Deformation modeling of aluminum: Stress relaxation, transient behavior, and search for microstructural correlations

Acta Metallurgica Vol. 29, pp. 41 Io 52

0001-6160 81 0101-0000502 00 0

¢) P~rgamon Press Lid 1981 Printed in Great Britain

DEFORMATION MODELING OF ALUMINUM: STRESS RELAXATION, TRANSIENT BEHAVIOR, AND SEARCH FOR MICROSTRUCTURAL CORRELATIONS'~ R. W. ROHDE and W. B. J O N E S Sandia National Laboratories,++ Albuquerque, NM 87185

and J. C. S W E A R E N G E N Sandia National Laboratories, Livermore, CA 94550, U.S.A.

(Received 17 March 1980; in revisedform 17 July 1980) Abstract--The inelastic behavior of high purity aluminum was measured by stress relaxation techniques following three mechanical histories: monotonic tensile loading, reversed cyclic loading, and elastic reloadings subsequent to an initial relaxation event. The data are used to test the validity of a model which had found previous success in describing simple stress relaxation of aluminum, stainless steels and solder alloys. The model is found to apply, in this work, to monotonic histories for strains greater than 0.6% and for the cyclic histories where total accumulated strains are greater than 0.5°~. However, the model failed to describe relaxation after either small monotonic straining (<0.10?o) or after elastic reloadings subsequent to an initial relaxation. Transmission electron microscopy of specimens supposedly in different material "states" failed to show any difference in dislocation sub-grain size or subboundary character. It is concluded that the model under discussion does not represent a complete deformation law for aluminum, and further, that searches for correlation between deformation law state variables and microstructural elements may be futile. General considerations for deformation law development are discussed in light of the experimental results. R~,wm6--Nous aeons d6termin6 le comportement in61astique de l'aluminium de haute purete par des techniques de relaxation de la contrainte, apr~s trois histoires m~caniques diff~rentes: traction monotone, charge cyclique attern/~e et remises en charge 61astique apr~s une relaxation. Nous utilisons ces r6sultats pour tester la validit~ d'un mod/~le qui a eu ant6rieurement un certain succ/~s pour d6crire une relaxation de contrainte simple dans l'aluminium, dans des aciers inoxydables et dans des alliages de soudure. Ce mod61e s'applique A notre cas pour une d6formation monotone sup~rieure fi 0,6°o et pour une d6formation cyclique, lorsque les d~formations totales cumul~'~s sont sup6rieures A 0,5 ,°~. Par contre, c¢ mod/:le 6choue pour une relaxation apr6s une petite d6formation monotone (<0,10°o) ou pour une remise en charge 61astique apr6s une premi6re relaxation. L'observation par microscopie ~lectronique en transmission d'6chantillons dont on pouvait penser qu'ils 6talent dans diff6rents "Stats" n'a pas permis de mettre en 6vidence une quelconque diff6rence dans la taille des sous-grains ou dans le caractere des sous-joints. Nous en d6duisons que le mod/~le en question ne repr~sente pas compl~tement la d6formation de raluminium eL qui plus est, que la recherche de corr61ations entre les variables d'6tat de la loi de la d6formation et les 616ments de la microstructure est peut-~tre vaine. Nous d~veloppons ensuite des consid6rations g~n6rales sur la loi de la deformation,/l la lumi6re des r6sultats exp~rimentaux. Z a ~ m m e n f a u m g - - D a s inelastische Verhalten hochreinen Aluminiums wurde mit Spannungsrelaxationstvchniken nach folgenden drei verschiedenen mechanischen Vorbehandlungen untersucht: Monotone Zugbelastung` Druck-Zug-Belastung und elastische Wiederbelastung nach einer unmittelbar vorausgegangenen Relaxation. Die Ergebnisse werden zur Priifung eines Modelles herangezogen, welches friiher bci der Beschreibung der einfachen Spannungsrelaxation yon Aluminium, nichtrostendem Stahl und L~tzinn Legierungen erfolgreich war. Das Modell kann bei der jetzigen Untersuchung angewendet werden auf monotone Vorbehandlungen bei Dehnungen gr~Ber als 0,6% und auf zyklische Vorbehandlung bei einer akkumulierten Dehnung yon gr6Ber 0,5%. Die Relaxation nach kleiner monotoner Dehnung (<0,10%) oder nach elastischen Wiederbelastungen, die auf eine urspriingliche Relaxation folgten, konnte mit dcm Model] jedoch nicht beschrieben werden. Elektronenmikroskopische Durchstrahlung von Proben mit verschiedenen Behandlungen ergaben keine Unterschiede in der SubkorngriSBe oder im Charakter der Subkorngrenzen. Es wird gefolgert, dab das betrachtete Modell keine vollst~ndige Beschreibung der Verformung in Aluminium darstellt, und weiterhin, dab die Suche nach tier Korrelation zwischen Zustandsvariablen des Verformungszustandes und mikrostrukturellen Elementen miSglicherweise vergeblich ist. Allgemeine Betrachtungen ftir die Entwicklung yon Verformungsgrundgesetzen werden anhand des experimentellen Materials diskutiert. T This work was supported by the U.S. Department of Energy (DOE), under Contract DE-ACO4-76-DP00789. ++A U.S. Department of Energy Facility. 41

42

ROHDEetaL:

DEFORMATION MODELING OF ALUMINUM

INTRODUCTION Economic, reliable and safe design of structural components for chemical processing and energy conversion systems places severe demands upon materials and procedures used to calculate and predict material performance. The complex load and thermal histories experienced in many structures make design prediction of the inelastic response of alloys a particularly difficult task. As a matter of practicality, laboratory experiments cannot be conducted for times representing lifetimes of real structures. Furthermore, all possible synergistic interactions which may be induced by various deformation and thermal histories cannot be duplicated in laboratory experiments. Thus, designers must predict long time behavior and synergistic interactions on the basis of limited, short term data. The increasing need to predict inelastic response induced by complex thermomechanical histories has given renewed impetus to the development of deformation models. An overview of work in this area is given in several recent compendia [1-3]. It is apparent that deformation models must meet at least two criteria in order to be most useful. The first arises from the fact that prediction of long term behavior by application of short-time data consists of extrapolation; and reliable, accurate extrapolation requires that deformation mechanisms be known over the regions of projection. Thus, models should have some physical basis so that areas or 'maps' of applicability are definable. 'Physical basis' refers, in this work, to the micromechanical level commonly associated with metallurgical microstructures. Second, since it is impossible to catalogue and follow all possible thermomechanical histories, models should be constructed to allow calculations of subsequent response based only on measurements of the current material state. A number of models attempting to meet these two requirements[4-13] have been proposed. In these models, inelastic behavior is functionally dependent upon the current microstructure; the function relating microstructure and mechanical properties contains one or more variables which attempt to quantitify microstructural 'state'. While precise description of a microstructure will require multiple state variables, in practice, most models contain only one. Moreover, that single state variable is often a direct function of the current flow stress. This work examines one model, a power law, for self-consistency and for ability to predict experimental data. The model examined here was originally proposed by Kocks [9] who demonstrated that it adecluately described some of the stress relaxation measurements of aluminum reported by Hart and Solomon [14]. Swearengen and Rohde [13-1 modified the model by introducing a back stress so that kinematic (i.e. Baushinger effects) as well as isotropic hardening could be treated. The same modifications also permitted description of stress relaxation behavior over a large range of homologous tempera-

tures[13]. This power law model was selected for examination because of its past successes, and, most important, because it embodies the features of a large class of models which treat inelastic strain rate in terms of competing effects of work-hardening and microstructural recovery. For a review of this class of deformation see Krieg [15]. As with all models, the model to be discussed is constantly evolving. Thus, Kocks recently proposed an alternative deformation law where strain softening plays a more dominant role[16]. The model discussed in this work, however, is easily applied, has reasonable past successes, and conveniently serves as a guide to requirements necessary to the success of the generic class of work-hardening/microstructural recovery models. Thus the conclusions of this work more generally reflect upon a large class of workhardening-recovery deformation models, and point out needs for further analytic development. Full development of the model is found elsewhere [131. Since this work is concerned with stress relaxation, only that special form of the model will be presented. All inelastic deformation is assumed to be rate-dependent, resulting from the stress-assisted, thermally-activated motion of dislocations through obstacles which become stronger or more numerous from work-hardening and weaker or fewer from recovery. For stress relaxation under uniaxial stress, the model becomes:

-]-=

(I-A) L

~r~ _1

+ AF(°" - a')']I" L al J)

(I)

with A=

SOoC (1 -

n/m)(1

-

al S00) 2 ~(1l/n*l/")'

(2)

The stress and plastic strain rate at the beginning of relaxation are specified by (71 and ~1 respectively, tr~ is the internal stress or 'back stress' which for consistency with assumptions in the model development must vary only with temperature. The internal stress may be assigned a value of zero for intermediate and high homologous temperatures (T > 0.3T~). 13 S is the combined compliance of the specimen and machine [ = l / E + l/M], where E is Young's modulus and M is the machine modulus. The material constants, given by 0o, C, m and n, must be independent of deformation history and microstructural state for the assumptions of the model to be met. 0o is the O K strain hardening coefficient; C is the recovery coefficient. Strain rate sensitivity under isostructural conditions is given by the parameter m. Likewise, n is the recovery rate exponent. When m >> n, the value of the constant n should be approximately that of the steady state creep exponent. Both m and n are expected to be temperature dependent and the microstructural state is defined by the internal

ROHDE et al.: DEFORMATION MODELING OF ALUMINUM stress cq and the aggregate state variable A. It should be noted that the microstructural elements undergoing evolution described by this model have to do with changes in the density and strength of obstacles to glide, not with changes in the glide elements themselves (i.e. dislocations). Then, by analogy to the Orowan equation for plastic strain rate, an assumption that the density of mobile dislocations remains constant is implicit in the model. As previously noted, the stress relaxation behaviors of stainless steels [17], aluminum [t8], and various solder alloys[13, 19] may be described quite accurately by the model. Furthermore, experiments which measured inelastic behavior after various deformation histories showed, in aluminum and stainless steels, that the constants m and n were independent of deformation history, as required for self-consistency of the model. No deformation history experiments were conducted in the solder alloys. Creep measurements performed on aluminum over a limited range of temperature and stress yielded an exponent for creep rate, n', which was the same as the recovery exponent n found from short time stress relaxation measurements[18]. This correspondence between n and n' satisfied one other requisite for self-consistency of the model. In the investigations of aluminum [18] sufficient data were obtained to allow prediction of the material response to multiple, consecutive stress relaxation events, wherein each subsequent stress relaxation was started at a lower stress than the preceeding one. The predictions of the model[18], agreed qualitatively with measurements of Hart and Solomon [14] on aluminum although they interpret their observations on the basis of a radically different model. While the cited observations are encouraging, the experiments were insufficient to demonstrate that the model represented a complete deformation law for the material. By 'complete' we mean applicable to general deformation paths in stress space. A particular problem lies in the evaluation of the parameter A. Equation (2) requires, at a constant initial plastic strain rate, th~it A be linearly proportional to the initial stress. Because of considerable scatter, previous data for aluminum stress relaxation could not demonstrate such correlation between stress and A [18]. The present work was stimulated by the desire to test the model more fully. High purity aluminum was selected as the experimental material because of the large amount of data available in the literature. Three goals were established. First, if possible, to reduce scatter in determining the value of A. This improvement was attempted by conducting relaxation experiments for times longer than in the previous work [18] (30 h rather than 30 rain), and by refining the numerical data analyses techniques. The second goal was to determine quantitatively the ability of the model to predict and describe multiple stress relaxation events, where subsequent relaxations are initiated following elastic reloadings. Thus, the 'reloading' experiments of

