Torsion Properties of High Density Polyethylene Foams YAOLIN ZHANG, DENIS RODRIGUE* AND ABDELLATIF A¨IT-KADI Department of Chemical Engineering and CERSIM Laval University, Quebec City, Canada G1K 7P4 ABSTRACT: High density closed-cell HDPE foams (450–950 kg/m3) were prepared by compression molding, and torsion rectangular tests were performed to measure their shear modulus in order to study: (1) the relationship between the modulus as a function of the density, and (2) to determine the effect of thin skins. Based on the assumption that the twist stiffness (product of shear modulus and moment of inertia) of the foams is the sum of the twist stiffness of the skin layers and the core part, several structural foam models are proposed. We found that structural foam models give better results than uniform foam models, indicating that thin skins have an important effect of the shear modulus of polymer foams. KEY WORDS: HDPE foams, torsion rectangular, shear modulus, sandwich structure.
INTRODUCTION
T
orsion properties of high-density polymer foams, like their flexural and tensile properties, are important for design purposes of structural materials, especially for long columnar parts. In previous studies [1–4], we first compared Young’s modulus with different models for uniform foams. It was found that the differential scheme and the simple square-power empirical equation [5] gave similar results and were best to predict the data in the range 0–55% of voids volume fraction. We also compared the flexural modulus and Young’s modulus of uniform and structural foams composed of a foamed core enclosed by *Author to whom correspondence should be addressed. E-mail:
[email protected]
JOURNAL OF CELLULAR PLASTICS Volume 39 — November 2003 0021-955X/03/06 0451–24 $10.00/0 DOI: 10.1177/002195503036234 ß 2003 Sage Publications
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unfoamed skins. It was found that very thin skins have a definite effect on flexural modulus that could be well represented by Gonzalez’s sandwich structure and Hobbs’ I-beam models. On the other hand, very thin skins were found to have little effect on tensile moduli. Based on the information gathered so far on the mechanical properties of polymer foams, a focus is made here on the shear modulus of polymer foams under small torsion rectangular deformation. It is also the objective of this paper to determine the effect of skin layers on the shear moduli of structural foams and to compare the measurements with models taken from the literature. We need to point out that very few reports on the torsion of polymer materials are available [6–10]. There is even less information when the polymer is foamed. Gibson and Ashby [11] developed a model for the shear modulus based on unit cells and micromechanical considerations for both open and closed-cell low-density materials. For high-density foams, Moore et al. [12] proposed an empirical expression for the shear modulus of polypropylene foams. Unfortunately, we did not find any models about the shear modulus of structural foams under torsion rectangular tests. Based on this limited amount of work available in the open literature, we propose here some models for the prediction of the shear modulus of structural foams. The models are finally compared with experimental data taken on high density HDPE foams prepared via a compression molding technique.
EXPERIMENTAL
Polymers and Sample Preparation Four HDPE with different melt index were used: J60-1700-173, A6070-162, and G60-110 are homopolymers from Solvay Polymers. HBW555-Ac is a high molecular weight polyethylene from Nova Chemicals. Their characteristics were given in the first part of this study [1]. The polyethylene-blowing agent compounds were blended using a laboratory internal mixer (Haake Rheomix) at 40–50 rpm and 150 C. A chemical blowing agent, azodicarbonamide (ACA, Sigma Chemicals), in concentrations between 1 and 3 wt.% was used. In each blend, 0.1 wt.% of an antioxidant (Irganox 1076) was added to prevent thermal oxidation during mixing and foaming processes. Foam plates with dimensions of 60 60 2.8–3.4 mm were obtained by a compression molding method. More details can be obtained in previous reports [1–4,13].
