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In-plane dynamic crushing of honeycomb. Part II: application to impact
International Journal of Mechanical Sciences 44 (2002) 1697 – 1714
In-plane dynamic crushing of honeycomb. Part II: application to impact A. H(onig1 , W.J. Stronge ∗ Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, UK Received 23 March 2001; received in revised form 15 April 2002
Abstract Finite element simulations were employed to analyse in-plane dynamic crushing of two di2erent hexagonal honeycombs (slenderness ratios L=t=38 and 167). The response of the honeycomb with the smaller slenderness ratio was studied for impact speeds up to 40:0 m=s which corresponds to a nominal strain rate for the specimen of 500 s−1 . Total plastically dissipated energy was used to quantify the e2ects of increasing strain-rate, since other measures showed a strong dependence on details of the 8nite element model. The simulations revealed a strong increase of total dissipated energy with increasing impact speed for velocities larger than the critical speed for wave trapping, vcr . One reason was a higher percent of cells collapsing in a symmetric crush mode (IV). Another reason for the larger total dissipation at higher crushing speeds was the greater irregularity in the folding pattern that developed. Experimental and calculated global force-time curves were compared for the honeycomb with the larger slenderness ratio at impact speeds 3:0 and 7:2 m=s and a satisfactory agreement was found. At these low impact speeds there was no increase of initial collapse or plateau stresses. A comparison of sequences of deformed mesh plots and high speed photos showed good correlation of the general distribution and modes of crushing. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction An advantage of ductile foams, cellular materials like wood and of ductile honeycombs is not only the small weight=strength ratio but also the ability to absorb energy during accidental impacts while limiting the crushing force [1]. This fact recently triggered many investigations in order to ∗
Corresponding author. Fax: +44-1223-332-662. E-mail address: [email protected] (W.J. Stronge). 1 Now at Department of Applied Mathematics, University of Bonn, Germany.
0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 2 ) 0 0 0 6 1 - 9
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understand dynamic properties of cellular materials (e.g. Refs. [2–8]). A main interest of these studies was the dynamic enhancement of initial crushing strength or plateau stress and how micro-inertia of cell walls contributed to the dynamic response. However, all the investigations were analytical or experimental and practically no numerical analysis of dynamic crushing of cellular material has been conducted. This is the approach of the present study, where we present results of 8nite element simulations of in-plane dynamic crushing of hexagonal honeycombs. An overview of rate e2ects in cellular material is presented in the monograph of Gibson and Ashby [1]. Also of interest is the work of Reid and Reddy [9] and Reid et al. [10]. These authors investigated experimentally and theoretically the inertia e2ects in a one-dimensional metal ring system subjected to impact. Later, Stronge and Shim [11] and Stronge [6] conducted static and dynamic tests with hexagonal- and square-packed arrays of thin-walled ductile metal tubes. Recently Reid and Peng [2] studied in a very systematic way the uniaxial dynamic crushing of cylindrical specimens composed of 8ve di2erent kinds of woods for impact velocities up to 300 m=s. They showed that the substantial enhancement of the initial crushing stress for wood loaded along the grain was due to micro-inertia inhibiting cell wall buckling. The dynamic enhancement of the crushing strength of specimens loaded across the grain, was correlated with a shock-type response. Also an increase of the plateau stress was measured. Similar results were reported by Harrigan et al. [4] who investigated experimentally aluminium honeycomb samples under out-of-plane loading for impact velocities up to 300 m=s. The increase of the plateau stress with increasing velocity resulted in signi8cant enhancement of the energy absorption capability of the honeycomb structures. Also Zhao and Gary [3] studied experimentally the crushing behaviour of aluminium honeycombs under impact loading. These authors used a split Hopkinson pressure bar to measure the dynamic stress–strain response and found for out-of-plane loading the same strong increase of plateau forces for higher loading rates as Harrigan et al. [4]. However for in-plane loading no strain-rate e2ects were detected up to impact speeds of 28 m=s which corresponded to a homogenised strain rate of 600 s−1 . The high strain rate compressive behaviour of two aluminium alloy foams (Alulight and Duocel) was recently investigated by Deshpande and Fleck [5] employing a split Hopkinson bar apparatus. Over the range of strain rates tested (10−3 s−1 –5 × 103 s−1 ) they found no strain-rate dependence of either the plateau stress or the densi8cation strain. In this paper we employ 8nite element simulation to analyse in detail dynamic in-plane crushing of two hexagonal honeycombs. One structure is identical to the honeycomb structure we studied in part I of this paper [12]. Some of the main results of that paper are reviewed in the following sections. The structure was also employed for several validation tests of the 8nite element model. A second structure was used in drop weight tests. Therefore the experimental results gave a further validation of the 8nite element simulation. While in part I the main concern was the initiation and location of the 8rst crush band, here we simulate the crushing of several bands and analyse the propagation of crush fronts for increasing applied impact speed. Since the calculated forces contain considerable high frequency variability due to complicated contact problems, we measure the inLuence of inertia by the total dissipated plastic energy and the total fraction of honeycomb cells deforming in a crush mode with large plastic dissipation. For the 8nite element simulation the program ABAQUS=Explict [13] was used.
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2. Simulation of crush front propagation 2.1. Description of 3nite element model The representative honeycomb material studied in this section is identical to the aluminium honeycomb structure used in part I; i.e. Al-5052-H39 material with nominal 11 mm cell size where ribbon and inclined cell wall length were, L = H = 5:5 mm and the thickness, t = 0:145 mm. The angle ◦ between the horizontal direction and the inclined cell wall was 0 = 30 . The material parameters of the aluminium cell walls were: Young’s modulus, Es = 68:97 GPa; yield stress Ys = 292:0 MPa; mass density, = 2700 kg=m3 . Further, since the main interests of the study in this section lay in inertia and dynamic e2ects, no strain hardening or strain-rate hardening were assumed. Results from part I showed that dynamic crushing can be e2ectively simulated by employing the 8nite element program ABAQUS=Explicit. For simulating crush band propagation with multiple contacts between cell walls which could have double sided contact, the possibilities of this program were quite limited. For example, there was no capability to simulate contact between lines (surfaces) consisting of beam-elements. The program does o2er the option of contacting surfaces for a class of shell-elements. In the following we describe some studies with a 8nite element model similar to the models described previously but with the cell walls discretised with shell-elements (S4R; 4-node doubly curved, reduced integration, hourglass control, 8nite membrane strain [13]) rather than beam elements. The model employs only one element in the out-of-plane direction (3-direction) of the honeycomb. As we show below, many degrees of freedom associated with the out-of-plane direction can be constrained so that the computational expense of the shell-element model was not much larger than that of the previous beam-element models used in part I. In order to estimate displacements in the out-of-plane direction, a characteristic cell for deformation mode IIIb (part I, Fig. 1) was discretised with 8 shell-elements along half of the length in the out-of-plane direction (bias factor 1.25 toward the outer edge). Since honeycomb panels of the described geometry and material (part I, Section 2) were commercially available with a depth of 15:9 mm (see for example Hexcel 3=8-5052-0.005 as used by Papka and Kyriakides [14]) this width was used throughout this investigation. In order to impose the periodicity conditions on the microsection the appropriate constraints were additionally introduced in the out-of-plane direction. The 8nite element calculation with this 3D model showed an increase of the peak stress to a value of 143 kPa in respect to the peak load for the 2D model of 122 kPa (part I, Fig. 4). At larger deLections the force–displacement curves of both models run roughly parallel to each other. It was checked that the 2- and 3-dimensional models were consistent by employing a 3-dimensional model with only one shell element along a width of 0:1 mm and with release of the additional periodicity constraints in the out-of-plane direction; additionally the Poisson’s ratio was set to zero. The force–displacement curve calculated with this model was close to the 2-dimensional results. The deformation state in the cell walls was nearly plane strain. This was demonstrated by employing a 8nite element model for the microsection with one shell-element along the depth of 15:9 mm, 8xed displacement in the 3-direction and 8xed rotation about the 1- and 2-directions for all nodes (constrained 3D model). The force–displacement curve of the full 3-dimensional model and the constrained 3D model were practically identical. One reason for this was that the vertical cell walls hardly deform because they have twice the thickness of the inclined walls. Therefore near the vertex points, the thicker walls enforce a strong constraint on the deformation of
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2 1 3
Fig. 1. Undeformed 8nite element mesh with soft layers at top and bottom of the honeycomb structure (constrained 3D model). Only the constrained degrees of freedom at the layers are shown. The out-of-plane direction is the 3-direction.
the inclined beams. Consequently, the following simulations were conducted with a constrained 3D model. A problem which emerges during the time of crushing is that both the contact between the colliding body and the honeycomb structure and the contacts that develop between di2erent cell walls during densi8cation occur mostly on a small scale in time and space. In order to calculate these contact forces, the 8nite element mesh has to be 8ne enough to resolve the transfer of contact forces. For example, when two cell walls come into contact during crushing at an impact speed larger than about 20 m=s, the material near the contact region deforms elastically or plastically and strong impulses travel through the thickness of the cell wall. All these events determine the details of the load transfer between the contacting walls. Therefore the 8nite element model with shell elements cannot accurately predict the response of the structure; the local contact response is in general too sti2. In order to calculate at least a rough estimate for forces, the 8nite element model shown in Fig. 1 was designed. This model has soft layers at top and bottom of the honeycomb structure with the following dimensions: length 88:1 mm, width 12:0 mm (here the out-of-plane width of the honeycomb structure was set to 10:0 mm) and thickness 2:0 mm. Each layer is discretised with eight C3D8R elements (8-node linear brick, reduced integration, hourglass control [13]); these are connected on one side to a contact surface. The layer material is modelled as elastic perfectly-plastic with a relatively small Young’s modulus, EL = 50:0 MPa, Poisson’s ratio, L = 0:3 and yield stress YL = 1:5 MPa and the mass density was chosen as L = 27 kg=m3 . In order to damp the high frequency oscillations of the honeycomb structure caused by sti2 contact response, sti2ness proportional damping with a Rayleigh damping factor RL = 10−6 s−1 was employed. The boundary conditions for the layers were set as following: 8xed vertical displacements for the nodes at the lower part of the bottom layer; prescribed vertical displacement for the nodes at the upper part of the top layer according the velocity ramping described before; 8xed displacement in the 1- and 3-direction at some nodes on the bottom and top layer prevent rigid body movements (see Fig. 1).