43

Hart and Solomon [14] were repeated, but on a material whose model parameters were to be determined in a complimentary set of experiments. The third experimental goal was to attempt correlation of microstructures, observed by means of transmission electron microscopy, with various "states' as indicated by the composite internal state variable A. The more severe tests described above and reported in this work highlight some of the deficiencies of the model. Scatter in the variable A was reduced but not eliminated. While parameters could be chosen to provide an excellent fit of the model to the relaxation data, the value of the recovery exponent, n, thus chosen, no longer equals the value of the creep exponent. Furthermore, the model does not predict multiple stress relaxation events quantitatively. The microscopic observations revealed no correlation of subgrain size or low angle boundary character with either the aggregate internal state variable, A, or with the stress relaxation behavior.

EXPERIMENTS Material

Specimens were machined from 16ram thick 99.99% pure aluminum cold rolled plate, Table 1 gives the chemical analysis. Specimens were oriented with the loading axis parallel to the final plate roiling direction. Gage diameter was 6.4mm, gage length was 25.4mm Prior to testing, the material was annealed 0.5 h at 573 K and furnace cooled, The average grain diameter was 0.25 ram. Tensile testing

Stress relaxation was measured by utilizing a servocontrolled electro-hydraulic machine modified for improved strain control stability. Relaxation events were conducted: (i) after simple monotonic tensile loading to some plastic strain, (ii) after ten complete cycles of reversed plastic strain and, (iii) after a small elastic strain increment (reloading) was applied subsequent to a previous relaxation event. A triangular control function was used for reversed cyclic strain control. Loads were measured to 0.2 N; strains were measured + 3 #m/m and controlled to +8 #m/m. Experiments were continued at least 30 h. Specimen temperature was maintained constant to 295 + 0.05 K by immersing them in a temperature controlled silicone oil bath. Data reduction

The data for strain and load as a function of time were digitized and stored for subsequent analysis. At short times when the relaxation rate was high, data were collected at 0.2s intervals. Longer intervals between collections (10 s) were used when the relaxation rate slowed. Strain rates during relaxation were obtained from the derivatives of curves which were piecewise fit using a second order polynomial function to the raw load-time relaxation measurements. The parameters m, n and A were obtained from data

44

ROHDE et al.: DEFORMATION MODELING OF ALUMINUM Table 1. Composition of high purity aluminum by emission spectrography Element Detected

Amount ppm

Mn Fe Mg Si Cu Ga

6 10 2 20 4 20

B

6

The stress relaxation data from samples strained monotonically to plastic strains of greater than 0.6~. or of samples cyclically deformed to cumulative total strains (added irrespective of sign) greater than 0.5~o could be described with identical values of m(= 100) and n(= 12) in equation (1). As expected A was a sample-to-sample variable. Typical results for samples deformed to these larger strains both cyclically and monotonically are shown in Fig. 1 along with curves showing the fits of equation 1 to the data. Measurements of two 'reload' stress relaxation events are I

I

I

I

o 9- I MONOIONI

1.70

t:J;~ " ~ ' ' ' ' ' ' - ° 12-1 CYCLIC

m = 100

=

n 12

j;~"'=

= A = 0.62 S

"

1.60

-11

I

m.l~

~ ~-

j

l. 55

20 10 15 20 20 1 5 2

Stress relaxation data

Samples for TEM examinations were taken from (i) virgin material, (ii) from specimens extended monotonically and cyclically to the same strains used for relaxation experiments and (iii) from specimens that had undergone stress relaxation. After unloading and removal from the test frame the specimens were stored in liquid nitrogen until each was prepared for I

Zn Cr Pb Zr Ni Ti Be V

RESULTS

Transmission microscopy

l

Sensitivity Limit, ppm

TEM examination. TEM specimens were prepared by spark machining wafers from the samples. These wafers were thinned in a solution of 49?/0 nitric acid, 2~o hydrochloric acid and 49~o methonol after which the foils were immediately examined.

giving strain rate as a function of stress by fitting that data with equation (1). At 295 K, the stress relaxation was best described by equation (1) with ai = 018. An iterative procedure was used to minimize the sum of the squared deviations of the data points to the fit line. Initially m and n were specified and A optimized; the initial m and n values were found from slopes of the data at high and low strain rates [13] respectively. Given these initial values of m and n, only 3 or 4 iterations were required to converge on optimum values for m, n and A.

