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International Journal of Impact Engineering 36 (2009) 73–80
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International Journal of Impact Engineering
journal homepage: www.elsevier .com/locate/ i j impeng
In-plane impact behavior of honeycomb structures randomlyfilled with rigid inclusions
Hiroaki Nakamoto, Tadaharu Adachi*, Wakako Araki 1
Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
a r t i c l e i n f o
Article history:Received 11 December 2007Received in revised form 4 April 2008Accepted 30 April 2008Available online 8 May 2008
Keywords:Honeycomb structureFinite element methodIn-plane impact behaviorInclusions
* Corresponding author. Tel.: þ81 3 5734 2156; faxE-mail address: [email protected] (T. Adac
1 Present address: Department of Mechanical Eng255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 3
0734-743X/$ – see front matter � 2008 Elsevier Ltd.doi:10.1016/j.ijimpeng.2008.04.004
a b s t r a c t
The in-plane impact behavior of honeycomb structures randomly filled with rigid inclusions was studiedby using the finite element method to clarify the effect of inclusions on the deformation process, meanstress, densification strain, and absorbed energy. The deformation processes of the models were dis-turbed by inclusions; shear bands were pinned, and the cell regions surrounded by inclusions wereshielded. Mean stress, densification strain, and absorbed energy per unit volume normalized by thevalues of the model without inclusions were found to be only dependent on the fraction of inclusions. Asthe volume fraction of inclusions increased, the normalized mean stress linearly increased and thenormalized densification strain linearly decreased. The normalized absorbed energy per unit volumecould be approximated by an inverted parabolic equation. The energy absorption of models with in-clusions having volume fractions from 0 to 0.25 was larger than that of the models without inclusions. Inparticular, honeycomb models with fractions of inclusion from 0.1 to 0.2 exhibited the maximum ab-sorbed energy. The model with a volume fraction larger than 0.4 could not be compressed because theinclusions in the model had already percolated before deformation. The in-plane impact behavior ofhoneycomb structures as energy absorbing materials can be designed by using the approximate equationand selecting the volume fraction of inclusions.
� 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Honeycomb structures are widely used in various engineeringfields as packaging and protective materials, core materials ofsandwich panels, building materials, heat-insulating materials, andsound-insulating materials. In particular, honeycomb structurescan absorb significant amounts of impact energy and are used toprotect passengers or cargos in vehicles. Because of this use, a lot ofstudies on the energy absorption characteristics of honeycombstructures have been conducted.
Several researchers have studied in-plane compression of hon-eycomb structures, in relation to the mechanical behavior of foammaterials. The mechanical properties have been analyzed by as-suming that the honeycomb structure is composed of an infiniterepetition of identical unit cells. Gibson and Ashby [1] suggestedmany unit cell models to analyze the mechanical properties.Structures composed of identical cells subjected to in-plane com-pression were also analyzed by using the homogenization theory
: þ81 3 5734 2893.hi).ineering, Saitama University,38-8570, Japan.
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[2,3], Landau theory [4], and limit analysis [5]. Full-scale finite el-ement model is often used to analyze the in-plane compressivebehavior of honeycomb structures composed of many uniform cells[6–10].
Actual honeycomb structures have various structural irregular-ities. The in-plane compressive behavior of honeycomb structureswith locally weakened cells, non-periodic cells, or imperfections incells has been theoretically and numerically analyzed. Silva et al.[11] discussed non-periodic honeycomb structures expressed asVoronoi patterns. Their elastic constants were found to vary byseveral percent compared with the regular one. Silva and Gibson[12] showed the effect of the imperfect cells on Voronoi honey-combs whereby the missing cell walls reduced the strength of thehoneycombs. Albuquerque et al. [13] showed that missing cell wallsdecreased the stiffness and strength of a honeycomb. They alsofound that the collapse stress and Young’s modulus stronglydepended on the volume fraction of the missing cell walls. Chenet al. [14] reported that various structural irregularities, i.e., wavycell walls, non-uniform wall thickness, missing cell walls, andnon-periodic cells, reduced the yield stress of the honeycombstructure. Wang and McDowell [15] discussed the effect of miss-ing cell walls on the in-plane properties, i.e., Young’s modulus,shear modulus, compressive strength, and shear strength, ofmetal honeycombs composed of square, triangular, or hexagonal
Fig. 1. Analytical model: (a) honeycomb structure with inclusions and (b) unit cell.
