Polymer Filled Honeycombs to Achieve a Structural

Material with Appreciable Damping

Gabriel Murray∗ , Farhan Gandhi† , and Eric Hayden‡

The Pennsylvania State University, University Park, Pennsylvania, 16802, USA

In this study lossy polymeric infills are introduced into metallic honeycombs with thegoal that the resulting filled honeycombs simultaneously have high stiffness and appreciableloss factor. A finite element analysis is conducted with the honeycomb walls modeled withbeam elements and the polymeric infill modeled with shell elements. Soft (10 MPa) andstiffer (100 MPa) fillers are used, and the filled honeycombs are oriented to deform alongthe x- and the y-directions. The paper presents a discussion on the cell geometries thatyield highest stiffness and loss factor, for loading in the different directions. When thehoneycomb is loaded the Poisson’s contraction of the cell walls in the transverse directionsqueezes and increases the strain energy in the polymer, to levels significantly greater thanthat in an isotropic polymer sheet subject to the same global strain. From the results in thestudy it was observed that filled Aluminum honeycombs can achieve a Young’s Modulus inthe range of 2-7 GPa while simultaneously having a loss factor in the range of 5− 10%.

Nomenclature

α Vertical wall length ratioβ Inclined wall thickness to inclined wall length ratioη Vertical wall thickness to inclined wall thickness ratioηmat Material loss factorηsys System loss factorθ Inclined wall angleEfill Young’s modulus of filler materialh Length of vertical wallsl Length of inclined wallt Thickness of inclined wallsUdiss Strain energy in dissipative elementsUtot Total Strain energy

I. Introduction

In general, metals have high stiffness, but very low damping. Aluminum and its alloys typically have aYoung’s Modulus around 70 GPa, but the Aluminum alloy 2024-T4, in particular, has a loss factor of

only 5 × 10−51 The high modulus makes Aluminum alloys suitable as a structural material, but the lowdamping can be problematic in many applications. On the other hand, elastomers and rubbers can havehigh damping levels, but their stiffness is typically much lower. As an example, Viton-B, a rubber that has ahigh loss factor of 0.878 at low frequencies,2 has a Young’s Modulus of only 29.3 MPa (which would be a verylow value for a structural material). With these highly-damped materials ineligible as structural materials,damping treatments (such as free-layer or constrained-layer treatments) often have to be incorporated whensignificant damping levels are required in constructions made of classical structural materials. Ideally, it∗Graduate Student, Department of Aerospace Engineering, University Park, PA, and AIAA Student Member.†Professor, Department of Aerospace Engineering, University Park, PA, and AIAA Member.‡Undergraduate Student, Department of Aerospace Engineering, University Park, PA, and AIAA Student Member.

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AIAA 2009-2119

Copyright © 2009 by Farhan Gandhi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

would very desirable if some high-modulus structural grade materials could also provide significant levelsof damping. An integral material offering, simultaneously, the high-modulus, high-damping features wouldthen eliminate the need for damping treatments in many structural dynamic and aeroelasctic problems. Thegoal of this paper is to examine whether this (high-modulus, high-damping combination) can be achievedthrough the introduction of highly damped polymeric infills in a honeycomb made of a structural material.

Previous work on filled honeycombs goes as far back as the late 1970’s with studies examining the effectof the infill on the honeycomb’s stiffness.3 An analytical solution to the filled honeycomb problem appearedadequate for soft infills, but showed poor correlation to data when the infill stiffness was increased. Morerecently, there have been studies on foam-filled honeycombs, with some effort focused on examining the filledhoneycomb’s elastic properties for use as a core material in a sandwiched structure,4,5 and others focusingon specific applications such as impact damage reduction,6 and vibration reduction.7 Of particular interestto the current efforts is this paper,8 which focused on obtaining damping from honeycombs with a polymericinfill. The emphasis in that study was on impact damage reduction and damping increase as a result of theinfill’s presence. In the current study, the focus is on trying to identify through design and analysis, filledhoneycombs with a stiffness at least in the few gigapascal range that can simultaneously provide a loss factorat least in the 5 − 10% range.

II. Analysis

In the present study the commercial Finite Element code ANSYS was used to model the honeycomb withthe polymeric infill. The honeycomb walls were modeled using (BEAM3) beam elements, and the infill wasmodeled using (SHELL63) shell elements. Figure 1 shows a unit cell of the honeycomb. The inclined walllength is l, the inclined wall thickness is t = βl, the vertical wall length is h = αl, the vertical wall thicknessis ηt, and the cell angle is θ. In the current simulations, the unit cell inclined wall length, l, is 0.0125 m,and its depth is 0.005 m. The non-dimensional parameters, α (vertical wall to inclined wall length ratio)and β (inclined wall thickness to length ratio), and the cell angle, θ, are varied in the current studies. Thenon-dimensional parameter, η, representing the vertical wall to inclined wall thickness ratio is held at 1 inthe current study.

