Keywords

1 Introduction

Stainless steel is commonly used in automobile industries because of its good mechanical properties, higher formability and increased resistance to corrosion [1]. Several research activities confirm that during transportation application, the material is exposed to different strain rates in the range of 102–103 [2]. Further, the mechanical properties of the stainless steel are strongly dependent on the loading rates [3]. Thus, it is important to characterize the deformation behaviour of the stainless steel sheets at different strain rates.

Generally, the quasi-static tensile test and the split Hopkinson pressure bar (SHPB) test have been used in most of the investigations to describe the material’s mechanical properties at low and high strain rates, respectively. It is difficult to attain intermediate strain rates with the normal test set-up. In the SHPB test set-up, the lowest strain rate of 600 s−1 is reported [4]. Recently, Grolleau et al. [5] modified the SHPB into a dynamic bulge testing device and performed a dynamic material test at intermediate strain rates. Further, Ramezani and Ripin [6] conclude that this approach is limited due to strain inhomogeneity because of the increased complexity in the setup.

In the last two decades, high energy rate forming devices have been widely used to obtain the bi-axial forming behaviour of the materials over a wide range of strain rates [7,8,9]. The shock tube facility has recently been used on a lab scale to study the dynamic response of thin sheets [10,11,12,13]. It is generally used to establish an impulsive loading environment for a small duration. The uniformity in loading and ease of handling make the shock tube facility advantageous to utilize as a dynamic testing device. Different levels of shock loading provide different loading conditions that help to study the dynamic mechanical behaviour of the material. Stoffel [10] used a shock tube during the investigation of the dynamic behaviour of the sheet metals. Justusson et al. [11] utilized the shock tube during the bi-axial forming of the aluminium sheets and validated it with the FE model results. Barik et al. [12] investigated the forming behaviour of aluminium alloy sheets using a shock tube and validated the forming outputs to the results obtained from FE simulation. They obtained the rate-dependent material parameters by the tensile test of the sheet deformed using the shock tube and incorporated during FE simulation. The predicted results matched quite well with the experimental outputs.

The tensile test of the deformed sheet is new and reliable to determine the rate-dependent mechanical properties. There has never been an attempt to use the shock tube to acquire the mechanical properties of the SS 304L sheet in the intermediate strain rate range. As a result, the shock tube is used in this study to perform impact loading on SS 304L sheet. During the experiment, a strain rosette is fixed on the sheet to capture the strain evolution during the deformation. The stress–strain data under that strain rate is obtained by the tensile test of the sample cut from the sheet deformed using a shock tube. Further, the tensile test data is validated with the flow stress curve obtained from the Cowper-Symonds model.

2 Experimental Methodology

2.1 Mechanical Properties of the Material

In the current investigation, SS 304L sheet of 1 mm thickness having chemical composition Cr %: 18.20, Ni %: 8.01, Mn %: 1.00, Si %: 0.36, P %: 0.031, C %: 0.056, S %: 0.002 is considered for the analysis. The tensile properties of the base sheet are characterized along 0°, 45° and 90° to the rolling direction in a universal testing machine (UTM) at 1 mm/min crosshead speed as per ASTM-E8 and represented in Fig. 1. ASTM-E517 is used to identify the plastic strain ratios (r) of the sheets. Table 1 depicts the mechanical properties of the base sheet.

Fig. 1
Three curves on engineering stress versus engineering strain plane for zero, 45, and 90 degrees to R D for S S 304 L base sheet. All curves depict similar trends.

Engineering stress and strain curves for SS 304L base sheet

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Table 1 Mechanical properties of the base sheet
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2.2 Shock Tube Experiment

In the present investigation, a shock tube facility as illustrated in Fig. 2 has been utilized to perform dynamic loading on sheets. The details about the experimental facility can be found elsewhere [12, 13]. The pressure difference created between the driver and the driven section of the shock tube helps during the sudden rupture of the diaphragm. It generates a shock wave, which propagates at a high velocity towards the end of the shock tube. After imparting the end of the shock tube, the shock wave reflects and generates higher pressure than the incident shock wave. The high-pressure field zone is created inside the shock tube for a short period. This impulsive environment can be used to deform a sheet in bi-axial mode at different strain rates by placing it at the end of the shock tube.

