Research Article Equivalent Elastic Modulus of Asymmetrical Honeycomb

International Scholarl Research Network ISRN Mechanical Engineering Volume, Article ID 57, pages doi:.5//57 Research Article Equivalent Elastic Modulus of Asmmetrical Honecomb Dai-Heng Chen and Kenichi Masuda Department of Mechanical Engineering, Toko Universit of Science, Kagurazaka, Shinjuku-ku, Toko 6-86, Japan Correspondence should be addressed to Kenichi Masuda, masudak@rs.kagu.tus.ac.jp Received 9 March ; Accepted 9 April Academic Editors: J. Botsis, A. Tounsi, and X. Yang Copright D.-H. Chen and K. Masuda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. The equivalent elastic moduli of asmmetrical heagonal honecomb are studied b using a theoretical approach. The deformation of honecomb consists of two tpes of deformations. The first is deformation inside the unit, which is caused b bending, stretching, and shearing of cell walls and rigid rotation of the unit; the second is relative displacement between units. The equivalent elastic modulus related to a direction parallel to one cell wall of the honecomb is determined from the relative deformation between units. In addition, a method for calculating other elastic moduli b coordinate transformation is described, and the elastic moduli for various shapes of heagon, which are obtained b sstematicall altering the regular heagon, are investigated. It is found that the maimum compliance C ma and the minimum compliance C min of elastic modulus C in one rotation of the (, ) coordinate sstem var as the shape of the heagon is changed. However, C ma takes a minimum and C min takes a maimum when the honecomb cell is a regular heagon, for which the equivalent elastic moduli are unrelated to the selected coordinate sstem, and are constant with C = C.. Introduction To date, honecomb materials consisting of regular heagonal cells or smmetrical heagonal cells [ ] have been the subject of etensive research. In the present stud, a general method is proposed for finding the equivalent elastic moduli for the two-dimensional (D) problem of honecomb consisting of an arra of heagonal cells, including asmmetrical heagonal cells. Moreover, the equivalent elastic moduli for several heagonal geometries are found using the proposed method, and a sstematic investigation is carried out into the effects of changes in geometr of heagonal cells on the equivalent elastic moduli of honecomb. Research into the equivalent elastic moduli of asmmetrical honecombs has been carried out b Overaker et al. [5], who proposed a method for finding the equivalent elastic moduli of an asmmetrical honecomb b fitting an equivalent strain field to satisf the displacements of each end of cell wall in one unit. In the present analsis, the method of Overaker et al. [5] is used although we simultaneousl attempt to find the strain field via a different approach. Specificall, b treating the deformation of the honecomb as the sum of the deformation of each cell wall in one unit and the relative displacement between each unit, the equivalent elastic modulus of the honecomb can be found from the relative displacement between units. The relative displacement between units is determined b the condition of the junctions between the cell wall ends of each adjacent unit after the deformation. As shown in Figure, the analzed model in the present stud is honecomb consisting of heagonal cells, and the honecomb core height is denoted b h. In order to form a honecomb b periodicall arraing heagonal cells, two opposing edges of the heagon must have the same length and be parallel. Here, the length and thickness are l, l,and l,andt, t,andt, respectivel, as shown in the figure; the internal angles formed b the heagon edges are γ, γ,and γ γ + γ + γ = π. () The cell wall material is homogeneous and isotropic with an elastic modulus of E s and Poisson s ratio of υ s.theaim of this analsis is to find the equivalent elastic modulus for thehonecombplaneproblem.specificall,wewanttofind the equivalent elastic modulus C ij (i, j =,, ), which is

