leading to a precipitate or a duplex microstructure, do so even more effectively. For this reason the source and sink mobility may dictate the behaviour of finegrained twophase superplastic alloys, which commonly show a sigmoidal stress/strainrate relation like that shown in Fig. 17.7.
A fine dispersion of a stable phase pins boundary dislocations, so that below a threshold stress their mobility is zero. The diffusional creep behaviour is then given by eqn. (17.33) with an appropriate (nonlinear) expression for M. But at a useful level of approximation we can think of the threshold stress as subtracting from the applied stress. Then the creeprate is given by the classical diffusional flow equation (eqn. (2.29)) with σ_{s} replaced by (σ_{s} – τ_{tr}); maps computed in this way are shown as Figs. 7.5 and 17.5. We anticipate that τ_{tr} is about onethird of the Orowan stress for boundary dislocations:
_{} 
(17.36) 
for the reasons given earlier in this section. Note that, because the boundary Burger's vector b_{b} is less than that of lattice, this threshold stress is lower, by a factor of perhaps 3, than that for powerlaw creep. Dispersionhardened materials show apparent threshold stresses of general order 10^{4} μ (Table 17.5), implying an obstacle spacing of about 3000 b_{b}.
Apparent threshold stresses are often observed to be temperaturedependent. This can be understood (Arzt et al., 1982) [45] as a superposition of discrete obstacles and an impurity, or solute, drag: the drag causes the stress at which the strainrate falls below the limit of resolution of the equipment to depend on temperature.
Fig. 17.6. A stress/temperature map for nickel of grain size 1 µm, with limited boundarydislocation mobility.
Fig. 17.7. A strainrate/stress map for nickel of grain size 1 µm, with limited boundarydislocation mobility.
Fig. 17.8. As Fig. 17.6, but for a grain size of 10 µm.
Fig. 17.9. As Fig. 17.7, but for a grain size of 10 µm.
17.4 THE EFFECT OF PRESSURE ON PLASTIC FLOW
In engineering design it is normal to assume that it is the shearing, or deviatoric, part of the stress field which causes flow (see Chapter 1). There is some justification for this: neither lowtemperature plasticity nor creep are measurably affected by pressures of less than K/100, where K is the bulk modulus. But when the pressure exceeds this value, the flow strength is increased and the creep rate is slowed. It is still the deviatoric part of the stress field which causes flow, but the material properties (such as _{}, ∆F, A and Q) have been altered by the pressure. Pressure must then be regarded, with temperature and shear stress, as an independent variable. In certain geophysical problems pressure is as important a variable as temperature: at a depth of 400 km below the earth's surface, for example, the pressure is about K/10, and its influence on material properties is considerable.
The influence of pressure on plastic flow has not been studied in anything like the same detail as that of temperature. But enough information exists to piece together a fairly complete picture. In this section we summarize the results required to incorporate pressure as an independent variable in computing deformation maps. Examples of their use are given in the final Case Study of this book (Section 19.8).
Effect of pressure on the ionic volume, lattice parameter and moduli 
In a linearelastic solid of bulk modulus K_{0}_{,} the atomic or ionic volume varies with pressure as:
_{}, 
(17.37) 
and the lattice parameter a (and the Burger's vector b) as:
_{}, 
(17.38) 
where Ω_{0}, K_{0} and a_{0} are the values at atmospheric pressure, p_{0}. Values of K_{0} for elements and compounds are tabulated by Huntington (1958) [49] and Birch (1966) [50].
To firstorder, the moduli increases linearly with pressure and decrease linearly with temperature. We write this in the form:
_{} 
(17.39) 
_{} 
(17.40) 
where p_{0} is atmospheric pressure, which, for almost all practical purposes we can ignore.
The coefficients in the square brackets are dimensionless. Table 17.6 lists means and standard deviations of the temperature and pressure coefficients for a number of cubic elements and compounds. Most are metals, though data for alkali halides and oxides are included. The coefficients are approximately constant; when no data are available for a specific material, these constant values may reasonably be used.
TABLE 17.6 Temperature and pressure coefficients of the moduli and yield strength
Coefficient 
Mean and 
Source of data 
_{} 
— 0.52 ± 0.1 
This book, Tables 4.1, etc. 
_{} 
— 0.36 ± 0.2 
Huntington(1958) [49] 
_{} _{} _{} 
1.8 ± 0.7
4.8 ± 1
5 ± 3 
Huntington (1958) [49]
Birch (1966) [50] 
_{} 
8 ± 2 
Richmond and Spitzig (1980) [51] 
Experiments on a number of steels (Spitzig et al., 1975, 1976 [52, 53]; Richmond and Spitzig, 1980 [51]) have characterized the effect of a hydrostatic pressure on the flow strength. To an adequate approximation it increases linearly with pressure.
_{} 
(17.41) 
