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The influence of anelasticity on the transient behaviour of superplastic Sn-Pb eutectic after stress and strain rate changes
THE INFLUENCE OF ANELASTICITY ON THE TRANSIENT BEHAVIOUR OF SUPERPLASTIC Sn-Pb EUTECTIC AFTER STRESS AND STRAIN RATE CHANGES J. H. SCHNEIBELt Department of Metallurgy & Science of Materials. University of Oxford. Parks Road. Oxford OX1 3PH, U.K. (Receit& 17 December 1979: in revise&firm I4 April 1980) Abatmct-The influence of anelasticity on the transient behaviour of superplastic Sn-38.lw/o Pb after stress changes and strain rate changes is examined, The transients following strain rate changes (e.g. loading with a constant strain rate. strain rate cycling, stress relaxation, dip test) are measured and the agreement with the transients calculated from independently determined plastic and anelastic data is found to be reasonable. The main assumptions involved in the calculations are the linear additivity of plastic and anelastic strains, and the representation of the plastic strain rate by a power law. Anelastic strains are seen to be important in all the transients investigated. The evaluation of the plastic properties of superplastic Sn-38.lw/o Pb from transient tests should. therefore. be approached with caution. R&u&Nous ttudions l’influence de I’anilasticitC sur le comportement transitoire de l’alliage superplastique Sn-38,l%Pb (en poids) apres des changemmts de contrainte et de vitesse de d&formation. Nous enregistrons les regimes transitoires qui suivent les changements de vitesse de d&formation (par exemple charge !! vitesse de dbformation constante. vitesse de deformation cyclique, relaxation de contrainte, saut de vitesse) et nous trouvons un accord raisonnable avec les rCgimes transitoires calculCs B partir de donntes de plasticit& et d’anblasticit6 dCterminCes indkpendamment. Les hypotheses principales de ces calculs consistent en I’additivitC Ii&ire des d&formations plastique et anllastiques et la reprisentation de la vitesse de d&formation plastique par une loi en puissance. Les dkformations anklastiques jouent un r6le important dans tous les rCgimes transitoires ttudib. On doit done Btre prudent lorsqu’on cherche & ivaluer Its propriktks plastiques du Sn-38,1%Pb superplastique & partir de 1’Ctudede rigimes transitoires. EinfluD der Anelastizitiit auf das ijbergangsverhalten der superplastischen Legierung Sn-38 Gew.-7; Pb nach Spannungs- und Dehngeschwindigkeitsspriingen wurde untersucht. Die Transienten nach Geschwindigkeitsspriingen (d.h. Belastung mit konstanter Dehngeschwindigkeit, Dehngcschwindigkeitsyklen, Spannungsrclaxation. ‘dip’-Test) werden gemessen; sic stimmen mit den aus unabhiingig gemessenen plaitischen und anelastischm Daten arechneten Transienten hinreichend W&t. Die wesentiichen Annahmen bei bieser Berechnung sind, daB plastische und anciastische Debnung linear add& wcrden kiinnen und daD die plastische ~hng~hwindi~cit mit eincm Potenzgesetz beschreiben werden kann. Die anelastische Dehnung ist bei alien untersuchten Transienten wichtig. Die Ermittlung der plastischm Eigenschaften der superplastischen Legierung Sn-38 Gew.-y0 Pb aus Transientenuntersuchungen sollte dahcr mit Vorsicht vorgenommen werden. Zunmmmfamng-Dtx
1. WIRODUCTZON Deformation may be conveniently anelastic and plastic &formation and plastic deformation are in
divided into etaatic, Cl]. Both anelastic
general time-depen dent; however, anelastic strains are recoverable whereas plastic strains are permanent. Anelastic deformation can contribute significantly to the strain rate transients following stress changes [I], and to the stress transients following strain rate changes. as for example in stress relaxation and in stress tran~ent dip t Now at : Departmentof MaterialsScience & Engineer-
ing, Massachusetts MA 02139, U.S.A.
institute
of Technology,
Cambridge.
tests [2]. The plastic properties of a material can be determined from the above transient tests only if anelastic deformation can be shown, to be negligible or otherwise if the evaluation takes the anelastic strains into account. Superplastic Sn-Pb eutccctic (i.e. Sn-38.1 w/o Pb) exhibits pronounced anelasticity with relaxation strengths sometimes larger than 100C3.4). Other superplastic materials, for example Sn-2 w/o Pb [33 and 2~22 w/o Al [SJ behave similarly. In the past, however, the plastic properties of superplastic materials have often been inferred from transient tests on the supposition that anelastic strains are negligible. This approach has been employed for the evaluation
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SCHNEIBEL:
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of stress relaxation tests [6,7]. strain transient dip tests [8] and strain rate change tests [9]. The main purpose of these transient tests is either to determine the internal stress below which plastic deformation does not proceed or to measure the plastic properties, in particular the strain rate sensitivity, at a constant sample structure. It is not always recognised that the internal stress may not only provide a threshold stress for plastic deformation but also a driving force for anelastic recovery (causing for example straightening of bowed-out dislocation segments). Anelastic readjustment of the sample structure and the strains associated with it may interfere with attempts to measure the plastic properties for a constant sample structure.. In order to assess the applicability of the transient tests mentioned above to superplastic materials it was attempted to estimate the influence of anelastic strains on transients in Sn-Pb eutectic more accurately than has hitherto been done. As will be seen this influence can be quite marked. Transient tests, therefore, do not necessarily provide a simple means of measuring the plastic properties of this alloy. This finding will also apply to transient test data obtained for other superplastic alloys with strong anelasticity as for example Zn-Al eutectoid [5, lo]. 2 EXPERIMENTAL
PROCEDURE
The G-38.1 w/o Pb alloy (eutectic composition) investigated was prepared from Sn and Pb of 99.999% purity by melting in air or in Argon and air-casting into a copper mould with dimensions 10 x 38 x 95 mm3. The ingots were rolled down from IO mm to thicknesses between 0.2 and 1.0 mm and tensile samples with a gauge length of 26 mm and a width of 6 mm were punched from the rolled sheet. Mean true grain sizes, L,,after different heat treatments were estimated from the plastic strain rates and the previously determined relationship between plastic strain rate and grain size (no difference was made between grain and phase boundaries). Stress change experiments were performed in silicone fluid employing a constant load creep rig with a strain resolution of f 10m5 and a temperature accuracy of +0.2 K. Since the sample elongations were smdl during each test the applied stress remained ap proximately constant, for a constant applied load. Constant strain rate tests were done employing an Instron tensile testing machine. The samples were immersed in a silicone fluid bath at (298 +, 0.5) K. In the load range employed the stiffness of the testing machine including the load cell and the pull rods was found to be 2.5 N/w. Since the sample elongations during the tests were small the constant extension rate of the machine resulted in an approximately constant true strain rate. A few accurately controlled stress relaxation tests were done in air in a creep rig equipped for stress relaxation in a manner similar to that described by
ALLOYS
Blum and Pschenitzka [ll]. The temperature of the whole loading train (length _ 300mm) including the load cell was kept at (298 & 0.05) K during the relaxations. Since many tests were usually performed with one sample a sufficiently long period of time was allowed to elapse after each test so that the anelastic strain rate became very small and did not interfere with the following test. 3. MEASURED AND CALCULATED TRA!!SIENTS 3.1. Stress changes 3.1.1. Loading behaciour. A sample was stressed at 328 K for 4 13 days with 0.36 MPa and subsequently unloaded in order to measure its elastic after-effect. i.e. its anelastic contraction. In Fig. 1 the anelastic strain thus obtained has been substracted from the combined plastic and anelastic strain obtained upon loading in order to find the plastic component of the strain during loading. (The elastic strains were subtracted from the experimental data using the stiffness of the creep rig employed and the elastic compliance of Sn-38.1 w/o Pb, 4.76.IO-’ MPa-’ [12]). Linear additivity of the plastic and anelastic strains has thus been assumed. Inspection of the plastic component of the deformation shows that either this assumption does not hold strictly or the plastic strain rate exhibits transient behaviour, i.e. it is not a unique function of the stress. However, the substraction of the anelasticity in Fig. 1 reduces the magnitude of the transient considerably. The measured loading transient may therefore be approximately represented by the sum of a plastic strain rate (which depends only on the applied stress) and an anelastic strain rate (which depends on the applied stress and on the time elapsed after its application). The physical processes responsible for the plastic and anelastic deformation of superplastic Sn-38. I wo Pb are not well understood to date. Mohamed and Langdon [13] conceded that the plastic deformation behaviour is not entirely consistent with any known mechanism of superplastic flow. The run-back of piled-up grain boundary dislocations has been proposed to explain the anelastic strains [3]. However. difficulties arise in explaining the large relaxation strengths observed. In view of these uncertainties the linear additivity of the plastic and anelastic strains cannot presently be verified in terms of the underlying mechanisms. From a mechanistic point of view it may even be possible that the separation into plasticity and anelasticity is artificial. that is. if anelastic and plastic deformation are closely interlinked. Such a situation is encountered in the conventional dislocation creep of pure metals as has been discussed by Nix and Ilschner [ 143. 3.1.2. Stress cycling. Since for superplastic Sn-38.1 w/o Pb the anelastically recovered strain. E,, is approximately proportional to the applied stress. u.