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Skin Thickness of the Foams The method is similar as previously used in an earlier study [4]. Two specimens were used to measure the skin thickness for each sample. A Spot-Insight digital camera and software from Diagnostic Instrument were used to take pictures via an Olympus SZ-6 stereomicroscope. Quantitative measurements were performed using Image-Pro Plus from Media Cybernetics. The procedure was: 1. Two photographs for skin layers were taken for each specimen. 2. The shortest distance from bubbles that are the closest to the skin from the surface was measured. 3. Averaging these distances of at least 20 bubbles to obtain the skin layer of each side of specimen. The standard deviation is also reported. 4. Divide the average skin thickness by the total thickness of each specimen to obtain the average ratio of skin to foam thickness. The results obtained were: 5.0 0.6%, 3.5 0.4%, 4.0 0.5%, and 3.8 0.5% respectively for J60-1700-173, A60-70-162, G60-110, and HBW555-Ac as previously reported (4). This gives a total average thickness of 4.0 0.5% for all foams. Torsion Measurements The samples were cut in rectangular shape as shown in Figure 1. Typical dimensions for length (L), width (B), and thickness (T) are 45, 8–10, and 2.0–3.4 mm, respectively [14]. Shear stress–shear strain curves were measured using a Rheometrics ARES rheometer with a transducer of 2000 g cm maximum torque. The following conditions were employed: room temperature, a shear strain rate of 0.004 s1, and a measurement time of 1 s.
Figure 1. Diagram of a torsion rectangular sample.
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SHEAR MODULUS FROM TORSION
The development of the models is presented in the Appendix. Shear modulus obtained by torsion rectangular (Equation (A-11)) is equivalent to the shear modulus of isotropic material by the elastic theory (Equation (A-7)). This means that if the beam is made of a material with uniform density and modulus, the modulus measured by torsion rectangular is the shear modulus, and if the beam is made of materials with different modulus (sandwich-like structure), the modulus measured with torsion test would be an average in relation with the structure of the beam.
SHEAR MODULUS MODELS FOR UNIFORM FOAMS
There are several parameters used in the models presented below, which are shear modulus of foam (f), shear modulus of matrix (m), elastic modulus of foam (Ef), elastic modulus of matrix (Em), foam density (f), matrix density (m), and void volume fraction ( f ). A detailed mathematical derivation is given in the Appendix. Gibson & Ashby Model From the Gibson & Ashby model proposed for low density closed cell foams [11], the shear modulus of uniform polyethylene foam can be obtained by: 2 f f 2 f ’ þ ð1 ’Þ ¼ ’2 ð1 f Þ2 þ ð1 ’Þ ð1 f Þ m m m
ð1Þ
where ’ is the fraction of solid material in the cell struts.
Differential Scheme for Two Phase Spherical Inclusion Composites McLaughlin [15] developed a differential scheme based on Boucher’s work for estimating the overall moduli of isotropic two-phase linearly elastic composites. Later, Farber and Farris [16] also developed a differential scheme using a different approach but obtained exactly the same differential equations for the shear and bulk moduli of a two-phase
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composite with spherical inclusions. With numerical integration and curve fitting, the shear modulus of uniform closed cell foam follows a power-law equation with normalized density in which the power index is related to the matrix Poisson ratio. For randomly oriented closed cell uniform polyethylene foams, the shear modulus is: 1:90 f f ¼ ð1 f Þ1:90 m m
ð2Þ
Moore Empirical Equation Moore et al. [12] found that shear moduli of polypropylene (Poisson ratio of 0.3 [17]) foams measured by the torsion of rectangular bars followed a square power-law relation with density as: 2 f f ¼ ð1 f Þ2 m m
ð3Þ
