Shear punch testing (SPT) was used to evaluate hot deformation constitutive parameters, and the results were compared with those of the conventional uniaxial tensile testing (UTT) method. Both tests were performed on a rolled Sn–5Sb alloy, as a model material, in the temperature range of 298–400K and under strain rates in the range of 5×10–4 to 1×10–2s–1. Reasonable agreement was found between the parameters obtained in both deformation modes for the power-law, exponential, and hyperbolic sine constitutive equations. The obtained stress exponents and activation energy values in shear deformation were almost the same as those found in the tensile deformation. Therefore, it can be concluded that the data provided by the easy-to-perform SPT can be used for the prediction of constitutive equations as well as deformation mechanisms of the material in the tensile deformation mode. Based on the power-law stress exponents in the range of 4.5–7.0 and activation energy values of about 54–59kJmol–1, dislocation climb mechanism controlled by the lattice diffusion could be suggested as the main controlling mechanism of the deformation of the alloy in both deformation modes.

Obtaining mechanical properties of materials from small samples by using miniature testing techniques seems quite attractive, especially for scarce, valuable and difficult-to-process materials [1]. Among different miniature testing methods, shear punch testing (SPT) has attracted much interest in recent years, due to several advantages including simplicity of the die and punching system, the possibility of using only very small amounts of material, and more importantly, the correlation of the obtained results with those of the conventional uniaxial tensile testing (UTT) method. Shear punch test is based on blanking operation [2], consisting of clamping a small thin sheet sample between die halves and driving a flat cylindrical punch through the sample. By plotting shear stress against normalized displacement, SPT curves that are similar to those obtained in uniaxial tensile tests, are acquired. Mechanical properties such as shear yield stress (SYS), ultimate shear strength (USS), and strain rate sensitivity (SRS) index have already been obtained from the SPT data at both room [3–5] and high temperatures [6,7]. It is now generally accepted that these data are well correlated with those of the UTT [8], and thus, attempts have been successfully made to predict tensile properties from the SPT results [9–11].

The shear punch testing method has been extensively used in the literature for the evaluation of mechanical properties of different metallic materials at room temperature, where the strain rate sensitivity is usually negligible. However, in recent years SPT has been successfully employed for the assessment of mechanical properties at elevated temperatures, and also, for measuring SRS, especially in fine-grained and nano-grained superplastic materials processed by severe plastic deformation processes, where only small amounts of materials are available [5,12–15]. Recently, SPT has been introduced as an appropriate tool for studying hot shear deformation behavior of metallic alloys. Accordingly, some reports have been published on the processing maps and constitutive analysis of Mg–Li [16,17] and Mg–Gd [18] alloys studied by this method.

It would be of some interest, however, to correlate the constitutive parameters obtained in the hot shear deformation studies by the SPT method, with those obtained under uniaxial deformation mode. This seems more vital when considering that the most important advantage of the SPT method is its simplicity and the fact that it saves time and material, as compared to the conventional testing methods. It is, therefore, the aim of the present investigation to study the correlation of hot deformation parameters of a Sn–5wt.% Sb alloy, as a model material, under the shear and uniaxial deformation modes.

2Experimental procedureRolled Sn–5wt.% Sb sheets with a nominal thickness of about 1mm were used in this investigation. The details of the material preparation and processing are explained elsewhere [19]. Tensile specimens were punched from the sheets along the rolling direction. The parallel gage length was 28mm long and 6.5mm wide. The specimens were pulled to fracture at the temperatures of 298, 320, 340, 370, and 400K and at initial strain rates of 5×10–4, 1×10–3, 5×10–3, and 1×10–2s–1 using an MTS universal tensile testing machine equipped with a three-zone split furnace. Load–extension curves were obtained over the whole gage length, from which the stress at the peak load was calculated as the flow stress needed for the calculation of the hot deformation parameters.