SCHNEIBEL:
TRANSIENT
B~HAVr~UR
IN SWPERPLAS-UC
f 529
ALLOYS
T: SW
0
0
1
2
3
and anelastl
L
5
t/la.
Fig. 1. The influence of aneiasticiry on the loading transient of a Sn-38.1 w/o Pb sample with a grain size of -8 pm. (6 = strain. r = time. T = test temperature). The sample was stressed at 328 K for w 13 days with 0.36 MPa, The anelastic strain as a function of the time was obtained from the elastic after-effect (i.e. the anelastic contraction after unloading\. The plastic strain upon loading was estimated by subtracting the anelastic strain from the loading strain.
for a given time after unloading [3]. it is convenient to define an anelastic compliance, J*(t) = Ir,(r)/ol. The corresponding anelastic compliance rate. j&), for a Sn-38.1 w/o Pb sample with a grain size of _ 1.5 pm is plotted in Fi8.2 (experiment A) as a function of the time after unloading. Stress change experiments were performed with the same sample (Fig. 3) and corrected for anelasticity assuming linear additivity of the plastic and anelastic strain rates. and using the data in Fig. 2 (experiment A) and Boltzmann’s superposition principle fl5]. It is seen that the subtraction of the anelasticity reduces the mamitude of the fransient behaviour. In particular. if removes the contradiction which would be found in Fig. 3 if anelasticity were neglected. namely. negative ptastic strain rates in response to positive applied stresses. 3.2. Strain rate changes In section 3.1, if has been demonstrated that the adding of plastic and anelastic strains provides a more realistic description of the mechanical behav-
Fig. 2. Anelastic compliance rates, j, as a function of the time after unloading. f. Experiment A: Prior to the relaxation al t = 0 a sample was loaded with 1.04 MPa for 3.6 ks. Experiment B: Prior to the relaxation a sample was loaded with 0.73 MPa for 4 days. Two different sums of exponentials have been fitted to curve B: (1) &f) = 11.2.10-‘ exp(-r/O.Ss) c 3.7*10-6 exp (-r,5s) + l.8~10~bexp(-r~50s)~MPa~’
5-j.
(‘1 J.(t) = iS.8*10-’ exp (-r/0.8s) + X3*10-* exp (-r;18s) + S*lO-‘expf-r.3OOs) + l.t*IO-’ exp(-r/4OOOs): MPa-' s-l, The values of the parameters of the Voigt elements in Fig. 4 are obtained from approximations (1f or (7) and are listed in Table I.
iour of Sn-38.1 w/o Pb than a description in terms of plasticity alone. With the condition of linear additivity of plastic and anelastic strains and the condition that the plastic strain rates are proportional to some power of the applied stress. the transients following strain rate changes may be calculated from the plastic and anelastic properties and compared to directly measured transients, The corresponding rheologicai model is shown in Fig. 4. It consists of a power-law dashpot describing the plastic defo~ation rate. $ (proportionality constant K, strain rate sensitivity nr). three Voigt elements which will be fitted to the anelastic properties of the material (compliances JOi. retaxation times TV. anelastic strains Eai*where I’ = I, 2. 3) and a spring with a compliance J, describing the efastic defo~ation of the sample and the testing
Fig. 3. The influence of anelasticity on the transients after stress changes. The loading was started at I = 0 with u = 1.1 MPa. At I = 330 s the stress was reduced by 0.56 MPa and at I = Ml s it was increased to I .1 MPa. The anelastic strain rate (see experiment A. Fig: 2) was substracted from the total, (plastic and anelastic) strain rate in order to obtain an estimate for the plastic strain rate.
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SCHNEIBEL:
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0,C stress, strain rote
1
‘I,’ Jo,
‘2’
2
Jo2
Fig. 4. Rheological model combining elasticity, anelasticity (3 Voigt-elements) and plasticity (non-linear viscous dashpot).
machine, f,. The elastic c~mpii~ce
is given
by:
J, = J,,, + J, + Jo0 = A/@.&) + J, + JoO,
(1)
where : J, = elastic compliance of the testing machine, 8, = elastic compliance of the sample (J, = 4.76*10-5 MPa-’ 1’121). Jo0 = compliance of that part of the anelasticity with relaxation times significantly smaller than those of the three Voigt elements in Fig. 4 (these short relaxation times are assumed to be 0, i.e. this part of the anelasticity is approximated by elastic behaviour), A = sample cross-section, S = stiffness of the testing machine (applied force/elastic deflection), L, = sample gauge length.
IN SUPERPLASTIC
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sums of exponentials, using the ‘subtraction of tails’method [ 163. Two different approximations, (1) and (2). for j, have been calculated (see caption to Fig. 2) from which sets of parameters Q, Joi (i = 1, 2, 3) describing J,,(r) can be found by integration (see Table 1). The shortest relaxation time (0.8 s) in approximation (2) for the compliance rate has been approximated by 0 s. In the calculations below approximations (1) or (2) were chosen depending on the duration of each transient experiment (i.e. approximation (1) for up to or IO0 s and approximation (2) for up to 4 10 ks). The anelasticity could be more accurately described by employing more than 3 Voigt elements. However, 3 Voigt elements are in many cases sufficient in order to demonstrate the effect of anelasticity on transients and a larger number of elements would make the mathematical treatment very tedious (compare equation (A5) for 3 Voigt elements). The parameters K and m characterizing the plastic properties were determined from creep experiments lasting sufficiently long for the anelastic strain rates to be negligible. For stresses below 5 10 MPa and strain rates between - to-‘OS-’ and _ 10vs s-l the strain rate sensitivity, m, was 0.41. For higher stresses higher strain rate sensitivities were found. The value of K depends on the particular strain rate sensitivity measured. In the experiments with a constant strain rate, K was found in each case from the appropriate value of m and the approximately constant flow stress at the end of the transient period. 3.2.2. Loading with a constant strain rate. Figure 5 shows the experimentally determined transient behaviour upon straining with a constant strain rate, &. Two transients have been calculated: one without and one with anelasticity. The calculation without anelasticity (i.e. a spring and a power-law dash-pot in series)
2:~~ co~culotlon(wtthoutonolosticltyf
The di~erential equation describing the rheological model in Fig. 4 (equation (AS)) and the initial values rquired for its numerical iteration during a succession of strain rate changes are derived in the Appendix. In section (a) below the plastic and anelastic parameters required to define the coefficients in equation (AS) will be determined. In sections (b) to (e) measured stress transients following particular strain rate changes will be presented and compared to the corresponding transients calculated by means of equation (AS) as well as to transients calculated without taking anelastic strains into account. It will be seen that the incorporation of anelasticity may substantially improve the agreement between measured and calculated transients. 3.2.1, Anelustic and plastic parameters employed for the calculation of the transients. The anelastic behaviour of the sample employed for the strain rate change tests is represented in FQ. 2 (experiment B). The anelastic compliance rate, J, may be approxi~t~ by
Fig. 5. Experimental and calculated transients for loading with a constant strain rate. i (t 2 0) = 2.24+10T6sT1. The numerical values required to solve equations (AS) fplasticity and anelasticity) and equation (2) (no anelasticity) are listed in Tables I and 2.
