a novel theory of effective mechanical properties of closed-cell foam materials

Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 26, No. 6, December, 2013 ISSN 0894-9166

A NOVEL THEORY OF EFFECTIVE MECHANICALPROPERTIES OF CLOSED-CELL FOAM MATERIALS��

Yuli Ma1,2 Xianyue Su1� R. Pyrz3 J. Ch. Rauhe3

(1The State Key Laboratory for Turbulence and Complex Systems (LTCS) and Department of Mechanics andAerospace Engineering, College of Engineering, Peking University, Beijing 100871, China)

(2Chinalco Research Institute of Science and Technology, Beijing 100082, China(3Department of Mechanical and Manufacturing Engineering, Aalborg University,

9220 Aalborg East, Denmark)

Received 21 January 2012, revision received 10 May 2012

ABSTRACT In this paper a new theory of effective mechanical properties of foam materials isproposed. A cell volume distribution coefficient is introduced to modify the original Gibson-Ashbyequations of effective mechanical properties of foam materials. The constants that influence theeffective modulus are replaced by the coefficient. Based on the modified distribution coefficient,the yield stress is also recalculated. Using X-ray microtomography, the internal structures of dif-ferent samples of polypropylene-nanoclay foam are obtained. The cell volume distributions ofthese samples are derived from the experiment by image analysis and the fitting curves are plot-ted. The distribution coefficient is acquired using the parameters from the theoretical model ofthe distribution curves. The results of the improved theory are compared with the experimentalvalues and show good fitting quality. It was found that the precision of the improved theory ishigh and the cell volume distribution has an impact on the effective mechanical properties thatwould lead to the optimization of the synthesis procedure.

KEY WORDS closed-cell foam, cell volume distribution, distribution coefficient, Young’s modulus,yield stress, distribution function

I. INTRODUCTIONFoam materials are widely used in car industry, aircraft materials and windmill turbines. These

materials have the advantage of light weight, high energy absorption, and wide range of mechanicalproperties so they are suitable for many different industrial and academic applications. It is well under-stood that the mechanism of foam materials depends on building materials as well as cell morphologyi.e. cell number, cell size and orientation distribution.

Since 1982 when Gibson and Ashby[1] discussed the structure and properties of cellular solids andderived the equations for the relationship between the bulky materials and the foam materials basedon numerous experimental results, there have been many researches about the foam mechanisms,especially in the area of experiment and computer simulation. Also, there are many discussions about

� Corresponding author. E-mail: [email protected]�� The study presented was supported by the National Natural Science Foundation of China under Grant No. 90916007,granted to the LTCS and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University.Y.L Ma is grateful to the support of Department of Mechanical and Manufacturing Engineering, Aalborg University,Denmark, and of the Department of Condensed Matter Physics, University of Valladolid. The authors acknowledge allthe participants of the project and their contributions.

· 560 · ACTA MECHANICA SOLIDA SINICA 2013

the mechanical properties of closed-cell non-polymer particles[2,3] and open-cell polymer foams[4–6]. Inrecent years, there appeared to be more in situ loading experiments on polymer closed-cell foams[7,8]

which determined the deformation mechanism of the foam structures.However, of all the researches on closed-cell polymer foams, most are focused on foams that have

cells of regular size and shape such as cubic or hexagon. The mechanism of such a kind of foam is widelydiscussed. As for the foams of irregular structures such as the wide range of cell volume distribution andarbitrary shapes, most of the researches focused on the characterization and simulation of certain typesof foam material. With the help of microtomography technique, the characterization of microstructureof closed-cell foam materials was developed rapidly. In 2002, Elmoutaouakkil et al.[9] presented themicrotomography characterizing the metal foams. In 2008, Stock[10] made a review on the advance ofmicrotomography applied to materials, especially on cellular materials. During the past decade, theresearch of microtomographic characterization on foam materials was widely presented[11–14]. In recentyears, Jeon et al.[15,16] did very detailed investigation on Al foams with finite element simulation. Teixeiraet al.[17] synthesized foams using advanced moulding technique and measured the foam mechanicalproperties. In the above research, the mechanical properties of the foams are also discussed but therelationship between mechanical properties and geometrical morphology is not thought of as a majorcause. In recent studies, Ma et al.[18] used X-ray microtomography to investigate the internal structureof polymer foam samples and found that the mechanical properties such as Young’s modulus and yieldstress have big differences while the cell volume distribution are different and the density difference isnot obvious. This has led to researches on what kind of influence the distribution of cell volumes wouldmake on the foam structures.

