On the Plastic Wave Propagation Along the Specimen Length in SHPB Test

In SHPB test, when the striker hits the incident bar in the axial direction, a trapezoidal stress impulse is generated and then propagates along the incident bar (Fig. 2). From an elastic wave analysis point of view, what composes the incident impulse and what the strain gages on the bar can record are the stress wave components with a longitude elastic velocity \( \sqrt {{{E_{{bar}} } \mathord{\left/ {\vphantom {{E_{{bar}} } {\rho_{{bar}} }}} \right. } {\rho_{{bar}} }}} \) in the axial direction. Kolsky and Davies did full frequency analysis on elastic stress wave propagation in SHPB test by using the Pochhammer and Chree equations [17‐21], which can successfully describe such wave motions in an elastic cylindrical medium [2, 3]. They concluded that wave components, which propagate along the bar with the elastic wave velocity \( \sqrt {{{E_{{bar}} } \mathord{\left/ {\vphantom {{E_{{bar}} } {\rho_{{bar}} }}} \right. } {\rho_{{bar}} }}} \), are only those with long wave lengths \( {r \mathord{\left/ {\vphantom {r {\Lambda }}} \right. } {\Lambda }}\kern1.5pt<\kern1.5pt0.1 \) (i.e. lower frequencies. Λ is wave length; r is the radius of bar) [2, 3]. So when the incident stress impulse travels along the elastic cylindrical bar, it’s distorted and dissipated. Finally, what strain gages record and what loads are applied to the specimen are only those wave components with long wave lengths. All other wave components with short wave lengths or high frequencies are filtered out.

In addition, from linear frequency analysis theory, for a step signal with a rising time T _r and a duration time Δt (shown in Fig. 3), we have the following relationship:

where T _min and T _max are the minimum and maximum period of wave components in the signal respectively. So by applying the above linear wave analysis result, the incident impulse has following properties:

And the rise time of incident impulse is around:

From equation (12), for the 19.7 mm diameter steel SHPB system in the Material and Impact Engineering Laboratory of the Faculty of Mechanical Engineering Chemnitz University of Technology, the rise time of the incident impulse should be: \( T_r \approx 11\mu s \). By contrast, a typical experimental incident signal recorded by strain gages is shown in Fig. 4, of which T _r is about 12 μs. The experimental results match the theoretical analysis very well.

It’s widely accepted that force equilibrium on the specimen in SHPB test is established in this rising time duration. So a maximum allowable specimen length is obtained: where c _sp is the elastic wave velocity in specimen; n is the number of times stress waves travel along the specimen’s length, to build the force equilibrium, which is believed to be three or four times.

Once force equilibrium is established, equations (3), (4) and (5) are immediately valid. When the specimen’s deformation is just passing the yield point of the specimen’s material, the following classic relationship is satisfied from one dimensional Hooke’s law: \( \sigma_y \) and E _sp are the yield stress and Young’s modulus of the specimen respectively.

At the yield point, if the strain rate \( \dot{\varepsilon } \) is known, the loading time on the specimen can be calculated as:

From Von Karman’s plastic wave theory, the velocity of the stress wave will immediately reduce from the elastic wave velocity to the plastic wave velocity, when the specimen yields and starts plastically deforming. So in the time duration t _y , the stress wave at least has to pass the whole specimen and reach the transmitted side of the specimen. The specimen length should fulfill the following equation:

Now applying the data processing method of SHPB and substituting equations (3), (5) and (15) into the above equation, we have:

From equation (4), when the specimen is yielding, the transmitted signal obeys the following relationship:

Furthermore, from one-dimensional Hooke’s law and equation (1), the strain of the incident impulse is:

Substituting equations (18), (19) into equation (17), and doing simplification and rearrangement, we obtain: where d _sp and d _bar are diameter of the specimen and bar respectively.

Now in equation (20), there are only material constants of specimen and bar in the right hand. Let

Finally we have:

Equation (23) shows there is a maximum allowable impact velocity (MAIV) of striker in the SHPB test. When the striker’s velocity is lower than MAIV, the condition for building up force equilibrium is fulfilled; so the specimen can be uniformly deformed in the SHPB test. On the other hand, if the striker’s velocity is higher than MAIV, incident side of specimen is always plastically deformed before the transmitted side of specimen yields; i.e. the specimen undergoes non-uniform plastic deformation, which may lead to the SHPB experiment being invalid.

In equation (23), both C ₁ and C ₂ are dimensionless parameters and have clear physical meaning. C ₁ represents the relative material property ρc, and C ₂ is the square of diameter ratio of specimen and bar, which describes the specimen’s relative geometrical dimension in one specific SHPB test. In addition, MAIV is proportional to the yield stress of the specimen material, which means MAIV will be rather low, if the test specimen’s yield stress is too low. For further discussion, relevant mechanical properties of the 19.7 mm diameter steel SHPB system in our Laboratory and several selected common engineering alloys are presented in Table 1.

