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Shear moduli for non-isotropic, open cell foams using a general elongated Kelvin foam model
International Journal of Engineering Science 47 (2009) 990–1001
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Shear moduli for non-isotropic, open cell foams using a general elongated Kelvin foam model Roy M. Sullivan a,*, Louis J. Ghosn b a b
Structures and Materials Division, NASA Glenn Research Center, Cleveland, OH 44135, USA Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, OH 44142, USA
a r t i c l e
i n f o
Article history: Received 20 February 2009 Received in revised form 28 April 2009 Accepted 5 May 2009 Available online 8 July 2009 Communicated by M. Kachanov Keywords: Foams Elastic constants Shear modulus Kelvin foam model Elongated cell
a b s t r a c t Equations for calculating the shear modulus of non-isotropic, open cell foams in the plane perpendicular to the rise direction and in a plane parallel to the rise direction are derived using an elongated Kelvin foam model. This Kelvin foam model is more general than that employed by previous authors as the size and shape of the unit cell are defined by specifying three independent cell dimensions. The equations for the shear compliances are derived as a function of three unit cell dimensions and the section properties of the cell edges. From the compliance equations, the shear modulus equations are obtained and written as a function of the relative density and two unit cell shape parameters. The dependence of the two shear moduli on the relative density and the two shape parameters is demonstrated. Published by Elsevier Ltd.
1. Introduction Over the last few decades, there have been numerous attempts to determine the elastic properties and strengths of open and closed-cell foams from a description of the foam microstructure and a measure of the cell dimensions. This endeavor requires selecting a suitable representative element to mimic the foam microstructure. One of the earliest attempts was by Gent and Thomas [1]. They represented the foam microstructure with a three-dimensional assemblage of thin threads. The threads were connected together at junction points, with an arbitrary number of threads joining at each junction point. They considered only axial deformation of the threads and developed an equation for the relative Young’s modulus as a function of the relative density. Gibson and Ashby [2] used a staggered arrangement of cubes to model the foam microstructure. For open cell foams, they assumed the cell edges were square in cross-section. They developed equations for the relative Young’s modulus and relative shear modulus as well as equations for the compression strength of the foam based on elastic buckling and plastic collapse of the cell edges. Many researchers have used a tetrakaidecahedron, a fourteen-sided polyhedron, to represent the foam microstructure [3– 8]. The tetrakaidecahedron foam model is known as a Kelvin foam model, after Thomson [3]. Zhu et al. [4] used an equi-axial Kelvin unit cell and assumed that the mechanical behavior of open cell foams could be simulated by treating the cell edges as structural elements possessing axial, bending and torsional rigidity. They applied the minimum potential energy theorem to the unit cell deformation and developed equations for the Young’s modulus, shear modulus and Poisson’s ratio in terms of the cell edge lengths, the edge cross-section properties and the solid material properties.
* Corresponding author. Tel.: +1 216 433 3249; fax: +1 216 433 8300. E-mail addresses: [email protected] (R.M. Sullivan), [email protected] (L.J. Ghosn). 0020-7225/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2009.05.005
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In some cases, the foam cells are elongated; as a result, these foams exhibit a non-isotropic mechanical behavior. Such is the case in spray-on foams where the cells are elongated in the rise direction due to the foaming and rising process. For these foams, an elongated Kelvin unit cell has been employed [6–8]. Some researchers have examined the effects of the irregular microstructure which occurs in real foams, using a representative volume consisting of an irregular arrangement of polyhedrons [9–11]. These representative foam volumes were generated from a random distribution of seed points and the use of Voronoi or Laguerre tessellation. A suitable representative volume requires a large number of cells [11]. This necessitates the use of a numerical solution such as the finite element method and prohibits the derivation of algebraic expressions for the effective foam elastic properties and strengths. Another refinement to modeling the mechanical behavior of foams is related to the assumed geometry of the elongated Kelvin unit cell. In the previous studies which utilized an elongated Kelvin foam model Refs. [6–8], the authors adopted a unit cell which could be described by specifying only two independent cell dimensions: one to specify the size of the unit cell and one to specify the elongation (or aspect) ratio. The elongated Kelvin unit cell is, however, a polyhedron with three independent dimensions. Recently, Sullivan et al. [12] has employed a general elongated Kelvin unit cell, one that is defined by specifying three independent dimensions. Using a regular arrangement of uniform size elongated Kelvin unit cells to represent the average foam microstructure, equations were derived for the Young’s modulus, Poisson’s ratios and tensile strengths for non-isotropic, open cell foams in the principle material directions. This more general Kelvin model accounts for an additional variation in the shape of the unit cell which was not present in the previous models. To quantify this additional shape variation, Sullivan et al. [12] introduced an additional shape parameter. They demonstrated the effect of this additional shape parameter on the stiffness and strength ratios in open cell foams. This paper is intended as a supplement to Sullivan et al. [12] as it uses the general elongated Kelvin unit cell to derive the equations for the shear moduli in two principle material planes: the plane perpendicular to the rise direction and a plane parallel to the rise direction. The equations are derived in a manner similar to Zhu et al. [4] and Sullivan et al. [12], by applying the minimum potential energy principle to the unit cell deformation. First, the equations for the shear compliances are derived as a function of three unit cell dimensions and the section properties of the cell edges. From the compliance equations, the equations for the shear moduli are derived and rewritten as a function of the two unit cell shape parameters and the foam relative density. In the final section, the shear moduli are plotted as a function of the relative density and the two shape parameters to demonstrate their dependence on these three quantities. 2. Description of the Kelvin unit cell The general elongated Kelvin unit cell is a fourteen-sided polyhedron (tetrakaidecahedron) constructed from eight hexagonal faces, four vertical rhombic faces and two horizontal square faces (Fig. 1). It has height H and width D. The vertical rhombic faces have sides of length L and the horizontal square faces have sides of length b. The hexagonal faces have four sides of length L and two sides of length b. The inclination angle h defines the orientation of the hexagonal faces with respect to the rise direction as well as the obtuse angle of the vertical diamond faces, 2h. In open-celled foams, all the solid mass is concentrated within the cell edges. The edges are the line segments formed by the intersection of multiple faces. Thus, the edges will be of length L or b. The edges are assumed to behave like struts, possessing axial, flexural and torsional rigidity. They have cross-sectional area A, bending moment of inertia I and torsion constant J. The Young’s modulus of the solid material is denoted as E and the shear modulus is G.
Fig. 1. Sketch of an elongated tetrakaidecahedron unit cell (Kelvin model).
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The size and shape of the general elongated Kelvin cell is defined by specifying any three of the five unit cell dimensions shown in Fig. 1. The cell height H and cell width D are related to the edge lengths L and b and the inclination angle h according to
H ¼ 4L sin h and D ¼ 2L cos h þ
pffiffiffi 2b:
ð1Þ
Alternatively, the size and shape of the Kelvin cell may be defined by specifying the value of either the cell height or the cell width and by specifying the value of two independent shape parameters R and Q. The parameter R is known as the aspect ratio of the cell. It is simply the ratio of the cell height to the cell width, R = H/D. Using the expressions in (1), the aspect ratio may be written as
R¼
4L sin h pffiffiffi : 2b
2L cos h þ
ð2Þ
The parameter Q was introduced by Sullivan et al. [12] and defined as Q = b/Lcosh. From Fig. 1 and the expression for D in Eq. it is easily recognized that the cell width is the sum of two dimensions: the diagonal of the horizontal square face pffiffiffi(1), 2b and two times the projection of a L-length edge onto a plane perpendicular to the rise direction (2Lcosh). The parameter Q is a measure of the ratio of these two dimensions. To illustrate the effect of the shape parameters, Kelvin unit cells with various combinations of values for R and Q are shown in Fig. 2. of the previous studies which used an elongated pMany ffiffiffi Kelvin foam model restricted the unit cell shape by specifying Q ¼ 2. Under this restriction, the diagonal of the horizontal square face is always one-half the cell width. The relative density c is the ratio of the foam density to the density of the solid material, c = q/qs. It may also be written in terms of volumes as c = Vs/V, where Vs is the volume occupied by solid matter and V is the total volume. The edges that form the perimeter of the vertical rhombic faces and those of length b that form the perimeter of the horizontal square faces are shared by adjacent cells. Thus, they contribute only half their cross-sectional area to the repeating unit. Assuming the crosssectional area is the same for all edges and constant along the edge length, then Vs = 16AL + 8Ab and thus
c¼
8Að2L þ bÞ HD2
:
Fig. 2. Sketches of various Kelvin unit cells illustrating the effect of the parameters R and Q on the unit cell shape.