Figure 1. Nondimensional parameters used to dimension honeycomb

This study examines the Young’s Modulus and the damping capability of the filled honeycomb for loadingin both the x- and the y-directions. The properties of the honeycomb can be adequately predicted byconsidering only the unit cell, provided the proper boundary conditions are imposed. Displacement controlis used to load the filled honeycomb in both the x- and the y-directions. The unit cell boundaries parallelto the loading direction are kinematically constrained so that all the nodes on the boundary move the sameamount perpendicular to the loading direction. An example is shown in Fig. 2 with the unit cell, comprisingof the honeycomb cell plus the infill, loaded in the y-direction. The bottom is constrained in the y-directiononly, and at the top, the displacement is uniform across the width of the unit cell. The boundaries parallelto the y-axis (highlighted in green) are constrained to have the same x-displacement. The four junctionswhere the inclined walls meet the vertical walls (marked with red stars on Fig. 2), are constrained to haveno rotation about the z-axis. The constraints discussed above properly account for the presence of adjacent

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xy

Figure 2. Y-direction filled honeycomb unit cell

cells in a full honeycomb. It should be noted that the honeycomb vertical wall thickness is set at half theactual thickness since it is shared by the adjacent cell, but with the neutral axis along the outer boundary.The Finite Element analysis allows the strain in the honeycomb walls and the polymeric infill to be properlycaptured. The latter is particularly important in the calculation of the filled honeycomb’s damping capability.to it and to be symmetric, that wall must be divided in half.

To assess the damping capability of honeycombs with the polymeric infill, the loss factor of such a filledhoneycomb is considered. The loss factor can be calculated in a couple of different ways. The first approachinvolves subjecting the filled honeycomb to a harmonic excitation and obtaining its stress/strain hysteresisloop. The orientation of, and the area enclosed by, the hysteresis loop yield the Complex Modulus of thefilled polymer, from which the Young’s Modulus, and the system loss factor, ηsys, are extracted.9 For thesimulation results provided in this paper, an excitation frequency of 10 Hz is used. The second approach isbased on the Modal Strain Energy (MSE) method.9 The system loss factor is calculated as:

ηsys = ηmatUdiss

Utot(1)

In the above equation, the strain energy ratio represents the energy in the dissipative elements (the polymericinfill, in the present case) as a fraction of the total energy in the structure over a given cycle, and ηmat isthe material loss factor of the dissipative component.

III. Results

The results in this paper are organized into a number of sub-sections. First, the 2D FEM analysisused in the study is compared against a higher-fidelity 3D analysis, as well as a simpler analytical model,and two methods to calculate the loss factor are compared. This is followed by a series of parametricstudies to determine the combinations of Young’s Modulus and loss factor achievable by using differenthoneycomb designs, orienting the honeycomb to deform along the x- or y-directions, and using polymericfillers of different stiffness. Emphasis is then placed on understanding the strain energy distribution withinthe polymeric infill for loading in x- and y-directions, how the honeycomb changes the strain energy, andhow the infill changes the deformation of the walls. Finally, by comparing some select cases, the possibilitiesin terms of combination of Young’s Modulus and loss factor are summarized.

A. Assessment of Model and Loss Factor Calculation

As stated in Section II, the honeycomb walls are modeled using beam elements while the polymeric filler ismodeled using plane-stress shell elements in the present study. One of the concerns with this approach has todo with potential through-the-thickness deformations of the polymeric infill as the honeycomb is subjected toin-plane strain, and its impact on the calculated stiffness and the loss factor. This stems from the polymericinfill being nearly incompressible (Poisson’s ratio close to 0.5), and the fact that the honeycomb cell areachanges during in-plane deformation. To assess the validity of the FEM model, its results are compared

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1.E+07

1.E+08

1.E+09

1.E+10

0 20 40 60 80

Theta (degrees)

You

ng's

Mod

ulus

of F

illed

Hon

eyco

mb

(Pa)

1E6 Pa-2D1E6 Pa-3D1E7 Pa-2D1E7 Pa-3D1E8 Pa-2D1E8 Pa-3D1E9 Pa-2D1E9 Pa-3D

Infill Stiffness

(a) Young’s Modulus

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70

theta (degrees)

En

erg

y ra

tio

(b) Strain energy ratio

Figure 3. Comparison between 3-dimensional and 2-dimensional finite element analyzes

to those obtained by modeling the system using 3D brick elements. Figure 3 shows a comparison of theYoung’s Modulus and Strain Energy Ratio results, versus cell angle, θ, using the two modeling approaches,for different values of filler modulii. From the figure, it is clear that both the modulus as well as the strainenergy ratio (a measure of the system loss factor), are accurately predicted using the approach of modelingthe honeycomb walls and the polymeric infill using beam and shell elements, respectively.

Next, a comparison of the results for system loss factor using the harmonic analysis method and theModal Strain Energy (MSE) method are presented. For the harmonic analysis method, the honeycomb issubjected to a 10 Hz excitation. Rigorously, the MSE method requires the calculation of the energy in thedissipative elements, as a percentage of the total energy, at the natural frequency, to obtain the modal lossfactor .9 In the present study, the energy ratio is calculated quasi-statically, and the resultant loss factor isreferred to as the system loss factor. Table 1 shows the percentage error in the loss factor calculated basedon the MSE method, relative to the harmonic analysis method, over a wide range of variation in honeycombparameters. The two methods compare well, with an average error well below 5% and maximum error below10%. While the harmonic analysis is used throughout this study in presenting calculated values of systemloss factor, the strain energy ratio offers an understanding of the interplay between the polymeric filler andthe honeycomb walls. Thus, it is valuable to establish the equivalence of the two approaches.

