Theoretical evaluation of tensile and flexural moduli of structural polymer foams

Abstract

In this paper, the tensile and flexural behaviors of structural polymer foams with integrated skin layer are studied. Several models are reviewed to show the dependence of Young’s modulus of cellular solids on void fraction. Effective flexural modulus of structural foam is computed based on the Euler–Bernoulli beam theory and results are compared with experimental data available for uniform density core. The effects of skin thickness, core relative density, and density transition profile on the elastic modulus are investigated for symmetrical structural foams. Lower and upper bounds of effective moduli are obtained to evaluate the role of structural foams in increasing specific mechanical strength of polymeric structures. The optimum thickness of solid skin is then estimated in terms of relative density of cellular core to get maximum specific tensile or flexural modulus. Density profile of structural foams with graded density core is also optimized to find the most possible value of specific flexural modulus. The present study is expected to provide some useful information for evaluation and designing of structural foams with the aim of material saving and improvement in mechanical properties.

Keywords

Structural foam tensile modulus flexural modulus density gradient structural optimization

Introduction

Ever-increasing consumption of plastic materials has been persuading modern industries to reduce the weight of manufactured plastic and composite products. So, weight reduction is one of the primary challenges, faced by manufacturing technologies in various industries such as automotive, transportation, floating devices, aerospace, construction, packaging, sporting goods, and furniture. Weight reduction not only allows less consumption of fuel and energy but also promotes performance and duration of servicing, as well as conserves the natural resources. A variety of processing techniques have been developed for light-weighting plastic-made components. One way is structural foam molding, which produces parts with a solid integral skin and a cellular core. The general process to produce structural foams is the low-pressure injection molding, where a physical or chemical blowing agent is mixed with melted polymer.

The resin is shot into the mold, but not completely filled or packed out. The blowing agent then expands to fill the mold cavities. Several factors are involved in controlling the morphology of structural foams, such as type and concentration of blowing agent, melt extrusion temperature, melt viscosity and elasticity of the polymer, pressure profiles in the extruder and extrusion die, and solubility and diffusivity of gas in the polymer. This technology offers more advantages besides weight reduction, over the conventional high-pressure injection molding, such as elimination of sink marks that allows molding large complex shapes with wall thickness from about 0.150 to 0.500 inch, lower internal stresses, and less warpage as a result of reduced melt viscosity, high stiffness to weight ratio, increased chemical resistance, superior impact resistance, and improved thermal and acoustical properties. On the other hand, since outer surface of the material remains unfoamed, the final product would have an excellent surface finish as well as improved flexural properties due to its sandwich-like structure.

To understand thermal and mechanical properties of structural foams, it is essential to know not only the properties of base material and apparent density of part but also the morphological structure, such as thickness of solid skin and the transition in density and cell size from the skin to the core. So, since 1970, numerous experimental or theoretical studies have been carried out to bridge their mechanical properties, specifically flexural modulus to structural parameters.^1–6

The primary models for flexural modulus were the same as those for uniform density foams which related mechanical behavior to the bulk relative density (ratio of overall reduced density to density of the base material) only, such as the rule of mixture,⁷ the power-law relationship,^8,9 and micromechanical methods originally developed for reinforced composite materials.^5,10–12 More accurate predictions were obtained by treating structural foams as three-layer sandwich structures and taking into account the skin-core morphology as well as the core relative density. Gonzalez¹³ assumed that the beam stiffness can be obtained from the sum of the uniform core and unfoamed skins stiffness. The I-beam model of Hobbs¹⁴ was based on the assumption that the stiffness of structural foam beam is equal to that of a one-component equivalent beam.