ð3Þ
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3. Derivation of the shear response 3.1. In-plane shear compliance In order to derive the equation for the foam in-plane shear modulus Gxy, we take advantage of the symmetry of the Kelvin unit cell and adopt the structural unit shown in red in Fig. 3 as the representative repeating unit. This repeating unit is a onequarter segment of the Kelvin foam unit cell shown in Fig. 1. We establish a Cartesian coordinate system xyz such that the zaxis is oriented in the elongation or rise direction (Fig. 3). The x-axis is parallel to a line normal to two of the vertical rhombic faces, and the y-axis is parallel to a line normal to the other two vertical rhombic faces. We also establish the xsysz coordinate system that is oriented with respect to the xyz system by a 45° counterclockwise rotation about the z-axis. The repeating structural unit is bounded by six symmetry planes: two planes parallel to the ysz plane, two planes parallel to the xsz plane, and two planes parallel to the xsys plane. Since the members AN, AO, BJ, BK, DM, DL, EP and EQ all lie within the xsys symmetry planes, they are shared by two adjacent unit cells. As such, we assume that within the unit cell, they possess a cross-sectional area A/2 and a bending moment of inertia of I/2. To induce a state of pure shear stress sxy and pure shear strain cxy in the unit cell, we apply the normal stresses s and s in the xs and ys directions, respectively, as shown in Fig. 4. We define the displacements of the unit cell with respect to the unit cell center at point C and assume that all displacements and rotations at point C are zero. The applied loading results in a unit cell displacement of 2u in the xs direction and 2u in the ys direction, where both displacements are symmetric about point C. There is no displacement of the unit cell in the z-direction (Fig. 4). Thus, the work done on the unit cell by the applied loading is pffiffiffi pffiffiffi pffiffiffi W ¼ 8L sin h 2L cos h þ b su. Furthermore, the principal strains are 2u=ð 2L cos h þ bÞ and 2u= 2L cos h þ b , so the pffiffiffi shear strain cxy is 4u= 2L cos h þ b . Any edge that is initially normal to a symmetry plane boundary must remain normal to it after the deformation. Thus, the points J, K, L, M, N, O, P, and Q do not rotate during the deformation. Furthermore, the half-edges AN, AO, BJ, BK, DL, DM, EQ, and EP must remain in their respective symmetry planes. Thus, the points A, B, D, and E translate within their respective symmetry planes and the only rotation of these points is about the z-axis. Points A and B undergo a clockwise rotation x about the z-axis, and points D and E undergo a counterclockwise rotation x. Since there are no stresses applied to the unit cell in the y direction, members AC and BC will experience no axial deformation. Therefore, points A and B will only translate in the x-direction. Point B translates in the x-direction by an amount v and point A translates by an amount v. Likewise, points D and E translate in the y-direction by an amount v and v, respectively. We note that as a result of similarity and symmetry, the deformed shape and the deformation energy of member BC are the same as those of members CD, AC, and CE. Likewise, the deformed shape and deformation energy of the members BK, DL, AN, and EP are equivalent and those of members BJ, DM, AO, and EQ are equivalent. Furthermore, the axial tension in member BK is equal to the axial compression in member BJ, and the bending moment in BJ is equal to the bending moment in BK. The deformation energy of this one-quarter unit cell is thus
U ¼ 4U BC þ 8U BK : As a result of the loading and symmetry boundary conditions imposed on the one-quarter unit cell, member BC experiences torsional and bending deformation and member BK experiences axial and bending deformation. The strain energy in members BC and BK are written in terms of the unit cell displacements u and v, and the rotation x as
Fig. 3. Sketch of the one-quarter model showing its reference with respect to the elongated Kelvin unit cell (point Q not shown; see Fig. 4).
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(a)
ys
τ u
y u
Q
B
E
P
τ
J K
ω
xs
C
ω N
τ
L
A
u
D
O
M
τ
x z
(b)
2
u
τ
O
A
B
J
τ
C
ys M
D
E
Q
Fig. 4. Sketch of the applied loading and deformed shape of the unit cell for derivation of the in-plane shear modulus Gxy. a) xsys plane. b) ysz plane.
GJ 2 EI x sin2 h þ 3 2L2 x2 cos2 h þ 6v 2 6v Lx cos h ; 2L L 2 EA v x2 ¼ u pffiffiffi þ EI : 2b 2b 2
U BC ¼ U BK
Applying the minimum potential energy theorem to the unit cell deformation, in the form of @(U W)/@u = @ (U W)/ @ v = @(U W)/@ x = 0, results in three independent equations written in terms of u, v, and x. The simultaneous solution of these three equations yields the expressions for u, v, and x. The expression for u is obtained as
3 2 GJ 2 3 2 L 2L þ 4b cos h þ h b sin pffiffiffi 6 b 7 EI 7 u ¼ sL sin h 2L cos h þ b 6 4EA þ 5: GJ 2 12EI 2L þ b cos2 h þ b sin h EI
pffiffiffi Since the shear strain is cxy ¼ 4u= 2L cos h þ b and since s = Gxycxy, then
h i9 8 2
ð4Þ
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z (rise direction)
τ
B
z (rise direction)
R
B H
C
S T′
T F
D′
E′
F Ω
C′
F ′ D′
τ
y
G′
y H′
G′
x
F′
D
τ
C′
G E
D
4L sin θ
B′
G
C
B′
H
R′ H′
τ 2 L cos θ + 2b
Fig. 5. Sketch of the Kelvin unit cell and the shear stress tractions applied to the unit cell boundaries for calculating the out-of-plane shear modulus Gyz.
3.2. Out-of-plane shear compliance Since the Kelvin unit cell is elongated in the z-direction, it is not possible to determine the shear modulus Gyz in the same manner in which the in-plane modulus Gxy was determined. We can not induce a state of pure strain cyz by applying a set of equal and opposite normal stresses to the unit cell in the yz plane at 45° from the y and z axes. Instead, we will use the full Kelvin unit cell and apply shear stress tractions directly to the unit cell boundaries as shown in Fig. 5. A shear stress s is applied in the +y direction along the top boundary plane (+z plane) and in the y direction along the bottom boundary plane (z plane). A shear stress s is also applied on the left and right boundary planes in the z and +z directions, respectively. The deformation of the unit cell that results from the shear tractions is shown in Fig. 6. The unit cell displacements are defined with respect to the point X, which is located at the center of the Kelvin unit cell. The points that lie on the left and right boundary planes and located at the mid-height of the unit cell (Points D, D0 , E, E0 ) do not rotate. They translate only in the z-direction. The points that lie on the top and bottom boundary planes and located at the mid-width of the unit cell (Points H, H0 , R, R0 ) also do not rotate. They translate only in the y-direction. All other points are free to translate in the yand z-directions. In addition, all other points experience either a clockwise rotation x1 or a counterclockwise rotation x2 about the x axis as illustrated in Fig. 6. There is no displacement of any of the points in the unit cell in the x-direction and there is no rotation of any of the points about the y or z axes. The deformed unit cell shown in Fig. 6 is in a state of pure shear strain cyz. The pure shear strain cyz is the sum of two simple shear strains, cyz = c0 + c00 , where c0 – c00 due to the elongated unit cell. From the deform shape shown in Fig. 6, it is easy to see that c0 = 2wD/D and c00 = 2vH/H, where vH is the y-direction displacement of Point H and wD is the z-direction displacement of Point D. In addition, we note that the work performed on the unit cell by the applied shear tractions is W = 2sD(HwD + DvH). Due to their similarity and the equivalence of their end point displacements, the L-length members that are aligned parallel to the xz plane, such as members CD, C0 D0 , GH and RS, will all have the same deformation energy. Likewise, all the L-length members that are aligned parallel to the yz plane (BC, B0 C0 , FG, GF0 , and ST) will have the same deformation energy and all the b-length members will have the same deformation energy. In addition, it should be noted that all members that lie along the unit cell boundary planes will contribute only half their deformation energy to the total deformation energy of the unit cell. Thus, the total deformation energy of the Kelvin unit cell U may be written
U ¼ 8ðU BC þ U BH þ U GH Þ: Note that member BC has axial and bending deformation energy, whereas members BH and GH have torsion and bending deformation energy. Since members CD and BH do not deform axially, vB = vH and wC = wD. Recognizing also that vC = vH vG, the strain energies UBC, UBH and UGH may be written in terms of the rotations x1 and x2, and the displacements vC, vH, wB and wD as
U BC ¼
EA 6EI 2 2 ½ðv H v c Þ cos h þ ðwD wB Þ sin h þ 3 ½ðwD wB Þ cos h ðv H v c Þ sin h 2L L þ
6EI L2
ðx1 x2 Þ½ðwD wB Þ cos h ðv H v c Þ sin h þ
2EI 2 ðx1 x1 x2 þ x22 Þ; L
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z τ ω2
H
H
B G
C
ω2
C′
ω1
ω1
ω1 ω2 ω2
Ω
wD
D′
γ′
τ
y
F
D
ω1
τ
ω1 ω1 ω2 H′
ω2
τ γ ′′
Fig. 6. Deformed shape of the Kelvin unit cell resulting from the out-of-plane shear stress.