Figure 5 shows the variation of MAIV as a function of the specimen’ normalized diameter by that of the bar d _sp/d _bar. Since normally the specimen’s diameter is smaller than that of the bar, the value of C ₂ should vary from zero to 1. In the figure, for soft materials (low yield and plastic flow stress compared with bar material): e. g. MgAM50, Al6063, and OFHC copper, MAIV isn’t sensitive to the variation of specimen diameter. Yield stress of the tested material becomes a dominant parameter, for example, the MAIV for OFHC copper after total annealing is only around 1 m/s to 2 m/s, which corresponds to a rather low yield stress 30 MPa. So it becomes quite understandable that very soft materials aren’t suitable for SHPB test because of their low yield stresses, no matter what the specimen geometry is. On the other hand, for hard materials (compared with bar material): Tungsten and Ti6Al4V, for which the value of ρc is equivalent to hard steel, thus, relative geometrical parameter becomes significant for MAIV. MAIV increases significantly with specimen diameter. MAIV varies from 20 m/s to 100 m/s for Tungsten and from 40 m /s to 90 m/s for Ti6Al4V. In such case specimen diameter becomes critical in the SHPB test and should be carefully chosen.

The 19.7 mm diameter SHPB in our laboratory was used to observe plastic wave propagation along specimens using a high speed digital camera (REDLAKE HGK100), which has a 50 µs frame rate and is assembled to focus on the specimen’s half cylindrical surface during the test. According to the above theoretical analysis and because of its low plastic wave velocity [16, 22, 23], OFHC copper was selected as experimental specimen material. The specimen geometry was chosen as 10–20 (mm) diameter-length cylinder. In the test, the specimen cylindrical surface was covered by a thin layer of white background and black dot pigments (shown in Fig. 6).

Both the oscilloscope for the strain gages on the bars and the camera are triggered by 5% increasing incident strain impulse and a software post trigger was also designed for the camera to enhance the camera’s frame resolution. So half full cylindrical surface photographs of the experimental specimen can be recorded simultaneously with the strain signals from the SHPB test, and half cylindrical strain field can be obtained from the photographs afterwards by an optical deformation and strain measurement software: ARAMIS V6.0.1-3, which is an optical technique to measure the deformation and strain of the surface of specimen and can recognize the surface structure of the measuring object in digital camera images and allocates coordinates to the image pixels.

The OFHC copper specimens are classified to two groups: as received in a cold work condition and after total annealing heat treatment. Static true stress-strain curves for these two types of OFHC copper specimens are shown in Fig. 7, and necessary technique coefficients for specimen and trigger in the experiment are shown in Table 2. For the OFHC copper specimens in the cold work condition, yield stress is around 300 M Pa and plastic stress almost remains constant regardless of plastic strain level, which is a typical non-strain hardening material and may lead to a very low plastic wave velocity (especially at the large plastic strain level) from Von Karman’s one dimensional plastic wave theory. By contrast, the OFHC copper specimens after total annealing heat treatment have a very low yield stress 30 M Pa and plastic stress increases with plastic strain level, which is a typical strain hardening material. The strain hardening effect is significant and plastic wave velocity reduces from the elastic value more moderately than OFHC copper specimens in the cold work condition, once specimen starts plastic deformation. For these two types of OFHC copper, theoretical plastic wave velocities, calculated from Von Karman’s theory, are shown in Fig. 8.

A schematic of the SHPB in our laboratory is shown in Fig. 9 and necessary parameters are shown in Table 3 to synchronize the photograph frames of the camera and the strain gage signals. By keeping the camera time scale static, and moving the strain gage signals recorded from the bars, we can synchronize the results from the camera and strain gage signals of the SHPB as:

Where, T _incid, T _refl and T _trans are the time duration to be shifted for those signals respectively; T _{p_trig} is the camera’s post trigger time. As an example, Figs. 10 and 11 show the recorded and shifted (with the mark of camera frames) SHPB signals of specimen #1 respectively. The half cylindrical strain fields of specimen #1 are shown in Fig. 12 with a 50 µs time step, which are obtained from photographs after imaging processing using software ARAMIS V6.0.1-3. In the software, the half cylindrical surface of the specimen is divided into 10 × 20 pixels, i. e. one pixel is approximately one square millimeter [Fig. 12(a)]. As the specimen is deformed, each pixel also moves and deforms. The software can track each pixel displacement, calculate the deformation, and display the whole half cylindrical strain field of the specimen. However, in the experiment some pixels in the strain field may be missed, especially for large plastic strain levels [Fig. 12(d), (e) and (f)], because part of the pigments on the cylindrical surface sometimes fly away during dynamic loading. Since our aim is to observe the propagation of plastic waves along the specimen length in an SHPB test and test Von Karman’s one-dimensional plastic wave theory, the variation of strain along the specimen length is the most interesting result. So the pixel path along the specimen length in the middle of the half cylindrical strain field is selected to represent the strain status of specimen, to minimize error from the camera’s depth of field.