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The thickness and the material parameters of the layers at the top and bottom were selected and roughly optimised according the following criteria: • Thickness of the layers was chosen as small as possible in order to reduce the acceleration forces due to the initial acceleration period; but it was also chosen large enough to allow for some damping. • The mass density was assumed small in order to reduce the initial acceleration forces and to facilitate the excitation of oscillations. However, there is a lower limit for the mass density, since the stable time step size [13] should be determined by the shell element length. • The Young’s modulus was set larger than the homogenised Young’s modulus of the honeycomb structure to prevent combined vibration of the layers and the honeycomb structure; but the value was chosen small enough to absorb and damp the strong impact impulses. The stable time step size is also a factor which determines an upper limit for sti2ness. • The yield stress was chosen in a range to limit some of the highest force peaks. Further, it was found using several test simulations that the discretisation of a layer along the length (1-direction) with eight elements is optimal in the sense that a 8ner discretisation reveals details which complicate the picture described before. For an impact velocity of 20:0 m=s the dissipated energy due to Rayleigh damping in the contact layers is less than 1:0% of the external work. 2.2. Crush front propagation For the investigation in this section, 8nite element models with an initial misalignment of the ◦ ◦ vertical walls in row 5 or 9 (impact site) of 1:0 and 4:0 were used, respectively. The model with the large misalignment in the top row was chosen in order to trigger mode III crush band initiation at impact. No friction was assumed between the soft contact layers and cell walls. The rise time of the prescribed velocity ramp was set, trise = 0:5 ms. The same period was also used for the investigation in part I, Section 4.2 and corresponds to a moderate initial transient response at an impact speed of 10 m=s. Figs. 2 and 3 show the force–time response at the impact surface for speeds, 10:0 and 30:0 m=s, respectively; these results were calculated with the 8nite element model having a misalignment in row 5. When the impact speed is increased from 10:0 to 30:0 m=s, the initial peak load increased from about 150 to 200 kPA. This e2ect was discussed in part I. At an impact speed of 10:0 m=s, the crush band initiated at the imperfect row. This is illustrated in Fig. 4a, where the deformed 8nite element mesh (initial imperfection in row 5) is displayed at time, 2:58 ms. After initiation the crush band propagated from the imperfect row of cells to the distal surface of the specimen (Fig. 4b, 5:16 ms). The larger peaks in the force–time curve of Fig. 2 that occur after 2:5 ms can be partly linked to events of sequential band densi8cation; although the exact correlation is diScult as the structure is crushed and changes in sti2ness. Not shown is the crushing pattern and the force–time response for the structure with an initial imperfection in row 9. At an impact speed of 10:0 m=s crushing initiated here at the impact surface and then at the distal surface. Finally, crushing of bands propagated from the distal to the impact surface. Due to the hard impact of the cell walls with the soft layer when the surface row of cell collapses, the force–time curve has a large peak at this time.
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Nom. Stress (kPa)
200
150
100
50
0 0
0.001
0.002
0.003 0.004 Time (sec)
0.005
0.006
Fig. 2. Temporal variation of nominal stress at impact surface for Al-5052-H39 and impact speed of 10:0 m=s. The 8nite ◦ element model has initial misalignment in row 5 of 1:0 . 600
Nom. Stress (kPa)
500 400 300 200 100 0 0
0.0004
0.0008
0.0012
0.0016
0.002
0.0024
Time (sec) Fig. 3. Temporal variation of nominal stress at impact surface for Al-5052-H39 and impact speed of 30:0 m=s. The 8nite ◦ element model has initial misalignment in row 5 of 1:0 .
The force–time curve in Fig. 3 for the impact speed, 30:0 m=s (imperfection in row 5) shows a strong transient response. The highest peak occurs at time, 0:47 ms before the end of the rise time; this is caused by impact of cell walls with the contact layer on the impact surface. After that, impacts between cell walls and the sudden collapse of cells excite heavy vibrations. This response may be exaggerated because the contact sti2ness between cell walls is too large as was discussed in the last section. A sequence of deformed mesh plots is displayed in Fig. 5a–e. For this large impact speed, crushing started at the impact surface in mode IVab (Fig. 5a–c) and it progressed through the honeycomb with rows collapsing sequentially in mode III or IV (Fig. 5d and e), irrespective of the initial imperfection in row 5. This pattern of crushing is similar to that of a 8nite element model
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v0
(b)
(a)
Fig. 4. Deformed 8nite element meshes at time: (a) 2:58 ms; and (b) 5:16 ms. Impact speed is 10:0 m=s and the model ◦ has initial misalignment in row 5 of 1:0 where crushing initiates.
(b)
(a)
(c)
2 3
(d)
1
(e)
Fig. 5. Sequence of deformed 8nite element meshes at time: (a) 0:46 ms; (b) 0:93 ms; (c) 1:39 ms; (d) 1:85 ms; and (e) ◦ 2:32 ms. Impact speed is 30:0 m=s and the model has initial misalignment in row 5 of 1:0 . Crushing starts at the impact surface in mode IVab.
with a large imperfection in row 9, as shown in Fig. 6a–e. However, the mode of initial crushing is IIIb due to the initial imperfection of the surface layer of cells. Fig. 4a proves that the critical impact speed for localisation at the impact surface is larger than a speed of 10:0 m=s. Despite this, the 2-dimensional beam-element model employed in part I clearly showed wave trapping near the impact surface at that impact speed. One reason for the di2erence is the out-of-plane e2ect (see last section). The yield stress increases by a factor 1:17 (di2erence between 2D and 3D microsection calculation) so that according to Eq. (18) in part I, the estimated critical speed is 8:9 m=s. Further 8nite element simulations indicate that the critical impact speed is
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◦
Fig. 6. Changes of initial crush mode calculated with initial misalignment in row 9 (top) of 4:0 . The sequence of deformed meshes are at time: (a) 0:46 ms; (b) 0:93 ms; (c) 1:39 ms; (d) 1:85 ms; and (e) 2:32 ms. Impact speed is 30:0 m=s.
7000 v= 2.0 m/s v=10.0 m/s v=20.0 m/s v=30.0 m/s v=40.0 m/s
Energy (Nmm)
6000 5000 4000 3000 2000 1000 0
0
10
20
30 40 50 60 Specimen shortening (%)
70
80
Fig. 7. Variation of totally dissipated plastic energy with specimen shortening =L for 8ve di2erent impact speeds. The ◦ results were calculated with an initial misalignment in row 9 (top) of 4:0 .
about 13:0 m=s. At this speed crushing starts at the impact surface and later jumps to the 8fth row where the initial imperfection was located. In order to estimate the e2ect of increasing impact speed on the energy absorption capacity of a speci8c volume of honeycomb, the dissipated plastic energy of the whole model with an out-of-plane thickness of 10:0 mm was calculated as a function of displacement of the impact surface. The result is displayed in Fig. 7. Here, we used the model with an initial imperfection in row 9 and varied the impact speed from 2:0 m=s up to 40:0 m=s. A similar result was found for an imperfection in the middle of the structure. The plot shows an increasing plastic dissipation with increasing impact speed for large specimen shortening (¿ 20%). While the curves for impact speeds, 2:0 and 10:0 m=s
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40 30/m/s
35
ENERGY (Nmm)
30 25 20 15 10 5 0 48
50
52
54
56
58
60
Cell wall indentification number
Fig. 8. Example of distribution of plastic dissipation. Each box represents the plastically dissipated energy of a shell element in a transverse cell wall in row 5. The cell walls are enumerated from left to right and each has 12 shell elements ◦ (144 in a row). Results were calculated with initial misalignment in row 5 of 1:0 and an impact speed of 30:0 m=s.
increase almost linearly with specimen shortening, the curves corresponding to a larger impact speed increase more rapidly. After crushing to a nominal 80% strain (specimen shortening), there is a di2erence of 65% in plastically dissipated energy between impact of 10:0 and 40:0 m=s. Some of the reasons for the higher dissipation at larger impact speeds can be explained by means of Fig. 8. Every box in this plot describes the plastically dissipated energy of a shell-element in the 8fth row of inclined cell walls; i.e. the 8fth row of inclined cell walls up from the distal surface in this 10 row specimen. The shell-elements are counted from left to right and the number of elements in each row is 144 (12 shell-elements per inclined cell wall). Since the element number in Fig. 8 is divided by 12, the x-axis gives the wall number—counted in the same way as the element number. For the simulation the model with an imperfection in row 5 was used and the dissipation was calculated at time 2:32 ms. The highest dissipation in each cell are here left or right of a vertex where in general the plastic hinges develop. The picture shows for example only one hinge at cell wall number 56 (mode III) and two hinges near number 51 (mode IV). Also there is substantial plastic deformation at the mid-span in cell wall 49. In contrast, at smaller impact speeds, the plastic deformation in the inclined cell walls toward the edges of the honeycomb was less because the structure crushed in mode III (approximately) and this allows a simple folding of the crushing bands of cells (Fig. 4b). A further reason for the increase of plastic dissipation with increasing impact speed is the increasing number of cells deforming in a pattern similar to mode IVab. Since that crushing mode has a larger dissipation than mode III (see part I), the global dissipation is elevated. To quantify this, the following fraction F of mode III deformation was de8ned: F:=
max(E pl ; E pl ) − min(E pl ; E pl ) L R L R pairs
max(ELpl ; ERpl )
;
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Frac. F of mode III
0.8 0.6 0.4 0.2 0 0
10
20
30 40 50 60 Specimen shortening (%)
70
80
Fig. 9. Volume fraction of honeycomb collapsing in antisymmetric mode III deformation rather than symmetric mode IV. The solid and dashed line correspond to impact speed (a) 10:0 m=s and (b) 30:0 m=s, respectively. The values were ◦ calculated with a 8nite element model with initial misalignment in row 5 of 1:0 .
