The reconstituted stress–strain curves and corresponding average hydrostatic pressure obtained through simulation, as shown in Figure 5, only consider the inertia effect (mainly lateral inertia confinement effect), ignoring the interface friction effect. The rising stage of the reconstituted stress–strain curves overlaps well with that of quasi-static curves before peak stress at different strain rates. As for the mean hydrostatic pressure curves in Figure 5, the point of peak stress is consistent with that of corresponding reconstituted curve, indicating a close relationship between hydrostatic pressure and compressive strength at a high strain rate. Thus, the enhancement of compressive strength for the reconstituted curve at a high strain rate may contribute to the lateral inertia confinement effect in the SHPB test.

Figure 6 manifests a clear trend that f_cl (the increment between quasi-static and reconstituted compressive strength) increases with the increase in the strain rate and decreases with the rise in porosity p. At a strain rate level (SRL) of 70/s, the increment in compressive strength caused by lateral inertia confinement f_cl drops from 5.9 MPa for a specimen with a porosity of 10% to 2 MPa for a specimen with a porosity of 40%. At a strain rate level of 100/s, the increment in compressive strength caused by lateral inertia confinement f_cl drops from 7.1 MPa for a specimen with a porosity of 10% to 2.4 MPa for a specimen with a porosity of 40%. At a strain rate level of 140/s, the increment in compressive strength caused by lateral inertia confinement f_cl drops from 8.8 MPa for a specimen with a porosity of 10% to 3 MPa for a specimen with a porosity of 40%. Thus, the lateral inertia confinement effect increases with strain rate; however, high porosity can weaken this inertia effect. Figure 7 also verifies this phenomenon from another perspective, in which the p_c values are almost the same for specimens with a porosity of 40% at a strain rate level from 70/s to 140/s, indicating a relatively small hydrostatic pressure and corresponding strength enhancement provoked by a higher strain rate for cellular concrete with high porosity. This phenomenon is consistent with the speculation in [2] that the inertia effect would be relatively less significant for cellular concrete since the confining pressure applied on the central core concrete may be smaller, contributing to the mass loss of the surrounding concrete induced by pores in the concrete. Thus, the mechanism of the crack-path-altering effect can be further confirmed to explain the higher dynamic increase factor for specimens with high porosity than for specimens with relatively low porosity at the selected high strain rate, which is found in the SHPB test for this kind of cellular concrete [2].

As shown in Figure 8, the time t = 187 μs is the moment that the stress wave begins to spread across the specimen. The stress nephogram of S22 (along the Y-axis, namely the radial stress nephogram along the Y-axis direction on the surface) at this moment can be roughly divided into three regions: the blue radial compressive stress region, the orange radial tensile stress region, and the green transition region. From the moment t = 187 μs to the moment t = 233 μs (the moment dynamic stress equilibrium is reached), the local radial compressive stress region and radial tensile stress region gradually expand from the incident end to the transmitted end. At the moment t = 233 μs of dynamic stress equilibrium, the local radial compressive stress region, the local radial tensile stress region, and the local transition region are uniformly distributed inside the specimen. At this time, the size and strength of each local stress region are basically the same, and there is no nonlinear distribution from the inside of the specimen (the upper-side line of each section in Figure 8) to the edge (bottom of each section in Figure 8). Therefore, this stage can be classified as the elastic development stage of radial stress. From the moment of dynamic stress equilibrium (t = 233 μs) to the moment of peak stress (t = 263 μs), the size and stress value of each region develop from a uniform distribution to a non-uniform distribution. The size and strength of the radial compressive stress region gradually shift to the center of the specimen, and the radial tensile stress region gradually shifts to the edge of the specimen. When t = 263 μs, the peak stress is reached, and the size and strength of the local radial compressive stress region show a decreasing distribution from the inside of the specimen to the edge. In this case, the local radial tensile stress region is mainly distributed in the region with a certain thickness on the edge of the specimen. This stage can be classified as the plastic development stage of radial stress.