0.28 0.57 0.33
0.72 0.68 0.53
0.92 0.38 0.51
0.77 0.48 0.29 0.86
0.45 0.66 0.12 0.24
inclusion-filled cell
empty cell
Fig. 2. Procedure of randomly filling inclusions into the cells of the honeycomb model.A random number is given to each cell. In this case, the reference number 40 is 0.5.Black cells are filled with inclusions because the number is less than 40.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–8074
cells. Zheng et al. [16] studied the deformation mode of Voronoihoneycombs and disordered honeycombs undergoing dynamiccrushing. Gan et al. [17] analyzed three-dimensional Voronoimodels under compressive loading. Zhu et al. [18] performeda low-strain analysis of compression of Voronoi honeycombs withperiodic boundary conditions. Li et al. [19] analyzed the dynamiccrushing behavior of honeycomb structures with irregular shapesand non-uniform cell wall thicknesses to ascertain the de-formation processes, plateau stress, densification strain energy,and stress wave propagation. Hwang et al. [20] suggested a lami-nate model to analyze the tensile, compressive, and bendingbehavior of foam materials having non-uniform cell sizes due tothe manufacturing process.
The effect of local strengthening of cells has also been analyzedby several researchers. Prakash et al. [21] studied local strength-ening of honeycomb structures by filling some cells with inclusionsand found that the Young’s modulus increased and densificationstrain decreased with increasing volume fraction of stiff inclusions.Chen et al. [22] indicated that clustered inclusions increasedYoung’s modulus and the bulk modulus of honeycomb structures.These studies indicate that the mechanical properties of honey-comb structures as energy absorbing materials can be designed bylocally strengthening cells.
In this study, the in-plane impact behavior of honeycombstructures randomly filled with inclusions was analyzed by usingthe finite element method (FEM) to clarify the effect of inclusionson the deformation process, mean stress, densification strain, andabsorbed energy. The effect of rigid inclusions on the in-planeimpact behavior, i.e., deformation process, mean stress, densifica-tion strain, and absorbed energy, were clarified by using the finiteelement method. From the results of these analyses, we consideredthe possibility of designing the energy absorption characteristics ofhoneycomb structures by using inclusions.
Table 1Material properties and geometry of honeycomb models
Model A
Cell wall properties Young’s modulus (GPa) 70Poisson’s ratio 0.33Density (kg/m3) 2700Yield stress, sys (MPa) 34
Unit cell Number of cells 20� 21 (Side length, l (mm) 5.00Wall thickness, h (mm) 0.10
Honeycomb Length, L (mm) 173.2Width, W (mm) 160.0
2. Analysis
2.1. Analytical model
2.1.1. Honeycomb structureAnalytical models were two-dimensional honeycomb structures
composed of equilateral hexagonal cells, as shown in Fig. 1a, whereW and L are the width and length of the models. The depth of themodel was the unit length. The length of every side of the unit cellin the model was l. The thickness of cell walls in the Y-direction was2h and the thickness of the other walls was h, because actualhoneycomb structures are manufactured by expanding a stack ofpartially bonded thin plates [10].
Three honeycomb models (A, B and C listed in Table 1) wereanalyzed. The mechanical properties of the cell walls in model Awere as follows: Young’s modulus of 70 GPa, Poisson’s ratio of 0.33,density of 2700 kg/m3, and yield stress, sys, of 34 MPa. The hon-eycomb of model A was composed of 21 rows by 20 columns of unitcells. l was 5 mm, and h was 0.1 mm. Thus, the length and width ofmodel A were 173.2 and 160.0 mm, respectively.