ISRN Mechanical Engineering Cell wall l Cell wall γ γ l l Cell wall Y θ t Cell wall γ t Y h Z t Cell wall X X Cell wall (a) (b) Figure : Geometr of honecomb: (a) honecomb plate and (b) heagonal cell. applicable to the relation of stress and strain in the - and -coordinate plane shown in Figure ε C C C σ ε = C C C σ. () ε C C C τ σ Cell wall l l Cell wall θ θ Cell wall l The shear strain ε used here is defined as a tensor, the engineering definition of which is γ with ε = γ /. Cell wall. Analsis of Elastic Moduli C, C, and C for the (, ) Coordinates with -Ais Parallel to Edge l Cell wall Cell wall σ Initiall, the -aisisparalleltocellwallinthecaseofθ = in Figure, and onl the -direction stress σ is considered to act. As shown in Figure,theanglesθ and θ are taken as those between edges l and l,andedgesl and l, respectivel θ = π γ, θ = π γ. () Figure : Geometr of heagon with an edge parallel to -ais (θ = ). from which, the forces T, T,andT are given b the following equations: T = σ h(l sin θ + l sin θ ),.. Force, Moment, and Displacement Acting on Each Cell Wall. Due to the stress σ acting in the -direction, the force T i (i = ) and moment M i (i = ) act on each cell wall,asshowninfigure. From the equilibrium of forces, we can obtain T + T = σ h(l sin θ +l sin θ ), T = σ h(l sin θ + l sin θ ), T = T + T, () T = σ hl sin θ, T = σ hl sin θ. The moments due to T i are given as follows: Here, M = M. M =, M = T l sin θ = σ hl sin θ l sin θ, M = T l sin θ = σ hl sin θ l sin θ. (5) (6)

ISRN Mechanical Engineering T δ δ M T δ δ T = T + T θ l M O M O M l θ l δ δ Figure : Mechanics of cell walls subjected to stress σ in the - direction. The -direction displacement δ i and -direction displacement δ i are found for each cell wall of the unit shown in Figure. The bold lines in Figure denote the cell walls, the thin lines denote the boundar of one unit; b arrangement of these units the honecomb is formed. The displacements of the cell walls are caused b bending deformation, shear deformation and tensile deformation of the cell walls, generated b each force and moment. Taking the junction of the three walls as the origin, the displacements in the -direction of the ends of cell walls,, and are given b the following equations: δ =, δ = T sin θ cos θ E s h ( l t l T cos θ sin θ E s ht, δ = T sin θ cos θ E s h ( l ) +(+υ s ) T l sin θ cos θ t + l T cos θ sin θ E s ht. ke s ht ) (+υ s ) T l sin θ cos θ ke s ht (7) Similarl, the displacements in the -direction of each cell wall are given as follows: δ = l T E s ht, δ = T sin θ E s h δ = T sin θ E s h ( l t ( l t ) +(+υ s ) T l sin θ ke s ht + l T co s θ E s ht, ) +(+υ s ) T l sin θ + l T co s θ. ke s ht E s ht (8) Figure : Displacements of ends of cell walls. Here, the sign of the displacement follows the coordinates showninfigure, andk is a correction coefficient related to shear deformation, which is taken as k = inthiswork (previous research has shown that results for k = agree well with those of numerical analsis b the finite element method [])... Analsis of Equivalent Elastic Moduli. Overaker et al. [5] proposed an elegant method for fiing the equivalent strain field, which satisfies the displacements of each cell wall found in the previous section. Figure shows the displacements of each wall end in a unit δ i and δ i, as well as the coordinates of each wall end ( i, i )(i =,, ). The displacements in the - and-directions of each wall end δ i and δ i,whichare found in (7) and(8),canbeseenasthoseduetotherigid bod displacements u and v, the rigid bod rotation ω, and the uniform strain field in the unit ε, ε,andε and are then described b the following equations: δ i = i ε + i ε + i ω + u, δ i = i ε + i ε i ω + v. (i =,, ) (9) Si unknowns, namel, ε, ε, ε, ω, u, and v, are determined b solving these equations. B using these strain fields obtained from (9), the equivalent elastic moduli C, C,andC can be found from the following equation. C = ε σ, C = ε σ, C = ε σ. () The strain field produced in the honecomb can also be determined from the relative displacements between units. In fact, the deformation of the whole honecomb is performed

