A sigmoidal model for superplastic deformation

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A sigmoidal model for superplastic deformation W Pan1, M Krohn1, S B Leen1 , T H Hyde1, and S Walløe2 1 School of Mechanical, Materials, and Manufacturing Engineering and Management, University of Nottingham, Nottingham, UK 2 Rolls-Royce plc, Derby, UK The manuscript was received on 1 October 2004 and was accepted after revision for publication on 31 May 2005. DOI: 10.1243/146442005X10355

Abstract: A new phenomenological model, designed to capture the sigmoidal nature of stress dependency on strain rate for superplastic deformation, is presented. The model is developed for the Ti–6Al–2Sn–4Zr–2Mo alloy using data obtained under controlled strain-rate tensile tests spanning a range of strain rates and temperatures, from 930 to 980 8C. The sigmoidal model performance is compared with that of a more conventional double-power law, strain, and strain-rate hardening model using time-dependent finite element and theoretical analyses. The primary intended application of the sigmoidal model is for more accurate simulation of the effects of strain-rate variation within test specimens and sheet during superplastic deformation. Analysis of this variation within two designs of tensile test specimens is presented to illustrate this aspect. Keywords: superplastic forming, sigmoidal model, Ti – 6AI – 2Sn – 4Zr – 2Mo 1

INTRODUCTION

Superplastic forming (SPF) technology is an important process in the manufacture of net shape parts for the aerospace and automotive industries. The ability of SPF materials to undergo large deformations is, however, offset by the requirements for high temperatures and controlled, typically, slow strain rates. The need for a controlled strain rate during SPF in order to optimize the material deformation and the widespread use of finite element (FE) modelling for process modelling of SPF mean that the constitutive model is an important ingredient in successful SPF. One of the simplest and most commonly used models takes the flow stress to be a product of power-law functions of strain rate and strain, on the basis of uniaxial test data, as presented for Ti–6Al– 4V alloy, for example, by Ghosh and Hamilton [1]. More complex models include the effects of grain size and growth, such as Argyris and St Doltsinis [2] and Chandra [3]. A more recent trend is the drive for mechanisms-based models such as those of Kim

Corresponding author: School of Mechanical, Materials, and

Manufacturing Engineering and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK. email: [email protected]

JMDA32 # IMechE 2005

and Dunne [4], who presented a constitutive equation which considered high temperature diffusion, grain boundary sliding, dislocation climb, and grain growth deformation mechanisms. To obtain the optimum values of the constitutive parameters, however, a complex four-stage least-squares technique is required, and detailed static and dynamic grain growth experimental data are required. The complexities involved in constitutive modelling of SPF, particularly for less common materials, and the common view that SPF is a complex process is in itself a disincentive to its more widespread use. The availability of phenomenological models, which cover the range of strain rates of interest and capture the correct salient behavioural trends with respect to conditions of interest, is an alternative to increasingly complex models. The sigmoidal model of the present paper is one such example, which mathematically captures the sigmoidal nature of the stress versus strain rate relationship. Although more simplified than mechanisms-based models, e.g. the sinh model of Kim and Dunne [4], this model has an advantage over power-law models for process modelling, as discussed subsequently. In the context of aeroengine manufacture, a lot of researches have been reported on the behaviour of Ti – 6Al – 4V, e.g. [1, 5], but significantly less attention has been devoted to other titanium SPF alloys, such

Proc. IMechE Vol. 219 Part L: J. Materials: Design and Applications

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W Pan, M Krohn, S B Leen, T H Hyde, and S Walløe

as Ti – 6Al – 2Sn – 4Zr – 2Mo (Ti-6242) [6– 8], which has significantly better creep resistance up to about 550 8C. This paper is primarily concerned with the tensile superplastic behaviour of Ti-6242. A future publication will address multi-axial behaviour. Other aspects relating to testing and modelling of tensile SPF behaviour are also discussed.