For honeycomb model B, the Young’s modulus and yield stresswere smaller than those of model A; Young’s modulus was 20 GPa,and sys was 10 MPa. Poisson’s ratio and the density were the sameas in model A. Although the number of unit cells in model B was thesame as in model A, the unit cell was larger, namely l¼ 7 mm andh¼ 0.15 mm. Therefore, model B had a side length of 242.5 mm anda width of 224.0 mm.
Model C was composed of 41 rows by 20 columns of unit cellsand had the same properties and unit cell geometry as that ofmodel A. The side length and width of model C were 173.2 and310 mm, respectively.
Model B Model C
20 700.33 0.332700 270010 34
410 cells) 20� 21 (410 cells) 20� 41 (800 cells)7.00 5.000.15 0.10242.5 173.2224.0 310.0
Fig. 3. Analysis conditions.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–80 75
2.1.2. Randomly filling inclusionsIn Prakash’s study [21], inclusions were filled into some cells to
strengthen the honeycomb locally. However, the arrangement ofthe inclusions was not discussed in terms of energy absorption. Inthis research, the inclusions were randomly arranged in eachhoneycomb model with various filling fractions. The ratio of thevolume number of the inclusion-filled cells to the total number ofcells was defined as the fraction of inclusions, 4.
The arrangement of inclusions was determined according tothe following procedure. First, a pseudo-random number rangingfrom 0 to 1, calculated by the linear congruential method [23],was assigned to every cell in a honeycomb model. If the randomnumber given individually to each cell was less than a referencevalue 40, the cell was filled with inclusions. For an example, thenumbers in Fig. 2 are random numbers assigned to each cell. Inthis case, 40 is 0.5. Black cells were filled with inclusions sincetheir numbers were less than 40. Cells with random numberslarger than 40 were kept hollow. 40 was approximately the sameas the fraction of inclusions, 4, in the honeycomb model. To givedifferent arrangements of inclusions to models with the samefraction of inclusions, we selected different initial numbers in thelinear congruential method. Different arrangements of inclusionsfor several values of 4 were produced for models A, B and C: 25for model A, 17 for model B and 9 for model C. In total, 51honeycomb models were analyzed. The inclusions were consid-ered to be rigid bodies having a density of 1 kg/m3 in theanalysis.
Fig. 4. Deformation process in mod
2.2. Analytical condition
To analyze the energy absorption of a honeycomb structurecrushed by a low-velocity impact, every honeycomb model wasfixed on the right side, and a rigid impactor having a mass of5.0 kg collided with the left side of the model with an impactvelocity of 5 m/s, as shown in Fig. 3. The width of the impactorwas much larger than that of the model. The apparent cross-sectional areas, W, of models A, B, and C were 160, 224, and310 mm2, respectively.
This problem was analyzed as a plane-stress problem by usingFEM with explicit time integration (RADIOSS, version. 4.4q). Eachcell wall of every model was discretized as three two-dimensionalEuler beam elements without transverse shearing deformation.Coulomb friction between the left side of the honeycomb modeland the impactor was considered to prevent local crushing at theends of the models. The friction constant was assumed to be 0.5.Contact deformation between the honeycomb model and rigid wallwas analyzed using the penalty method. Compressive stress, s, wasdefined as the reaction force of the impactor divided by the ap-parent cross-section area of the honeycomb model. The strain ofthe honeycomb model, 3, was taken as the displacement of theimpactor after the collision divided by the side length of the hon-eycomb structure.
3. Numerical results
3.1. Honeycomb model A
3.1.1. Deformation processes of model AFig. 4 shows the deformation process of model A without in-
clusions. First, cells collapsed in the oblique directions; namely,shear bands occurred and grew from the corners of the impact sideof the honeycomb at 3¼ 0.14, and more shear bands occurred re-peatedly. When the strain of the honeycomb was 3¼ 0.58, thegrowth of shear bands ceased because the shear deformation in thetriangle region near the fixed side of the honeycomb was restricted.After that, the cells in the regions were compressed. The honey-comb structure was completely densified at 3¼ 0.80.
