Elastic properties of green expanded perlite particle ...

Powder Technology 310 (2017) 329–342

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Elastic properties of green expanded perlite particle compacts Haleh Allameh-Haery a, Erich Kisi a,⁎, Jubert Pineda b, Laxmi Prasad Suwal b, Thomas Fiedler a a b

Discipline of Mechanical Engineering, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia ARC Centre of Excellence for Geotechnical Science and Engineering, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e

i n f o

Article history: Received 26 August 2016 Received in revised form 5 January 2017 Accepted 21 January 2017 Available online 25 January 2017 Keywords: Expanded perlite particles Young's modulus Poisson's ratio Elastic wave Analytical modelling

a b s t r a c t This paper describes an experimental and analytical study characterizing the elastic properties of packed beds of expanded perlite. Elastic moduli of packed beds of expanded perlite particles (expanded siliceous volcanic glass) were investigated by elastic wave velocity measurement along the axial direction. By adopting an isotropic model for the medium, the elastic moduli Poisson's ratio and Young's modulus were measured. Young's modulus increased nonlinearly with increasing bulk density. Poisson's ratio did not show a large variation with density and which may be understood in terms of the fabric of the medium (double porosity structure of the packed beds). During compaction to achieve different densities, some crushing of particles into smaller particles and platy debris occurred. Analyses were based on both the raw compaction densities and densities modified by removal of debris from consideration on the assumption that it is non-structural. Four analytical models were applied to predict elastic moduli of packed beds of expanded perlite particles within the porosity range 84-95%. Models were assessed on their ability to successfully predict elastic moduli of these highly porous bodies for both cases: using the raw compact density and the modified density. It was found that the Wang (Minimum Solid Area) model was able to estimate Young's modulus and the Gibson and Ashby model was reasonable for the average behaviour of both elastic moduli. The best agreement was found for the Phani model with our modified shape factor. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Perlite is a glassy volcanic rock of silicic or rhyolithic composition typically formed by the hydration of obsidian. Upon rapid controlled heating within the softening temperature range 760–1100 °C, the combined water in perlite grains is converted to pressurized steam that causes expansion of perlite to 4–20 times its original volume. The expanded form of perlite (EP particles) has low density, high porosity and offers excellent thermal and acoustic insulating properties, chemical inertness, physical resilience, fire resistance, and water retention properties [1]. In a previous study [2], a novel lightweight foam core composite was developed by embedding a high volume fraction of EP particles in a matrix of epoxy resin, and subsequently compacting the mixture to different target densities. Containing a high volume fraction of EP particles, EP/epoxy foams had the structure of a packed bed of EP particles with epoxy resin filling the interstices. Compression tests were conducted on the EP/epoxy foams and their compressive properties were found to be independent of particle size but highly affected by foam density. Attempts were made to measure and predict the elastic modulus of packed beds of EP particles and their contribution to the resulting EP/epoxy foam's elastic modulus. However, due to the limitations and inappropriateness of conventional quasi-static testing ⁎ Corresponding author. E-mail address: [email protected] (E. Kisi).

http://dx.doi.org/10.1016/j.powtec.2017.01.045 0032-5910/© 2017 Elsevier B.V. All rights reserved.

methods (e.g., compression tests) for powdered materials [3–6], the accuracy of the measured Young's modulus was questionable. Using Voigt and Reuss models, an upper and a lower bound were estimated for the Young's modulus of EP particles in EP/epoxy foams. It was found that EP particles show Reuss-like behaviour similar to metals but atypical of non-plastic materials. In this study, the elastic properties of packed beds of EP particles were characterized by means of elastic wave propagation (compression and shear) along the axial (compaction) direction for a wide range of compaction densities. By adopting an isotropic model, the Young's modulus and Poisson's ratio are used to characterise the elastic response of the medium. In addition, five analytical models were used to study the elastic moduli-porosity relations and to predict the elastic moduli of the EP particles measured experimentally. This approach is opposite to what has been frequently adopted in metallurgy where the properties of sintered products are estimated from those of the powder. It will help to estimate the properties of EP/epoxy foams having different volume fractions of EP particles in order to produce foams with different porosities. 2. Experimental procedure 2.1. Material EP particles, supplied by Industrial Processors Limited (INPRO), were sieved in the size range 2-2.8 mm. The chemical composition provided