2

EXPERIMENTAL DETAILS

Figures 1(a) and (b) show the two types of test specimens used for the tensile tests. The Type I specimen, with a gauge length of 49 mm and a total length of 175 mm, is comparatively long and typical of the type used for non-superplastic sheet testing. The Type II specimen, with a gauge length of 25 mm and a total length of 126 mm, is comparatively short and typical of the type used for superplastic ductility. In the present study, all specimens are machined from 3 mm thick hot-rolled sheet. The testing employs a Mayes screw-operated machine with three-zone split furnace of 100 mm diameter and 250 mm length. A K-type sheathed thermocouple, which is mounted touching the centre of the specimen, controls the central zone of the furnace through a Eurotherm PID temperature controller. Upper and lower thermocouples, which are mounted touching the extremities of the specimen, read onto a temperature indicator.

Fig. 1

(a) Type I and (b) Type II tensile test specimens for SPF testing


Fig. 2

Extensometer arrangement used for strain measurement of the Type I specimen (furnace not shown)

Potentiometers on the upper and lower zones were adjusted to equalize the temperature across the specimen. The maximum permissible operating temperature is 1000 8C. Figure 2 shows the extensometer arrangement used for the Type I specimen. A plate-type extensometer, with Nimonic 80A load arms to connect each linear variable displacement transducer (LVDT) to specimen measurement locations, is used. The +12.5 mm LVDTs are mounted outside the furnace. Specially designed wire-eroded Nimonic 115 beam springs are used to effect clamping of the load arms to the specimen. The alloy composition is shown in Table 1. It is a microduplex alloy consisting of a close-packed hexagonal a-phase and a body-centred cubic b-phase. Microstructural definition was achieved using a variant of Kroll’s reagent (5 per cent HF, 10 per cent HNO3, and H2O), and the grain size was established using the visual-based linear intercept approach. At room temperature, the a-phase dominates the structure and the mean a grain size was determined to be 10.46 mm in the transverse direction and 9.53 mm in the longitudinal direction, for the as-received material. In addition, at room temperature, the yield stress (at 0.2 per cent strain) is 900 MPa, the ultimate stress is 1000 MPa, and the elongation is 10 per cent. The material is isotropic at room temperature. The annealing condition JMDA32 # IMechE 2005

Sigmoidal model for superplastic deformation

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Table 1 Composition (wt%) of the Ti – 6Al – 2Sn– 4Zr – 2Mo material C

Fe

N

Al

Mo

Zr

Sn

O

Si

Ti

0.008

0.028

0.002

5.839

1.920

4.050

1.964

0.091

0.073

Balance

is: duplex anneal for 30 min at 900 8C plus air cooling, followed by 15 min at 900 8C plus air cooling. The advantage of the Type I specimen is that available extensometry, designed for non-SPF high temperature testing, can be employed. The disadvantage is that it can only be used up to engineering strains of 30 per cent, because of the specimen versus furnace length. Consequently, the Type II specimens are employed to measure ductility. Figure 3 shows a flowchart of the strain-rate control procedure for the Type I specimen. For the Type II specimen tests, extensometry is not normally used, because of the short gauge length and the very large deformations. It is normally assumed that the strain and strain rate within the gauge length is homogeneous and this is further discussed subsequently. The required crosshead speed variation for constant strain rate, 1_ 0 , is v(t) ¼ l0 1_ 0 e(_10 t)

(1)

where l0 is the initial gauge length and t is the time in seconds. 3

CONSTITUTIVE MODELS

3.1

General

A commonly used phenomenological model for SPF behaviour is the power-law strain-rate and strain-hardening model, as follows

s ¼ k 0 1_ m 1n d p

(2)

where k0 , m, n, and p are material parameters. m is commonly referred to as the strain-rate sensitivity and n as the strain-hardening index. s, 1_ , and 1 are the instantaneous material flow stress, strain rate, and strain, respectively. d is some average measure of the current grain size, for which a separate evolution equation is required, based on