Modified Empirical Equation It has been shown in the past that an empirical square power-law equation between the normalized Young’s modulus and normalized density of polymer foams exists [5], and combine the Poisson ratio from differential scheme to obtain the shear modulus of uniform random oriented closed cell polyethylene foam as: 8 >
: 1:016 ð1 f Þ2
ð0 f 0:12Þ ð4Þ ð0:12 f 1Þ
SHEAR MODULUS MODELS FOR STRUCTURAL FOAMS
To determine the mechanical properties of structural foams, it is necessary to take into account the effect produced by the nonuniform density across the foam section: i.e. skin and core density. This is important because foams having the same overall density will have different mechanical properties based on the density difference between
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the skin and core, skin thickness, and other morphological parameters like cell size, cell density, and cell distribution. For theoretical calculations, Gonzalez [18] assumed that the transition zone in structural foams, between the skin layer and core zone, is much smaller than the core zone. A structural foam structure is simplified as a three-layer panel like a sandwich structure and assumed that the bending stiffness (product of elastic modulus by moment of inertia) of such a beam is the sum of the bending stiffness of the core and the skins. Several expressions of average shear modulus of structural foam were proposed based on similar assumption as in Gonzalez approach: the twist stiffness of structural foams (product of shear modulus by moment of inertia) is the sum of the twist stiffness of the core and the skins. Several parameters were used in structural models, which are skin thickness of each side (s), core part thickness (c), structural foam thickness (f ¼ c þ 2s), shear modulus of structural foam (SF), shear modulus of matrix (m), and void volume fraction ( f ). (1) Model I: Combination for Sandwich Structure and the Differential Scheme The average shear modulus of polyethylene structural foam is: 3 3 SFI c c f 1:90 ¼1 þ 1 f m f f c
ð5Þ
(2) Model II: Combination of Sandwich Structure and Moore Empirical Equation 3 3 SFII c c f 2 ¼1 þ 1 f m f f c
ð6Þ
(3) Model III: Combination of Sandwich Structure and Modified Empirical Equation The average shear modulus of polyethylene structural foam is: 8 3 3 c 1:34 c f 2 > > > 1 þ 1 f SFIII < 1:167 þ 0:173ð1f Þ f f c ¼ > m 3 3 > > ;f f 2 : 1 c þ 1:016 c 1 f
f
c
0 f f 0:12 c
0:12 f f 1 c
ð7Þ
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RESULTS AND DISCUSSION
Thickness of HDPE Foam Skins With a compression molding process, the foams usually have a very thin skin layer on each side. An average thickness of 4.0 0.5% for all foams was obtained giving an average relative core thickness (c/f) of 96.0 0.5%. Torsion Properties The shear modulus of HDPE foams as a function of density is shown in Figure 2. Because the foam modulus is related to the matrix modulus, a normalized modulus (ratio of foam modulus to unfoamed polymer modulus) and normalized density (ratio of foam density to unfoamed polymer density) are used in order to eliminate the relative effect of the unfoamed polymer matrix on the foam. A plot of the normalized data is shown in Figure 3 and the transformation produces a single curve. Comparison of Shear Models All the models described contain simplifying assumptions on the structure and the distribution of the voids that are most likely unrealistic. Their validity must be judged, at least partially, in terms of how closely they can predict the experimental data.
Figure 2. Shear modulus as a function of density.
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Figure 3. Normalized shear modulus as a function of normalized density.
Figure 4. Comparison of the normalized shear modulus as function of normalized density for uniform foam models: : Gibson & Ashby model (’ ¼ 0.8), : Gibson & Ashby model (’ ¼ 0.9); ——: Differential scheme; — —: Moore’s empirical equation and Gibson & Ashby model (’ ¼ 1.0), and — —: Modified empirical equation.
