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Experiments and full-scale numerical simulations of in-plane crushing of a honeycomb
PII:
Acta mater. Vol. 46, No. 8, pp. 2765±2776, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00453-9
EXPERIMENTS AND FULL-SCALE NUMERICAL SIMULATIONS OF IN-PLANE CRUSHING OF A HONEYCOMB S. D. PAPKA and S. KYRIAKIDES Research Center for Mechanics of Solids, Structures and Materials, The University of Texas at Austin, WRW 110, Austin, TX 78712, U.S.A. (Received 30 September 1997; accepted 10 November 1997) AbstractÐThe in-plane mechanical behavior of honeycombs has been widely used as a two-dimensional model of the behavior of more complicated space ®lling foams. This paper deals with the mechanisms governing in-plane crushing of hexagonal aluminum honeycombs. Finite size honeycomb specimens are crushed quasi-statically between parallel rigid surfaces. The force±displacement response is initially sti and elastic but this is terminated by a limit load instability. Localized crushing involving narrow zones of cells is initiated and subsequently crushing spreads through the material while the load remains relatively constant. When the whole specimen is crushed the response stiens again. It has been found that although the crushing patterns that develop during the load plateau vary from specimen to specimen (in¯uenced by geometric imperfections and by specimen size) the underlying cell collapse mechanism is common to all specimens. As a result, the level of the stress plateau and its extent in strain are quite repeatable. The crushing process is simulated numerically by full-scale FE models in which the geometric characteristics of the actual cells are used. The manufacturing of the honeycomb involves cold expansion of specially bonded aluminum sheets. This is simulated numerically in order to reproduce the material changes and residual stresses introduced to the aluminum by the process. The expanded honeycomb is then crushed as in the experiments. It is demonstrated that once the key geometric, material and processing parameters are incorporated in the models, the simulations reproduce the experimental results both qualitatively as well as quantitatively. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Two-dimensional cellular materials with a regular and periodic microstructure such as honeycombs have proven to be useful in understanding the mechanical behavior of cellular materials (e.g. [1± 15]). Common to many cellular materials is a compressive response characterized by three regimes: An initial relatively sti and essentially elastic regime, a relatively ¯at extended stress plateau regime, and after signi®cant crushing a ®nal regime in which the material stiness recovers signi®cantly. In extensive studies of in-plane crushing of aluminum honeycombs with hexagonal cells and a polycarbonate honeycomb with circular cells [11, 12], it was demonstrated that the plateau part of such responses is caused by an instability which leads to progressive localized crushing of narrow zones of cells. While the instability spreads through the material, the average stress remains relatively constant. When all of the cells have collapsed, the densi®ed material regains stiness. Crushing experiments of this type using ®nite size honeycomb specimens were also simulated numerically through large scale FE models. It was demonstrated that provided the geometric characteristics of the microstructure and
the mechanical properties of the base material are modeled accurately, several of the major features of the crushing responses can be reproduced with accuracy. In the ®rst of these studies involving aluminum hexagonal cell honeycomb [11], we pointed out that the process through which the honeycomb is manufactured introduces geometric imperfections and changes in material properties which tend to aect the in-plane properties. The eect of such imperfections was illustrated in analyses involving characteristic cells. Such analyses can not account for the spatial distribution of such imperfections and, in addition, tend to yield only approximate estimates of some of the properties of interest. Thus, full scale simulations of the crushing process are necessary for quantitative accuracy. Imperfections were not introduced in the ®nite size honeycomb specimens used in the full simulation of the crushing process. Rather, the specimens were assumed to have perfectly hexagonal cells with virgin aluminum properties. As a result, the full scale simulations reproduced the events qualitatively but the predicted responses exhibited some quantitative dier-
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
Fig. 1. Honeycomb cell geometric parameters. (a) Laminate of periodically bonded aluminum sheets. (b) Perfect hexagonal cell. (c) Cell with imperfect bond width expanded such that y = 308. (d) Underexpanded cell (y>308) with imperfect bond width.
ences from the measured results (see Fig. 11(a) in [11]). To remove any reservations that arise from these discrepancies between experiments and predictions, the subject is revisited. In what follows we will demonstrate that once the geometric and material property issues mentioned above are appropriately incorporated in the full scale models, quantitatively accurate predictions can be produced. The procedures used in both the experiments and the analysis are essentially the same as those followed in [11]. To avoid repetition the reader will often be directed
to the original paper for details about such procedures. 2. EXPERIMENTS
2.1. Eect of manufacturing process on cell geometry The honeycombs used in the experiments were made from sheets of Al-5052-H39 of two thicknesses 0.0057 and 0.0047 in (144.8 and 119.4 mm) using a special process involving the following steps (Hexcelcorp.). The aluminum sheets come in rolls containing many meters of the material. The long
Fig. 2. (a) Photograph showing details of a cell in an expanded honeycomb (t = 0.0057 in 144.8 mm). (b) Calculated cell geometry of expanded honeycomb.
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
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Table 1. Mean values, standard deviations and extreme values of geometric parameters of honeycomb cells t in (mm) Variable Mean value Max d() Min d() SD
0.0057 (144.8)
0.0047 (119.4)
l in (mm)
l
m
l in (mm)
l
0.217 (5.51) +0.0044 (0.11) ÿ0.0057 (0.14) 0.002 (0.05)
0.837 +0.022 ÿ0.025 0.013
0.885 +0.016 ÿ0.007 0.006
0.216 (5.49) +0.0033 (0.08) ÿ0.0022 (0.05) 0.002 (0.05)
0.848 +0.018 ÿ0.021 0.010
sheets go through a set of tensioners and rollers which print on them at regular intervals bond lines of speci®c width. The sheets are cut to predetermined lengths and are stacked in a way that places the bond lines in adjacent sheets half a period out of phase as shown in Fig. 1(a). The assembled stacking is then placed in a press with heated platens and taken through a cycle of transverse pressure and temperature to cure the bond lines. The stacking is subsequently sliced to the required honeycomb thickness and the slices are expanded by transverse tension to form the hexagonal honeycomb [Fig. 1(b)]. Two sides of each cell of honeycomb produced in this fashion have double wall thickness. As a result, in-plane crushing is highly anisotropic [15]. The expansion process introduces residual stresses in the cell walls and changes in mechanical properties of the material. Another consequence of this process is that the corners of the cells are somewhat rounded as shown in Fig. 2. Even more importantly the ®nal shapes of the cells are not perfect hexagons. The ®nal geometry of the cells can be aected by the accuracy with which each of the three major steps of the manufacturing process are carried out. For example, the printing of the epoxy strips on the original sheet metal is crucial to the ®nal geometry. Any deviation from the ideal values, in either the positioning of each strip or in its width, aects the ®nal geometry of the cells. The stacking of the sheets can also aect the ®nal geometry. To produce perfectly hexagonal cells, the epoxy lines in alternate sheets must be exactly half a period out of phase. Although the cutting and placement of the sheets is monitored by electronic sensors, small errors due to the tolerance of the instruments is unavoidable. Finally, the expansion process plays an important role in the ®nal shape of the cells. In order for the ®nished product to have the nominal value of cell size c, the honeycomb must be expanded past this value to compensate for the springback that occurs on unloading (see Section 3.1). Pinpointing the exact value of net displacement at which unloading must commence to guarantee optimal results is dicult. As a result, the honeycombs tend to be somewhat under- or overexpanded. A set of detailed measurements of the actual geometry of the cells of our honeycombs were conducted using a low ampli®cation microscope in conjunction with the NIH Image Analysis program.