1.80

Not Detected

n= 12 =;?63

I

A-

If/'

I

-10

-9

I

I

.

I

-8 -7 -6 LOG STRAtaRATE (s"1)

I

-5

-4

Fig. 1. Comparison of stress relaxation behavior for a sample monotonically loaded to 0.59~ plastic strain with a sample loaded to lO cycles of reversed total strain of ±0.025~o per cycle prior to relaxation. Solid lines, model fit.

ROHDE et al.:

DEFORMATION MODELING OF ALUMINUM I

I

I

I

I

45

I

1 I: m= i00, n= 12, A= 0.62 6 2: m= 500, n= 12, A= 0.6.5 f 3: m-800, n-12, A= 0. 58

2

A

1. lO

/

o,

1.60

1.55

-11

-10

i

i

I

I

I

-9

-8

-7

-6

-5

LOG STRAINRATE

-4

(s -1)

Fig. 2. Comparison of stress relaxation of a sample monotonically loaded to 0.59°0 plastic strain (1), relaxed, then reloaded an additional 0.013~o plastic strain (2), relaxed, then reloaded an additional 0.005% plastic strain (3). Solid lines, model fit. shown in Fig. 2. These data were generated in the following m a n n e r : the specimen was monotonically extended to a large plastic strain (% = 0.6Yo), then allowed to relax, producing the curve labeled 1. Subsequent to that relaxation, the specimen was reloaded to a stress less than the beginning stress of the previous relaxation. This produced a small plastic strain (ep = 0.013~o) a n d a relaxation behavior labeled 2. Subsequent to this relaxation, a n o t h e r relaxation event was imposed at a n additional plastic strain, ep = 0.005Yo; shown as 3. Two features of these data are important. First, the initial plastic strain rates for the 'reload' relaxations are considerably less than the initial plastic strain rate of the first relaxation. In this series of experiments, the total strain rate (Jr) was held constant at J r = 3 x 10 -5 s -1. The first relaxation was initiated well into the plastic region of the

stress-strain curve: the 'reload' relaxations on nearly elastic portions. Thus the plastic strain rates (Jp = J r - J , ) j u s t prior to relaxation in these three instances were variable. Since plastic strain rate is continuous [20], the plastic strain rates just before a n d just after relaxation is initiated are equal, a n d the relaxation events shown in Fig. 2 start at different plastic strain rates. The second feature of the data worth noting is that the 'reload' relaxation curves cross over the initial relaxation data. In view of the observations of Hart a n d Solomon [14] a n d the previous predictions of the model under discussion [18], this result was unexpected. Nonetheless, it is real; identical results were obtained on two other specimens, and Li [ 2 t ] has recently observed similar behavior in a l u m i n u m a n d stainless steels. To determine if this 'crossover' were an effect related to initial plas-

Table 2. Model parameters for stress relaxation of aluminum Test No. I'~ 8-I 8-R 9-I 9-R1 9-R2 10-I 10-RI 10-R2 11-1 11-2 12-C1 12-C2

al (MPa) 60.4 56.9 61.1 53.5 49.5 59.4 53.5 51.0 48.8 55.0 58.7 58.7

~l (s- 1) 1.6 0.7 2.7 0.2 0.08 0.7 0.1 0.09 0.3 1.8 1.0 1.1

x x x × x x x x x x x ×

10 -5 10 -5 10 -5 10 -5 10 -5 10 -s 10 -5 10-s 10 -s 10 -5 10 -5 10 -5

ep (O/o)

m

n

A

0.82 0.03 0.59 0.013 0.005 1.5 0.025 0.005 0.07 0.15 0.5 0.7

100 300 100 500 800 100 500 800 300 200 100 100

12 12 12 12 12 12 12 12 12 12 12 12

0.65 0.56 0.62 0.65 0.58 0.64 0.65 0.62 0.62 0.53 0.63 0.65

("1 = initial relaxation. R = reload relaxation: plastic strain is given using the initial relaxation strain as reference,

C = cyclic loading: 10 cycles of reversed strain to the total indicated strain.

R O H D E et al.:

46

1.80

DEFORMATION MODELING OF ALUMINUM I

l

I

I

I

I

-9

-8

-1

-6

I -5

m = 300 n - 12 A = 0.62 A

1.10

,7, S

1.60

1.55

I -10

-11

-4

LOG STRAIN RATE (s-l) Fig. 3. Stress relaxation of sample monotonically loaded from a virgin state to 0.07°0 plastic strain. Compare with relaxation upon 'elastic" reloading curve 2, Fig. 2.

tic strain rate or to initial stress, two other experiments were conducted: (i) A monotonic straining to 1.5% plastic strain at a lower total strain rate than that described above followed by two 'reload' relaxation measurements; and (ii) monotonic loading to very small plastic strains before relaxation. The results of these measurements as well as those previously discussed are given in Table 2. Additionally, Fig. 3 shows data taken for monotonic loading to a small (0.07~o) strain. We note that while the parameter m for large strain deformation (% > 0.6~) is independent of history

O. 76 O. 72 O. 68

l

•

Initial

A

Reload

(m = 100), it does vary (m > 100) in experiments begun at small total plastic strains, Fig. 3, and in the 'reload' events, Fig. 2. It is impossible, in these two cases, to fit the data with a value for m of 100. Best fits to the data are with m values given in the table and shown in the figures. In contrast to this deformationdependent behavior of m, the parameter n was found to be constant in all experiments (n = 12). The new iterative method of data reduction discussed in the 'Experiments" section, coupled with relaxation data covering 5 to 6 orders of magnitude in plastic strain rate allows a more precise determination

I

I

I

D Cyclic

A

A

•

[]

0.64

D

~C

0.60 A A

0.56

O. 52 0.48

I

1.50

160

i

I

i

170

I

i

180 a!