In this paper, a new theory of the mechanism of closed-cell foam materials is proposed. Based on theequations of foam theory from Gibson and Ashby, we introduce a cell volume distribution coefficientto determine the influence on the Young’s modulus and yield stress from the geometrical morphologyof the cell volumes such as cell size distribution. The distribution coefficient was dependent on variousmaterials and cell distributions. We used the X-raymicrotomography technique to make the investigationof internal structures of polypropylene-nanoclay foams. After that, 3D models of foam samples wereprocessed and the cell volume distributions were calculated. A burr distribution was applied to performfitting the cell volume distribution. The distribution coefficient was calculated from the parametersof the fitting curves. Then the equations of effective Young’s modulus and yield stress were modifiedand the constants were recalculated. The final results of both foam theories by Gibson and Ashby andby improved coefficient of cell volume distribution were obtained and compared with the experimentalvalues. The results showed that the improved theory with cell volume distribution fit the experimentalvalues better, and the mechanical properties could be optimized by changing the synthesis conditionto obtain idealized geometrical morphology of the foam structure.

II. THEORETICAL PRINCIPLE2.1. Gibson-Ashby Theory of Effective Mechanical Properties for Closed-cell Foams

In cellular solid mechanics, Gibson and Ashby[1] provided the following equations of effective Young’smodulus of close cell foams based on the experiments of various experimental data:

E∗

Es= C1φ

2

(ρ∗

ρs

)2

+ C′1(1− φ)ρ∗

ρs+

p0(1− 2ν∗)

Es(1− ρ∗/ρs)(1)

where E∗ is the effective modulus of foam, Es the bulky modulus, φ the proportion of cell edge to wholesolid part which makes 1−φ the proportion of cell surface part, ρ∗ the effective density of foam, ρs thebulky density, p0 the fluid pressure in closed-cell foam cells, ν∗ the equivalent Poisson’s ratio of cellularstructure, and C1 and C′1 the constants determined by geometrical parameters. The constants C1 andC′1 are commonly defined as C1 = 1 and C′1 = 1 based on various experimental results of regular periodiccell structure foams or average foam parameters if the size and shape do not show much diversity[19–22].

If the fluid pressure in the cells is near atmospheric pressure, p0/Es will be much smaller than E∗/Es

and can be neglected. Then Eq.(1) is rewritten as

E∗

Es= C1φ

2

(ρ∗

ρs

)2

+ C′1 (1− φ)ρ∗

ρs(2)

Vol. 26, No. 6 Yuli Ma et al.: Novel Theory of Mechanical Properties of Closed-Cell Foam Materials · 561 ·

In this case, the effective modulus is determined by φ and relative density. For regular-shaped-cellfoam, the parameter φ can be precisely calculated based on geometrical morphology. For common closed-cell foams, Gibson and Ashby provided the value of φ ranging from 0.6 to 0.8 according to differenttypes of materials. As for the irregular foams studied, it is difficult for the parameter φ to preciselydetermine, so we approximate it from the image of microtomography images in the following chapters.

During the loading experiment, when the applied pressure is increased to a certain value σ∗pl, thedeformation procedure enters the plastic process, the material begins to collapse and cannot be recovered.For the plastic collapse process, the Gibson-Ashby theory yields:

σ∗pl

σys= C3

(φ

ρ∗

ρs

)3/2

+ C4 (1− φ)ρ∗

ρs+

p0 − pat

σys(3)

where σys is the bulky material yield stress, and pat the atmosphere pressure. The coefficients C3 andC4 are usually set as 0.3 and 0.4 based on the experimental data[23–25]. This equation determines thecritical yield stress of the plastic process. Under the plastic process, the stress of the material remainsσ∗pl with the strain increasing to a value for the whole structure to become dense and bulky as the cellsare compressed and collapse and the solid parts get stuck together. The strain is called densificationstrain and is defined by the following equation[26]:

εD = 1− 1.4

(ρ∗

ρs

)(4)

From εD, the stress of the materials increases significantly with hardly any strain increase. The limitof the strain is

εL = 1−

(ρ∗

ρs

)(5)

which is obvious since the whole material has becomee bulky and no more cell void can be found.