where ELpl and ERpl are the plastically dissipated energy in the element to the left and right of a thick ribbon cell wall. The sum in the equation runs over element pairs where at least one element of the pair was deformed plastically. For pure mode III collapse the value of F is therefore 1 while in mode IV, F = 0. The results in Fig. 9 show, that for global strain larger than 30% the amount of collapse in antisymmetric mode III is about 0:25 smaller at the impact speed of 30 m=s than it is at an impact speed v0 = 10 m=s. 3. Comparison with experiment In order to further validate the numerical calculations and apply them to another problem, we compare in this section measurements from drop weight tests on an aluminium honeycomb with results from 8nite element simulations. 3.1. Description of experiment and 3nite element model The dropped weight is a heavy mass, m = 2:5 kg, which is guided by four rails. During impact tests, photos of the deformed honeycomb structure were taken with a high speed camera. Further, two data loggers were used to record the acceleration of the striking mass and the force acting on the distal surface. The data logger had a sample frequency of 25 kHz. Two impact speeds were studied, 3:0 and 7:2 m=s. While 7:2 m=s was the largest velocity achievable with the drop tower apparatus, the velocity of 3:0 m=s was the smallest velocity for reasonable simulations with ABAQUS=Explict. For the dynamic crush experiments, an aluminium honeycomb panel (1.8-3=4-25(3003)) was used ◦ which had L = 11:5 mm; H = 5:75 mm; t = 0:069 mm; = 35:0 and depth of 50:0 mm. The honeycomb specimen had 8 columns and 13 rows; in comparison with our earlier investigation, this large specimen was chosen in order to reduce the e2ect of boundary imperfections that are caused by partly damaged cell walls.
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Similar to the previous calculations, a density of s = 2700 kg=m3 and a Young’s modulus of Es = 68:97 GPa were assumed for the Al 3003 material. The yield behaviour of the aluminium alloy was measured by Breton [15] using small sheets of material taken from the honeycomb panel. The material has a relative sharp transition to yielding with a yield stress of approximately 240:0 MPa. Shortly after yielding the specimens broke due to necking and a shear instability. Therefore no post-yield behaviour could be measured. In the following simulation an ideal plastic material model was employed with a yield stress, Ys = 240:0 MPa. Also no strain-rate dependency was assumed. However, we will show the e2ects of strain hardening and strain-rate hardening are small for a realistic strain-hardening modulus and strain-rate dependency. For these material properties we calculate a critical impact speed for the uniaxial stress case, vcr = 8:64 m=s, using Eq. (18) in part I. This is larger than the largest experimental velocity and therefore no localisation at the impact side should be expected. Preliminary experimental tests with a pneumatic launch device that produced higher impact speeds indicated that the critical impact speed was about 13 m=s. The 8nite element model for the honeycomb specimen was similar to the constrained 3D model used in the last section. Through the thickness, the honeycomb model had only one shell-element and the vertical and horizontal cell walls were discretised with six and 16 shell-elements, respectively. All degrees of freedoms corresponding to the out-of-plane direction were 8xed and also a Poisson’s ratio, = 0, was assumed. The depth of the model was set to 10:0 mm. Since the imperfections of the honeycomb specimen could not be determined exactly, the model has an imperfection in rows ◦ ◦ 7 and 13 with misalignments of = 1:0 ; −1:0 , respectively. Therefore crushing at these positions was equally favoured. The soft bottom and top contact layers had a length of 190:7 mm, depth of 12:0 mm and thickness of 2:0 mm and consisted of 10 brick elements, each. The layer material was chosen as elastic with Youngs’ modulus, EL = 1:6 MPa, Poisson’s ratio, L = 0:3, Rayleigh damping factor, RL = 5:0 × 10−6 s−1 and density, L = 1:35 kg=m3 . The drop weight was modelled as a rigid mass (concentrated mass point). Finally, the vertical displacements of the nodes at the top of the impact layer were constrained to be equal to the vertical displacement of the rigid mass. Both, the mass and the contact layer on the impact surface had the same initial velocity, called impact velocity v, in the following discussion. The other boundary conditions for the honeycomb structure and the contact layers are identical to those described in the previous section. 3.2. Comparison: 3nite element simulations and experiments An essential parameter of the model is the coeScient of Coulomb friction, between the contact layers and the honeycomb. This point was analysed with some preliminary studies which showed that the strong barrelling of the specimen seen during the experiment is absent in the case of very low friction ( ¡ 0:1). Because of the sharp edges at the periphery of the honeycomb specimen, a value of Coulomb friction, = 0:3 was assumed. Figs. 10 and 11 show a comparison of the calculated and measured force–time response at the distal surface for impact speeds 3:0 and 7:2 m=s, respectively. The measured plateau force for both velocities is about 5:0 kPa; i.e. smaller than the value (6:24 kPa) predicted by Eq. (5) in part I and the value obtained from the 8nite element simulations.
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Nom. Stress (kPa)
12 10 8 6 4 2 0 0
0.01
0.02
0.03 Time (sec)
0.04
0.05
Fig. 10. Comparison of nominal stress vs time curves for 1.8-3=4-25(3003) and impact speed 3:0 m=s. The solid line was calculated with the 8nite element model and the dashed line represents the experimental curve. The forces were determined at the distal surface.
14 FEM Exp.
Nom. Stress (kPa)
12 10 8 6 4 2 0 0
0.003
0.006
0.009
0.012
0.015
0.018
Time (sec)
Fig. 11. Comparison of nominal stress vs time curves for 1.8-3=4-25(3003) and impact speed 7:2 m=s. The solid line was calculated with the 8nite element model and the dashed line represents the experimental curve. The forces were determined at the distal surface.
Impact against the top layer on the honeycomb excites a very strong longitudinal wave, which travels in a short time through cell walls from the impact to the distal surface of the honeycomb. This causes a very high peak force. As already discussed in the last section, this peak is too large due to the relatively simple 8nite element model and problems with the contact algorithm. A similar situation occurs when after about 3:0 ms the bending wave reaches the distal surface (Figs. 10 and 11). The response predicted by the 8nite element model is too sti2 and violent oscillations are excited (solid line in Figs. 10 and 11). Except for these points, there is satisfactory agreement between the
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8
Nom. Stress (kPa)
7 6 5 4 3 2
FEM Exp.
1 0 0
5
10
15 20 25 Specimen Shortening [%]
30
35
Fig. 12. Comparison of nominal stress vs specimen shortening =L curves for 1.8-3=4-25(3003). The dashed line represents a static experimental curve. The solid line was calculated with the 8nite element model for impact speed 3:0 m=s with x-axes shifted by 6:0.
8 7
Nom. Stress (kPa)
6 5 4 3 2
FEM Exp.
1 0 0
5
10
15 20 25 Specimen Shortening [%]
30
35
Fig. 13. Comparison of nominal stress vs specimen shortening =L curves for 1.8-3=4-25(3003). The dashed line represents a static experimental curve. The solid line was calculated with the 8nite element model for impact speed 7:2 m=s with x-axes shifted by 17:0.
calculated force–time curves and the experimental measurements; the calculated force–time curves compare satisfactory to the experimental measurements and the simulation yields good estimates for the beginning of the densi8cation period where the force increases sharply. To compare the response more directly, the simulated macroscopic nominal stress–strain curves for the impact speeds 3:0 and 7:2 m=s were plotted (solid line) in Figs. 12 and 13, respectively. Additionally the measured static nominal stress–strain curve taken from Breton’s report [15] was included (dashed line). Since the calculated load at the impact surface was too noisy, we used the load at the distal surface. The proximal end displacement of each calculated curve is then shifted by the amount of macroscopic nominal strain corresponding to the time when the bending wave had
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Fig. 14. Comparison of sequences of photos and deformed 8nite element plots for impact speed 7:2 m=s. The photos were taken at time: (a) 4:39 ms; (c) 11:06 ms; and (e) 15:5 ms after impact. The deformed 8nite element meshes correspond to time: (b) 4:44 ms; (d) 11:11 ms; and (f) 15:56 ms.
reached the distal surface (time after the initial second largest force peak). Comparing the results, we 8nd that the plateau stress has not increased signi8cantly with impact speed. However, the transient response in the beginning is much stronger at the higher impact speed. Since as expected there were no distinctive di2erences between the experimental deformation pattern for impact velocities of 3:0 and 7:2 m=s, the crush band development for the higher speed was analysed in more detail. Fig. 14a, c and e show three photos of a deformed specimen
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Fig. 15. Comparison of photo and deformed 8nite element plot for impact speed 3:0 m=s. The photo and the deformed 8nite element mesh plot correspond to time: (a) 26:2 ms; and (b) 27:0 ms, respectively.