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
Table 1. Numeri~I values for the Voigt elements in Fig. 4 and for Jo0 in equation (I). obtained from different ap proximations of experiment B in Fig. 2 Approximation of experiment B in Fig. 2
i
(1)
0 f
2 3 0 (2)
Joi/MPa- ’
Qls
0
0
6. IO-’ 1.85.1o-s 9*10-*
0.5 5 50
4.64.10-* 4.14~10-5
1.5*10-’
: 3
4.8*10-*
0 18 300 4000
solving the equation :
&olves
J, 6 + (u,‘K)“~ = &,,
(2)
where Jo0 = 0 [compare equation (l)]. The transient including anelasticity was calculated by means of equation (AS). The data employed for the calculations are listed in Tables 1 and 2, and the caption to Fig. 5. A perfect agreement between the calculated transient incorporating anelastic deformation, and the experiment cannot be expected since the assumption of the linear additivity of plastic and anelastic strains implies a plastic transient (see Fig. 1). The initial plastic strain rate in the experiment is then likely to be higher than the steady-state rate employed for the calculation. This might explain why the measured stresses are lower than the calculated ones. The incorporation of the anelasticity in the model nevertheless allows a much more accurate prediction of the loading transient than an approach based only on elasticity and plasticity (see Fig 5). 3.2.3. Strain rate cycling. The strain rate cycling behaviour of G-38.1 w/o Pb is also significantly influenced by anefasticity. A typical experimental stress vs time relationship is shown in Fig. 6 and compared with the transients calculated with equation (AS). For the strain rate increase shown, the calculated stresses are smaller than the measured ones, Probably the strain rate sensitivity has been underestimated. However, the general shape of the calculated curve (i.e. the values of G(t)) is in much better agreement with the experiment than the shape of the transients without anelasticity [equation (211.This is seen
IN SUPERPLASTIC
1531
ALLOYS
for the strain rate reduction at t = 82s in Fig. 6. Hedworth and Stowell[9] suggested that after a strain rate change there is an initially linear change in the stress (i.e. ir = const.). They evaluated strain rate sensitivities using the difference between the stress immediately prior to a strain rate change and the stress for which the transient after the strain rate change deviates from linearity. Since Fig. 6 suggests that the shape of the transient is significantly influenced by anelasticity. the deviation of the initial transient from linearity should not be used to measure a parameter characterizing the plastic deformation of Sn-38.1 w/o Pb, namely, the strain rate sensitivity. 3.2.4. Stress relaxation. The results of two stress relaxation experiments performed after straining for different periods of time are shown in Fig. 7. The calculated relaxation curves start with noticeably smaller values of llrj than the experimental ones. This is due to the fact that the fastest relaxation time in approximation (2) in Fig. 2 (T iii: 0.8 s) has been set to 0. The anelastic compliance Jo0 thus is approximated by an elastic compliance. According to equation (1) the total elastic compliance of the system therefore increases and the relaxation rate decreases. The oscillations in the calculated curves are due to the approximation of the anelasticity by a sum of exponentials. Neither the present experiments nor the calculations indicate a plateau region which has been found experimentally in the superplastic Sn-Pb eutectic [63 and has been calculated with the anelasticity characterized by only one relaxation time [17]. Carefully controlled stress relaxation experiments (the whole loading train including the load cell, with an overall length of * 300mm. was held at a temperature of (298 f 0.05 K) down to strain rates of approximately 5*10-” S-I failed to establish a plateau region but resulted in relaxations similar to those in Fig, 7. The reason for this disagreement with Geckinli and Barrett’s data is not known. The effect of the anelasticity on the relaxations in Fig. 7 is not as strong as predicted. Whereas the average slope of the calculated curves in Fig. 7 is quite different for different durations of the previous straining (i.e. m - 0.29 and m 5 O-35), the duration of the previous straining does not seem to have much in particular
Table 2. Numerical values employed for the calculations represented in Figs S-9
Fig. Compliances (see Table I) 0 (f = O)/MPa Cl
0 =
0)
l,3
0 =
0)
f.3
tt f
0)
&/MPa-’ K m
5
6
7
8
9
(;I
(1) 10.4
(2) 3.9
:: 0 3.91*10-’ 402 MPas*,” 0.41
6.22. 1.92. lo-’ IO-’ 8.87 *IO-” 4.5*10-:’ 1380MPasO,” 0.49
8 0 3.19-10-5 480 MPasO.*’ 0.41
(1) 10 6. IO-’ 1.85*10-4 8.98‘ 10-L 4.76. 1o-5 600 MPas*,*’ 0.41
6*10-’ 1.85. lo-’ 8.55. IO-’ 4.69. 1O-s 600 MPas*,‘l 0.41
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SCHNEIBEL:
TRANSIENT
BEHAWOUR
IN SUPERPLASTIC
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onelosticityk
Fig. 6. Experimental and calculated transients during strain rate cycling. Prior to the first strain rate change shown (at t = 0) the sample has been deformed with a stress of s 10.4 MPa for -_ 150 s. The strain rates employed are indicated. The data needed for the calculations are found in Tables I and 2.
influence on the slope of the measured curves. However, an important resuit of the calculations is born out by the experiments, namely, that anelasticity reduces the slope of the relaxation curves in a log c vs log 6 plot. This can be seen by comparing the experimental and calculated curves with the broken lines in Fig. 7 which have been evaluated from equation (2) for i 0 = 0. The anelasticity should therefore not be neglected during a stress relaxation test and the slope in the corresponding log u vs log ir plot should not immediately be identified with the strain rate sensi-8
-7
-5
‘i
tivity. This view is in disagreement with Murty’s [ 181 results. He did not find an appreciable difference between conventionally evaluated data (neglecting anelasticity) found from stress relaxation tests with Sn-Pb and data generated by strain rate cycling. However, he compared his data only over approximately 1 decade in the strain rate. The influence of anelasticity on stress relaxation tests may therefore have been overlooked. 3.2.5. Stress transient dip tests. A series of stress transient dip tests [19] was performed. In one case the previously strained sample was unloaded as rapidly as possible (by applying a high negative strain rate for a short period of time) and, immediately afterwards, the strain rate was set equal to 0 (Fig. 8). 5 T = 298K.
onelast~c~ty)
coiculotion~wtthout -3
anelostfctty) -2
fmstontoneous
-1
unloading)
logt-~/(MPo c’1 0
P \D
-
Mow
i
unloodlng)
1
,without colculatl~n fwlthout
’ 4
-7
I
-3
onelosticttyf
I
-2 togr-b/W.tPo S&l
I
onelosticity
1
I:
-I
after long loading relaxation 7. Stress Fig. (i = 8.2.1O-6s-’ for 1330s) and after short loading (i = 2.0. lo- 5 s- 1for 38s). The calculated times needed for the relaxation to various stresses are indicated. The iteration of equation (AS) was started with the constant strain rate-loading prior to the relaxations (at r = 0 in Table If, The data needed for the calculations are listed in Tables 1 and 2.
Fig. 8. Experimental and calculated reloading behavior. After a stress reduction from 10-o MPa the total strain (of testing machine and sample) was kept constant. The strain rate employed in the case of siow unloading was e = -2.2*10-‘s-‘. In the experiment 5s were needed for the unloading, in the corresponding calculation 7 s. The other numerical values needed for the calculation are found in Tables I and 2.
SCHNEIBEL:
TRANSIENT BEHAVIOUR IN SUPERPLASTIC ALLOYS
1533
a
6 0
5 t/s
10
0
5
10
1/S
Fig. 9. Experimental and calculated dip tests. After stress reductions from an initial stress of 10 MPa the total strain (of testing machine and sample) was kept constant. The numerical values needed for the iteration of equation (A5). starting at t = 0. are shown in Tabies 1 and 2. The anefastic strains prior IO the partial unload&s
were evaluated from the stress history prior to the unloading.
