The dynamic axial crush response of circular cell polycarbonate honeycombs was studied for 3-cell and 7-cell specimens experimentally and through finite-element (FE) simulation. The experiments were conducted using two loading methods: (i) the wave loading device (WLD) method and (ii) the direct impact method (DIM). The specimens were subjected to crush velocities of about 12 000 mm s−1 in the WLD method and 5000 mm s−1 in the DIM. The two methods were used to obtain a fairly wide range of input velocities. The collapse sequence and displacement information of the specimens were captured using a high-speed camera. The mode of collapse was through progressive concertina-diamond fold formation over a fairly constant state of load, which is referred to as the crush load. The crushing was simulated using an explicit FE analysis using ABAQUS, with geometrically imperfect 3-cell and 7-cell honeycomb models that incorporated the rate-dependent properties of polycarbonate. The FE results were found to agree well with the experimental results in terms of overall force–displacement plots, thus providing a basis to extract energy absorption estimates from the models and to draw comparisons between the 3-cell and 7-cell response behaviour. Moreover, the dynamic crush results were compared against a quasi-static axial crush response to demonstrate the presence of rate effects.
Light-weight honeycomb structures are commercially manufactured from metals, such as aluminium and stainless steel, or from synthetic polymers, such as polycarbonate, polypropylene and aramid. They are widely used as the core material in sandwich panels owing to their high out-of-plane stiffness to weight ratio. They are also capable of absorbing large amounts of energy when subjected to static or impact loading and are hence used as energy absorption devices in the automotive and aerospace engineering industries (McFarland 1964; Goldsmith & Sackman 1992; Gibson & Ashby 1997). A variety of cell shapes, sizes and cell wall thicknesses are now available for commercial use. These honeycombs fall into the broad category of cellular solids, and their mechanics, including random microstructures, were described in Ostoja-Starzewski (2008). In the present study, the axial crush response of circular cell polycarbonate honeycombs is the subject of investigation.
Structurally, the constituent repeating unit, referred to as the cell, can be classified as a thin circular shell with radius-to-wall thickness ratio R/t≫1. Therefore, the mechanics of energy absorption of the basic repeating unit has close association with the classical problem of shell buckling (Brush & Almroth 1975). Naturally, the understanding of honeycomb structural response is more complicated than that of its basic repeating shell unit. The axial direction, which is parallel to the cell generators, also referred to as the out-of-plane direction of the honeycomb panel, exhibits a much higher stiffness when compared with the in-plane direction. Consequently, the macroscopic characterization of a honeycomb structure does not conform to an isotropic model. Several analytical and experimental methods to determine anisotropic honeycomb material properties were reviewed by Schwingshackl et al. (2006).
A key parameter that characterizes the crashworthiness and that gives a measure of the energy absorption capacity of these structures is the axial crush load. The crush load (also referred to as the plateau load in some studies, for example, Papka & Kyriakides 1994) is defined as a state of near constant load where progressive collapse of the structure occurs. The effect of crush load of circular polycarbonate honeycombs with respect to scaling in terms of the number of cells per specimen under quasi-static axial loading was experimentally studied by Mellquist & Waas (2002). They concluded that scaling in terms of number of cells had no significant effect on the axial crush load per cell. Wierzbicki & Abramowicz (1983) studied the axial crushing of metallic tubes and reported that the mean crush force depends on the thickness (t) of the shell as t5/3. Wilbert et al. (2011) studied the quasi-static axial crush response of hexagonal cell Al-5052-H39 honeycomb panels, both experimentally and through explicit finite-element (FE) simulations. They reported that the fold initiation starts at the centre of the specimen and the specimen progressively crushes towards either ends. Wu & Jiang (1997) studied a static and dynamic axial crush response of metallic honeycombs and reported that the crush load was proportional to the impact velocity. Baker et al. (1998) studied the static and dynamic response of high-density metal honeycombs. They reported that the dynamic plateau stress was about 50 per cent higher than the quasi-static value. Vural & Ravichandran (2003) studied the dynamic compressive response of balsa wood (a naturally occurring cellular material) along the grain direction using a modified split-Hopkinson pressure bar (SHPB). They showed that initial failure stress was highly sensitive to loading rate and that the plateau stress was insensitive to the strain rate. Recently, Hou et al. (2011) used a nylon SHPB system with bevelled ends of different angles to study the combined shear–compression dynamic response of hexagonal honeycombs. They concluded that for a given angle of loading, there was strength enhancement when compared with the quasi-static loading. Hong et al. (2008) conducted quasi-static and dynamic crush tests of 5052-H38 honeycomb specimens under out-of-plane inclined loads. They reported that as the impact velocity increased, the normal crush strength increased, but the shear strength was relatively unaffected. Xue & Hutchinson (2006) proposed a continuum constitutive model to simulate the dynamic strengthening behaviour of square honeycomb cores during multi-axial dynamic loading. Their model takes into account inertial resistance, inertial stabilization of webs and material strain-rate dependence. Mohr & Doyoyo (2006) developed a constitutive model (finite strain, orthotropic and rate independent) for metallic honeycombs based on the plateau stress.