taken at 4:39, 11:06 and 15:5 ms after impact. For comparison with these high speed photographs, Fig. 14b, d and f show calculated honeycomb deformation 8elds at similar times after impact (4:44; 11:11 and 15:56 ms). A comparison of the photo and the calculated deformation of the mesh at time 4:4 ms (Fig. 14a and b) shows a close agreement, with more deformation of cells near the top. This pattern is consistent with the fact that at this time, the bending wave has returned half-way back from the distal to the impact surface. At 11:06 ms the photo shows two developing crush bands which cross in row 4 (Fig. 14c). The honeycomb has the barrelled shape which was also shown by Papka and Kyriakides [16]. These authors argued that barrelling due to friction at the external surfaces is the source of the initiation of the two crossed crush bands. The same features were seen in the simulation of Fig. 14d; although here the crushing started in row 2. Interesting also is the strong deformation and rotation of the left and right cells in the bottom row, while cells in the centre of that row are nearly undeformed. A similar pattern can be found in the corresponding photo from the experiment. The next comparison, for impact speed v = 7:2 m=s, is shown in Fig. 14e and f at a time just before densi8cation. Here the photo and the deformed mesh plot show no crushing at the top and bottom but there is crushing in the middle of structure. For slow speed impact, v = 3:0 m=s, the deformation was similar to the pattern apparent at the higher speed but somewhat more localised in the central rows. This is shown in Fig. 15a and b where the photo at time 26:2 ms and the deformed 8nite element mesh at time 27:0 ms are displayed; at that time the specimen shortening was about the same as in Fig. 14c and d. A distinctive feature apparent in the high speed photo (Fig. 15a) but not occurring in the simulation was the shear band that touched the left upper corner. This band was partly blocked by the collapse of horizontal crush bands and this could be the reason that the experimental force–time curve (Fig. 10, broken line) did not decrease as it had in the simulation (e.g. at time 27 ms). In order to estimate the e2ect of strain hardening and strain-rate hardening, the 8nite element simulation for the impact speed v = 7:2 m=s, was repeated until a time of 11:11 ms, including these e2ects. We found that a post-yield strain-hardening modulus h=E=100, the same as that employed by Papka and Kyrikides [14], made practically no di2erence in the force–time curve or the deformed mesh plot. Strain-rate hardening of the aluminium alloy was modelled by adopting the Cowper– Symonds visco-plastic strain-rate power law [13] with parameters D = 6500 s−1 and p = 4:0. These
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parameters were taken from Ref. [17] and represent the relatively small strain-rate dependency of aluminium alloy. (Holt et al. [18] have shown athermal behaviour for heat treated Al alloys at strain rates smaller then 1000 s−1 .) Using this model the simulation for an impact speed 7:2 m=s gave a small increase of the force. For example at the time 11:11 ms the force was about 11% larger than the calculated force obtained from the 8nite element model with no strain-rate dependency. This is consistent with experimental results of Zhao and Gray [3], who found no strain-rate e2ects during in-plane crushing of an aluminium honeycomb at a macroscopic nominal strain rate of 780 s−1 . Papka and Kyriakides [16] showed that, due to the manufacturing process (expansion) the initiation and plateau stress can be higher than the values predicted with a 8nite element model not including this process. In one case they calculated a di2erence in the initiation and plateau stress of 14% and 8%, respectively. The reason are strain-hardening and slightly rounded cell corners. However, the specimens of the used honeycomb (1.8-3=4-25(3003)) had relatively large defects due to processing and sample cutting and it is probable that these defects have a much stronger e2ect on the response. We found partly damaged cell walls and vertexes at the periphery of the specimen with small parts remaining from cut o2 cell walls. Further the honeycomb panels were quite irregular as received ◦ and a scatter in the wall lengths of up to ±0:3 mm and variation in the angle of up to 10 were estimated. 4. Conclusion Finite element simulations were employed to analyse in-plane dynamic crushing of two di2erent hexagonal honeycombs which had wall slenderness ratios L=t = 38 and 167 for single thickness cell walls. Crush band initiation and wave trapping in the honeycomb structure with the smaller slenderness ratio was investigated in the companion paper [12]. The honeycomb with the larger ratio was also used in drop weight experiments. The honeycomb structures were discretised by shell-elements with contact surfaces on both sides; this is necessary for the analysis to extend into the range of cell densi8cation. Studies showed that in the inclined cell walls the strain in the out-of-plane direction was small. The reason is that the out-of-plane width is several times (3–5) larger than the cell wall length. Furthermore, the vertices introduce constraint on out-of-plane displacements because the vertical ribbon walls deform very little. Therefore the 8nite element model had all displacements in the out-of-plane direction constrained to zero in order to represent this plane-strain condition. Since the local load transfer between colliding cell walls or between cell walls and the boundary surfaces (impact and distal surface) for impact speeds ¿ 10 m=s could not be resolved in detail by shell-elements, a 8nite element model with soft layers at top and bottom of the honeycomb structure was employed. These layers were necessary to partially damp initial transient shocks arising from an over-sti2 contact response of the 8nite element model. The total reaction forces calculated with small damping in the soft layers were in most cases reasonable. The response of the honeycomb with the smaller slenderness ratio was studied for impact speeds up to 40:0 m=s which corresponds to a macroscopic nominal strain rate for the specimen of 500 s−1 . For impact speeds ¿ 10 m=s the impact of adjacent cell walls and the impact of cell walls with the soft boundary layer generated strong vibrations and large amplitude waves in the structure. Therefore maximum forces could not be used to characterise the response at di2erent impact speeds; these
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forces were also strongly sensitive to details of the analytical model. As a less sensitive measure, we used the total energy dissipated by plastic deformation in the honeycomb. For impact speeds larger than the critical speed for wave trapping, vcr , the simulations showed an increase in total plastic dissipation with increasing impact speed. For example, at 80% global strain the dissipated energy resulting from impact at a speed of 40:0 m=s is 65% larger than the dissipated energy from impact at a speed of 10:0 m=s. One reason was a higher percent of cells collapsing in a symmetric crush mode (IV) which dissipates more plastic energy. Another reason for the larger total dissipation at higher crushing speeds is greater irregularity in the mode of crushing (folding pattern). For the simulation of the honeycomb with a larger slenderness ratio, at the top and bottom of the specimen a Coulomb friction coeScient, = 0:3 was assumed. This gave a predicted deformation 8eld with the same strong barrel shaped form as that seen in the experiments. The calculated force– time curves at the distal surface, calculated for impact speeds 3:0 and 7:2 m=s, compared quite well with the corresponding experimental curves. For impact speeds below the speed for wave trapping there was no increase of initial collapse or plateau stress with increasing impact speed. A comparison of sequences of deformed mesh plots and high speed photos showed good correlation of the general distribution and mode of crushing. Acknowledgements The authors are grateful for 8nancial support from EPSRC, Grant reference GR=K54137, for helpful discussions with Professor J.R. Willis and for experimental assistance of S. Wearn. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties, 2nd ed. Cambridge, UK: Cambridge University Press, 1997. [2] Reid SR, Peng C. Dynamic uniaxial crushing of wood. International Journal of Impact Engineering 1997;19:531–70. [3] Zhao H, Gary G. Crushing behaviour of aluminium honeycombs under impact loading. International Journal of Impact Engineering 1998;21:827–36. [4] Harrigan JJ, Reid SR, Peng C. Inertia e2ects in impact energy absorbing materials and structures. International Journal of Impact Engineering 1999;22:955–79. [5] Deshpande VS, Fleck NA. High strain rate compressive behaviour of aluminium alloy foams. International Journal of Impact Engineering 2000;24:277–98. [6] Stronge WJ. Dynamic crushing of elastoplastic cellular solids. In: Jono M, Inoue T, editors. Proceedings of the Sixth International Conference of Mechanical Behaviour of Materials, vol. 1. Oxford: Pergamon Press, 1991. p. 377–87. [7] Klintworth JW. Dynamic crushing of cellular solids. PhD dissertation, University of Cambridge, UK, 1988. [8] Klintworth JW, Stronge WJ. Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs. International Journal of Mechanical Sciences 1988;30:273–92. [9] Reid SR, Reddy TY. Experimental investigation of inertia e2ects in one-dimensional metal ring systems subjected to impact—I: 8xed-ended systems. International Journal of Impact Engineering 1983;1:85–106. [10] Reid SR, Bell WW, Barr RA. Structural plastic shock model for one-dimensional ring systems. International Journal of Impact Engineering 1983;1:175–91. [11] Stronge WJ, Shim VPW. Dynamic crushing of a ductile cellular array. International Journal of Mechanical Sciences 1987;29:381–406.
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[12] H(onig A, Stronge WJ. In-plane dynamic crushing of hexagonal honeycomb I: crush band initiation and wave trapping. International Journal of Mechanical Sciences, PII: S0020-7403(02)00060-7. [13] ABAQUS Standard and Explicit User’s Manuals, Version 5.7. Providence, RI, USA: Hibbit, Karlson and Sorensen, Inc., 1997. [14] Papka SD, Kyriakides S. In-plane compressive response and crushing of honeycomb. Journal of Mechanics and Physics of Solids 1994;42:1499–532. [15] Breton N. Research report. Engineering Department, University of Cambridge, UK, 1997, unpublished. [16] Papka SD, Kyriakides S. Experiments and full-scale numerical simulations of in-plane crushing of a honeycomb. Acta Materialia 1998;46:2765–76. [17] Tam LL, Calladine CR. Inertia and strain-rate e2ects in a simple plate-structure under impact loading. International Journal of Impact Engineering 1991;11:349–77. [18] Holt DL, Babcock SG, Green SJ, Maiden CJ. Transactions of the American Society of Metals 1967;60:152.