I

190

a

200

( MPa )

( l-nlm ) !~1 (11m + 11n) ) Fig. 4. Plot of the aggregate state variable A against initial relaxation conditions.

ROHDE et al.:

DEFORMATION MODELING OF ALUMINUM

47

of the parameter A. Values for this parameter are given for each experiment in Table 2. Those values are accurate to _+0.02; that is, a variation of 0.03 will result in a significant change of the goodness to fit of the model to the experimental results. The parameter A is intended to reflect the current state of the material through its functional relationship to the flow stress (gl) at a specified plastic strain rate {~1). Inspection of equation 2 shows that at constant temperature A should be linearly proportional to the function b l / ( 1 - n/m) ~1/~ lJ,,). A plot of A versus values for that function is shown in Fig. 4. Despite the improvements in determination of A, considerable deviation from linear proportionality still exists. Since the scatter is greater than the precision of determination of A from a single experiment, the probability, given m and n values, of correctly predicting a relaxation curve from the information in Fig. 4 is not good.

found in material deformed to even smaller plastic strains. Similarly no differences in microstructures could be found in specimens which were relaxed 30 h. Thus, the wide differences in observed relaxation behavior, as indicated by values of m and A in Table 2 and shown in Figs 2 & 3, are not reflected as discernable differences in dislocation substructure.

Electron microscopy

Mechanical measurements

The objective for this phase of the study was to correlate dislocation substructures observed by TEM with material states as defined by the model and reflected by the current flow stress or the parameter A. As in most similar studies, no effort was made here to lock in the exact dislocation distribution under load prior to unloading. Accordingly, caution must be exercised in interpreting the resulting substructures. However, any cell or subgrain structure present does not change in size with unloading nor does the character of the cell of subgrain boundaries (i.e. the 'tightness' or 'looseness') change. The distribution of dislocations within cells or subgrains is expected to change with unloading and is further expected to be altered in the thin film TEM specimens as frec dislocations are allowed to escape to the foil surface. An additional concern with aluminum is whether sufficient recovery occurs at room temperature to significantly alter the substructure. The work of Alden [7] and Hasegawa and Kocks [22] indicates that room temperature annealing of aluminum alters neither cell or subgrain size nor the boundary character. Three specimens will be discussed: (i) annealed, untested material; (ii) monotonically deformed to 1.5°o strain, trl = 59.4 MPa, A = 0.64, C = trl/(1 - n/m) (~(11/"+~/"~) = 207 M P a ; (iii) cyclically deformed for 10 cycles at +0.025"~0 strain, trl = 58.7 MPa, A = 0.65. C = 193 MPa. In this matrix, the states of samples (ii) and (iii) should be equal and both should be different from sample (i). Figure 5 shows transmission electron micrographs of the samples: there are no clear differences present. Annealing aluminum results in a microstructure which consists of numerous subgrains 2.5/~m in size. Straining these specimens to small plastic strains either monotonically or cyclically does not reduce cell size, or obviously sharpen cell boundaries. It is assumed that if no differences were evident in these very dissimilar cases, none would be

These experiments were designed to yield more precise measurements of the parameters n, m and A. The precision on n is about + 1.0, that is, deviation greater than 1.0 will alter the goodness of fit of the model. The n value of 12 determined here from stress relaxation testing is less than the value of 20 found by previous measurements on identical material using identical experimental techniques [18], e x c e p t that in the present case measurements were made for much longer times (30 h. rather than 0.5 h). With these more accurate results the present data cannot be satisfactorily modeled with n' = 20. Creep experiments [18] on this material at 298 K gave a steady-state creep exponent n' of 20. This model assumes that moderate temperature recovery processes are extensions of their high temperature counterparts. Thus, steady state creep and stress relaxation in the region dominated by recovery (i.e. defining n) are controlled by the same (recovery) mechanism. Hence, self-consistency requires for m >> n that n {relaxation) = n' (creep). The present measurements indicate this equality is not satisfied. Additionally these findings stress the importance of long time relaxation measurements to provide accurate data for modeling. The value of m is precise to about 10~o, and this parameter is required by the model to be a material constant, (i.e., invariant with deformation history). It is evident from examination of Table 2 that this requirement is not satisfied; m is history dependent, becoming very large for 'reload' and small plastic strain experiments. The model cannot be fit to the small strain data with the large strain m value of 100. If the requirement that m be a material constant is empirically removed, m can be taken as a state variable with some functional dependence upon some measure of state. Often such state measures are taken to be the flow stress or plastic strain rate. Regression analysis indicated that there was no correlation between m and t71 or ~ . Weak correlation between m

DISCUSSION To this point we have cast experimental results in terms of a specific work-hardening/recovery deformation model While this is consistent with the main goal of this work, the data have more general applicability to equation of state modeling. Thus, in this section we will consider the significance of the experimental measurements both in terms of the power law model and in terms of the philosophy of equation of state modeling.