The foregoing literature review indicates a gap in analytical and computational studies related to the deformation response of honeycombs at high loading rates, validated by experimental data, with respect to assessing energy absorption and collapse. The studies of Hong et al. (2008) and Hou et al. (2011) on aluminium hexagonal cell honeycombs show an increase in energy absorbing capacity at elevated crush rates, but similar findings with respect to circular cell honeycombs and for direct axial crush at elevated rates are absent. A central goal of this study is to extend the findings reported in Mellquist & Waas (2002) to the elevated strain rate regime and to understand the dynamic crush response of a non-metallic honeycomb material with a view to increasing the specific energy absorption (energy absorbed per unit weight). To this end, we note that hexagonal and square cell honeycombs are essentially flat plates arranged vertically as a collection, whereas the circular cell is a vertical thin shell, with curvature, which provides a rich plethora of folding dynamics during the axial collapse stage. While the circular shell has been studied in isolation in many prior studies (Brush & Almroth 1975), the interaction of many connected shells and the significance of tailoring this interaction to exploit energy absorption has not been studied before in the dynamic regime. This is an area that requires further study and understanding because of its practical significance related to use in the design of efficient energy absorbing devices in the automotive and aerospace engineering sectors.
We present experimental and associated numerical results of the axial dynamic crush response of 3-cell and 7-cell polycarbonate honeycombs with emphasis on the crush load, fold initiation, fold progression and mode of collapse. The deformation response of specimens with larger number of cells can be gleaned from the 7-cell specimen results, as will be discussed later. To the best knowledge of the authors, this study is the first of its kind concerning the dynamic out-of-plane crush response of circular polycarbonate specimens over a range of crush velocities. The features of axial collapse in circular cell honeycombs are important for development of a macroscopic continuum theory (Xue & Hutchinson 2006), which can adequately predict collapse stresses in honeycombs of given physical and material properties. Even though localization of deformation may preclude such a task, a continuum model is indispensable in the analysis of sandwich panels that may contain thousands of cells. To this end, we discuss the dynamic axial crushing of honeycombs, which is carried out using two methods, each providing a unique set of loading characteristics and facilitating a range of input crush velocity: (i) the wave loading device (WLD) method and (ii) the direct impact method (DIM). From the details to follow, the loading from the WLD set-up is primarily owing to stress waves, particularly during initial stages of loading, whereas the loading in the DIM set-up is from inertia loads.
2. Test preparation
(a) Honeycomb dimensions
3-cell and 7-cell circular honeycomb specimens made out of polycarbonate were used in this study (figure 1). An accurate measurement of physical properties of the polycarbonate material and geometric properties of the honeycomb cell, respectively, is crucial to correctly characterize the dynamic crush response of the honeycomb panels and also to create FE models. First, samples were prepared with 7-cell specimens cast in cylindrical moulds using epoxy resin and hardener at room temperature. The top face of the hardened sample was polished carefully with dry emery papers of increasing grit sizes in a polishing wheel and was finally polished with a diamond paste with a particle size of 5 μm. Several samples were prepared and their micro-section of the cell was studied under an optical microscope (figure 2). As can be seen in figure 2, the cells in the honeycomb panel are in contact with each other not at a point, but over a finite length, which we refer to as the cell adhesion length (l). Variations in the cell dimensions were observed across the samples and the mean value appended with the standard deviation is provided. Cell wall thickness (t) was 78±3 μm, the double wall thickness (tD) was 166±4 μm, the cell adhesion length (l) was 245±23 μm and the cell radius (R) was 2.026±0.025 mm. The length (L) of the specimen along the axial direction, measured separately, was 25.4 mm.