Reloading caused by the anelastic strain stored during the previous straining is observed. The reloading causes plastic deformation, and at a later stage conventional stress relaxation (i.e. a decrease in the load) is observed. Equation (AS) was successively solved starting with the negative strain rate employed in the experiment above and with appropriate values of the anelastic strains of the three Voigt elements found from the previous loading history (‘slow unloading’ in Fig. 8). As would be expected from the plastic transient implied in Fig. I the predicted stresses increase to higher values than the measured ones. However. the stress rates immediately after the stress drop are similar. Another calculated curve presented in Fig. 8 (‘instantaneous unloading’) demonstrates that the rate with which the unloading occurs is important. For instantaneous unloading. much higher reloadingstresses are predicted than for slow unloading. In terms of the model in Fig. 4 the ‘fastest’ Voigt element employed (T, = 0.5 s) does not contract during instantaneous unloading and thus causes very high reioading. For slow unloading rates (in the calculated case z 7 s were needed for complete unloading). however, the strain of the ‘fastest’ Voigt element is nearly in phase with the actual stress and does therefore not contribute much to the reloading. In this case. the Voigt element with 7, = 0.5 s only introduces an additional. approximately elastic compliance and causes therefore the reloading due to the Voigt elements with 12 = Ssand ‘ts = 50 s to be less pronounced. In Fig. 9 the measured stress vs. time behaviour after partial unloading is shown, together with computed curves. The stress increments calculated for ap proximately the first second after the completion of the partial unloadings may be caused by an overestimate of the anelastic strain rates for short times. After longer periods of time. however, the predicted curves are in reasonable agreement with the experimental ones. in particular for unioadings by less than - 50°b of the previously applied stress. A comparison with the curves calculated from equation (2) (‘no aneiasticity’ in Fig. 9) shows that the calculation incorporat-
ing anelasticity indeed constitutes a major inprovement. The present calculations substantiate. in somewhat greater detail, Geckinii and Barrett’s [6] result. namely. that the influence of anelastic strains on dip tests performed with superplastic Sn-Pb eutectic should not be neglected. These dip tests may measure an apparent internal stress which is largely caused by anelastic recovery. 4. SUMMARY
AND CONCLUSIONS
(1) The pronounced anelasticity found in superpiastic Sn-38.1 w/o Pb influences the transient behaviour after stress changes and strain rate changes. (2) Stress change experiments performed with Sn-38.1 w/o Pb indicate that the transient behaviour can be better described by the linear addition of plastic and anelastic strain rates than by an approach neglecting anelasticity completely. (3) The transients following strain rate changes may be approximately calculated if a power-law is assumed for the plastic strain rate and if 3 Voigt eiements are fitted to the anelastic data. The corresponding 4th order differential equation and the initial values required for its solution are derived. (4) Transients measured after strain rate changes performed with Sn-38.1 w/o Pb are compared with transients calculated from the independently measured plastic and anelastic properties. Loading with a constant strain rate. strain rate cycling and stress relaxation (also after complete and partial unioadings. i.e. dip tests) are examined and the agreement between the experiments and the theory is found to be reasonable. The incorporation of anelastic strains constitutes a major improvement as compared to an approach based only on plasticity and elasticity. In agreement with experiments and calculations the apparent strain rate sensitivities measured in a stress relaxation test are smaller than the strain rate sensitivities determined from steady-state creep experiments. (51 It is concluded that, in the investigated Sn-Pb alloy. transient tests to assess the plastic properties
1534
SCHNEIBEL:
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should be considered with care since the anelastic strains during these tests usually cannot be neglected. An accurate analysis of transient tests would require more knowledge than is presently available about the mechanisms of plasticity and anelasticity in this material and, in particular, about the relative contributions of those mechanisms to the sample strain.
IN SUPERPLASTIC
Equations (Al) and (AZ) define the differential equation for an anelastic solid with three discrete relaxation times (see for example Nowick and Berry (15) ). Evaluation ofi, and its higher derivatives from equation (A3) and adding of the resulting expressions to the differential equation for the anelastic solid. with suitable weighting factors, results in the differential equation describing the model in Fig. 4: + a,& + alii + asa + a4F + aso’-‘d
a# Acknowledgements-The author would like to acknowledge financial support by the German Academic Exchange Service and the provision of laboratory facilities by Professor Sir Peter Hirsch F.R.S.
REFERENCES
ALLOYS
+ agun-W
+ ala”-%’
+ a9an-“(iii
+ alOan-lij
+ a#-‘li
= -i + &,i + ha< + bsz-.
WI
where:
G. J. Lloyd and R. J. McElroy, Acra merall. 22, 339 (1974). G. J. Lloyd and R. J. McElroy, Phil. Mag. 32, 231 (1975). J. H. Schneibel and P. M. Hauledine, Proceedings of the 5th International Conference on the Strength of Metals and Alloys, Aachen, West Germany (edited by P. Haasen, V. Gerold and G. Kostorz), Vol. 1. p. 259. Pergamon Press (1979). 4. C. Homer and B. Baudelet. Scripra meroll. 11, 185 (1977). 5. D. J. Eastgate, Thesis, Oxford (1978). 6. A. E. Geckinli and C. R. Barrett, Scripra moral/. 8, 11.5 (1974). D. A. Woodford, Metoll. Trans. A7, 1244 (1976). ;I: R. Horiuchi. A. B. El-Sebai and M. Otsuka. Phvsica starus solidi (a) 21, KS9 (1974). 9. J. Hedworth and M. J. Stowell, J. Marer. Sri. 6, 1061 (1971). 10 K. Nuttall, J. Inst. Me&s 99, 266 (1971). 11. W. Blum and F. Pschenitzka. Z. Merallk. 67.62 (1976). 12. B. Subrahmanyam, Trans. Japan Inst. Merals 13, 89 (1972). 13. F. A. Mohamed and T. G. Langdon. Phil. Mag. 32,697 (1975). 14. W. D. Nix and B. Iischner, Proceedings ofrhe 5th Inrernational Confirence on the Strengrh of Merals and Alloys, Aachen, West Germany (edited by P. Haasen, V. Gerold and G. Kostorz), Vol. 3, p. 1503. Pergamon Press (1979). 15. A. S. Nowick and B. S. Berry. Anehstic Relaxorion in Crvsralline Solids. Academic Press. New York (1972). 16. R.-G&la and C. A. Wert, Acra metal/. 14, 1095 (1966). 17. J. H. Schneibel and P. M. Hazzledine. Scripra merall. 11, 953 (1977). 18. G. S. Murty, J. Marer. Sci. 8, 611 (1973). 19. G. B. Gibbs, Phil. MQ~. 13,317 (1966).
n = I/m, a0
=
(A6)
K-l/m
(A7)
bl 3: tl +
72 +
b2 = tlf2
+ ?,t3 + ~~7~.
bs =
t3.
w3)
(A9)
~1~2~3.
WOf
a; = J, + JO! + Jo2 + Jos. at =
T3(J, +
a3 =
+
Jot
r*(J,
T~TJ(J. +
a4 =
+ +
+
102) +
Jo2) + 502) +
r,r,(J,
+
(All) Tt(Jt+
(71 + ~2fdJ,
Jo,)
7NO3. +
6412)
JOI)
(A13)
Jo313
(A14)
~~f273L
Qs = aobln,
(AJ5)
Qb = aob2nfn - 1).
(A16)
Q7 = aobsn(n - l)(n - 2),
(A17)
as = aOb2n.
(A181
a9 %O
=
3aobsn(n
=
aobsn.
(b) ~niria~ t&es.
-
1).
(A19) (A201
From equation (A2) follows:
&,i = (Jota - e*r)/Tj* i = L2.3.
fA211
From equations (Al), (A3) and (A4). one obtains for a constant applied strain rate i. = *(to < I & rl) and an initial stress Q. = a(ro):
Since & = Fe = 0. differentiating of equations (Atl) and (A22) leads to:
APPENDIX
& - ( Jor&o - i&q,
The diflrenrial equarion for rhe rheological model shown in Fig. 4 and rhe initial wlues needed for irs irerarire solution
i = 1.2.3,
(A=)
(a) Di#erenriQl equorion. The equations for the individual elements of the model shown in Fig. 4 are: Jru = 4,
(Al)
JegQ = e*i + r&r* i = 192.3
a = K$*
a, = -
6421
,il F,
+ aont(n -
I)&‘Qo
J..
1.2.3.
643) + a~-‘iio}
3
e=e,+
[
I:Qd-f,. ,=I
where the notations are as previously.