Fig. 2
A 3-dimensional display of a shock tube consists of a high-pressure gas inlet, driver sections, a digital pressure gauge, a flat sheet, a fixture unit, and a diaphragm.

Illustration of the shock tube experimental facility

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During the shock tube experiment, 31.02 ± 0.68 bar of bursting pressure is attained. The detailed instrumentation in the shock tube is mentioned in the previous work of Barik et al. [12]. The pressure–time signals acquired from the pressure transducers positioned in the driven part of the shock tube are illustrated in Fig. 3. The experimentally obtained Mach number (Ms), incident pressure and reflected pressure is 1.94 ± 0.02, 4.32 ± 0.82 bar and 19.02 ± 0.55 bar, respectively.

Fig. 3
A 2-line graph displays incident pressure and reflected pressure for sensor 1 and 2 on a time scale. Incident and reflected pressures noted are 4.32 plus or minus 0.82 bar and 19.02 plus or minus 55 bar respectively.

Pressure–time history obtained from the experiment

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It is observed that SS 304L becomes rate-dependent when the rate of loading reaches a threshold limit [3]. SHPB has been used in many studies to characterize the mechanical properties of SS 304L sheets at high strain rates [14]. During the high strain rate forming, Cowper-Symonds (CS) flow stress model is also used, which takes into account the rate-dependent effect during material forming. The Cowper-Symonds (CS) flow stress model is given by

$$\sigma = \sigma_{qs} \left( {1 + \left( {\frac{{\dot{\overline{\varepsilon }}}}{{C_{m} }}} \right)^{\frac{1}{p}} } \right)$$
(1)

where \(\sigma_{qs}\) is the quasi-static constitutive behaviour of the sheet, σ is the dynamic flow stress, \(\dot{\overline{\varepsilon }}\) is the effective strain rate, Cm and P are the strain rate factors that scale the material’s initial quasi-static stress to represent the dynamic mechanical behaviour. In this study, the dynamic mechanical properties of the sheet are identified by the tensile test of the sheet deformed using a shock tube. The tensile sample is cut from the deformed sheet’s mid location along 0° to the RD (Fig. 4), and then it is tested in UTM at 1 mm/min crosshead speed.

Fig. 4
The two images represent a square sheet on the T D versus R D scale from which a dumbbell-shaped tensile test sample is derived after the deformation of the sheet with the help of a shock tube.

Tensile test sample obtained from the sheet deformed using the shock tube

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2.3 Strain Rate Evolution Measurement

During this study, the strain rosette is used to measure the strain rate evolution during the shock wave-based deformation analysis. Generally, in many high strain rate forming experiments, a high-speed 3D digital image correlation (DIC) system has been utilized to determine the transient variation of the forming parameters [11]. However, it is witnessed that the strain gauge can also be used to measure the strain rate and the results have a good agreement with the results obtained from DIC [15]. Thus, a strain rosette is attached at the centre of the sheet (Fig. 5) to quantify the in-plane strain as well as strain rate along 0°, 45° and 90° to the RD. Details about the strain rosette can be found elsewhere [12]. The strain rosette is connected by the Wheatstone quarter bridge circuit (Fig. 5). When the pressure developed by the shock wave generates impact loading on the sheet, the strain gauge’s resistance changes and it causes instability in the Wheatstone bridge.

Fig. 5
An image depicts the strain gauge fixed at the center of the sheet and is connected to the Wheatstone bridge. The setup consists of a DC voltage supply, voltage amplifier, oscilloscope, and other components.

Strain gauge mounted on the specimen during the experiment

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The output voltage from the Wheatstone bridge circuit is obtained in millivolts. It is difficult to predict the strain outputs from the measured signal. Thus, the voltage outputs are amplified in INA 128, DC voltage amplifier and then captured in an oscilloscope. The voltage outputs can be converted into strain rate signals. The strain rate signals obtained from the strain rosette along 0°, 45° and 90° to the rolling direction are represented in Fig. 6. The effective strain rate (\(\dot{\overline{\varepsilon }}\)) can be obtained using Hill’s 1948 yield criterion [16], which takes into account the peak strain rate along 0° and 90° during the calculation. Hill’s 1948 yield criterion is expressed as

$$\begin{aligned} &\dot{\overline{\varepsilon }}{}^{2} = \frac{(G + H)}{{(FG + FH + GH)^{2} }}\\&\left[ {F^{2} \left( {G + H} \right)\dot{\varepsilon }_{1}^{2} + G^{2} \left( {F + H} \right)\dot{\varepsilon }_{2}^{2} + H^{2} \left( {F + G} \right)\dot{\varepsilon }_{3}^{2} } \right] \end{aligned}$$
(2)
Fig. 6
A line graph depicts the strain rate versus the time curve for 0, 45, and 90 degrees to a rolling direction. Maximum peak is observed for 0 and 45 degrees R D at 0.001 seconds.

