A New Set-Up to Investigate Plastic Deformation of Face Centered Cubic Metals in High Strain Rate Loading

This paper presents a novel set-up designed to investigate plastic deformation of a metal at high strain rates. The set-up is similar to conventional split Hopkinson pressure bar but the striker bar is eliminated and instead of it a spherical projectile, accelerated to high velocity with a two-stage light-gas gun, impacts and penetrates in a steel plate attached to the input bar. This results in propagation of plane waves in the bars and the sample. Impacts were carried out by aluminum spherical projectiles with diameter of 3.5 mm and velocity between 2 and 2.5 km/s onto front plates with thickness 10 mm and diameter of 55mm; the thickness of samples was between 3 and 7 mm. The experimental results were compared with 3D finite element simulation. Also, the effect of projectile velocity and sample thickness were investigated through experimental tests.


Introduction
Constitutive equations associate flow stress to effective parameters such as strain, strain rate and temperature (Meyers, 1994).Flow stress in metals logarithmically increases in the ranges of less than 10 3 s -1 or 10 4 s -1 (dependent on the material) however for the strain rate larger than 10 3 s -1 or 10 4 s -1 , flow stress goes up much more dramatically.Follansbee and Kocks (1998) and Follansbee and Gray (1991) demonstrated that the flow stress in OFHC copper starts to rise from the strain rate of 10 3 s -1 .
In the usual split Hopkinson pressure bar (SHPB) system, a sample is placed between two elastic pressure bars (input and output bar), made of high strength material, so that they remain elastic even though the sample itself may be taken to plastic strains.SHPB test is based on two assumptions: on one hand, force is in equilibrium on both sides of the sample.The required time and number of wave transits in the sample to satisfy this assumption depends on the sample length, relative impedance and sample to bars area (Ravichandran & Subhash, 1994;Yang & Shim, 2005).On the other hand, the sample deforms at constant volume.Impact experiments were carried out at the CISAS Hypervelocity Impact Facility that is based upon a two-stage light as gun (Angrilli, Pavarin, De Cecco, & Francesconi, 2003;Pavarin & Francesconi, 2004;Francesconi, Pavarin, Bettella, & Angrilli, 2008) with the ability to accelerate projectiles up to 100 mg at a maximum speed of 6 km /s.The impact facility has been used in the past to investigate the impact behavior of various materials for different structures at different impact conditions (Colombo, Arcaro, Francesconi, Pavarin, Rondini, & Debei, 2003;Nagao, Kibe, Daigo, Francesconi, & Pavarin, 2005;Francesconi, Pavarin, Giacomuzzo, & Angrilli, 2006;Francesconi et al., 2008;Pavarin et al., 2008;Higeshide, Nagao, KIbe, Francesconi, & Pavarin, 2009;Francesconi, Giacomuzzo, Kibe, Nagao, & Higashide, 2012;Francesconi, 2013;Francesconi, Giacomuzzo, Barilaro, Segato, & Sansone, 2013;Francesconi, Giacomuzzo, Branz, & Lorenzini, 2013).For this study, 3.5 mm spherical projectiles were launched at velocities in the ranges of 2 to 2.5 km/s.The impact angle was 0° for all the tests.In each experiment, an aluminum projectile was launched onto the front plate creating a crater on it and producing pressure waves that were transmitted to the input bar.In order to obtain plane waves in the bar, the front plate thickness, the input bar diameter and the impact velocity were selected according to the results of the numerical simulations presented later in section 3.2.
Strain gauges SR4 VISHAY MICRO-MEASUREMENT with a gage length of 3mm and gage factor of 2.08 were used on pressure bars.The signals from the strain gauges on the input and output bars were amplified with two 2310 Vishay amplifiers with 100 kHz bandwidth.The output signals were acquired by an oscilloscope with sampling rate equal to 250 MS/s.For each test, stress ( , strain rate ( and strain ( in the sample were calculated using the following equations (Field et al., 2004): (1) Where E is the Young's modulus of the bar material, is the elastic wave speed in the bar material which is 5850 m/s for steel 4340 and is the sample thickness.Table 2 summarizes the test parameters for different projectile velocities (V), front plate thickness (h f ), sample thicknesses (h s ), penetration depth (d h ).

Modeling of Problem
The experimental tests described in section 2 were reproduced with Abaqus 3D.Abaqus is standard and successful software to simulate plastic deformation of FCC metals in high strain rate loadings (Liu & Sun, 2014;Li & Ramesh, 2007;Verleysen & Degrieck, 2006).
An elastic model was used for modeling input and output bars made of steel 4340.Johnson-Cook plasticity model (Johnson & Cook, 1983) was used for modeling plastic deformation of projectile, front plate and sample.
Foresaid constitutive model has much success because of its simplicity and the availability of parameters for various materials of interest (Khan & Liang, 1999;Armstrong & Walley, 2008).The Johnson-Cook plasticity model is: where A is the static strength, B the strain-hardening modulus, the normal strain, C the rate sensitivity coefficient, m the thermal-softening exponent, n the strain-hardening exponent, the strain rate, the reference strain rate, the normal strain, T the current temperature, T 0 the room temperature, and T m the melting temperature.Table 3 reports the mechanical properties of steel 4340, Al 1100 and OFHC copper as well as the coefficients of Johnson-Cook plasticity model for OFHC copper (Johnson & Cook, 1983;Khan & Liang, 1999;Armstrong & Walley, 2008;Fathipour, Zoghipour, Tarighi, & Yousefi, 2012).
Table 3. Coefficients in the Johnson-Cook plasticity model (Johnson & Cook, 1983;Khan & Liang, 1999;Armstrong & Walley, 2008), Johnson-Cook damage model (Fathipour, Zoghipour, Tarighi, & Yousefi, 2012;Johnson & Cook, 1985) and Mie-Gruneisen EOS for different metals (Corbett, 2006;Steinberg, 1996)  In addition to the plasticity model, the Johnson-Cook damage model was used to simulate failure in both of front plate and projectile.This model is appropriate to predict initiation of damage in ductile materials experiencing large pressures, strain rates and temperatures (Johnson & Cook, 1985).The model is: Coefficients of D 1 to D 5 are reported in Table 3 (Fathipour, Zoghipour, Tarighi, & Yousefi, 2012;Johnson & Cook, 1985).D is the damage parameter and failure occurs when D=1.  * is stress triaxiality and is defined as ratio of effective to hydrostatic stress.D 1 to D 5 are material-dependent parameters.Johnson-Cook damage model was used for simulation of projectile penetration to the front target.An element removes from the mesh when damage parameter (D) reaches the ultimate value.
The Mie-Grüneisen EOS was used to describe the volumetric behavior of Al 1100 and steel 4340: Where is hydrostatic pressure, ρ is the initial density, Γ is the Gruneisen's gamma at reference state, and e are:

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Table 1 .
Materials and size of spherical projectile, front plate, pressure bars and specimen

Table 2 .
Test parameters.Uncertainly are ± 50 m/s for V, ± 0.05 mm for h f , h s and d h