Deformation dynamics at low and ambient temperatures

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Bull. Mater. Sci.. Vol. 10, Nos 1 & 2, March lq~, pp. 133 154. ~('~ Printed in India.

Deformation dynamics at low and ambient temperatures

P R O D R I G U E Z a n d S K RAY

Metallurgy Programme, lndira Gandhi (_'entre fi)r Atomic Research, Kalpakkam 603 102, India

Abstract. The variation of tensile yield stress at a constanl strain rate as a function of temperature for well-annealed pure metals show, with increasing temperatures, a rather sharp drop in yield stress (luw temperature regime), followed by the intermediate temperature regime where yield stress decreases more slowly land the ratio of yield stress to shear modulus remains more or less constant), which in turn is followed by the high temperature regime where the yield stress drops again rather sharply. The paper discusses the phenomenological framework for studying deformation dynamics in the low and intermediate temperature regimes, The approach adopted is the well-known state variable approach, where the evolutionary nature of deformation struclurc is described by one or more structure variables such lh;~l the current values of mechamcal variables and structure variables together completely define the current state of deformation. A critical analysis of expenme~ltal results available suggest that at least for deformation at low strain rates, stress-rate is probably not a state variable of dcfi)rmati~m. Thus deformation is most conveniently studied in terms of TASRA (thermally activated strain rate analysis) where the stress, plastic strain rate, temperature and slructurc are interrelated thcough a Gibb's free energy for thermal acuvation by an Arrhcmus equation. The stress-dependence of Gibb's free energy and its maximum ,,alue then form the basis of Identifying the rate-controlling obstacles. The need for careful experimentation and systematic analysis is illustrated by the example of low temperature deformation of hard hop metals. Modelling for the evolution of deh)rmation structure is also touched LIpon.

Keywords. l)eformati~n dynamics: time-independent dcformatit)n; thermally-activated strain rate analysis; hard hop metals, d).namic strain ageing.

1. Introduction

1.1 State variable approach to dqJ~)rmation dynamics study

O n plastically d e f o r m i n g a metallic material, its flow properties change, the simplest examples being the w o r k - h a r d e n i n g in uniaxial t e n s i o n or the p r i m a r y creep b e h a v i o u r in u n i a x i a l creep. In the state variable a p p r o a c h to the d e f o r m a t i o n d y n a m i c s study, it is p o s t u l a t e d that it is possible to identify a set of state variables of d e f o r m a t i o n , such t h a t their values at a n y i n s t a n t d u r i n g d e f o r m a t i o n c o m p l e t e l y describe the i n s t a n t a n e o u s m e c h a n i c a l responses, a n d there is no a d d i t i o n a l history d e p e n d e n c y implied or explicit. T h e a p p r o a c h , a p a r t from the pedagogical interest, is of t r e m e n d o u s practical significance in that if the efficacy of such a n a p p r o a c h is d e m o n s t r a t e d then it becomes practicable to predict the b e h a v i o u r of structures subjected to complex l o a d i n g histories from a limited n u m b e r of l a b o r a t o r y tests.

However, there are no a priori guidelines for identifying these variables: we must rely u p o n e x p e r i m e n t a l evidences for this purpose. In this paper, wc shall be c o n c e r n e d with n o n - r e v e r s e d uniaxial loading; for this type of loading, it is generally recognized that flow stress or, plastic strain rate i:p a n d (absolute) t e m p e r a t u r e T c o n s t i t u t e the m e c h a n i c a l variables of d e f o r m a t i o n . It is also generally agreed that, in a d d i t i o n , it is necessary to i n d u c t one or m o r e variables to describe the d e f o r m a t i o n structure. In 133

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134 P Rodriguez and S K Ray

fact, Swearengen et al (t976) showed that at least one variable pertaining to the internal structure of the material is necessary to describe inelastic deformation. The study of deformation dynamics thus consists of two aspects: (i) identifying the state variables of deformation and studying their inter-relation at constant deformation structure and (ii) describing the evolution of the deformation structure under different constraints, e.g. tension (ideally, constant ~:pl, or creep (ideally, constant a), or stress relaxation (ideally constant total strain F,,). 'There are several models, with different approaches towards defining the structure variables and the assumed interactions of these with the mechanical variables, which have been proposed;

some of these are referenced in a later section.

It is stressed that phenomenological modelling of deformation relies upon successful interpretation of carefully designed experiments. In this paper, we have chosen two somewhat controversial areas of deformation dynamics study to illustrate this point. We are mainly concerned with the inter-relation of state variables at a constant deformation structure; in the concluding section, we also touch upon the study of evolution of deformation structure.

1.2 Thermally-activated strain rate analysis (TASRA)

Thermally activated strain rate analysis (TASRA) is the most widely used formalism for studying deformation dynamics at constant deformation structure. Several excellent reviews are available on this topic (Kocks et al 1975; Rodriguez 1979, for example). The study of the subject is essentially the application of Eyring's absolute rate theory (Eyring 1936) to the process of plastic deformation. The motivation for this approach is easily seen from figure I, which schematically shows the variation of yield stress with temperature at a constant kp. Three regimes of deformation are identifiable with increasing temperature; at low temperatures stress decreases rapidly, at intermediate temperatures the fall is more gradual and (approximately) parallels the variation of shear modulus/a with temperature and at still higher temperatures it again decreases rapidly. These three regimes are dependent upon the material and the strain rate. A plot like figure 1 is useful for identifying the low, intermediate and high temperature regimes of deformation for a given material. It is generally agreed that deformation in the low temperature regime is thermally activated. The relevant formulation is

ep =/:o e x p ( - AG/kT).

~~

CONS]AN'I S I R A I N R A ] E

I E M P E R A T U R E t K

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Figure 1. Schematic variation of yield stress with temperature for a well-annealed metal.

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Deformation dynamics at low and ambient temperatures 135 Here ko is the pre-exponential factor, AG the Gibb's free energy for thermal activation and k the Boltzmann constant. (In the following discussion, we have used tensile stress tr and tensile strain rate e.p rather than shear stress r and shear strain rate respectively. The conversions are readily achieved using the Taylor factor M: a = Mr and kp = ~/M. The value of M is 3"06 for well-annealed polycrystalline fcc metals, and has been taken to be equal to 2 for polycrystalline hcp metals). The basic idea for this formalism is explained in figure 2. The dislocations are propelled forward by the applied stress tr, which is sufficient for the dislocations to travel across the obstacle- free region o f the glide plane, but is not fully sufficient for the dislocations to move past the rate-controlling obstacles which have a characteristic mechanical threshold stress of # > 0 . The dislocations thus require a characteristic waiting time at the obstacles before these obstacles are overcome with the help of thermal activation.