(b)

(c)

Fig. 5. Transmission electron micrographs of aluminum after various histories; (a) annealed material; (b) monotonically loaded to 1.5% plastic strain; (c) cyclically loaded 10 cycles +0.025? / total strain.

(a)

>

©

©

Z

>.

©

ROHDE et al.: DEFORMATION MODELING OF ALUMINUM

significant changes in mobile dislocation density are expected, i.e., at plastic strains just beyond the proportional limit. Inclusion of a term or terms which would treat variation of mobile dislocation density would greatly increase the complexity of both the model and experimental procedures required to establish model parameters. Thus, before such modifications are attempted the assumption that changes in mobile dislocation density are responsible for the inappropriate variation in m must be verified. In addition it is necessary to assess the need for fitting all aspects of experimental data at the expense of considerable additional complexity in models. From the pragmatic standpoint, perhaps the most significant problem with the power law model is the scatter in the state parameter A shown in Fig. 4. This scatter means that for any value of A selected there could be considerable variation between predicted and measured curves. From a design standpoint, this scatter may be acceptable. From a physical standpoint the inability to predict data to greater precision may reflect significant errors in modeling and certainly make extrapolation of data hazardous. More work is needed to determine if this scatter is a consequence of measurement error, material variability, or modeling difficulties.

and plastic strain (p was found, but strain is an illdefined state variable. An alternative explanation of the need for m to vary with mechanical history may be to consider the initial relaxation at small plastic strains and upon 'reloading' to be anelastic. In this case the reloading is merely transient behavior and not appropriately considered in the model; thus, m may be held constant. However, this leads to considerable discrepancy between predictions and the actual data. An example of the inability of the model to fit reloading data with constant m is shown in Fig. 6, where predictions are made of two 'reload' relaxations using as model parameters n = 12, m = 100 and A values taken as a function of initial conditions from Fig. 4. The disagreement between the model and the data is significant, but it is important to note that this recovery, work-hardening based model does correctly predict crossing of the initial relaxation curve by the subsequent 'reload' relaxation curves. This prediction is in graphic contrast to models which assume a constant material state during relaxation (i.e., no recovery)[4,14]. Such models require that relaxation curves, when plotted as in Fig. 6, never intersect [4]. Finally, some models permit structural changes to occur during relaxation, but also require a discontinuous change in plastic strain rate which we find at variance with other experiments as referenced above [20]. As is typical in most equation of state models, the power law model implicitly treats the mobile dislocation density as constant. This assumption has not been established. In fact, a number of recent reports [23-25] indicate that mobile dislocation density does vary during relaxation. The parameter m was found to vary widely in exactly those instances where 1.80

I

I

Transmission electron microscopy

In the study of the deformation of pure metals, transmission electron microscopy has been performed with basically two different goals. Most previous work has been conducted on polycrystalline samples in which no effort was made to retain details of the substructure present at the imposed stress. The objectives of these previous studies have been to characterize the substructures that evolve under various defor-

I

I

I

I

~.l.?0 v

,% o

n

1.60

1.55 -11

I/

II

-i0

t

-9

I

I

-8 -7 -6 LOGSTRAINRATE (s"t)

I

I

-5

-4

Fig. 6. Stress relaxation experimental data, [] (curve 1 initial relaxation, curves 2 & 3 relaxations subsequent to 'elastic" reloadings) compared to predictions of the model; rn is held constant at 100. Compare with possible fits if ra is allowed to vary, Fig. 2.

A.M. 2 9 1 - - 0

49

50

ROHDE et al.:

DEFORMATION MODELING OF ALUMINUM

marion histories and to examine them for the presence of any correlations between cell or subgrain size and stress level. Such correlations are numerous in the literature and have been reviewed recently by Thompson [26]. Fewer studies have been performed in which the details of the dislocation interactions in the stressed condition have been examined. This work usually involved neutron irradiation of specimens while under load to pin all of the dislocations [27, 28]. In none of these studies has the purpose been to examine the substructure for indications of what feature or features could be unambiguously associated with the evolutionary internal state variables of physically based deformation models. The first question to be addressed is: do the substructural features which have been found to exhibit empirical correlations with mechanical behavior (cell or subgrain size for example) meet the requirements for being correlated to evolutionary state variables? In all deformation models relaxation behavior may be used as a probe of the material 'state'. In particular with the power law model presented here A and or1 should reflect the current internal state. In addition to the annealed condition, electron microscopy was conducted on a monotonically deformed specimen and a cyclicly deformed specimen, both of which had A ~- 0.65 and t~1 ~- 59 MPa. In this way a comparison of the substructures with annealed material would allow the important "state' determining features to be identified. In fact, the annealed, the monotonically deformed, and the cyclicly deformed specimens all had the same subgrain size and the same subgrain boundary character. Listed in Table 2 are several monotonic and cyclically deformed specimens which have been subjected to less strain than those examined by TEM which have values of A and tr I much different than 0.65 and 59 MPa respectively. Yet these specimens would have substructures no different than those in Fig. 5. In other words, nothing in the subgrain size or subgrain boundary character reflects the measured differences in relaxation behavior which should be a direct reflection of material state. The annealed material contained numerous dislocations, certainly enough so that only a small fraction of the total dislocations need to be mobile to accommodate the imposed strain rates or the strains accumulated during relaxation. It would clearly take much greater deformations than imposed here to markedly alter the microstructure. Our work and the work of Hart [4, 14] shows that stress relaxation behavior does not change with prestrain for plastic strains much greater than 0.6°J0, so no history dependence need be considered at large strains. Similar results are seen in the work of Miller and Sherby [29] where the steady state creep behavior of aluminum is quite accurately modeled (large strains) while the same model represents creep transients (small strains) rather poorly. Based on the above observations, we conclude that the large strain deformation behavior of aluminum,