To predict the behavior of structural foams with a higher accuracy, some authors have introduced the spatial change of foam core density from the skin to the core center. For example, modified I-beam model of Hobbs¹⁴ and I-beam model of Wasserstrass and Throne¹⁵ assumed a linear transition zone within the foam core. Barzegari and Rodrigue³ compared these and more complicated density profiles^16,17 with a new continuous and continuously differentiable profile and showed that their proposed model had minimum deviation from the experimental data. For the special case of asymmetric structural foams, where the skins have different thicknesses, all the studies conducted so far reveal that the apparent flexural modulus is higher when the thicker skin side is under loading.^18–21

This paper was conducted to evaluate and optimize the response of structural foams under tensile and flexural loadings using some of the most well-known theoretical and empirical models. Assuming uniform density distribution in the core region, effective flexural Young’s modulus of structural foam is computed based on the Euler–Bernoulli beam theory and results are compared with available experimental data. The skin thickness of structural foam beams with symmetric rectangular section is then optimized in order to get maximum effective tensile or flexural stiffness to weight ratio. Real structural foams with graded density core are also studied to find the optimum density profile that yields the most possible value of specific flexural modulus.

Models for uniform foams

A considerable number of theoretical and semi-theoretical approaches on predicting the modulus of uniform density foams have been reported in the literature.⁵ A category of models consider the polymer-gas system as an open network of interconnecting polymeric girders. Meinecke and Clark²² in a review article on the mechanical properties of foams concluded that the girder network models are only applicable to foams at density levels below those experienced in structural foam applications. The second category includes micromechanics-based approaches by assuming that particles (voids) have negligible mechanical properties and cannot support shear or tension/compression.^{5,10–12,23–25} They include semi-empirical models with arbitrary constants or expressions evaluated from experimental data and purely empirical relationships derived from a curve fit analysis of experimental data. The density-squared relationship^8,9 is of this last category. Some reliable models of the last two categories are reviewed in the remaining of this section.

Kerner model

Kerner²³ developed a model for a multi-particulate-reinforced composite based on the assumption that the reinforcing particles are spherical in shape, randomly distributed, and suspended in, and bonded to, a uniform elastic medium. Assuming the total change in volume in the composite will be the sum of the volume changes of its constituents, Kerner obtained bulk and shear moduli of a multi-phase composite. Modification of this model for uniform density foams results in the elastic modulus of foamed polymer as

E_{f} = \frac{2 (5 ν - 7) ρ_{r} E_{p}}{15 (1 - ρ_{r}) ν 2 + (12 - 2 ρ_{r}) ν + 13 ρ_{r} - 27}

(1)

where

ρ_{r}

is the foam relative density which is equal to the volume fraction of its solid phase.

Rusch models

Rusch^26,27 presented some relations for elastic properties of high-density foams in terms of volume fraction, based on the theoretical results of Kerner²³ and MacKenzie.²⁸

In the first model, air bubbles were assumed as reinforcing particles with zero shear modulus. Using an empirical relationship for the volume dependence of foam Poisson’s ratio, Rusch obtained Young’s modulus as a function of the volume fraction of polymer

E_{f} = \frac{12 ρ_{r} E_{p}}{23 - 11 ρ_{r}}

(2)

Rusch’s second model assumed that the shear modulus should approach zero as the volume fraction of polymer approaches zero. Using this approximation, Rusch determined Young’s modulus of a cellular solid as a function of its relative density

E_{f} = \frac{4 ρ_{r} (2 ρ_{r} + 1) E_{p}}{- 2 ρ_{r}^{2} + ρ_{r} + 13}

(3)

Ogorkiewicz and Sayigh model

Ogorkiewicz and Sayigh²⁹ modeled the foam structure as a hollow cube in a cube of base material. Assuming strains along any section normal to the direction of the tensile stresses are constant, Young’s modulus of uniform density foam was obtained as

E_{f} = E_{p} - \frac{(1 - ρ_{r}) E_{p}}{1 - (1 - ρ_{r}) (1 / 3) + (1 - ρ_{r}) (2 / 3)}

(4)

Lee and Westmann model

Based on the theoretical model of Lee and Westmann³⁰ for the case of a spherical gas-filled inclusion, Young‘s modulus is given by

E_{f} = \frac{2 ρ_{r} E_{p}}{- 3 (1 + ν) ρ_{r} + 3 ν + 5}

(5)

which was in general agreement with experimental data for gas-filled phenolic foam.