The development of plastic strain in the two stages can be directly reflected from the time t = 187 μs of the stress wave entering the specimen to the time t = 233 μs of dynamic stress equilibrium (the first stage) and from the time t = 233 μs of dynamic stress equilibrium to the time t = 263 μs of peak stress being achieved (the second stage). In ABAQUS, PEEQ (PE represents plastic strain component and EQ represents equivalent quantity in ABAQUS) is the parameter used to describe the accumulation of strain in a material during plastic deformation. When the equivalent plastic strain PEEQ is greater than 0, it indicates that the material has yielded and entered the plastic deformation stage. In the first stage (Figure 9), the PEEQ of each part of the specimen is basically zero, and the specimen is basically in the elastic stage. In the second stage, after the dynamic stress equilibrium is achieved, the equivalent plastic strain PEEQ begins to originate and develop. The plastic strain is mainly generated at the edge and then propagates to the inside of the specimen. It is due to the radial inertia generated by the plastic expansion of the edge plastic zone and then propagates to the inside and the edge of the specimen; the size and strength of the local radial compressive stress region show a decreasing distribution from the inside of the specimen to the edge at the moment t = 263 μs that peak value of stress is reached.

Figure 9 directly confirms the “two-stage” development mechanism, and the effect of lateral inertia confinement can be divided into two stages: the elastic development stage before the dynamic stress equilibrium is achieved and the plastic development stage after the dynamic stress equilibrium moment. In the elastic development stage, the lateral inertia confinement develops with a uniform distribution in each local area, and the lateral inertia confinement is small. In the plastic development stage, the local regions of the lateral inertia confinement develop a decreasing distribution from the inside of the specimen to the edge, and the lateral inertia confinement is larger. Since the plastic development stage is the stage where the peak stress is reached and the strain rate level is determined [2], the increase in compressive strength caused by the lateral inertia confinement effect is mainly attributable to the propagation of the lateral confinement to the interior of the specimen, which starts from the plastic expansion of the plastic zone at the edge of the specimen.

Figure 10 demonstrates that, in the elastic development stage of the lateral inertia confinement, with the increase in the strain rate, the radial compressive stress region and stress value gradually increase, while the radial tensile stress region and its value gradually decrease. As can be seen from Figure 11, the radial compressive stress region transfers to the interior of the specimen in the plastic development stage with the increase in the strain rate, while the radial tensile stress region transfers to the edge of the specimen. The higher the degree of the transfer developed, the stronger the lateral inertia confinement effect caused by plastic expansion. Figure 10 and Figure 11 reveal the internal mechanism of the lateral inertial confinement effect increasing with the increase in the strain rate when the porosity is the same.

As can be seen from Figure 12, at the same strain rate level, the radial compressive stress region has a more significant decreasing distribution trend from the inside of the specimen to the edge with a decrease in porosity. Meanwhile, the compressive stress value in the radial compressive stress region increases significantly with a decrease in porosity. Figure 12 reveals the internal mechanism of the lateral inertial confinement effect decreasing with the increase in porosity at the same strain rate level.

As can be concluded from the stress and strain analysis, in an SHPB test of cellular concrete materials, the lateral inertia confinement caused by axial inertia cannot be explained by the elastic theory, but the mechanism should be analyzed by the plastic-flow-related theory. When concrete materials enter the plastic stage, the strain increases rapidly with the increase in load, and the axial strain will cause the rapid growth of transverse strain. The rapid growth of strain results in a larger radial inertia confining pressure since the derivative of strain rate with respect to time is the amount related to inertia. The plastic expansion of the plastic zone inside the specimen generates transverse inertia, and the radial confining pressure will simultaneously propagate to the inside and edge of the specimen, as shown in Figure 13.

Figure 13a shows a specimen subjected to impact loading in the horizontal direction, in which the upper end of the vertical direction is the symmetry axis of the specimen (the central end in the radial direction), the lower end is the free surface at the edge of the specimen, and the orange band is an area where plastic deformation occurs. After the local area enters the plastic deformation phase (as shown by the orange band), the plastic zone becomes the disturbance source of the confining pressure wave inside the specimen, and the radial confining pressure generated there propagates to the central end and the free edge simultaneously (as shown by the red arrow in Figure 13a). The stress of the confining wave is doubled after reflection at the central end and unloaded to zero after reflection at the free end. Thus, the distribution curve of radial confining pressure with the radial position at a certain moment as shown in Figure 13b is formed. When most or even all regions enter the stage of plasticity, the radial confining pressures generated by disturbance sources in each plastic zone are superimposed on each other, and finally, the radial confining pressure as shown in Figure 13c is distributed in a parabolic form along with the radial position. That is, the lateral confinements produced by the inertia effect in the specimen show a parabolic decreasing trend from the center to the edge.