which is history independent, does not represent a critical test of proposed material laws. This region is not interesting simply because it is described trivially. We feel that the true test of a material law, considering aluminum, will be adequate description of the small strain, history dependent deformation region. The large strain behavior of aluminum might be anticipated from studies of other wavy slip metals (like aluminum) by comparing their response under cyclic vs monotonic deformation. At higher cyclic strain amplitudes the flow stress of these metals is independent of prior state and varies inversely with dislocation cell size[30,31]. Some authors suggest that the monotonic and cyclic states of wavy slip metals are analogous [31-33]; others indicate that the states are equal when cell sizes [34-35] or long-range stress fields [32] are the same. Even with large strain deformations, cell or subgrain substructures cannot be unambiguously related to mechanical behavior. Lukas and Klesnil [30, 31] have studied the cyclic deformation of pure copper (wavy slip) and Cu-30 °/ Zn iplanar slip} and examined the substructures produced. Their results for Cu showed that cyclic stress range for a given plastic strain range did not depend on history, even though an annealed specimen cycled to a low strain range would develop a banded or veined structure while a previously deformed specimen which contained a cell structure would retain the cell structure at the lower cyclic strain range. For the Cu-30)o Zn alloy the opposite was observed. In this case, the cyclic stress range was history dependent even though all of the histories examined produced very similar planar arrays of dislocations. The role of recovery processes at room temperature in determining both the mechanical behavior and the substructure is not clear. This study indicates that some mechanical recovery occurs during stress relaxation. Figure 7 is a tracing of a load-strain measurement including an initial relaxation and two 'reload' events. It is evident that deformation subsequent to the final relaxation occurs at a lower flow stress than would be inferred from extrapolation of the loadstrain curve obtained previous to relaxation. Similar observations have been made by Bradley, et al. [36] and Alden [37]. This evidence of recovery during relaxation of aluminum indicates that proposed state variable models must include that phenomenon. Alden [37] seeks to explain his observations in terms of changes in number and arrangement of dislocations; recovery is alleged to result from rearrangement. One objection to Alden's model concerns the issue of discontinuity of plastic strain rate which is predicted. We present elsewhere our case for continuity of ~p[20]. Hasegawa and Kocks [22] have examined in pure aluminum the effects of recovery on the reloading stress-strain response and have found that times of up to 4h at room temperature have almost no effect on the reloading behavior. They did observe for periods of up to 2h at 453 K or 4h at 403 K that the loose cell walls evolve into well defined

ROHDEetaI.:

DEFORMATION MODELING OF ALUMINUM I

I

I

I

I

51

I

2O0O

1000 o

o

a 2

I 6

I

I

10

12

14

PERCENTTOTALSTRAIN

Fig. 7. Tracing of stress-strain plot of a sample showing initial relaxation and two subsequent relaxations after 'elastic' reloading. Note that the extrapolation of the curve prior to relaxation lies above the curve obtained after the relaxation events. subgrain boundaries, and that higher recovery temperatures were necessary to cause subgrain coarsening. In a study with similar objectives to this work, Turner [38] has recently observed difficulties in trying to correlate state variables with microstructures. Using stress-strain curves to probe the mechanical state of 304 stainless steel as a function of deformation history at 0.34 and 0.5 times the melting temperature (T,,). Turner found that cyclic loading changed the microstructure, but that this change was not reflected in a change of mechanical state. This result is exactly opposite the behavior observed here and elsewhere at lower temperatures, as described above. The conclusion which can be drawn from all of the comparisons discussed here is that such substructural characteristics as cell or subgrain size and sub-boundary character do not contain sufficient information to be considered as representative of the internal state of the material. SUMMARY For reasonably large plastic strains (~p ,~ 0.60~o) stress relaxation in aluminum is history independent and easily modeled. Thus, large strain behavior does not represent a critical test of deformation laws. In the power law analysis, the parameters m and n are constant as required. The aggregate internal state variable, A, exhibits scatter, however, and it seems f Note added in proof: The referenced creep experiments [18] were performed over a limited range of stresses. More recent creep measurements conducted over a much larger stress range and for longer times yielded a creep exponent n' = 14. This indicates that the creep exponent and the stress relaxation constant, n, are equal as required by the model.

unlikely that this merely reflects measurement precision. At small plastic strains, however, (Ep ~ 0.05~o) the behavior of aluminum is not well modeled. Stress relaxation curves 'cross over', indicating that relaxation is not a constant state process in the material. Thus, it must appropriately be modeled by considering microstructural changes, such as work hardening and recovery. Unfortunately, in this small strain region, the material constant m in the power law model is ill-behaved and must be permitted to vary with strain in order to allow the model to fit the data. It is not clear whether the inability to model small strain behavior results because these low strains are truly anelastic, and thus merely representative of transient phenomena, or instead if the inability to describe the data is a result of a more fundamental modeling error such as the implicit assumption that the mobile dislocation density is constant. The material constant n is indeed invariant in all cases; however, it does not, as assumed in development of the model, equal the steady state creep power law exponent. Again, it is not clear if this is a measurement error in creep experiments or a more fundamental modeling problemf. Transmission electron microscopy observations indicate that the microstructure of the annealed aluminum is sufficiently dislocated to accommodate the plastic deformations imposed in these experiments without observable alteration in subgrain size or in the low angle boundary character. Earlier success in relating stress levels to dislocation cell or subgrain sizes in tensile, cyclic and creep tests made these features prime candidates as microstructural state variables. Turner [38] has attempted in Type 304 stainless steel to correlate changes in cell size and boundary character to elevated temperature stressstrain response and has observed microstructural