Square power-law model

Moore et al.³¹ compared experimental data of 14 different foams with Kerner’s,²³ Lederman’s,³² and Ogorkiewicz and Sayigh’s ²⁹ theoretical models. They found that the following empirical square relationship had a very good agreement with their experimental results for thermoplastic foams

E_{f} = ρ_{r}^{2} E_{p}

(6)

Properties of structural foams

For theoretical analysis, the structural foam can be considered as a three-layer sandwich panel having solid (unfoamed) skins on both sides and a foamed core section, as shown in Figure 1.

Figure 1.

Geometry of a structural foam beam with rectangular cross-section.

Tensile modulus

Assuming a general distribution function $E (y)$ for Young’s modulus of structural foam along its thickness, the average tensile modulus can be obtained from¹

E_{T} = \frac{\int_{0}^{h} E (y) d y}{h}

(7)

Flexural modulus

Following the Euler–Bernoulli beam theory for the general case of a nonhomogeneous prismatic bar with arbitrary cross-section under pure bending, Stokes³³ has defined the effective flexural modulus $E_{F}$ as that required to produce an equal bending moment for an equal deformation of a homogeneous beam of the same dimensions. Thus, in the case of rectangular cross-section with uniform thickness h we have

E_{F} = \frac{12 \int_{0}^{h} E (y) (y - y_{0}) 2 d y}{h 3}

(8)

where y₀ is the position of natural axis that can be obtained from

y_{0} = \frac{\int_{0}^{h} E (y) (y - h / 2) d y}{\int_{0}^{h} E (y) d y}

(9)

Properties of structural foams with uniform core

Tensile modulus

For the case of structural foams with uniform core of thickness $h_{c}$ and skin layers of equal thickness $h_{s}$ , equation (7) results

E_{T} = \frac{E_{c} h_{c} + 2 E_{s} h_{s}}{h_{c} + 2 h_{s}}

(10)

Specific tensile modulus is defined as the ratio of tensile modulus to overall relative density $ρ_{f}$ , which is readily obtained from the rule of mixtures

ρ_{f} = (ρ_{c} h_{c} + 2 h_{s}) / (h_{c} + 2 h_{s})

(11)

By defining thickness ratio $α = h_{c} / h_{s}$ the specific tensile modulus $E_{ST}$ will be obtained as

E_{ST} = \frac{E_{T}}{ρ_{f}} = \frac{E_{c} α + 2 E_{s}}{(ρ_{c} α + 2)}

(12)

$E_{ST}$ can be evaluated by letting $ρ_{c} = ρ_{r}$ , $E_{c} = E_{f}$ and $E_{s} = E_{p}$ .

Normalized specific tensile modulus $(E_{ST} / E_{s})$ computed by different models is presented in Table 1. The corresponding contour plots in terms of density ratio and thickness ratio are shown in Figure 2 for $ν = 0.3$ .

Figure 2.

Contours of normalized specific tensile modulus predicted by different models: Kerner model (a), Rusch’s first model (b), Rusch’s second model (c), Ogorkiewicz and Sayigh model (d), Lee and Westmann model (e), and square power-law model (f).

Table 1.

Normalized specific tensile modulus.