m 0.893 +0.0136 ÿ0.0175 0.008
In general, we found the placement of the bond lines to be very accurate p but their lengths to be less than the value l = c/ 3 required for perfectly hexagonal cells. In addition, the cells were found to be under-expanded. To help us describe the geometric imperfections we de®ne the following parameters. Let ll (l>0) be the actual length of the bond line. De®ne p c 0 2 ÿ ll 3 1 that is, the width of a cell with l $ 1 if it is expanded so that the angle of 308 shown in Fig. 1(c) is preserved. Let mc' be the actual cell width (Fig. 1(d) Ð note that this de®nition of m is dierent from the de®nition in [11]). The lengths of the bond lines, the other cell sides and the cell widths were measured for a large number of cells of the honeycomb sheets from which the specimens that were subsequently crushed were extracted. These were used to calculate values for the parameters l, l and m. The mean values and standard deviations of the measurements along with the maximum and minimum deviations for each variable for the two honeycombs considered here are listed in Table 1. The numbers clearly show that the cells of both honeycombs have bond lines which are shorter than the ideal value and at the same time are underexpanded. The standard deviations of the three
Fig. 3. Geometry of honeycomb specimens used in experiments and analysis.
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
variables listed are quite small which indicates that the manufacturing process is actually rather accurate. More importantly however, the maximum and minimum deviations are signi®cantly larger than the standard deviations. These values are important because it is cells with such extreme distortions that trigger collapse. We also point out that the description of imperfections aorded through the three parameters used here is not complete. Inadvertent rotation of some walls, undue peeling of the bonds and some out of plane distortions of the cell walls can not be described by just these three variables. Finally, the reader is reminded that the fact that l and m are less than 1 does not necessarily have an adverse eect on the honeycomb in its customary use. Honeycombs are typically loaded laterally and thus, what are considered to be ``imperfections'' in the present use where the loading is in the plane, do not adversely impact the honeycomb's normal performance. 2.2. Crushing experiments Several honeycomb specimens cut from the sheets analyzed were crushed between two rigid platens under displacement control in a standard, screwtype testing machine. In [11] we had pointed out that events associated with the crushing of such honeycombs depend on the size of the specimen. At the same time, the specimen size only had a modest eect on the main parameters of the corresponding load±displacement responses. The specimens crushed in [11] had 9 rows and 6 columns because at this size crushing spreads through the specimen
in a reasonably clean row-by-row manner. To illustrate more complex features of the spreading of collapse in the present study we used specimens with 15 rows by 10 columns of cells (see Fig. 3). As in [11] the vertical walls at the top and bottom of the specimen were bent over to smoothen the contact between the honeycomb and platens and to prevent the premature collapse of cells in the two end rows. A computer-operated data acquisition system was used to monitor the load and displacement (d) during the crushing test while the deformation of the whole specimen was recorded on a synchronized video system. Selected deformed con®gurations were later reproduced from this recording for presentation purposes. The average stress (s = force/initial cross sectional area)±average strain (d/L0 d) response recorded in a crushing experiment of a honeycomb with thickness of 5.7 10ÿ3 in (144.8 mm) is shown 0 and in Fig. 4(a). Figure 4(b) shows the initial seven deformed con®gurations of the specimen which correspond to the numbered points marked on the response. Initially, the deformation of the honeycomb is elastic and the cells deform symmetrically about vertical axes through their centers (e.g. 1 ). The initial modulus of the specimen, E*, is
341 psi (2.35 MPa). (This is based on the net shortening of the specimen and, as a result, to some degree is specimen dependent. More accurate measurement of the modulus requires local measurements of shortening as described in [12].) At a stress of approximately 16 psi (110 kPa) the
Fig. 4(a). Caption on facing page.
Fig. 4. (a) Load±displacement response from a crushing experiment on aluminum honeycomb. (b) Sequence of deformed con®gurations corresponding to response in (a).
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
Table 2. Elastic moduli, initiation and propagation stresses, and extents of strain plateaus measured in ®ve honeycomb crushing experiments t in (mm) 0.0057 0.0057 0.0057 0.0047 0.0047
(144.8) (144.8) (144.8) (119.4) (119.4)
E* psi (MPa) 341 309 303 156 155
(2.35) (2.13) (2.09) (1.08) (1.07)
sI psi (kPa)
sP psi (kPa)
17.68 (121.9) 16.49 (113.7) 16.40 (113.1) 10.35 (71.4) 10.19 (70.3)
17.22 16.23 15.79 9.79 9.80
sÿ d response becomes progressively softer until a local load maximum develops at a stress of 17.68 psi (121.9 kPa, initiation stress, sI) and a strain of 6.8%. Beyond this point, deformation starts to localize to a narrow zone of cells near the 2 , the center of the specimen. By con®guration cells in this region are seen to be deforming in an asymmetric shear-type mode which spreads from the center outwards towards the free edges of the specimen. One of the consequences of testing a larger specimen is the ``barreling'' seen in con®gur2 ± 4 . This barreling tends to induce the ations 3 . criss-cross pattern of collapse seen in Once this pattern develops, crushing spreads from the collapsed zones to neighboring cells which have been weakened and destabilized. However, collapse is also in¯uenced by geometric imperfections and by deformation incompatibilities caused by the collapse sequence which stien particular clusters of cells. As a result, although the shear type mode of collapse reported in [11] is still the underlying mechanism of spreading of deformation in the specimen, the events are less orderly than in the smaller speci5 and 6 . mens as clearly seen in con®gurations For the same reason, the stress undulations that occur during the propagation of collapse are more randomly placed and their amplitudes and periods are more irregular. When essentially all the cells have collapsed the stress increases sharply. The extent of the undulating stress plateau (DeP=net strain from initiation stress to end of stress plateau) is 59.7% while the average value of the stress (propagation stress, sP) is 17.22 psi (118.8 kPa). The major problem parameters recorded are listed in Table 2 along with the results from two additional experiments on honeycomb specimens of the same size. Small variations are observed in the values of E*, sI and sP whereas DeP is very repeatable. Such dierences can be attributed to small imperfections unique to each specimen which result in dierences in the sequence of collapse of the cells. For the same reason, the positions and amplitudes of stress undulations on the stress plateaus were dierent in the three experiments. Crushing experiments on honeycombs of the same size made from the same material but with thickness of 4.7 10ÿ3 in (119.4 mm) produced qualitatively similar responses. However, this thinner honeycomb exhibited less barreling and collapse of horizontal rows of cells was favored over inclined zones of localized deformation (closer to sequence
(118.8) (111.9) (108.9) (67.5) (67.6)
DeP% 59.7 60.9 59.7 63.1 61.7
of events reported in [11]). The problem variables recorded in two experiments on this honeycomb are listed in Table 2. In this case, all variables are seen to be repeatable.