Strain rate developed during the shock tube-based forming of flat sheet

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where \(\dot{\overline{\varepsilon }}\) is the effective strain rate, \(\dot{\varepsilon }_{1}\) is the strain rate along 0° to RD, \(\dot{\varepsilon }_{2}\) is the strain rate along 90,° and \(\dot{\varepsilon }_{3}\) is the strain rate along the thickness direction. Identification of \(\dot{\varepsilon }_{3}\) is difficult, and it is neglected during \(\dot{\overline{\varepsilon }}\) calculation. Hill’s constant (F, G and H) can be correlated to the plastic anisotropy parameters (r0 and r90) by the relations F = r0; G = r90; H = r0r90 [16]. The values of r0 and r90 are identified experimentally as mentioned in Table 1. The effective strain rate (\(\dot{\overline{\varepsilon }}\)) calculated experimentally is 34.26 s−1.

3 Results and Discussion

3.1 Determination of Rate-Dependent Mechanical Properties

The tensile test results obtained from the deformed sheet are compared to the base sheet (Fig. 7). Table 2 depicts the mechanical properties of the deformed sheet. It is observed that both the yield stress and the ultimate tensile stress of the material rise as compared to the base sheet. The material’s strength coefficient (K) increases significantly as a result of the action of strain hardening. However, after the deformation, the strain hardening coefficient (n) does not vary as much.

Fig. 7
3 line curves on true stress versus true strain scale for base sheet, Cowper Symond model, and deformed sheet. Strain is zero till stress equals 300 Mega Pascal, after which it gradually increases.

Comparison of the stress–strain curves obtained from the deformed sheet to the Cowper-Symonds model

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Table 2 Comparison of tensile properties of the deformed sheet with the base sheet
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Furthermore, the identified rate-dependent tensile properties are validated with the dynamic stress–strain curve obtained from the CS model. During the calculation, the quasi-static flow stress data (\(\sigma_{qs}\)) obtained from the base sheet along 0° to RD is considered. The values of Cm = 17,772 s−1 and P = 3.16 for SS 304L sheets are obtained from literature [17]. The effective strain rate (\(\dot{\overline{\varepsilon }}\)) of deformation is calculated experimentally as 34.26 s−1 (reported in Sect. 2.3). The results illustrate that rate-dependent flow stress data obtained from the CS model slightly deviated from the data obtained from the tensile test of the deformed sheet. The correlation coefficient (R) between both the flow stress curves is obtained as 0.963, which is an acceptable limit. It confirms that the approach of obtaining the dynamic mechanical properties by the tensile test of the deformed sheets is reliable and can be used to predict the dynamic forming behaviour of the sheet. A similar approach can be implemented in another rate-dependent forming processes to identify the rate-dependent mechanical properties.

4 Conclusions

The present work aims to identify the rate-dependent mechanical properties of a 1-mm-thick SS 304L sheet. For this analysis, a tensile sample is cut from the mid location of the sheet deformed using the shock tube and tested in UTM. The rate-dependent flow stress–strain data is also validated with the Cowper-Symonds flow stress model. The following conclusions are drawn from the results.

  1. i.

    The tensile test data obtained from the deformed sheet has a reasonable correlation with the Cowper-Symonds model. Thus, the same procedure can be followed during the identification of mechanical properties in other high strain rate forming processes.

  2. ii.

    The yield strength and ultimate tensile strength obtained from the deformed sheet are higher than the base sheet. The strength coefficient (K) of the material raises significantly after deformation because of the strain hardening phenomenon. However, there is less variation in the strain hardening coefficient.

  3. iii.

    During the shock tube-based experiment, the strain rosette can able to acquire the strain rate of deformation. The peak strain rate confirms the rate of loading during the experiment.