(b) Polycarbonate elastic properties and static axial crush load
Next, the elastic modulus of the polycarbonate material was determined using the measured cell dimensions. A single cell was carefully isolated from the honeycomb panel and was mounted to a table-top uniaxial compression test set-up. This cell was subjected to displacement-controlled compression loading in the in-plane direction to obtain the stiffness data. Such a test was carried out on 10 samples. The experiments gave a consistent measure of the stiffness measuring 24.1±2.3 N mm−1. The factors that amount to the uncertainty in stiffness measurements arise from uncertainty in the cell dimensions (radius and thickness) and also due to the presence of geometric imperfections. A cell subjected to this type of loading is one where the boundary conditions of contact change with the extent of deformation. Therefore, finding an analytical solution form to match the experimentally obtained stiffness is difficult. The problem configuration was simulated using FEs using the commercially available software ABAQUS/Standard in performing the modulus extraction. The value of the elastic modulus of polycarbonate was assumed to be the same in the in-plane as well as in the out-of-plane direction. Different trial elastic modulus values were used to match the honeycomb cell stiffness from this analysis to the honeycomb cell stiffness that was experimentally measured. The elastic modulus (E) backed out from this analysis was 2330±222 N mm−2. Next, we conducted axial static crush experiments on 3-cell and 7-cell specimens. The specimens were crushed at a rate of 0.033 mm s−1 using an INSTRON machine. The crushing occurred via concertina-diamond folds, the folds propagating from one end of the specimen to the other. The plateau load obtained for the 3-cell specimen was 44±4 N, whereas for the 7-cell, the value was 120±7 N. The crush loads obtained will be used to compare with the dynamic crush experiments, in order to determine whether loading rate influences the crush response.
(c) Measurement of experimental data
The dynamic force sensor is a PCB Piezotronics model 208C02. The force sensor was calibrated by the manufacturer. In the experiments that are described in the following sections, the data from the force sensor were acquired at the rate of 1.6×106 samples per second. The raw data were first filtered to remove the high-frequency noise such that only the frequencies above the upper limit of the design frequency (90 kHz) of the sensor were eliminated. These data are called raw data in the load response plots to follow. The raw data were additionally smoothened using the basic third-order Savitzky–Golay averaging filter in MATLAB. These data, which we refer to as filtered data, will be plotted alongside the raw data. The raw data will be used to compute estimates of crush/plateau loads and plateau stresses. The discharge time constant of the force sensor was greater than 120 s, a duration that is orders of magnitude higher than the time scale of the crush event. The displacement information and mode of collapse were obtained using a high-speed camera. Images were collected at the rate of 50 000 frames per second. The images acquired had a spatial resolution of 0.33 mm and 0.2 μs temporal resolution, which was adequate in obtaining the displacement information through pixel measurements.
3. Experiments: wave loading device method
(a) Set-up and procedure
A WLD was used to study the dynamic crush response of 3-cell and 7-cell honeycomb specimens. The experimental set-up of the WLD is shown in figure 3. The transmitter bar far end was held against a thick rigid immovable steel plate to prevent its movement when the honeycomb specimen was being loaded. The specimen was carefully mounted between the incident bar and the force sensor that was instrumented on the transmitter bar. The end faces of the specimen were positioned at the geometrical centre of the incident bar face and the end-cap face to avoid misalignment during impact. The striker bar was fired from the gas gun at a known pressure (1.379–6.895 MPa range), and it impacted the incident bar head on. A compressive stress pulse travelled from the impacted end along the length (Ls) of the incident bar with a longitudinal stress wave velocity cs. When the stress pulse reached the specimen end of the incident bar, the pulse reflected back as a tensile pulse in the incident bar and a part of it travelled as a compressive pulse in the specimen. The specimen was thus loaded in compression. The time taken to crush the entire specimen (L=24.5 mm) was much longer compared with the crush duration owing to the stress wave. Limited crush length (approx. 1.3 mm) was attained in the first period of crushing owing to stress waves. The time taken for the next cycle of crushing was equal to the roundabout trip of the stress wave, i.e. time taken to cover a distance equal to 2Ls. The motion of the incident bar was found to be a ramp–rest sequence (figure 4). The plot in figure 4 was obtained by tracking the position of the incident bar end in contact with the honeycomb, by performing pixel measurements of the high-speed images obtained. The cross-sectional diameter of the hardened steel bars was 12.7 mm. This dimension is much smaller compared with the length (Ls=1828.8 mm) of the incident bar. The diameter of the hardened steel striker bar was 12.7 mm and had a length of 304.8 mm. One-dimensional wave theory states that the fundamental longitudinal wave velocity (cs) in a material depends on the elastic modulus (Es) and density (ρs), and is given by the relation (see Kolsky 2003) 3.1where the elastic modulus (Es) of the bar material is 210 GPa and the mass density (ρs) is 7800 kg m−3. From equation (3.1), the stress wave velocity was calculated to be 5225 m s−1. The time interval (ΔT1−D) between successive tensile stress waves to reflect from the specimen end of the incident bar was 0.70 ms. This time interval agreed very well with the time interval (ΔTEXP≈0.68 ms) between two successive loading events measured from the experiment (figure 4). Note that this time interval is independent of the gas gun firing pressure or the striker bar velocity. This time interval depends on the stress wave velocity, which is a function of the physical properties of the incident bar material. However, the velocity of impact of the striker bar has a direct relation with the velocity of crushing during the ramping motion of the incident bar. The ramp–rest motion ceased to exist as time progressed because the stress waves in the incident bar gradually waned out owing to damping losses and owing to wave interference resulting from multiple reflections in the incident bar. Thereafter, the motion of the bar was mainly owing to rigid-body motion with lower crush velocities, as seen clearly in figure 4. Therefore, the one-dimensional wave theory is sufficient to satisfactorily explain the mechanics of ramp–rest motion that was encountered in this study. Note that one is not concerned with the wave interactions in the incident bar beyond the first reflection at the specimen end. In other words, the incident bar mainly serves as a projectile with the ramp–rest loading profile. Also, from the high-speed images, we observed that the transmitter bar did not move when the specimen was being loaded. This observation can be explained owing to two reasons: (i) the rigid, immovable plate acts as a momentum trap, thus absorbing any stress waves that get transmitted through the specimen, and (ii) the high impedance mismatch at the specimen–transmitter bar interface curtails the ability of the stress waves to pass from the specimen into the transmitter bar. Moreover, note that the intensity of the initial stress pulse that was transmitted through the specimen, up to this point, was significantly reduced owing to a high impedance mismatch at the incident bar–specimen interface. Hence, unlike in a regular SHPB set-up, the transmitter bar plays no active role in the WLD experiment, except for restraining the honeycomb in the out-of-plane direction. In summary, the success of the WLD method is due to the use of a momentum trap (via the thick, immovable plate) and the fact that the SHPB material (hardened steel) has a high impedance mismatch with the test specimen (polycarbonate). Therefore, in effect, the load that the specimen experiences is that arising from the stress waves (wave loading) in the incident bar.
When the 3-cell and 7-cell specimens were subjected to crush velocities (vc≈11 000–12 000 mm s−1 range) in the WLD set-up, a characteristic deformation response was observed. The load–time plot for the 3-cell specimen is shown in figure 5, and the images of the corresponding deformation sequence are shown in figure 6. For the 7-cell specimen, the load–time plot is shown in figure 7 and the corresponding images during the initial stages of deformation are shown in figure 8. A steep near-linear rise was seen initially and the load rose to a maximum value, referred to as the peak load. Up to this point, the deformation in the specimen was purely axial, and the stiff resistance to the moving bar gave rise to the initial peak load. Thereafter, the first localized axisymmetric concertina-diamond folding began, accompanied by a drop in the load. The preference of fold initiation immediately after the peak load is attained depends on the inherent geometrical imperfections present. Fold initiation was typically observed at both ends of the specimen. The folds continued to form as long as the incident bar was in motion. Also, transient elastic folds were observed near the ends of the specimens, which recovered back to the undeformed state in the rest period. After the end of the initial ramp step, the bar was momentarily at rest. Here, the load recorded fluctuated about a mean value with smaller amplitude owing to the residual stress waves in the specimen. No folds were formed during the rest period. However, the specimen exhibited hoop-like or breathing vibration modes (these are clearly visible in high-speed movies of the experiment). During the second ramp step, the load momentarily rose with fold formation and dropped at the end of the ramp step. This process continued until the specimen was completely crushed as new folds were formed progressively along the length of the honeycomb cell. The load recorded, beyond the initial phase where fold formation takes place, is referred to as the crush load. The crushing process was observed to occur either (i) from one end, with folds formed up to the other end, or (ii) simultaneously at both ends. The deformation characteristics explained above hold for both the 3-cell and 7-cell specimens. For the 3-cell specimens, the peak load recorded was approximately 150 N and the crush load measured was 54±14 N compared with the static crush load of 44±4 N. For the 7-cell specimens, the peak load recorded was 335 N and the crush load measured was 159±22 N compared with the static crush load of 120±7 N. Here, the mean crush load estimate was calculated as the integral average for the loads recorded over successive ramp sections, except for the first ramp movement (corresponding to the initial linear response). The uncertainty indicates standard deviation in this load level across various experimental trials.