3. ANALYSIS
The crushing process is simulated numerically using a FE model developed within the nonlinear FE code ABAQUS. The cell sides are discretized with three-noded, quadratic beam elements (B22). These elements are based on kinematics which allow for large rotations of the normals, ®nite membrane strains and have a Timoshenko-type correction for shear deformations. Quadratic deformable surfaces are used for contact between the walls of the collapsing cells. Single thickness sides are discretized with seven elements each with a distribution biased towards the ends which undergo localized bending. Double thickness sides undergo limited deformation and, as a result, are represented with a single element. Nine Simpson integration points are used through the thickness. The elastic±plastic behavior of the Al-5052-H39 sheet metal is modeled as a J2-¯ow theory solid with isotropic hardening. The mechanical properties used came from uniaxial tension tests on strips cut from the sheets used to manufacture the honeycombs. The elastic moduli (E) and the yield stresses (s0) of the two materials are given in Table 3. The stress±strain response is idealized as bilinear with a post-yield modulus of E/100. The size of the specimens analyzed is the same as that of the test specimens (15 10 cells). In order to make the geometry of the cells in the model as close to that of the actual cells as possible, we use the mean of the measured values of the geometric parameters t, l, l and m given earlier in Table 1. In other words, all the cells are made to have the same deviations from the perfect geometry. The additional, randomly distributed deviations in these variables alluded to earlier, are neglected. Table 3. Elastic moduli and yield stress of Al-5052-H39 sheet metals from which the honeycombs are manufactured t in (mm)
E msi (GPa)
s0 ksi (MPa)
0.0057 (144.8) 0.0047 (119.4)
10.0 (68.97) 10.0 (68.97)
42.34 (292) 38.80 (267.6)
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
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Fig. 5. Simulation of expansion process for 15 10 cells specimen. (a) Traction stress vs cell width response. (b) Deformed con®gurations during process.
3.1. Simulation of honeycomb expansion We start by simulating the expansion process. In the initial (unstressed) con®guration, the model consists of 20 strips stacked together as shown in Fig. 5 0 ). The strips are connected to their neighbors on ( either side via the double wall thickness elements which are alternately placed along their lengths. The nodes of the double wall elements on the outer strip on the LHS are ®xed in the horizontal direction while those of the outer strip on the RHS are prescribed equal horizontal displacements incrementally. Figure 5 shows a plot of the traction stress se (net force/initial area of the walls being pulled) required for the expansion, vs the instantaneous width of the cell, 2D, normalized by the variable c' de®ned in equation (1) [for this case c' = 0.437 in (11.1 mm)]. As the honeycomb expands, the material yields and the response softens. At higher
values of D, membrane tension starts to develop in 1 ± 3 ) and the rotated cell sides (see con®gurations the response progressively stiens. The honeycomb is expanded beyond the required width so that when the load is released the width of the cells is 4 . exactly mc'. The ®nal con®guration is shown in The cell angle y de®ned in Fig. 1(d) is given by p 3 ÿ1 : 2 m y cos 2 In this case m = 0.885 and, as a result, y 1408. The dimensions of the honeycomb cells formed in this fashion have the geometric characteristics listed in Table 1. A scaled drawing of a cell with these characteristics is shown in Fig. 1(d). To demonstrate that the process faithfully reproduces some of the ®ner details of the actual honeycomb in Fig. 2(b) we include an expanded view of half of a cell from
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
the formed honeycomb. The cell corners are seen to have ``knees'' of similar geometry to that of the actual section of honeycomb shown in the photograph in Fig. 2(a). 3.2. Simulation of honeycomb crushing In addition to the geometric changes, the cold forming process introduces changes to the mechanical properties of the base material and leaves behind residual stresses. To capture these eects the honeycomb model is ®rst expanded and then crushed between two parallel, rigid surfaces (Fig. 3). One of the surfaces is ®xed and the second is translated vertically by prescribing incrementally its vertical displacement d. Before crushing begins the double walls seen protruding out at the top and 4 in Fig. 5 are bent bottom ends in con®guration over, as was done in the actual tests, in order to provide smoother points of contact with the rigid surfaces. Despite this, friction with the rigid surfaces still plays some role in the crushing process and was included in the analysis. Coulomb friction with a coecient of 0.2 was used for contact between the angled ends of the specimen and the rigid surfaces. The calculated sÿ d response is drawn in dashed line in Fig. 6(a) which also includes two of the experimental responses drawn in solid lines. The 0 and nine deformed con®gurations undeformed q from the simulation are shown in Fig. 6(b). They correspond to the numbered points marked on the response in Fig. 6(a). Initially, the response is sti with a modulus of EÃ* = 316 psi (2.18 MPa Ð
based on the net end-displacement as in the experiment). The deformation of the cells is symmetric about vertical axes through their centers. The rigid surfaces at the top and bottom of the specimen restrain the deformation of the end rows of cells to some degree. At the same time, the two free vertical edges can deform more than the rest of the specimen. Thus, the deformation has some initial biases which result in barreling of the specimen and more cell deformation in its center. Indeed, this initial bias in deformation is sucient to cause the central cells to destabilize ®rst by switching to the sheartype mode of deformation seen in the experiments 2 ). (In problems of plastic buck(see con®guration q ling like this one, a small imperfection is usually needed to help initiate the instability. Here, numerical noise and round-o errors left behind from the manufacturing process were sucient to initiate the instability and, as a result, no external disturbance was necessary.) The central row of cells collapses from the center outwards and, in the process, destabilizes the neighboring rows. Eventually, contact of the walls of the collapsed cells arrest local deformation in the central row and collapse continues in the two neigh3 ). This spreading of deformation boring rows (q from the collapsing row to its neighbor(s) continues 4 , rows 6±9 have collapsed. and, by con®guration q Subsequently, the lower of the collapse fronts stops moving and collapse ceases to be symmetric about the center of the specimen. Rows 5 and 4 collapse 5 ). At this stage, the stiening eect of next (see q the upper rigid surface starts to be felt and the load
Fig. 6(a) Caption on facing page..
Fig. 6. (a) Comparison of two experimental and a calculated load±displacement crushing responses of an aluminum honeycomb with t = 0.0057 in (144.8 mm). (b) Sequence of calculated deformed con®gurations corresponding to response in (a).