The dislocations then move for a period of time of flight to sweep across the glide plane to the next rate-controlling obstacles where once again they are held up.

Usually the waiting time is much larger than the time of flight so that the free energy of thermal activation AG required determines ep. The pre-exponential factor ko is proportional to the product of the density of such sites for thermal activation, the average area of the slip plane swept out per successful thermal activation, the attempt frequency and the Burgers vector b. In general, both AG and ~o are functions of stress and structure.

There is some controversy about the rate-controlling deformation mechanism in the intermediate regime. Noting that the temperature dependence of stress in this regime (approximately) parallels that of shear modulus, Seeger (1957) postulated that this regime of deformation is athermal. According to this interpretation, the total stress field that a dislocation sees when moving in the glide plane consists of a stress field tru due to long range barriers, with those from short range barriers superimposed (el figure 2). Now, the long range barriers cannot be overcome by thermal activation; in effect, for the dislocations to be mobile at all, a~>tr,. Even though the peak stresses involved for short range obstacles are high, their effects spread over smaller distances; as a result, these can be overcome by thermal activation. The total flow stress tr is the sum of two components, the athermal component tru and the thermal component a*. In this interpretation, deformation in the intermediate temperatures is athermal, a * = 0 and ~r=a~. Obviously, this interpretation is critically dependent upon the experimentally determined

PPLIED STRESS

f f * f f P / S H A D E D AREA IS - f - - - ~ PROPORTIONAL TO ZIG

DISTANCE ALONG SLIP PLANE

Figure 2. Stresses experienced by a dislocation moving on the slip plane.

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temperature dependence of a. Thus it has been suggested (see e.g. Sastry and Tangri 1975) that this regime of deformation might also be thermally activated but with a low temperature sensitivity (visualizing this regime as an extrapolation from the low temperature regime of deformation in figure 1) so as to appear athermal. Also, there are several other difficulties for the athermal deformation interpretation for the intermediate temperature regime (e.g. Sastry and Tangri 1975).

The methodology for TASRA analysis can be carried out either in the applied stress or in the effective stress interpretation. The parameters of interest are the magnitudes of AG O , the activation energy at zero stress and the activation area, which is the stress dependence of activation energy, h~ the effective stress formalism, activation area is defined as

A* = - (M/b)(~AG/Oa*). (2)

The activation area in the applied stress formalism can be similarly defined. TASRA study thus aims at experimentally evaluating these parameters and their stress and temperature dependencies and comparing these values with the corresponding theoretical predictions for different deformation models. Several deformation models have been theoretically analysed. Rodriguez (1976) has given a comprehensive bibliography. Table l, taken from Rodriguez (1976), gives a summary of these models.

However, the experimentally determined quantities are Q*, the apparent activation enthalpy, and A,*, the apparent activation area, defined by

Q* = - k T 2 (d In ~p/c~ (a*/m) )(O (a*/M)/~T), (3a) and

A *= (k T/b) (~ In ;:p/d (a* / M) ). (3b)

Clearly A* equals true activation area only if ko is independent of stress. Similarly, to obtain AG from Q*, two corrections are to be applied. Rodriguez (1979) provided a

Table I. Characteristics of some possible thermally activated deformation mechanisms (Rodriguez 1976).

Mechanism A * / b 2 Other characteristics

Peierls-Nabarro (P-N) 1 -102 A* is independent of strain. Number of theoretical models that predict variation of A t t and A* with a are available.

A t l ° is related to the energy required to nucleate a

double-kink.

1 102 A* depends on concentration of point defects; dependent on strain only if the point density is altered by deformation.

102 104 A* decreases and e* increases with strain. Many theoretical models have been proposed.

1 A H 0 is equal to the activation energy for diffusion.

Theoretical models predict stress and temperature dependence of strain rate.

10 10 z Many theoretical models available, but none seems to be adequate.

10 103 ,~ t t ° is equal to the self-diffusion energy. A* decreases with strain as the jog spacing decreases. Theory is well- developed.

> 10 "~ A* decreases as the dipole density increases with strain.

Point defect interaction Dislocation interaction Climb

Cross slip

Movemcnt of jogged scrcw dislocations

Dislocation dipoles

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D e f o r m a t i o n d y n a m i c s at low a n d a m b i e n t temperatures i 37 comprehensive discussion on this topic. Firstly, eo might depend upon stress and/or temperature, or might be constant. Writing AG = A H - TAS,

A n = Q* - kT 2 (c3 ln~: o/dT), (4)

which provides for this correction. Secondly, the obstacles might be inelastic, when AS = 0, or could interact elastically with dislocations, when AS#0. For systematic investigations into probable rate-controlling mechanism(s), it is essential to explore all these possibilities. For inelastic obstacles obeying Friedel (1964) statistics (this is the case for obstacles separate from the glide dislocation e.g. solute atoms, forest dislocations, point defects etc.). Conrad et al (1975) derived

A G = A H = Q* - a A * a * / M , (5)

where a = ( T / l ~ ) ( d # / d T ) . The thermodynamic analysis of linear elastic obstacle controlled glide has been successfully developed (Surek et al 1973, 1975; Sastry et al 1976). The back stress exerted by linear elastic obstacles on dislocation is independent of a*, and AG°oc/~. If ko is either constant or a function of (tr*/ll) only, then

A G = (Q* + ot bA * ( a * / M ) ) / ( 1 - a ). (6)

Equation (6) is clearly significant as this allows calculations of AG from experimentally determined quantities. Once A G is known, k0 can be calculated by using (2), and the postulated dependence of ~o on (a*//0, or its constancy can be verified. Equations corresponding to (3H6) have been derived for the applied stress formalism as well.