ISRN Mechanical Engineering b the relative displacements between each unit. Here, we consider a part of honecomb consisting of three units, as showninfigure5, in which the three units are denoted as units,, and, counterclockwise from the lower left, and thecellwalljointsofeachunitareo, O,andO.Denote the relative displacements of O and O with respect to O b U and V,andU and V, respectivel, as shown in Figure 5(b). Thus, the following equations can be obtained from the relation between the strain field produced in the honecomb and the displacements of O and O with respect to O : U L L L ε V L L L ε =, () U L L L ε V L L L ω where L and L are the distances in the -and-directions between O and O,whileL and L are the distances between O and O L = l sin θ + l sin θ, L = l cos θ l cos θ, L = l sin θ, L = l + l cos θ. () From (), the strain field can be obtained as a function of the relative displacements between units as follows: ε = L U + L U ( L L + L L ), ε = L V + L V ( ), L L + L L ε = L U + L V + L U + L V ( L L + L L ), ω = L U L V + L U L V ( L L + L L ). () The relative displacements between units are determined b the condition of the junctions between the wall ends of each adjacent unit. In order to consider the connection between each cell wall after deformation, Figure 5(b) also shows the displacements of points A, B, andc. PointA of unit is on cell wall, and the displacement of points A with respect to point O in the respective - and-direction, U A and V A,areequaltothecellwalldeformationitself U A = δ, V A = δ. () Point B ofunitisoncellwallandpointc of unit is on cell wall. Therefore, the displacements of points B and C with respect to point O in the respective -and-direction, U i and V i (i = B, C), are calculated b adding the relative displacements between the units to the displacements due to the cell wall deformation itself U B = δ + U, V B = δ + V, U C = δ + U, V C = δ + V. (5) Since points A, B, and C are the same point prior to deformation, as shown in Figure 5(a), the displacement of points A, B, andc after deformation must be the same and the condition that U A = U B = U C and V A = V B = V C holds true. From this condition, the relative displacements U and V,andU and V of O and O with respect to O are given as follows: U = δ δ, V = δ δ, U = δ δ, V = δ δ. (6) B substituting () and(6) into(), the strain field can be obtained, and then, each equivalent elastic modulus C, C,andC can be determined from ()... Calculation of Elastic Modulus Matri. In the previous section, we presented a method for finding the three equivalent elastic moduli for the directions parallel to a cell wall constituting the heagon cell; however, these are onl three of the nine components of the elastic modulus described in (). To epress the elastic characteristics of a heagonal honecomb, it is necessar to know all nine components. In this section, using the three equivalent elastic moduli relating to the directions parallel to a cell wall, the nine components of the honecomb equivalent elastic modulus C, C C are derived. Since the approach described above allows the three equivalent elastic moduli for the direction parallel to an cell wall to be found, the three elastic moduli can be found for each direction of cell walls,, and, respectivel. Specificall, as shown in Figure 6, we take the (α, β ) coordinates based on cell wall, the (α, β ) coordinates based on cell wall and the (α, β ) coordinates based on cell wall, in which the β-ais is set to be parallel to the cell wall. Thus, C, C, and C in the (α, β ) coordinates, C, C, andc in the (α, β ) coordinates and C, C,andC in the (α, β ) coordinates can be found for each coordinate sstem (the prime superscripts of the coordinate sstem correspond with those of the elastic moduli). However, the nine components of the elastic modulus C, C C to be found are attached to the (, ) coordinates of Figure 6. Thus, we transform coordinates from the (, ) coordinate sstem to the (α, β) coordinate sstem. Here, we suppose an angle θ between the (, )

ISRN Mechanical Engineering 5 O V L L U O δ δ C δ O L L O δ A O O B δ δ V U (a) (b) Figure 5: Condition of junctions between wall ends: (a) a part of honecomb consisting of three units and (b) displacement of cell wall ends A, B,andC. coordinate sstem and the (α, β) coordinate sstem. In the (α, β ) coordinate sstem, θ is θ = θ,forthe(α, β ) coordinate sstem, it is θ = θ +(π γ ), and for the (α, β ) coordinate sstem, it is θ = θ +(π γ )+(π γ ). For eample, b transforming the stress and strain in the (, ) coordinate sstem to the stress and strain in the (α, β ) coordinate sstem, the following equation can be obtained from (): ε α C C C σ α ε β = [T] C C C [T] σ β. (7) ε α β C C C τ α β The coordinate transformation matri [T]isgivenbelow: [T] = cos θ sin θ sinθ cos θ sin θ cos θ sinθ cos θ. sin θ cos θ sin θ cos θ cos θ sin θ (8) However, the stress-strain equations in the (α, β ) coordinate sstem are epressed b the following equation: ε C C C α σ α β β = C C C σ β. (9) ε α β C C C τ α β Since both (7)and(9) are the same, the following equation is obtained: C C C C C C C C C = [T] C C C [T]. () C C C C C C As stated above, C, C,andC are known and from (), the can be epressed as functions of the components C C,whicharetobefound C = C cos θ ( C +C )cos θ sin θ + (C + C C )cos θ sin θ + (C C ) cos θ sin θ + C sin θ, C = C cos θ + (C +C )cos θ sin θ + (C + C +C )cos θ sin θ + (C +C ) cos θ sin θ + C sin θ, C = C cos θ + (C C + C )cos θ sin θ + ( C + C C + C )cos θ sin θ + (C C C ) cos θ sin θ C sin θ. () Similarl, b transforming the (, ) coordinate sstem to the (α, β )and(α, β ) coordinate sstems, C, C,andC and C, C,andC can be epressed as functions of C C. Therefore, b solving these nine simultaneous equations, the nine components, C C,canbedetermined B using this method, the honecomb equivalent elastic components are found. For eample, for a cell thickness of