Model	$E_{ST} / E_{s}$
Kerner	$\frac{2 [(- 15 ν 2 + 5 α ν - 2 ν - 7 α + 13) ρ_{c} + 15 ν 2 + 12 ν - 27]}{(ρ_{c} α + 2) [(- 15 ν 2 - 2 ν + 13) ρ_{c} + 15 ν 2 + 12 ν - 27]}$
Rusch’s first model	$\frac{2 [(6 α - 11) ρ_{c} + 23]}{(ρ_{c} α + 2) [23 - 11 ρ_{c}]}$
Rusch’s second model	$\frac{2 [(4 α - 2) ρ_{c}^{2} + (2 α + 1) ρ_{c} + 13]}{(ρ_{c} α + 2) (- 2 ρ_{c}^{2} + ρ_{c} + 13)}$
Ogorkiewicz and Sayigh	$\frac{(α + 2) [ρ_{c} - (1 - ρ_{c}) (1 / 3) + (1 - ρ_{c}) (2 / 3)] + 2 (1 - ρ_{c})}{(ρ_{c} α + 2) [1 - (1 - ρ_{c}) (1 / 3) + (1 - ρ_{c}) (2 / 3)]}$
Lee and Westmann	$\frac{2 [(α - 3 ν - 3) ρ_{c} + 5 + 3 ν]}{(ρ_{c} α + 2) [5 + 3 ν - 3 (1 + ν) ρ_{c}]}$
Square power-law model	$\frac{α ρ_{c}^{2} + 2}{(ρ_{c} α + 2)}$

It is obvious that although all models possess similar behaviors, Ogorkiewicz and Sayigh model gives the maximum values for $E_{ST}$ , whereas square power-law model gives the minimum values among the other models.

Flexural modulus

For the structural foam with rectangular cross-section shown in Figure 1 the specific flexural modulus can be written as

E_{SF} = \frac{E_{F}}{ρ_{f}} = \frac{E_{c} α 3 + E_{s} [6 α (α + 2) + 8]}{(ρ_{c} α + 2) (α + 2) 3}

(13)

Normalized specific flexural modulus $(E_{SF} / E_{s})$ from each model is presented in Table 2. The contours of normalized flexural modulus for $ν = 0.3$ are shown in Figure 3.

Figure 3.

Contours of normalized specific flexural modulus predicted by different models: Kerner model (a), Rusch’s first model (b), Rusch’s second model (c), Ogorkiewicz and Sayigh model (d), Lee and Westmann model (e), and square power-law model (f).

Table 2.

Normalized specific flexural modulus.

Model	$E_{SF} / E_{s}$
Kerner	$\frac{[(15 ν 2 + 2 ν - 13) A + 2 (7 - 5 ν) α 3] ρ_{c} - (15 ν 2 + 12 ν - 27) A}{(ρ_{c} α + 2) (α + 2) 2 [(15 ν 2 + 2 ν - 13) ρ_{c} - 15 ν 2 - 12 ν + 27]}$
Rusch’s first model	$\frac{(12 α 3 - 11 A) ρ_{c} + 23 A}{(ρ_{c} α + 2) (α + 2) 2 (23 - 11 ρ_{c})}$
Rusch’s second model	$\frac{2 (- A + 4 α 3) ρ_{c}^{2} + (A + 4 α 3) ρ_{c} + 13 A}{(ρ_{c} α + 2) (α + 2) 2 (- 2 ρ_{c}^{2} + ρ_{c} + 13)}$
Ogorkiewicz and Sayigh	$\frac{[(1 - ρ_{c}) (2 / 3) - (1 - ρ_{c}) (1 / 3)] (α 3 - A) + ρ_{c} α 3 + A}{(ρ_{c} α + 2) (α + 2) 2 [1 - (1 - ρ_{c}) (1 / 3) + (1 - ρ_{c}) (2 / 3)]}$
Lee and Westmann	$\frac{[3 (1 + ν) A - 2 α 3)] ρ_{c} - (5 + 3 ν) A}{3 (ρ_{c} α + 2) (α + 2) 2 [(1 + ν) ρ_{c} - 5 - 3 ν)]}$
Square power-law model	$\frac{α 3 ρ_{c}^{2} + A}{(ρ_{c} α + 2) (α + 2) 2}$

$A = 2 (3 α 2 + 6 α + 4) .$

As observed, the results of different models for flexural modulus are closer to each other than those for the tensile modulus. This is because the coefficient of $E_{c}$ in equation (12) is $\frac{α}{(ρ_{c} α + 2)}$ , whereas the coefficient of $E_{c}$ in equation (13) is $\frac{α 3}{(ρ_{c} α + 2) (α + 2) 3}$ . Consequently, the effect of $E_{c}$ on the normalized specific flexural modulus of structural foam is less than that on the tensile modulus. This means that one has less concern about which model to select for computation of flexural modulus.