pffiffiffi GJ 2 EI x þ ð6w2B 3 2bwB x2 þ b2 x22 Þ; 4b 2 b3 GJ 2EI 2 ¼ x21 cos2 h þ 3 3v 2C 3v C Lx1 sin h þ L2 x21 sin h : 2L L
U BH ¼ U GH
Once again, we apply the minimum potential energy theorem to the unit cell deformation, this time with respect to the six unit cell rotations and displacements x1, x2, vC, vH, wB and wD, i.e. @(U W)/@ x1 = @(U W)/@ x2 = @(U W)/@ vc = = 0. This results in six independent equations written in terms of the two rotations and the four displacements. The simultaneous solution of these six equations yields the expressions for x1, x2, vC, vH, wB and wD. The displacements vH and wD may be written as
hpffiffiffi i 8 9 2 2 4
ð5aÞ hpffiffiffi i 8 2 9 2 2 4
9 8 h i2 pffiffiffi 2 > > b = 1 L EA ; : 12EI 1 þ sin h þ 2 b sin h ðEIÞ2 þ 2 þ 2 b cos2 h EIGJ þ ðcos2 hÞðGJÞ2 > L L
ð6Þ
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where
k1 ¼
DkA þ HkD 2L
k2 ¼
DkB þ HkE 2L
k3 ¼
DkC þ HkF : 2L
Note that when b = L and h = p/4, the unit cell is equi-axial. Under these conditions, both Eqs. (4) and (6) reduce to
pffiffiffi pffiffiffi 4 1 2 2L2 2L 8EI þ GJ ; ¼ þ G EA 6EI 5EI þ GJ which is the expression obtain by Zhu et al. [4] for isotropic open cell foams.
4. Expressions for the Shear Moduli in terms of R, Q and c We now seek to obtain expressions for the in-plane and out-of-plane shear modulus in terms of the relative density c and the two shape parameters R and Q. Using R = H/D and the expression for H in equation (1), equation (3) may be rearranged to obtain
A L2
3
¼
8c sin h ð2 þ b=LÞR2
ð7Þ
:
Defining the constant C1 as the ratio of the edge cross-sectional area to the square of the characteristic cross-section dimension, A = C1r2, we can rearrange equation (7) to obtain (r/L)2 = 8csin3h/(C1R2(2 + b/L)). Furthermore, defining the constant C2 through the relation 12I = C1C2 r4, we can also write
12I L4
6
¼
C 2 64c2 sin h : C 1 ð2 þ b=LÞ2 R4
ð8Þ
Table 1 Summary of the equations for the relative Young’s moduli, Poisson’s ratios and relative tensile strengths that result from the general Kelvin foam model written as a function of R, Q and c (from Sullivan, et al. [14]). e 5 RT 8Q Relative Young’s modulus
Ex Ey ¼ E E
ð2Q þ TÞ2
e 3 Rð8 þ QTÞ 8Q c þ C 1 Tð32Q 3 þ Qe 2 R2 TÞ C2 ð2Q þ TÞ e 5 R3 T Q
Ez E
ð2Q þ TÞ2
c2
e 5 R3 Q c þ 4 C1 T 2 C2 ð2Q þ TÞ 8Q 4C 1 Q 2
Poisson’s Ratio
mxy
mxz = myz
mzx = mzy
rsxx rsyy ¼ s r rs
rszz rs
e 3R C2 Q c 2Q þ T
!
e3 e 2 R2 TÞC 1 þ 8C 2 Q Rð8 þ QTÞ c ð32Q 3 þ Q Tð2Q þ TÞ ! e3 e TC 1 4C 2 Q R c 4Q Tð2Q þ TÞ e3 e 2 R2 TÞC 1 þ 8C 2 Q Rð8 þ QTÞ c ð32Q 3 þ Q Tð2Q þ TÞ ! e3 e R2 C 1 4C 2 Q R c Q 2 T ð2Q þ TÞ 8C 1 þ
Relative tensile strength
c2
e 5 R3 2C 2 Q T 2 ð2Q þ TÞ
c
e 2:5 R1:5 c1:5 4C 3 Q Rð2Q þ TÞ1:5
e 1:5 0:5 0:5 pffiffiffiffiffiffi e R þ 16C 3 Q R c C1 Q Tð2Q þ TÞ0:5
e 2:5 R1:5 c1:5 C3 Q ð2Q þ TÞ1:5
e 2:5 R1:5 c0:5 pffiffiffiffiffiffi C 3 Q C1 þ Tð2Q þ TÞ0:5
!
!
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pffiffiffi Eq. (2) may be combined with the expression Q = b/Lcosh to obtain tan h ¼ 2 þ 2Q R=4. This leads to
4 cos h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 16 þ Q
eR Q sin h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 16 þ Q
pffiffiffi e ¼ 2 þ 2Q . where Q Also, if the solid material is isotropic, we have
b 4Q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; L e 2 R2 16 þ Q
bJ GJ ¼ ; EI 2ð1 þ mÞ
ð9Þ
ð10Þ
where m is the Poisson’s ratio of the solid material and bJ ¼ J=I. The values of J and I are given in Mills [13] for a variety of cross-section shapes. Using (10) and substituting the expressions in (9) into equations (4), (6), (7) and (8) leads to
e 5R 4Q Gxy ¼ E
ð2Q þ TÞ2 e 3R 16Q Q C c þ 1 T3 ð2Q þ TÞ C2
c2
! bJ 1þm ! b e 2 R2 J T 3 þ 32Q þ Q Q 1þm
e 2 R2 T 3 þ 128Q þ Q Q
Fig. 7. Relative shear moduli Gxy/E and Gyz/E plotted versus aspect ratio R for various values of Q. (a) c = 0.01. (b) c = 0.05.