6 ISRN Mechanical Engineering Cell wall Cell wall β θ α Cell wall θ θ = θ Figure 6: (α, β) coordinates based on each cell wall. t = t = t =.5l, Poisson s ratio of υ s =., the honecomb equivalent elastic moduli for a heagon with parameters of l /l = l /l =, γ =, γ =,and γ = can be determined.76.58.55 [C] =.58.6.6. () E s.75.66 8.568 Moreover, for a heagon with parameters of l /l =.5, l /l =.6, γ = 5.7, γ = 5,andγ = 9., the following honecomb equivalent elastic moduli are calculated:..5.8 [C] =.5.75.79. () E s..85 5.7 It can be seen from these results that the smmetr of the elastic moduli holds C = C, C = C, C = C.. Effects of Geometr on Elastic Moduli β α β α () In order to investigate whether the geometr of heagonal cell affects each of the equivalent elastic moduli, the equivalent elastic moduli are found for various heagons that deviates from the regular heagon, which is taken as a basic geometr here. For the following investigation, in order to observe the effects due to changes in the cell geometr, each cell wall thickness of the basic regular heagon is taken to be the same, t = t = t =.866l. Here, l is the length of one edge of the regular heagon. Figure 7(a) shows heagon A BCD EF (geometr ), which is formed from the regular heagon ABCDEF b fiing edges BC and EF and moving onl points A and D in the - direction b Δ and Δ, respectivel. Each equivalent elastic modulus corresponding to the heagonal cell of geometr shown in Figure 7(a) is shown in Figure 8. Here, with the elastic modulus of a regular heagonal cell C regular taken as the standard, the elastic modulus C ij along the vertical ais is compared with C regular. In the figure, the following is observed. () The elastic modulus C, which epresses the magnitude of the -direction strain due to the stress in the -direction, is a maimum for Δ =, that is, the regular heagon, and C decreases with increasing Δ. When Δ/( l/), C is not but converges to C /C regular =.88, because heagon A BCD EF becomes parallelogram A BD E when Δ/( l/) =, as shown in Figure 7(b). For the parallelogram A BD E, the elastic modulus C is C = l/(te s ). () The elastic modulus C, which epresses the magnitude of the -direction strain due to the stress in the -direction, appears not to be strongl influenced b the change in geometr due to the movement of points A and D in the -direction; for each Δ, C remains nearl constant. () The elastic modulus C, which epresses the magnitude of the shear strain due to the stress in the - direction, for the case of Δ =, that is, for the regular heagon, is zero due to smmetr. As Δ increases and the geometr deviates from that of a regular heagon, C increases; however, in the vicinit of about Δ/( l/) =.6, C decreases, because shear deformation due to the stress σ decreases, as the geometr approaches that of a parallelogram. In order to investigate the resultant deformation due to C and C, we consider the displacement of the upper end U and U of a honecomb plate in geometr under a tensile stress σ,asshowninfigure9(a). For a plate length of L under σ, the displacement is given as follows: U = σ C L, U = σ C L. (5) Figure 9(b) shows the ratio of the compliance U/(σ L)of the plate to the compliance C regular of the regular heagon. Here, U = U + U is the displacement of the upper end of the plate. In Figure 9(b), it can be seen that the comprehensive compliance due to C and C increases as Δ increasesandreachesamaimuminthevicinitof Δ/( l/) =.5. As Δ further increases, when geometr deviates from the regular heagon greatl, the compliance conversel becomes smaller. () The elastic modulus C, which epresses the magnitude of the -direction strain due to stress in the -direction, is alwas C <. For Δ =, C is a maimum and becomes zero when Δ/( l/) =.

