Theoretical and semi-empirical models

We first review a selection of results that are available for periodic cellular solids. The results illustrate the basic mechanisms of deformation, and provide benchmark tests for the FEM. These results, gathered from the literature, also provide useful comparisons with the FEM results for random models discussed in subsequent sections.

First consider a simple cubic array of uniformly spaced intersecting aligned struts. From elementary considerations, the Young's modulus along one of the principal axis is given by E /E_s ≈ 1/3 ( ρ / ρ _s ). The linear dependence of modulus on density is typical of model foams that contain 'straight-through' struts that traverse the extent of the sample); longitudinal compression or tension being the only mode of deformation [Christensen, 1986, Warren & Kraynik, 1988, Zhu et al., 1997].

Since most foams do not contain straight-through struts, beam bending comes into play [Ko, 1965, Gibson & Ashby, 1982, Warren & Kraynik, 1988, Zhu et al., 1997]. This is most easily seen in the unit cell of the simple open-cell model shown in Figure 1(b). The cell is a modification of the model shown in Figure 5.6 of Gibson and Ashby's  [Gibson, 1988] book, since the original model could not be periodically extended. The solid fraction is

(2)

where the approximation is for thin beams d w, L, and the length scale parameters are shown in Figure 1(a). Now consider the deformation of the cell if it is compressed along the x-axis (100 axis) by a force f. The Young's modulus is E =  (f / L² ) / ( δ L / L) where δ L is the total deformation. From thin beam theory [Landau & Lifshitz, 1959], the deformation of a thin beam of half-length wwith clamped ends due to an applied force f is $\delta L= f w^3/ 24 E_s I_m$ L = fw 3 /24E_sI _m. Here I_m is the principal moment of inertia, with I_m=d ⁴/12 for a square beam of width d. The deformation is halved because two beams intersect at the point of application, but as this occurs twice in the unit-cell the total deformation is the same as that for a single beam. Thus in the thin-beam limit we have,

$$\frac {E_{100}}{E_s} \approx \frac {2 d^4}{w^3 L} \approx C \... ...;\; C= 2 \frac{L^3}{w^3} \left( 3 + 18 \frac w L \right)^{-2}.$$ (3)

The quadratic dependence of the modulus on density is typical of foams where beam bending is the principle mechanism of deformation. Ko [Ko, 1965] demonstrated this behaviour for the model shown in Figure 1(c) loaded in the $\langle 111 \rangle$ direction. Note that the calculation leading to equation (3) is illustrative only. In reality, the beam ends are not clamped, so the pre-factor is only an approximation.

Figure 1: Periodic models of open-cell solids. (a) The parameters used to define a simple model. (b) A 3-D version of the simple model. (c) An open cell foam considered by Ko [Ko, 1965]. The foam is stressed in the $\langle 111 \rangle$ direction indicated. (d) A unit cell of the tetrakaidecahedral model

(a)

(b)

(c)

(d)

Zhu et al [Zhu et al., 1997] and Warren and Kraynik  [Warren & Kraynik, 1997] derived analytic results for the open cell tetrakaidecahedral model (Figure 1d) packed in a body-centred cubic array. The results provide a useful check of the FEM (see § 3), and demonstrate incompressible behaviour (ν     0.5) at low densities. The results of Zhu et al. for the Young's modulus and Poisson's ratio for strain parallel to an axis are,

$$\frac{E_{100}}{E_s}= \frac 23 C_Z \left(\frac\rho\rho_s\right... ...2 \left( \frac{1-C_Z(\rho/\rho_s)}{1+C_Z(\rho/\rho_s)} \right)$$ (4)

where $C_Z=8\sqrt{2}I/A^2$depends on the cross-sectional area A and the second moment of the area I. For equilateral triangles C_Z =1.09 [Zhu et al., 1997], and for cylindrical beams C_Z =0.900. Note that the Poisson's ratio depends on orientation. The notation ν ₁₂ corresponds to expansion measured in the $\langle 010 \rangle$ or $\langle 001 \rangle$ directions. As mentioned above, the foam is relatively stiff under uniform compression, with the bulk modulus given by K / E_s = 1/9 ρ / ρ_s.

We now review semi-empirical and analytic results for random foams. The most commonly used result for open-cell foams is [Gibson & Ashby, 1988]

$$\frac EE_s \approx C \left( \frac\rho\rho_s \right)^2;\;\; \nu \approx \frac13$$ (5)

where the pre-factor C ≈  1 and Poisson's ratio have been empirically determined. This semi-empirical formula broadly describes data obtained for many different types of foams.

There have also been several methods proposed to derive analytic predictions for isotropic foams. A typical result, which performs an isotropic average of randomly placed long thin (i.e. straight-through) struts, has been derived by Christensen (1986),

$$\frac EE_s = \frac16 \left( \frac\rho\rho_s \right);\;\; \nu = \frac14.$$ (6)

Christensen notes that the results are equivalent to those of Gent and Thomas [Gent & Thomas, 1963]. In the low density limit, the same results have been derived for a rotationally averaged simple cubic structure [Warren & Kraynik, 1988]. The absence of bending in these models is indicated by the linear dependence of the Young's modulus on density.

Warren and Kraynik [Warren & Kraynik, 1998] have derived analytic results for the properties of a foam comprised of isotropically oriented tetrahedrally arranged struts. The geometry can be visualised as a node located at the centre of a tetrahedron with equilateral faces, the four struts (separated by an angle of 109.5º) connecting the central node to the vertices. There are eight nodes of this type adjacent to the central node in Figure 1(c). The results are

$$\frac EE_s = \frac{C_W p^2(11+4 C_W p)}{10+31 C_W p+4 C_W^2 ... ... \frac 12 \frac{(1-C_Wp)(10+8C_Wp)}{10+31 C_W p + 4 C_W^2 p^2}$$ (7)

where p = ρ / ρ_s and $C_W=18I/\sqrt3A^2$. For struts of equilateral triangular cross-section C_W =1, while for a circular cross-section C_W ≈  0.827. As expected from the definition of the model, beam bending is the primary mode of deformation for uniaxial compression. However, equations (7) imply K / E_s = 1/9 p indicating that bending is not activated under pure compression. Like the tetrakaidecahedral model, the Poisson's ratio of the model therefore tends to 0.5 (indicating incompressible behaviour) at low densities.