Comparison with Uniform Foam Models Figure 4 shows a comparison of the normalized modulus as function of the normalized density for uniform foam models. It can be seen that these predictions seem to underestimate the experimental data. For the Gibson & Ashby model, one needs to determine the value of the parameter ’. In their book, the value of ’ for low density closed cell foam
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High Density Polyethylene Foams Table 1. Average deviation as a function of ’. ’ 0.80 0.82 0.84 0.86 0.88 0.90 0.91 0.92
Deviation (%)
’
Deviation (%)
18.9 18.6 18.3 17.8 17.4 16.9 16.6 16.3
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
16.0 15.7 15.4 15.1 14.8 14.5 14.2 13.9
was around 0.8–0.9. For high density polyethylene foam, ’ needs to be determined. When the normalized density (f /m) reaches unity, the normalized modulus reaches 1 ’ þ ’2 . This gives f less than m when ’ is less than 1. For high density closed cell foam, the expansion does not extend up to the wall formation, and it is possible to set the value 1 for ’. In this case, the Gibson & Ashby model simplifies to the Moore empirical equation. The average deviations as function of ’ are shown in Table 1. It can be seen that increasing the value of ’ from 0.8 to 1.0 decreases the average deviation from 18.9 to 13.9%. This also suggested that for high density foams, it is reasonable to set ’ ¼ 1. The other three models (Moore empirical equation, differential scheme and modified empirical equation) give similar predictions. The average deviations are found to be 11.0, 13.9, and 12.8% for differential scheme, Moore empirical equation and modified empirical equation models. Since all the models underestimate the data, we suspect that the small skin must have an effect on the modulus. Comparison of Structural Foam Models Structural foam models were then used to evaluate the effect of skin thickness on shear modulus. In each case, the calculations were made using m ¼ 0.34 as described earlier. The average deviations of each model at different skin thickness ratio are presented in Table 2. Values for c /f were taken between 94 and 97% to determine the effect of this parameter. The results show that the c /f with the minimum deviation is obtained around 95.5%, which is in agreement with the optical microscopy measurements of 96%. From Figure 5, it can be seen that these models fit reasonably well the data. The average deviations were found to be 5.9, 6, and 6.4% for the Structural Model I of Equation (5), the Structural Model II of Equation (6), and the Structural Model III of
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Table 2. Average deviation of structural foam models for different skin thickness (%). c/f (%) Equation
97.0
96.5
96.0
95.5
95.0
94.5
94.0
5 6 7
5.7 7.6 6.8
5.2 6.7 6.1
5.0 6.0 5.5
5.0 5.5 5.1
5.3 5.1 5.0
5.8 5.0 5.1
6.4 5.2 5.5
Figure 5. Comparison of the normalized shear modulus as function of normalized density for structural foam models: —: Model I; : Model II, and — —: Model III.
Equation (7), respectively. It is not possible to determine which is best, and we believe that they are equally able to represent the data over the void fraction under study. Comparison Between Uniform and Structural Foam Models By comparing Figures 4 and 5, it can be seen that structural foam models give more reasonable prediction than uniform foam models. Because the shear modulus measured is the average shear modulus of the sample, its value is related to the twist stiffness, which is affected by the skin thickness, core thickness, skin shear modulus, core shear modulus, and foam morphology. The structural foam models give a better prediction than the uniform models because these models consider the effect of skins. The differences of the average deviation for structural foam models with the corresponding uniform foam models
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High Density Polyethylene Foams Table 3. Average deviation of different models for the normalized shear modulus (%).
Uniform model Structural model Difference
Differential Scheme
Moore Equation
Modified Empirical
11.0 5.9 5.1
13.9 6.9 7.0
12.8 6.4 6.4
are presented in Table 3. We can see that structural foam models have much lower average deviations than the uniform models, indicating that very thin skin layers for HDPE foams have a definite effect on the shear modulus. CONCLUSIONS
The shear modulus of closed-cell HDPE foams were measured in torsion rectangular. Based on the assumption that the twist stiffness is the sum of the twist stiffness of the skin and core parts, several structural foam models were proposed. Experimental data were compared with the proposed uniform foam and structural foam models. For the shear modulus, the Gibson & Ashby model deviated from 18.9 to 13.9% when ’ changes from 0.8 to 1.0. The differential scheme, Moore’s empirical square power-law models, and modified empirical equation were found to deviate 11.0, 13.9, and 12.8% respectively. On the other hand, the structural foam models were found to fit more closely the data with an average deviation of 5.9, 6.9, and 6.4% for the combined differential scheme, Moore’s empirical equation and modified empirical equation, respectively. These lower deviation values indicate that for the shear properties of polymer foams, even very thin skins have a definite effect of the modulus in torsion rectangular tests. ACKNOWLEDGMENTS
The authors would like to acknowledge the support of the following contributors: Natural Sciences and Engineering Research Council of Canada (NSERC) and Fonds pour la Formation de Chercheurs et l’Aide ` la Recherche (FCAR) of Quebec. Thanks go also to Dr. James M. a Killough of BP Solvay Polyethylene North America Technical Center and Sarah Marshall of Nova Chemicals for the HDPE samples used in this study. We also want to mention the tragic loss of one of the authors (AAK) to a car accident in the course of this study.