2.2. Improved Theory of Mechanical Properties Based on Cell Volume Distribution

The theory of cellular materials by Gibson and Ashby is widely discussed and fits well the existingcellular materials. However, the equations of Gibson and Ashby are based on cellular materials of similarcell size and periodic cell structure that lead to C1 and C′1 equal to 1. Furthermore, the constants arederived from experimental curve fitting methods so the fitting quality of the equation is rough. Gibsonand Ashby also mentioned that the results of C1 and C′1 might have error due to the diversity of geometricmorphology[1]. When the error is small, the constants are considered robust. Also, the fitting constantsare based on all kinds of foam materials so it fit well in general scale, but for certain detailed material itsprecision needs to be refined. In recent studies[18], under different synthesis conditions, the samples withclose density and porosity show a wide range of cell volumes and the mechanical properties are greatlydifferent from regular cellular foams. The comparison is given in Table 1 which shows that under theGibson-Ashby theory, the Young’s modulus has big errors compared to the experimental values. Also,all the values from Gibson-Ashby theory exhibit a similarity that around 100 MPa the experimentalvalues differ significantly. If we just changed the constant value of C1 and C′1, because the density andvolume fraction of all the samples are similar, the result of the Gibson-Ashby theory will still remainat the same level and errors still occur. Furthermore, if we use experimental data to calculate φ, itwould be larger than 1 which is obviously impossible. So a new method has to be found to express theinfluence of cell volumes distribution on the mechanical properties such as Young’s modulus and yieldstress.

Since it is difficult to build a system capable of precisely identifying the mechanism of irregularcells, we focus on the statistic data of the foam structure. Based on Eq.(2), we introduce a distributioncoefficient λ that represents the impact of cell volume distribution to replace C1 and C′1, and then theEq.(1) is rewritten as

E∗

Es= (λφ)2

(ρ∗

ρs

)2

+ (1− λφ)ρ∗

ρs(6)

In this equation, λ is not a constant, but a function related to cell volume distribution. After thisimprovement, the part to solve for the equation became λφ, so φ could be approximately assumed from


Table 1. Young’s modulus values of experiment value and Gibson-Ashby theory of foam samples studied

Sample Experiment Value[18] Gibson-Ashby theory (C1 = 1, C′1 = 1) Error (%)

Sample 1 42.1 90.71 115.46Sample 2 46.6 89.69 92.46Sample 3 58 97.40 67.92Sample 4 76.5 98.00 28.10Sample 5 115 120.33 4.64Sample 6 77.9 98.87 26.92

the image of structures sincebecause we considered λ as the core part of the function. Also, as λ is basedon the coefficients of distribution of cell volume, its expression is different from various materials. Inthe following chapter λ is solved as a function of Burr distribution parameters with the polymer foammaterial we made. It is the main purpose of this paper to derive λ from the existing materials.

Since λφ is obtained from Eq.(6), it is substituted into Eq.(3) to yield

σ∗pl

σys= C3

(λφ

ρ∗

ρs

)3/2

+ C4 (1− λφ)ρ∗

ρs+

p0 − pat

σys(7)

where C3 and C4 are constants and need to be determined. For regular cellular foams, the values of C3

and C4 are 0.3 and 0.4 as shown in Eq.(4), respectively. However, as the cell size distribution has animpact on the yield stress, the constant might be different for different compositions of materials.

Also, with the density modification coefficient[1 + (ρ∗/ρs)

1/2]

included in Eq.(7), a more precise

expression of yield stress is obtained

σ∗pl

σys= C1

(λφ

ρ∗

ρs

)3/2[1 + C2

(ρ∗

ρs

)1/2]

+ C3 (1− λφ)ρ∗

ρs+

p0 − pat

σys(8)

Then more accurate yield stress of foam materials are calculated from the equation.This improved theory fits better foam materials with a wide range of cell size bridging the modulus

and yield stress gap between regular uniform cell size foams and foams with different cell size distribution.