Fig. 3 Flowchart for the control procedure of the Type I specimen



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observed grain growth behaviour, including typically both static grain growth, due to temperature alone, and dynamic grain growth, due to material deformation effects [2]. Grain growth effects are not included in the present study, so that the grain size effect is lumped with the constant k0 to give a new constant k ¼ k 0 d p . Typically, a series of constant strain-rate tests are carried out at fixed temperatures for a range of strain rates to determine the constants k, m, and n, which are then dependent on temperature, strain rate, and grain size. An alternative approach, which reduces the number of tests required, is to employ the strain-rate jump technique. The approach of equation (2), although attractive for its simplicity, has some disadvantages. In commercial FE modelling of SPF processes, the rationale adopted is to determine the time variation of forming pressure required to maintain the strain rate within a specified range of the optimum value, for a given forming temperature and initial grain size. However, the strain rate can vary significantly through the part and forming cycle, from the initial flat blank to the final complex geometrical shape, so that it is not clear which strain rate should be used to determine the correct instantaneous value of pressure. One solution to this is to employ the instantaneous maximum value of strain rate within the forming blank. However, the power-law constitutive behaviour of equation (2) only provides discrete values of the material constants, corresponding to specific (tested) target strain rates, so that the target strainrate material constants are normally used to define the constitutive behaviour, assuming (incorrectly) that all points deform at this strain rate. The other important disadvantage of the powerlaw model is that the predicted strain-rate sensitivity (m-value), which has a first-order relationship with superplastic ductility [1], is very sensitive to fluctuations in measured stress, which, as shown subsequently, are common. Consequently, as also shown subsequently, it is not always clear how physically realistic the m-values obtained from test data are. 3.2

Sigmoidal model

Figure 4 shows the generic relationship between stress and strain rate for metallic materials at a given high temperature, for example, as discussed by Pilling and Ridley [9] and Hertzberg [10]. The general shape of this curve, which is referred to as sigmoidal, can be represented by the following equation log (s) ¼

a cþ

eb( log (_1=_1a ))

(3)


Fig. 4

Phenomenological curve for stress versus strain rate, with maximum m-value within superplastic behaviour region (after Hertzberg [10])

where a, b, c, and 1_ a are temperature-dependent constants. Of particular interest is the region of high strain-rate sensitivity generally referred to as the superplastic region. This equation is employed here to define the constitutive behaviour of Ti-6242 for constant temperatures but for variable strain rate. This approach therefore overcomes the first disadvantage of power-law phenomenological models, which are only applicable for discrete strain rates. In addition, the sigmoidal approach overcomes the second limitation by fitting the material constants to a logarithmic scatter plot of strain rate against stress for a given temperature, with the data obtained from tests across a range of target strain rates and for different strain intervals. The rationale is that the fitted constants are then more generally representative of the material behaviour.

3.3

Determination of material constants for sigmoidal model

The process of determining the material constants of equation (3) at a given temperature is as follows. 1. Figure 5 shows typical measured stress –strain responses for two different controlled strain-rate tests, corresponding to target strain rates of 1_ a and 1_ b , respectively, where 1_ b . 1_ a . The complete strain range is divided into N 2 1 equal intervals, limited by strain levels 11 , 12 , . . . , 1N . For each target strain rate, the average flow stress corresponding to each strain interval is determined, i.e. s 1a , s 2a , . . . , s Na 1 . 2. For the discrete times t1a , t2a , . . . , tNa (Fig. 6) associated with the strain intervals from Step 1, average a a a strain rates, 1_ 1 , 1_ 2 , . . . , 1_ N 1 , are determined for all target strain rates. JMDA32 # IMechE 2005


Fig. 5

153

Illustration of strain interval definitions for two constant strain-rate tests

3. Assemble the instantaneous average stresses and corresponding strain rates as a logarithmic a scatter plot, log (s ia ) versus log (1_ i ), as shown in Figs 7(a) and (b) at 930 and 950 8C, for example. 4. An objective function, F, is established

F¼

N 1 X

(yi yiexp )2

(4)

i¼1

where yi and yexp correspond, respectively, to the i theoretical and experimental-derived values of instantaneous logarithmic stresses. The material constants are then chosen as the values of a, b, c, and 1_ 0 , which give a minimum value of F, as described in the next step. 5. An initial guess is made for the ranges of values for the parameters a, b, c, and 1_ a . Then, F is minimized within these ranges of values. This procedure gives optimal values of the material constants corresponding to a relative minimum value for F. For example, at a temperature of 930 8C, the defined value ranges of a, b, c, and 1_ a are 1 –100, 0.2 – 20, 0.5 – 50, and 26.0 to 22.5,

Fig. 7

Logarithmic scatter plot of instantaneous average strain rate against instantaneous average stress at (a) 930 8C and (b) 950 8C

respectively, with a minima of F ¼ 0.090 determined within the F range of (0.09 2 21.887). The corresponding curve fit is shown in Fig. 7(a), whereas Fig. 7(b) shows the test data and curve fit at 950 8C. The strain-rate sensitivity of the material at a particular strain rate is determined as follows m¼

3.4

@ log s abeb ¼ @ log 1_ ½c þ eb

log (_1=_1a ) log (_1=_1a ) 2

(5)

Determination of material constants for power-law model

In this case, the constants are determined only at discrete strain rates for each temperature. The process is as follows for each temperature.