The predictions of different models are compared with available experimental data³⁴ and the results are reported in Table 3. As observed, the results of Lee and Westmann model are the closest ones to the experimental results in smaller densities, whereas the prediction of Ogorkiewicz and Sayigh model is the most accurate one in the largest density.

Table 3.

Experimental data and theoretical predictions for the normalized specific flexural modulus of structural foams.

Method	$ρ_{f}$ ( $ρ_{c}$ ,α)
0.791 (0.696,4.4)	0.813 (0.712,3.70)	0.845 (0.768,4.03)	0.859 (0.797,4.55)
Experiment³⁴	0.972	1.002	1.000	1.058
Kerner	1.072	1.079	1.041	1.033
Rusch’s first model	1.077	1.082	1.053	1.036
Rusch’s second model	1.068	1.076	1.048	1.031
Ogorkiewicz and Sayigh	1.104	1.105	1.076	1.059
Lee and Westmann	1.033	1.047	1.017	0.996
Square power-law model	1.052	1.064	1.039	1.022

Structural foam with maximum specific tensile modulus

The optimization problem is defined as finding the thickness ratio $α *$ and relative core density $ρ_{c}^{*}$ for which the ratio of tensile modulus to overall foam density gets maximized. The inequality constraints are $0 \leq α *$ and $0 \leq ρ_{c}^{*} \leq 1$ . Thus, the problem can be represented by

\partial (- E_{ST}) / \partial α |_{α = α *} = λ_{1}

\partial (- E_{ST}) / \partial ρ_{c} |_{ρ_{c} = ρ_{c}^{*}} = λ_{2} + λ_{3}

(14)

where

λ_{1}

λ_{2}

and

λ_{3}

are nonnegative Lagrange multipliers such that

λ_{1} α * = 0

λ_{2} ρ_{c}^{*} = 0

, and

λ_{3} (ρ_{c}^{*} - 1) = 0

Substituting Kerner formula from Table 1 gives the boundaries of problem domain as the global optimum, that is, $ρ_{c}^{*} = 0$ , $ρ_{c}^{*} = 1$ , and $α * = 0$ . This result can be confirmed according to Figure 2(a).

The second and third optimum conditions correspond to a solid beam with specific modulus of unity, whereas the first condition corresponds to a hollow-core panel. These conditions are found the same for other theoretical models, as can be seen in Figure 2(b) to (f). By directly substituting any of these optimum conditions in equation (12) one obtains

(\frac{E T}{ρ f}) max = E s

(15)

This means that the specific tensile modulus of structural foam does not exceed that of a solid beam of the same material. As can be seen in Figure 2(a) to (f), for the case of hollow-core panel, even the thickness of its skins or the distance between them does not impact on its specific tensile modulus.

Structural foam with minimum specific tensile modulus

To obtain the minimum condition for $E_{T}$ it is more convenient to be stated in terms of β which is equal to $1 / α$

E_{ST} = \frac{E_{c} + 2 E_{s} β [3 + 6 β + 4 β 2]}{(ρ_{c} + 2 β) (1 + 2 β) 2}

(16)

Thus, the optimization problem is defined as finding $β *$ and relative density $ρ_{c}^{*}$ for which the ratio of tensile modulus to overall foam density gets minimized. The inequality constraints are $0 \leq β *$ and $0 \leq ρ_{c}^{*} \leq 1$ . Thus, the problem can be represented by

\partial (E_{ST}) / \partial β |_{β = β *} = λ_{1} \partial (E_{ST}) / \partial ρ_{c} |_{ρ_{c} = ρ_{c}^{*}} = λ_{2} + λ_{3}