Table 1. Comparison of Loss Factors Calculated using two Methods

α 0.5 1 2 1 1 1β 0.1 0.1 0.1 0.05 0.1 0.2

Direction X X X Y Y Y15o 0.06% 0.05% 0.11% −15o 0.55% 0.05% 0.10%20o 0.06% 0.06% 0.18% −20o 0.71% 0.10% 0.03%25o 0.04% 0.09% 0.35% −25o 0.90% 0.09% 0.05%30o 0.07% 0.16% 0.59% −30o 1.18% 0.17% −0.05%35o 0.10% 0.31% 0.99% −35o 1.55% 0.28% 0.02%40o 0.17% 0.59% 1.59% −40o 2.06% 0.35% 0.12%45o 0.43% 0.97% 2.39% −45o 2.81% 0.49% 0.14%50o 0.76% 1.61% 3.41% −50o 3.95% 0.71% 0.04%55o 1.31% 2.50% 4.60% −55o 5.73% 1.07% 0.07%60o 2.20% 3.66% 5.96% −60o 8.69% 1.70% 0.20%

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A previous study 3 presented analytical expressions for the stiffness of a filled honeycomb based on twokey simplifying assumptions: (1) the elastic bending of the honeycomb walls is not influenced by the filler,and (2) in the calculation of the energy in the polymeric filler the strain in the filler is assumed to beconsistent with the honeycomb walls undergoing rigid-body translations and rotations about pinned joints(the elastic bending of the walls is not considered in the calculation of filler strain). These assumptions ledto satisfactory predictions of the filled honeycomb stiffness for soft fillers, but the results deteriorated as themodulus of the polymeric filler increased. Figure 4 shows Young’s Modulus and Strain Energy Ratio (energyin infill as a fraction of the total energy) results for filled honeycombs, versus filler modulus. Results arepresented for filled honeycombs with different cell angles, and for loading in the x-direction (Figs. 4a and4b) as well as in the y-direction (Figs. 4c and 4d). As reported in,3 it is observed that the analytical modelpredicts the filled honeycomb modulus well for low infill stiffness (10 MPa and less), but the errors increasefor higher infill stiffness values. However, it is observed, that even for these soft fillers, the strain energyratio (Figs. 4b and 4d) is not predicted well by the analytical method. Since the system loss factor is theproduct of the strain energy ratio and the material loss factor of the polymeric infill (which was taken to be1 in the simulations), this amounts to the analytical method being unable to adequately predict the systemloss factor - even for soft fillers.

10

100

1000

10000

0.1 1 10 100 1000

Infill Young's Modulus (MPa)

Fill

ed H

oney

com

b E

last

ic M

odul

us (

Mpa

)

θ=15, analyθ=15, FEMθ=30, analyθ=30, FEMθ=45, analyθ=45, FEMθ=60, analyθ=60, FEM

(a) Young’s Modulus, x-direction

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

0.1 1 10 100

Infill Young's Modulus (MPa)

Str

ain

Ene

rgy

Rat

io (

%)

(b) Strain energy ratio, x-direction

10

100

1000

10000

100000

0.1 1 10 100 1000

Infill Young's Modulus (MPa)

Fill

ed H

oney

com

b E

last

ic M

odul

us (

MP

a)

θ=15, analyθ=15, FEMθ=30, analyθ=30, FEMθ=45, analyθ=45, FEMθ=60, analyθ=60, FEM

(c) Young’s Modulus, y-direction

0

10

20

30

40

50

60

70

80

0.1 1 10 100

Infill Young's Modulus (MPa)

Str

ain

Ene

rgy

Rat

io (

%)

(d) Strain energy ratio, y-direction

Figure 4. Comparison between analytical solution3 (dashed lines) and finite element results (solid lines)

Based on the above discussion it is clear that the analytical model method3 is inadequate for calculationof the loss factor of filled honeycombs. And that a 2D Finite Element Analysis is adequate (3D brick elementsare not required), and that the system loss factor calculated based on harmonic analysis and the modal strainenergy approach show good overall agreement.

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Table 2. Geometric parameter values

Parameter Valuesα [0.5 0.65 0.85 1 1.3 1.8 2]β [0.05 0.065 0.085 0.1 0.13 0.18 0.2]η [0.5 1 2]θ [15 20 25 30 35 40 45 50 55 60]

B. Honeycomb with Efill = 10 MPa

A soft polymeric filler, whose modulus is 10 MPa and material loss factor is ηmat = 1, is examined in thissection. The filled honeycomb is loaded in both the x- and the y-directions. The nominal honeycomb cellparameters used in the simulations are: α = 1, β = 0.1, and wall thickness ratio η = 1. The variation in cellparameters considered in the study are presented in Table 2.

1. Loading in the x-direction

α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(a)

α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

2

3

4

5

6

7

8

9

10

11

12x 10

8

(b)

Figure 5. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in x-direction. Efill = 10MPa

For a filled honeycomb loaded in the x-direction, Figure 5 shows the variation in its loss factor andYoung’s Modulus, with variation in wall length ratio, α, and cell angle, θ. It is observed that the loss factoris highest for large values of θ and α. But the opposite is true for the Young’s Modulus, which is highestfor lower values of θ and α. The relationship between low values of θ and a high Young’s Modulus can beeasily explained. For lower θ, and loading in the x-direction, the inclined walls are more aligned with theload and hence provide higher stiffness. Similarly, a lower α implies that the load-bearing inclined walls aremore closely spaced. The loss factor is seen to vary between 0.1 and 0.4 in Fig. 5a.