In-plane dynamic crushing of honeycomb. Part II: application to impact A. H(onig1 , W.J. Stronge ∗ Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, UK Received 23 March 2001; received in revised form 15 April 2002
Abstract Finite element simulations were employed to analyse in-plane dynamic crushing of two di2erent hexagonal honeycombs (slenderness ratios L=t=38 and 167). The response of the honeycomb with the smaller slenderness ratio was studied for impact speeds up to 40:0 m=s which corresponds to a nominal strain rate for the specimen of 500 s−1 . Total plastically dissipated energy was used to quantify the e2ects of increasing strain-rate, since other measures showed a strong dependence on details of the 8nite element model. The simulations revealed a strong increase of total dissipated energy with increasing impact speed for velocities larger than the critical speed for wave trapping, vcr . One reason was a higher percent of cells collapsing in a symmetric crush mode (IV). Another reason for the larger total dissipation at higher crushing speeds was the greater irregularity in the folding pattern that developed. Experimental and calculated global force-time curves were compared for the honeycomb with the larger slenderness ratio at impact speeds 3:0 and 7:2 m=s and a satisfactory agreement was found. At these low impact speeds there was no increase of initial collapse or plateau stresses. A comparison of sequences of deformed mesh plots and high speed photos showed good correlation of the general distribution and modes of crushing. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction An advantage of ductile foams, cellular materials like wood and of ductile honeycombs is not only the small weight=strength ratio but also the ability to absorb energy during accidental impacts while limiting the crushing force [1]. This fact recently triggered many investigations in order to ∗
Corresponding author. Fax: +44-1223-332-662. E-mail address: [email protected] (W.J. Stronge). 1 Now at Department of Applied Mathematics, University of Bonn, Germany.
0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 2 ) 0 0 0 6 1 - 9
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understand dynamic properties of cellular materials (e.g. Refs. [2–8]). A main interest of these studies was the dynamic enhancement of initial crushing strength or plateau stress and how micro-inertia of cell walls contributed to the dynamic response. However, all the investigations were analytical or experimental and practically no numerical analysis of dynamic crushing of cellular material has been conducted. This is the approach of the present study, where we present results of 8nite element simulations of in-plane dynamic crushing of hexagonal honeycombs. An overview of rate e2ects in cellular material is presented in the monograph of Gibson and Ashby [1]. Also of interest is the work of Reid and Reddy [9] and Reid et al. [10]. These authors investigated experimentally and theoretically the inertia e2ects in a one-dimensional metal ring system subjected to impact. Later, Stronge and Shim [11] and Stronge [6] conducted static and dynamic tests with hexagonal- and square-packed arrays of thin-walled ductile metal tubes. Recently Reid and Peng [2] studied in a very systematic way the uniaxial dynamic crushing of cylindrical specimens composed of 8ve di2erent kinds of woods for impact velocities up to 300 m=s. They showed that the substantial enhancement of the initial crushing stress for wood loaded along the grain was due to micro-inertia inhibiting cell wall buckling. The dynamic enhancement of the crushing strength of specimens loaded across the grain, was correlated with a shock-type response. Also an increase of the plateau stress was measured. Similar results were reported by Harrigan et al. [4] who investigated experimentally aluminium honeycomb samples under out-of-plane loading for impact velocities up to 300 m=s. The increase of the plateau stress with increasing velocity resulted in signi8cant enhancement of the energy absorption capability of the honeycomb structures. Also Zhao and Gary [3] studied experimentally the crushing behaviour of aluminium honeycombs under impact loading. These authors used a split Hopkinson pressure bar to measure the dynamic stress–strain response and found for out-of-plane loading the same strong increase of plateau forces for higher loading rates as Harrigan et al. [4]. However for in-plane loading no strain-rate e2ects were detected up to impact speeds of 28 m=s which corresponded to a homogenised strain rate of 600 s−1 . The high strain rate compressive behaviour of two aluminium alloy foams (Alulight and Duocel) was recently investigated by Deshpande and Fleck [5] employing a split Hopkinson bar apparatus. Over the range of strain rates tested (10−3 s−1 –5 × 103 s−1 ) they found no strain-rate dependence of either the plateau stress or the densi8cation strain. In this paper we employ 8nite element simulation to analyse in detail dynamic in-plane crushing of two hexagonal honeycombs. One structure is identical to the honeycomb structure we studied in part I of this paper [12]. Some of the main results of that paper are reviewed in the following sections. The structure was also employed for several validation tests of the 8nite element model. A second structure was used in drop weight tests. Therefore the experimental results gave a further validation of the 8nite element simulation. While in part I the main concern was the initiation and location of the 8rst crush band, here we simulate the crushing of several bands and analyse the propagation of crush fronts for increasing applied impact speed. Since the calculated forces contain considerable high frequency variability due to complicated contact problems, we measure the inLuence of inertia by the total dissipated plastic energy and the total fraction of honeycomb cells deforming in a crush mode with large plastic dissipation. For the 8nite element simulation the program ABAQUS=Explict [13] was used.
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2. Simulation of crush front propagation 2.1. Description of 3nite element model The representative honeycomb material studied in this section is identical to the aluminium honeycomb structure used in part I; i.e. Al-5052-H39 material with nominal 11 mm cell size where ribbon and inclined cell wall length were, L = H = 5:5 mm and the thickness, t = 0:145 mm. The angle ◦ between the horizontal direction and the inclined cell wall was 0 = 30 . The material parameters of the aluminium cell walls were: Young’s modulus, Es = 68:97 GPa; yield stress Ys = 292:0 MPa; mass density, = 2700 kg=m3 . Further, since the main interests of the study in this section lay in inertia and dynamic e2ects, no strain hardening or strain-rate hardening were assumed. Results from part I showed that dynamic crushing can be e2ectively simulated by employing the 8nite element program ABAQUS=Explicit. For simulating crush band propagation with multiple contacts between cell walls which could have double sided contact, the possibilities of this program were quite limited. For example, there was no capability to simulate contact between lines (surfaces) consisting of beam-elements. The program does o2er the option of contacting surfaces for a class of shell-elements. In the following we describe some studies with a 8nite element model similar to the models described previously but with the cell walls discretised with shell-elements (S4R; 4-node doubly curved, reduced integration, hourglass control, 8nite membrane strain [13]) rather than beam elements. The model employs only one element in the out-of-plane direction (3-direction) of the honeycomb. As we show below, many degrees of freedom associated with the out-of-plane direction can be constrained so that the computational expense of the shell-element model was not much larger than that of the previous beam-element models used in part I. In order to estimate displacements in the out-of-plane direction, a characteristic cell for deformation mode IIIb (part I, Fig. 1) was discretised with 8 shell-elements along half of the length in the out-of-plane direction (bias factor 1.25 toward the outer edge). Since honeycomb panels of the described geometry and material (part I, Section 2) were commercially available with a depth of 15:9 mm (see for example Hexcel 3=8-5052-0.005 as used by Papka and Kyriakides [14]) this width was used throughout this investigation. In order to impose the periodicity conditions on the microsection the appropriate constraints were additionally introduced in the out-of-plane direction. The 8nite element calculation with this 3D model showed an increase of the peak stress to a value of 143 kPa in respect to the peak load for the 2D model of 122 kPa (part I, Fig. 4). At larger deLections the force–displacement curves of both models run roughly parallel to each other. It was checked that the 2- and 3-dimensional models were consistent by employing a 3-dimensional model with only one shell element along a width of 0:1 mm and with release of the additional periodicity constraints in the out-of-plane direction; additionally the Poisson’s ratio was set to zero. The force–displacement curve calculated with this model was close to the 2-dimensional results. The deformation state in the cell walls was nearly plane strain. This was demonstrated by employing a 8nite element model for the microsection with one shell-element along the depth of 15:9 mm, 8xed displacement in the 3-direction and 8xed rotation about the 1- and 2-directions for all nodes (constrained 3D model). The force–displacement curve of the full 3-dimensional model and the constrained 3D model were practically identical. One reason for this was that the vertical cell walls hardly deform because they have twice the thickness of the inclined walls. Therefore near the vertex points, the thicker walls enforce a strong constraint on the deformation of
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2 1 3
Fig. 1. Undeformed 8nite element mesh with soft layers at top and bottom of the honeycomb structure (constrained 3D model). Only the constrained degrees of freedom at the layers are shown. The out-of-plane direction is the 3-direction.
the inclined beams. Consequently, the following simulations were conducted with a constrained 3D model. A problem which emerges during the time of crushing is that both the contact between the colliding body and the honeycomb structure and the contacts that develop between di2erent cell walls during densi8cation occur mostly on a small scale in time and space. In order to calculate these contact forces, the 8nite element mesh has to be 8ne enough to resolve the transfer of contact forces. For example, when two cell walls come into contact during crushing at an impact speed larger than about 20 m=s, the material near the contact region deforms elastically or plastically and strong impulses travel through the thickness of the cell wall. All these events determine the details of the load transfer between the contacting walls. Therefore the 8nite element model with shell elements cannot accurately predict the response of the structure; the local contact response is in general too sti2. In order to calculate at least a rough estimate for forces, the 8nite element model shown in Fig. 1 was designed. This model has soft layers at top and bottom of the honeycomb structure with the following dimensions: length 88:1 mm, width 12:0 mm (here the out-of-plane width of the honeycomb structure was set to 10:0 mm) and thickness 2:0 mm. Each layer is discretised with eight C3D8R elements (8-node linear brick, reduced integration, hourglass control [13]); these are connected on one side to a contact surface. The layer material is modelled as elastic perfectly-plastic with a relatively small Young’s modulus, EL = 50:0 MPa, Poisson’s ratio, L = 0:3 and yield stress YL = 1:5 MPa and the mass density was chosen as L = 27 kg=m3 . In order to damp the high frequency oscillations of the honeycomb structure caused by sti2 contact response, sti2ness proportional damping with a Rayleigh damping factor RL = 10−6 s−1 was employed. The boundary conditions for the layers were set as following: 8xed vertical displacements for the nodes at the lower part of the bottom layer; prescribed vertical displacement for the nodes at the upper part of the top layer according the velocity ramping described before; 8xed displacement in the 1- and 3-direction at some nodes on the bottom and top layer prevent rigid body movements (see Fig. 1).