Fig. 6

Illustration of time interval definitions for determination of average instantaneous strain rates corresponding to strain intervals of Fig. 5


1. Referring to Fig. 5, for each test, determine the average flow stresses, e.g. s 1a , s 2a , . . . , s Na 1 for 1_ a , corresponding to each of the N21 strain intervals. 2. Determine the average instantaneous strain a a a rates 1_ 1 , 1_ 2 , . . . , 1_ N 1 corresponding to the strain Proc. IMechE Vol. 219 Part L: J. Materials: Design and Applications

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intervals. The same procedure is applied to all test strain rates. 3. For a given pair of constant strain-rate tests, e.g. 1_ a and 1_ b , a set of strain-rate sensitivity values, mi, corresponding to each of the strain intervals, is calculated as follows mi ¼

log (s ib =s ia ) b a log (1_ =1_ ) i

(6)

i

where i ¼ 1, 2, . . . , N 1. These mi-values are assumed to correspond to the average of the associated average strain rates from the 1_ a and 1_ b tests a b for the same i interval, i.e. [(1_ i þ 1_ i )=2]. An average m-value is then obtained from this set of mi-values using the following equation m¼

N 1 X i¼1

This approach is applied to the pairs of constant strain-rate tests, e.g. 1_ a and 1_ b , and again uses the same strain intervals, due to the presence of the log 1 strain term on the left-hand side of equation (8). This gives rise to log [(1i þ 1iþ1 )=2] a and log (s ia ) m log (1_ i ) data points and similar b to log [(1i þ 1iþ1 )=2] and log (s ib ) m log (1_ i ) data points giving 2 (N 2 1) data points, from which values of n and log(k) are obtained from the slope and intercept, respectively.

mi (N 1)

(7)

It is not clear now for which strain rate, i.e. 1_ a or 1_ b , this average m-value corresponds to. The solution adopted here is to associate two m-values with each test strain rate, one each calculated using the test data from a higher strain rate and a lower strain rate. In process modelling, the instantaneous flow stress for a given instantaneous strain rate is then calculated as the average from the two individual flow stresses calculated using the two m-values. 4. Determine the strain-hardening exponent, n, and the constant multiplier, k. Equation (2) is re-expressed as follows log k þ n log 1 ¼ log s m log 1_

(8)

n and k are then obtained by applying linear regression to a plot of log 1 versus log s m log 1_ .

Fig. 8

4

4.1

EXPERIMENTAL RESULTS AND MATERIAL CONSTANTS Experimental results

Figures 8 to 11 show the measured stress –strain curves from the constant strain-rate tests using the Type I specimens at temperatures of 930, 950, 965, and 980 8C, for three different strain rates in each case of approximately 1:6 104 , 3:6 104 , and 5.0 1024 s21. This choice of temperature range and strain rates is based on the findings of Cope et al. [7] who characterized the superplastic behaviour of Ti-6242 over the temperature range 820 – 970 8C for a range of strain rates, in a comparative study with Ti – 6Al – 4V. The results show that the flow stress increases with increasing strain rate and generally also with decreasing temperature. Strain hardening occurs in all tests and, in general, increases with decreasing strain rate and increasing temperature. Figures 12 and 13 show the stress – strain curves from the Type II specimen tests at temperatures of 930 and 950 8C, where the instantaneous strain presented is based on the current crosshead displacement, controlled according to equation (1). Figure 14 shows the variation of failure strain with strain rate at different temperatures including single test data points at 965 and 980 8C. It is clear