4. Experiments: direct impact method
(a) Set-up and procedure
In order to observe the behaviour of honeycombs at lower crush velocities, the specimens were impacted directly with the striker bar (figure 9). We refer to this experimental procedure as the DIM. The DIM was used to achieve crushing in the honeycomb cell by a striker bar moving at a uniform velocity. The specimens were bonded to the end cap of the force sensor at its geometrical centre, and positioned directly in front of the gas gun. The striker bar was fired from the gas gun, and it crushed the sample head-on with uniform velocity (vc≈4000–5000 mm s−1). Various firing pressures were used as inputs for striker bar firing. A piezoelectric force sensor was instrumented on the transmitter bar. The far end of the transmitter bar was fastened tightly to a rigid, immovable steel plate to avoid any movement of the bar before the specimen had crushed completely. The DIM produces lower crush velocities than the WLD. Between the WLD and DIM, a disparate range of crush velocities can be obtained.
During the initial stages of deformation of the 3-cell and 7-cell specimens loaded by the DIM, the load rose nearly linearly during the stage where the deformation was dominated by axial motion with little or no radial deformation. For the 3-cell specimens, it was observed that when the first fold initiated at the crushing end, there was a prominent peak at the end of the initial rise (figure 10). In contrast, there was no peak at the end of the initial rise when the fold formation initiated at the far end of the specimen (figure 11). For the 7-cell specimens, the load plot and the images from the high-speed camera are provided in figure 12. Here, there was no prominent peak at the end of the initial region. Because collapse owing to fold formation results in a drop in the crush load, the effect of fold formation at a particular end has an effect on the presence or absence of a peak in the load history that is recorded by the force sensor at the far end of the specimen. If folding initiates at the far end of the specimen, the drop in load is instantly captured by the force sensor, as the far end is in contact with it. Thus, we see no prominent peak in this case. During crushing at high velocities, the specimen is not in dynamic equilibrium. If the fold formation starts at the impact end, the effect of this collapse does not reflect immediately at the far end of the specimen, which is still stiff. Therefore, we see a prominent peak at the end of the initial region if fold initiation takes place at the impacted end of the specimen. Soon after the initial rise, fold formation occurred, causing the load to drop. This pattern was observed for both the 3-cell and 7-cell specimens. Thereafter, as the striker bar was crushing the specimen, the folding progressed from the fold initiation end to over the length of the specimen. The crush load measured for the 3-cell specimens was 51±7 N (compared with the static crush load of 44±4 N), while that for the 7-cell specimen was 157±15 N compared with the static crush load of 120±7 N). The mean crush load value is calculated as the integral average of points in the plateau region and the uncertainty is the standard deviation of the crush load.
5. Finite-element model
The FE analysis was conducted using the commercially available software ABAQUS to simulate the dynamic axial crush response of the 3-cell and 7-cell honeycombs in the out-of-plane direction. The 3-cell and 7-cell honeycomb specimens were modelled as uniform circular shells with the average measured cell dimensions. In the honeycomb specimens, each cell is in contact with its neighbour through outer wall-to-wall adhesion. We assume that during dynamic crushing, there is no delamination at the contact surfaces. Therefore, the FE model considers the contact site to be composed of a single unit of adhesion length (l) 250 μm with double wall thickness (tD) 160 μm. A uniform FE mesh was generated using linear four-noded S4R elements, with the elements having an aspect ratio approximately 1. A convergence study (with respect to the plateau load) was conducted, and the results presented use the converged mesh. The model contained 125 elements along the axial length and 60 elements along the circumference. The 3-cell model contained 22 696 elements and the 7-cell model contained 52 959 elements.