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB 2773
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
starts to increase. When the load reaches a high enough level, the 12th row, which until this point was away from the two collapse fronts, gets destabilized and two new collapse fronts develop (see [6]) albeit at a relatively higher load than earlier events. With this new development, and because the outer fronts are close to the rigid surfaces, crushing stops being limited to horizontal rows and we see the development of inclined zones of localized deformation which connect the hitherto unconnected col7 and q 8 ). By con®guration q 9 , all lapsed zones (q cells are essentially crushed and the response of the densi®ed material recovers signi®cant stiness. A note regarding coexisting multiple crushing fronts. In the four structures which can develop propagating instabilities discussed by Kyriakides in [16], the propagation loads were signi®cantly lower than the loads required to initiate the instabilities. Thus, once initiated the instabilities propagated at either one or two fronts. In the present problem, the initiation and propagation stresses do not dier signi®cantly. Therefore, if for any reason the average stress increases to a value close to sI, a new initiation event can take place as was observed 6 and q 7 . Multiple here between con®gurations q propagating fronts were also reported in a recent study of the evolution of phase transformations in NiTi shape memory alloy strips [17]. The underlying micromechanical mechanisms involved in the two instabilities are very dierent. However, in both problems the reason for coexisting multiple fronts is the relatively small dierence between sI and sP. As was reported in Section 2, the sequence of collapse events seen in any one experiment is never exactly the same as that of any other experiment primarily because of the in¯uence of small randomly distributed imperfections. At the same time, the resultant variation in the major parameters of the crushing response is relatively small. Although the expansion processes was modeled faithfully, random imperfections were not introduced. Thus, even though the sequence of crushing events seen in the simulation is close to those seen in experiments, it does not exactly represent that of any particular experiment. On the other hand, the calculated response in Fig. 6(a) is seen to be, in the main, in good agreement with the two experimental responses. As expected, the predicted initiation stress (s^ I ) is somewhat higher than the experimental values because this is the variable most aected by small initial geometric imperfections. By contrast, the calculated propagation stress (s^ P ) of 17.17 psi
(118.4 kPa) is in excellent agreement with the corresponding measured values. The same can be said for the extent of the plateaus. Thus, overall, the simulation is very successful. For comparison purposes a crushing simulation was also conducted for a honeycomb specimen with the same geometric parameters as those used above except that the expansion process was bypassed. Thus, the honeycomb is initially stress free and the material is at the virgin state. In addition, the sharply rounded knees present at the corners of the expanded honeycomb (Fig. 2) are now absent. The crushing response is similar to the one discussed above and will not be presented here. The major problem variables are compared to those of the expanded honeycomb in Table 4. The elastic modulus is somewhat higher (as predicted via the micromodel in [11]), while the initiation and propagation stresses are, respectively, 14% and 7.6% lower. The extent of the stress plateau is also somewhat lower. Similar calculations have been performed for the thinner honeycomb (t = 4.7 10ÿ3 in ÿ119.4 mm). Once again, the mean values of t, l, l and m given in Table 1 were used. The ®rst specimen was crushed following simulation of the expansion process and the second was initially stress free. The calculated response for the expanded honeycomb is compared to one of the experimental ones in Fig. 7. The major parameters are summarized in Table 4. As was the case in the experiments, the numerical simulations also exhibited crushing sequences con®ned mainly to horizontal rows of cells. The order of crushing of the various rows is dierent from the experiment and, thus, the stress undulations in the response occur at dierent strain values than in the experimental response. However, the main problem variables are once more in good agreement with the measured values in Table 2. The dierence between results of the expanded and the stress free honeycombs has the same general trends as those reported above for the thicker honeycomb. Finally a word about the eect of specimen size on the variables presented. This issue was explored for the honeycomb with t = 5.7 10ÿ3 in (144.8 mm) by crushing several specimens of the following sizes: 9 6, 15 10 and 21 14 (rows columns of cells). All specimens were crushed in the ribbon direction. The crushing of the smaller size specimens involved mainly horizontal rows of cells whereas the larger specimens exhibited signi®cant barreling of the sides and developed the criss±cross patterns of localized collapse seen in Fig. 4(b) but
Table 4. Elastic moduli, initiation and propagation stresses, and extents of strain plateaus predicted t in (mm)
l
m
0.0057 0.0057 0.0047 0.0047
0.837 0.837 0.848 0.848
0.885 0.885 0.893 0.893
(144.8) (144.8) (119.4) (119.4)
Expanded Yes No Yes No
EÃ* psi 316 353 176 196
(2.18) (2.43) (1.21) (1.35)
s^ I psi
s^ P psi
D^eP %
19.00 (131.0) 16.35 (112.8) 11.90 (82.1) 10.29 (71.0)
17.17 (118.4) 15.86 (109.4) 10.21 (70.4) 9.66 (66.6)
62.3 60.8 64.3 61.4
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
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Fig. 7. Comparison of an experimental and a calculated load±displacement crushing response of an aluminum honeycomb with t = 0.0047 in (119.4 mm).
with this pattern being more pronounced. In all cases the mechanism of collapse was essentially the same shear-type mode discussed here and in [11]. The elastic modulus calculated based on the net end-displacement varied with specimen size. By contrast and despite the dierences in the crushing patterns mentioned above, the variation of the other variables was rather modest and of the order seen in the results in Table 2.
4. CONCLUSIONS
The work presented here is a continuation of the study reported by the same authors in [11]. Results from additional honeycomb crushing experiments are reported involving specimens of larger size than those reported in [11]. The results con®rm that such honeycombs develop a shear type instability which localizes into a narrow band of cells. Under displacement controlled loading, the crushed zone spreads until the whole specimen is consumed. The sequence of events through which crushing spreads in the specimens is aected by the size of the specimen. In smaller (and thinner wall) specimens, crushing is along horizontal rows of cells. Larger specimens exhibited barreling of the sides which is partly due to friction between the ends of the specimen and the rigid platens used to crush it. Criss±cross patterns of localized crushing were common in larger specimens. Despite these dierences, the propagation stress and extent of the stress plateau were found to be relatively unaected. The initiation stress, which is more sensitive to initial geometric imperfections, exhibited somewhat larger variation. Detailed measurements of the geometry of cells of several honeycombs showed that the cells were in general under-expanded and that the lengths of the bonded sides were shorter than the values of the
corresponding ideal hexagonal cells. The mean values of the parameters measured de®ne the dimensions and shapes of the cells in an average sense. These values were used in full scale simulations of the expansion process followed by crushing of the specimen; that is, the specimen crushed in the simulations included in an average sense the geometric and material changes and the residual stresses introduced by the manufacturing process. Both parts of the simulation matched the experiments well. The simulated crushing resulted in events which were somewhat dierent from those seen in any particular experiment but these dierences did not have any signi®cant eect on the propagation stresses or on the strain extent of the stress plateaus. Additional, randomly distributed small geometric imperfections present in the actual test specimens were not accounted for in the numerical models. The main eect of this deviation of the models from the actual specimens was that the predicted initiation stresses were somewhat higher than the corresponding experimental values. AcknowledgementsÐThe ®nancial support of the Air Force Oce of Scienti®c Research under Grant No. F49620-95-1-0154 and of the University of Texas at Austin is acknowledged with thanks. We are also grateful to H-K&S Inc. for making ABAQUS available under academic license and to Hexcel Corporation for providing the honeycombs tested.
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
5. Gibson, L. J. and Ashby, M. F., Cellular Solids: Structure and Properties, 2nd edn, Cambridge University Press, Cambridge, England, 1997. 6. Klintworth, J. W. and Stronge, W. J., Int. J. Mech. Sci., 1988, 30, 273. 7. Klintworth, J. W. and Stronge, W. J., Int. J. Mech. Sci., 1989, 31, 359. 8. Zhang, J. and Ashby, M. F., Int. J. Mech. Sci., 1992, 34, 491. 9. Warren, W. E. and Kraynik, A. M., Mech. Mater., 1987, 6, 27. 10. Warren, W. E., Kraynik, A. M. and Stone, C. M., J. Mech. Phys. Solids, 1989, 37, 717. 11. Papka, S. D. and Kyriakides, S., J. Mech. Phys. Solids, 1994, 42, 1499.