(A4)
Ii
i=
(A26)
Equations (A21)-(A26) define the initial values of citro)= 60. = ii0 and g(rO) = Z. if initial values of a(to) = ao.
ii
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
g(re G r Q I,) = 40 and s&r = ro) (i = 1.2.3) are provided. With these initial values equation (AS) can now be integrated between I = to and r = r,,. From the values of u”‘(t,) = da/dt’(t,) = a:” (i = O-3) immediately prior to a strain rate change at f = I,. the values of cni (i = 1.2.3) at this moment can be found by solving a system of 3 linear equations in iai. The first equation of the system is obtained by rewriting equation (A22):
ii, i,i =
io -
J,&, -
aoa;.
(~27)
IN SUPERPLASTIC
Sjmilarly. differentiating and elimination of
ALLOYS of equations
1535 (A28) and (A29),
f F#al i=, results in the third equation for & :
j,
aailTf
In the relation obtained from equation (AZ)
one rentaces (with eauations (All. (A3). and (A411the left-
,,=,, &,e,,y:‘
-
&
I’
= - J,~J - aona;-‘ti
(~29)
and obtains: 1 I 1 \ ii &‘ri = @I, Jei/r,)Bt + J$t +
aonal-‘6, . (A301
From the system of linear equations defined by equations (A27), (A30) and (A31), one calculates &,(I = t,) (i = 1,2,3) and obtains c,,~ (i = 1,2,3) from equation (AZ). The anelastic strains G,,~(i = 1.2, 3) thus found define, via equations (A21) to (A26). the new initial values of #I (i = l-2,3) for the integration of equation (A5). after a strain rate change at t = 1%.The flow stress u can thus be calculated-successively from equation (A$) for any succession of strain rate changes.
1. WIRODUCTZON Deformation may be conveniently anelastic and plastic &formation and plastic deformation are in
divided into etaatic, Cl]. Both anelastic
general time-depen dent; however, anelastic strains are recoverable whereas plastic strains are permanent. Anelastic deformation can contribute significantly to the strain rate transients following stress changes [I], and to the stress transients following strain rate changes. as for example in stress relaxation and in stress tran~ent dip t Now at : Departmentof MaterialsScience & Engineer-
ing, Massachusetts MA 02139, U.S.A.
institute
of Technology,
Cambridge.
tests [2]. The plastic properties of a material can be determined from the above transient tests only if anelastic deformation can be shown, to be negligible or otherwise if the evaluation takes the anelastic strains into account. Superplastic Sn-Pb eutccctic (i.e. Sn-38.1 w/o Pb) exhibits pronounced anelasticity with relaxation strengths sometimes larger than 100C3.4). Other superplastic materials, for example Sn-2 w/o Pb [33 and 2~22 w/o Al [SJ behave similarly. In the past, however, the plastic properties of superplastic materials have often been inferred from transient tests on the supposition that anelastic strains are negligible. This approach has been employed for the evaluation
1527
SCHNEIBEL:
TRANSIENT
BEHAVIOUR IN SUPERPLASTIC
of stress relaxation tests [6,7]. strain transient dip tests [8] and strain rate change tests [9]. The main purpose of these transient tests is either to determine the internal stress below which plastic deformation does not proceed or to measure the plastic properties, in particular the strain rate sensitivity, at a constant sample structure. It is not always recognised that the internal stress may not only provide a threshold stress for plastic deformation but also a driving force for anelastic recovery (causing for example straightening of bowed-out dislocation segments). Anelastic readjustment of the sample structure and the strains associated with it may interfere with attempts to measure the plastic properties for a constant sample structure.. In order to assess the applicability of the transient tests mentioned above to superplastic materials it was attempted to estimate the influence of anelastic strains on transients in Sn-Pb eutectic more accurately than has hitherto been done. As will be seen this influence can be quite marked. Transient tests, therefore, do not necessarily provide a simple means of measuring the plastic properties of this alloy. This finding will also apply to transient test data obtained for other superplastic alloys with strong anelasticity as for example Zn-Al eutectoid [5, lo]. 2 EXPERIMENTAL
PROCEDURE
The G-38.1 w/o Pb alloy (eutectic composition) investigated was prepared from Sn and Pb of 99.999% purity by melting in air or in Argon and air-casting into a copper mould with dimensions 10 x 38 x 95 mm3. The ingots were rolled down from IO mm to thicknesses between 0.2 and 1.0 mm and tensile samples with a gauge length of 26 mm and a width of 6 mm were punched from the rolled sheet. Mean true grain sizes, L,,after different heat treatments were estimated from the plastic strain rates and the previously determined relationship between plastic strain rate and grain size (no difference was made between grain and phase boundaries). Stress change experiments were performed in silicone fluid employing a constant load creep rig with a strain resolution of f 10m5 and a temperature accuracy of +0.2 K. Since the sample elongations were smdl during each test the applied stress remained ap proximately constant, for a constant applied load. Constant strain rate tests were done employing an Instron tensile testing machine. The samples were immersed in a silicone fluid bath at (298 +, 0.5) K. In the load range employed the stiffness of the testing machine including the load cell and the pull rods was found to be 2.5 N/w. Since the sample elongations during the tests were small the constant extension rate of the machine resulted in an approximately constant true strain rate. A few accurately controlled stress relaxation tests were done in air in a creep rig equipped for stress relaxation in a manner similar to that described by
ALLOYS
Blum and Pschenitzka [ll]. The temperature of the whole loading train (length _ 300mm) including the load cell was kept at (298 & 0.05) K during the relaxations. Since many tests were usually performed with one sample a sufficiently long period of time was allowed to elapse after each test so that the anelastic strain rate became very small and did not interfere with the following test. 3. MEASURED AND CALCULATED TRA!!SIENTS 3.1. Stress changes 3.1.1. Loading behaciour. A sample was stressed at 328 K for 4 13 days with 0.36 MPa and subsequently unloaded in order to measure its elastic after-effect. i.e. its anelastic contraction. In Fig. 1 the anelastic strain thus obtained has been substracted from the combined plastic and anelastic strain obtained upon loading in order to find the plastic component of the strain during loading. (The elastic strains were subtracted from the experimental data using the stiffness of the creep rig employed and the elastic compliance of Sn-38.1 w/o Pb, 4.76.IO-’ MPa-’ [12]). Linear additivity of the plastic and anelastic strains has thus been assumed. Inspection of the plastic component of the deformation shows that either this assumption does not hold strictly or the plastic strain rate exhibits transient behaviour, i.e. it is not a unique function of the stress. However, the substraction of the anelasticity in Fig. 1 reduces the magnitude of the transient considerably. The measured loading transient may therefore be approximately represented by the sum of a plastic strain rate (which depends only on the applied stress) and an anelastic strain rate (which depends on the applied stress and on the time elapsed after its application). The physical processes responsible for the plastic and anelastic deformation of superplastic Sn-38. I wo Pb are not well understood to date. Mohamed and Langdon [13] conceded that the plastic deformation behaviour is not entirely consistent with any known mechanism of superplastic flow. The run-back of piled-up grain boundary dislocations has been proposed to explain the anelastic strains [3]. However. difficulties arise in explaining the large relaxation strengths observed. In view of these uncertainties the linear additivity of the plastic and anelastic strains cannot presently be verified in terms of the underlying mechanisms. From a mechanistic point of view it may even be possible that the separation into plasticity and anelasticity is artificial. that is. if anelastic and plastic deformation are closely interlinked. Such a situation is encountered in the conventional dislocation creep of pure metals as has been discussed by Nix and Ilschner [ 143. 3.1.2. Stress cycling. Since for superplastic Sn-38.1 w/o Pb the anelastically recovered strain. E,, is approximately proportional to the applied stress. u.
SCHNEIBEL:
TRANSIENT
B~HAVr~UR
IN SWPERPLAS-UC
f 529
ALLOYS
T: SW
0
0
1
2
3
and anelastl
L
5
t/la.