2. Time-independent deformation (TID) formalism of Alden

2.1 Motivations behind T I D theory

In the TASRA formalism, the time rate of change of stress b does not play any role in deformation kinetics. On the other hand in the time-independent deformation (TID) theory of Alden (1972, 1982), b is an important variable of deformation. Alden (1982) discussed in detail the motivations behind this theory. In the thermal stress formalism of TASRA, deformation in the intermediate temperature range is considered to be athermal; in the absence of softening of the structure due to dynamic recovery processes, because of work-hardening tr, would continually increase with deformation. Now, as for athermal deformation tr must at least equal a,, it stands to reason that continued deformation cannot be sustained unless tr also increases; in other words a positive stress rate is necessary to sustain mechanical (as distinct from thermal) activation of strain. Moreover, in this formalism, the limiting value of a* = 0 is not excluded in principle, and in fact has been claimed to have been obtained experimentally (Abe et al 1977). Again, the highly successful recovery creep theories (pertaining to high temperature regime of deformation) assume that straining is athermal but thermally activated softening of structure occurs leading to the observed strong temperature dependence for the steady state when the work- hardening and softening due to recovery match each other (see e.g. Weertman 1955;

McLean 1968; Alden 1969). Direct evidence of athermal strain in creep including delay times on stress decrease and instantaneous (loading) strains on stress increase

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138 P Rodriguez and S K Ray

either during (Mitra and McLean 1967; Alden and Clark 1975) or at the start of a creep test (Garofalo 1964) also supports the TID formalism and contradicts the TASRA formalism. Alden (1982) suggested that probably general deformation is a mixture of both time-independent and time-dependent deformations; short range obstacles, high temperature, low stress rate and low stress would augment the time- dependent deformation while opposite factors would augment the time-independent deformation.

For mechanically activated plastic strain, Alden (1972, 1977) suggested

.~p=(b+ rA)AR/Or, b >10, (7a)

and e~ = rA AR/Oy, b < 0, (7b)

where Or is the work-hardening rate in the absence of recovery, r a a recovery parameter involving both loss and rearrangement of dislocations and AR an area function.

2.1 Experimental verification of TID theory

In the previous section, we quoted some experimental findings which motivated TID theory. To the extent these experimental findings may be criticised, TID theory would become suspect; Ray (1984) discussed some of these experimental findings and remarked that the conclusions drawn may not be unequivocal. One of the experimental verification for this TID theory that has been usually adopted (Abe et al 1976; Holbrook et al 1981; Alden 1982; Alden and Gibeling 1982; Ray 1984;

Gibeling and Alden 1984, 1986) involves transition from tension (b~0) to relaxation (o'<0), henceforth called TR tests. This test is carried out by instantaneously halting the crosshead motion during tension or compression; as a is high enough to cause plastic flow, kp > 0, but as the total strain rate is zero (ignoring deformation in the machine, see below), plastic strain rate is equal and opposite to the elastic strain rate (ignoring specimen anelasticity). Thus stress relaxes continuously, resulting in progressively decreasing kp values; as a result, the total plastic strain accumulated during a typical stress relaxation test is very small. For analysing the applicability of TID theory, kp immediately before and after such transitions are computed from the measured load rate values immediately before and after transition; for TID theory to be valid, the ratio R = (~p for relaxation)/(~, for tension), at the transition, forpositive b in tension, must be less than unity. In order to compute ~p and b values from the experimentally determined load rate values, it is necessary to consider the effects of the finite stiffness of the testing machine. For this purpose, it is assumed that the machine behaves like a linear elastic spring with constant compliance CM, in series with the specimen (e.g. Hart 1967), so that the total deformation controlled is the sum of the deformations of the machine and of the specimen. The total elastic compliance of the load train Cr is then the sum of CM and that of the specimen gauge-length, C~p. Now, CT values are determined by separate experiments.

Two remarks are pertinent here. Firstly, as it will become obvious from the following, the applicability of TID theory can be examined by employing tests with either upward or downward change of strain rate. However, as far as the authors are aware, only Ray et al (1988) have used the upward change of strain rates for this purpose. Secondly, as shown by Ray (1984), independent measurement of CT, with all the incident uncertainties can, in principle, be obviated by taking the deformation

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Deformation dynamics at low and ambient temperatures 139 dependence of C r into account. Let L 0, Ao and C w be the initial gauge length, cross- sectional area and specimen compliance respectively, and E be the Young's modulus of the specimen material at the test temperature. Then C~ ° = Lo/AoE. If A.;c is the change (either upward or downward) in crosshead speed, and AP the corresponding change in load rate, then, in the absence of anelasticity, and assuming continuity of e v at the transition, within experimentally available accuracies (see e.g. Ray et al 1988):

C r = A.~/A P= C M + C°p(l + x / L o)2 (8)

where x is the measured crosshead displacement. This equation shows that (i) a plot of Cr vs (1 + x / L o)2 should be a straight line and (ii) the slope of this line must equal Lo/AoE , which can be independently calculated. These two consistency criteria involved in (8) make it suitable for examining whether TID theory is valid, without an independent determination Of Cr: equation (8) is violated if TID theory is applicable.

However, even if TID theory is inapplicable, inadequate data resolution may lead to an apparent violation of (8) (see Holbrook et al 1981; Gibeling et al 1984). Also specimen anelasticity effects have been ignored in deriving (8). Ray et a1(1988)used a simple rheologicai model to quantitatively examine the possible role of anelasticity effects on the interpretation of data from such tests. They asstmled that the specimen behaviour can be represented by an elastic element (strain eel an anelastic element (strain eo) and a plastic element (strain %) in series. With the constitutive equation for the anelastic element given by ~', + fit:° = ~,a/E + 6b/E, where fl, ), and 6 are constants, and taking into account the continuity of ep at the transition point, they obtained (A before a variable indicates its change on effecting change of ~:)

A~,+ flA~, = flA?~p + (1 + ),) A b / E + (fl + 3) A~/E. (9) If however A e a = 0, a simpler relation is obtained:

Cr : (CM + L 2 / E V ) / [ I - (a/E)ib/fl)] (10)

with L = Lo + x. Equation (10) predicts that Cr is independent of anelasticity if 6 = 0, i.e. anelastic strain rate is independent of stress rate, as argued by Gibeling et al (1984) from qualitative arguments. This equation also shows that for 6/fl>O, with increasing deformation (i.e. with increasing L and a), measured Cr should increase.

Also, as a/E<< 1, the variation in CT would be quite small unless 3/fl is very large.

There is another experimental artefact in strain rate change tests that need to be considered for a meaningful analysis of the experimental data. In these tests, when a crosshead speed change is effected, because of the inertia of the testing system, a finite time elapses before the crosshead acquires the new constant speed, (e.g. Saimoto et al 1981; Ray 1984; Gibeling and Alden 1986). For upward change of crosshead speed, this transition manifests itself as a region of load time trace curved upwards (P>0), provided adequate load and time resolutions have been used. For TR tests, this effect is not prominent in the load time trace (Ray 1984). But for either kind of tests, mis effect can be seen clearly by accurately measuring x through the transition.

Apparently, the duration of this transition period depends upon several factors, including the stiffness of the testing system and the initial and final crosshead speeds.

