CHAPTER 17

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FURTHER REFINEMENTS: TRANSIENT BEHAVIOUR; VERY HIGH AND VERY LOW STRAIN RATES; HIGH PRESSURE

 

17.1 Transient Behaviour and Transient Maps

• Elastic deformation

• Low-Temperature plasticity

• Power-law creep

• Diffusional flow

• Construction of transient maps

17.2 High Strain-Rates

• Phonon and electron drag, and the relativistic limit

• Power-law breakdown

• Adiabatic shear

• Deformation maps extended to high strain-rates

17.3 Very Low Stresses

• Threshold stresses for power-law creep

• Diffusional flow at low stresses

• The threshold stress in pure materials

• Mobility-controlled diffusional flow in alloys and compounds

17.4 The Effect of Pressure on Plastic Flow

• Effect of pressure on the ionic volume, lattice parameter and moduli

• Effect of pressure on low-temperature plasticity

• The effect of pressure on creep

• Incorporation of pressure dependence into deformation maps

References for Chapter 17

 

The maps shown so far in this book are based on simplified rate-equations which describe behaviour at steady structure or at steady state (Chapters 1 and 2), ignoring the effects of work-hardening 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.

 

17.1 TRANSIENT BEHAVIOUR AND TRANSIENT MAPS

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            When strains are small, as they are in service life of most engineering structures, the steady-state approximation is a poor one.  At ambient temperatures, metals work-harden, so that the flow stress (at a given strain-rate) changes with strain.  At higher temperatures most materials show primary or transient creep as well as steady-state 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 non-steady flow the state variables change with time or strain:

Si = Si(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(σs, 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 work-hardening, and both transient and steady state creep (Ashby and Frost, 1976) [1].

            In doing this, we lose some of the generality of the steady-state maps.  The strain-rate (which is used as the dependent variable in the steady-state 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.

 

Elastic deformation

            A stress σ_s produces an elastic strain:

(17.5)

Since we are now concerned with strain (not strain-rate), this elastic contribution must be added to the plastic strain to calculate the total strain.

 

Low-temperature plasticity

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            Polycrystal stress-strain curves can, in general, be fitted to a work-hardening 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

 

Power-law creep

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             When an intrinsically soft material such as a metal is loaded, dislocations are generated and the material usually work-hardens.  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 co-workers (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 creep-rate 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 co-workers (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 steady-state creep-rate, 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 strain-rate, eqn. (17.9) becomes:

(17.10)

where  is the steady-state strain-rate (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.

 

Diffusional flow

            In a pure, one-component 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 steady-state 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 steady-state strain-rate by diffusional flow and thus is identical with the rate used to construct the steady-state 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.

 

Construction of transient maps

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             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 10⁴ s (about 3 hours); the second after a time of 10⁸ s (about 3 years), the third after 10⁹ 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: 10⁴, or 10⁸ or 10⁹ 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 low-temperature plasticity; the spacing of the strain contours reflects work-hardening.  The power-law 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 10⁴ s (about 3 hours).

 

 

Fig. 17.2. As Fig. 17.1, but for a time of 10⁸ s (about 3 years).

 

 

Fig. 17.3. As Fig. 17.1, but for a time of 10⁹ s (about 30 years).

 

17.2 HIGH STRAIN-RATES

            Under impact conditions, and in many metalworking operations (Chapter 19, Section 19.3), strain-rates are high.  They lie in the range 1/s to 10⁶/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 power-law 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 power-law breakdown coefficient α', can be determined.

 

 

Phonon and electron drag, and the relativistic limit

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            A moving dislocation interacts with, and scatters, phonons and electrons. If no other obstacles limit its velocity, a force σ_sb 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 phonon-drag 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/m² at 4.2 K to about 10^-4 Ns/m² at room temperature.  Using the Orowan equation (eqn. (2.2)), we find:

The high strain-rate experiments of Kumar et al. (1968) [12], Kumar and Kumble (1969) [13] and of Wulf (1979) [14] allow the difficult term ρb²μ/B_p to be evaluated; in all three sets of experiments the result is close to 5 x 10⁶ /s at room temperature.  Combining these results gives an approximate rate-equation 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 strain-rate, exists which we take to be 10⁶ 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.

 

Power-law breakdown

            The transition from pure power-law creep to glide-controlled 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 power-law 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 power-law breakdown parameter

 

Adiabatic shear

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            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 strain-rate 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.  Work-hardening, or an increase in strain rate, raises σ_y.  Treatments of diffuse necking (Considère, 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, strain-rate 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 strain-rate 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: work-hardening is just offset by the fall in strength caused by heating.  If heating is uniform:

dq = CpdT= σydε

                                              or

(17.19)

If work-hardening is described by a power-law:

σy = Kεm

(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 temperature-dependent 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 work-hardening 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, heat-treated 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 strain-rate 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_pR²)/(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 left-hand 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 strain-rate 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 strain-rate must exceed the critical strain-rate 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.

 

Deformation maps extended to high strain-rates

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            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 strain-rate 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 strain-rates above 10² /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 strain-rate and temperature (Fig. 17.4); it displays most effectively the region in which high strain-rate effects are unimportant.  The same information can, of course, be cross-plotted 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 drag-controlled 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 commercial-purity titanium

 

 

TABLE 17.4 Apparent threshold stresses for creep in pure metals