12. Papka, S. D. and Kyriakides, S., Int. J. Solids Struct., in press. 13. Papka, S. D. and Kyriakides, S., in Material Instabilities in Solids, Proc. IUTAM Symposium, ed. R. de Borst and E. van der Giessen, Wiley, Chichester, England, 1997. 14. Prakash, O., Bichebois, P., Brechet, Y., Louchet, F. and Embury, J. D., Philos. Mag. A, 1996, 73, 739. 15. Triantafyllidis, N. and Schraard, M. W., J. Mech. Phys. Solids, in press. 16. Kyriakides, S., in Advances in Applied Mechanics, Vol. 30, ed. T. Wu and J. W. Hutchinson, Academic Press, Boston, MA, 1993, pp. 67. 17. Shaw, J. A. and Kyriakides, S., Acta mater., 1997, 45, 683.
Acta mater. Vol. 46, No. 8, pp. 2765±2776, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00453-9
EXPERIMENTS AND FULL-SCALE NUMERICAL SIMULATIONS OF IN-PLANE CRUSHING OF A HONEYCOMB S. D. PAPKA and S. KYRIAKIDES Research Center for Mechanics of Solids, Structures and Materials, The University of Texas at Austin, WRW 110, Austin, TX 78712, U.S.A. (Received 30 September 1997; accepted 10 November 1997) AbstractÐThe in-plane mechanical behavior of honeycombs has been widely used as a two-dimensional model of the behavior of more complicated space ®lling foams. This paper deals with the mechanisms governing in-plane crushing of hexagonal aluminum honeycombs. Finite size honeycomb specimens are crushed quasi-statically between parallel rigid surfaces. The force±displacement response is initially sti and elastic but this is terminated by a limit load instability. Localized crushing involving narrow zones of cells is initiated and subsequently crushing spreads through the material while the load remains relatively constant. When the whole specimen is crushed the response stiens again. It has been found that although the crushing patterns that develop during the load plateau vary from specimen to specimen (in¯uenced by geometric imperfections and by specimen size) the underlying cell collapse mechanism is common to all specimens. As a result, the level of the stress plateau and its extent in strain are quite repeatable. The crushing process is simulated numerically by full-scale FE models in which the geometric characteristics of the actual cells are used. The manufacturing of the honeycomb involves cold expansion of specially bonded aluminum sheets. This is simulated numerically in order to reproduce the material changes and residual stresses introduced to the aluminum by the process. The expanded honeycomb is then crushed as in the experiments. It is demonstrated that once the key geometric, material and processing parameters are incorporated in the models, the simulations reproduce the experimental results both qualitatively as well as quantitatively. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
Two-dimensional cellular materials with a regular and periodic microstructure such as honeycombs have proven to be useful in understanding the mechanical behavior of cellular materials (e.g. [1± 15]). Common to many cellular materials is a compressive response characterized by three regimes: An initial relatively sti and essentially elastic regime, a relatively ¯at extended stress plateau regime, and after signi®cant crushing a ®nal regime in which the material stiness recovers signi®cantly. In extensive studies of in-plane crushing of aluminum honeycombs with hexagonal cells and a polycarbonate honeycomb with circular cells [11, 12], it was demonstrated that the plateau part of such responses is caused by an instability which leads to progressive localized crushing of narrow zones of cells. While the instability spreads through the material, the average stress remains relatively constant. When all of the cells have collapsed, the densi®ed material regains stiness. Crushing experiments of this type using ®nite size honeycomb specimens were also simulated numerically through large scale FE models. It was demonstrated that provided the geometric characteristics of the microstructure and
the mechanical properties of the base material are modeled accurately, several of the major features of the crushing responses can be reproduced with accuracy. In the ®rst of these studies involving aluminum hexagonal cell honeycomb [11], we pointed out that the process through which the honeycomb is manufactured introduces geometric imperfections and changes in material properties which tend to aect the in-plane properties. The eect of such imperfections was illustrated in analyses involving characteristic cells. Such analyses can not account for the spatial distribution of such imperfections and, in addition, tend to yield only approximate estimates of some of the properties of interest. Thus, full scale simulations of the crushing process are necessary for quantitative accuracy. Imperfections were not introduced in the ®nite size honeycomb specimens used in the full simulation of the crushing process. Rather, the specimens were assumed to have perfectly hexagonal cells with virgin aluminum properties. As a result, the full scale simulations reproduced the events qualitatively but the predicted responses exhibited some quantitative dier-
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
Fig. 1. Honeycomb cell geometric parameters. (a) Laminate of periodically bonded aluminum sheets. (b) Perfect hexagonal cell. (c) Cell with imperfect bond width expanded such that y = 308. (d) Underexpanded cell (y>308) with imperfect bond width.
ences from the measured results (see Fig. 11(a) in [11]). To remove any reservations that arise from these discrepancies between experiments and predictions, the subject is revisited. In what follows we will demonstrate that once the geometric and material property issues mentioned above are appropriately incorporated in the full scale models, quantitatively accurate predictions can be produced. The procedures used in both the experiments and the analysis are essentially the same as those followed in [11]. To avoid repetition the reader will often be directed
to the original paper for details about such procedures. 2. EXPERIMENTS
2.1. Eect of manufacturing process on cell geometry The honeycombs used in the experiments were made from sheets of Al-5052-H39 of two thicknesses 0.0057 and 0.0047 in (144.8 and 119.4 mm) using a special process involving the following steps (Hexcelcorp.). The aluminum sheets come in rolls containing many meters of the material. The long
Fig. 2. (a) Photograph showing details of a cell in an expanded honeycomb (t = 0.0057 in 144.8 mm). (b) Calculated cell geometry of expanded honeycomb.
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
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Table 1. Mean values, standard deviations and extreme values of geometric parameters of honeycomb cells t in (mm) Variable Mean value Max d() Min d() SD
0.0057 (144.8)
0.0047 (119.4)
l in (mm)
l
m
l in (mm)
l
0.217 (5.51) +0.0044 (0.11) ÿ0.0057 (0.14) 0.002 (0.05)
0.837 +0.022 ÿ0.025 0.013
0.885 +0.016 ÿ0.007 0.006
0.216 (5.49) +0.0033 (0.08) ÿ0.0022 (0.05) 0.002 (0.05)
0.848 +0.018 ÿ0.021 0.010
sheets go through a set of tensioners and rollers which print on them at regular intervals bond lines of speci®c width. The sheets are cut to predetermined lengths and are stacked in a way that places the bond lines in adjacent sheets half a period out of phase as shown in Fig. 1(a). The assembled stacking is then placed in a press with heated platens and taken through a cycle of transverse pressure and temperature to cure the bond lines. The stacking is subsequently sliced to the required honeycomb thickness and the slices are expanded by transverse tension to form the hexagonal honeycomb [Fig. 1(b)]. Two sides of each cell of honeycomb produced in this fashion have double wall thickness. As a result, in-plane crushing is highly anisotropic [15]. The expansion process introduces residual stresses in the cell walls and changes in mechanical properties of the material. Another consequence of this process is that the corners of the cells are somewhat rounded as shown in Fig. 2. Even more importantly the ®nal shapes of the cells are not perfect hexagons. The ®nal geometry of the cells can be aected by the accuracy with which each of the three major steps of the manufacturing process are carried out. For example, the printing of the epoxy strips on the original sheet metal is crucial to the ®nal geometry. Any deviation from the ideal values, in either the positioning of each strip or in its width, aects the ®nal geometry of the cells. The stacking of the sheets can also aect the ®nal geometry. To produce perfectly hexagonal cells, the epoxy lines in alternate sheets must be exactly half a period out of phase. Although the cutting and placement of the sheets is monitored by electronic sensors, small errors due to the tolerance of the instruments is unavoidable. Finally, the expansion process plays an important role in the ®nal shape of the cells. In order for the ®nished product to have the nominal value of cell size c, the honeycomb must be expanded past this value to compensate for the springback that occurs on unloading (see Section 3.1). Pinpointing the exact value of net displacement at which unloading must commence to guarantee optimal results is dicult. As a result, the honeycombs tend to be somewhat under- or overexpanded. A set of detailed measurements of the actual geometry of the cells of our honeycombs were conducted using a low ampli®cation microscope in conjunction with the NIH Image Analysis program.