Experimental Type I stress–strain curves at 930 8C for different strain rates




Fig. 9


that temperature has a significant effect on the superplastic ductility from the 1 1023 s21 test results, suggesting an optimum temperature of 930 8C with respect to elongation to failure over the range of temperature considered. This trend suggests further increase in ductility at lower temperatures. However, although it is not clear what will happen at 900 8C, it is well known that SPF alloys display an optimum temperature and strain rate for elongation, either side of which the elongation will reduce. The optimum forming temperature for Ti-6242 is known to be close to 900 8C [7]. Figure 14 also suggests an optimum strain rate of 4 1024 s21. However, the figure suggests that it may be possible to use higher strain rates, depending on the required maximum deformation. There is a more dramatic reduction in ductility with reduction in strain rate below the optimum value. The Type II specimens showed significant alpha-case formation due to (a) the absence of any protective gases, e.g. argon, (b) the high temperatures, and (c) the long duration of the tests to failure (due to slow strain

Fig. 10


155

rates). However, it has been shown previously [11] that alpha case mostly affects ductility. In addition, the ratio of created alpha case over specimen dimensions is favourable. In particular, the effect of alpha-case formation is to reduce the measured ductility, due to the brittle nature of the alpha-case. Consequently, it seems reasonable to suggest that the trends of Fig. 14 may still be valid, viz. optimum strain rate and temperature, even without alpha-case formation (as would be the case in forming operations), but that the presented ductilities are conservative. For similar strain rates and temperatures, under protective conditions with argon, Cope et al. [7] reported significantly higher ductilities in the range 600 –800 per cent.

4.2

Material constants

On the basis of procedures described in section 3, using the Type I and Type II specimens, the material parameters can be calculated. The most important



156


Fig. 11


parameters are the strain-rate sensitivity and the failure strain, which are typically employed to choose the optimal process window for manufacturing process optimization. Figure 15 shows the variation of strain-rate sensitivity, as calculated using the power-law model (equations (2), (6), and (7)), with strain rate at different temperatures. Note that as discussed in section 3.4, these m-values have been obtained using equation (7) and therefore represent an average value over the complete strain range. These results suggest that Ti-6242 shows good superplastic formability, which is generally associated with m-values of greater than about 0.3. The power-law model indicates values of greater than 0.6 for all temperatures, but for specific strain-rate ranges, which are dependent on temperature in a nonsystematic way, i.e. no single optimum strain rate for all temperatures is suggested. Figures 16(a) to (d) show comparisons between the sigmoidal model and the power-law model in terms of m-value versus strain rate; the sigmoidal curve fit of Fig. 7 is representative of all temperatures. The most important distinction is that the sigmoidal model m is a continuous function of strain rate. In

general, the m-value from the power-law model is seen to lie either side of the sigmoidal model data. For the 950 8C case, the two methods are in good agreement, with the extrapolated sigmoidal curve suggesting a peak m-value at a strain rate lower than the lowest test strain rate. For other temperatures, the peak m-values from the two models do not coincide and for the sigmoidal model this is generally outside the test range, particularly at 930 and 950 8C. On the basis of consideration of the strain-rate sensitivity and failure strains, the optimal SPF temperature is 930 8C and the optimal strain rate is about 4 1024 s21.

Fig. 12

Fig. 13

Experimental Type II stress –strain curves at 930 8C for different strain rates


5 5.1

VALIDATION OF MATERIAL MODEL FE modelling of tensile tests

Figure 17 shows the three-dimensional FE model of the (uniform section) gauge length of the Type I tensile test specimen, which was analysed using the ABAQUS FE code [12]. Eight-noded, reduced

Experimental Type II stress –strain curves at 950 8C for different strain rates



157

follows the conventional approach for viscoplastic material behaviour in ABAQUS [12]. For the powerlaw model, the following incremental equation was implemented D1cr eq ¼

s 1=m n eq 1eq Dt k

(10)

where the equivalent strain, 1eq, is updated each increment. For the sigmoidal model, the corresponding incremental equation is R D1cr eq ¼ 10 Dt

Fig. 14

Variation of superplastic ductility with strain rate for Type II specimens at different temperatures

integration brick elements were employed, as recommended for large plastic deformation simulations. The simulations employed a boundary displacement user subroutine to implement controlled strain-rate conditions, using the following equation for time variation of applied displacement, u(t) u(t) ¼ l0 e1_ o t l0

(9)

where l0 is the initial gauge length and 1_ 0 is the target true strain rate. A homogeneous stress and strain state is thus induced throughout the FE model. The constitutive equations were implemented using the creep user subroutine with large deformation theory and using equivalent quantities based on an equivalent von Mises stress, seq, and the work-conjugate equivalent strain, 1eq. This

where 1 a R ¼ log 1_ a ln c b log seq

Power-law strain-rate sensitivity (m) versus 1_ at different temperatures


(12)

Figures 18(a) and (b) show FE models of the Type I and Type II specimens including the fillet regions. These models were developed to study the effect of specimen geometry on the strain-rate distributions during the tensile test.