ð11Þ
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Fig. 8. Shear modulus ratio versus aspect ratio R for various values of Q and two relative densities: 1% and 5%.
and
e 7 R3 T 8Q ð2Q þ TÞ2 2
Gyz ¼ E e 5 Rð4 þ Q e R2 Þ2 4Q cþ ð2Q þ TÞ
4C 1 T 4C1 2 C 2 48T
bJ
c2 þ C2
1þm
!2
1þm
!2
bJ
þ ð128Q þ 2T 3 Þ
bJ 1þm
bJ 1þm
;
3
ð12Þ
þ C3 5 3
e 2 R2 T þ 16Q Q e 2 R2 Þ5 þ ð2T 3 þ 2 Q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 . Expressions for C1, C2 and C3 in terms of R; Q ; Q e and T are listed in Appendix B. where T ¼ 16 þ Q For the sake of completeness, the equations for the relative Young’s moduli, Poisson’s ratios and relative tensile strengths which result from the general Kelvin foam model are listed in Table 1. These equations were presented in Sullivan, et al. [14]. Note that the expressions for the relative shear moduli in Eqs. (11) and (12) as well as the equations for the relative Young’s moduli in Table 1 are written in the form f1c2/(f2c + f3) where f1 and f2 are functions of the cell shape parameters and f3 is a function of the shape parameters and the edge cross-section properties. This same functional relationship between the relative moduli and the relative density has been proposed by previous authors (e.g. Refs. [4,7]). 5. Results The variation of the relative shear moduli Gxy/E and Gyz/E with the two shape parameters and the relative density c is illustrated in Fig. 7. The relative shear moduli are plotted as a function of R and Q for a relative density of 1% in Fig. 7a and for a relative density of 5% in Fig. 7b. In both plots, three-cusp hypocycloid (Plateau border) cross-sections were assumed ðbJ ¼ 0:6Þ. We also assumed that m = 0.3. Note the strong dependence of the relative shear moduli on the relative density. Note also that, for a constant relative density, the out-of-plane relative shear modulus Gyz/E appears to be a stronger function of the shape parameter Q than the shape parameter R, whereas the in-plane relative shear modulus Gxy/E is a stronger function of R than Q. The shear modulus ratio Gyz/Gxy is plotted in Fig. 8 as a function of R, Q and c. Again, Plateau border cross-sections were assumed. The shear modulus ratio does not appear to be a strong function of the relative density. It is an increasing function of the aspect ratio R and a decreasing function of the shape parameter Q. In contrast, the Young’s modulus ratio Ez /Ey and the tensile strength ratio rszz =rsyy were both shown, in Sullivan et al. [12], to be increasing functions of Q. Finally, we note that although there is an infinite number of combinations of R and Q in which the material is isotropic in its shear response (Gxy = Gyz), pffiffiffithere is only one combination of R and Q which results in both Ez/Ey = 1 and Gyz/Gxy = 1. This occurs when R = 1 and Q ¼ 2. 6. Concluding remarks The equations for the shear moduli have been derived for non-isotropic, open-cell foams using a general elongated Kelvin model. This elongated Kelvin model is more general than that employed by previous authors as it uses a unit cell with three
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independent dimensions. As such, this model accounts for all shape variations which are possible with a fourteen-sided polyhedron unit cell. The equations for the shear modulus in the plane perpendicular to the rise direction and in a plane parallel to the rise direction have been derived as a function of the relative density and two unit cell shape factors: the aspect ratio R and an additional shape factor Q, which was introduced by the authors in a previous publication. The dependence of the shear moduli on the relative density and the two shape factors has been demonstrated. The inplane shear modulus appears to be a stronger function of the aspect ratio than the shape factor Q, whereas the out-of-plane shear modulus is a stronger function of Q than the aspect ratio. Both shear moduli are a strong function of the relative density. Acknowledgement The authors are grateful for funding from the External Tank Project under NASA’s Space Shuttle Program. Appendix A. Expressions for kA, kB, kC, kD, kE, and kF in terms of h and b/L
pffiffiffi pffiffiffi b 2 2 2 2 kA ¼ 2 cos3 hð1 þ sin hÞ þ ð16 cos3 h sin h þ 2 cos4 h 48 cos h sin h þ 16 2 sin hÞ L 2 pffiffiffi b 2 2 þ 4 2 sin hð4 þ cos hÞ ; L 2 h pffiffiffi ib pffiffiffi pffiffiffi b 2 2 2 kB ¼ 4 cos3 h þ 2 2 þ 14 2 sin h þ 4 cos3 hð1 4 sin hÞ þ 2 2 cos2 hðcos2 h sin hÞ ; L L pffiffiffi b 2 kC ¼ 2 cos5 h þ 2 cos2 hð1 þ sin hÞ ; L pffiffiffi b pffiffiffi 2 2 kD ¼ 2 cos h sin hð1 þ sin hÞ cos h sin h 8 2 40 cos h 2 cos2 h þ 16 cos3 h L 2 3 4 pffiffiffi b b b 2 2 3 þ 2 cos h sin hð10 sin h 6Þ 2 sin hð1 5 sin hÞ þ 8 sin h ; L L L 2 pffiffiffi b pffiffiffi b kE ¼ 4 cos2 h sin h þ 8 cos h sin h 2 cos3 h 2 þ 10 2 cos3 h sin h L L 3 4 b b þ 2 sin hð4 þ 3 cos2 hÞ þ 8 cos2 h sin h ; L L 3 pffiffiffi b b kF ¼ 2 cos4 h sin h 2 cos3 h sin h þ 4 cos2 h sin h : L L e and T Appendix B. Expressions for C1, C2 and C3 in terms of R; Q ; Q
h
pffiffiffi
i
e 2 R2 T 2 þ 128Q 3 Q e 2 R2 T; C1 ¼ 8ð1 þ Q 2 ÞT 4 þ 64Q ð2 2 QÞ þ 8 Q
pffiffiffi
e 2 R2 112Q 2 þ 32 2Q þ 16 T 4 64Q ð2 þ 7Q 2 6 Q e ÞT 3 C2 ¼ ð1 þ 8Q 2 ÞT 6 þ 32Q 3 T 5 þ 3 Q e ð16 þ Q 2 Q e R2 ÞT þ 2048Q 4 Q e 2 R2 ; þ 256Q Q pffiffiffi e 2 R2 60Q 2 þ 8 2Q T 4 2048Q ð1 þ Q 2 ÞT 3 C3 ¼ 2ð1 þ 8Q 2 ÞT 6 þ 128Qð1 þ Q 2 ÞT 5 þ 4 Q h pffiffiffi i e 2 R2 2 Q e 2 R2 þ 16Q 2 2 Q T 2 512Q Q e 2 R2 ð2Q 2 þ Q e ÞT þ 256Q 4 Q e 4 R4 : þQ References [1] [2] [3] [4]
A.N. Gent, A.G. Thomas, The deformation of foamed elastic materials, Journal of Applied Polymer Science 1 (1) (1959) 107–113. L.J. Gibson, M.F. Ashby, The mechanics of three-dimensional cellular materials, Proceedings of the Royal Society London A 382 (1982) 43–59. W. Thomson (Lord Kelvin), On the division of space with minimum partitional area, Philosophical Magazine 24 (1887) 503–514. H.X. Zhu, J.F. Knott, N.J. Mills, Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells, Journal of the Mechanics and Physics of Solids 45 (3) (1997) 319–343. [5] W.E. Warren, A.M. Kraynik, Linear elastic behavior of a low-density Kelvin foam with open cells, Journal of Applied Mechanics 64 (1997) 787–794. [6] A.G. Dement’ev, O.G. Tarakanov, Model analysis of the cellular structure of plastic foams of the polyurethane type, Mekhanika Polimerov. 5 (1970) 859– 865. Trans. Polymer Mechanics 6 (1970) 744–749. [7] L. Gong, S. Kyriakides, W.Y. Jang, Compressive response of open-celled foams, Part I: Morphology and elastic properties, International Journal of Solids and Structures 42 (2005) 1355–1379.