(17)

where

λ_{1}

λ_{2},

and

λ_{3}

are nonnegative Lagrange multipliers such that

λ_{1} β * = 0

λ_{2} ρ_{c}^{*} = 0

, and

λ_{3} (ρ_{c}^{*} - 1) = 0

By applying Kerner formula from Table 1, the optimal solution can be found at $λ_{3} = 0$ , $β * = 0$ , and $ρ_{c}^{*} = 0$ . These conditions are found identical for other theoretical models. Therefore, by substituting $β = 0$ in equation (16) and finding the limit of the result as $ρ_{c}$ approaches zero, the optimal solution can be obtained as

(E ST) min = lim ρ c \to 0 \frac{E c (ρ c)}{ρ c}

(18)

Minimum values of normalized specific tensile modulus are presented in Table 4. As observed, the power-law model results in zero modulus to weight ratio that seems unrealistic. Figure 4 shows the variation of normalized specific tensile modulus in terms of

ρ_{c}

β = 0

Figure 4.

Diagram of normalized specific tensile modulus vs. foam core density at $β = 0$ .

Table 4.

Minimum value of specific tensile modulus.

Model	$E_{SF} / E_{s}$
Kerner	$\frac{2 (7 - 5 ν)}{(- 15 ν 2 - 12 ν + 27)}$
Rusch’s first model	$\frac{12}{23}$
Rusch’s second model	$\frac{4}{13}$
Ogorkiewicz and Sayigh	$\frac{2}{3}$
Lee and Westmann	$\frac{2}{(5 + 3 ν)}$
Square power-law model	0

Structural foam with maximum specific flexural modulus

Definition of the optimization problem is similar to that for tensile modulus, except that $E_{ST}$ in equation (14) is replaced by $E_{SF}$ . Substituting Kerner formula from Table 2 gives the global optimum at $λ_{1} = 0$ , $λ_{3} = 0$ , $ρ_{c}^{*} = 0$ , and $α * \to \infty$ . These conditions are the same for other theoretical models, as can be seen in Figure 3(b) and (f). Thus, by substituting $ρ_{c} = E_{c} = 0$ in equation (12) and then finding the limit of result as α approaches infinity we obtain

(\frac{E F}{ρ f}) max = 3 E s

(19)

This means that the specific flexural modulus of a hollow-core panel is three times that of a solid beam of the same material.

Structural foam with minimum specific flexural modulus

The optimization problem is defined in a similar manner to that for tensile modulus, except that $E_{ST}$ in equation (17) is replaced by $E_{SF}$ . Optimum conditions are the same as for tensile modulus which provides the same result as in equation (18), that is, the data on Table 4 are applicable for the flexural modulus, too.

Optimum skin thickness for maximum specific flexural modulus

The optimization problem is defined as finding the positive ratio $α *$ for a given $ρ_{c}$ for which the ratio of normalized flexural modulus $E_{SF}$ to overall foam density gets maximized. This requires that

\partial (E_{SF}) / \partial α * = 0

(20)

By substituting the foam core modulus from any of equations (1) to (6) one could analytically obtain the corresponding $α *$ . Figure 5 shows optimum values of α in terms of $ρ_{c}$ estimated from any of aforementioned models.

Figure 5.

Dependence of optimal core to skin ratio on relative core density estimated by different models.

Structural polymer foams with graded density

Density profile

Rodrigue³⁵ proposed a generalized Fourier series to predict the flexural modulus of symmetric structural polymer foams. He represented the density variation from the surface of the foam to its center in the following general form

ρ_{r} (ξ) = 1 - (1 - ρ_{0}) [\frac{1 + \cos (π ξ n)}{2}] m

(21)

where

ρ_{0}

is the relative density at the center of foam core and ξ is the dimensionless position related to distance x from the center of foam core

ξ = \frac{x}{(h_{c} / 2 + h_{s})}

(22)

Here, m, n, and $ρ_{0}$ are treated as fitting parameters which are to be adjusted, based on the experimental data.