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The thickness and the material parameters of the layers at the top and bottom were selected and roughly optimised according the following criteria: • Thickness of the layers was chosen as small as possible in order to reduce the acceleration forces due to the initial acceleration period; but it was also chosen large enough to allow for some damping. • The mass density was assumed small in order to reduce the initial acceleration forces and to facilitate the excitation of oscillations. However, there is a lower limit for the mass density, since the stable time step size [13] should be determined by the shell element length. • The Young’s modulus was set larger than the homogenised Young’s modulus of the honeycomb structure to prevent combined vibration of the layers and the honeycomb structure; but the value was chosen small enough to absorb and damp the strong impact impulses. The stable time step size is also a factor which determines an upper limit for sti2ness. • The yield stress was chosen in a range to limit some of the highest force peaks. Further, it was found using several test simulations that the discretisation of a layer along the length (1-direction) with eight elements is optimal in the sense that a 8ner discretisation reveals details which complicate the picture described before. For an impact velocity of 20:0 m=s the dissipated energy due to Rayleigh damping in the contact layers is less than 1:0% of the external work. 2.2. Crush front propagation For the investigation in this section, 8nite element models with an initial misalignment of the ◦ ◦ vertical walls in row 5 or 9 (impact site) of 1:0 and 4:0 were used, respectively. The model with the large misalignment in the top row was chosen in order to trigger mode III crush band initiation at impact. No friction was assumed between the soft contact layers and cell walls. The rise time of the prescribed velocity ramp was set, trise = 0:5 ms. The same period was also used for the investigation in part I, Section 4.2 and corresponds to a moderate initial transient response at an impact speed of 10 m=s. Figs. 2 and 3 show the force–time response at the impact surface for speeds, 10:0 and 30:0 m=s, respectively; these results were calculated with the 8nite element model having a misalignment in row 5. When the impact speed is increased from 10:0 to 30:0 m=s, the initial peak load increased from about 150 to 200 kPA. This e2ect was discussed in part I. At an impact speed of 10:0 m=s, the crush band initiated at the imperfect row. This is illustrated in Fig. 4a, where the deformed 8nite element mesh (initial imperfection in row 5) is displayed at time, 2:58 ms. After initiation the crush band propagated from the imperfect row of cells to the distal surface of the specimen (Fig. 4b, 5:16 ms). The larger peaks in the force–time curve of Fig. 2 that occur after 2:5 ms can be partly linked to events of sequential band densi8cation; although the exact correlation is diScult as the structure is crushed and changes in sti2ness. Not shown is the crushing pattern and the force–time response for the structure with an initial imperfection in row 9. At an impact speed of 10:0 m=s crushing initiated here at the impact surface and then at the distal surface. Finally, crushing of bands propagated from the distal to the impact surface. Due to the hard impact of the cell walls with the soft layer when the surface row of cell collapses, the force–time curve has a large peak at this time.
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Nom. Stress (kPa)
200
150
100
50
0 0
0.001
0.002
0.003 0.004 Time (sec)
0.005
0.006
Fig. 2. Temporal variation of nominal stress at impact surface for Al-5052-H39 and impact speed of 10:0 m=s. The 8nite ◦ element model has initial misalignment in row 5 of 1:0 . 600
Nom. Stress (kPa)
500 400 300 200 100 0 0
0.0004
0.0008
0.0012
0.0016
0.002
0.0024
Time (sec) Fig. 3. Temporal variation of nominal stress at impact surface for Al-5052-H39 and impact speed of 30:0 m=s. The 8nite ◦ element model has initial misalignment in row 5 of 1:0 .
The force–time curve in Fig. 3 for the impact speed, 30:0 m=s (imperfection in row 5) shows a strong transient response. The highest peak occurs at time, 0:47 ms before the end of the rise time; this is caused by impact of cell walls with the contact layer on the impact surface. After that, impacts between cell walls and the sudden collapse of cells excite heavy vibrations. This response may be exaggerated because the contact sti2ness between cell walls is too large as was discussed in the last section. A sequence of deformed mesh plots is displayed in Fig. 5a–e. For this large impact speed, crushing started at the impact surface in mode IVab (Fig. 5a–c) and it progressed through the honeycomb with rows collapsing sequentially in mode III or IV (Fig. 5d and e), irrespective of the initial imperfection in row 5. This pattern of crushing is similar to that of a 8nite element model
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v0
(b)
(a)
Fig. 4. Deformed 8nite element meshes at time: (a) 2:58 ms; and (b) 5:16 ms. Impact speed is 10:0 m=s and the model ◦ has initial misalignment in row 5 of 1:0 where crushing initiates.
(b)
(a)
(c)
2 3
(d)
1
(e)
Fig. 5. Sequence of deformed 8nite element meshes at time: (a) 0:46 ms; (b) 0:93 ms; (c) 1:39 ms; (d) 1:85 ms; and (e) ◦ 2:32 ms. Impact speed is 30:0 m=s and the model has initial misalignment in row 5 of 1:0 . Crushing starts at the impact surface in mode IVab.
with a large imperfection in row 9, as shown in Fig. 6a–e. However, the mode of initial crushing is IIIb due to the initial imperfection of the surface layer of cells. Fig. 4a proves that the critical impact speed for localisation at the impact surface is larger than a speed of 10:0 m=s. Despite this, the 2-dimensional beam-element model employed in part I clearly showed wave trapping near the impact surface at that impact speed. One reason for the di2erence is the out-of-plane e2ect (see last section). The yield stress increases by a factor 1:17 (di2erence between 2D and 3D microsection calculation) so that according to Eq. (18) in part I, the estimated critical speed is 8:9 m=s. Further 8nite element simulations indicate that the critical impact speed is
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◦
Fig. 6. Changes of initial crush mode calculated with initial misalignment in row 9 (top) of 4:0 . The sequence of deformed meshes are at time: (a) 0:46 ms; (b) 0:93 ms; (c) 1:39 ms; (d) 1:85 ms; and (e) 2:32 ms. Impact speed is 30:0 m=s.
7000 v= 2.0 m/s v=10.0 m/s v=20.0 m/s v=30.0 m/s v=40.0 m/s
Energy (Nmm)
6000 5000 4000 3000 2000 1000 0
0
10
20
30 40 50 60 Specimen shortening (%)
70
80
Fig. 7. Variation of totally dissipated plastic energy with specimen shortening =L for 8ve di2erent impact speeds. The ◦ results were calculated with an initial misalignment in row 9 (top) of 4:0 .
about 13:0 m=s. At this speed crushing starts at the impact surface and later jumps to the 8fth row where the initial imperfection was located. In order to estimate the e2ect of increasing impact speed on the energy absorption capacity of a speci8c volume of honeycomb, the dissipated plastic energy of the whole model with an out-of-plane thickness of 10:0 mm was calculated as a function of displacement of the impact surface. The result is displayed in Fig. 7. Here, we used the model with an initial imperfection in row 9 and varied the impact speed from 2:0 m=s up to 40:0 m=s. A similar result was found for an imperfection in the middle of the structure. The plot shows an increasing plastic dissipation with increasing impact speed for large specimen shortening (¿ 20%). While the curves for impact speeds, 2:0 and 10:0 m=s
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1705
40 30/m/s
35
ENERGY (Nmm)
30 25 20 15 10 5 0 48
50
52
54
56
58
60
Cell wall indentification number
Fig. 8. Example of distribution of plastic dissipation. Each box represents the plastically dissipated energy of a shell element in a transverse cell wall in row 5. The cell walls are enumerated from left to right and each has 12 shell elements ◦ (144 in a row). Results were calculated with initial misalignment in row 5 of 1:0 and an impact speed of 30:0 m=s.
increase almost linearly with specimen shortening, the curves corresponding to a larger impact speed increase more rapidly. After crushing to a nominal 80% strain (specimen shortening), there is a di2erence of 65% in plastically dissipated energy between impact of 10:0 and 40:0 m=s. Some of the reasons for the higher dissipation at larger impact speeds can be explained by means of Fig. 8. Every box in this plot describes the plastically dissipated energy of a shell-element in the 8fth row of inclined cell walls; i.e. the 8fth row of inclined cell walls up from the distal surface in this 10 row specimen. The shell-elements are counted from left to right and the number of elements in each row is 144 (12 shell-elements per inclined cell wall). Since the element number in Fig. 8 is divided by 12, the x-axis gives the wall number—counted in the same way as the element number. For the simulation the model with an imperfection in row 5 was used and the dissipation was calculated at time 2:32 ms. The highest dissipation in each cell are here left or right of a vertex where in general the plastic hinges develop. The picture shows for example only one hinge at cell wall number 56 (mode III) and two hinges near number 51 (mode IV). Also there is substantial plastic deformation at the mid-span in cell wall 49. In contrast, at smaller impact speeds, the plastic deformation in the inclined cell walls toward the edges of the honeycomb was less because the structure crushed in mode III (approximately) and this allows a simple folding of the crushing bands of cells (Fig. 4b). A further reason for the increase of plastic dissipation with increasing impact speed is the increasing number of cells deforming in a pattern similar to mode IVab. Since that crushing mode has a larger dissipation than mode III (see part I), the global dissipation is elevated. To quantify this, the following fraction F of mode III deformation was de8ned: F:=
max(E pl ; E pl ) − min(E pl ; E pl ) L R L R pairs
max(ELpl ; ERpl )
;
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Frac. F of mode III
0.8 0.6 0.4 0.2 0 0
10
20
30 40 50 60 Specimen shortening (%)
70
80
Fig. 9. Volume fraction of honeycomb collapsing in antisymmetric mode III deformation rather than symmetric mode IV. The solid and dashed line correspond to impact speed (a) 10:0 m=s and (b) 30:0 m=s, respectively. The values were ◦ calculated with a 8nite element model with initial misalignment in row 5 of 1:0 .