ISRN Mechanical Engineering 7 A A ( /)l A l/ F l B F B l l (/)l E C E C D Δ D D (a) (b) Figure 7: Geometr of heagonal cell: (a) horizontal movement of point A and D and (b) parallelogram (Δ = l/). Cij/C regular 5 C C C C C C.5 Δ/( /)l Figure 8: Equivalent elastic moduli for geometr shown in Figure 7. In addition, the ratio of C and C is Poisson s ratio υ, υ = C /C.Figure shows the change in Poisson s ratio υ with changing Δ. Poisson s ratio υ for Δ = is υ =.97 (as the tensile deformation and the shear deformation of the cell wall are also taken into consideration in the present research, in addition to the bending deformation of the cell wall, υ =.97; however, as indicated b Gibson et al. [], υ = when onl bending deformation of the cell wall is considered). Near Δ/( l/) =.9 Poisson s ratio reaches its maimum value of about.. (5) For geometr of the heagonal cell, we also investigate the maimum value C ma and the minimum value C min of the elastic moduli C in one rotation of the (, ) coordinate aes, which are shown in Figure. When Δ = ; that is, when geometr is a regular heagon, C ma is at a minimum, and C min is at a maimum; both equal the elastic modulus of regular heagonal cell C regular.whenδ, the compliance C min for a certain direction becomes small, however, the compliance C ma for other direction becomes large. That is, when deviating the cell form from a regular heagonal cell, the rigidit of the honecomb can increase for a certain specific direction; however, direction for which the rigidit becomes small also eists. For the regular heagon cell, it is found that the equivalent elastic moduli are unrelated to the selected coordinate sstem, and the compliance of arbitrar direction is alwas the same as follows: C = C = C regular C = C regular for -ais parallel to a cell wall, for -ais parallel to a cell wall, C = C = C = C =, C = C C. (6) It is not dependent on whether the tensile or the shear deformation is taken into the analsis of equivalent elastic modulus that (6) holds.equation(6) isbasedonthe characteristic smmetr of the regular heagon. That is, using the smbols shown in Figure 6, foraregularheagon, we have θ =, θ = π/, θ = π/, C i = C i = C i (i =,, ). (7)

8 ISRN Mechanical Engineering.5 σ L U U U U/(σL)/C regular.5.5 Δ/( /)l (a) (b) Figure 9: Resultant deformation due to C and C : (a) displacement of plate subjected to stress σ and (b) compliance U/(σ L). 6 A C ma/c regular, C min/c regular C min C ma.5 Δ/( /)l Figure : Change in Poisson s ratio υ with changing Δ for geometr..5 Δ/( /)l Figure : Change in C ma and C min with changing Δ for geometr. B substituting (7)into(), (6)canbeobtained. (6) The elastic modulus C, which epresses the magnitude of the shear strain due to the shear stress, is a minimum when Δ = ; however, as the geometr approaches that of a parallelogram, C becomes larger, since shear deformation is generated easil. Net, we consider the heagonal cell AB C DE F,which is referred to as geometr here and is formed from the regular heagon b fiing points A and D, and moving points B, C, E, and F in the -direction, as shown in Figure (a). The nonzero elastic moduli for geometr (C = C = from left-right smmetr) are shown in Figure. For geometr, points B and C,aswellas points E and F, converge when Δ = l/, transforming the heagon into rhomboid AB DE.However,whenΔ = l/, the three points A, B,andF and the three points C, D, ande form straight lines, transforming the heagon to rectangle B C E F. In Figure, whenδ changes from the rhomboid to the rectangle, the following is observed. () C becomes large; that is, C increases from the value of