Fig. 1. The influence of aneiasticiry on the loading transient of a Sn-38.1 w/o Pb sample with a grain size of -8 pm. (6 = strain. r = time. T = test temperature). The sample was stressed at 328 K for w 13 days with 0.36 MPa, The anelastic strain as a function of the time was obtained from the elastic after-effect (i.e. the anelastic contraction after unloading\. The plastic strain upon loading was estimated by subtracting the anelastic strain from the loading strain.
for a given time after unloading [3]. it is convenient to define an anelastic compliance, J*(t) = Ir,(r)/ol. The corresponding anelastic compliance rate. j&), for a Sn-38.1 w/o Pb sample with a grain size of _ 1.5 pm is plotted in Fi8.2 (experiment A) as a function of the time after unloading. Stress change experiments were performed with the same sample (Fig. 3) and corrected for anelasticity assuming linear additivity of the plastic and anelastic strain rates. and using the data in Fig. 2 (experiment A) and Boltzmann’s superposition principle fl5]. It is seen that the subtraction of the anelasticity reduces the mamitude of the fransient behaviour. In particular. if removes the contradiction which would be found in Fig. 3 if anelasticity were neglected. namely. negative ptastic strain rates in response to positive applied stresses. 3.2. Strain rate changes In section 3.1, if has been demonstrated that the adding of plastic and anelastic strains provides a more realistic description of the mechanical behav-
Fig. 2. Anelastic compliance rates, j, as a function of the time after unloading. f. Experiment A: Prior to the relaxation al t = 0 a sample was loaded with 1.04 MPa for 3.6 ks. Experiment B: Prior to the relaxation a sample was loaded with 0.73 MPa for 4 days. Two different sums of exponentials have been fitted to curve B: (1) &f) = 11.2.10-‘ exp(-r/O.Ss) c 3.7*10-6 exp (-r,5s) + l.8~10~bexp(-r~50s)~MPa~’
5-j.
(‘1 J.(t) = iS.8*10-’ exp (-r/0.8s) + X3*10-* exp (-r;18s) + S*lO-‘expf-r.3OOs) + l.t*IO-’ exp(-r/4OOOs): MPa-' s-l, The values of the parameters of the Voigt elements in Fig. 4 are obtained from approximations (1f or (7) and are listed in Table I.
iour of Sn-38.1 w/o Pb than a description in terms of plasticity alone. With the condition of linear additivity of plastic and anelastic strains and the condition that the plastic strain rates are proportional to some power of the applied stress. the transients following strain rate changes may be calculated from the plastic and anelastic properties and compared to directly measured transients, The corresponding rheologicai model is shown in Fig. 4. It consists of a power-law dashpot describing the plastic defo~ation rate. $ (proportionality constant K, strain rate sensitivity nr). three Voigt elements which will be fitted to the anelastic properties of the material (compliances JOi. retaxation times TV. anelastic strains Eai*where I’ = I, 2. 3) and a spring with a compliance J, describing the efastic defo~ation of the sample and the testing
Fig. 3. The influence of anelasticity on the transients after stress changes. The loading was started at I = 0 with u = 1.1 MPa. At I = 330 s the stress was reduced by 0.56 MPa and at I = Ml s it was increased to I .1 MPa. The anelastic strain rate (see experiment A. Fig: 2) was substracted from the total, (plastic and anelastic) strain rate in order to obtain an estimate for the plastic strain rate.
1530
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
0,C stress, strain rote
1
‘I,’ Jo,
‘2’
2
Jo2
Fig. 4. Rheological model combining elasticity, anelasticity (3 Voigt-elements) and plasticity (non-linear viscous dashpot).
machine, f,. The elastic c~mpii~ce
is given
by:
J, = J,,, + J, + Jo0 = A/@.&) + J, + JoO,
(1)
where : J, = elastic compliance of the testing machine, 8, = elastic compliance of the sample (J, = 4.76*10-5 MPa-’ 1’121). Jo0 = compliance of that part of the anelasticity with relaxation times significantly smaller than those of the three Voigt elements in Fig. 4 (these short relaxation times are assumed to be 0, i.e. this part of the anelasticity is approximated by elastic behaviour), A = sample cross-section, S = stiffness of the testing machine (applied force/elastic deflection), L, = sample gauge length.
IN SUPERPLASTIC
ALLOYS
sums of exponentials, using the ‘subtraction of tails’method [ 163. Two different approximations, (1) and (2). for j, have been calculated (see caption to Fig. 2) from which sets of parameters Q, Joi (i = 1, 2, 3) describing J,,(r) can be found by integration (see Table 1). The shortest relaxation time (0.8 s) in approximation (2) for the compliance rate has been approximated by 0 s. In the calculations below approximations (1) or (2) were chosen depending on the duration of each transient experiment (i.e. approximation (1) for up to or IO0 s and approximation (2) for up to 4 10 ks). The anelasticity could be more accurately described by employing more than 3 Voigt elements. However, 3 Voigt elements are in many cases sufficient in order to demonstrate the effect of anelasticity on transients and a larger number of elements would make the mathematical treatment very tedious (compare equation (A5) for 3 Voigt elements). The parameters K and m characterizing the plastic properties were determined from creep experiments lasting sufficiently long for the anelastic strain rates to be negligible. For stresses below 5 10 MPa and strain rates between - to-‘OS-’ and _ 10vs s-l the strain rate sensitivity, m, was 0.41. For higher stresses higher strain rate sensitivities were found. The value of K depends on the particular strain rate sensitivity measured. In the experiments with a constant strain rate, K was found in each case from the appropriate value of m and the approximately constant flow stress at the end of the transient period. 3.2.2. Loading with a constant strain rate. Figure 5 shows the experimentally determined transient behaviour upon straining with a constant strain rate, &. Two transients have been calculated: one without and one with anelasticity. The calculation without anelasticity (i.e. a spring and a power-law dash-pot in series)
2:~~ co~culotlon(wtthoutonolosticltyf
The di~erential equation describing the rheological model in Fig. 4 (equation (AS)) and the initial values rquired for its numerical iteration during a succession of strain rate changes are derived in the Appendix. In section (a) below the plastic and anelastic parameters required to define the coefficients in equation (AS) will be determined. In sections (b) to (e) measured stress transients following particular strain rate changes will be presented and compared to the corresponding transients calculated by means of equation (AS) as well as to transients calculated without taking anelastic strains into account. It will be seen that the incorporation of anelasticity may substantially improve the agreement between measured and calculated transients. 3.2.1, Anelustic and plastic parameters employed for the calculation of the transients. The anelastic behaviour of the sample employed for the strain rate change tests is represented in FQ. 2 (experiment B). The anelastic compliance rate, J, may be approxi~t~ by
Fig. 5. Experimental and calculated transients for loading with a constant strain rate. i (t 2 0) = 2.24+10T6sT1. The numerical values required to solve equations (AS) fplasticity and anelasticity) and equation (2) (no anelasticity) are listed in Tables I and 2.