For example, for the results described below, this transient period was about 0"15 s.

Consider tests with upward change of strain rate; P values required for C T computation are determined by drawing tangents to the segments of load-time plot on either side of the transient period (characterized by fi> 0). Now, if ~p strongly

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140 P Rodriguez and S K Ray

depends upon stress, then kp (and thus ca) changes significantly during this transient period. In effect, the average anelastic strains for the segments used for/3 calculation would be different, effectively violating the condition A~o = 0. Equation (9), which is valid even if A~a ~-0, in effect, provides for a specific variation of ~p with time during this transition period through the flA~p term. Thus, if use of (9) and (10) leads to different conclusions, it must reflect a strong stress dependence of kp.

2.3 Experimental results (Ray et al 1988)

We now present some experimental results to illustrate the ideas put forth above. The two materials tested were high purity annealed aluminum (mean linear intercept grain size 0-100 mm), and a solution annealed stainless steel of grade AISI 316 (mean linear intercept grain size 0-060 mm). The experimental details will be found elsewhere (Ray 1984). Briefly, the tests were conducted in a floor model (Instron 1195 unit). Different test temperatures (with adequate temperature stabilities over the gauge portions of the specimens) were obtained by using a high temperature furnace or an oil bath as appropriate. Appropriate load cells and the lnstron load suppression unit were used to enhance load resolution. For many of the tests reported below, the load data were recorded at high time resolution by using an 8 bit transient signal recorder and subsequently plotted at a slower rate in a X-t plotter;

for these tests the effective chart speed was 3000 mm.min-

2.3a Continuity of kp in SS316" The strain rate change (SRC) tests involved a change of nominal strain rate from 3.17 x 10 -5 s -1 to 3.17 x 10 -4 s -1 for SS 316.

Another type of strain rate change tests, called post-relaxation tension (PRT) tests, was also carried out; these tests involved deforming the specimen to chosen levels of strain at a nominal strain rate of 3"17 x 10 -4 s -~, followed by stress relaxation for various periods (1000-2000 s) and then straining the specimen again at 3.17 x 10 -4 s-l; C T values are measured from the relaxation to tension transition.

The difference of SRC and PRT tests lies in the fact that as plastic strain rate during stress relaxation decreases very rapidly, the ep value at the instant of strain rate transition in PRT tests can be taken to be very close to zero. The SRC test data were collected using the transient sigr..al recorder and plotted as described earlier.

Figure 3 shows the results. In all these figures the best fit straight line with fixed slope C ° ' theoretically calculated for the two sets of data is also shown. The temperature- dependent E values reported by Hoke (1977) were used for this purpose. These results show that the SRC and PRT techniques give similar results within the experimental accuracies involved, though CT from PRT test results seem to be consistently higher than those from SRC tests. Now, if TID theory is applicable, as b is positive for both crosshead speeds, using (7a),

A "~,=(A~/Or + 1/E) A b, (11)

indicating that in the presence of appreciable plastic strain rates, the measured CT values should be higher than elastic compliance because of the AR/O r term. Thus the higher CT values for PRT data (despite lower ~p at the transition) in the present case is probably due to the lower time resolution used in the PRT tests.

2.3b Continuity of ~p in At: The strain rate change tests involved an upward

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Deformation dynamics at low and ambient temperatures 141

0 . 1 2 ~ G

1.0 1.2 I-4 1.6 1-8 2.0 2.2

( o ) 3 0 0 K

I z

0.16 0.12

0.16 0.12

o.2d

0.16

0.21 0.17

1.0

- ®

; ee_ et~..~e... ..e . . . .

- - a

.ta~.t~.-m-rr- --m-~ ~ - :o

- _ _

O ® 0 0 0

1.2 1./, 1.6

(1 • x / t o 12

(b) 1023 K

(c) 1073 K

(d) 1123K

(e) 1173K

( f ) 1223K

Figure 3. Variation of C,, with (1 +x/Lo) 2 for SS 316 for temperatures between 300 and 1223 K, determined from both PRT tests (open circles) and SRC tests (open squares).

Transient signal recorder was used to gather latter set of data. (after Ray et al 1988).

change of nominal strain rate from 6"56 × 10 -5 s - t to 6-56 x 10 - 4 s - 1 . The transient signal recorder-plotter combination was used for these tests. According to the 0-2%

yield stress data of Mukherjee et al (1965), the athermal regime of deformation for high purity AI is above 150 K. Therefore, these tests were conducted at several temperatures above 300K. The results from these tests are shown as C r vs (1 + x/Lo) 2 plots in figure 4. Cr decreases systematically with increasing deformation.

This trend is qualitatively consistent with inadequate data resolution, which would result in higher apparent CT value (see Holbrook et al 1981). However, as the experiments have been conducted using high data resolutions, this explanation is not fully satisfactory. Now, the results could also be satisfactorily interpreted in terms of the applicability of TID theory. From (11), the apparent Cr values should be higher than the ideal value, and also, as A R decreases with increasing deformation, so should the measured Cr value. Both these predictions are qualitatively borne out by the deformation dependence of the measured Cr seen in figure 4. The flattening out of these plots are also in qualitative agreement with the expected saturation bchaviour of Ate.

Let us now examine the possibility that anelasticity effects, rather than the applicability of TID theory, is responsible for the observed behaviour, lntcrpretation of the results of figure 4 ( a - b ) according to (10) would suggest that 6/[~ is negativc, and of very large magnitude. In the absence of an independent assessmcnt of thc

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142 P Rodriguez and S K Ray

!.6 ALUMINUM

1 , / EXPERIMENTAL RESULTS: 338K ®

" t L \

,. c, FRO, F,,:

0.8

1.0 1.2 1.4 1.6 1.8

(I • x / t o )2 11 OX/La) l

1., I.t, 1.6

I . I [ 1.0

ALUMINUM

1.0 EXPERIMENTAL RESULTS 398K.

I ~ 438K®D'9

[ ~ C! FROM FIT r = O - -

~ 0 . 8 11.7 •

, I \ ° . . .

o, t ....

0 $ ~ O.IL

1.0 ~ 1.2 1./, 1.6

( 1 • x / t , )2

Figure 4. Variation of C.r with (1 + x/Lo) 2 for AI at different temperatures determined by SRC tests. Transient signal recorder was used to collect the data. Smooth lines correspond to least square fit with assumption 6:~0 and y =0; X indicates calculated value of C r from least square fit with assumption I,:/:0 and 6 = 0 (after Ray et al 1988).

anelastic constants (see below), we cannot comment upon this prediction. For using (9), ~p should be expressed in terms of experimentally determined quantities. Ray et a!