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by the supplier is presented in Table 1. Fig. 1 shows two SEM images taken from (a) the outer surface and (b) the mid-cross section of an expanded perlite particle. It can be seen that the outer surface is covered with both closed and open pores and has a froth-like structure. Fig. 1b reveals that internally the expanded perlite has a structure mainly composed of closed cells sealed off from their neighbours by membranous walls [7] which are almost uniform in size. Tapped density of the perlite particles was measured using a tapping device with a graduated measuring cylinder of 100 ml. After every 20 taps, the cylinder was rotated to minimize any possible separation of the mass during tapping down. Five hundred taps were conducted for each density measurement and the average of five measurements was found to be 0.086 g/cm3 (Standard deviation: 0.0033 g/cm3). Particle density was also measured via the wax- immersion method (ASTM C914-95) and found to be 0.183 g/cm3 (Standard deviation: 0.010 g/cm3). 2.2. Sample preparation A cylindrical compaction mould made of Al was used to produce samples of packed EP particles 45 mm ± 0.015 in height and 80 mm in diameter. Perlite particles were introduced into the mould assisted by an Al sleeve of the same diameter to avoid spillage. The target compaction densities were achieved by controlling the mass of EP particles within a constant volume. To ensure homogeneity and minimize variations in density, specimens were compacted in several layers (up to five). The compaction process was carried out in a 50 kN computer-controlled load frame using a constant displacement rate of 1.0 mm/min. Lubricant oil was used to minimize friction between the piston and the walls of the mould. Fig. 2 illustrates the compaction stress as a function of density of the EP particle bed. Solid perlite was prepared by grinding 27 g of perlite particles with a mortar and pestle and sieving to b 250 μm. The powder was transferred into a mould and uniaxially pressed into pellets at 146 MPa. The pellets, resting on a bed of alumina powder, were heated at 10 °C/min to the sintering temperature of 1100 °C and held for 12 h. Subsequently the samples were cooled at 10 °C/min to 450 °C and held for an hour before cooling at the same rate to 300 °C when the furnace was turned off and the samples left to cool slowly. Sintered pellets were cut and polished into samples of 30 mm in diameter and 16 mm high. Densities of the prepared sintered pellets were measured at room temperature using the Archimedes method from which the value of 2.3 g/cm3 (standard deviation 0.05 g/cm3) was found. 2.3. Determination of dynamic moduli

Fig. 1. SEM images showing (a) external structure of a perlite particle; (b) the internal structure of a perlite particle.

displacements are perpendicular to the direction of the wave propagation. In an isotropic linear elastic body, in which the propagating wave does not interact with the boundary of the medium, compression (or P-wave) (CP) and shear (CS) wave velocities are given by: CS ¼

Elastic stress waves propagate in materials by inducing infinitesimal elastic deformation. Hence, wave propagation equations which are based on elasticity theory can be used to measure elastic moduli of a material, if elastic wave velocities and densities are measured independently [9,10]. Two wave types propagate in extended elastic solids: (i) compression waves in which particle displacements are in the direction of the wave propagation, and (ii) shear waves in which particle

rffiffiffi μ ρ

sffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M λ þ 2μ CP ¼ ¼ ρ ρ

Table 1 Chemical composition of expanded perlite [8]. Constituent

Percentage present

Silica Aluminium Oxide Ferric Oxide Calcium Oxide Magnesium Oxide Sodium Oxide Potassium Oxide Titanium Oxide Heavy Metals Sulphate

74.0% 14.0% 1.0% 1.3% 0.3% 3.0% 4.0% 0.1% Trace Trace

Fig. 2. Compaction stress versus the density of the packed bed of EP particles.

ð1Þ

ð2Þ

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where M is constrained modulus, ρ is density of the material, λ and μ are Lamé constants. The Lamé constants are related to the stiffness tensor, Cijkl, by:    C ijkl ¼ λδij δkl þ μ δik δjl þ δil δjk εkl ;

ð3Þ

where δ is the Kronecker delta function defined as:  δij ¼

0 for i≠j ; 1 for i ¼ j

i; j ¼ 1; 2; 3:

Young's modulus (E) and Poisson's ratio (ν) can be expressed as a function of the density, compression wave velocity and shear wave velocity of the material as follows:  E¼