Fig. 5 shows the deformation process of honeycomb model Awith 4¼ 0.1. The pinning effect and shielding effect were observed[21]; shear bands were pinned by the inclusions, and the regionbetween the inclusions remained almost undeformed. At 3¼ 0.14,the growth of shear bands near the impact side was disturbed bythe pinning effect. As the honeycomb structure became shorter, theinclusions connected from the impact side to the fixed side at
el A without inclusion, 4¼ 0.
Fig. 5. Deformation process in model A with fraction of inclusions, 4¼ 0.1.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–8076
3¼ 0.66; namely, the percolation of inclusions occurred. At thistime, the honeycomb structure was densified, and the cell regionssurrounded by the inclusions could not be collapsed. The de-formation process of model A with 4¼ 0.4 is shown in Fig. 6. Al-though the deformation process of this model was similar to that inFig. 5, the inclusions were percolated even with a slight strain of3¼ 0.08.
3.1.2. Compressive stress–strain curves of model AFig. 7 shows the compressive stress–strain curves of model A for
fractions ranging from 4¼ 0 to 0.4. A stress plateau appeared ineach curve after the elastic range with a sharp peak. The plateau isattributed to shear band growth. Once further growth of the shearbands was prevented, the stress plateau disappeared and thecompressive stress gradually increased because of compression ofthe remaining cells at 3¼ 0.58 for 4¼ 0, 3¼ 0.43 for 4¼ 0.05,3¼ 0.34 for 4¼ 0.10, and 3¼ 0.24 for 4¼ 0.20. The inclusions dis-turbed the shear band growth, so collapsing the honeycomb wasdifficult, and the densification strain decreased and stress increasedsignificantly.
The densification strain, 3D, decreased with increasing 4. Thedensification strain was defined as the strain when the peak stressafter crushing reached 100 kPa for models A and C, and 25 kPa formodel B. As shown in Fig. 7, the densification strain was approxi-mately linear with respect to the fraction of inclusions, 4. As 4increased, the mean stress sM from initiation to the densificationstrain increased: sM¼ 11.9 kPa for 4¼ 0, sM¼ 15.3 kPa for 4¼ 0.05,sM¼ 18.2 kPa for 4¼ 0.1, sM¼ 26.8 kPa for 4¼ 0.2, sM¼ 29.1 kPafor 4¼ 0.3, and sM¼ 55.8 kPa for 4¼ 0.4.
3.2. Honeycomb model B
3.2.1. Deformation processes of model BThe mechanical properties and geometry of the cell wall in
model B were different from those in model A. Young’s modulus
Fig. 6. Deformation process i
was 20 GPa and the yield stress was 10 MPa. The side length of theunit cell in model B was 7 mm, and the cell wall thickness was0.15 mm. The length of model B was 242.5 mm, and the width was224.0 mm. The numbers of unit cells, Poisson’s ratio, and densitywere the same as those of model A. Fig. 8 shows the deformationprocess of model B with 4¼ 0.1. The deformation processes ofmodel B were very similar to those of model A. As can be seen inthe deformation process of model B with 4¼ 0.1, the shear bandswere pinned by the inclusions, and the honeycomb deformeduntil percolation of inclusions occurred. The cells surrounded bythe inclusions did not collapse, and the densification straindecreased.
3.2.2. Compressive stress–strain curves of model BFig. 9 shows the compressive stress–strain curves for honey-
comb model B for different fractions of inclusions. As 4 in-creased, the mean stress, sM, increased. Model B had a lowermean stress than that of model A because of its lower yieldstress of the cell wall. Densification strain was governed bypercolation of inclusions; hence, it was determined geometricallyand not by the material properties. In particular, the densifica-tion strain for the model without inclusions was 0.8, the same asfor model A without inclusions. Thus, dependence of the densi-fication strain on the inclusions was approximately the same asthat of model A.
3.3. Honeycomb model C
3.3.1. Deformation processes of model CThe deformation processes of model C were similar to those of
models A and B. Fig. 10 shows the deformation process of modelC without inclusions. In this model, shear bands occurred fromthe corners of the honeycomb, and they occurred repeatedly.They ceased when the strain was 3¼ 0.58, and the honeycomb
n model A with 4¼ 0.4.