Figure 6 shows the variation in the filled honeycomb’s loss factor and Young’s Modulus, with variation inwall thickness to length ratio, β, and cell angle, θ. While increasing wall thickness (higher β) increases theYoung’s Modulus, it reduces the filled honeycomb’s loss factor. Evidently, lower wall thicknesses, especiallyin combination with high values of θ, result in a higher percentage of the system’s strain energy in the lossypolymeric infill.

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β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a)

β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.5

1

1.5

2

2.5

3

3.5x 10

9

(b)

Figure 6. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in x-direction. Efill = 10MPa

For loading in the x-direction, the loads in the honeycomb are carried through the inclined walls andthe vertical walls are not load-bearing. Thus the thickness of the vertical walls (or the parameter η) has noeffect on the Young’s Modulus or the loss factor of the filled honeycomb, and no results are presented here.

2. Loading in the y-direction

For a filled honeycomb loaded in the y-direction, Figure 7 shows the variation in its loss factor and Young’sModulus, with variation in wall length ratio, α, and cell angle, θ. In this case it is observed that both the lossfactor and the Young’s Modulus attain their largest values for high θ and α. The high values of θ result in ahigh Young’s Modulus in the y-direction as the inclined walls are more aligned with the load and there aremore load-bearing honeycomb walls per unit length in the x-direction. Higher α implies a higher percentageof vertical walls over a certain length in the y-direction, and these provide the highest possible stiffness inthe y-direction. The loss-factor is seen to vary between 0.1 and 0.28 in Fig. 7a. Even though the maximumloss factors are lower than those seen in Fig. 5a, for loading in the x-direction, it is possible to simultaneouslyhave a higher modulus.

Figure 8 shows the variation in the filled honeycomb’s loss factor and Young’s Modulus, with variationin wall thickness to length ratio, β, and cell angle, θ. As was the case when the filled honeycomb was loadedalong the x-direction (Fig. 6), thicker walls (higher β) result in a higher Young’s Modulus but a lower lossfactor.

The thickness of the vertical walls (parameter η) again has no effect on the modulus or the loss factor(results not shown). Although the vertical walls do carry load when the filled honeycomb is loaded in they-direction, the y-direction modulus is dependent on the bending of the inclined walls, not stretching inthe vertical walls. Thicker or thinner vertical walls do not influence the bending of the inclined walls andconsequently the strains in the polymeric filler. Hence it is reasonable that no change is observed in the lossfactor either.

3. Comparison of filled honeycombs loaded in the x- and y-directions

Figure 9 shows variation in Young’s Modulus and loss factor, as a function of β, for loading in x- and y-directions (Figs. 9a and 9b, respectively). Results shown are for a value of α = 1, and for two values of θ,

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α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

(a)

α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

9

(b)

Figure 7. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in y-direction. Efill = 10MPa

β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

(a)

β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

9

(b)

Figure 8. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in y-direction. Efill = 10MPa

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0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

β

Loss

Fac

tor

0.05 0.1 0.15 0.20

1000

2000

3000

4000

5000

6000

7000

8000

You

ng’s

Mod

ulus

(M

Pa)

Young’s Modulus, θ=15o

Young’s Modulus, θ=60o

Loss Factor, θ=15o

Loss Factor, θ=60o

(a)

0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

β

Loss

Fac

tor

0.05 0.1 0.15 0.20

1000

2000

3000

4000

5000

6000

7000

8000

You

ng’s

Mod

ulus

(M

Pa)

Young’s Modulus, θ=15o

Young’s Modulus, θ=60o

Loss Factor, θ=15o

Loss Factor, θ=60o

(b)

Figure 9. Filled honeycomb’s Loss Factor and Young’s Modulus for loading in x-direction (a), and y-direction(b). Efill = 10 MPa

15o (which gives high-stiffness in the x-direction) and 60o (which gives high stiffness in the y-direction). βis the parameter varied as it represents a direct trade off between stiffness and loss factor.

If the filled honeycomb is oriented to deform along the x-direction, it can be observed from Fig. 9a thatfor θ = 15o, a 1 GPa modulus can be achieved along with a good loss factor of nearly 0.1. A higher modulusof 2 GPa could be realized with the corresponding loss factor reduced to nearly 0.05. If the filled honeycombis oriented to deform along the y-direction, the performance is significantly improved. From Fig. 9b, it isobserved that for θ = 60o, a 1 GPa modulus yields a loss factor of around 0.32, a 2 GPa modulus yields aloss factor in excess of 0.15, a 3 GPa modulus yields a loss factor of around 0.1, a 4 GPa modulus yields aloss factor of around 0.07, and a 5 GPa modulus corresponds to a loss factor around 0.05.

It is known that the 30o unfilled honeycomb with α = 1 is isotropic (for any β), and introducing apolymeric infill keeps it so - the filled honeycomb modulus and loss factor in the x- and y-directions remainequal. However, for higher or lower values of θ, the filled honeycomb (like the unfilled honeycomb) can behighly orthotropic. For example, a filled honeycomb with a 60o cell angle which has a high modulus in they-direction would have a very low modulus in the x-direction. This has to be taken into consideration if thefilled honeycomb is being used in plate-like structures rather than beam-like structures.