ρC 2S

3C 2P −4C 2S



C 2P −C 2S

C 2 −2C 2S ν ¼ P 2 C 2P −C 2S

ð4Þ

ð5Þ

When the characteristic dimensions of a body is not large compared with the wavelength, these equations no longer hold; the velocities become dependent upon frequency [11]. In the case of an unconstrained cylindrical bar (i.e. uniaxial conditions) with radius much smaller than the wave length, the compression wave has a different velocity called the longitudinal velocity (or rod velocity) CL, given by [11]: CL ¼

sffiffiffi E ρ

ð6Þ

However, the stiffness that controls shear waves in cylindrical bars is the same as the one in infinite medium (i.e. Eq. (1)) [12]. Consequently, Poisson's ratio is expressed as: ν¼

 1 CL −1 2 CS

ð7Þ

In the current study, the above equations are used to characterise elastic properties of the samples by assuming pores are air-filled voids which are randomly distributed and oriented throughout the medium. Hence, the porous solid body exhibits isotropic elastic behaviour in a statistical sense. 2.4. Experimental setup Two wave propagation systems were used in this study. In both cases, elastic wave velocities were measured along the cylinder axis of the specimens. Elastic wave velocities of pore-free (solid) perlite samples were measured in an aluminium frame schematically shown in Fig. 3a. The frame consisted of a mobile beam, allowed to move vertically via a frictionless roller bearing, a stationary beam and a pair of aluminium rams attached to the mobile and stationary beams within which the sample was placed. Rams were designed to have the same height as the transducer and a groove held the transducer tightly with the ram surface flush with the active element of the transducer which had the same diameter as the solid perlite sample. A constant contact pressure is applied in this setup, by adding 5 kg dead weight on the top of part A, in order to improve the coupling between the specimen and the transducers which in turn enhances the quality of the output signals. To measure the properties of EP particle compacts, a compaction rig was designed to include transducers and particles. It consisted of a pair of aluminium rams located inside the bottom and top of the cylindrical aluminium die enclosing compacted EP particles. Rams were designed

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in the same way for the first set-up and had a clearance of 0.05 mm to the wall of the mould. A schematic representation of the set-up is shown in Fig. 3b. Elastic wave velocity measurements were carried out at constant contact pressure, equal to the average compaction stress at each density. The pressure was applied by a computer-controlled Shimadzu testing machine (Fig. 2). In both setups, two pairs of piezoelectric transducers with a nominal frequency of 1 MHz were used to measure compression and shear wave velocities in the axial direction. A thin layer of silicone grease was used to improve the coupling between the specimen and transducers. The experimental setup is similar to the system used by Arroyo et al. for testing clayey rocks [13]. It includes a programmable function generator (Thurlby Thandar® TTi – TG4001) to generate and transmit elastic waves as well as a digital oscilloscope (Tektronix® TDS 1001C-EDU) to acquire both input and output signals. The output signal was amplified by a pre-amplifier (OLYMPUS® Panametrics 5656C, with a gain of 40 dB). Signal stacking was applied to both input and output waves to minimize random noise. Sine pulses with apparent frequency ranging from 5 to 40 kHz (100 V peak-to-peak) were used as the input signal. Elastic wave velocities were estimated as the ratio of travel length to travel time. The recorded input and output signals were used to estimate the travel time by considering the time delay between the starting of the input signal and the starting of output signal (details explained in [14]). The travel length is the thickness of the samples and was determined using callipers. Examples of input and output signals for low and high density EP particle compacts are given in Fig. 4. There, the arrival times for Pwaves (tP) and S-waves (tS) are indicated by vertical dashed lines. The arrival time tP was clearly identified as the ‘common first break time’ in the output signals, irrespective of the sample density. Two sets of input frequencies were used to determine the arrival time tS depending on the density of the EP particle compacts. Input frequencies up to 40 kHz were used in dense samples whereas good quality output signals were obtained in low density samples (b 150 g/cm3) using frequencies up to 15–20 kHz. A step pulse of 10 Hz was also applied which, in combination with the sine pulses, helped to check the arrival time of the Swave. The determination of the arrival time tS required extra care due to the influence of low-energy P-waves, which travel faster than S-waves, on the output signals. This phenomenon, known as the near-field effect e.g., [12,13], is strongly frequency-dependent and may leads to misleading estimates of tS if the arrival time of the P-wave is not available. From inspection of Fig. 4b and d, it is clear that the arrival time of the S-wave corresponds to the first important break time whereas the small disturbance (‘bump’) observed earlier in the output signals (clearer in low density samples) is consistent with the arrival time tP previously estimated in Fig. 4a and c. 3. Results and discussion 3.1. Elastic moduli The elastic wave velocities within sintered perlite samples and the two elastic constants E and ν, derived therefrom using Eqs. (6) and (7) are presented in Table 2. Using microscope images taken from the polished surface of samples and an image processing technique (MATLAB, MathWorks, MA), the porosity of the sintered samples was determined to be about 4%. It has been shown that for porosity b 5%, there is a linear relationship between elastic moduli and porosity [15–18]. Within this range (0-5%), elasticity theory successfully predicts the zero-porosity properties from the elastic moduli of porous solids, by the following relations [19]: E ¼ E0 ð1−θE P Þ;