III. MATERIAL PREPARATION3.1. Synthesis of Polypropylene-Nanoclay Foam Samples

The polypropylene-nanoclay foammaterialswere synthesized using the improved compressionmould-ing technique[27,28]. The nano-composite capsuleswere prepared fromhighmolecular strengthpolypropy-lene containing 10% Polybond 3150 compatibilizer, 5% Cloisite 15A clay, 5% chemical blowing agent,0.1% of antioxidant and 0.5% estearic acid using a twin screw extruder, all by weight. The capsules wereput into a cylindrical mould with heating devices attached to its shells. The capsules were heated up tothe decomposition temperature of the blowing agent. During the stage the capsules were constrainedto expand by the application of external pressure through a piston placed inside the mould. After thewhole blowing agent had decomposed, the piston was removed in a preset way to release the pressure.The foaming procedure took place during the movement of the piston. This procedure allows controllingindependently the density and cell sizes of foams.

After the samples were made, we use an X-ray Microtomography scanner to obtain the internalstructure of the samples. In X-ray microtomography, the samples were rotated in the chamber ofscanner to receive the X-ray beam and had back project images by X-ray attenuation through thesolid parts. After the samples wererotated 180 degrees and all the project images were obtained, areconstruction procedure was processed and the back project images were used to create cross sectionlayers of the structure. Figure 1 shows the typical cross section image of a foam material with a widerange of cell volume. From these layers, we had a visual acquaintance with the internal structures ofthe samples.


Fig. 1. Typical cross section image of foam material withwide range of cell size.

Fig. 2. 3D structure model of foam material.

3.2. Material Characterization

After we obtained the 3D structure images, we used an image processing software Aphelion to acquirethe information from the images. The advantage of this software was that it could process 3D modelsand perform numerical analysis based on experimental images such as calculating porosity and eachcell volume. Figure 2 shows the 3D structure of certain foam materials. Because the foam material hasa very high porosity, it was difficult to see the hollow cells. So when we processed the image, we madethe void cells solid and the cell walls hollowed as shown in Fig.2 to make it clearer to be viewed. Weused a special processing method to calculate the cell volume of each cell[18]. This method was basedon the image analysis called watershed segmentation[29]. It helped clarify the boundary of the cells andcount the voxels of cells more precisely.

After we obtained the cell volume data from the image analysis, we used it to calculate the distributionfunction and made fitting curves. This is discussed in the following chapter.

In the present paper, we used Skyscan 1072 X-ray Microtomographic scanner to run the test. Theresolution was set at 3 micrometer/pixel to get a detailed image of small cells. The images of sampleswere set in 1.5×1.5×1.5 mm3 cube. The cross section images contained 500 layers of 500×500 pixelimage. The reconstruction and segmentation of the 3D model was processed by self-made Aphelionprogram[18].

From the synthesis experiment we obtained the following parameters.Bulky Polypropylene (PP)+Nanoclay Properties:Density ρs = 900 kg/m3

Young’s modulus Es = 1.8 GPaYield stress σys = 33 MPaTable 2 shows the properties of different porousPP+Clay foamsamples.The propertiesweremeasured

using the Standard Test Method for compressive properties of rigid cellular plastics (ASTM 1621).

IV. ANALYSIS AND DISCUSSION4.1. Analysis of Fitting Distribution of Cell Volumes

Based on the X-ray microtomography image analysis, we find the following volume distributions ofthe six different types of samples of PP+clay foam. The distributions were shown in Table 3.