0 0.25 0.50 0.75 1.000
30
60
90
120
150
Strain
Com
pres
sive
str
ess
[kP
a]
ϕ = 0.40
ϕ = 0.30
ϕ = 0.20
ϕ = 0.05
ϕ = 0
Fraction of inclusions ϕ = 0.10
Fig. 7. Compressive stress–strain curves of honeycomb model A with different frac-tions of inclusions.
0 0.25 0.50 0.75 1.000
10
20
30
40
ϕ = 0.30
ϕ = 0.20
ϕ = 0.10
ϕ = 0.05
ϕ = 0
Fraction of inclusions
Com
pres
sive
str
ess
[kP
a]
Strain
Fig. 9. Compressive stress–strain curves of honeycomb model B with different frac-tions of inclusions.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–80 77
structure densified at 3¼ 0.80. The densification strain of thehoneycomb without inclusions was independent of the geo-metrical and material properties.
Figs. 11 and 12 show the deformation processes in model C with4¼ 0.1 and 0.3. The effect of inclusions evident in these figures wasvery similar to those in models A and B, namely, pinning of theshear bands, percolation of inclusions, and shielding of cell regionssurrounded by inclusions.
3.3.2. Compressive stress–strain curves of model CFig. 13 shows compressive stress–strain curves of model C for
various fractions of inclusions. The curves were similar to thoseof models A and B; namely, they have an elastic range witha sharp peak, a plateau region, and a densification region. ModelC had similar compressive stress to that of model A, because ithad the same yield stress and geometry of the unit cell as inmodel A.
The effects of inclusions on the mean stress and the densifica-tion strain were also similar to those of models A and B; as thefraction of inclusions increased, the mean stress increased anddensification strain decreased.
Fig. 8. Deformation process
4. Discussion
4.1. Mean stress
The effect of inclusions on the mean stress, sM, was discussed.The mean stress was normalized by the plastic collapse stress, spl,of honeycombs made of elasto-plastic materials. The plastic col-lapse stress is given by the following equation [24].
spl ¼23
�hl
�2
sys (1)
The relationship between the normalized mean stress, sM/spl, and4 is plotted in Fig. 14. Each point is distributed along one straightline for 4< 0.35. The normalized mean stress is only dependent on4, regardless of the cell material or size. The normalized meanstress increased linearly as 4 increased to 4¼ 0.35. Beyond4¼ 0.35, its values became quite scattered, and it was infinitelydispersed at 4¼ 0.4.
From these results, the relationship between the normalizedmean stress and fraction of inclusions can be expressed with thefollowing approximate equation fitted by the least-squares method.
in model B with 4¼ 0.1.
Fig. 10. Deformation process in model C without inclusion, 4¼ 0.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–8078
sMð4Þs
¼ 8:14þ 1:5 at 0 � 4 � 0:35 (2)
pl4.2. Densification strain
The densification strains for the models without inclusions, 3D0,could be approximated as 0.8 regardless of the material or geo-metrical properties. Thus, the densification strains were normal-ized to 3D0¼ 0.80. Fig. 15 shows the relationship between the
Fig. 11. Deformation process
normalized densification strain, 3D/3D0, and 4 of every analyzedhoneycomb model. The normalized densification strain of eachmodel was also dependent on only 4. It decreased linearly withincreasing 4 and came close to zero at 4¼ 0.4.
The relationship between the normalized densification strainand fraction of inclusions can be approximated as the followinglinear equation by using the least-squares method for every plot.
3Dð4Þ3D0
¼ �2:34þ 1:0 at 0 � 4 � 0:40 (3)
in model C with 4¼ 0.1.
Fig. 12. Deformation process in model C with 4¼ 0.3.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–80 79
Eq. (3) corresponded well with Prakash’s experimental results [21],so this equation seems valid.
4.3. Absorbed energy per unit volume
Absorbed energy, UD, can be evaluated by multiplying Eq. (2)with Eq. (3). The energy per unit volume of honeycomb structureswithout inclusions (4¼ 0) is UD ¼ sMð0Þ3Dð0Þ ¼ 1:5spl3D0.