5.2

Comparison between FE and test results

Figures 19(a) and (b) show a comparison between the FE-predicted stress –strain responses, using the sigmoidal model, and the actual test data at temperatures of 930 and 950 8C and for a range of strain rates, using the model of Fig. 17; the results for other temperatures show similar quality of correlation. The measured strain rate, as per Fig. 6, for example, was employed as input to control the specimen deformation for a more meaningful comparison. This is not possible with the power-law model. Figure 20 shows a comparison between (a) the FE power-law predictions, (b) the theoretically predicted power-law response, and (c) the measured test data at 950 8C. The results for other temperatures and strain rates show a similar quality of modelexperiment correlation. The power-law FE and theoretical responses use the target strain rate.

5.3

Fig. 15

(11)

Effect of specimen geometry

FE modelling of the different geometries of test specimen can furnish valuable insight for interpretation of test data. The Type II specimen, without extensometry, is widely used for determination of superplastic material properties [1, 2], because of the need for large extensions (hence the short original length) under controlled strain-rate conditions Proc. IMechE Vol. 219 Part L: J. Materials: Design and Applications

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Fig. 16

Comparison between sigmoidal and power-law models in terms of strain-rate sensitivity (m) versus strain rate at (a) 930 8C, (b) 950 8C, (c) 965 8C, and (d) 980 8C

(hence the use of small fillet radii). The assumption behind the use of this type of specimen is that deformation is uniformly concentrated within the gauge length due to the small fillet radius and the large ratio of specimen gauge width to specimen non-gauge width, so that it can be assumed that gauge length deformation (strain rate) can be calculated directly from crosshead displacement. In many cases, the crosshead speed is controlled according to equation (1) to account for increasing specimen gauge length; however, in some cases, the approach used is simply that of constant crosshead speed, which is clearly not satisfactory. Watanabe et al. [13], who have recently discussed the issue of appropriate shape and geometry of the test specimens for SPF property determination, as well as the appropriate loading procedure for development of an ISO standard for evaluation of tensile properties of metallic superplastic materials, concluded with a recommendation for constant crosshead speed. The justifications for this were (a) the fact that most of the stress –strain and stress– strain rate data presented in the literature have been obtained under Proc. IMechE Vol. 219 Part L: J. Materials: Design and Applications

constant crosshead velocity conditions and (b) that such a condition is easily performed using conventional tensile testing machines. The FE results of Figs 21 and 22 show two problems that can occur with this type of approach;

Fig. 17

Three-dimensional FE model of gauge length of Type I tensile test specimen



159

Fig. 18 FE models of tensile test specimens including fillet geometry for (a) Type I and (b) Type II specimens

one is possible non-validity of the assumption of uniform gauge length deformation and the other is non-validity of the assumption that deformation is concentrated within the gauge length. Figure 21 shows the variation of creep strain rate distribution along the gauge length at the specimen x ¼ 0 midplane (Fig. 18(b)) at a series of different times for a target strain rate of 5 1025 s21, thus corresponding to approximate true strain levels of between 10 and 200 per cent. This result was obtained using the model of Fig. 18(b), which controls the specimen deformation rate using equation (1), via a boundary user subroutine, but does not include the loading pin and hole in the specimen. The strain rate within the gauge length (4.45 1025 s21) is initially reasonably uniform and close to the target strain rate, giving an error of 11 per cent (inclusion of the hole will increase this error). With increasing specimen deformation, e.g. 200 per cent, the strain rate within the gauge length has become significantly non-uniform, so that near the y ¼ 0 mid-plane, the strain rate has increased to 7.53 1025 s21 giving 50 per cent error, whereas the strain rate towards the fillet end of the gauge length has decreased significantly to 1.43 1025 s21 giving 71 per cent error. For the Type I specimen (Fig. 1(a)), the same FE analysis can be performed using the corresponding mesh (Fig. 18(a)). The strain rate within the Type I gauge length, which is the same as the length of the simulated Type II mesh, is found to be uniformly distributed, with a maximum relative error in strain rate of ,0.1 per cent. These deformation inhomogeneities cause ambiguity with respect to flow stress definition. JMDA32 # IMechE 2005