m 0.893 +0.0136 ÿ0.0175 0.008
In general, we found the placement of the bond lines to be very accurate p but their lengths to be less than the value l = c/ 3 required for perfectly hexagonal cells. In addition, the cells were found to be under-expanded. To help us describe the geometric imperfections we de®ne the following parameters. Let ll (l>0) be the actual length of the bond line. De®ne p c 0 2 ÿ ll 3 1 that is, the width of a cell with l $ 1 if it is expanded so that the angle of 308 shown in Fig. 1(c) is preserved. Let mc' be the actual cell width (Fig. 1(d) Ð note that this de®nition of m is dierent from the de®nition in [11]). The lengths of the bond lines, the other cell sides and the cell widths were measured for a large number of cells of the honeycomb sheets from which the specimens that were subsequently crushed were extracted. These were used to calculate values for the parameters l, l and m. The mean values and standard deviations of the measurements along with the maximum and minimum deviations for each variable for the two honeycombs considered here are listed in Table 1. The numbers clearly show that the cells of both honeycombs have bond lines which are shorter than the ideal value and at the same time are underexpanded. The standard deviations of the three
Fig. 3. Geometry of honeycomb specimens used in experiments and analysis.
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
variables listed are quite small which indicates that the manufacturing process is actually rather accurate. More importantly however, the maximum and minimum deviations are signi®cantly larger than the standard deviations. These values are important because it is cells with such extreme distortions that trigger collapse. We also point out that the description of imperfections aorded through the three parameters used here is not complete. Inadvertent rotation of some walls, undue peeling of the bonds and some out of plane distortions of the cell walls can not be described by just these three variables. Finally, the reader is reminded that the fact that l and m are less than 1 does not necessarily have an adverse eect on the honeycomb in its customary use. Honeycombs are typically loaded laterally and thus, what are considered to be ``imperfections'' in the present use where the loading is in the plane, do not adversely impact the honeycomb's normal performance. 2.2. Crushing experiments Several honeycomb specimens cut from the sheets analyzed were crushed between two rigid platens under displacement control in a standard, screwtype testing machine. In [11] we had pointed out that events associated with the crushing of such honeycombs depend on the size of the specimen. At the same time, the specimen size only had a modest eect on the main parameters of the corresponding load±displacement responses. The specimens crushed in [11] had 9 rows and 6 columns because at this size crushing spreads through the specimen
in a reasonably clean row-by-row manner. To illustrate more complex features of the spreading of collapse in the present study we used specimens with 15 rows by 10 columns of cells (see Fig. 3). As in [11] the vertical walls at the top and bottom of the specimen were bent over to smoothen the contact between the honeycomb and platens and to prevent the premature collapse of cells in the two end rows. A computer-operated data acquisition system was used to monitor the load and displacement (d) during the crushing test while the deformation of the whole specimen was recorded on a synchronized video system. Selected deformed con®gurations were later reproduced from this recording for presentation purposes. The average stress (s = force/initial cross sectional area)±average strain (d/L0 d) response recorded in a crushing experiment of a honeycomb with thickness of 5.7 10ÿ3 in (144.8 mm) is shown 0 and in Fig. 4(a). Figure 4(b) shows the initial seven deformed con®gurations of the specimen which correspond to the numbered points marked on the response. Initially, the deformation of the honeycomb is elastic and the cells deform symmetrically about vertical axes through their centers (e.g. 1 ). The initial modulus of the specimen, E*, is
341 psi (2.35 MPa). (This is based on the net shortening of the specimen and, as a result, to some degree is specimen dependent. More accurate measurement of the modulus requires local measurements of shortening as described in [12].) At a stress of approximately 16 psi (110 kPa) the
Fig. 4(a). Caption on facing page.
Fig. 4. (a) Load±displacement response from a crushing experiment on aluminum honeycomb. (b) Sequence of deformed con®gurations corresponding to response in (a).
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB 2769
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
Table 2. Elastic moduli, initiation and propagation stresses, and extents of strain plateaus measured in ®ve honeycomb crushing experiments t in (mm) 0.0057 0.0057 0.0057 0.0047 0.0047
(144.8) (144.8) (144.8) (119.4) (119.4)
E* psi (MPa) 341 309 303 156 155
(2.35) (2.13) (2.09) (1.08) (1.07)
sI psi (kPa)
sP psi (kPa)
17.68 (121.9) 16.49 (113.7) 16.40 (113.1) 10.35 (71.4) 10.19 (70.3)
17.22 16.23 15.79 9.79 9.80
sÿ d response becomes progressively softer until a local load maximum develops at a stress of 17.68 psi (121.9 kPa, initiation stress, sI) and a strain of 6.8%. Beyond this point, deformation starts to localize to a narrow zone of cells near the 2 , the center of the specimen. By con®guration cells in this region are seen to be deforming in an asymmetric shear-type mode which spreads from the center outwards towards the free edges of the specimen. One of the consequences of testing a larger specimen is the ``barreling'' seen in con®gur2 ± 4 . This barreling tends to induce the ations 3 . criss-cross pattern of collapse seen in Once this pattern develops, crushing spreads from the collapsed zones to neighboring cells which have been weakened and destabilized. However, collapse is also in¯uenced by geometric imperfections and by deformation incompatibilities caused by the collapse sequence which stien particular clusters of cells. As a result, although the shear type mode of collapse reported in [11] is still the underlying mechanism of spreading of deformation in the specimen, the events are less orderly than in the smaller speci5 and 6 . mens as clearly seen in con®gurations For the same reason, the stress undulations that occur during the propagation of collapse are more randomly placed and their amplitudes and periods are more irregular. When essentially all the cells have collapsed the stress increases sharply. The extent of the undulating stress plateau (DeP=net strain from initiation stress to end of stress plateau) is 59.7% while the average value of the stress (propagation stress, sP) is 17.22 psi (118.8 kPa). The major problem parameters recorded are listed in Table 2 along with the results from two additional experiments on honeycomb specimens of the same size. Small variations are observed in the values of E*, sI and sP whereas DeP is very repeatable. Such dierences can be attributed to small imperfections unique to each specimen which result in dierences in the sequence of collapse of the cells. For the same reason, the positions and amplitudes of stress undulations on the stress plateaus were dierent in the three experiments. Crushing experiments on honeycombs of the same size made from the same material but with thickness of 4.7 10ÿ3 in (119.4 mm) produced qualitatively similar responses. However, this thinner honeycomb exhibited less barreling and collapse of horizontal rows of cells was favored over inclined zones of localized deformation (closer to sequence
(118.8) (111.9) (108.9) (67.5) (67.6)
DeP% 59.7 60.9 59.7 63.1 61.7
of events reported in [11]). The problem variables recorded in two experiments on this honeycomb are listed in Table 2. In this case, all variables are seen to be repeatable.