where ELpl and ERpl are the plastically dissipated energy in the element to the left and right of a thick ribbon cell wall. The sum in the equation runs over element pairs where at least one element of the pair was deformed plastically. For pure mode III collapse the value of F is therefore 1 while in mode IV, F = 0. The results in Fig. 9 show, that for global strain larger than 30% the amount of collapse in antisymmetric mode III is about 0:25 smaller at the impact speed of 30 m=s than it is at an impact speed v0 = 10 m=s. 3. Comparison with experiment In order to further validate the numerical calculations and apply them to another problem, we compare in this section measurements from drop weight tests on an aluminium honeycomb with results from 8nite element simulations. 3.1. Description of experiment and 3nite element model The dropped weight is a heavy mass, m = 2:5 kg, which is guided by four rails. During impact tests, photos of the deformed honeycomb structure were taken with a high speed camera. Further, two data loggers were used to record the acceleration of the striking mass and the force acting on the distal surface. The data logger had a sample frequency of 25 kHz. Two impact speeds were studied, 3:0 and 7:2 m=s. While 7:2 m=s was the largest velocity achievable with the drop tower apparatus, the velocity of 3:0 m=s was the smallest velocity for reasonable simulations with ABAQUS=Explict. For the dynamic crush experiments, an aluminium honeycomb panel (1.8-3=4-25(3003)) was used ◦ which had L = 11:5 mm; H = 5:75 mm; t = 0:069 mm; = 35:0 and depth of 50:0 mm. The honeycomb specimen had 8 columns and 13 rows; in comparison with our earlier investigation, this large specimen was chosen in order to reduce the e2ect of boundary imperfections that are caused by partly damaged cell walls.
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Similar to the previous calculations, a density of s = 2700 kg=m3 and a Young’s modulus of Es = 68:97 GPa were assumed for the Al 3003 material. The yield behaviour of the aluminium alloy was measured by Breton [15] using small sheets of material taken from the honeycomb panel. The material has a relative sharp transition to yielding with a yield stress of approximately 240:0 MPa. Shortly after yielding the specimens broke due to necking and a shear instability. Therefore no post-yield behaviour could be measured. In the following simulation an ideal plastic material model was employed with a yield stress, Ys = 240:0 MPa. Also no strain-rate dependency was assumed. However, we will show the e2ects of strain hardening and strain-rate hardening are small for a realistic strain-hardening modulus and strain-rate dependency. For these material properties we calculate a critical impact speed for the uniaxial stress case, vcr = 8:64 m=s, using Eq. (18) in part I. This is larger than the largest experimental velocity and therefore no localisation at the impact side should be expected. Preliminary experimental tests with a pneumatic launch device that produced higher impact speeds indicated that the critical impact speed was about 13 m=s. The 8nite element model for the honeycomb specimen was similar to the constrained 3D model used in the last section. Through the thickness, the honeycomb model had only one shell-element and the vertical and horizontal cell walls were discretised with six and 16 shell-elements, respectively. All degrees of freedoms corresponding to the out-of-plane direction were 8xed and also a Poisson’s ratio, = 0, was assumed. The depth of the model was set to 10:0 mm. Since the imperfections of the honeycomb specimen could not be determined exactly, the model has an imperfection in rows ◦ ◦ 7 and 13 with misalignments of = 1:0 ; −1:0 , respectively. Therefore crushing at these positions was equally favoured. The soft bottom and top contact layers had a length of 190:7 mm, depth of 12:0 mm and thickness of 2:0 mm and consisted of 10 brick elements, each. The layer material was chosen as elastic with Youngs’ modulus, EL = 1:6 MPa, Poisson’s ratio, L = 0:3, Rayleigh damping factor, RL = 5:0 × 10−6 s−1 and density, L = 1:35 kg=m3 . The drop weight was modelled as a rigid mass (concentrated mass point). Finally, the vertical displacements of the nodes at the top of the impact layer were constrained to be equal to the vertical displacement of the rigid mass. Both, the mass and the contact layer on the impact surface had the same initial velocity, called impact velocity v, in the following discussion. The other boundary conditions for the honeycomb structure and the contact layers are identical to those described in the previous section. 3.2. Comparison: 3nite element simulations and experiments An essential parameter of the model is the coeScient of Coulomb friction, between the contact layers and the honeycomb. This point was analysed with some preliminary studies which showed that the strong barrelling of the specimen seen during the experiment is absent in the case of very low friction ( ¡ 0:1). Because of the sharp edges at the periphery of the honeycomb specimen, a value of Coulomb friction, = 0:3 was assumed. Figs. 10 and 11 show a comparison of the calculated and measured force–time response at the distal surface for impact speeds 3:0 and 7:2 m=s, respectively. The measured plateau force for both velocities is about 5:0 kPa; i.e. smaller than the value (6:24 kPa) predicted by Eq. (5) in part I and the value obtained from the 8nite element simulations.
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Nom. Stress (kPa)
12 10 8 6 4 2 0 0
0.01
0.02
0.03 Time (sec)
0.04
0.05
Fig. 10. Comparison of nominal stress vs time curves for 1.8-3=4-25(3003) and impact speed 3:0 m=s. The solid line was calculated with the 8nite element model and the dashed line represents the experimental curve. The forces were determined at the distal surface.
14 FEM Exp.
Nom. Stress (kPa)
12 10 8 6 4 2 0 0
0.003
0.006
0.009
0.012
0.015
0.018
Time (sec)
Fig. 11. Comparison of nominal stress vs time curves for 1.8-3=4-25(3003) and impact speed 7:2 m=s. The solid line was calculated with the 8nite element model and the dashed line represents the experimental curve. The forces were determined at the distal surface.
Impact against the top layer on the honeycomb excites a very strong longitudinal wave, which travels in a short time through cell walls from the impact to the distal surface of the honeycomb. This causes a very high peak force. As already discussed in the last section, this peak is too large due to the relatively simple 8nite element model and problems with the contact algorithm. A similar situation occurs when after about 3:0 ms the bending wave reaches the distal surface (Figs. 10 and 11). The response predicted by the 8nite element model is too sti2 and violent oscillations are excited (solid line in Figs. 10 and 11). Except for these points, there is satisfactory agreement between the
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8
Nom. Stress (kPa)
7 6 5 4 3 2
FEM Exp.
1 0 0
5
10
15 20 25 Specimen Shortening [%]
30
35
Fig. 12. Comparison of nominal stress vs specimen shortening =L curves for 1.8-3=4-25(3003). The dashed line represents a static experimental curve. The solid line was calculated with the 8nite element model for impact speed 3:0 m=s with x-axes shifted by 6:0.
8 7
Nom. Stress (kPa)
6 5 4 3 2
FEM Exp.
1 0 0
5
10
15 20 25 Specimen Shortening [%]
30
35
Fig. 13. Comparison of nominal stress vs specimen shortening =L curves for 1.8-3=4-25(3003). The dashed line represents a static experimental curve. The solid line was calculated with the 8nite element model for impact speed 7:2 m=s with x-axes shifted by 17:0.
calculated force–time curves and the experimental measurements; the calculated force–time curves compare satisfactory to the experimental measurements and the simulation yields good estimates for the beginning of the densi8cation period where the force increases sharply. To compare the response more directly, the simulated macroscopic nominal stress–strain curves for the impact speeds 3:0 and 7:2 m=s were plotted (solid line) in Figs. 12 and 13, respectively. Additionally the measured static nominal stress–strain curve taken from Breton’s report [15] was included (dashed line). Since the calculated load at the impact surface was too noisy, we used the load at the distal surface. The proximal end displacement of each calculated curve is then shifted by the amount of macroscopic nominal strain corresponding to the time when the bending wave had
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Fig. 14. Comparison of sequences of photos and deformed 8nite element plots for impact speed 7:2 m=s. The photos were taken at time: (a) 4:39 ms; (c) 11:06 ms; and (e) 15:5 ms after impact. The deformed 8nite element meshes correspond to time: (b) 4:44 ms; (d) 11:11 ms; and (f) 15:56 ms.
reached the distal surface (time after the initial second largest force peak). Comparing the results, we 8nd that the plateau stress has not increased signi8cantly with impact speed. However, the transient response in the beginning is much stronger at the higher impact speed. Since as expected there were no distinctive di2erences between the experimental deformation pattern for impact velocities of 3:0 and 7:2 m=s, the crush band development for the higher speed was analysed in more detail. Fig. 14a, c and e show three photos of a deformed specimen
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Fig. 15. Comparison of photo and deformed 8nite element plot for impact speed 3:0 m=s. The photo and the deformed 8nite element mesh plot correspond to time: (a) 26:2 ms; and (b) 27:0 ms, respectively.