(1988) proposed a formalism to derive from (9),

A~.,+ fl A~,= [1 +y] Ab/E+[fl+(S] A'~/E+(flkpm/a)Ab, (12) where m = (~3 In kp/c9 In a) at constant temperature and deformation structure, i.e. the isostructural strain rate sensitivity. Analysis of the experimental data using (12) pertains to the idealization that the experimentally determined load rate P for the higher extension rate equals the load rate at the ideal transition point that would be obtained by extrapolating the two constant .~ segments in the transition regime, but anelasticity effects are present such that ~, at the ideal transition point, where the two extrapolated constant 5¢ segments intersect, is discontinuous. Least square fits of the experimental data subject to the condition Cu >0, with temperature dependent E values of AI (Sutton 1953), and m values reported by Ray (1984), were carried out to compute the optimal values of the anelasticity parameters. The details of the

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Deformation dynamics at low and ambient temperatures 143 computation technique, as well as the results can be found in Ray et al (1988). A careful perusal of the results, despite the scatter in the experimental data, and the experimental uncertainty associated with the value of rn used, led to the conclusion that [:t is of the order of 0' 1 to 1 s while y is a small positive number; also the results showed insensitivity to ,5 values. Nir et al (1976) examined anelasticity in both mono- and polycrystals of Ai at 298 K in the absence of plasticity by instantaneous change of stress between two fixed levels; their results show that grain boundary anelasticity obeys a linear relation, with fl=0-09+0.01 s and 7=0.25. The results shown in figures 3 and 4 point to a strong anelasticity effect in A1, but a weaker anelasticity effect in stainless steel. Comparison of/3 values calculated here with those reported by Nir et al (1976) suggests grain boundary as the important source of this anelasticity. The strength of grain boundary anelasticity increases linearly with increasing stacking fault energy (Cordea and Spretnak 1966); this factor, coupled with the possibility that in SS 316, grain boundary locking due to solute segregation might effectively reduce the anelasticity from this source could explain the experimental observations. Also, high/3 values reported for other anelasticity mechanisms by Nir et al (1976), and by Gibeling and Nix (1981) suggest that dislocation back flow processes or flexing of dislocation sub-boundaries are not responsible in the present instance. It is interesting to note that according to Gibeling and Nix, anelastic strains from two different sources may not be additive. As Nir et al (1976) or Gibeling and Nix (1981) did not employ tests with non-zero b, 6 values for these mechanisms are not known and we are unable to theoretically predict the possible role of anelasticity in terms of (10).

The success of (12) in interpreting the data raises an important question: whether the observed anelasticity effect is only an apparent one and whether the successful description is solely due to an empirical (and reasonable) modelling for the variation of kp during the transition period through the flA ~p term. Intuitive support in favour of this argument is provided by (12): higher the magnitude of m, higher is the magnitude of A ~p. And, as noted above, the apparent effect of anelasticity on compliance variation in the present instance was predominantly through the term (13mep/o) A a.

Now, for AI, over the test temperatures, m values ranged between 60 and 100 whereas for SS at 300 K, m < 0 (due to dynamic strain ageing, henceforth called DSA), m = 24 at 973 K and m = 10 at 1123 K (Ray 1984). It would then seem quite likely that the apparent anelasticity effects would be negligible for SS 316 for the testing temperature chosen, but important for the present AI data. If this interpretation is valid, no physical significance is to be attached to the calculated values of the anelastic constants; in particular the reasonable agreement of the calculated fl values with those reported in the literature could be fortuitous. Ray and Rodriguez (1987) used a simple analytical model for the increase of ~', with stress during the transient period, assuming constancy of deformation structure, and ignoring anelasticity effects;

equation (8) was used for verifying the internal consistency of the computed results.

This model was partially successful in the sense that the computed CT values showed the expected deformation dependence, but some of the data points were beyond the statistically expected scatter band. Thus, the tentative results do support the contention that the anomalous variation of Cr with deformation is due to the increase of ;:p during the transition period rather than anelasticity effects, but a more rigorous analysis (using computer simulation), as well as experiments specifically aimed at evaluating 6, are necessary before an unequivocal answer can be reached.

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3. Thermally-activated deformation in hard hcp metals at low temperatures 3.1 Introduction

Extensive investigations have been carried out in the last two decades on the rate- controlling deformation mechanism of metals and solid solution alloys at low temperatures. Rodriguez (1976) summarized the findings, and also provided an extensive bibliography. It is now generally agreed that the low temperature deformation in fcc metals is controlled by intersection of dislocations. Also, the temperature dependence of yield and flow stresses is generally found to be altered by both substitutional and interstitial solutes; interaction of dislocations with randomly distributed solute atoms is considered to be rate-controlling in such situations. In spite of the large volume of experimental data available, there has been no consensus on the rate-controlling deformation mechanism in bcc metals and their solid solu- tions. Essentially three rate-controlling mechanisms have been discussed in the literature: (a) inherent lattice or Peierls-Nabarro hardening; (b) dislocation impurity interaction and (c) recombination of screw dislocations that are dissociated off the glide plane. It is generally agreed that for soft hcp metals like magnesium, cadmium and zinc (with higher-than-ideal c/a ratios) which predominantly slip on the basal system, interaction of glide and forest dislocations is the rate-controlling process (Risebrough and Teghtsoonian 1967; Sastry et al 1969, 1970). In the case of hard hcp metals (with lower-than-ideal c/a ratios) like zirconium, titanium and hafnium which deform mostly by prism slip, there is a controversy about the rate- controlling mechanism (s). These metals can take a large amount of interstitial elements like oxygen and nitrogen in solution; and the temperature dependence and strain rate sensitivity of yield and flow stresses of these metals are found to increase with increasing interstitial solute concentrations (Rodriguez and Arunachalam 1968;

Tyson 1968). Several authors (Conrad 1967; Rodriguez and Arunachalam 1968;

Tyson 1968; Soo and Higgins 1968a, b; Dasgupta and Arunachalam 1968; Conrad and Jones 1970; Tiwari et al 1972; Okazaki and Conrad 1972, 1973; Conrad et al 1973; Moskalenko and Putsova 1974) believe that this thermal hardening arises from dislocation-interstitial solute atom interactions. Another viewpoint (Levine 1966;

Sastry et al 1971; Sastry and Vasu 1972) is that the Peierls-Nabarro barrier is the rate-controlling obstacle, which is increased by interstitial additions.