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NOMENCLATURE
L ¼ length B ¼ width T ¼ thickness f ¼ shear modulus of foam m ¼ shear modulus of matrix SF ¼ shear modulus of structural foam f ¼ density of foam m ¼ density of matrix ’ ¼ fraction of solid material in the cell struts f ¼ void volume fraction c ¼ thickness of core zone f ¼ thickness of structural foam s ¼ thickness of one skin m ¼ matrix Poisson ratio APPENDIX
Shear Modulus from Torsion of Isotropic Materials According to Cauchy’s approach [19], an elastic body subjected to an external stress should follow several sets of equations such as equilibrium equations, compatibility equations and stress–strain relationship equations. From these equations, it is possible to obtain the response of a material under imposed conditions. At the surface of an elastic body, the stress components must also satisfy the equilibrium conditions, which are called boundary conditions. For a rectangular bar made of linearly elastic, homogeneous and isotropic material under torsion, the lateral surfaces of the bar are free of external load and the end planes of the bar (z ¼ 0 and L), which are perpendicular to the lateral surfaces, are subjected to the forces. When body forces are negligible, the following sets of equations apply: (a) Equilibrium equations [20,21]: 8 @ @yx @zx x > > > @x þ @y þ @z ¼ 0 > > < @ @y @zy xy þ þ ¼0 > @x @y @z > > > > : @xz þ @yz þ @z ¼ 0 @x @y @z
ðA-1Þ
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where i is tensile stress in the i-direction, ij shear stress on the i-face and in the j-direction. (b) Stress–strain relations [21,22]: 8 1 > > x ðy þ z Þ "x ¼ > > E > > > 1 > > y ðx þ z Þ "y ¼ > > > E > > > 1 > > z ðx þ y Þ < "z ¼ E 1 > > xy ¼ xy > > > > > 1 > > > ¼ > > xz xz > > > > 1 > : yz ¼ yz
ðA-2Þ
where "i is elongation strain in the i-direction, ij distortion strain in the ij plane, E elastic modulus, and shear modulus. (c) Boundary Conditions [20, 21]: On the lateral surfaces (direction cosines l, m, n ¼ l, m, 0): 8 < Px ¼ lx þ mxy ¼ 0 Py ¼ lxy þ my ¼ 0 : Pz ¼ lxz þ myz ¼ 0
ðA-3Þ
where l is cos(N, x) for the angle between N and x-axis, m is cos(N, y) for the angle between N, and y-axis, and n is cos(N, z) for the angle between N and z-axis. On end planes (z ¼ 0 and L; direction cosines l, m, n ¼ 0, 0, 1) xz , yz prescribed functions
ðA-4Þ
According to Hsu [22], there are two general approaches to the solution of stresses in the theory of elasticity: direct and inverse approaches. The direct approach is to integrate the equilibrium equations to obtain the stresses. Because there is a lack of general methods to adopt this approach, it is only used in some special cases. The inverse approach assumes the displacement components (u, v, w) and substitutes them into the compatibility equations to get the strain components first, and then obtain the stress components from the strains. The force can be obtained from the equilibrium equations and