Fig. 16 depicts the relationship between the normalized absor-bed energy per unit volume, UD/1.5spl3D0, and 4. The models filledwith inclusions had higher absorbed energy than the modelswithout inclusions up to 4¼ 0.25. In particular, for 4 from 0.1 to 0.2,the honeycomb models exhibited 1.3 times larger absorbed energy
0 0.25 0.50 0.75 1.000
30
60
90
120
150
ϕ = 0.30ϕ = 0.20
ϕ = 0.10
ϕ = 0.05
ϕ = 0
Fraction of inclusions
Strain
Com
pres
sive
str
ess
[kP
a]
Fig. 13. Compressive stress–strain curves of honeycomb model C with different frac-tions of inclusions.
than that of the models without inclusions. For 4> 0.25, theabsorbed energies were lower than that of the models withoutinclusions, and UD/1.5spl3D0 decreased with increasing 4,approaching zero at 4¼ 0.4. Thus, the range from 4¼ 0 to 0.25gives the effective fraction of inclusions to increase the absorbedenergy of honeycomb structures.
The relationship between the normalized absorbed energy perunit volume and the fraction of inclusions can be approximatelyexpressed as Eq. (4), and the curve fit is shown in Fig. 16.
UDð4Þ1:5spl3D0
¼ �12:442 þ 3:14þ 1:0 at 0 � 4 � 0:35 (4)
0 0.25 0.50 0.75 1.000
5
15
Fraction of inclusions,
Model A
Model B
Model C
Eq. (2)
10
Nor
mal
ized
mea
n st
ress
, M
/ pl
Fig. 14. Relationship between the normalized mean stress and fraction of inclusions.
0 0.25 0.50 0.75 1.000
0.25
0.50
0.75
1.00
1.25
Fraction of inclusions,
Nor
mal
ized
den
sifi
cati
on s
trai
n, ε
εD
/D
0
Model A
Model B
Model C
Eq. (3)
Fig. 15. Relationship between the densification strain and fraction of inclusions.
H. Nakamoto et al. / International Journal of Impact Engineering 36 (2009) 73–8080
As 4 increases, the normalized mean stress linearly increases,normalized densification strain decreases linearly, and normalizedabsorbed energy per unit volume is parabolic to 4. These variationsare mathematically expressed in Eqs. (2–4). The in-plane impactbehavior of honeycomb structures as energy absorbing materialscan be designed by using these equations and by selecting thefraction of inclusions.
5. Conclusion
The in-plane impact behavior of honeycomb structures ran-domly filled with inclusions was analyzed by using FEM to clarifythe effect of inclusions on the deformation process, mean stress,densification strain, and absorbed energy per unit volume.
The deformation processes of the models were disturbed by theinclusions; shear bands were pinned, and the cell region
0 0.25 0.50 0.75 1.000
0.5
1.0
1.5
Fraction of inclusions,
Model A
Model B
Model C
Eq. (4)
Nor
mal
ized
abs
orbe
d en
ergy
per
unit
vol
ume,
UD
/1.5
pl
D0
Fig. 16. Relationship between the normalized absorbed energy per unit volume andfraction of inclusions.
surrounded by the inclusions was shielded. The mean stress, den-sification strain, and absorbed energy per unit volume normalizedto those of the honeycomb model without inclusions were onlydependent on the fraction of inclusions, 4. As 4 increased, thenormalized mean stress linearly increased and the normalizeddensification strain linearly decreased. The normalized absorbedenergy per unit volume followed an inverted parabolic curve. Theenergy absorption of the honeycombs for the range from 4¼ 0 to0.25 was found to be larger than that of the honeycombs withoutinclusions. In particular, for 4 from 0.1 to 0.2, the honeycombsexhibited 1.3 times more energy absorption than that of the hon-eycombs without inclusions. The honeycomb model with 4> 0.4cannot be compressed because it percolated before deformationwhen 4 was greater than 0.4.
The in-plane impact behavior of honeycomb structures as en-ergy absorbing materials can be designed by using the approximateequations for mean stress, densification strain, and absorbedenergy per unit volume, and by selecting the fraction of inclusions.
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