C. Honeycomb with Efill = 100 MPa

In this section, a stiffer polymeric filler, whose modulus is 100 MPa is considered. Polymers with higherYoung’s Modulus typically have a lower material loss factor,1 and the value assumed for this stiffer polymericfiller is ηmat = 0.35 (down from ηmat = 1, for the 10 MPa filler). The stiffer polymeric filler is expected toyield a higher Young’s Modulus for the filled honeycomb - certainly a desirable feature. It is more difficultto predict whether the system loss factor is larger or smaller than that of the honeycomb with the 10 MPafiller. The percentage of strain energy in the stiffer filler could be significantly greater than the softer filler,but this is negated somewhat by the lower material loss factor. As with the honeycomb with the softer filler,the honeycomb with the 100 MPa filler is also loaded in both the x- and the y-directions. The nominalhoneycomb cell parameters, and their range of variations, used in the simulations are the same as those forthe 10 MPa filler.

1. Loading in the x-direction

Figure 10 shows the variation in loss factor and Young’s Modulus, with variation in wall length ratio, α,and cell angle, θ, for the honeycomb with the stiffer filler loaded in the x-direction. As with the softer filler(Fig. 5), the loss factor is highest for large values of θ and α, whereas the Young’s Modulus, is highest forlower values of θ and α. The maximum value of the loss factor is seen to be around 0.3, compared to a valueof 0.4 for the honeycomb with the softer infill. However, the minimum value of the loss factor is higher (0.16,

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compared to 0.1 for the 10 MPa filler). The maximum values of Young’s Modulus with the stiffer infill areabout twice as high as those observed for the honeycomb with the softer infill. Therefore, a stiffer infill maybe an appropriate choice if higher stiffness is important, while still being able to provide an adequate lossfactor.

Table 3. Detailed Values of stiffness for variation in β and θ in Pascals

θ β = 0.05 β = 0.065 β = 0.085 β = 0.160 1.17E+08 1.21E+08 1.29E+08 1.39E+0855 1.15E+08 1.20E+08 1.31E+08 1.43E+0850 1.14E+08 1.21E+08 1.36E+08 1.52E+0845 1.17E+08 1.26E+08 1.45E+08 1.66E+08

Figure 11 shows the variation in loss factor and Young’s Modulus, with variation in wall thickness tolength ratio, β, and cell angle, θ. In the case of the softer infill (Fig. 6), reduction in β resulted in a rapidincrease in loss factor, albeit at a reduced Young’s Modulus. For the stiffer infill, the system loss factorcannot exceed 0.35, so reduction in β does not have as dramatic an effect on increasing the loss factor. Sincethe filled honeycomb loss factor at lower wall thicknesses in combination with high values of θ (upper leftof Fig. 11a) approaches the material loss factor, it can be concluded the majority of the system’s strainenergy is in the polymeric infill (or strain energy ratio approaches unity). In such cases it is also observedin Table 3 that the Young’s Modulus of the filled honeycomb approaches the filler modulus. With the filledhoneycomb stiffness dominated by the polymeric infill stiffness and the strain energy primarily in the infill,further reduction in wall thickness, β, will not affect the system stiffness or loss factor.

α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

(a)

α

θ

0.5 1 1.5 215

20

25

30

35

40

45

50

55

60

0.5

1

1.5

2

2.5x 10

9

(b)

Figure 10. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in x-direction. Efill = 100MPa

2. Loading in the y-direction

Figure 12 shows the variation in loss factor and Young’s Modulus, with variation in wall length ratio, α,and cell angle, θ, for the honeycomb with the stiffer filler loaded in the y-direction. Although the range ofloss factors are similar to those observed for the soft infill (Fig. 7), unlike the soft infill the maximum lossfactors are no longer observed at high θ and α values. Rather, they are observed at low values of θ and α.

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β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.05

0.1

0.15

0.2

0.25

0.3

(a)

β

θ

0.05 0.1 0.15 0.215

20

25

30

35

40

45

50

55

60

0.5

1

1.5

2

2.5

3

3.5

4x 10

9

(b)

Figure 11. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in x-direction. Efill = 100MPa

This implies that the optimum cell geometry for high modulus and high loss factor differ significantly, andtradeoff is required to achieve adequate stiffness and damping. The maximum Young’s Modulus of the filledhoneycomb (at high α and θ) is about twice as large as that achieved with the softer filler (compare Figs. 7and 10).

Figure 13 shows the variation in the filled honeycomb’s loss factor and Young’s Modulus, with variationin wall thickness to length ratio, β, and cell angle, θ. Again, higher β increases the filled honeycomb’sYoung’s Modulus, especially at high cell angles, θ. Reduction in wall thickness increases the loss factor, butnot as dramatically as was seen for the softer infill (Fig. 8). This is because the lower material loss factor ofthe polymeric filler limits the maximum loss factor the filled honeycomb can reach. At low values of b, theloss factor is already approaching the theoretical maximum of 0.35, as was seen when the honeycomb withstiffer infill was loaded in the x-direction (Fig. 11). The majority of the strain energy is in the polymericfiller, and further improvement in damping capability is not possible.

3. Comparison of filled honeycombs loaded in the x- and y-directions

For the stiffer infill, variation in Young’s Modulus and loss factor, as a function of β, for loading in x- andy-directions is shown in Figs. 14a and 14b, respectively. Results shown are for a value of α = 1 and fortwo values of θ, 15o (which gives high-stiffness in the x-direction) and 60o (which gives high stiffness in they-direction).