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the surface forces from the boundary conditions. This approach is quite straightforward and used in most cases. In order to solve the torsion problem for noncircular cross sections, Saint-Venant [22–24] developed a ‘‘semi-inverse’’ method, in which some features of the displacements u, v, and w are first assumed, and the remaining features are then determined by satisfying all the equations of the theory of elasticity. Based on Navier’s derivation of torsion formulae for circular section [22], two deformation assumptions were made: (1) the shape of a cross section must remain unchanged after twist, and (2) a plane section must remain plane after twist with no warping. These assumptions were found to be valid for circular sections. For noncircular bars, warping of the cross sections occurs after twisting, but the first assumption seems to be also valid for noncircular cross section bars. Saint-Venant also made two assumptions to describe the displacement components for noncircular sections: (1) the shape of the cross sections remains unchanged after twisting, and (2) the warping of the cross sections is identical throughout the length of the noncircular bar. From these two assumptions, the displacement components were obtained as: 8 > u ¼ zy > > > L < ðA-5Þ v ¼ þ zx > L > > > :w ¼ ðx, yÞ L where u is the displacement in the x-direction, v the displacement in the y-direction, w is the displacement in the z-direction, the twist angle at z ¼ L, and (x,y) the warping function of (x,y). Based on these assumption and equilibrium equations, stress–strain equations and boundary conditions, the solution for the torsion of rectangular bars was obtained as [22–24]: " # 1 1 X 32 T 3 B 1 2T X 1 n B M¼ tanh L 4 n4 B n¼1, 3, 5, n5 2T n¼1, 3, 5 " # ! 1 1 X T 3 B 1 192 T X 1 n B 1
4 1 5 tanh ¼ ¼ , L 3
B n¼1, 3, 5, n5 2T n4 96 n¼1, 3, 5 L ¼ 12k1 I0 L
¼ k1 T 3 B
ðA-6Þ
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where M is the torque, is the shear modulus, L is the length of beam, is the twist angle at z ¼ L, and I0 is the moment of inertia for a rectangular bar given by: Z
T=2
x2 ðB dxÞ
I0 ¼
ðA-6aÞ
T=2
where k1 is a numerical factor depending on the magnitude of the ratio B/T as: " # 1 1 192 T X 1 n B 1 5 tanh ðA-6bÞ k1 ¼ 3
B n¼1, 3, 5, n5 2T We can thus rewrite the above equations for the shear modulus as: ¼
ML k1 BT 3
ðA-7Þ
Comparison of Shear Modulus from Torsion Tests The stress () and strain () were determined by the following Equations [14]: ¼
ð3 þ 1:8ðT=BÞÞM BT2
T T2 1 0:378 2 ¼ L B
ðA-8Þ
ðA-9Þ
where M is the torque and is shear angle of the motor. For torsion measurements with rectangular bars, the shear modulus is obtained from the ratio of stress over strain. Equations ((A-8), (A-9)) give: ð3 þ 1:8ðT=BÞÞM=BT2 ML 3 þ 1:8ðT=BÞ ¼ ¼ ¼ ðT=LÞð1 0:378ðT=BÞ2 Þ BT3 1 0:378ðT=BÞ2
ðA-10Þ
Equation (A-10) can now be written as: ¼
ML k0 BT 3
ðA-11Þ
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Table 4. Values for k1 of Equation (A-7) and k0 of Equation (A-11). B/T
k1
k0
k0 /k1
B/T
k1
k0
k0 /k1