Table 3 is a basic demonstration of cell volume distribution since it only contains porosity, meanvolume, Xi volume, and volume range of middle 50%, which is (X75 −X25). The concept of Xi is thati% of the cells have volume smaller than X . It is simple and clear to show the range of cell volumes.However, if we want to obtain detailed information about the cell volume distribution, we need to figure


Table 2. Synthesis properties of six samples of PP+clay foam samples

Blowing Foaming FoamingDensity

Young’s YieldSample agent temperature pressure

(kg/m3)modulus strength

content (%) (◦C) (Tons) (MPa) (MPa)

Sample 1 5 190 15 175.7 42.1 0.54Sample 2 5 190 7 174.47 46.6 0.61Sample 3 5 190 3 183.6 58 0.9Sample 4 5 190 1.2 184.3 76.5 0.98Sample 5 2.5 190 3 182.46 115 2Sample 6 2.5 190 1.2 158.2 77.9 1.14

Table 3. Structure parameters derived from the image analysis of X-ray pictures. The values of Xi (×10−03 mm3) mean thati% of the cells have volume smaller than X × 10−03 mm3

Sample Porosity (±0.01) (%)Volume (×10−03 mm3) Cell volume range of

Mean X25 X50 (Median) X75 mid 50% (×10−03 mm3)

Sample 1 88 8.08 1.01 2.76 9.06 8.05Sample 2 89 10.3 0.636 1.88 7.19 6.55Sample 3 85 15.4 0.869 2.53 14.8 13.931Sample 4 86 20.3 0.822 2.08 5.57 4.748Sample 5 80 8.62 0.969 2.01 4.15 3.181Sample 6 85 14.3 0.562 1.75 5.47 4.908

out the precise portion of the cell size, so we imported the distribution function and use fitting curvesto do the research.

After comparing with all kinds of fitting distribution, we found that Burr distribution was wellsuited to all the samples. Burr distribution was suggested by Burr as a number of forms of cumulativedistribution functions useful for fitting data. The principal aim in choosing one of these forms ofdistributions is to facilitate the mathematical analysis to which it was subjected, while attaining areasonable approximation. The probability density function (PDF) of Burr distribution is

f (x) =αk (x/β)

α−1

β [1 + (x/β)α]k+1

(9)

where k, α and β were the continuous constants of the distribution.The cumulative distribution function (CDF) of Burr distribution is

F (x) = 1−

[1 +

(x

β

)α]−k

(10)

In order to make sure the Burr distribution well described the actual cell volume distribution, weused Kolmogorov-Smirnov test to measure the fitting accuracy of the distribution. This test was usedto decide if a sample came from a hypothesized continuous distribution. It was based on the empiricalcumulative distribution function (ECDF). Assuming that we had a random sample x1 ∼ xn from thedistribution with CDF F (x). The empirical CDF was denoted by

Fn(x) =1

n[Number of observations ≤ x] (11)

The Kolmogorov-Smirnov statistic (D) was based on the largest vertical difference between thetheoretical and the empirical cumulative distribution functions

D = max1≤i≤n

[F (xi)−

i− 1

n,

i

n− F (xi)

](12)

Also, there were null and alternative hypotheses which were defined as:H0: the data follow the specified distributionHA: the data do not follow the specified distribution


The hypothesis regarding the distributional form was rejected at the chosen significance level (η) ifthe test statistic, D, was greater than the critical value obtained from a table. The fixed values of η(0.01, 0.05 etc.) were generally used to evaluate the null hypothesis (H0) at various significance levels.A value of 0.05 was typically used for most applications, however, in some critical industries, a lowerη value might be applied. In this case, value 0.01 was applied.

Table 4 shows the corresponding constants of burr distributions of all samples. D and P and η arethe parameters used in the Kolmogorov-Smirnov test. From the table, we have found that none of thefitting distributions is rejected at η = 0.01. This means that Burr distribution shows the good fittingquality of the cell volume distribution of each sample.