taken at 4:39, 11:06 and 15:5 ms after impact. For comparison with these high speed photographs, Fig. 14b, d and f show calculated honeycomb deformation 8elds at similar times after impact (4:44; 11:11 and 15:56 ms). A comparison of the photo and the calculated deformation of the mesh at time 4:4 ms (Fig. 14a and b) shows a close agreement, with more deformation of cells near the top. This pattern is consistent with the fact that at this time, the bending wave has returned half-way back from the distal to the impact surface. At 11:06 ms the photo shows two developing crush bands which cross in row 4 (Fig. 14c). The honeycomb has the barrelled shape which was also shown by Papka and Kyriakides [16]. These authors argued that barrelling due to friction at the external surfaces is the source of the initiation of the two crossed crush bands. The same features were seen in the simulation of Fig. 14d; although here the crushing started in row 2. Interesting also is the strong deformation and rotation of the left and right cells in the bottom row, while cells in the centre of that row are nearly undeformed. A similar pattern can be found in the corresponding photo from the experiment. The next comparison, for impact speed v = 7:2 m=s, is shown in Fig. 14e and f at a time just before densi8cation. Here the photo and the deformed mesh plot show no crushing at the top and bottom but there is crushing in the middle of structure. For slow speed impact, v = 3:0 m=s, the deformation was similar to the pattern apparent at the higher speed but somewhat more localised in the central rows. This is shown in Fig. 15a and b where the photo at time 26:2 ms and the deformed 8nite element mesh at time 27:0 ms are displayed; at that time the specimen shortening was about the same as in Fig. 14c and d. A distinctive feature apparent in the high speed photo (Fig. 15a) but not occurring in the simulation was the shear band that touched the left upper corner. This band was partly blocked by the collapse of horizontal crush bands and this could be the reason that the experimental force–time curve (Fig. 10, broken line) did not decrease as it had in the simulation (e.g. at time 27 ms). In order to estimate the e2ect of strain hardening and strain-rate hardening, the 8nite element simulation for the impact speed v = 7:2 m=s, was repeated until a time of 11:11 ms, including these e2ects. We found that a post-yield strain-hardening modulus h=E=100, the same as that employed by Papka and Kyrikides [14], made practically no di2erence in the force–time curve or the deformed mesh plot. Strain-rate hardening of the aluminium alloy was modelled by adopting the Cowper– Symonds visco-plastic strain-rate power law [13] with parameters D = 6500 s−1 and p = 4:0. These
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parameters were taken from Ref. [17] and represent the relatively small strain-rate dependency of aluminium alloy. (Holt et al. [18] have shown athermal behaviour for heat treated Al alloys at strain rates smaller then 1000 s−1 .) Using this model the simulation for an impact speed 7:2 m=s gave a small increase of the force. For example at the time 11:11 ms the force was about 11% larger than the calculated force obtained from the 8nite element model with no strain-rate dependency. This is consistent with experimental results of Zhao and Gray [3], who found no strain-rate e2ects during in-plane crushing of an aluminium honeycomb at a macroscopic nominal strain rate of 780 s−1 . Papka and Kyriakides [16] showed that, due to the manufacturing process (expansion) the initiation and plateau stress can be higher than the values predicted with a 8nite element model not including this process. In one case they calculated a di2erence in the initiation and plateau stress of 14% and 8%, respectively. The reason are strain-hardening and slightly rounded cell corners. However, the specimens of the used honeycomb (1.8-3=4-25(3003)) had relatively large defects due to processing and sample cutting and it is probable that these defects have a much stronger e2ect on the response. We found partly damaged cell walls and vertexes at the periphery of the specimen with small parts remaining from cut o2 cell walls. Further the honeycomb panels were quite irregular as received ◦ and a scatter in the wall lengths of up to ±0:3 mm and variation in the angle of up to 10 were estimated. 4. Conclusion Finite element simulations were employed to analyse in-plane dynamic crushing of two di2erent hexagonal honeycombs which had wall slenderness ratios L=t = 38 and 167 for single thickness cell walls. Crush band initiation and wave trapping in the honeycomb structure with the smaller slenderness ratio was investigated in the companion paper [12]. The honeycomb with the larger ratio was also used in drop weight experiments. The honeycomb structures were discretised by shell-elements with contact surfaces on both sides; this is necessary for the analysis to extend into the range of cell densi8cation. Studies showed that in the inclined cell walls the strain in the out-of-plane direction was small. The reason is that the out-of-plane width is several times (3–5) larger than the cell wall length. Furthermore, the vertices introduce constraint on out-of-plane displacements because the vertical ribbon walls deform very little. Therefore the 8nite element model had all displacements in the out-of-plane direction constrained to zero in order to represent this plane-strain condition. Since the local load transfer between colliding cell walls or between cell walls and the boundary surfaces (impact and distal surface) for impact speeds ¿ 10 m=s could not be resolved in detail by shell-elements, a 8nite element model with soft layers at top and bottom of the honeycomb structure was employed. These layers were necessary to partially damp initial transient shocks arising from an over-sti2 contact response of the 8nite element model. The total reaction forces calculated with small damping in the soft layers were in most cases reasonable. The response of the honeycomb with the smaller slenderness ratio was studied for impact speeds up to 40:0 m=s which corresponds to a macroscopic nominal strain rate for the specimen of 500 s−1 . For impact speeds ¿ 10 m=s the impact of adjacent cell walls and the impact of cell walls with the soft boundary layer generated strong vibrations and large amplitude waves in the structure. Therefore maximum forces could not be used to characterise the response at di2erent impact speeds; these
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forces were also strongly sensitive to details of the analytical model. As a less sensitive measure, we used the total energy dissipated by plastic deformation in the honeycomb. For impact speeds larger than the critical speed for wave trapping, vcr , the simulations showed an increase in total plastic dissipation with increasing impact speed. For example, at 80% global strain the dissipated energy resulting from impact at a speed of 40:0 m=s is 65% larger than the dissipated energy from impact at a speed of 10:0 m=s. One reason was a higher percent of cells collapsing in a symmetric crush mode (IV) which dissipates more plastic energy. Another reason for the larger total dissipation at higher crushing speeds is greater irregularity in the mode of crushing (folding pattern). For the simulation of the honeycomb with a larger slenderness ratio, at the top and bottom of the specimen a Coulomb friction coeScient, = 0:3 was assumed. This gave a predicted deformation 8eld with the same strong barrel shaped form as that seen in the experiments. The calculated force– time curves at the distal surface, calculated for impact speeds 3:0 and 7:2 m=s, compared quite well with the corresponding experimental curves. For impact speeds below the speed for wave trapping there was no increase of initial collapse or plateau stress with increasing impact speed. A comparison of sequences of deformed mesh plots and high speed photos showed good correlation of the general distribution and mode of crushing. Acknowledgements The authors are grateful for 8nancial support from EPSRC, Grant reference GR=K54137, for helpful discussions with Professor J.R. Willis and for experimental assistance of S. Wearn. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties, 2nd ed. Cambridge, UK: Cambridge University Press, 1997. [2] Reid SR, Peng C. Dynamic uniaxial crushing of wood. International Journal of Impact Engineering 1997;19:531–70. [3] Zhao H, Gary G. Crushing behaviour of aluminium honeycombs under impact loading. International Journal of Impact Engineering 1998;21:827–36. [4] Harrigan JJ, Reid SR, Peng C. Inertia e2ects in impact energy absorbing materials and structures. International Journal of Impact Engineering 1999;22:955–79. [5] Deshpande VS, Fleck NA. High strain rate compressive behaviour of aluminium alloy foams. International Journal of Impact Engineering 2000;24:277–98. [6] Stronge WJ. Dynamic crushing of elastoplastic cellular solids. In: Jono M, Inoue T, editors. Proceedings of the Sixth International Conference of Mechanical Behaviour of Materials, vol. 1. Oxford: Pergamon Press, 1991. p. 377–87. [7] Klintworth JW. Dynamic crushing of cellular solids. PhD dissertation, University of Cambridge, UK, 1988. [8] Klintworth JW, Stronge WJ. Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs. International Journal of Mechanical Sciences 1988;30:273–92. [9] Reid SR, Reddy TY. Experimental investigation of inertia e2ects in one-dimensional metal ring systems subjected to impact—I: 8xed-ended systems. International Journal of Impact Engineering 1983;1:85–106. [10] Reid SR, Bell WW, Barr RA. Structural plastic shock model for one-dimensional ring systems. International Journal of Impact Engineering 1983;1:175–91. [11] Stronge WJ, Shim VPW. Dynamic crushing of a ductile cellular array. International Journal of Mechanical Sciences 1987;29:381–406.
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[12] H(onig A, Stronge WJ. In-plane dynamic crushing of hexagonal honeycomb I: crush band initiation and wave trapping. International Journal of Mechanical Sciences, PII: S0020-7403(02)00060-7. [13] ABAQUS Standard and Explicit User’s Manuals, Version 5.7. Providence, RI, USA: Hibbit, Karlson and Sorensen, Inc., 1997. [14] Papka SD, Kyriakides S. In-plane compressive response and crushing of honeycomb. Journal of Mechanics and Physics of Solids 1994;42:1499–532. [15] Breton N. Research report. Engineering Department, University of Cambridge, UK, 1997, unpublished. [16] Papka SD, Kyriakides S. Experiments and full-scale numerical simulations of in-plane crushing of a honeycomb. Acta Materialia 1998;46:2765–76. [17] Tam LL, Calladine CR. Inertia and strain-rate e2ects in a simple plate-structure under impact loading. International Journal of Impact Engineering 1991;11:349–77. [18] Holt DL, Babcock SG, Green SJ, Maiden CJ. Transactions of the American Society of Metals 1967;60:152.