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
Table 1. Numeri~I values for the Voigt elements in Fig. 4 and for Jo0 in equation (I). obtained from different ap proximations of experiment B in Fig. 2 Approximation of experiment B in Fig. 2
i
(1)
0 f
2 3 0 (2)
Joi/MPa- ’
Qls
0
0
6. IO-’ 1.85.1o-s 9*10-*
0.5 5 50
4.64.10-* 4.14~10-5
1.5*10-’
: 3
4.8*10-*
0 18 300 4000
solving the equation :
&olves
J, 6 + (u,‘K)“~ = &,,
(2)
where Jo0 = 0 [compare equation (l)]. The transient including anelasticity was calculated by means of equation (AS). The data employed for the calculations are listed in Tables 1 and 2, and the caption to Fig. 5. A perfect agreement between the calculated transient incorporating anelastic deformation, and the experiment cannot be expected since the assumption of the linear additivity of plastic and anelastic strains implies a plastic transient (see Fig. 1). The initial plastic strain rate in the experiment is then likely to be higher than the steady-state rate employed for the calculation. This might explain why the measured stresses are lower than the calculated ones. The incorporation of the anelasticity in the model nevertheless allows a much more accurate prediction of the loading transient than an approach based only on elasticity and plasticity (see Fig 5). 3.2.3. Strain rate cycling. The strain rate cycling behaviour of G-38.1 w/o Pb is also significantly influenced by anefasticity. A typical experimental stress vs time relationship is shown in Fig. 6 and compared with the transients calculated with equation (AS). For the strain rate increase shown, the calculated stresses are smaller than the measured ones, Probably the strain rate sensitivity has been underestimated. However, the general shape of the calculated curve (i.e. the values of G(t)) is in much better agreement with the experiment than the shape of the transients without anelasticity [equation (211.This is seen
IN SUPERPLASTIC
1531
ALLOYS
for the strain rate reduction at t = 82s in Fig. 6. Hedworth and Stowell[9] suggested that after a strain rate change there is an initially linear change in the stress (i.e. ir = const.). They evaluated strain rate sensitivities using the difference between the stress immediately prior to a strain rate change and the stress for which the transient after the strain rate change deviates from linearity. Since Fig. 6 suggests that the shape of the transient is significantly influenced by anelasticity. the deviation of the initial transient from linearity should not be used to measure a parameter characterizing the plastic deformation of Sn-38.1 w/o Pb, namely, the strain rate sensitivity. 3.2.4. Stress relaxation. The results of two stress relaxation experiments performed after straining for different periods of time are shown in Fig. 7. The calculated relaxation curves start with noticeably smaller values of llrj than the experimental ones. This is due to the fact that the fastest relaxation time in approximation (2) in Fig. 2 (T iii: 0.8 s) has been set to 0. The anelastic compliance Jo0 thus is approximated by an elastic compliance. According to equation (1) the total elastic compliance of the system therefore increases and the relaxation rate decreases. The oscillations in the calculated curves are due to the approximation of the anelasticity by a sum of exponentials. Neither the present experiments nor the calculations indicate a plateau region which has been found experimentally in the superplastic Sn-Pb eutectic [63 and has been calculated with the anelasticity characterized by only one relaxation time [17]. Carefully controlled stress relaxation experiments (the whole loading train including the load cell, with an overall length of * 300mm. was held at a temperature of (298 f 0.05 K) down to strain rates of approximately 5*10-” S-I failed to establish a plateau region but resulted in relaxations similar to those in Fig, 7. The reason for this disagreement with Geckinli and Barrett’s data is not known. The effect of the anelasticity on the relaxations in Fig. 7 is not as strong as predicted. Whereas the average slope of the calculated curves in Fig. 7 is quite different for different durations of the previous straining (i.e. m - 0.29 and m 5 O-35), the duration of the previous straining does not seem to have much in particular
Table 2. Numerical values employed for the calculations represented in Figs S-9
Fig. Compliances (see Table I) 0 (f = O)/MPa Cl
0 =
0)
l,3
0 =
0)
f.3
tt f
0)
&/MPa-’ K m
5
6
7
8
9
(;I
(1) 10.4
(2) 3.9
:: 0 3.91*10-’ 402 MPas*,” 0.41
6.22. 1.92. lo-’ IO-’ 8.87 *IO-” 4.5*10-:’ 1380MPasO,” 0.49
8 0 3.19-10-5 480 MPasO.*’ 0.41
(1) 10 6. IO-’ 1.85*10-4 8.98‘ 10-L 4.76. 1o-5 600 MPas*,*’ 0.41
6*10-’ 1.85. lo-’ 8.55. IO-’ 4.69. 1O-s 600 MPas*,‘l 0.41
1532
SCHNEIBEL:
TRANSIENT
BEHAWOUR
IN SUPERPLASTIC
ALLOYS
onelosticityk
Fig. 6. Experimental and calculated transients during strain rate cycling. Prior to the first strain rate change shown (at t = 0) the sample has been deformed with a stress of s 10.4 MPa for -_ 150 s. The strain rates employed are indicated. The data needed for the calculations are found in Tables I and 2.
influence on the slope of the measured curves. However, an important resuit of the calculations is born out by the experiments, namely, that anelasticity reduces the slope of the relaxation curves in a log c vs log 6 plot. This can be seen by comparing the experimental and calculated curves with the broken lines in Fig. 7 which have been evaluated from equation (2) for i 0 = 0. The anelasticity should therefore not be neglected during a stress relaxation test and the slope in the corresponding log u vs log ir plot should not immediately be identified with the strain rate sensi-8
-7
-5
‘i
tivity. This view is in disagreement with Murty’s [ 181 results. He did not find an appreciable difference between conventionally evaluated data (neglecting anelasticity) found from stress relaxation tests with Sn-Pb and data generated by strain rate cycling. However, he compared his data only over approximately 1 decade in the strain rate. The influence of anelasticity on stress relaxation tests may therefore have been overlooked. 3.2.5. Stress transient dip tests. A series of stress transient dip tests [19] was performed. In one case the previously strained sample was unloaded as rapidly as possible (by applying a high negative strain rate for a short period of time) and, immediately afterwards, the strain rate was set equal to 0 (Fig. 8). 5 T = 298K.
onelast~c~ty)
coiculotion~wtthout -3
anelostfctty) -2
fmstontoneous
-1
unloading)
logt-~/(MPo c’1 0
P \D
-
Mow
i
unloodlng)
1
,without colculatl~n fwlthout
’ 4
-7
I
-3
onelosticttyf
I
-2 togr-b/W.tPo S&l
I
onelosticity
1
I:
-I
after long loading relaxation 7. Stress Fig. (i = 8.2.1O-6s-’ for 1330s) and after short loading (i = 2.0. lo- 5 s- 1for 38s). The calculated times needed for the relaxation to various stresses are indicated. The iteration of equation (AS) was started with the constant strain rate-loading prior to the relaxations (at r = 0 in Table If, The data needed for the calculations are listed in Tables 1 and 2.
Fig. 8. Experimental and calculated reloading behavior. After a stress reduction from 10-o MPa the total strain (of testing machine and sample) was kept constant. The strain rate employed in the case of siow unloading was e = -2.2*10-‘s-‘. In the experiment 5s were needed for the unloading, in the corresponding calculation 7 s. The other numerical values needed for the calculation are found in Tables I and 2.
SCHNEIBEL:
TRANSIENT BEHAVIOUR IN SUPERPLASTIC ALLOYS
1533
a
6 0
5 t/s
10
0
5
10
1/S
Fig. 9. Experimental and calculated dip tests. After stress reductions from an initial stress of 10 MPa the total strain (of testing machine and sample) was kept constant. The numerical values needed for the iteration of equation (A5). starting at t = 0. are shown in Tabies 1 and 2. The anefastic strains prior IO the partial unload&s
were evaluated from the stress history prior to the unloading.
Reloading caused by the anelastic strain stored during the previous straining is observed. The reloading causes plastic deformation, and at a later stage conventional stress relaxation (i.e. a decrease in the load) is observed. Equation (AS) was successively solved starting with the negative strain rate employed in the experiment above and with appropriate values of the anelastic strains of the three Voigt elements found from the previous loading history (‘slow unloading’ in Fig. 8). As would be expected from the plastic transient implied in Fig. I the predicted stresses increase to higher values than the measured ones. However. the stress rates immediately after the stress drop are similar. Another calculated curve presented in Fig. 8 (‘instantaneous unloading’) demonstrates that the rate with which the unloading occurs is important. For instantaneous unloading. much higher reloadingstresses are predicted than for slow unloading. In terms of the model in Fig. 4 the ‘fastest’ Voigt element employed (T, = 0.5 s) does not contract during instantaneous unloading and thus causes very high reioading. For slow unloading rates (in the calculated case z 7 s were needed for complete unloading). however, the strain of the ‘fastest’ Voigt element is nearly in phase with the actual stress and does therefore not contribute much to the reloading. In this case. the Voigt element with 7, = 0.5 s only introduces an additional. approximately elastic compliance and causes therefore the reloading due to the Voigt elements with 12 = Ssand ‘ts = 50 s to be less pronounced. In Fig. 9 the measured stress vs. time behaviour after partial unloading is shown, together with computed curves. The stress increments calculated for ap proximately the first second after the completion of the partial unloadings may be caused by an overestimate of the anelastic strain rates for short times. After longer periods of time. however, the predicted curves are in reasonable agreement with the experimental ones. in particular for unioadings by less than - 50°b of the previously applied stress. A comparison with the curves calculated from equation (2) (‘no aneiasticity’ in Fig. 9) shows that the calculation incorporat-
ing anelasticity indeed constitutes a major inprovement. The present calculations substantiate. in somewhat greater detail, Geckinii and Barrett’s [6] result. namely. that the influence of anelastic strains on dip tests performed with superplastic Sn-Pb eutectic should not be neglected. These dip tests may measure an apparent internal stress which is largely caused by anelastic recovery. 4. SUMMARY
AND CONCLUSIONS
(1) The pronounced anelasticity found in superpiastic Sn-38.1 w/o Pb influences the transient behaviour after stress changes and strain rate changes. (2) Stress change experiments performed with Sn-38.1 w/o Pb indicate that the transient behaviour can be better described by the linear addition of plastic and anelastic strain rates than by an approach neglecting anelasticity completely. (3) The transients following strain rate changes may be approximately calculated if a power-law is assumed for the plastic strain rate and if 3 Voigt eiements are fitted to the anelastic data. The corresponding 4th order differential equation and the initial values required for its solution are derived. (4) Transients measured after strain rate changes performed with Sn-38.1 w/o Pb are compared with transients calculated from the independently measured plastic and anelastic properties. Loading with a constant strain rate. strain rate cycling and stress relaxation (also after complete and partial unioadings. i.e. dip tests) are examined and the agreement between the experiments and the theory is found to be reasonable. The incorporation of anelastic strains constitutes a major improvement as compared to an approach based only on plasticity and elasticity. In agreement with experiments and calculations the apparent strain rate sensitivities measured in a stress relaxation test are smaller than the strain rate sensitivities determined from steady-state creep experiments. (51 It is concluded that, in the investigated Sn-Pb alloy. transient tests to assess the plastic properties
1534
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
should be considered with care since the anelastic strains during these tests usually cannot be neglected. An accurate analysis of transient tests would require more knowledge than is presently available about the mechanisms of plasticity and anelasticity in this material and, in particular, about the relative contributions of those mechanisms to the sample strain.