Table 4. Parameters of Burr distribution of all samples

Sample No. k α β D P fit η

Sample 1 1.61 0.92 0.0058 0.02563 0.28139 0.2Sample 2 0.656 1.12 0.00116 0.03113 0.26793 0.2Sample 3 0.76 0.95 0.002 0.04589 0.11148 0.1Sample 4 0.7 1.05 0.00136 0.05355 0.10649 0.1Sample 5 1.05 1.21 0.0021 0.048 0.012 0.01Sample 6 1.66 0.66 0.0051 0.06059 0.01216 0.01

Based on the Burr distribution function, Fig.3 shows the cumulative density of the cell volume ofall the samples. The horizontal axis is set logarithmic to improve the visual clarity. From the figure,we have found that the volume distribution well fit the Xi parameter in Table 2. For example, by thepoint of X25, sample 6 has the smallest volume while sample 1 has the biggest one. Also, by the point ofX75, sample 5 has the smallest volume and sample 3 has the biggest one, as is the quantity of volumes.These results are exactly the same as those derived from Table 2. This is proof of the robustness ofBurr distribution on the foam cell volume distribution, and can be used to make further investigationon the mechanical properties.

Fig. 3. The cumulative density of all six samples using Burrdistribution fitting curve.

Fig. 4. Result of Young’s Modulus based on Gibson-Ashbytheory and modified distribution theory compared with ex-perimental values.

4.2. Effective Properties Discussion Using the Improved Theory

As we mentioned in the improved theory chapter, the sample we studied had a wide range of cell size,so the equation of mechanical properties had a distribution coefficient λ. In the samples we synthesized,this coefficient was the function of all the parameters of Burr distribution, which were k, α and β.


We noticed that all k, α and β were involved in the distribution function as power functions. Weassumed the coefficient had the following relation equation

λ = exp(ak + bα + cβ + d) (13)

where a, b, c and d are constants.Based on Eqs. (6) and (13), we use the least square regression method to calculate the four constants

using the Burr distribution parameters. In the procedure, it is assumed that φ from samples 1∼4 equals0.9 and from samples 5∼6 equals 0.8. This is mainly based on the porosity of the samples investigated.The results of calculation are as follows:

a = −0.578, b = 7.10× 10−03, c = 137.586, d = 0.353

After the calculation, we made a comparison between the regular foam theory by Gibson and Ashbyand improved theory with cell size distribution to the experimental values of Young’s modulus. Theresults are shown in Table 5. Figure 4 gave a clearer comparison of the change between Gibson-Ashbytheory and improved distribution theory.

Table 5. Comparison of Young’s modulus of Gibson-Ashby theory with improved distribution theory

Sample Experiment value Gibson-Ashby theory Error (%) Distribution theory Error (%)

Sample 1 42.1 90.71 115.46 41.90 −0.49Sample 2 46.6 89.69 92.46 49.55 6.32Sample 3 58 97.40 67.92 54.94 −5.27Sample 4 76.5 98.00 28.10 71.42 −6.64Sample 5 115 120.33 4.64 111.52 −3.02Sample 6 77.9 98.87 26.92 80.01 2.71

From Table 5 and Fig.4, we know that the improved theory based on cell volume distributiongreatly increase the accuracy compared with the experimental results. The error between the resultsof experiment and improved theory is acceptable. It is obvious that the result of the original theoryis always higher than the experimental values and has little difference when the synthesis conditionchanged. This is mainly because the inhomogeneous and irregular structure has led to anisotropy inmicro-structure, so the modulus is lower than the homogeneous ones as described by the Gibson-Ashbytheory . According to Table 3, we calculated the cell volume range of the middle 50% cells. This showsthat the smaller the cell volume range is, the closer the improved theory is to the Gibson-Ashby theory.This also proved the theory that the homogeneous structure has the highest average Young’s modulusin all three directions of the same material. Also, the equation of Gibson-Ashby theory was more focusedon densities so the close densities lead to a similar result, showing that for this kind of foam material,the distribution function is well suited, and we can further predict the mechanical properties of differentsynthesis conditions leading to a wide range of cell volumes.

After we got λ, we further looked into the theory of yield stress solution. From Eq.(8), we knowthat the yield stress is derived from three different powers of λφρ∗/ρs, which are equal to 2, 1.5 and 1,respectively. So Eq.(8) without the fluid pressure can be rewritten as

σ∗pl

σys= C1

(λφ

ρ∗

ρs

)2

+ C2

(λφ

ρ∗

ρs

)3/2

+ C3λφρ∗

ρs+ C4 (14)

From Eq.(14), we know that the problem is changed to finding the best C1, C2, C3 and C4 fittingthe equation.