One controversial aspect of low temperature deformation of hard hcp metals is the variation of the experimentally determined activation energy with temperature at constant strain rate. Some researchers (Conrad 1967; Conrad and Jones 1970; Sastry et a11971; Sastry and Vasu 1972; Okazaki and Conrad 1972, 1973; Conrad et a11973) have been able to obtain a linear plot between A H, O or Q* and T(cf equation (1)) in the range between 0 and 600 K; on the other hand, manyothers (Levine 1966; Soo and Higgins 1968a, b; Dasgupta and Arunachalam 1968; Samuel and Rodriguez 1973;

Moskalenko and Putsova 1974) have detected breaks or deviations from linearity in the activation enthalpy vs temperature plots. Various possibilities for such a deviation like "two stages of thermal activation (Reed-Hill 1967; Samuel and Rodriguez 1973), variation in the pre-exponential term with stress and temperature (Reed-Hill 1967; Samuel and Rodriguez 1973) and occurrence of dynamic strain ageing (Reed-Hill 1967; Samuel and Rodriguez 1973; Moskalenko and Putsova 1974) have been suggested and examined.

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Deformation dynamics at low and ambient temperatures 145 As mentioned in § 1.2, it is imperative to examine the different possibilities, namely k0 may or may not be constant, and A S may or may not be zero, when establishing the rate-controlling deformation mechanism (s). However, these refinements in A G computation did not remove the break in the A G versus T plots for Zr (Rodriguez 1976), Zircaloy-2 (Ray et al 1977) and a Zr-l.6 wt% Cu alloy (Rodriguez and Ray 1978). In the next section, we discuss the application of a rather novel experimental technique (Ray et al 1977; Rodriguez 1978), coupled with careful analysis of the data, for elucidating the deformation mechanism of a hard hcp metals, namely Ti.

3.2 The experimental technique

A novel experimental technique, devised by Rodriguez (1978) was adopted for this study. This technique, called the temperature cycling-cum-stress relaxation (TS) technique, is designed to combine the benefits of both temperature cycling and stress relaxation, to enable determination of both Q* and A* at a constant temperature and deformation structure. Stress relaxation technique, introduced in §2.2, has emerged as one of the most potent tools of deformation dynamics study; several reviews of this technique are now available, see e.g. Rodriguez (1979)and Ray (1984).

It may be recalled that this technique involves arresting the crosshead motion during deformation; as the stress is sufficiently high, plastic strain continues, but as the total strain is held constant, elastic strain (and thus stress) continually decreases; thus, plastic strain accumulates at continually decreasing rates, and the total plastic strain accumulated during stress relaxation is very small. Now, if plastic strain increment is an adequate measure of change of structure (this is likely to be true at low temperatures, so that a simultaneous thermal softening of the structure does not occur) and in the absence of dynamic strain ageing, then in effect it can be assumed that the entire relaxatior~ event occurs at a constant deformation structure. This technique then makes available the relation between kp and ~ at the test temperature, at constant deformation structure, over large ranges of these variables. Provided a r is constant during the stress relaxation period, a* approaches zero asymptotically, requiring, in principle, infinite time for its determination. Rodriguez (1968a) devised a method called decremental unloading to circumvent this problem; in this technique, instead of continually following relaxation till a* approaches zero, the specimen is intermittently unloaded by small extents, and the subsequent relaxation studied to see if a* has approached acceptably small values. Thus, provided deformation structure is constant during the entire period of relaxation, this technique can also yield the values of a* and a~; note, however, that application of decrementai unloading method also requires absence of thermal softening and dynamic strain ageing during stress relaxation (Rodriguez 1968b, 1979).

The TS testing technique involves the following steps (figure 5): A specimen is first deformed at a nominal strain rate of k to a convenient strain in the macroscopic flow region at a high temperature T. The crosshead movement is then stopped and stress relaxation is allowed to occur for about 1000 s following which tr* and a t are determined by decremental unloading method. Immediately thereafter, the specimen is brought to a lower temperature T 1 by suitably changing the bath surrounding the specimen cage. Once the specimen, the cage and the entire grip assembly have attained the test temperature equilibrium, the specimen is again strained at the nominal strain rate ~:, but the crosshead movement is arrested just before macro-

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. , 0 0 0 . c

A

ua/ ~-, zx~p I \

/

S T R A I N E OR TIME t

Figure 5. Schematic illustration of combined stress relaxation and temperature cycling (TS) test procedure (after Rodriguez 1978).

scopic flow occurs. The point of arrest of crosshead has to be carefully chosen so that two opposing conditions are optimized: (i) the plastic strain rate attained by the specimen has increased (from zero at the start of the pulling) to a large value sufficiently close to k, and (ii) additional plastic strain suffered by the specimen due to this straining is as small as possible, and negligible compared to the original plastic strain accumulated at temperature T. Stress relaxation is again followed for about 1000 s, and a* and au at this temperature evaluated. The temperature is now lowered to a new temperature T2 and the whole sequence of operation repeated at this temperature. The sequence of steps is repeated at progressively lower temperatures T3, T4,.. etc.

3.3 Results of TS test on Ti

The results for a commercial purity Ti ( O + N + C 1975 ppm and H 68 ppm by wt) in annealed condition (mean linear intercept grain diameter 0.062 mm) are given below;

the details of the material preparation, testing technique etc are indicated in Rodriguez (1976). The nominal strain rate chosen for the tests was 6-6 × 10 -4 s-1; the TS tests was started at ep value of 0-0488 at 415 K, and carried out for several temperatures down to 77 K. The total increase of ep during the TS test campaign was 0.0042. The stress relaxation data were analysed to obtain kp- a relations at each test temperature; the results are shown in figure 6; between 196 and 350 K, log kp varies linearly with a while at lower as well as higher temperatures, the variation shows negative curvature. Data from these plots were used for TASRA by obtaining (a) (~ga/OT)~, and (~ log "ep/Oa)r , from which the activation parameters were calculated (cf equations (3}-(6)). Figure 7 shows the stress vs temperature plots for three different strain rates derived from these experiments; the dashed line in the figure represents the expected change in tr u for # = C66. a* = a - a t values calculated are shown in figure 8 for the three strain rates. The apparent activation areas evaluated from these experiments are plotted against stress in figure 9. It is apparent that A*

does not very monotonically with stress (or temperature): a hump in A* is noticed between 196 K and 300 K, and a plateau between 300 K and 400K, which is followed by an abrupt rise in A* with temperature above 400 K.