ð8aÞ

G ¼ G0 ð1−θG P Þ;

ð8bÞ

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Fig. 3. Schematic representation of experimental set-up for measuring wave velocity in (a) pore-free perlite samples (b) Packed beds of EP particles.

ν ¼ ν0 ð1−θv P Þ;

ð8cÞ

C L ¼ C L0 ð1−θL P Þ;

ð28dÞ

C S ¼ C S0 ð1−θS P Þ;

ð8eÞ

where θE ¼ 1=18ð29 þ 11ν0 Þ; θG ¼ 5=3; θv ¼ 5=9 þ 11ν 0 =18−1=ð18ν0 Þ;

θL ¼ 1=2 θE þ 2θv ν 20 ð2−ν 0 Þ=½ð1−ν 0 Þð1 þ ν0 Þ:ð1−2ν0 Þ−1 ; θS ¼ 1=3; The resulting zero porosity values of density ρ0, longitudinal wave velocity, shear wave velocity, Poisson's ratio ν0, shear modulus G0 and Young's modulus E0 of sintered perlite are shown in row 2 of Table 2. Moreover, the corresponding values from solid obsidian, measured by Manghnani and co-workers [20], are also presented. It can be seen that the measured wave velocity values for perlite with 4% porosity is closer to obsidian than the ones for pore-free perlite. This might be due to the obsidian structure which is, by definition, vesicle-poor [21] (i.e. b2.5% porosity [22]), though the percentage of the porosity was not reported by Manghnani and co-workers [20]. Comparison with the measured values for obsidian provides a benchmark for confidence in the measured values for sintered perlite, especially Poisson's ratio. The compression and shear wave velocities of packed beds of EP particles at different densities are shown in Fig. 5. As can be seen, both compression and shear wave velocities reach a plateau for densities higher

than 0.2 g/cm3. At lower densities, particles have more space available to move or rotate. At higher densities the relative movement of particles is restrained as the particles reach an optimum structural arrangement (i.e. the particle fabric). Further compression reduces pore volumes between the EP particles by fracturing particles which produces both smaller cellular EP particles and fine platy debris such as that shown in Fig. 6. Because of the closed cellular EP structure, microscopic analysis conducted on EP particles in a previous study [23], indicated that debris does not form inside EP particles i.e. the debris is restricted to the interparticle spaces. The use of static density in Eqs. (1)-(2) and (4)-(6) can be justified only if the debris are mechanically linked to particles [24]. If the debris particles are free moving, they should not be taken into account in the evaluation of the dynamic compact density. Consequently, further investigation of the debris was undertaken. The debris concentration was investigated by separating the EP particles and debris particles and measuring their relative masses at each compact density. Mechanical separation was ineffective without causing more particle fracture and debris. Therefore, EP particle compacts were immersed in water for 24 h to disperse. The dispersion resulted in two phases: a top phase comprising EP particles floating on the surface and sediment. The sediment was considered to be 100% debris due to its higher density (note the solid density of perlite in Table 2). Beakers containing the EP particles and sediment were placed in an oven at 120 °C for 24 h. The debris size was measured and found to be b212 μm which is used here to discriminate between particles and debris. To be sure all the debris had separated from the EP particles, the samples were sieved and the fraction bellow 212 μm was added to the debris. Fig. 7a illustrates the weight percentage of debris as the EP