where (for pure iron and five steels) the dimensionless constant in square brackets has values in the range 6 to 11. The effect is far too large to be accounted for by the permanent volume expansion associated with plastic flow, and must be associated instead with a direct effect of pressure on the motion of dislocations.
This pressuredependence can be accounted for almost entirely by considering the effect of pressure on the activation energies ∆F_{p} and ∆F and the strengths _{} and _{} which appear in eqns. (2.9) and (2.12). It is commonly found (see Kocks et al., 1975 [10], for a review) that the activation energy for both obstacle and latticeresistance controlled glide scales as μb^{3} and the strengths _{} scale as μ. For the b.c.c. metals, for example, the activation energy is close to 0.07 μb^{3} and the flow stress at absolute zero is close to 0.01 μ (Table 5.1, Chapter 5). As already described, both μ and b depend on pressure, μ increasing more rapidly than b^{3} decreases. Pressure, then, has the effect of raising both the activation energies (∆F) and the strengths (_{}).
At absolute zero the shear stress required to cause flow is simply _{}. Using eqn. (17.39) for the modulus, and neglecting p_{0}, we find by inserting the above proportionalities into eqn. (2.9) and inverting:
_{} 
(17.42) 
which has the form of eqn. (17.41), with:
_{} 
(17.43) 
Values of both dimensionless quantities are listed in Table 17.6 for variety of materials. The calculated values range from 2 to 9 compared with the measured coefficient of 6 to 11, but the measurements, of course, were made at room temperature—about 0.2 T_{M} for many of the listed materials. When a correction for this is made (Ashby and Verrall, 1977) [54], closer agreement is obtained.
There are other contributions to the pressuredependence of lowtemperature plasticity, but they appear to be small. The presence of a dislocation expands a crystal lattice, partly because the core has a small expansion associated with it (about 0.5 Ω per atom length) and partly because the nonlinearity of the moduli causes any elastic strainfield to produce an expansion (Seeger, 1955 [55]; Lomer, 1957 [56]; Friedel, 1964 [57]). It is this second effect which is, in general, the more important. The volume expansion per unit volume of uniformly strained material is roughly:
_{} 
(17.44) 
where ∆E^{el} is the elastic energy associated with the strainfield. If the activation energy which enters the rateequations is largely elastic in origin (as it appears to be) then during activation there is a small temporary increase in volume, ∆V. A pressure further increases the activation energy for glide by the amount p∆V. But when this contribution is compared with that caused by the change in moduli with pressure (eqn. (17.42)) it is found to be small. The influence of pressure on the lowtemperature plasticity, then, is adequately described by eqn. (17.41). A pressure of 0.1 K_{0} roughly doubles the flow strength.
There have been a limited number of creep tests in which pressure has been used as a variable; they have been reviewed by McCormick and Ruoff (1970) [58]. Typical of them are the observations of Chevalier et al. (1967) [59], who studied the creep of indium under pressure. When the pressure was switched between two fixed values the creeprate changed sharply but reversibly, returning to its earlier value when the additional pressure was removed.
When creep is glidecontrolled, pressure should influence it in the way described in the last subsection. When, instead, it is diffusioncontrolled, the main influence of pressure is through its influence on the rate of diffusion. Pressure slows diffusion because it increases the energy required for an atom to jump from one site to another, and because it may cause the vacancy concentration in the solid to decrease. The subject has been extensively reviewed by Lazarus and Nachtrieb (1963) [60], Girifalco (1964) and Peterson (1968) [61]; detailed calculations are given by Keyes (1963) [62].
The application of kinetic theory to selfdiffusion by a vacancy mechanism (see, for example, Shewmon, 1963 [63]) gives, for the diffusion coefficient:
_{} 
(17.45) 
where α is a geometric constant of the crystal structure, and α is the lattice parameter (weakly dependent on pressure in the way described by eqn. (17.38)). The important pressuredependencies are those of the atom fraction of vacancies, n_{υ}, and the frequency factor, Γ. In a pure, onecomponent system, a certain atom fraction of vacancies is present in thermal equilibrium because the energy (∆G_{f} per vacancy) associated with them is offset by the configurational entropy gained by dispersing them in the crystal. But in introducing a vacancy, the volume of the solid increases by V_{f}, and work pV_{f} is done against any external pressure, p. A pressure thus increases the energy of forming a vacancy without changing the configurational entropy, and because of this the vacancy concentration in thermal equilibrium decreases. If we take:
∆G_{f} = ∆G_{f0} + pV_{f} , 
(17.46) 
where the subscript "0" means "zero pressure", then:
n_{υ}_{ }= exp {(∆G_{f0} + pV_{f})/kT} 
(17.47) 
A linear increase in pressure causes an exponential decrease in vacancy concentration.