Q, AH, AG and ko values were evaluated from these data for three strain rates, 2× 10 -4, 2× 10 -5 and 2x 10 -6 s -1 using equations (4)-(6). As there is an uncertainty about the appropriate shear modulus to be used (Ray et al 1977; Rodriguez

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Deformation dynamics at low and ambient temperatures 147

T

,~J,I o.

lo-?

~ T i (~p,s-' l

O 2 x 10 -4 / x 2 x 10 -s / A 2 x 10 -6 ]

)}

t523K

Ti

!

/

36BK

333K 300K

i0 -!

.~d ~73K

1 0 - 7 J l I 1

1,5 /,9 53

Tl

7 7 K

lt, l I Itll

,~ ~o 2's 3'0 is /,~ /,'5 ,o "'6o 6'3 6's 679o,2

0-, kg/mm 2 0", kg/mm 2

(a) (b)

Figure 6. Log~p vs ~r plots computed from TS experiments on Ti (after Rodriguez 1976).

9 0

~E 8°

ETo

W 3O

10

100. 200 300 400 500 TEMPERATURE,K

Figure 7. Stress for indicated {.p values and a u in Ti from TS experiments; dashed line indicates au(TSI523x(#r/,asz31; (after Rodriguez 19761.

and Ray 1978) several monocrystalline elastic moduli, as well as polycrystalline moduli computed from these by suitable averaging procedures, have been used. The results are shown in figure 10. AG results are depicted for the two moduli having the highest and lowest temperature dependencies (C44 and C66 respectively); the results for the other moduli were between the vah~es for these two moduli. These results are

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(a) GO

e,,q

E E SO

c~ 40 .1I

e..

u1 30 u~

I.Q e,~ 2c

i r a ,

10 ILl - - 0

t,-.,

I,d 2 0 U.

la.

I./.I 10

Ep

® 2 x 1()'4se¢ °t x 2 x 10"Ssec "t

- ¢_1

(b)

1 ¢8 P Rodriguez and S K Ray

o o

T E M P E R A T U R E , K

Figure g. Effective stress o* in Ti from TS experiments, a. derived using o* = ~r- tru (TS)sz3 x 0£r//zs23~). b. derived using tr* : a - tr~ (TS); (after Rodriguez 1976).

411

~ E 30

2C

10

Ti TEHP, K

® 77

& 19G V 273

• 298

" 333 e 369 x 523

I I 1 I I

20 &O 60 80 100

STRESS, k g / r a m 2

Figure 9. Activation areas evaluated from results depicted in figure 6 plotted against stress (after Rodriguez 1976).

similar to those obtained for Ti using more conventional techniques (Rodriguez 1976).

The results in figure 10 show that the variation of activation energy (for either type of obstacle, with or without stress-dependent pre-exponential factor) shows a non- monotonic variation with temperature; as noted by several authors (loc cit). Such a non-monotonic variation cannot be the result of a single rate-controlling mechanism.

Moreover, if it be considered that the activation energy results point to two different

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Deformation dynamics at low and ambient temperatures 149 1.5 (o)~p:2x10 "4s -I ;

® Q / / I

• ~ 6 ~ c , , ) / ~ . ~ j - ~ r j /

! 2 . , o

~ 0 . 5

r,

w o

cZZ 1.5 Ic] ~p =2xlO- s ~

~1-0

~_~0.5

0 0 100 200 300 /,00 500 600

TEMPERATURE,K

Figure 10. Activation energy vs temperature for Ti from TS experiments (after Rodriguez 1976),

rate-controlling mechanisms, one below about 300 K, and the other above about 400 K, then we should expect two segments in the activation energy-temperature plots for these two temperature regimes, both extrapolating to zero activation energy at 0 K. Clearly, the lower temperature segments of these plots do extrapolate to zero activation energy at 0 K, but the higher temperature segments do not. Therefore, we can rule out the possibility (Samuel and Rodrigucz 1973: Reed-Hill 1967) of a change in rate-controlling deformation mechanism at about 3()0 K.

We may note here that results qualitatively similar to those reported above have been obtained by applying TS technique to Zr (Rodriguez 1976) and Zr-l.6 Cu alloy (Rodriguez and Ray 1978): moreover, these results are similar to those determined by using conventional techniques to Ti, Zr, Zr-l.6 Cu and zircaloy-2 (loc oil). These results strongly support the view point that the anomalie,~ in the TASRA in Ti are due to dynamic strain ageing. Extcnsivc studies of dynamic strain ageing (DSA)due to interstitial solutes in Zr and Ti have been carried out by Reed-Hill and coworkers (Ramachandran and Reed-Hill 1970: Santhanam et al 1970; Santhanam and Reed- hill 1971: Garde ct a/ 1972: Garde ct a/ 1975). In both Zr and Ti, they have established the occurrence of two regions of dynamic strain ageing: the first one around 300 K (for strain rates of the order of 10 5 s t thc actual temperature region observed by them is - 106 to 400 K) was attributed to hydrogen, and the second one in the region 450 to 800 K was attributed to the combined effects of oxygen, nitrogen and carbon. The anomalies in the TASRA results shown in figure 10 coincide with the DSA due to hydrogen. The various manifestations of DSA like serrated yielding, anomalous variations of work-hardening as well as TASRA parameters, minima in

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tensile ductility, etc have recently been reviewed by Rodriguez (1984); all these anomalies have been observed in different types of tests on Ti, Zr a~ well as Zr-l.6 Cu (Rodriguez 1976), corroborating the conclusion that the anomaly in TASRA results observed in the range 200 to 450 K is due to DSA, believed to be due to hydrogen, as suggested by Reed-Hill and coworkers (Ramachandran and Reed-Hill 1970; Santha o nam et al 1970; Garde et al 1972, 1975; Santhanam and Reed-Hill 1971). We may note here that from both TS as well as conventional tests, the extent of anomalous behaviour in TASRA has been found to be stronger in Ti than in Zr, consistent with the observation (Garde et al 1975) that the DSA is much weaker in Zr than in Ti.