If the honeycomb with the stiffer infill is oriented to deform along the x-direction, it can be observed fromFig. 14a that for θ = 15o, a 1 GPa modulus can be achieved along with a good loss factor of nearly 0.2 (twicethat achieved with the softer infill, Fig. 9a). Similarly, a higher modulus of 2 GPa could be realized witha corresponding loss factor of about 0.1 (again twice that achieved with the softer infill). Young’s moduliiof 3 GPa, or 4 GPa, are also achievable with higher loss factors than possible with the softer infill (0.05compared to 0.015 for the 3 GPa and 0.04 compared to under 0.01 for 4 GPa).

If honeycomb with the stiffer infill is oriented to deform along the y-direction, it can be observed fromFig. 14 that for θ = 60o, a 3 GPa modulus can be achieved along with a loss factor of nearly 0.15 (about 50%greater than that achieved with the softer infill, Fig. 9b). With the stiff infill and the honeycomb oriented todeform along the y-direction, a Young’s Modulus as high as 6 GPa could be realized with a correspondingloss factor of about 0.1. Amongst Figs. 9 and 14, this appears to be the best combination of filled honeycomb

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α

θ

0.5 1 1.5 215

20

25

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35

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45

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55

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0.12

0.18

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θ

0.5 1 1.5 215

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40

45

50

55

60

0.5

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1.5

2

2.5

3

3.5

4

4.5x 10

9

(b)

Figure 12. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in y-direction. Efill = 100MPa

β

θ

0.05 0.1 0.15 0.215

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25

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0.05 0.1 0.15 0.215

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40

45

50

55

60

1

2

3

4

5

6

7x 10

9

(b)

Figure 13. Filled honeycomb’s loss factor (a), and Young’s Modulus (b), for loading in y-direction. Efill = 100MPa

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0.05 0.1 0.15 0.20

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0.2

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0.7

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Loss

Fac

tor

0.05 0.1 0.15 0.20

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2000

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You

ng’s

Mod

ulus

(M

Pa)

Young’s Modulus, θ=15o

Young’s Modulus, θ=60o

Loss Factor, θ=15o

Loss Factor, θ=60o

(a)

0.05 0.1 0.15 0.20

0.1

0.2

0.3

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Fac

tor

0.05 0.1 0.15 0.20

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You

ng’s

Mod

ulus

(M

Pa)

Young’s Modulus, θ=15o

Young’s Modulus, θ=60o

Loss Factor, θ=15o

Loss Factor, θ=60o

(b)

Figure 14. Filled honeycomb’s Loss Factor and Young’s Modulus for loading in x-direction (a), and y-direction(b). Efill = 100 MPa

modulus and loss factor, suggesting that stiffer infills, with honeycomb oriented to deform along the y-axismay be the best prospect.

D. Strain Energy in the Polymeric Infill

To understand why the filled honeycombs tend to yield a higher loss factor (for a given modulus) whenoriented to deform along the y-direction, the strain energy distribution in the polymeric infill is carefullyexamined. For a 10 MPa infill, the strain energy in the polymeric filler in a single cell is shown for the15o honeycomb deformed in the x-direction (Fig. 15a) and the 60o honeycomb deformed in the y-direction(Fig. 15b). The Young’s Modulus of both honeycombs is the same value, 3 GPa, obtained by adjusting thewall thickness, β. And the honeycombs are strained to different levels in the x- and y-directions such thatstrain energy in the cell walls is the same for both cases. In other words, the strain energy distributionin the polymeric infill in Figs. 15a and 15b correspond to honeycombs having the same Young’s Modulusin the loading direction and strained so that the strain energy in the deformed cell walls is identical. Itis interesting to note that for the honeycomb deformed in the x-direction (Fig. 15a), the strain energy inthe polymeric infill is low in the proximity of the vertical walls and increases towards the central region ofthe cell. However, in over half of the infill, the strain energy appears to be about an order of magnitudelower than the peak strain energy levels in the central regions of the cell. For the honeycomb deformedin the y-direction (Fig. 15b), the strain energy in the polymeric infill is at its highest between the verticalwalls and remains high almost everywhere else except very close to the wall junctions. Besides having amore favorable strain energy distribution in the polymeric infill when the honeycomb is loaded along they-direction, the magnitudes are higher as well (note the differences in scale between Figs. 15a and 15b). Forboth configurations the high strain energy in the infill appears to be due to the mismatch in the Poisson’sratio between the honeycomb and the polymeric infill (νxy for honeycomb in Fig. 15a is around 3, and νyx

for the honeycomb in Fig. 15b is around 6, as opposed to νfiller being around 0.45). This results in theinfill getting squeezed in the transverse direction in the central regions of the cell. Figures 15c and 15d showsimilar results for the 100 MPa filler. While the distributions are qualitatively similar to the 10 MPa filler,the strain energy levels are approximately 10 times larger (compare the scale on Figs. 15a and 15c, and onFigs. 15 and 15d).