IN SUPERPLASTIC
Equations (Al) and (AZ) define the differential equation for an anelastic solid with three discrete relaxation times (see for example Nowick and Berry (15) ). Evaluation ofi, and its higher derivatives from equation (A3) and adding of the resulting expressions to the differential equation for the anelastic solid. with suitable weighting factors, results in the differential equation describing the model in Fig. 4: + a,& + alii + asa + a4F + aso’-‘d
a# Acknowledgements-The author would like to acknowledge financial support by the German Academic Exchange Service and the provision of laboratory facilities by Professor Sir Peter Hirsch F.R.S.
REFERENCES
ALLOYS
+ agun-W
+ ala”-%’
+ a9an-“(iii
+ alOan-lij
+ a#-‘li
= -i + &,i + ha< + bsz-.
WI
where:
G. J. Lloyd and R. J. McElroy, Acra merall. 22, 339 (1974). G. J. Lloyd and R. J. McElroy, Phil. Mag. 32, 231 (1975). J. H. Schneibel and P. M. Hauledine, Proceedings of the 5th International Conference on the Strength of Metals and Alloys, Aachen, West Germany (edited by P. Haasen, V. Gerold and G. Kostorz), Vol. 1. p. 259. Pergamon Press (1979). 4. C. Homer and B. Baudelet. Scripra meroll. 11, 185 (1977). 5. D. J. Eastgate, Thesis, Oxford (1978). 6. A. E. Geckinli and C. R. Barrett, Scripra moral/. 8, 11.5 (1974). D. A. Woodford, Metoll. Trans. A7, 1244 (1976). ;I: R. Horiuchi. A. B. El-Sebai and M. Otsuka. Phvsica starus solidi (a) 21, KS9 (1974). 9. J. Hedworth and M. J. Stowell, J. Marer. Sri. 6, 1061 (1971). 10 K. Nuttall, J. Inst. Me&s 99, 266 (1971). 11. W. Blum and F. Pschenitzka. Z. Merallk. 67.62 (1976). 12. B. Subrahmanyam, Trans. Japan Inst. Merals 13, 89 (1972). 13. F. A. Mohamed and T. G. Langdon. Phil. Mag. 32,697 (1975). 14. W. D. Nix and B. Iischner, Proceedings ofrhe 5th Inrernational Confirence on the Strengrh of Merals and Alloys, Aachen, West Germany (edited by P. Haasen, V. Gerold and G. Kostorz), Vol. 3, p. 1503. Pergamon Press (1979). 15. A. S. Nowick and B. S. Berry. Anehstic Relaxorion in Crvsralline Solids. Academic Press. New York (1972). 16. R.-G&la and C. A. Wert, Acra metal/. 14, 1095 (1966). 17. J. H. Schneibel and P. M. Hazzledine. Scripra merall. 11, 953 (1977). 18. G. S. Murty, J. Marer. Sci. 8, 611 (1973). 19. G. B. Gibbs, Phil. MQ~. 13,317 (1966).
n = I/m, a0
=
(A6)
K-l/m
(A7)
bl 3: tl +
72 +
b2 = tlf2
+ ?,t3 + ~~7~.
bs =
t3.
w3)
(A9)
~1~2~3.
WOf
a; = J, + JO! + Jo2 + Jos. at =
T3(J, +
a3 =
+
Jot
r*(J,
T~TJ(J. +
a4 =
+ +
+
102) +
Jo2) + 502) +
r,r,(J,
+
(All) Tt(Jt+
(71 + ~2fdJ,
Jo,)
7NO3. +
6412)
JOI)
(A13)
Jo313
(A14)
~~f273L
Qs = aobln,
(AJ5)
Qb = aob2nfn - 1).
(A16)
Q7 = aobsn(n - l)(n - 2),
(A17)
as = aOb2n.
(A181
a9 %O
=
3aobsn(n
=
aobsn.
(b) ~niria~ t&es.
-
1).
(A19) (A201
From equation (A2) follows:
&,i = (Jota - e*r)/Tj* i = L2.3.
fA211
From equations (Al), (A3) and (A4). one obtains for a constant applied strain rate i. = *(to < I & rl) and an initial stress Q. = a(ro):
Since & = Fe = 0. differentiating of equations (Atl) and (A22) leads to:
APPENDIX
& - ( Jor&o - i&q,
The diflrenrial equarion for rhe rheological model shown in Fig. 4 and rhe initial wlues needed for irs irerarire solution
i = 1.2.3,
(A=)
(a) Di#erenriQl equorion. The equations for the individual elements of the model shown in Fig. 4 are: Jru = 4,
(Al)
JegQ = e*i + r&r* i = 192.3
a = K$*
a, = -
6421
,il F,
+ aont(n -
I)&‘Qo
J..
1.2.3.
643) + a~-‘iio}
3
e=e,+
[
I:Qd-f,. ,=I
where the notations are as previously.
(A4)
Ii
i=
(A26)
Equations (A21)-(A26) define the initial values of citro)= 60. = ii0 and g(rO) = Z. if initial values of a(to) = ao.
ii
SCHNEIBEL:
TRANSIENT
BEHAVIOUR
g(re G r Q I,) = 40 and s&r = ro) (i = 1.2.3) are provided. With these initial values equation (AS) can now be integrated between I = to and r = r,,. From the values of u”‘(t,) = da/dt’(t,) = a:” (i = O-3) immediately prior to a strain rate change at f = I,. the values of cni (i = 1.2.3) at this moment can be found by solving a system of 3 linear equations in iai. The first equation of the system is obtained by rewriting equation (A22):
ii, i,i =
io -
J,&, -
aoa;.
(~27)
IN SUPERPLASTIC
Sjmilarly. differentiating and elimination of
ALLOYS of equations
1535 (A28) and (A29),
f F#al i=, results in the third equation for & :
j,
aailTf
In the relation obtained from equation (AZ)
one rentaces (with eauations (All. (A3). and (A411the left-
,,=,, &,e,,y:‘
-
&
I’
= - J,~J - aona;-‘ti
(~29)
and obtains: 1 I 1 \ ii &‘ri = @I, Jei/r,)Bt + J$t +
aonal-‘6, . (A301
From the system of linear equations defined by equations (A27), (A30) and (A31), one calculates &,(I = t,) (i = 1,2,3) and obtains c,,~ (i = 1,2,3) from equation (AZ). The anelastic strains G,,~(i = 1.2, 3) thus found define, via equations (A21) to (A26). the new initial values of #I (i = l-2,3) for the integration of equation (A5). after a strain rate change at t = 1%.The flow stress u can thus be calculated-successively from equation (A$) for any succession of strain rate changes.