For the sake of completeness, we indicate how incidence of DSA would affect TASRA results. It is now well recognized (Brindley and Worthington 1970; Baird 1971) that when DSA occurs there is an additional contribution aj to the flow stress (as shown in figure I l (a)) from one or more of the following sources (cf Rodriguez 1984); (i) from the drag force exerted by the solutes on the dislocations; (ii) from the increased work-hardening due to greater dislocation multiplication and (iii) due to the reduction in mobile dislocation density. For a given strain rate, aj manifests itself only in a limited temperature range between Tt and T 2, and is maximum at an intermediate temperature Tp, at which the velocities of the strain ageing solutes and dislocations match; this temperature range gets shifted to higher temperatures at higher strain rates. Irrespective of the magnitudes of the different contributions to aj, it is evident that the experimentally determined magnitudes of ((3cr/?T)would be greater at temperatures above Tp, and smaller at temperatures below Tp, than the

b

IEMPERAIURE

(a)

b

s-_1

, , , 2 ",,,1 ffid kocJ ~p

(b)

Figure I I. a. Schematic illustration of the influence of dynamic strain ageing on the flow stress vs temperature varxation; full curve is the variation in the absence of dynamic strain ageing and the dotted curves indicate the contribution from strain ageing (after Rodriguez 1984). b. Schematic illustration showing the influence of dynamic slrain ageing on stress- strain rate variation (1) for the case of strong strain ageing and (2) for mild strain ageing (after Rodriguez 1984).

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Deformation dynamics at low and ambient temperatures 151 corresponding magnitudes in the absence of DSA. The stress-strain rate variation would also be altered by DSA in an analogous way as shown in figure 11 (b). Here, assuming constancy of structure and mobile dislocation density, only the contribution from the solute drag force is considered. The full curve shows the expected variation in a with log kp for a thermally activated process; the curve has a positive curvature consistent with the requirement that A* should decrease monotonically with stress. For a very strong interaction giving large solute drag stresses, a becomes a multivalued function of kp leading to a negative strain rate sensitivity and repeated yielding (Nabarro 1967). A mild interaction on the other hand can lead to a linear log~p vs a plot, as seen for temperatures between 196 K and 350 K in figure 6 ( a - b ) . Outside this temperature range, these plots show negative curvatures as expected. The linear variation of log ~,p with stress leads to a constant value for Aa*

and contributes to anomalies in calculated AH and AG values.

4. Kinetics for structure change

We have so far paid attention to the constitutive relation for deformation at constant deformation structure. In general, structure evolves with deformation, and as indicated earlier, different types of tests are carried out to quantify this change. Also, to the extent that plastic deformation is responsible for the observed mechanical behaviour under different modes of testing, it should be possible to inter-relate data obtained by these tests. Recently there has been a significant progress in deformation kinetic modelling (Kocks 1976, 1987; Miller 1976; Lagneborg 1978; Rohde and Swearengen 1982; Estrin and Mecking 1984; Prinz and Argon 1984; Gottstein and Argon 1984: Nix et al 1985; Nabarro 1986; Klepaczko and Chiem 1986; Kocks 1987).

We here touch upon this aspect.

There are basically two different ways of tackling the problem; neither of these two approaches tries to solve the problem completely but attempts at obtaining adequate description of deformation response. One of these approaches is best exemplified by the work of Estrin and Mecking (1984). Suppose, for the deformation behaviour to be studied, total dislocation density p, provides adequate description;

this is expected to be the case for low temperature deformation, and in the quasi- steady state, i.e. after the initial sharp transients associated with change of control parameter (e.g. a for creep) for the test has died down. Now, there is enough evidence to suggest that for such deformation aoz#b x/P," Then, we need to model the evolution of p, in different testing conditions, and compare the predicted behaviour with experimental results. Estrin and Mecking (1984) have used their formulation for comparing tensile work-hardening and primary creep behaviours of copper. For different regimes of deformation and different temperatures and strain rates other microstructural information like mobile dislocation density, grain size, solute concentration, precipitation, etc. will have to be quantitatively incorporated in the description. Several such models are discussed in the papers cited above.

The other approach (see Ray 1984 for details) attempts at interrelating test data from different modes of testing in a self-consistent manner without specifying, a priori, the relevant microstructural information. Thus, it can be shown that irrespective of the specific variable, one such parameter describes the structure adequately if the relation between tensile work-hardening and stress at a given strain rate and temperature is unique; this consistency criterion can be used to determine if indeed

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one parameter, for example p~ in the Estrin-Mecking formulation, does describe lhe deformation structure, without any modelling for the evolution of Pt during deformation. It has been shown by adopting this technique that the single parameter description is inadequate at low strain rates and/or high tempcratures. For the regime of deformation over which single parameter description is wflid, it becomes possible to inter-relate test data from differentmodes of testing by using additional assumptions which can be rigorously verified by appropriately designed experiments.

Ray (1984) for example used this technique to speculate upon the expected creep and tensile behaviour from the observed stress relaxation behaviour. The methodology for this approach is thus complementary to the first approach, in that instead of theoretically modelling for the evolution of structure parameter, experimental information from one mode of testing is used to predict the bchaviour in other modes.

The main attraction of this approach is that the assumptions made in each step of its development are experimentally verifiable. Development of this approach to in- corporate larger number of parameters has yet, at least to the authors" knowledge, not been attempted.

5. Conclusions

In this paper, a critical assessment of the theoretical and phenomenological formalisms available for studying detbrmation dynamics has been made. It is con- cluded that a state variable approach to deformation is valid, with a, [;p, 7" and deformation structure being the variables necessary and sufficient to formulate the constitutive relation, provided there is no anelasticity or strain ageing. The experimental measurements and the evaluation of various deformation parameters from these data, however, involves so many assumptions, that the investigators should ask the question "Do we measure what we think we measure'?" Careful execution of experiments and systematic interpretation of results therefore assume a special importance. We have given examples of how a systematic theoretical investigation along with carefully executed experiments are necessary to rule out the possibility that anelasticity is responsible for the apparent applicability of TID theory in some fcc metals, and how a number of experimental techniques could be combined in a novel way to establish that the anomaly in TASRA results of Ti is indeed due to dynamic strain ageing and not due to some of the assumptions made in calculating activation energies. While significant progress has been made in understanding deformation kinetics at constant structure, the quantitative analysis for the inter- relation of mechanical test data and the evolutionary structural parameterls) is far from complete. Two typical approaches to this problem are discussed.

References

Abe K, Yoshlnaga l | aild Morozunli S 1976 7)'an~. Jim. lnsl. Met. 4(I 393 Abe K, Yoshinaaa [! and Morozumi S 1977 Tr,n.~. Jim. Inst. Met. 18 479 Alden T H 1969 .,h ta Metall 17 1435

Alden T H 1972 Phdo~s. ~1a,q. 25 785 Alden T H 1977 Met. 7)'an~. A8 1675

A l d e n T tt I t)~2 m ,~h'chantcal towing fi)r d~/brmation model development, (eds) R W R o h d e and J ("

S~.', carCllgCll ..\S1M STP 765 ( Philadelphia: A S I M ) p. 29