ISRN Mechanical Engineering 9 A A F F l A B B Δ F F l B B Δ E E D C C E E D D C C (a) (b) Figure : (a) Geometr of heagonal cell; (b) geometr of heagonal cell. 5 5 C Cij/C regular C C C ma/c regular, C min/c regular C min C ma C min C.5.5 Δ/l Figure : Equivalent elastic moduli for geometr shown in Figure (a)..5.5 Figure : Change in C ma and C min with changing Δ for geometr. Δ/l C /C regular =. for the rhomboid to C /C regular = 9. for the rectangle. () The elastic moduli C, C,and C (absolute values) each become smaller. Namel, these elastic moduli decrease from the values of C /C regular =., C /C regular =.9 and C /C regular =.56 for the rhomboid to C /C regular =., C /C regular =.67, and C = for the rectangle. The maimum value C ma and the minimum value C min of the elastic modulus C in one rotation of the (, ) coordinate aes are shown in Figure. C ma and C min take a minimum and a maimum, respectivel, when Δ =, that is, when geometr is a regular heagon, which is similar to the case of geometr. Lastl, we consider a heagonal cell A B C D E F, referred to as geometr here, which is formed from the regular heagon b moving the upper edge FAB and lower edge CDE to the upper and lower sides b Δ in the -direction, respectivel, as shown in Figure (b). The nonzero elastic moduli for geometr (C = C = from left-right smmetr) are shown in Figure 5. As shown in the figure, C decreases. This is because the length of the

ISRN Mechanical Engineering Cij/C regular 5 C C C.5.5 Figure 5: Equivalent elastic moduli for geometr shown in Figure (b). C ma/c regular, C min/c regular 5 C min Δ/l C ma C.5.5 Δ/l C min Figure 6: Change in C ma and C min with changing Δ for geometr. cell walls parallel to the -ais increases with increasing Δ; these cell walls onl undergo tensile deformation, and the amount of deformation is small compared to the bending deformation. Moreover, due to the increase in the length of one unit in the -direction, the force acting at the sloping cell walls due to stress σ increases, thus increasing C. Furthermore, even if Δ changes, the elastic modulus C remains constant, maintaining a value of C /C regular =.97. When Δ changes, the force acting at the sloping cell walls due to the stress σ does not change because the width of one unit does not change in the -direction. Therefore, the deformation of the sloping cell walls is the same and the equivalent strain in the -direction, ε,alsoremainsthe same. The maimum value C ma and the minimum value C min of the elastic modulus C in one rotation of the (, ) coordinate aes are shown in Figure 6, fromwhich it is seen that C ma is at a minimum and C min is at a maimum when Δ =, that is, when geometr is a regular heagon, as in the cases of geometr and geometr.. Conclusions In this research, the equivalent elastic moduli of asmmetrical heagonal honecomb are studied b using a theoretical approach. The deformation of honecomb consists of two tpes of deformations. The first is deformation inside the unit, which is caused b bending, stretching, and shearing of cell walls and rigid rotation of the unit; the second is relative displacement between units. The relative displacements between units are determined b condition of the junctions between wall ends of each adjacent unit, and the equivalent elastic modulus related to a direction parallel to one cell wall of the honecomb is determined from the relative deformation between units. In addition, using the three equivalent elastic moduli relating to the directions parallel to the cell wall, the nine components of the honecomb equivalent elastic modulus C, C C are derived b coordinate transformation. Using the proposed calculation equation, the elastic moduli for various shapes of heagon, which are obtained b sstematicall altering the regular heagon, are investigated. It is found that the maimum compliance C ma and the minimum C min of elastic modulus C in one rotation of the (, ) coordinate sstem var as the shape of the heagon is changed. However, C ma takes the minimum and C min takes the maimum when the honecomb cell is a regular heagon, for which the equivalent elastic moduli are unrelated to the selected coordinate sstem and are constant with C = C and C = C C. References [] I. G. Masters and K. E. Evans, Models for the elastic deformation of honecombs, Composite Structures, vol. 5, no., pp., 996. [] W. E. Warren and A. M. Kranik, Foam mechanics: the linear elastic response of two-dimensional spatialll periodic cellular materials, Mechanics of Materials, vol. 6, no., pp. 7 7, 987. []L.J.Gibson,M.F.Ashb,G.S.Schajer,andC.I.Robertson, The mechanics of two-dimensional cellular materials, Proceedings of The Roal Societ of London, Series A, vol. 8, no. 78, pp. 5, 98. [] D. H. Chen and S. Ozaki, Analsis of in-plane elastic modulus for a heagonal honecomb core: Effect of core height and proposed analtical method, Composite Structures, vol. 88, no., pp. 7 5, 9. [5] D. W. Overaker, A. M. Cuitiño, and N. A. Langrana, Elastoplastic micromechanical modeling of two-dimensional irregular conve and nonconve (re-entrant) heagonal foams, Journal of Applied Mechanics, Transactions ASME, vol.65,no.,pp. 78 757, 998.

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