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particle compact density increases. This is also demonstrated by the particle size distribution measured after compaction which is shown in Fig. 7b. It can be seen that as compact density increases the percentage of larger particles decreases due to breakage and the amount of debris increases. This post compaction investigation revealed that the mass of EP particles that resisted crushing at densities higher than 0.2 g/cm3 remained relatively constant. This is very strongly correlated with the plateau region of the compression wave and shear wave velocity versus density curves suggesting that only the EP particles contribute to sound wave propagation and the debris plays little role. This was further validated by investigating the evolution of inter-particle porosity as compact density increases. To this end, the density of different EP particles size ranges was measured via the wax immersion method. In this method, wax was melted in a 5 ml measuring cylinder and kept at constant temperature for each measurement. EP particles of known mass and size were immersed in the melted wax and the particle density was measured through the change in wax volume. The results are shown in

333

Fig. 8a. Using the results shown in Figs. 7b and 8a, inter-particle porosity (excluding debris) within the total sample volume was calculated and is presented in Fig. 8b. It can be seen that the inter-particle porosity does not change significantly as compact density increases. This can be explained by the breaking of the original EP particles into smaller EP particles as compact density increases. The smaller particles created new inter-particle porosity the increase of which was offset by simultaneous production of debris. Considering the debris mass fraction at each compact density (Fig. 7a) and the results shown in Fig. 8a, the volume percentage of debris with respect to the total sample volume as well as with respect to the inter-particle porosity volume was calculated (Fig. 8c). As can be seen debris do not occupy a significant portion of the total sample volume and inter-particle porosity volume due to the very high density of debris compared with the density of EP particles. Therefore, the raw compact densities were modified by subtracting the debris mass from the total mass of EP particles and new densities were calculated. The modified results are plotted along with the experimental results in Fig. 5. It should be noted that both of these results are

Fig. 4. Examples of (a) P-wave and (b) S-wave signals in low density (0.12 g/cm3) and (c) P-wave and (d) S-waves in high density (0.3 g/cm3) EP particle compacts.

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Fig. 4 (continued).

useful. In practical situations, it is not usually feasible to separate the debris from the EP particles. The raw or experimental results however are readily measured and reflect the effective state of the granular body. Hence, these results are of most practical interest. However, if the pure EP particle properties are desired for more in-depth studies, the modified results are appropriate. The Young's modulus of packed beds of EP particles, corresponding to the wave velocities in Fig. 5, calculated based on both raw compact density and modified density using Eq. (4) is shown in Fig. 9a. Fig. 9a also illustrates the normalized value of Young's modulus (E/E0) versus

Table 2 Elastic properties of solid phase of expanded perlite particle. For comparison, the corresponding data for obsidian by Manghnani and co-workers [20] are presented.

4% porosity Perlite 0% porosity Perlite Obsidian Ref. [20]

ρ(g/cm3)

CL (m/s)

CS (m/s)

G (GPa)

E (GPa)

ν

2.312 2.411 2.357

5616.44 5836.75 5775

3655.1 3704.5 3608

30.9 33.1 30.7

72.93 78.33 75.16

0.181 0.183 0.180

the compact porosity. Both curves, calculated from experimental and modified wave velocity results show that the Young's modulus of packed beds of EP particles increases as the density of the compact increases. In addition, these two curves provide an upper and lower bound for Young's modulus of EP particle compacts. In the authors' view, the lower bound is closer to the true value considering the minor mechanical role of the debris. Compression wave velocity measurements (Fig. 5a) and Eq. (2) were used to compare the constrained moduli for packed EP particles to those obtained in an earlier mechanical testing study [23] which used the unloading path after loading to peak stresses equal to 10, 15 and 20% of the maximum compaction stress (Fig. 10). Although the very different strain levels applied in elastic wave propagation and mechanical tests are known to give different results [25–28], the comparison of the elastic moduli shown in Fig. 10 provides an indication for the expected variation of the stiffness in EP compacts measured in different ways. Also shown are constrained moduli modified using the modified form of compact density. For the most part, the mechanical test results lie in between the two wave speed determined moduli which could

