Evaluation of the anisotropic mechanical properties of reinforced polyurethane foams
Introduction
Cellular materials are widely used in the energy and transport industries as lightweight structural materials, most notably as the core material in structurally-efficient sandwich panels. Even in nonstructural applications, like packaging or insulation, the mechanical performance and integrity of these materials can be critical. Reinforcing polymer foams with short-fiber or particulate additives is a potential route to improve the mechanical properties, and reduce the weight and cost of these materials. Polyurethane (PU) foams are excellent candidates for targeting mechanical improvement via reinforcement because the mechanical properties of PU foams are relatively poor, and yet the cost and availability compare favorably with alternative foams and natural products (e.g. polyvinylchloride foams and balsa wood).
The mechanical properties of cellular materials are highly dependent upon the cellular structure of the foam, as well as the properties of the solid material making up the foam, both of which may be influenced by reinforcing additives. One of the most important features of the cellular structure in terms of mechanical properties is the void fraction, which is typically characterized by the relative density (ρf/ρs) – defined as the ratio of the density of the foam to that of the bulk material of which the foam is constituted. In foams with a low relative density (ρf/ρs < 0.4), many of the mechanical properties can be related to the relative density according to a power law of the form [1]:where P is the mechanical property of interest, ρ is density, the parameters C and n depend on the property of interest and the particulars of the foam (including the foam microstructure and deformation mode) [2], [3], [4], and the subscripts s and f indicate the properties of the fully dense solid and of the foam, respectively.
The exponent, n, in Eq. (1) typically ranges from 1 < n < 2 for the elastic moduli. A value of n = 2 corresponds to a bending-dominated deformation mode, which is typical of open-cell foams with no cell walls. A value of n = 1 corresponds to stretch-dominated deformation, as might occur in a lattice with members oriented in the direction of loading. Intermediate values of n are typical in closed-cell foams, which have cell walls that undergo stretching and struts that undergo bending.
Another important attribute for the mechanical properties of cellular materials is the cell shape. In both synthetic and natural cellular materials it is typical for the cell shape to be elongated, leading to anisotropic material properties [1]. The cells of polymer foams tend to be elongated in the direction of foaming (also referred to as the foam rise direction), as shown in Fig. 1(a). Mechanical models based on an elongated unit cell have been developed to capture this anisotropic behavior. Huber and Gibson [5] considered a rectangular unit cell (Fig. 1(b)) with a cell shape anisotropy ratio, R, defined as:where h and l are the dimensions of the unit cell parallel and perpendicular to the direction of elongation, respectively, as shown in Fig. 1(a). This rectangular-cell model predicts transversely isotropic material properties that may be calculated using Eq. (1) with an additional term that is related to the shape anisotropy:where f(R) is one of several functions of the shape anisotropy ratio of the unit cell, which are tabulated for the moduli in different material directions in Table 1. According to this model, the mechanical properties increase in the direction of foaming (1) and decrease in all other directions as the shape anisotropy, R, increases. Using Eq. (3), the cell shape anisotropy may be calculated by taking the ratio of the foam moduli in different directions, for example:where E1, E2, and E3 are the foam moduli in the material directions defined in Fig. 1(a).
The tetrakaidecahedron introduced by Kelvin [6], shown in Fig. 1(c), is an alternative cell geometry that is a closer representation of the cellular structure observed in polymer foams than the rectangular cell of Huber & Gibson. In addition to the term R, the Kelvin cell requires a second term, Q, to uniquely define the geometry [7]. The impact of varying the parameter Q on the cell geometry is illustrated in Fig. 1(c). The full set of equations for the transversely isotropic material properties as functions of Ps, ρf, ρs, R, and Q are published elsewhere [7], [8]. The equivalent expression to Eq. (4) using this alternative cellular geometry is:where , , and for a hypocycloid cross-section [7]. Whereas the properties of the solid do not appear in Eq. (4), the relative density is included in Eq. (5).