3. ANALYSIS
The crushing process is simulated numerically using a FE model developed within the nonlinear FE code ABAQUS. The cell sides are discretized with three-noded, quadratic beam elements (B22). These elements are based on kinematics which allow for large rotations of the normals, ®nite membrane strains and have a Timoshenko-type correction for shear deformations. Quadratic deformable surfaces are used for contact between the walls of the collapsing cells. Single thickness sides are discretized with seven elements each with a distribution biased towards the ends which undergo localized bending. Double thickness sides undergo limited deformation and, as a result, are represented with a single element. Nine Simpson integration points are used through the thickness. The elastic±plastic behavior of the Al-5052-H39 sheet metal is modeled as a J2-¯ow theory solid with isotropic hardening. The mechanical properties used came from uniaxial tension tests on strips cut from the sheets used to manufacture the honeycombs. The elastic moduli (E) and the yield stresses (s0) of the two materials are given in Table 3. The stress±strain response is idealized as bilinear with a post-yield modulus of E/100. The size of the specimens analyzed is the same as that of the test specimens (15 10 cells). In order to make the geometry of the cells in the model as close to that of the actual cells as possible, we use the mean of the measured values of the geometric parameters t, l, l and m given earlier in Table 1. In other words, all the cells are made to have the same deviations from the perfect geometry. The additional, randomly distributed deviations in these variables alluded to earlier, are neglected. Table 3. Elastic moduli and yield stress of Al-5052-H39 sheet metals from which the honeycombs are manufactured t in (mm)
E msi (GPa)
s0 ksi (MPa)
0.0057 (144.8) 0.0047 (119.4)
10.0 (68.97) 10.0 (68.97)
42.34 (292) 38.80 (267.6)
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
2771
Fig. 5. Simulation of expansion process for 15 10 cells specimen. (a) Traction stress vs cell width response. (b) Deformed con®gurations during process.
3.1. Simulation of honeycomb expansion We start by simulating the expansion process. In the initial (unstressed) con®guration, the model consists of 20 strips stacked together as shown in Fig. 5 0 ). The strips are connected to their neighbors on ( either side via the double wall thickness elements which are alternately placed along their lengths. The nodes of the double wall elements on the outer strip on the LHS are ®xed in the horizontal direction while those of the outer strip on the RHS are prescribed equal horizontal displacements incrementally. Figure 5 shows a plot of the traction stress se (net force/initial area of the walls being pulled) required for the expansion, vs the instantaneous width of the cell, 2D, normalized by the variable c' de®ned in equation (1) [for this case c' = 0.437 in (11.1 mm)]. As the honeycomb expands, the material yields and the response softens. At higher
values of D, membrane tension starts to develop in 1 ± 3 ) and the rotated cell sides (see con®gurations the response progressively stiens. The honeycomb is expanded beyond the required width so that when the load is released the width of the cells is 4 . exactly mc'. The ®nal con®guration is shown in The cell angle y de®ned in Fig. 1(d) is given by p 3 ÿ1 : 2 m y cos 2 In this case m = 0.885 and, as a result, y 1408. The dimensions of the honeycomb cells formed in this fashion have the geometric characteristics listed in Table 1. A scaled drawing of a cell with these characteristics is shown in Fig. 1(d). To demonstrate that the process faithfully reproduces some of the ®ner details of the actual honeycomb in Fig. 2(b) we include an expanded view of half of a cell from
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
the formed honeycomb. The cell corners are seen to have ``knees'' of similar geometry to that of the actual section of honeycomb shown in the photograph in Fig. 2(a). 3.2. Simulation of honeycomb crushing In addition to the geometric changes, the cold forming process introduces changes to the mechanical properties of the base material and leaves behind residual stresses. To capture these eects the honeycomb model is ®rst expanded and then crushed between two parallel, rigid surfaces (Fig. 3). One of the surfaces is ®xed and the second is translated vertically by prescribing incrementally its vertical displacement d. Before crushing begins the double walls seen protruding out at the top and 4 in Fig. 5 are bent bottom ends in con®guration over, as was done in the actual tests, in order to provide smoother points of contact with the rigid surfaces. Despite this, friction with the rigid surfaces still plays some role in the crushing process and was included in the analysis. Coulomb friction with a coecient of 0.2 was used for contact between the angled ends of the specimen and the rigid surfaces. The calculated sÿ d response is drawn in dashed line in Fig. 6(a) which also includes two of the experimental responses drawn in solid lines. The 0 and nine deformed con®gurations undeformed q from the simulation are shown in Fig. 6(b). They correspond to the numbered points marked on the response in Fig. 6(a). Initially, the response is sti with a modulus of EÃ* = 316 psi (2.18 MPa Ð
based on the net end-displacement as in the experiment). The deformation of the cells is symmetric about vertical axes through their centers. The rigid surfaces at the top and bottom of the specimen restrain the deformation of the end rows of cells to some degree. At the same time, the two free vertical edges can deform more than the rest of the specimen. Thus, the deformation has some initial biases which result in barreling of the specimen and more cell deformation in its center. Indeed, this initial bias in deformation is sucient to cause the central cells to destabilize ®rst by switching to the sheartype mode of deformation seen in the experiments 2 ). (In problems of plastic buck(see con®guration q ling like this one, a small imperfection is usually needed to help initiate the instability. Here, numerical noise and round-o errors left behind from the manufacturing process were sucient to initiate the instability and, as a result, no external disturbance was necessary.) The central row of cells collapses from the center outwards and, in the process, destabilizes the neighboring rows. Eventually, contact of the walls of the collapsed cells arrest local deformation in the central row and collapse continues in the two neigh3 ). This spreading of deformation boring rows (q from the collapsing row to its neighbor(s) continues 4 , rows 6±9 have collapsed. and, by con®guration q Subsequently, the lower of the collapse fronts stops moving and collapse ceases to be symmetric about the center of the specimen. Rows 5 and 4 collapse 5 ). At this stage, the stiening eect of next (see q the upper rigid surface starts to be felt and the load
Fig. 6(a) Caption on facing page..
Fig. 6. (a) Comparison of two experimental and a calculated load±displacement crushing responses of an aluminum honeycomb with t = 0.0057 in (144.8 mm). (b) Sequence of calculated deformed con®gurations corresponding to response in (a).