E. The Role of the Honeycomb in Amplifying the Strain Energy in the Polymer

In the previous section it was suggested that the high strain energy in the polymeric infill was due to mismatchin the Poisson’s ratios of the honeycomb and the infill. To verify this hypothesis, the filled honeycomb and asection of an isotropic sheet of the infill polymer were both subject to the same global strain. The isotropicsheet thickness is slightly lower than that of the filled honeycomb so that the volume of the polymer is the

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(a) X-direction, 10 MPa infill (b) Y-direction, 10 MPainfill

(c) X-direction, 100 MPa infill (d) Y-direction, 100 MPa infill

Figure 15. Comparison of strain energy for both the x- and y-direction and for the 10 MPa and 100 MPa infill

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same. Figure 16a shows the strain energy distribution when the filled honeycomb and the sheet polymer areloaded in the x-direction (to the same global strain level), and Fig. 16b shows similar results for loading inthe y-direction. The strains, and therefore the strain energy, in the polymer sheet are uniform, and the strainenergy levels are significantly lower than those experienced by the polymeric filler (compare Fig. 16a to 16b,and Fig. 16c to 16d). From these figures it is clear that the energy in the polymeric infill is substantiallyhigher than that in a polymeric sheet undergoing the same global strain. And this is primarily due to thehoneycomb walls squeezing the polymer in the central region of the cell, during deformation.

F. Influence of Infill Stiffness on Honeycomb Wall Deformation

One of the limitations of the analytical model3 was that the infill had no effect on the deformation of thehoneycomb walls. Figure 17 shows the bending deformation of the inclined walls of the honeycomb with the10 MPa and the 100 MPa infills, when loaded in the x-direction (Fig. 17a) and in the y-direction (Fig. 17b).The undeformed inclined walls a-b in Fig. 17a and c-d in Fig. 17b are also marked on Fig. 1, for clarity. Forboth the soft and the stiffer infills, the honeycomb global strain is the same. Clearly, the bending in thehoneycomb walls is greater with the softer infill (red line) than with the stiffer infill (blue line). In effect,the stiffer infill pushes back on the honeycomb walls. Consequently, not only is there more strain energyin a stiffer infill than a softer infill (as seen in Section D), but there is also less energy in the honeycombwalls with the stiffer infill, than with the softer infill. Both these factors combine to increase the infill strainenergy ratio, when the infill is stiffer.

However, it should be noted that if the infill were to become too stiff, approaching the stiffness of theunfilled honeycomb, the ability of the honeycomb walls to squeeze the infill in the transverse direction wouldstart to diminish. Then, the high strain energy in the infill would start to reduce. This feature is, in fact,observed in Fig. 4d for the 60o honeycomb (solid red line) where the strain energy in the infill starts toreduce when the infill stiffness increases beyond a point.

G. Comparison of Select Configurations

Sections B.3 and C.3 considered honeycombs with soft and stiff fillers, oriented to deform along the x- orthe y-directions (Figs. 9 and 14). For the highest possible stiffness while simultaneously realizing loss factorsin the range of 5 − 10%, low cell angle is preferable for a honeycomb loaded in the x-direction (θ = 15owasselected), while a higher cell angle is preferable for a honeycomb loaded in the y-direction (θ = 60o wasselected). However, the cell wall lengths were kept the same (α = 1). It is known, however, that forhoneycombs loaded in the x-direction, lower values of α result in a higher Young’s Modulus. On the otherhand, for honeycombs loaded in the y-direction, higher values of α increase the Young’s Modulus. In thissection, honeycombs loaded along the x- and y-directions, with both soft (10 MPa) and stiff (100 MPa) infillare considered, and using the ”best” values of the cell parameters (α, β, θ) from the range considered, theYoung’s Modulus of the filled honeycombs are compared corresponding to loss factors of 5% and 10%. Theseresults are presented in Table 4. The observations drawn from Table 4 are in many ways similar or equivalentto those in Sections B.3 and C.3. For either infill stiffness, corresponding to a given honeycomb loss factor,a higher modulus is achieved if the honeycomb is oriented to deform along its y-axis. The honeycomb’sYoung’s Modulus always decreases as its loss factor increases. It is noted that by introducing a polymericinfill into a honeycomb a loss factor of 5% (large compared to most standard structural materials) can berealized while simultaneously achieving a Young’s Modulus of up to nearly 7 GPa. While this value ofYoung’s Modulus is about one tenth that of Aluminum (the material from which the honeycomb is made forthe current simulations), it may be sufficient for certain applications, or the filled honeycomb might be usedas a layer in a multi-layered structure.

IV. Conclusions

The goal of this study was to examine whether through proper design metallic honeycombs filled withlossy polymers could simultaneously achieve high structural stiffness and significant damping (measuredby the loss factor). The realization of such a system yields an integrally damped structural material andpotentially eliminates the need for auxiliary damping treatments. A 2D Finite Element Analysis was usedwith honeycomb walls modeled with beam elements and the polymeric infill modeled with shell elements. The

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(a) X-direction, 10 MPa infill (b) X-direction, 10 MPa sheet

(c) Y-direction, 10 MPa infill (d) Y-direction,10 MPa sheet

Figure 16. Comparison of strain energy for both the x- and y-direction for filled honeycomb and the polymersheet

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(a) X-direction (b) Y-direction

Figure 17. Comparison of inclined wall deformation with 10 MPa (red) infill and 100 MPa (blue) infill

Table 4. Young’s Modulus and loss factor for select configurations

Direction Efill (MPa) ηsys Young’s Modulus α β θ

x 10 5% 3.312 GPa 0.5 0.148 15o

x 10 10% 1.966 GPa 0.5 0.1179 15o

y 10 5% 6.868 GPa 2 0.1956 60o

y 10 10% 4.505 GPa 2 0.1569 60o

x 100 5% 6.889 GPa 0.5 0.1917 15o

x 100 10% 4.178 GPa 0.5 0.1392 15o

y 100 10% 5.468 GPa 2 0.1203 60o

results from this analysis showed good correlation with those obtained using a 3D Finite Element Analysis.From the results presented in this paper, the following conclusions can be drawn.