For the OFHC copper specimens in the cold work condition, Fig. 13 shows the strain distribution of the defined pixel path along its normalized length. Solid curves are from frame 3 to frame 9 of specimen #3, which corresponds to the red dashed-dot line in Fig. 11. As the post trigger is designed to increase the resolution of the camera, the corresponding results (dashed lines) of specimens #1, #2, #4 and #5 are inserted between the curves of frame 3 and frame 4 of specimen #3 with a 10 µs time interval. In the figure, initially the specimen is plastically deformed from the incident side to the transmitted side, and the plastic strain level decreases with distance to the incident end of specimen. It clearly shows the existence and propagation of one dimensional plastic wave along the specimen length. But when the strain level is around 4% to 5%, the plastic deformation becomes rather uniform along the specimen length. Afterwards, the plastic strain at the specimen’s two ends is gradually smaller than that in its middle, when the plastic strain level goes larger. This trend is accumulated and becomes more and more severe, until the end of loading, which finally leads a barrel shaped deformed specimen, because the lubricant between the bars and specimen is squeezed out and the side confinement effect of the friction between specimen and bars becomes significant.

For the OFHC copper specimens after total annealing heat treatment, the SHPB strain signals synchronized with camera frame mark for specimen #9 are shown in Fig. 14. The strain distribution of the defined pixel path along the normalized specimen length is also shown in Fig. 15, obtained by the same method used for the previous OFHC copper specimens in the cold work condition (Fig. 13). Compared with OFHC copper in a cold work condition, similar material plastic behavior is obtained for the specimens after total annealing heat treatment, although it is a strain hardening material (Fig. 15).

The simulation is based on the SHPB in our laboratory and performed by commercial software ABAQUS Explicit 6.7-1, which takes wave propagation effect into account [24]. The whole virtual assembly in ABAQUS consists of three parts: incident bar, transmitted bar and the specimen. The only difference from the real setup in our laboratory (Fig. 9) is that a step stress impulse is applied at the end of the incident bar in the axial direction instead of the striker with length L _s and velocity V. The impulse loading on the incident bar in the axial direction is shown in Fig. 16, which is equivalent to a 618 mm striker with a velocity of 20 m/s, and the amplitude and duration of which can be calculated from equations (1) and (2).

In the simulation, an elastic steel model is defined for the bars and two elastic-plastic models are defined for the two types of OFHC copper: in the cold work condition and after total annealing (each type OFHC copper is simulated separately) by using the material model in Fig. 7. Three types of constraints are defined for the three parts to assemble the whole setup: Parallel Face, Face to Face and Coaxial. Both of the contacts (between incident bar and specimen; between transmitted bar and specimen) are defined by penalty contact method, of which tangential behavior is Frictionless and normal behavior is Hard Contact. The dimensions and technique parameters in the simulation are given in Table 4.

Since here our task is to investigate the specimen’s plastic deformation and the effect of plastic wave propagation, failure of specimen’s material model wasn’t considered in the simulation (actually these two types of OFHC copper specimens do not fail in the real SHPB test). On the other hand, strain rate effects on material behavior of the specimens is also ignored; i.e. strain rate hardening was not included in material model for the sake of constant nominal strain rate in the SHPB test.

To compare with results from the digital camera, a nodal point path (shown in Fig. 17) is also defined along the specimen length (z axis direction) for each simulation, which corresponds to the pixel path in the previous experimental half cylindrical strain field. The historical plastic strain distribution in the z axis direction is recorded and plotted as normalized specimen length in Figs. 18 and 19 for these two types of OFHC copper specimens with a 10 µs time interval. Since numerical simulation strictly obeys Von Karman’s plastic wave theory, there is a very low plastic wave velocity for OFHC copper specimens in the cold work condition (Fig. 8). So in Fig. 18, the specimen’s deformation is quite non-uniform along its length and the plastic strain on the specimen incident side is always larger than that in specimen transmitted side. This non-uniform strain distribution becomes more and more severe as the strain lever becomes larger, and finally the cross section of the specimen changes from a rectangular shape to a trapezoidal shape. By contrast, there is a relatively high plastic wave velocity (400 m/s to 600 m/s) for OFHC copper specimens after total annealing heat treatment. The propagation and reflection of plastic waves along the specimen length are clearly shown in Fig. 19, thus the plastic strain distribution along specimen length is much more uniform than that for OFHC copper in the cold work condition.