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[8] M. Ridha, V.P.W. Shim, L.M. Yang, An elongated tetrakaidecahedral cell model for fracture in rigid polyurethane foam, Key Engineering Materials 306– 308 (2006) 43–48. [9] H.X. Zhu, J.R. Hobdell, A.H. Windle, Effects of cell irregularity on the elastic properties of open-cell foams, Acta Material 48 (2000) 4893–4900. [10] A.P. Roberts, E.J. Garboczi, Elastic properties of model random three-dimensional open-cell solids, Journal of the Mechanics and Physics of Solids 50 (2002) 33–55. [11] S. Kanaun, O. Tkachenko, Representative volume element and effective elastic properties of open cell foam materials with random microstructures, Journal of Mechanics of Materials and Structures 2 (8) (2007) 1607–1628. [12] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, A general tetrakaidecahedron model for open-celled foams, International Journal of Solids and Structures 45 (2008) 1754–1765. [13] N.J. Mills, Micromechanics of polymeric foams, in: Third Nordic Meeting on Materials and Mechanics, Aalborg, Denmark, 2000, pp. 45–76. [14] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, Application of an elongated Kelvin model to space shuttle foams, AIAA Journal of Spacecraft and Rockets 46 (2) (2009) 411–418.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Shear moduli for non-isotropic, open cell foams using a general elongated Kelvin foam model Roy M. Sullivan a,*, Louis J. Ghosn b a b
Structures and Materials Division, NASA Glenn Research Center, Cleveland, OH 44135, USA Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, OH 44142, USA
a r t i c l e
i n f o
Article history: Received 20 February 2009 Received in revised form 28 April 2009 Accepted 5 May 2009 Available online 8 July 2009 Communicated by M. Kachanov Keywords: Foams Elastic constants Shear modulus Kelvin foam model Elongated cell
a b s t r a c t Equations for calculating the shear modulus of non-isotropic, open cell foams in the plane perpendicular to the rise direction and in a plane parallel to the rise direction are derived using an elongated Kelvin foam model. This Kelvin foam model is more general than that employed by previous authors as the size and shape of the unit cell are defined by specifying three independent cell dimensions. The equations for the shear compliances are derived as a function of three unit cell dimensions and the section properties of the cell edges. From the compliance equations, the shear modulus equations are obtained and written as a function of the relative density and two unit cell shape parameters. The dependence of the two shear moduli on the relative density and the two shape parameters is demonstrated. Published by Elsevier Ltd.
1. Introduction Over the last few decades, there have been numerous attempts to determine the elastic properties and strengths of open and closed-cell foams from a description of the foam microstructure and a measure of the cell dimensions. This endeavor requires selecting a suitable representative element to mimic the foam microstructure. One of the earliest attempts was by Gent and Thomas [1]. They represented the foam microstructure with a three-dimensional assemblage of thin threads. The threads were connected together at junction points, with an arbitrary number of threads joining at each junction point. They considered only axial deformation of the threads and developed an equation for the relative Young’s modulus as a function of the relative density. Gibson and Ashby [2] used a staggered arrangement of cubes to model the foam microstructure. For open cell foams, they assumed the cell edges were square in cross-section. They developed equations for the relative Young’s modulus and relative shear modulus as well as equations for the compression strength of the foam based on elastic buckling and plastic collapse of the cell edges. Many researchers have used a tetrakaidecahedron, a fourteen-sided polyhedron, to represent the foam microstructure [3– 8]. The tetrakaidecahedron foam model is known as a Kelvin foam model, after Thomson [3]. Zhu et al. [4] used an equi-axial Kelvin unit cell and assumed that the mechanical behavior of open cell foams could be simulated by treating the cell edges as structural elements possessing axial, bending and torsional rigidity. They applied the minimum potential energy theorem to the unit cell deformation and developed equations for the Young’s modulus, shear modulus and Poisson’s ratio in terms of the cell edge lengths, the edge cross-section properties and the solid material properties.
* Corresponding author. Tel.: +1 216 433 3249; fax: +1 216 433 8300. E-mail addresses: [email protected] (R.M. Sullivan), [email protected] (L.J. Ghosn). 0020-7225/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2009.05.005
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In some cases, the foam cells are elongated; as a result, these foams exhibit a non-isotropic mechanical behavior. Such is the case in spray-on foams where the cells are elongated in the rise direction due to the foaming and rising process. For these foams, an elongated Kelvin unit cell has been employed [6–8]. Some researchers have examined the effects of the irregular microstructure which occurs in real foams, using a representative volume consisting of an irregular arrangement of polyhedrons [9–11]. These representative foam volumes were generated from a random distribution of seed points and the use of Voronoi or Laguerre tessellation. A suitable representative volume requires a large number of cells [11]. This necessitates the use of a numerical solution such as the finite element method and prohibits the derivation of algebraic expressions for the effective foam elastic properties and strengths. Another refinement to modeling the mechanical behavior of foams is related to the assumed geometry of the elongated Kelvin unit cell. In the previous studies which utilized an elongated Kelvin foam model Refs. [6–8], the authors adopted a unit cell which could be described by specifying only two independent cell dimensions: one to specify the size of the unit cell and one to specify the elongation (or aspect) ratio. The elongated Kelvin unit cell is, however, a polyhedron with three independent dimensions. Recently, Sullivan et al. [12] has employed a general elongated Kelvin unit cell, one that is defined by specifying three independent dimensions. Using a regular arrangement of uniform size elongated Kelvin unit cells to represent the average foam microstructure, equations were derived for the Young’s modulus, Poisson’s ratios and tensile strengths for non-isotropic, open cell foams in the principle material directions. This more general Kelvin model accounts for an additional variation in the shape of the unit cell which was not present in the previous models. To quantify this additional shape variation, Sullivan et al. [12] introduced an additional shape parameter. They demonstrated the effect of this additional shape parameter on the stiffness and strength ratios in open cell foams. This paper is intended as a supplement to Sullivan et al. [12] as it uses the general elongated Kelvin unit cell to derive the equations for the shear moduli in two principle material planes: the plane perpendicular to the rise direction and a plane parallel to the rise direction. The equations are derived in a manner similar to Zhu et al. [4] and Sullivan et al. [12], by applying the minimum potential energy principle to the unit cell deformation. First, the equations for the shear compliances are derived as a function of three unit cell dimensions and the section properties of the cell edges. From the compliance equations, the equations for the shear moduli are derived and rewritten as a function of the two unit cell shape parameters and the foam relative density. In the final section, the shear moduli are plotted as a function of the relative density and the two shape parameters to demonstrate their dependence on these three quantities. 2. Description of the Kelvin unit cell The general elongated Kelvin unit cell is a fourteen-sided polyhedron (tetrakaidecahedron) constructed from eight hexagonal faces, four vertical rhombic faces and two horizontal square faces (Fig. 1). It has height H and width D. The vertical rhombic faces have sides of length L and the horizontal square faces have sides of length b. The hexagonal faces have four sides of length L and two sides of length b. The inclination angle h defines the orientation of the hexagonal faces with respect to the rise direction as well as the obtuse angle of the vertical diamond faces, 2h. In open-celled foams, all the solid mass is concentrated within the cell edges. The edges are the line segments formed by the intersection of multiple faces. Thus, the edges will be of length L or b. The edges are assumed to behave like struts, possessing axial, flexural and torsional rigidity. They have cross-sectional area A, bending moment of inertia I and torsion constant J. The Young’s modulus of the solid material is denoted as E and the shear modulus is G.
Fig. 1. Sketch of an elongated tetrakaidecahedron unit cell (Kelvin model).
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The size and shape of the general elongated Kelvin cell is defined by specifying any three of the five unit cell dimensions shown in Fig. 1. The cell height H and cell width D are related to the edge lengths L and b and the inclination angle h according to
H ¼ 4L sin h and D ¼ 2L cos h þ
pffiffiffi 2b:
ð1Þ
Alternatively, the size and shape of the Kelvin cell may be defined by specifying the value of either the cell height or the cell width and by specifying the value of two independent shape parameters R and Q. The parameter R is known as the aspect ratio of the cell. It is simply the ratio of the cell height to the cell width, R = H/D. Using the expressions in (1), the aspect ratio may be written as
R¼
4L sin h pffiffiffi : 2b
2L cos h þ
ð2Þ
The parameter Q was introduced by Sullivan et al. [12] and defined as Q = b/Lcosh. From Fig. 1 and the expression for D in Eq. it is easily recognized that the cell width is the sum of two dimensions: the diagonal of the horizontal square face pffiffiffi(1), 2b and two times the projection of a L-length edge onto a plane perpendicular to the rise direction (2Lcosh). The parameter Q is a measure of the ratio of these two dimensions. To illustrate the effect of the shape parameters, Kelvin unit cells with various combinations of values for R and Q are shown in Fig. 2. of the previous studies which used an elongated pMany ffiffiffi Kelvin foam model restricted the unit cell shape by specifying Q ¼ 2. Under this restriction, the diagonal of the horizontal square face is always one-half the cell width. The relative density c is the ratio of the foam density to the density of the solid material, c = q/qs. It may also be written in terms of volumes as c = Vs/V, where Vs is the volume occupied by solid matter and V is the total volume. The edges that form the perimeter of the vertical rhombic faces and those of length b that form the perimeter of the horizontal square faces are shared by adjacent cells. Thus, they contribute only half their cross-sectional area to the repeating unit. Assuming the crosssectional area is the same for all edges and constant along the edge length, then Vs = 16AL + 8Ab and thus
c¼
8Að2L þ bÞ HD2
:
Fig. 2. Sketches of various Kelvin unit cells illustrating the effect of the parameters R and Q on the unit cell shape.