Numerous studies have reported improvements in the mechanical properties of polymer foams reinforced with short fibers [9], [10], [11], [12], [13], particles [14], [15], [16], [17], [18], and nano-particles [19], [20], [21], [22], but relatively few have made use of cellular models to interpret the results of mechanical tests and to develop predictive tools. Barma et al. [15] related the foam stiffness (Ef) to the solid stiffness (Es) and cell size in particle-reinforced foams at the same density. Saint-Michel et al. [16] modeled reinforced foams with higher relative densities (ρf/ρs > 0.3) as a porous composite filled with closed, isolated, spherical voids. Zhang et al. [20] used a Mori-Tanaka model to account for carbon nanotube reinforcement and cellular voids [23]. Goods et al. [14] used a cellular model in the form of Eq. (1) to describe the foam modulus (Ef) of PU foams reinforced with metal particles, along with the Kerner equation to account for changes in the solid modulus (Es); others have taken a similar approach with various composite models to estimate Es for different materials [13], [17], [21], [22]. The effect of additives on mechanical anisotropy has been reported in several studies on chopped aramid and glass fibers [10], [11], [12], but was only qualitatively attributed to a combination of cell shape (R) and preferential fiber-alignment. Sorrentino et al. [18] reported mechanical anisotropy in foams reinforced with iron particles aligned in a magnetic field, which the authors attributed wholly to the reinforcement and not to cell shape. Nano-scale fillers are known to influence the foaming process by inducing bubble nucleation [19] and have been reported to affect the cell shape [24], yet despite the potential influence of nanoparticles on cellular structure, mechanical anisotropy is often overlooked in the analysis of reinforcement in nanocomposite foams.
A complete picture of the effect of reinforcement on the mechanical properties of composite foams requires mechanical characterization in multiple material directions, and the consideration of these factors related to cellular structure that may be affected by reinforcing additives. In this paper, low-density polyurethane foams (ρf/ρs < 0.2) are characterized in tension and compression in two principal material directions and in shear using a modified Arcan testing fixture [25]. Power-law relationships between the in-plane moduli and density are established for pure PU foams to compensate for density effects in the comparison between composite and pure PU foams. The modulus data are used in conjunction with cellular material models to make predictions about the cellular structure of the foams that are compared with microscopic observations. Changes in the moduli of composite foams are attributed to changes in the cellular structure (relative density, and cell shape) and to changes in the solid properties (ρs, Es), and a definition for the degree of foam reinforcement is proposed that is independent of cellular structure and that takes into account the tradeoff between stiffness and density in the mechanical performance of foams.
Section snippets
Foam preparation
Rigid, closed-cell PU foams and the precursor components for producing these foams (methylene diphenyl diisocyanate and polyol blends) were obtained from the industrial producer, Recticel1. Pure PU foams (with no particle reinforcement) were obtained from the manufacturer in a range of densities (ρf = 128.0, 153.8, 163.4 kg m−3), and were also produced in the lab under the same conditions as composite foams (ρf = 144.5 kg m−3).
Mechanical characterization of pure PU foams
Representative tensile, compressive, and shear stress–strain curves for pure PU foam are shown in Fig. 5 (ρf = 128.0 kg m−3). The tensile and compressive moduli in the rise direction of the foam are higher than those in the transverse direction, indicating orthotropic material properties due to an elongated cell shape in the rise direction. The moduli are plotted as a function of density for all pure PU foams in Fig. 6, and power law curves in the form of Eq. (3) are fitted to the experimental data.
Conclusions
Composite PU foams with glass-fibers and hybrid nano-particles were fully characterized in the plane parallel to foaming. The moduli of composite foams were normalized to the trends with density established for pure PU foams. The normalized moduli of composite foams in the foam rise direction increased by 4–26%, but increased less or even decreased by as much as 40% in the transverse direction. These divergent trends were explained by the increased cell shape anisotropy, which was predicted
Acknowledgments
This work was conducted and financially supported through EU FP7-NMP-2007-2.1-1 large scale integrated project ‘‘NanCore’’, Grant Agreement No. 214148, and the Department of Mechanical and Manufacturing Engineering, Aalborg University, Denmark. The authors gratefully acknowledge Thomas Sørensen Quaade for his technical assistance, especially with electron microscopy.
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