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB 2773
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PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
starts to increase. When the load reaches a high enough level, the 12th row, which until this point was away from the two collapse fronts, gets destabilized and two new collapse fronts develop (see [6]) albeit at a relatively higher load than earlier events. With this new development, and because the outer fronts are close to the rigid surfaces, crushing stops being limited to horizontal rows and we see the development of inclined zones of localized deformation which connect the hitherto unconnected col7 and q 8 ). By con®guration q 9 , all lapsed zones (q cells are essentially crushed and the response of the densi®ed material recovers signi®cant stiness. A note regarding coexisting multiple crushing fronts. In the four structures which can develop propagating instabilities discussed by Kyriakides in [16], the propagation loads were signi®cantly lower than the loads required to initiate the instabilities. Thus, once initiated the instabilities propagated at either one or two fronts. In the present problem, the initiation and propagation stresses do not dier signi®cantly. Therefore, if for any reason the average stress increases to a value close to sI, a new initiation event can take place as was observed 6 and q 7 . Multiple here between con®gurations q propagating fronts were also reported in a recent study of the evolution of phase transformations in NiTi shape memory alloy strips [17]. The underlying micromechanical mechanisms involved in the two instabilities are very dierent. However, in both problems the reason for coexisting multiple fronts is the relatively small dierence between sI and sP. As was reported in Section 2, the sequence of collapse events seen in any one experiment is never exactly the same as that of any other experiment primarily because of the in¯uence of small randomly distributed imperfections. At the same time, the resultant variation in the major parameters of the crushing response is relatively small. Although the expansion processes was modeled faithfully, random imperfections were not introduced. Thus, even though the sequence of crushing events seen in the simulation is close to those seen in experiments, it does not exactly represent that of any particular experiment. On the other hand, the calculated response in Fig. 6(a) is seen to be, in the main, in good agreement with the two experimental responses. As expected, the predicted initiation stress (s^ I ) is somewhat higher than the experimental values because this is the variable most aected by small initial geometric imperfections. By contrast, the calculated propagation stress (s^ P ) of 17.17 psi
(118.4 kPa) is in excellent agreement with the corresponding measured values. The same can be said for the extent of the plateaus. Thus, overall, the simulation is very successful. For comparison purposes a crushing simulation was also conducted for a honeycomb specimen with the same geometric parameters as those used above except that the expansion process was bypassed. Thus, the honeycomb is initially stress free and the material is at the virgin state. In addition, the sharply rounded knees present at the corners of the expanded honeycomb (Fig. 2) are now absent. The crushing response is similar to the one discussed above and will not be presented here. The major problem variables are compared to those of the expanded honeycomb in Table 4. The elastic modulus is somewhat higher (as predicted via the micromodel in [11]), while the initiation and propagation stresses are, respectively, 14% and 7.6% lower. The extent of the stress plateau is also somewhat lower. Similar calculations have been performed for the thinner honeycomb (t = 4.7 10ÿ3 in ÿ119.4 mm). Once again, the mean values of t, l, l and m given in Table 1 were used. The ®rst specimen was crushed following simulation of the expansion process and the second was initially stress free. The calculated response for the expanded honeycomb is compared to one of the experimental ones in Fig. 7. The major parameters are summarized in Table 4. As was the case in the experiments, the numerical simulations also exhibited crushing sequences con®ned mainly to horizontal rows of cells. The order of crushing of the various rows is dierent from the experiment and, thus, the stress undulations in the response occur at dierent strain values than in the experimental response. However, the main problem variables are once more in good agreement with the measured values in Table 2. The dierence between results of the expanded and the stress free honeycombs has the same general trends as those reported above for the thicker honeycomb. Finally a word about the eect of specimen size on the variables presented. This issue was explored for the honeycomb with t = 5.7 10ÿ3 in (144.8 mm) by crushing several specimens of the following sizes: 9 6, 15 10 and 21 14 (rows columns of cells). All specimens were crushed in the ribbon direction. The crushing of the smaller size specimens involved mainly horizontal rows of cells whereas the larger specimens exhibited signi®cant barreling of the sides and developed the criss±cross patterns of localized collapse seen in Fig. 4(b) but
Table 4. Elastic moduli, initiation and propagation stresses, and extents of strain plateaus predicted t in (mm)
l
m
0.0057 0.0057 0.0047 0.0047
0.837 0.837 0.848 0.848
0.885 0.885 0.893 0.893
(144.8) (144.8) (119.4) (119.4)
Expanded Yes No Yes No
EÃ* psi 316 353 176 196
(2.18) (2.43) (1.21) (1.35)
s^ I psi
s^ P psi
D^eP %
19.00 (131.0) 16.35 (112.8) 11.90 (82.1) 10.29 (71.0)
17.17 (118.4) 15.86 (109.4) 10.21 (70.4) 9.66 (66.6)
62.3 60.8 64.3 61.4
PAPKA and KYRIAKIDES: IN-PLANE CRUSHING OF A HONEYCOMB
2775
Fig. 7. Comparison of an experimental and a calculated load±displacement crushing response of an aluminum honeycomb with t = 0.0047 in (119.4 mm).
with this pattern being more pronounced. In all cases the mechanism of collapse was essentially the same shear-type mode discussed here and in [11]. The elastic modulus calculated based on the net end-displacement varied with specimen size. By contrast and despite the dierences in the crushing patterns mentioned above, the variation of the other variables was rather modest and of the order seen in the results in Table 2.
4. CONCLUSIONS
The work presented here is a continuation of the study reported by the same authors in [11]. Results from additional honeycomb crushing experiments are reported involving specimens of larger size than those reported in [11]. The results con®rm that such honeycombs develop a shear type instability which localizes into a narrow band of cells. Under displacement controlled loading, the crushed zone spreads until the whole specimen is consumed. The sequence of events through which crushing spreads in the specimens is aected by the size of the specimen. In smaller (and thinner wall) specimens, crushing is along horizontal rows of cells. Larger specimens exhibited barreling of the sides which is partly due to friction between the ends of the specimen and the rigid platens used to crush it. Criss±cross patterns of localized crushing were common in larger specimens. Despite these dierences, the propagation stress and extent of the stress plateau were found to be relatively unaected. The initiation stress, which is more sensitive to initial geometric imperfections, exhibited somewhat larger variation. Detailed measurements of the geometry of cells of several honeycombs showed that the cells were in general under-expanded and that the lengths of the bonded sides were shorter than the values of the
corresponding ideal hexagonal cells. The mean values of the parameters measured de®ne the dimensions and shapes of the cells in an average sense. These values were used in full scale simulations of the expansion process followed by crushing of the specimen; that is, the specimen crushed in the simulations included in an average sense the geometric and material changes and the residual stresses introduced by the manufacturing process. Both parts of the simulation matched the experiments well. The simulated crushing resulted in events which were somewhat dierent from those seen in any particular experiment but these dierences did not have any signi®cant eect on the propagation stresses or on the strain extent of the stress plateaus. Additional, randomly distributed small geometric imperfections present in the actual test specimens were not accounted for in the numerical models. The main eect of this deviation of the models from the actual specimens was that the predicted initiation stresses were somewhat higher than the corresponding experimental values. AcknowledgementsÐThe ®nancial support of the Air Force Oce of Scienti®c Research under Grant No. F49620-95-1-0154 and of the University of Texas at Austin is acknowledged with thanks. We are also grateful to H-K&S Inc. for making ABAQUS available under academic license and to Hexcel Corporation for providing the honeycombs tested.
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