Fig. 19

Comparison between FE result and test data for Type I specimen using sigmoidal model at (a) 930 8C and (b) 950 8C


160


Fig. 20

Comparison between FE, theoretical and test data for Type I specimen using the double power-law model at 950 8C for two different strain rates

Figure 22 shows the second problem that can occur with Type II specimens, namely, that the loading hole in the specimen always deforms significantly, so that the assumption of deformation being concentrated within the gauge length, as justifying use of the crosshead speed for deformation control, is not generally valid for superplastic deformation. The FE model used to predict the deformed shapes of Fig. 22 simulates contact between the specimen

Fig. 21

Variation of creep strain rate distribution along gauge length (y-direction) for Type II specimen at specimen x ¼ 0 at different times during tensile test with target strain rate of 5 105 s1


hole and the loading pin, assuming zero friction. Figure 23 shows the corresponding post-tested specimen with the enlarged hole. For an increase in gauge length of 24 mm, the original hole diameter dimension of 12 mm increases to a distance of

Fig. 22

FE-predicted deformed and undeformed shapes of upper half of Type II tensile test specimen



161

constitutive models, which give stress as a continuous function of strain rate or vice versa. Additional work is required to incorporate more detailed aspects such as grain growth effects and strain-hardening effects and to investigate implementation for multi-axial forming and validation. ACKNOWLEDGEMENT The authors wish to thank Rolls-Royce plc for funding and permission to publish, as well as Brian Webster, Barry Holdsworth, and William Robotham (University of Nottingham) for technical support, advice, and helpful discussions. Fig. 23

Comparison of the (a) deformed and (b) original Type II specimen, showing deformation of hole

REFERENCES 15.5 mm, giving an error in documented strain, on the basis of crosshead movement, of 7 per cent, for example, at a true strain of 74 per cent. For the Type II specimen, to minimize the effects of hole deformation and non-uniform gauge length deformation, one possible simple solution is to reduce the specimen thickness within the gauge length, so that the deformation will be better concentrated in the gauge length. 6

CONCLUSION

Controlled strain-rate tensile testing on the superplastic behaviour of Ti –6Al – 2Sn – 4Zr – 2Mo under different temperatures between 930 and 980 8C suggest an optimum forming temperature of 930 8C and an optimum strain rate of about 4.0 1024 s21. The tests included material constant determination up to strain levels of 40 per cent for (a) a combined strain-rate and strain-hardening, power-law material model and (b) a novel sigmoidal model for the stress versus strain rate relationship in superplastic deformation. Theoretical and FE implementations of both models have been successfully validated against the measured test data. The benefits of the sigmoidal model are (a) it is a reasonably simple formulation, based directly on the generic high temperature stress– strain rate behaviour of metals and (b) it gives stress as a continuous function of strain rate. Real specimens, even simple tension specimens, and especially the types widely used to characterize superplastic material behaviour, can experience significant variations in strain rate, even within the gauge length, during supposedly constant strain-rate testing. More realistic modelling of such behaviour for more reliable failure prediction will require JMDA32 # IMechE 2005

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APPENDIX Notation a, b, c, 1_a F

0

k, k l0 m mi n N t, T v x, y

material parameters the sum of the squares of the difference between theoretical and experimental results material parameters initial specimen length (mm) strain-rate sensitivity material strain-rate sensitivity over different strain ranges strain-hardening exponent number of strain-range segments time (s) crosshead speed variables


strain rate (s21) equivalent strain true strain component true strain rate component average strain rate for different strain rate test initial strain rate (s21) tensile stress von Mises equivalent stress tensile stress under different strain rate

1_ 1eq 1i 1_ i a b 1_ i , 1_ i 1_ 0 s seq sia , sbi

Subscripts i

1, 2, . . . , N 2 1