(b) Eigenbuckling analysis
Shell structural response is influenced by unintended geometrical imperfections, such as out-of-roundness, non-uniform wall thickness and uncertain boundary conditions that produce non-uniformities in load. Among these, geometrical imperfections play a major role, as is well established in the shell buckling literature (Brush & Almroth 1975; Babcock 1983). Therefore, our model must account for geometric imperfections present in the honeycomb specimens. The imperfect geometry is modelled as perfect cylinders that have been perturbed by a linear combination of the eigenbuckling modes of the honeycomb. Mathematically, if the buckling mode shapes are represented as un(xl,yl)=ϕn(xl,yl), vn(xl,yl)=ψn(xl,yl) and wn(xl,yl)=χn(xl,yl) (where u, v and w denote displacements along the axial, circumferential and radial directions, respectively), then the location of a point on the perfect shell mid-plane with coordinates (Xl,Y l,0) is perturbed to new coordinates , where , and . The shell local axes, xl, yl and zl are indicated in figure 13, and the symbols Ai, Bj and Ck are the amplitudes of the perturbations. Linear buckling analysis was conducted in ABAQUS/Standard to extract the eigenbuckling mode shapes for the 3-cell and 7-cell models. The purpose was to use these modes to seed the perfect geometry to arrive at a model with geometrical imperfections, as the specimens used in the experiment are not geometrically perfect and also their initial shape cannot be measured with sufficient ease and accuracy. The boundary conditions (figure 13) imposed here are as follows. The impact end surface of the cell was constrained in translation along the in-plane direction (x and y), and the far end surface of the cell was constrained in displacement along all three translational degrees of freedom (x-, y- and z-directions). The Lanczos algorithm was used as it is most suited for solving large sparse generalized eigenvalue problems (Morris 1990) such as linear buckling problems. Note that all analyses reported here have used only a single eigenmode to perturb the initial geometry and therefore, strictly, one must not expect this to resemble geometrical imperfections that may be present in the real structure. For every eigenmode chosen to seed the perfect geometry, we define the imperfection amplitude (δ). If a specimen of wall thickness t is seeded with a mode shape of maximum radial amplitude (), the imperfection amplitude is defined as 5.1The upper limit of imperfection amplitudes are obtained from the deviation from the perfect circular geometry (15% of wall thickness) in the radial direction of the specimen. The maximum values of δ are chosen such that the initial linear slope of the crush response is not significantly different from that measured from the experiment. Two eigenmodes for each of the 3-cell and 7-cell specimens were chosen as the seeding modes to perturb the perfect geometry (figure 14).
6. Dynamic crush simulation
The dynamic crush FE simulation of 3-cell and 7-cell specimens was carried out in ABAQUS/Explicit. The displacement–time information for the WLD and DIM cases was taken from the experiment using the high-speed images. The explicit integration algorithm is ideal for solving large problems because the cost of computation increases linearly with problem size.
Polycarbonate is a strain-rate-dependent material, and this dependency is incorporated in the numerical crush simulation. Mulliken & Boyce (2006) have provided the rate-dependent compressive behaviour of polycarbonate for strain rates . From their work, the values of true yield stress (σy) for various compressive strain rates were tabulated and used in the dynamic crush simulations. The initial yield stress (σ0) was taken to be 66 MPa under static loading (). Note that from the plot in figure 15, we can see two distinct regions: region I () and region II (). In region II, we see that the yield strength increases much faster with strain rate when compared with that in region I. Polycarbonate does not exhibit a strong strain-hardening response (for ϵ≤0.4, which is much larger than the local strain values at cell walls during crushing). Therefore, for a given strain rate, the response past the elevated yield stress is modelled as perfectly plastic. The viscoelastic properties are not modelled because we assume that the stress relaxation effects are small to negligible owing to the small time duration in which the crushing takes place. A friction coefficient of μ=0.31 was used for both polycarbonate–polycarbonate contact and polycarbonate–steel contact to simulate sliding behaviour between these two types of surfaces during crushing. For each type of imperfections discussed in the previous section, simulations were conducted for imperfection amplitudes ranging from δ=0.1 per cent up to 5 per cent. The imperfection was applied to the shell geometry at the start of the ABAQUS/Explicit run. In the experiment, the specimen ends were held by frictional contact, and this effect is neglected in the FE model. The boundary conditions used in the dynamic run were as follows: only axial movement was allowed and no rotation constraints were applied on the impact face, whereas the far end of the specimen was only constrained in translation. For the crush simulations, the incident bar (WLD test) and the striker bar (DIM test) were modelled with a coarse mesh using hexahedral solid elements (C3D8R) with material properties of hardened steel. In the WLD simulation, the striker bar was not modelled because: (i) the displacement information from the high-speed camera is known and (ii) the wave propagation effects of the specimen are not significant owing to high-impedence mismatch between the hardened steel incident bar and the polycarbonate honeycomb at the interface.
In the WLD simulation, the input for the displacements prescribed for the incident bar end was a sequence of near-perfect ramp–rest inputs. The actual ramp displacement measured from the experiments strictly did not vary linearly with time. Moreover, the displacement–time input for various experiments varied slightly in displacement values, but the duration of the ramp for each loading cycle was consistent. The start and end points of each ramp were averaged, and only values at the end points were provided in the displacement–time input for the 3-cell and 7-cell FE simulations. This method was followed to reduce the run time in the ABAQUS/Explicit simulations. A representative crush velocity of 5000 mm s−1 was used as input to the striker bar for all the 3-cell and 7-cell FE models simulating the DIM experiments. The centre node along each contact strip was constrained against out-of-plane motion to prevent beam-like global buckling. This point constraint is valid as it does not over-constrain the structure and does not interfere with the physics of axial crushing.