1 1.2 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0.141 0.166 0.196 0.229 0.249 0.263 0.273 0.281 0.287 0.291 0.295 0.298
0.130 0.164 0.198 0.232 0.253 0.266 0.276 0.283 0.289 0.293 0.297 0.300
0.922 0.987 1.012 1.015 1.013 1.011 1.009 1.008 1.007 1.006 1.006 1.005
7 8 9 10 15 20 25 30 40 60 80 100
0.303 0.307 0.310 0.312 0.319 0.323 0.325 0.326 0.328 0.330 0.331 0.331
0.305 0.308 0.311 0.313 0.320 0.323 0.323 0.327 0.328 0.330 0.331 0.331
1.004 1.004 1.003 1.003 1.002 1.002 1.002 1.001 1.001 1.001 1.000 1.000
where, k0 ¼
1 0:378ðT=BÞ2 3 þ 1:8ðT=BÞ
ðA-11aÞ
Comparing Equation (A-7) and Equation (A-11), the only difference is a numerical constant (k1 or k0 ) which is function of the B/T ratio and given in Table 4. We can see that the ratio of k0 and k1 is very close to 1 when the ratio B/T is higher than 1.2, indicating that both equations are equivalent. Gibson & Ashby Model Gibson and Ashby derived the shear modulus of low-density closedcell foams as: ( ) f 3 2 f 2 f ’ þ ð1 ’Þ ðA-12Þ Em 8 m m where ’ is the fraction of solid material in the cell struts. The foam modulus is composed of two terms: the first term gives the contribution of cell struts and the second term accounts for cell walls. For homogeneous, isotropic and elastic materials, the shear and Young’s moduli are related by: ¼
E 2ð1 þ Þ
ðA-13Þ
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Substitution of Equation (A-13) into Equation (A-12) gives: ( ) f 3ð1 þ m Þ 2 f 2 f ’ þ ð1 ’Þ 4 m m m
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ðA-14Þ
For polyethylene, a Poisson ratio of 0.34 is taken and Equation (A-14) becomes: 2 f f 2 f ’ þ ð1 ’Þ m m m
ðA-15Þ
Differential Scheme The differential equations for the shear and bulk moduli of a twophase composite with spherical inclusions are: 8 dc 15c ð1 c Þð1 I =c Þ > > ¼ > < df ð7 5c þ 2ð4 5c ÞI =c Þð1 f Þ > dKc K I Kc > >
¼ : df 1 þ ðK I Kc Þ=ðKc þ ð4=3Þc Þ ð1 f Þ
ðA-16Þ
where and K are the shear and bulk moduli respectively, indices c and I refer to the composite and inclusion respectively, and c is the Poisson ratio of the composite. These equations constitute a coupled system which can be solved by numerical methods with the following boundary conditions: f ¼ 0,
c ¼ m ðmatrix shear modulusÞ Kc ¼ Km ðmatrix bulk modulusÞ
f ¼ 1,
c ¼ I ðinclusion shear modulusÞ Kc ¼ K I ðinclusion bulk modulusÞ
For foams, I ! 0 and KI ! 0, Equation (A-16) becomes: 8 5f ð3Kf þ 4f Þ df 15f ð1 f Þ > > ¼ ¼ < ð7 5f Þð1 f Þ ð9Kf þ 8f Þ ð1 þ f Þ df Kf Kf ð3Kf þ 4f Þ > dKf > : ¼ ¼ 4f ð1 þ f Þ df ð1 Kf =ðKf þ ð4=3Þf ÞÞð1 f Þ
ðA-17Þ
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where, f ¼ 0,
f ¼ m ðmatrix shear modulusÞ Kf ¼ Km ðmatrix bulk modulusÞ
f ¼ 1,
f ¼ 0 ðvoids shear modulusÞ Kf ¼ 0 ðvoids bulk modulusÞ
Knowing that for isotropic materials, the Poisson ratio is related to the bulk and shear moduli by: ¼ ð3K 2Þ=2ð3K þ Þ (see [25]). To solve the differential system of equations, Euler’s method was used as [26]: dy ¼ f ðt, yÞ subject to y ¼ y0 dt
when t ¼ t0
ðA-18Þ