Overall foam density $ρ_{f}$ can be obtained by the integration of equation (21)

ρ_{f} = 1 - (1 - ρ_{0}) \int_{0}^{1} [\frac{1 + \cos (π ξ n)}{2}] m d ξ

(23)

Optimum density profile

Assuming the square power-law relation between the local density and the flexural modulus, contours of normalized specific flexural modulus for different values of foam core density are given in Figure 6. As can be seen, there is a reverse relation between the maximum value of normalized specific flexural modulus and the value of relative density at the center of foam core.

Figure 6.

Contours of normalized specific flexural modulus for different values of foam core density: 0.2(a), 0.4(b), 0.6(c), and 0.8(d).

The optimization problem is defined as finding the values of m and n for a specific $ρ_{0}$ for which the specific flexural modulus get maximized. Since integral in equation (23) has no analytical solution, optimization needs to be carried out by any numerical method. The particle swarm optimization method, a nature-inspired evolutionary algorithm, is a suitable option for this purpose. The details of this method are presented by Clerc.³⁶ Optimization results show that the global optimum is obtained when m tends to infinity, which means an abrupt density change from $ρ_{0}$ to unity. The optimum values of n and corresponding values of foam density are listed in Table 5. Figure 7 shows optimum density profiles for different values of foam core density. With regard to abrupt change in density transition zone, the values of core to skin thickness ratio are estimated from density profiles and listed in Table 5. As expected, these values agree with the data in Figure 5. Figure 8 shows the maximum values of normalized specific flexural modulus in terms of $ρ_{0}$ .

Figure 7.

Optimum density profiles for different values of foam core density.

Figure 8.

Maximum values of normalized specific flexural modulus in terms of $ρ_{0}$ .

Table 5.

Optimum values of density profile parameters and corresponding foam relative density for a given core density.

$ρ_{0}$	n	$h_{c} / h_{s}$	$ρ_{f}$
0.1	60	5.7	0.34
0.2	44	3.8	0.47
0.3	37	2.9	0.58
0.4	32	2.5	0.67
0.5	28	2.2	0.74
0.6	26	1.9	0.80
0.7	24	1.7	0.86
0.8	23	1.6	0.91
0.9	22	1.5	0.96

Conclusion

A study was carried out on different models to predict the tensile and flexural moduli of structural polymer foams with uniform or graded density core. The effects of skin thickness, core void fraction, and transition profile on the flexural modulus were investigated for symmetrical structural foams. It was concluded that the dependence of both tensile and flexural moduli on the thickness ratio decreased with increasing relative density of foam core. On the other hand, the effect of relative density on tensile and flexural moduli decreased with increasing thickness ratio.

The structure of structural foam beams was optimized seeking the maximum specific tensile and flexural moduli. The results showed that the tensile modulus per weight of structural foam was at most equal to that of a solid beam made of the same material. Such case corresponded to a hollow-core panel, that is, structural foam with zero-density core. A hollow-core panel was also predicted to have the most amount of specific flexural modulus up to three times of that in a solid beam made of the same material.

Although the specific tensile modulus of hollow-core panel was shown to be independent of its skins thickness or the distance between them, its flexural modulus strongly depended on its cross-section geometry, so that the greater the distance between skins or the lower their thickness, the larger the specific flexural modulus. The minimum tensile and flexural moduli were found equal and observed when the relative core density was close to, but not exactly, zero and core to skin thickness ratio tends to infinity. In such case, every model had its own prediction of tensile or flexural modulus, but the power-low model seemed not to give a realistic result, since it predicted zero normalized specific modulus for vanished value of foam core density. The optimal skin thickness corresponding to maximum flexural modulus was computed and found to be in inverse relation with the foam core density. For structural polymer foams with graded density core, the optimum density profile concerned with maximum flexural modulus was obtained in the form of a step function. Although such abrupt change may not be provided in practice, it can give a reference to what the real density profile is desired to be. On the whole, the purpose of this study was to evaluate different theoretical and empirical models for estimating the tensile and flexural moduli of structural foams, as well as to predict the optimal structural foam with optimum specific mechanical properties. The present study is expected to guide the manufacturer toward the ideal foam structure, which can result in material saving and improvement in mechanical properties.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interest

None declared.

Appendix 1

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