The 1-mm thick slices of the sheets were ground to a thickness of about 0.8mm, from which disks of 15mm in diameter were punched for the SPT. A shear punch fixture with a 3.175mm diameter flat cylindrical punch and 3.225mm diameter receiving-hole was used (Fig. 1). After locating the specimen in the fixture, the assembly of the specimen and fixture were accommodated by the split furnace. After application of the load, the applied load, P, was measured automatically as a function of the punch displacement and the data were acquired by a computer so as to determine the shear stress of the tested material using the relationship [20]:

where t is the specimen thickness and d is the average of the punch and die diameters.Schematic representation of the shear punch die assembly [5].

The tensile and shear strain rates, ε˙ and γ˙, respectively, are related to cross head speed, v, by the following equations:

where l0 is the initial gage length of the tensile samples and w is the clearance between die and punch in the SPT fixture [3]. All SPT tests were performed at the same temperatures and strain rates employed in the tensile tests. This excludes any possibility of strain rate and temperature effects on the level of flow curves. Both tensile samples and the SPT fixture were held for 20min in furnace to establish thermal equilibrium in the testing arrangement before the test was started.3Results and discussion3.1Tensile and shear deformation behaviorThe tensile and SPT curves of the alloy, obtained at 298K and under strain rates in the range of 5×10–4 to 1×10–2s–1, are shown in Fig. 2. Similar curves were obtained at the other test temperatures of 320, 340, 370 and 400K. As can be seen in Fig. 2, the strength of the material shows a great dependency on the strain rate in both tensile and shear punch tests, where the peak stress values increase with increasing the applied strain rate, demonstrating a positive sensitivity to strain rate. Furthermore, it is clear that at a given strain rate and temperature, the strength of the material in tensile deformation is greater than the corresponding shear strength. There are different types of constitutive equations for describing the relation between strain rate, flow stress and the temperature of deformation, which would be studied in details in the following section.

3.2Constitutive analysisIn the hot deformation studies, it is important to find appropriate constitutive equations for correlating the flow stress of materials with the strain rate at different test temperatures. Such equations can be used for predicting flow stress of the material at different strain rates or vice versa, for determining the appropriate strain rate for deformation of the material to achieve a specific flow stress level. Whenever, the strain rate and temperature of deformation are both in appropriate ranges, the constitutive equations are valid. In this regard, the relationship between strain rate, flow stress (σ) and temperature (T) can be described by the power-law, exponential and hyperbolic sine functions:

where A1, n1, A2, β, A3, n3 and α are constants, Q1, Q2 and Q3 are the activation energy values, and R is the universal gas constant. At low stresses (ασ<0.8), the hyperbolic sine equation, Eq. (6), can be approximated to a power relation (Eq. (4)), while at high stresses (ασ>1.2), it reduces to an exponential relation (Eq. (5)). Accordingly, at low stresses, n3≅n1, while at high stresses α≅βn3. In hot deformation studies, the common procedure is to estimate the α-value as α≅βn1, and this approximation is used in the present study.The constitutive equations described above can be simply modified for evaluating hot shear deformation behavior in the SPT method by replacing ε˙ with γ˙ and σ with τ, as described in our previous publications [17]. This conversion can be carried out using the von Mises criterion for a state of pure shear of kinematically hardening materials, which presents σ=3τ and ε=(1/3)γ. Hence, the modified constitutive equations for the hot shear deformation analysis can be rewritten as

where A′1, n′1, A′2, β′, A′3, n′3 and α′ are constants. Also, Q′1, Q′2 and Q′3 are activation energies found in the shear deformation. Similar to the α constant, the α′ constant can be approximated as α′≅β′/n′1.Due to the constancy of activation energy at a given temperature, the values of n1, n′1, β, β′, n3 and n′3 parameters can be obtained from the following equations:

Also, at constant strain rates, the activation energy values of the hyperbolic sine function, Q3 and Q′3, can be obtained from the following equations:

The relationships between strain rate, peak stress and temperature are shown in Figs. 3 and 4 for the tensile and shear punch tests, respectively. Parameters of the hyperbolic sine constitutive equations (Eqs. (6) and (9)) can be calculated from such plots according to Eqs. (10)–(17), and then, it would be very interesting to compare the obtained parameters for the tensile and shear punch tests. As can be inferred from Figs. 3a and 4a, the power-law stress exponent values can be calculated at each temperature by plotting the strain rate against stress on a log–log scale. The stress exponents were in the ranges of 4.6–7.0 and 4.5–7.0 in tensile and shear punch tests, respectively. It can be observed that the obtained stress exponents in tensile test (n1) and SPT (n′1) are very close to each other at different test temperatures. This similarity between the power-law stress exponents found in the tensile and shear punch tests seems reasonable by assuming the linear relation between σ and τ (σ=Kτ, where K is a constant) and also, considering the fact that similar strain rates have been used in both tests. Although the value of constant K does not affect the power-law stress exponent values, it should be noted that this constant is usually around 1.77, when comparing the yield stresses, and around 1.80, when comparing the peak stresses [3]. However, plotting the obtained UTS values against the USS values in Fig. 5 demonstrates that the K constant is 2.12 in the present study. The K constant greater than the theoretical von Mises ratio of 1.73 is believed to be caused by friction, bending, and stretching of the materials in the deformation zone of the SPT [21]. These effects may be intensified at high temperatures, while all previous studies [3,9–11] on the relationship between tensile and shear punch tests have been carried out at room temperature.

The β and βʹ constants in Eqs. (12) and (13) can be calculated by plotting the strain rate versus stress on a semi-logarithmic scale, as shown in Figs. 3b and 4b, respectively. Again, considering the linear relationship between σ and τ, it is anticipated that the βʹ constant is equal to Kβ, which seems to be almost true when comparing the β and βʹ constants at different test temperatures. The ratio of the average β value to average βʹ is about 2.3, which is almost comparable with the 2.12 value found for the K constant in Fig. 5.

After calculation of the n1 and β constants at all test temperatures, the average α constant was calculated to be around 0.031 according to the α≅β/n1 relation. As discussed above and was also shown in Figs. 3a and 4a, the obtained n1 values in tensile test are almost equal to n′1 values found in the SPT. Accordingly, it is expected that the α′ constant in the SPT is equal to Kα, considering the relationship between β and βʹ constants. Such approximation seems to be valid according to the calculated average α′ constant which is about 0.071. Therefore, according to the linear relationship between stresses in tensile and shear deformations, it can be concluded that the n1, β and α constants in the tensile constitutive equations can be estimated form the corresponding n′1, βʹ and α′ values in the SPT, where n1=n′1, β=β′/K and α=α′/K.

The variations of the strain rate with sinhασ and sinhα′τ plotted on log–log scales are shown in Figs. 3c and 4c. It can be observed that the n3 and n′3 constants are in the ranges of 3.9–4.5 and 4.0–4.5, respectively. Average n3 and n′3 values of about 4.26 and 4.23 were respectively, used for the calculation of the Q3 and Q′3 activation energies according to Eqs. (16) and (17). The variations of sinhασ and sinhα′τ with the reciprocal of temperature are plotted on a semi-log scale and at constant strain rates in Figs. 3d and 4d, respectively. As can be observed, average activation energy value of about 54.0kJmol–1 has been obtained in the tensile test (Fig. 3d), which is almost close to the activation energy value of 58.9kJmol–1calculated for the SPT method (Fig. 4d).

The A3 and A′3 constants in Eqs. (6) and (9) can be obtained by plotting ε˙exp(Q/RT) and γ˙exp(Q′/RT) against sinhασ and sinhα′τ on log–log scales, respectively. Such plots are shown in Fig. 6 for comparison and also for the validation of linear relationships between the supposed parameters. An average activation energy value of 56.4kJ mol–1 was assumed in both testing conditions. It can be observed that all data points obtained at different test temperatures can be fitted to single lines in both tests:

According to these equations, A′3/A3≅∼1.07, which is almost close to 1. This value seems reasonable, since at equal strain rates and activation energies:

which should be around 1, since n3≅n′3 and ασ≅α′τ. It should be noted here that the A3 and A′3 constants can be considered as material dependent constants, similar to the A1 constant in the power-law equation (Eq. (4)), which depends on material properties such as stacking fault energy [22,23]. However, this material dependency would not affect the relation between the A3 and A′3 constants (Eq. (17)), since here the same material has been used in both tests.According to the obtained results, it is concluded, therefore, that the constitutive hyperbolic sine equation found in the SPT can be used for the prediction of the corresponding constitutive equation in tensile testing. For such conversions of the constitutive parameters, it should be noted that α≅α′/K, n3≅n′3, Q3≅Q′3 and A3≅A′3, as long as the SPT and tensile test are considered in similar strain rate ranges. Accordingly, and in order to validate the applicability of the shear punch testing method for the assessment of constitutive parameters, the estimated constitutive equation for the tensile test from the SPT data is plotted in Fig. 6b (the solid line) and compared with the experimental tensile data points. It is clear that the predicted line has acceptable correlation with the experimental tensile data points. This means that the tensile constitutive equation can be obtained by performing shear punch tests on small thin samples. It should be noted here that other investigators have also reported the possibility of prediction hot deformation behavior by the mathematical models [24,25]. However, the novelty of the present study is the prediction of hot tensile deformation behavior from the experimental results obtained in the shear deformation mode. This capability of the SPT method can be very useful and important especially for scarce, valuable, and hard to process materials. It is also interesting to note that the mass of the SPT specimens is only about 14% of that of the specimens used in this investigation for the tensile tests.

Due to the similarity of the constitutive parameters found in the tensile and shear punch tests, the deformation mechanisms can also be identified from these constitutive parameters. In this regard, and under the experimental conditions where the power-law creep relation is valid, different mechanisms have been proposed for various combinations of stress exponents and activation energies. Additionally, there is a substantial contiguity between creep and hot working, so that the controlling deformation mechanism could be essentially similar [26]. According to the theories of dislocation climb-controlled creep, the stress exponent has the value in the range 4–6 and the activation energy has the value of the activation energy of lattice self-diffusion [27]. Therefore, the power-law stress exponents, n1 and n′1, which were found to be, respectively, in the ranges of 4.6–7.0 and 4.5–7.0 (Figs. 3a and 4a), and the activation energy of about 54–59kJmol–1, which is very close to the estimated activation energy for lattice diffusion in Sn (60.3kJmol–1[28]), suggest that dislocation climb controlled by lattice diffusion could be the dominant deformation mechanism in the studied alloy in the temperature range of 298–400K. The obtained power-law stress exponents and activation energies found in this study are comparable to those reported previously for the Sn–5Sb alloy obtained by different methods. Power-law stress exponents in the range of 3.4–7.1 and activation energy of 55kJmol–1 have been reported for a wrought Sn–5Sb alloy [19]. Accordingly, due to the similarity of the power-law stress exponents and activation energy values found in tensile and shear punch tests, the constitutive parameters found in the SPT can also be used for the identification of deformation mechanism of the material in the investigated ranges of temperature and strain rate.

4ConclusionsThe correspondence between hot deformation behaviors under uniaxial and shear modes was assessed for a Sn–5Sb alloy. This was achieved by the shear punch testing, as a localized testing technique, and the results were compared with those obtained by the conventional tensile testing technique. The following conclusions were made:

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Constitutive parameters calculated from the SPT data can be simply converted to the corresponding parameters in the tensile deformation mode.

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While the power-law and hyperbolic sine stress exponents are almost identical in both deformation modes, the β and α constants in the exponential and hyperbolic sine equations are proportional to the corresponding values in the SPT. The calculated activation energy values were also found to be close to each other in the tensile and shear deformation modes.

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Due to the similarity of the power-law stress exponents and activation energy values, obtained in both deformation modes, dislocation climb mechanism controlled by the lattice diffusion was suggested as the main controlling mechanism of the deformation in both cases, showing the capability of SPT for the identification of hot deformation mechanisms.

The authors declare no conflicts of interest.

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