52

ROHDE et al.: DEFORMATION MODELING OF ALUMINUM

changes that were not reflected in changes in the mechanical behavior. In this work, we have observed differences in relaxation response in pure A1 that were not reflected in any change in subgrain size or boundary character. Therefore, the search via T E M for correlations between state variables of mechanical models and the dislocation cell or subgrain size or the character (i.e. the 'tightness' or 'looseness') of the boundaries seems futile. The microstructural 'state' of a material can then only be determined by examining the distribution of dislocations and barriers as they exist under the loading conditions imposed. The substructure present after unloading does not contain sufficient information to uniquely define the 'state' of the material in the loaded condition. At present, we do not believe that substructure has been correlated adequately with internal state variables in any material, whether wavy or planar in slip character.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

REFERENCES 1. A. S. Argon, (Ed.) Constitutive Equations in Plasticity, M.I.T. Press, (1975). 2. K. G. Valanis, (Ed.) Constitutive Equations in ViscoplasticiD': Phenomenological and Physical Aspects, A.S.M.E. (1976). 3. S. Nemat-Nasser, (Ed.) Proceedings of the Workshop in Applied Thermoviscoplasticity, Technological Institute, Northwestern University (1975). 4. E. W. Hart, J. Engn9 Mater Technol. 193 (1976). 5. A, K. Miller, J. Engng Mater. Technol. 97 (1976). 6. A, R. S. Ponter and F. A. Leckie, J. EngnO Mater. Technol. 47 (1976). 7. T. H. Alden, Phil. Mag. 25, 785 (1976). 8. U. F. Kooks, J. Engng Mater. Technol. 76 (1976). 9. U. F. Kocks, In Proceedings of the Workshop on Applied Thermoviscoplasticity, (Edited by S. NematNasser) p. 244. Technological Institute, Northwestern University (1975), 10. S. J. Chang, Proceedings of the Southeastern Conference on Theoretical and Applied Mechanics, Nashville, Tennessee (1978). 11. J. C. Swearengen, R. W. Rohde and D. L. Hicks, Acta raetall. 24, 969 (1976). 12. R. D. Krieg, J. C. Swearengen, and R. W. Rohde In Inelastic Behavior of Pressure Vessel and Piping Corn-

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

ponents, (Edited by T. V. Chang and E. Krempl), p. 15. ASME, PVP-PB-028, (1978). J. C. Swearengen and R. W. Rohde, MetalL Trans. A 8, 577 (1977). E. W. Hart and H. D. Solomon, Acta metalL 21, 295 (1976). R. D. Krieg, Proc. 4th International Conference on Structural Mechanics in Reactor Technology, (1977). Paper M6/4. U. F. Kooks, 107th Annual Meetin 9 of the Metallurgical Society of AIME, Feb. 26-Mar. 2, 1978, p. 250. J. C. Swearengen and R. W. Rohde, J. Engn9 Mater. Technol. 100, 221 (1978). R. W. Rohde, and J. C. Swearengen, Stress Relaxation Testing, (Edited by A. Fox,) p. 21. ASTM STP 676, (1979). R. W. Rohde and J. C. Swearengen, J. Engng Mater. Technol. 102, 207 (1980). J. A. Holbrook, R. W. Rohde and J. C. Swearengen, Sandia Laboratories, Albuquerque, N.M. 87185, to be published. C. Y. Li, Materials Science Dept., Cornell University, Private Communication. T. Hasegawa and U. F. Kocks, Acta metall. 27, 1705 (1979). J. Moteff, Progress Report, Jan. 1, 1974-Dec. 31, 1975, Cincinnati Univ., Mar. 15, 1976, K. Okazaki, Y. Aono and M. Kagawa, Acta metall. 24, 1121 (1976). M. A. Meyers, J. R. C. Guimaraex, and R. R. Avillez, Metall. Trans. A 10, 33 (1979). A. W. Thompson, Metall. Trans. A 8, 833 (1977). H. Mughrabi, Phil. Mag. 18, 1211 (1969). J. C. Crump, III and F. W. Young, Jr, Phil. Mag. 17, 381 (1968). A. K. Miller and O. D. Sherby, Acta metall. 26, 298 (1978). C. E. Feltner and C. Laird, Acta metall. 15, 1633 (1967). P. Lukas and M. Klesnil, Mater. Sci. Engng. 11, 345 (1973). R. C. Daniel and G. T. Home, Metall. Trans. 2, 1161 (1971). P. O. Kettunen, Phil. Mag. 16, 2531 (1967). H. Abdel-Raouf and A. Plumtree, Metall. Trans. 2, 1251 0971). H. Abdel-Raouf and A. Plumtree, Metall. Trans. 2, 1863 (1971). W. L. Bradley, W. Renfroe and D. K. Matlock, Scripta metall. 10, 905 (1976). T. H. Alden, Metall. Trans. 4, 1047 (1973). A. P. L. Turner, MetalL Trans. A 10, 225 (1979).