ð3Þ
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993
3. Derivation of the shear response 3.1. In-plane shear compliance In order to derive the equation for the foam in-plane shear modulus Gxy, we take advantage of the symmetry of the Kelvin unit cell and adopt the structural unit shown in red in Fig. 3 as the representative repeating unit. This repeating unit is a onequarter segment of the Kelvin foam unit cell shown in Fig. 1. We establish a Cartesian coordinate system xyz such that the zaxis is oriented in the elongation or rise direction (Fig. 3). The x-axis is parallel to a line normal to two of the vertical rhombic faces, and the y-axis is parallel to a line normal to the other two vertical rhombic faces. We also establish the xsysz coordinate system that is oriented with respect to the xyz system by a 45° counterclockwise rotation about the z-axis. The repeating structural unit is bounded by six symmetry planes: two planes parallel to the ysz plane, two planes parallel to the xsz plane, and two planes parallel to the xsys plane. Since the members AN, AO, BJ, BK, DM, DL, EP and EQ all lie within the xsys symmetry planes, they are shared by two adjacent unit cells. As such, we assume that within the unit cell, they possess a cross-sectional area A/2 and a bending moment of inertia of I/2. To induce a state of pure shear stress sxy and pure shear strain cxy in the unit cell, we apply the normal stresses s and s in the xs and ys directions, respectively, as shown in Fig. 4. We define the displacements of the unit cell with respect to the unit cell center at point C and assume that all displacements and rotations at point C are zero. The applied loading results in a unit cell displacement of 2u in the xs direction and 2u in the ys direction, where both displacements are symmetric about point C. There is no displacement of the unit cell in the z-direction (Fig. 4). Thus, the work done on the unit cell by the applied loading is pffiffiffi pffiffiffi pffiffiffi W ¼ 8L sin h 2L cos h þ b su. Furthermore, the principal strains are 2u=ð 2L cos h þ bÞ and 2u= 2L cos h þ b , so the pffiffiffi shear strain cxy is 4u= 2L cos h þ b . Any edge that is initially normal to a symmetry plane boundary must remain normal to it after the deformation. Thus, the points J, K, L, M, N, O, P, and Q do not rotate during the deformation. Furthermore, the half-edges AN, AO, BJ, BK, DL, DM, EQ, and EP must remain in their respective symmetry planes. Thus, the points A, B, D, and E translate within their respective symmetry planes and the only rotation of these points is about the z-axis. Points A and B undergo a clockwise rotation x about the z-axis, and points D and E undergo a counterclockwise rotation x. Since there are no stresses applied to the unit cell in the y direction, members AC and BC will experience no axial deformation. Therefore, points A and B will only translate in the x-direction. Point B translates in the x-direction by an amount v and point A translates by an amount v. Likewise, points D and E translate in the y-direction by an amount v and v, respectively. We note that as a result of similarity and symmetry, the deformed shape and the deformation energy of member BC are the same as those of members CD, AC, and CE. Likewise, the deformed shape and deformation energy of the members BK, DL, AN, and EP are equivalent and those of members BJ, DM, AO, and EQ are equivalent. Furthermore, the axial tension in member BK is equal to the axial compression in member BJ, and the bending moment in BJ is equal to the bending moment in BK. The deformation energy of this one-quarter unit cell is thus
U ¼ 4U BC þ 8U BK : As a result of the loading and symmetry boundary conditions imposed on the one-quarter unit cell, member BC experiences torsional and bending deformation and member BK experiences axial and bending deformation. The strain energy in members BC and BK are written in terms of the unit cell displacements u and v, and the rotation x as
Fig. 3. Sketch of the one-quarter model showing its reference with respect to the elongated Kelvin unit cell (point Q not shown; see Fig. 4).
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(a)
ys
τ u
y u
Q
B
E
P
τ
J K
ω
xs
C
ω N
τ
L
A
u
D
O
M
τ
x z
(b)
2
u
τ
O
A
B
J
τ
C
ys M
D
E
Q
Fig. 4. Sketch of the applied loading and deformed shape of the unit cell for derivation of the in-plane shear modulus Gxy. a) xsys plane. b) ysz plane.
GJ 2 EI x sin2 h þ 3 2L2 x2 cos2 h þ 6v 2 6v Lx cos h ; 2L L 2 EA v x2 ¼ u pffiffiffi þ EI : 2b 2b 2
U BC ¼ U BK
Applying the minimum potential energy theorem to the unit cell deformation, in the form of @(U W)/@u = @ (U W)/ @ v = @(U W)/@ x = 0, results in three independent equations written in terms of u, v, and x. The simultaneous solution of these three equations yields the expressions for u, v, and x. The expression for u is obtained as
3 2 GJ 2 3 2 L 2L þ 4b cos h þ h b sin pffiffiffi 6 b 7 EI 7 u ¼ sL sin h 2L cos h þ b 6 4EA þ 5: GJ 2 12EI 2L þ b cos2 h þ b sin h EI
pffiffiffi Since the shear strain is cxy ¼ 4u= 2L cos h þ b and since s = Gxycxy, then
h i9 8 2
ð4Þ
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z (rise direction)
τ
B
z (rise direction)
R
B H
C
S T′
T F
D′
E′
F Ω
C′
F ′ D′
τ
y
G′
y H′
G′
x
F′
D
τ
C′
G E
D
4L sin θ
B′
G
C
B′
H
R′ H′
τ 2 L cos θ + 2b
Fig. 5. Sketch of the Kelvin unit cell and the shear stress tractions applied to the unit cell boundaries for calculating the out-of-plane shear modulus Gyz.
3.2. Out-of-plane shear compliance Since the Kelvin unit cell is elongated in the z-direction, it is not possible to determine the shear modulus Gyz in the same manner in which the in-plane modulus Gxy was determined. We can not induce a state of pure strain cyz by applying a set of equal and opposite normal stresses to the unit cell in the yz plane at 45° from the y and z axes. Instead, we will use the full Kelvin unit cell and apply shear stress tractions directly to the unit cell boundaries as shown in Fig. 5. A shear stress s is applied in the +y direction along the top boundary plane (+z plane) and in the y direction along the bottom boundary plane (z plane). A shear stress s is also applied on the left and right boundary planes in the z and +z directions, respectively. The deformation of the unit cell that results from the shear tractions is shown in Fig. 6. The unit cell displacements are defined with respect to the point X, which is located at the center of the Kelvin unit cell. The points that lie on the left and right boundary planes and located at the mid-height of the unit cell (Points D, D0 , E, E0 ) do not rotate. They translate only in the z-direction. The points that lie on the top and bottom boundary planes and located at the mid-width of the unit cell (Points H, H0 , R, R0 ) also do not rotate. They translate only in the y-direction. All other points are free to translate in the yand z-directions. In addition, all other points experience either a clockwise rotation x1 or a counterclockwise rotation x2 about the x axis as illustrated in Fig. 6. There is no displacement of any of the points in the unit cell in the x-direction and there is no rotation of any of the points about the y or z axes. The deformed unit cell shown in Fig. 6 is in a state of pure shear strain cyz. The pure shear strain cyz is the sum of two simple shear strains, cyz = c0 + c00 , where c0 – c00 due to the elongated unit cell. From the deform shape shown in Fig. 6, it is easy to see that c0 = 2wD/D and c00 = 2vH/H, where vH is the y-direction displacement of Point H and wD is the z-direction displacement of Point D. In addition, we note that the work performed on the unit cell by the applied shear tractions is W = 2sD(HwD + DvH). Due to their similarity and the equivalence of their end point displacements, the L-length members that are aligned parallel to the xz plane, such as members CD, C0 D0 , GH and RS, will all have the same deformation energy. Likewise, all the L-length members that are aligned parallel to the yz plane (BC, B0 C0 , FG, GF0 , and ST) will have the same deformation energy and all the b-length members will have the same deformation energy. In addition, it should be noted that all members that lie along the unit cell boundary planes will contribute only half their deformation energy to the total deformation energy of the unit cell. Thus, the total deformation energy of the Kelvin unit cell U may be written
U ¼ 8ðU BC þ U BH þ U GH Þ: Note that member BC has axial and bending deformation energy, whereas members BH and GH have torsion and bending deformation energy. Since members CD and BH do not deform axially, vB = vH and wC = wD. Recognizing also that vC = vH vG, the strain energies UBC, UBH and UGH may be written in terms of the rotations x1 and x2, and the displacements vC, vH, wB and wD as
U BC ¼
EA 6EI 2 2 ½ðv H v c Þ cos h þ ðwD wB Þ sin h þ 3 ½ðwD wB Þ cos h ðv H v c Þ sin h 2L L þ
6EI L2
ðx1 x2 Þ½ðwD wB Þ cos h ðv H v c Þ sin h þ
2EI 2 ðx1 x1 x2 þ x22 Þ; L
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z τ ω2
H
H
B G
C
ω2
C′
ω1
ω1
ω1 ω2 ω2
Ω
wD
D′
γ′
τ
y
F
D
ω1
τ
ω1 ω1 ω2 H′
ω2
τ γ ′′
Fig. 6. Deformed shape of the Kelvin unit cell resulting from the out-of-plane shear stress.