(b) Simulation results and discussion: wave loading device method
In the initial stages of loading, the deformation in the 3-cell and 7-cell specimens was purely axial and the load rose linearly to a peak. This type of behaviour agreed well with that seen in the experiments. Thereafter, folds appeared on either side and the load dropped. The folds disappeared at the far end of the specimen and the folding progressed at the impacted end. Fold formation took place as long as the impacted end was in motion and the load rose again in the process. During the rest period, the cells were shown to exhibit hoop-like modes, similar to the ones observed in the experiment. Here, the load fluctuated about a value lower than that when the impacted end was in motion. When the next ramp step occurred, the load rose again with more fold formation. The load dropped again when the impacted end was at rest. This cycle continued, and the simulations ran for a crush length of approximately 10 mm. For the 3-cell model, the crush load obtained from the simulation was 50.45±12 N when compared with the mean crush load of 54 N measured from the experiment. The images from the simulations for 3-cell and 7-cell models are shown in figure 16. For the 7-cell model, the crush load obtained was 158.8±12.2 N compared with the experimental mean crush load of 159 N. The load–time and displacement–time plots for the 7-cell specimen for varying imperfection amplitudes are shown in figure 17. The average and standard deviation calculation for the plateau load from FE simulations was performed only during the ramping motion of the incident bar. The mode of collapse was by concertina-diamond fold formation, which was similar to the mode of collapse seen in the experiments. It is not possible to do a side-by-side comparison of the experiment and the simulation because the FE analysis uses a rendition, based on initial eigenmode shape geometrical imperfections, of the actual model. It was observed from the simulation results that the type and amplitude of imperfections had no effect on the collapse mode and little effect on the peak and crush loads. In addition, irrespective of the severity of imperfections with regard to the type of modes considered, the fold initiation took place at either end soon after the peak load was attained. These observations are encouraging and lend credence to the method of analysis that has been adopted.
(c) Simulation results and discussion: direct impact method
During the initial stages of loading, the load rose to a peak (maximum) load. The deformation in the specimen was only axial up to this point. Folding initiated near the ends of the specimen and the load dropped quickly. New folds were formed owing to the constant rate of crushing, and the load stabilized at an approximately constant level, which is referred to as the crush load. In the 3-cell model, it was observed that the impact end of the specimen that showed localized deformation, recovered elastically nearly to the original shape. The load decreased for a while immediately after the peak and stabilized at the crush load. The folding progressed from the region where fold initiation took place towards both sides of the specimen. The images from the simulations for 3-cell and 7-cell models are shown in figure 18. The load–time and displacement–time plots for a 7-cell model for various imperfection amplitudes are shown in figure 19. For the 7-cell model, fold initiation took place at both ends, and the specimen progressively crushed from these ends. The crush load observed for the 3-cell specimen was 50.54±8.4 N, compared with the experimentally measured mean crush load of 51.5 N. For the 7-cell specimen, the crush load obtained from the simulation was 148±14.2 N compared with the crush load value of 157 N seen in the experiments. Hence, the estimate of crush loads during progressive collapse is close to that measured from the experiment.
It is noted that the difference in peak collapse load between simulation and experiment is large, especially in the case of the DIM study. The peak loads measured from the experiment varied significantly among specimens (from no noticeable peak, to about 190 N, as seen in figure 12 for the 7-cell specimen), whereas the crush load was consistent. In contrast, in the WLD case with 7-cells, the variation in the measured peak load was somewhat consistent between 330 and 400 N. One of the reasons for this difference in the peak value is due to unknown and unintended geometrical imperfections present in the sample. Furthermore, in the DIM experiment, there is uncertainty as to the angle of the striker bar with respect to its ‘straightness’ in trajectory. This reduction in the peak collapse load owing to misalignment in loading (as shown by Wilbert et al. (2011) for hexagonal aluminium honeycomb system) is another reason for large differences in the initial peak load value.