For each subinterval [ti, ti þ 1], a small step size (h ¼ ti þ 1 ti) is used to discretize the equations: yðti þ 1 Þ yi þ ðti þ 1 ti Þfi
ðA-19Þ
Equation (A-17) can now be written as: 8 > > > < i þ 1 i þ ð f i þ 1 f i Þ
5i ð3Ki þ 4i Þ ð9Ki þ 8i Þð1 þ fi Þ
> K ð3Ki þ 4i Þ > > : Ki þ 1 Ki þ ðfi þ 1 fi Þ i 4i ð1 þ fi Þ
ð0 fi 1Þ ðA-20Þ ð0 fi 1Þ
By taking a small time step (105), the shear modulus was found to follow a power-law relation in the form of: f ð1 f Þn m
ð1:77 n 2:09Þ
ðA-21Þ
where n is the power-law index. The results are presented in Table 5. The Poisson ratio was found to follow the relation: ( f ¼
m ðað1 f Þ þ bÞ
ð0 f 0:12Þ
k
ð0:12 f 1Þ
ðA-22Þ
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Table 5. Relationship between the parameters of Equations (A-21, A-22) and Poisson ratio. m n a b k
0.05 2.09 3.330 4.329 0.070
0.10 2.06 1.139 2.139 0.114
0.15 2.04 0.388 1.380 0.157
m n a b k
0.33 1.91 0.485 0.515 0.311
0.34 1.90 0.508 0.492 0.319
0.36 1.89 0.549 0.451 0.336
0.20 2.00 0 0.2 0.2
0.22 1.99 0.109 0.891 0.217
0.24 1.97 0.201 0.799 0.234
0.26 1.96 0.279 0.721 0.251
0.28 1.95 0.348 0.652 0.268
0.30 1.93 0.408 0.592 0.285
0.32 1.92 0.462 0.539 0.302
0.38 1.87 0.587 0.413 0.353
0.40 1.86 0.621 0.379 0.370
0.42 1.84 0.651 0.349 0.387
0.44 1.82 0.678 0.322 0.404
0.46 1.81 0.702 0.298 0.421
0.48 1.79 0.724 0.276 0.438
0.495 1.77 0.738 0.262 0.450
k is the foam’s Poisson ratio for normalized densities less than 0.88.
where a and b are constants related to the Poisson ratio of the matrix, and k is foam’s Poisson ratio for normalized densities not higher than 0.88, which is constant. The results are also presented in Table 5. Modified Empirical Equation It has been shown in the past that an empirical square power-law equation between the normalized Young’s modulus and normalized density of polymer foams exists [5]: 2 Ef f ¼ ð1 f Þ2 ðA-23Þ Em m We can modify this equation through substitution of Equations (A-13), (A-22) into Equation (A-23) to get an expression between the normalized shear modulus and the normalized density of polymer foams as: 8 1 þ m > > ð1 f Þ2 ð0 f 0:12Þ > < 1 þ m fað1 f Þ þ bg f ðA-24Þ m > 1 þ m > 2 > ð1 f Þ ð0:12 f 1Þ : 1 þ k For polyethylene foams, the expressions become: 8
> > 1 f > < 1 þ m að1 ðf =c Þf Þ þ b c
f m > > > 1 þ m f 2 > : 1 f 1 þ k c
0
f f 0:12 c
0:12
f f 1 c ðA-35Þ
For polyethylene, it becomes: 8 1:34 f 2 > > > > < 1:167 þ 0:173ð1 f Þ 1 c f f 2 m > > > > : 1:016 1 f f c
f 0 f 0:12 c f 0:12 f 1 c
ðA-36Þ
The average normalized shear modulus of the beam is then obtained after substitution of Equations (A-35) into Equation (A-28) to give the structural model III as: 8 3 3 c 1 þ m c f 2 f > > > 1 þ 1 f 0 f 0:12 > < f 1 þ m fað1 ðf =c Þf Þ þ bg f c c
SFIII ¼ 3 > m > 1 þ m c 3 f 2 > > :1 c þ 1 f f 1 þ k f c
0:12
f f 1 c ðA-37Þ
High Density Polyethylene Foams
473
For polyethylene, Equation (A-37) becomes:
SFIII m
8 3 3 c 1:34 c f 2 f > > þ 1 f 0 f 0:12 1 > < f c c 1:167 þ 0:173ð1 f Þ f ¼ 3 3 2 > > f f > : 1 c þ 1:016 c 1 f 0:12 f 1 f f c c ðA-38Þ
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