1. Simple analytical models that adequately predicted the Young’s Modulus of a filled polymer when thepolymeric infill was soft, did a poorer job of predicting the loss factor.

2. When a soft 10 MPa lossy infill was introduced in the Aluminum honeycomb, it was observed thatorienting the honeycomb to deform along the y-direction was preferable. When the honeycomb wasoriented to deform along the x-direction, high Young’s Modulus was obtained at low cell angle, θ, andwall length ratio, α, but high loss factor required the opposite. On the other hand, when the honeycombis oriented to deform along the y-direction, the Young’s Modulus and loss factor are simultaneouslyimproved for high θ and α. In both directions, higher wall thickness, β, increased the modulus butdecreased the loss factor.

3. When a stiffer 100 MPa infill (with a lower material loss factor of 0.35) was introduced into theAluminum honeycomb, observations for loading in the x-direction were similar to those correspondingto the softer infill, except that higher values of Young’s Modulus were achieved for comparable levelsof loss factor. When the honeycomb is oriented to deform in the y-direction, high Young’s Modulusis still observed for large cell angle, θ, and wall length ratio, α, but high loss factor now requires theopposite.

4. An overall assessment reveals that a filled honeycomb design with a Young’s Modulus close to 7 GPa(about one tenth that of the virgin material, but still stiff enough to be of interest as a structuralmaterial), and a loss factor of nearly 5% (orders of magnitude higher than that of the virgin material)is indeed realizable. If a larger loss factor of 10% is required, it is still possible to simultaneouslyachieve a Young’s Modulus of 2 GPa and greater.

5. The high loss factor of the filled honeycomb can be attributed to the significant strain energy inthe polymeric infill due to the Poisson’s mismatch between the honeycomb and the infill. When amonolithic polymer sheet of the same material is strained to the same degree, the strain energy levels

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are much lower. When the filled honeycomb is extended in the x-direction the inclined cell walls squeezethe polymer in the y-direction, and when extended in the y-direction, the vertical cell walls squeezethe polymer in the x-direction. For loading in the x-direction, the strain energy within the polymericinfill in the cell is maximum in the central region and decreases outward towards the vertical walls. Onthe other hand, for loading in the y-direction, the infill strain energy remains close to the maximumover the entire cell.

6. The softer 10 MPa infill provides less resistance to the deforming cell walls, but the stiffer 100 MPainfill pushes back on the cell walls which show lower deformations.

Future work will focus on experimental validation, examining a wider range of cell geometries, and explor-ing designs to maximize the filled honeycomb’s loss factor (even at the expense of a lower Young’s Modulus)through maximization of strain energy amplification in the polymeric infill due to Poisson’s mismatch.

Acknowledgments

The authors would like to acknowledge and thank the National Defense Science and Engineering Graduate(NDSEG) Fellowship, who has funded and made possible the lead author’s graduate studies.

References

1Lazan, B. J., Damping of Materials and Members in Structural Mechanics, Pergamon Press, 1968.2Jones, D. I. G., Handbook of Viscoelastic Vibration Damping, John Wiley & Sons, 2001.3El-Sayed, F. K. A., Jones, R., and Burgess, I. W., “A Theoretical Approach to the Deformation of Honeycomb Based

Composite Materials,” Composities, Vol. 10, No. 4, 1979, pp. 209–214.4Beyle, A., Akpan, N., and Ibeh, C., “Mechanics of foam-filled honeycombs,” Society of Plastics Engineers Annual Technical

Conference 2006, ANTEC 2006 - Conference Proceedings, Vol. 2, 2006, pp. 690–694.5Kuwabara, A., Ozasa, M., Shimokawa, T., Watanabe, N., and Nomoto, K., “Basic mechanical properties of balloon-type

TEEK-L polyimide-foam and TEEK-L filled aramid-honeycomb core materials for sandwich structures,” Advanced CompositeMaterials: The Official Journal of the Japan Society of Composite Material , Vol. 14, No. 4, 2005, pp. 343–363.

6Vaidya, U., Ulven, C., Pillay, S., and Ricks, H., “Impact damage of partially foam-filled co-injected honeycomb coresandwich composites,” Journal of Composite Materials, Vol. 37, No. 7, 2003, pp. 611–626.

7.Takemiya, H., “Field vibration mitigation by honeycomb WIB for pile foundations of a high-speed train viaduct,” SoilDynamics and Earthquake Engineering, Vol. 24, No. 1, January 2004, pp. 69–87.

8Hao, H., Joe, C.-R., and Kim, D.-U., “Mechanical Behavior of Rubber Filled Multifunctional Honeycomb Sandwich Com-posite,” Sandwich Structures 7: Advancing with Sandwich Structures and Materials, edited by O. T. Thomsen, E. Bozhevolnaya,and A. Lyckegaard, Springer, 2005, pp. 671–680.

9Mead, D., Passive Vibration Control , Wiley, 1998.

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