(d) Variation of local strain rates and plateau stress with crush velocity
We also compared the variation of local strain rates with the crush velocity for the 3-cell and 7-cell models. When the honeycomb crushing occurs, the strain rates experienced by the honeycomb walls are different for varying crush velocities. From the explicit FE simulations that were carried out on the 3-cell and 7-cell models, several locations were monitored on the cell walls where severe fold formation occurred. For a given model and crush velocity, strain rates were found to not differ by a significant amount in magnitude. At a typical such location, the time histories of these quantities were averaged, and the variation of (averaged) strain rates and plateau stress with various crush velocities are provided in figure 20, along with the maximum amplitudes of strain rates over the time history. The results show that with an increase in crush velocity, strain rate and crush loads increase. At such locations, the bending and membrane strains were found to increase with increase in crush velocity. Furthermore, the model predictions are seen to agree with the experimental data corresponding to the crush velocity of 5000 mm s−1. It is also observed that for a given crush velocity, the strain rate values for 3-cell and 7-cell models are not significantly different, suggesting that results for specimens with a larger number of cells would be similar. We have verified this with DIM simulations of 3-cell, 4-cell, 7-cell, 13-cell and 19-cell models that were each crushed at a velocity of 5000 mm s−1. A plot showing normalized values (i.e. load per cell) of the crush and peak loads for the mentioned cell numbers is shown in figure 21. It is observed that the plateau load per cell for the 7-cell model is higher than that for the 3-cell model by approximately 40 per cent. For the 13-cell and 19-cell specimens, the plateau load per cell is approximately 8 per cent and approximately 20 per cent, respectively, higher than the 7-cell specimen, suggesting that this load asymptotes to a constant value of approximately 24 N, as can be seen in figure 21.
With reference to figure 20, we see that for an increase in crush velocities, the normalized plateau stress (calculated as Pplateau/NAE, where Pplateau is the average plateau load, A=2πRt is the true contact area, E is the static Young modulus of polycarbonate and N is the number of cells in the specimen) slightly increases, indicating that rate effects do play a role in the axial crush response. The normalized plateau stress calculated from the static 3-cell and 7-cell experiments are also provided in figure 20 to enable comparison with the dynamic crush experiment DIM. We observe that the normalized plateau stress values for static crush experiments are lower than the DIM experiment (at a crush velocity of 5000 mm s−1) and FE predictions provided, thus indicating the presence of rate effects. More specifically, comparing the dynamic crushing (at 5000 mm s−1 to the static crushing), figure 20 shows only a slight increase for the 3-cell specimen, i.e. an increase of about 5 per cent compared with an increase of about 10 per cent for the 7-cell specimen. The comparatively higher increase in load for the 7-cell specimen is due to the higher degree of lateral constraint that is provided to the cells as the number of cells increase.
The crush response of 3-cell and 7-cell circular cell honeycombs was studied using two methods—the WLD method and the DIM. The crush velocities seen in the WLD method are higher (approx. 12 000 mm s−1) compared with those in the DIM (approx. 5000 mm s−1). The collapse of 3-cell and 7-cell specimens occurred over a constant state of crush load at these crush velocities. Mean crush loads for the specimens measured from the WLD experiments were slightly higher compared with those measured from DIM experiments. Moreover, higher crush load levels were measured in the DIM experiment for 3-cell and 7-cell honeycombs compared with corresponding static crush experiments, thus clearly showing the presence of rate effects in the crush response. The peak load measured from the WLD experiment was higher than that from the DIM. This is evident because the deceleration force upon impact increases when the crush velocity is larger. The crush initiation took place at one or both ends of the specimen. The mode of collapse of the honeycombs was through localized concertina-diamond fold formation that progressively propagated from the specimen ends until the entire specimen had crushed. A series of FE simulations were conducted using eigenbuckling modes to approximate the imperfect geometry of the honeycomb. These simulations satisfactorily captured deformation features observed in the experiment, thus providing a meaningful way to estimate the energy absorption of clusters of cells. Beyond the stiff initial response, the load at which progressive collapse took place was found to be fairly close to that measured from the experiments. Moreover, the plateau load occurring with the concertina-diamond mode of collapse was found to be insensitive to the type and amount of geometrical imperfections.
The authors are grateful for the financial sponsorship of the Army Research Office (ARO) and the Army Research Laboratories (ARL), Aberdeen Proving Ground, MD, USA. The encouragement and support of Dr R. Athenian (ARO) and Dr C. Yen (ARL) are gratefully acknowledged. The authors also thank Amit Salvi, Mark Pankow and Eugene Kheng of the Composite Structures Laboratory, Department of Aerospace Engineering, University of Michigan, for their assistance during the course of this work.
- Received December 16, 2011.
- Accepted April 23, 2012.
- This journal is © 2012 The Royal Society