pffiffiffi GJ 2 EI x þ ð6w2B 3 2bwB x2 þ b2 x22 Þ; 4b 2 b3 GJ 2EI 2 ¼ x21 cos2 h þ 3 3v 2C 3v C Lx1 sin h þ L2 x21 sin h : 2L L
U BH ¼ U GH
Once again, we apply the minimum potential energy theorem to the unit cell deformation, this time with respect to the six unit cell rotations and displacements x1, x2, vC, vH, wB and wD, i.e. @(U W)/@ x1 = @(U W)/@ x2 = @(U W)/@ vc = = 0. This results in six independent equations written in terms of the two rotations and the four displacements. The simultaneous solution of these six equations yields the expressions for x1, x2, vC, vH, wB and wD. The displacements vH and wD may be written as
hpffiffiffi i 8 9 2 2 4
ð5aÞ hpffiffiffi i 8 2 9 2 2 4
9 8 h i2 pffiffiffi 2 > > b = 1 L
ð6Þ
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where
k1 ¼
DkA þ HkD 2L
k2 ¼
DkB þ HkE 2L
k3 ¼
DkC þ HkF : 2L
Note that when b = L and h = p/4, the unit cell is equi-axial. Under these conditions, both Eqs. (4) and (6) reduce to
pffiffiffi pffiffiffi 4 1 2 2L2 2L 8EI þ GJ ; ¼ þ G EA 6EI 5EI þ GJ which is the expression obtain by Zhu et al. [4] for isotropic open cell foams.
4. Expressions for the Shear Moduli in terms of R, Q and c We now seek to obtain expressions for the in-plane and out-of-plane shear modulus in terms of the relative density c and the two shape parameters R and Q. Using R = H/D and the expression for H in equation (1), equation (3) may be rearranged to obtain
A L2
3
¼
8c sin h ð2 þ b=LÞR2
ð7Þ
:
Defining the constant C1 as the ratio of the edge cross-sectional area to the square of the characteristic cross-section dimension, A = C1r2, we can rearrange equation (7) to obtain (r/L)2 = 8csin3h/(C1R2(2 + b/L)). Furthermore, defining the constant C2 through the relation 12I = C1C2 r4, we can also write
12I L4
6
¼
C 2 64c2 sin h : C 1 ð2 þ b=LÞ2 R4
ð8Þ
Table 1 Summary of the equations for the relative Young’s moduli, Poisson’s ratios and relative tensile strengths that result from the general Kelvin foam model written as a function of R, Q and c (from Sullivan, et al. [14]). e 5 RT 8Q Relative Young’s modulus
Ex Ey ¼ E E
ð2Q þ TÞ2
e 3 Rð8 þ QTÞ 8Q c þ C 1 Tð32Q 3 þ Qe 2 R2 TÞ C2 ð2Q þ TÞ e 5 R3 T Q
Ez E
ð2Q þ TÞ2
c2
e 5 R3 Q c þ 4 C1 T 2 C2 ð2Q þ TÞ 8Q 4C 1 Q 2
Poisson’s Ratio
mxy
mxz = myz
mzx = mzy
rsxx rsyy ¼ s r rs
rszz rs
e 3R C2 Q c 2Q þ T
!
e3 e 2 R2 TÞC 1 þ 8C 2 Q Rð8 þ QTÞ c ð32Q 3 þ Q Tð2Q þ TÞ ! e3 e TC 1 4C 2 Q R c 4Q Tð2Q þ TÞ e3 e 2 R2 TÞC 1 þ 8C 2 Q Rð8 þ QTÞ c ð32Q 3 þ Q Tð2Q þ TÞ ! e3 e R2 C 1 4C 2 Q R c Q 2 T ð2Q þ TÞ 8C 1 þ
Relative tensile strength
c2
e 5 R3 2C 2 Q T 2 ð2Q þ TÞ
c
e 2:5 R1:5 c1:5 4C 3 Q Rð2Q þ TÞ1:5
e 1:5 0:5 0:5 pffiffiffiffiffiffi e R þ 16C 3 Q R c C1 Q Tð2Q þ TÞ0:5
e 2:5 R1:5 c1:5 C3 Q ð2Q þ TÞ1:5
e 2:5 R1:5 c0:5 pffiffiffiffiffiffi C 3 Q C1 þ Tð2Q þ TÞ0:5
!
!
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pffiffiffi Eq. (2) may be combined with the expression Q = b/Lcosh to obtain tan h ¼ 2 þ 2Q R=4. This leads to
4 cos h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 16 þ Q
eR Q sin h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 16 þ Q
pffiffiffi e ¼ 2 þ 2Q . where Q Also, if the solid material is isotropic, we have
b 4Q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; L e 2 R2 16 þ Q
bJ GJ ¼ ; EI 2ð1 þ mÞ
ð9Þ
ð10Þ
where m is the Poisson’s ratio of the solid material and bJ ¼ J=I. The values of J and I are given in Mills [13] for a variety of cross-section shapes. Using (10) and substituting the expressions in (9) into equations (4), (6), (7) and (8) leads to
e 5R 4Q Gxy ¼ E
ð2Q þ TÞ2 e 3R 16Q Q C c þ 1 T3 ð2Q þ TÞ C2
c2
! bJ 1þm ! b e 2 R2 J T 3 þ 32Q þ Q Q 1þm
e 2 R2 T 3 þ 128Q þ Q Q
Fig. 7. Relative shear moduli Gxy/E and Gyz/E plotted versus aspect ratio R for various values of Q. (a) c = 0.01. (b) c = 0.05.