It is the nature of the metallic bond that the metal tends to maintain a fixed volume per free electron. If a vacancy is created by removing an ion from the interior and placing it on the surface, the number of free electrons is unchanged, and the metal contracts. For this reason, the experimentally measured values of V_{f} for metals are small: about _{}Ω_{0} where Ω_{0} is the atomic volume. Strongly ionic solids can behave in the opposite way: the removal of an ion exposes the surrounding shell of ions to mutual repulsive forces. The vacancy becomes a centre of dilatation, and V_{f} is large: up to _{}Ω_{0} where Ω_{0} is the volume of the ion removed. There are no data for oxides or silicates, but when the bonding is largely covalent one might expect the closepacked oxygen lattice which characterizes many of them to behave much like an array of hard sphere. Forming a vacancy then involves a volume expansion of Ω_{0}, the volume associated with an oxygen atom in the structure.
There is a complicating factor. In a multicomponent system vacancies may be stabilized for reasons other than those of entropy. Ionic compounds, for instance, when doped with ions of a different valency, adjust by creating vacancies of one species or interstitials of the other to maintain charge neutrality; pressure will not, then, change the vacancy concentration significantly. Oxides may not be stoichiometric, even when pure, and the deviation from stoichiometry is often achieved by creating vacancies on one of the sublattices. The concentration of these vacancies is influenced by the activity of oxygen in the surrounding atmosphere, so that the partial pressure of oxygen determines the rates of diffusion. For these reasons it is possible that the quantity V_{f} in eqn. (17.46) could lie between 0 and _{}Ω_{0}.
The jump frequency, too, depends on pressure. In diffusing, an ion passes through an activated state in which its free energy is increased by the energy of motion, ∆G_{m}. The frequency of such jumps is:
Γ = v exp (–∆G_{m}/kT), 
(17.48) 
where v is the vibration frequency of the atom in the ground state (and is unlikely to depend on pressure). In moving, the ion distorts its surroundings, temporarily storing elastic energy. If all the activation energy of motion is elastic, then (by the argument leading to eqn. (17.44)) it is associated with volume expansion,
V_{m} = 3∆G_{m}/2μ 
per unit volume. Taking the activation energy for motion to be 0.4 of the activation energy of diffusion, we find, typically, V_{m} = 0.2–0.4 Ω_{0}, where Ω_{0} is the volume of the diffusing ion. Experimentally, V_{m} is a little smaller than this, suggesting that the activation energy is not all elastic.
Assembling these results we find:
D = D^{0}(1 – 2p/3K) exp (–pV*/kT) ≈ D^{0} exp (–pV*/kT) 
(17.49) 
where D^{0} = α(a_{0})^{2}v exp – (∆G_{f} + ∆G_{m})/kT) is the diffusion coefficient under zero pressure, and:
V* = V_{f} + V_{m} for intrinsic diffusion; V* = V_{m} for extrinsic diffusion. 
Because experiments are difficult, there are few measurements of V*, and these show much scatter. They have been reviewed by Lazarus and Nachtrieb (1963) [60], Keyes (1963) [62], Girifalco (1964) [61], Goldstein et al. (1965) [64], Brown and Ashby (1980) [65] and Sammis and Smith (1981 [66]). Some results are summarized in Table 17.7: they lie between 0 and 2Ω_{0}, where Ω_{0} is the volume of the diffusing ion.
When creep is diffusioncontrolled, the creeprate should scale as the diffusion coefficient. The powerlaw creep equation (eqn. (2.21)) then becomes:
_{} 
(17.50) 
where μ and b are the values at the pressure p. The diffusional flow equation (eqn. (2.29)), similarly, becomes:
_{} 
(17.51) 
where Ω is the atomic or ionic volume at the pressure p.
When pressures are large the creeprate depends strongly on pressure: a pressure of 0.1 K_{0} reduces the creep rate, typically, by a factor of 10. The principal contribution is that of the term involving V*, so that the "activation volume" for creep, defined by:
_{} 
should be close to that for diffusion, V*. Table 17.7 shows that this is so.
TABLE 17.7 Activation volumes for diffusion and creep
Material 
Structure 
V*/Ω_{0} for diffusion 
V*_{cr}/Ω_{0} for creep 
Pb 
f.c.c. 
0.8 ± 0.1 
0.76 
Al 

— 
1.35

Na 
b.c.c. 
0.4 ± 0 2 
0.41 
K 

— 
0.54

In 
h.c.p. 
— 
0.76 
Zn 

0.55 ± 02 
0.65 
Cd 

— 
0.63

AgBr 
Rock salt 
1.9 ± 0 5 
1.9 




Sn 
Tetragonal 
0.3 ± 0 1 
0.31 
P 

0.5 ± 01 
0.4 
Data from Lazarus and Nachtrieb (1963), Goldstein et al. (1965) and McCormick and Ruoff(1970). 
Incorporation of pressure dependence into deformation maps
The information summarized above allows the effect of pressure on material properties to be incorporated into deformation maps. The quantities b, Ω, μ are calculated for the given pressure, using eqns. (17.37) to (17.39). Diffusion coefficients are corrected for pressure using eqn. (17.49), or (alternatively) the creep equations are modified to eqns. (17.50) and (17.51). Finally, the flow stress for lowtemperature plasticity is replaced by that given in eqn. (17.41). The final Case Study (Section 19.8) shows maps computed in this way.
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