We used the linear fitting method to solve the problem. After the calculation, we found the bestsolution was

C1 = −10.88, C2 = 6.84, C3 = 1.92, C4 = −0.14

Table 6 shows the comparison of yield stress of experimental values with Gibson-Ashby theory andthe improved distribution theory. The visual comparison is plotted in Fig.5.


Table 6. Comparison of yield stress of Gibson-Ashby theory with improved distribution theory

Sample Experiment value Gibson-Ashby theory Error (%) Distribution theory Error (%)

Sample 1 0.54 0.99 82.74 0.61 13.75Sample 2 0.61 0.98 60.22 0.69 12.94Sample 3 0.9 1.05 16.46 0.82 −8.85Sample 4 0.98 1.05 7.51 1.04 5.88Sample 5 2 1.18 −40.91 1.92 −4.20Sample 6 1.14 0.99 −13.50 1.22 6.64

According to Table 6 and Fig.5, we know that the Gibson-Ashby foam theory could not describe the big difference ofyield stress between the samples (the triangle points) be-cause all the six samples’ yield stresses were relatively closeto each other (the circle points). The Gibson-Ashby theoryonly considered relative density and solid portions of celledges which were not significantly different among all thesesamples. As shown in Table 6, the values of Gibson-Ashbytheory are all around 1.0∼1.1 MPa, which led to major errorcompared to experimental values. On the other hand, theimproved distribution theory distinguished the difference ofstructures of samples with a close density and solid por-tion of cell edge. The result shows that the improved theorymatched the experimental values more accurately. The av-erage error percentage is not as low as the Young’s modulusbecause the values of yield stress are quite small. So the erroraccounts for a relatively bigger portion. With the different

Fig. 5 Result of yield stress based on Gibson-Ashby theory and modified distribution theorycompared with experimental values.

blowing agent added (samples 1-4 and samples 5-6), the yield stress differs significantly showing thatthe inhomogeneous structure would have better properties than the regular ones defined by Gibson andAshby if the synthesis condition is properly set.

From the results, we know that the yield stress is higher than the originally regular foam structure(samples 5 and 6) if the synthesis condition is optimized. This is an important discovery because theindustry usually preferred regular and homogeneous foams. From now on, the diversity of cell volumewill become an interesting part of foam properties and should be put more emphasis on.

V. CONCLUSIONSThe original equations of Young’s modulus and yield stress by Gibson and Ashby of foam materials are

based on homogeneous cell foams ofidentical cell sizes. It brings about big errorsmeeting the experimentalvalues of actual foam materials with a wide range of cell volumes. In this paper an improved theory thatmodifies the mechanical properties of foam materials based on cell volume distribution is presented. Anew distribution coefficient λ is introduced to describe the impact of different foam structures on themechanical properties of foam materials. Furthermore, several constants of yield stress equation arerecalculated according to the new theory.

We used the newly synthesized Polypropylene-Nanoclay foam material as the testing samples. Fromdifferent synthesis conditions, the samples exhibit similar densities but widely different mechanicalproperties. With the help of X-ray microtomography technique, we obtained the internal structures ofall the samples, and figured out the cell volumes of each sample. We used Burr distribution functionsto approach the actual cell volume diversity. The result of Kolmogorov-Smirnov test showed goodrobustness of the approach. Based on the parameters of Burr distribution, the distribution coefficientλ of the improved theory was derived. The constants of yield stress equation were also calculated.

From the results, we found that the values from improved distribution theory had good fitting qualityfor the experimental values of mechanical properties. The different distribution of cell volumes had astrong influence on the mechanical properties. The Young’s modulus of samples of irregular cell sizewas always smaller than the regular cell size samples under the same density condition, but the yield


stress was different and had a relationship with cell structure. The result gave an implication that withproper synthesis condition, the sample with a wide range of cell sizes could have better performancethan the sample with the same cell size and regular geometric structure. This provides an optimizationmethod for synthesizing the foam with irregular structures to get better yield performance in industrialapplications.

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a novel theory of effective mechanical properties of closed-cell foam materials

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