ð11Þ
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Fig. 8. Shear modulus ratio versus aspect ratio R for various values of Q and two relative densities: 1% and 5%.
and
e 7 R3 T 8Q ð2Q þ TÞ2 2
Gyz ¼ E e 5 Rð4 þ Q e R2 Þ2 4Q cþ ð2Q þ TÞ
4C 1 T 4C1 2 C 2 48T
bJ
c2 þ C2
1þm
!2
1þm
!2
bJ
þ ð128Q þ 2T 3 Þ
bJ 1þm
bJ 1þm
;
3
ð12Þ
þ C3 5 3
e 2 R2 T þ 16Q Q e 2 R2 Þ5 þ ð2T 3 þ 2 Q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 R2 . Expressions for C1, C2 and C3 in terms of R; Q ; Q e and T are listed in Appendix B. where T ¼ 16 þ Q For the sake of completeness, the equations for the relative Young’s moduli, Poisson’s ratios and relative tensile strengths which result from the general Kelvin foam model are listed in Table 1. These equations were presented in Sullivan, et al. [14]. Note that the expressions for the relative shear moduli in Eqs. (11) and (12) as well as the equations for the relative Young’s moduli in Table 1 are written in the form f1c2/(f2c + f3) where f1 and f2 are functions of the cell shape parameters and f3 is a function of the shape parameters and the edge cross-section properties. This same functional relationship between the relative moduli and the relative density has been proposed by previous authors (e.g. Refs. [4,7]). 5. Results The variation of the relative shear moduli Gxy/E and Gyz/E with the two shape parameters and the relative density c is illustrated in Fig. 7. The relative shear moduli are plotted as a function of R and Q for a relative density of 1% in Fig. 7a and for a relative density of 5% in Fig. 7b. In both plots, three-cusp hypocycloid (Plateau border) cross-sections were assumed ðbJ ¼ 0:6Þ. We also assumed that m = 0.3. Note the strong dependence of the relative shear moduli on the relative density. Note also that, for a constant relative density, the out-of-plane relative shear modulus Gyz/E appears to be a stronger function of the shape parameter Q than the shape parameter R, whereas the in-plane relative shear modulus Gxy/E is a stronger function of R than Q. The shear modulus ratio Gyz/Gxy is plotted in Fig. 8 as a function of R, Q and c. Again, Plateau border cross-sections were assumed. The shear modulus ratio does not appear to be a strong function of the relative density. It is an increasing function of the aspect ratio R and a decreasing function of the shape parameter Q. In contrast, the Young’s modulus ratio Ez /Ey and the tensile strength ratio rszz =rsyy were both shown, in Sullivan et al. [12], to be increasing functions of Q. Finally, we note that although there is an infinite number of combinations of R and Q in which the material is isotropic in its shear response (Gxy = Gyz), pffiffiffithere is only one combination of R and Q which results in both Ez/Ey = 1 and Gyz/Gxy = 1. This occurs when R = 1 and Q ¼ 2. 6. Concluding remarks The equations for the shear moduli have been derived for non-isotropic, open-cell foams using a general elongated Kelvin model. This elongated Kelvin model is more general than that employed by previous authors as it uses a unit cell with three
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independent dimensions. As such, this model accounts for all shape variations which are possible with a fourteen-sided polyhedron unit cell. The equations for the shear modulus in the plane perpendicular to the rise direction and in a plane parallel to the rise direction have been derived as a function of the relative density and two unit cell shape factors: the aspect ratio R and an additional shape factor Q, which was introduced by the authors in a previous publication. The dependence of the shear moduli on the relative density and the two shape factors has been demonstrated. The inplane shear modulus appears to be a stronger function of the aspect ratio than the shape factor Q, whereas the out-of-plane shear modulus is a stronger function of Q than the aspect ratio. Both shear moduli are a strong function of the relative density. Acknowledgement The authors are grateful for funding from the External Tank Project under NASA’s Space Shuttle Program. Appendix A. Expressions for kA, kB, kC, kD, kE, and kF in terms of h and b/L
pffiffiffi pffiffiffi b 2 2 2 2 kA ¼ 2 cos3 hð1 þ sin hÞ þ ð16 cos3 h sin h þ 2 cos4 h 48 cos h sin h þ 16 2 sin hÞ L 2 pffiffiffi b 2 2 þ 4 2 sin hð4 þ cos hÞ ; L 2 h pffiffiffi ib pffiffiffi pffiffiffi b 2 2 2 kB ¼ 4 cos3 h þ 2 2 þ 14 2 sin h þ 4 cos3 hð1 4 sin hÞ þ 2 2 cos2 hðcos2 h sin hÞ ; L L pffiffiffi b 2 kC ¼ 2 cos5 h þ 2 cos2 hð1 þ sin hÞ ; L pffiffiffi b pffiffiffi 2 2 kD ¼ 2 cos h sin hð1 þ sin hÞ cos h sin h 8 2 40 cos h 2 cos2 h þ 16 cos3 h L 2 3 4 pffiffiffi b b b 2 2 3 þ 2 cos h sin hð10 sin h 6Þ 2 sin hð1 5 sin hÞ þ 8 sin h ; L L L 2 pffiffiffi b pffiffiffi b kE ¼ 4 cos2 h sin h þ 8 cos h sin h 2 cos3 h 2 þ 10 2 cos3 h sin h L L 3 4 b b þ 2 sin hð4 þ 3 cos2 hÞ þ 8 cos2 h sin h ; L L 3 pffiffiffi b b kF ¼ 2 cos4 h sin h 2 cos3 h sin h þ 4 cos2 h sin h : L L e and T Appendix B. Expressions for C1, C2 and C3 in terms of R; Q ; Q
h
pffiffiffi
i
e 2 R2 T 2 þ 128Q 3 Q e 2 R2 T; C1 ¼ 8ð1 þ Q 2 ÞT 4 þ 64Q ð2 2 QÞ þ 8 Q
pffiffiffi
e 2 R2 112Q 2 þ 32 2Q þ 16 T 4 64Q ð2 þ 7Q 2 6 Q e ÞT 3 C2 ¼ ð1 þ 8Q 2 ÞT 6 þ 32Q 3 T 5 þ 3 Q e ð16 þ Q 2 Q e R2 ÞT þ 2048Q 4 Q e 2 R2 ; þ 256Q Q pffiffiffi e 2 R2 60Q 2 þ 8 2Q T 4 2048Q ð1 þ Q 2 ÞT 3 C3 ¼ 2ð1 þ 8Q 2 ÞT 6 þ 128Qð1 þ Q 2 ÞT 5 þ 4 Q h pffiffiffi i e 2 R2 2 Q e 2 R2 þ 16Q 2 2 Q T 2 512Q Q e 2 R2 ð2Q 2 þ Q e ÞT þ 256Q 4 Q e 4 R4 : þQ References [1] [2] [3] [4]
A.N. Gent, A.G. Thomas, The deformation of foamed elastic materials, Journal of Applied Polymer Science 1 (1) (1959) 107–113. L.J. Gibson, M.F. Ashby, The mechanics of three-dimensional cellular materials, Proceedings of the Royal Society London A 382 (1982) 43–59. W. Thomson (Lord Kelvin), On the division of space with minimum partitional area, Philosophical Magazine 24 (1887) 503–514. H.X. Zhu, J.F. Knott, N.J. Mills, Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells, Journal of the Mechanics and Physics of Solids 45 (3) (1997) 319–343. [5] W.E. Warren, A.M. Kraynik, Linear elastic behavior of a low-density Kelvin foam with open cells, Journal of Applied Mechanics 64 (1997) 787–794. [6] A.G. Dement’ev, O.G. Tarakanov, Model analysis of the cellular structure of plastic foams of the polyurethane type, Mekhanika Polimerov. 5 (1970) 859– 865. Trans. Polymer Mechanics 6 (1970) 744–749. [7] L. Gong, S. Kyriakides, W.Y. Jang, Compressive response of open-celled foams, Part I: Morphology and elastic properties, International Journal of Solids and Structures 42 (2005) 1355–1379.
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[8] M. Ridha, V.P.W. Shim, L.M. Yang, An elongated tetrakaidecahedral cell model for fracture in rigid polyurethane foam, Key Engineering Materials 306– 308 (2006) 43–48. [9] H.X. Zhu, J.R. Hobdell, A.H. Windle, Effects of cell irregularity on the elastic properties of open-cell foams, Acta Material 48 (2000) 4893–4900. [10] A.P. Roberts, E.J. Garboczi, Elastic properties of model random three-dimensional open-cell solids, Journal of the Mechanics and Physics of Solids 50 (2002) 33–55. [11] S. Kanaun, O. Tkachenko, Representative volume element and effective elastic properties of open cell foam materials with random microstructures, Journal of Mechanics of Materials and Structures 2 (8) (2007) 1607–1628. [12] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, A general tetrakaidecahedron model for open-celled foams, International Journal of Solids and Structures 45 (2008) 1754–1765. [13] N.J. Mills, Micromechanics of polymeric foams, in: Third Nordic Meeting on Materials and Mechanics, Aalborg, Denmark, 2000, pp. 45–76. [14] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, Application of an elongated Kelvin model to space shuttle foams, AIAA Journal of Spacecraft and Rockets 46 (2) (2009) 411–418.