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consensus among researchers about the influence of porosity on Poisson's ratio. While some consider Poisson's ratio as a constant [29], others claim it to be a function of porosity [30–32]. Such debates may arise from the fact that the Poisson's ratio is usually derived as a function of two other elastic moduli (i.e. Young's and shear moduli) whose relative dependence on porosity may intensify or supress the dependence of Poisson's ratio on porosity. The evolution of Poisson's ratio in the low density compacts seems to be affected by pore character rather than solid phase material properties. The compacted bed of EP particles has a double-porosity structure; intra-particle porosity within the individual particles and inter-particle porosity between the perlite particles. Thus, there are two phenomena which contribute to Poisson's ratio of the sample: Poisson's effect of unit cells within individual particles and Poisson's effect of the whole particle. This suggests that in the low density samples (ρ b 0.15 g/cm3), the bulk elastic response may be dominated by the Poisson effect of cells as well as the compliance of the inter-particle contact region. Initially contact areas are relatively small and well separated, hence localised lateral deformation (Poisson effect) has a greater probability of intruding into the porous region between particles rather than straining the adjacent particle. Therefore, overall lateral deformation is reduced resulting in suppression of the Poisson's ratio. Further increase in density reduced the porosity between particles and increased the contact size along with increasingly more solid phase material involved in load transfer throughout the bulk. Consequently, the lateral deformation in both unit cells and individual particles contributes to the observed Poisson's ratio. 4. Mathematical models for prediction for elastic properties of porous bodies

Fig. 5. The (a) Compression wave and (b) Shear wave velocities as a function of compact density and modified density.

be considered as upper and lower bounds for constrained moduli of EP particle compacts. For a compacted density of 0.1 g/cm3, the constrained modulus determined from mechanical tests is higher than those determined by elastic wave tests. Additional mechanisms such as friction and other inelastic processes during mechanical tests could be responsible for this behaviour; however, further investigation would be required to verify this hypothesis. Fig. 8b shows Poisson's ratio of EP particle compacts as a function of compact density as well as the normalized Poisson's ratio ( ν/ν0) as a function of porosity. As the calculation of Poisson's ratio is independent of density (Eq. (5)), the values obtained by experimental and modified wave velocities resulted in the same Poisson's ratios. Poisson's ratio does not show as large a variation with density as Young's modulus does within this range of compact density (or porosity). It increases up to density 0.25 g/cm3 followed by a decreasing trend however the changes are small (b 10% over the whole range). There is no general

Fig. 6. Formation of platy and fine particles as a result of brittle crushing of cell walls.

A large number of empirical and semi-empirical models have been developed explaining the porosity dependence of material properties. The majority of these models are successful in predicting elastic properties within the porosity range of b38% (e.g., [29,33,34]) and some assume vanishing of elastic properties at porosity of 50% (i.e. as a result of an implicit assumption of symmetry between pore and solid phase [35–38]). However, this is not valid as many granular and porous materials have measurable elastic properties for porosities N 50% (e.g., bodies made of hollow spherical particles [39]). Consequently, models have far been developed which cover the other end of porosity spectrum (N50%). In this study, four of these models were used to predict elastic properties of EP particle compacts of different densities. It is noteworthy that when choosing a model, three criteria were considered: (i) interpretation of physical parameters of the model (ii) whether standard relations of linear elasticity hold between all moduli (iii) the ability of the model to satisfy boundary conditions. The boundary conditions include prediction of solid phase properties at zero porosity and the presence of a critical porosity, or percolation limit, at which particles no longer form a continuous network. Hence stiffness (i.e. E, G, K) goes to zero and the Poisson's ratio approaches an asymptotic value. This value has been found to be independent of Poisson's ratio of the solid phase but dependent on the geometry of the solid phase at the critical porosity [40]. Experimental and numerical investigations have shown that regardless of the solid phase Poisson's ratio, the Poisson's ratio of a porous body asymptotically approaches a fixed value which has been identified differently to be 0.2 [41–43] or 0.25 [30,44]. Dunn [32] analytically investigated four different pore shapes as a function of the aspect ratio of a spheroid and found pore shape is another factor affecting the asymptotic behaviour of Poisson's ratio. It was found that the asymptotic value for spherical and needle shaped pores is 0.2 while for disk-shaped pores and penny shaped cracks it is 0. With respect to experimental and analytical findings, the critical Poisson's ratio is considered to lie in the range 0 ≤ νcr ≤ 0.25 at critical porosity, Pcr. For a granular material, Pcr falls within the range of gravity induced packing states Ptap b Pcr b Pgreen, where Ptap and Pgreen are porosities of the “tapped” and “as-poured” packing states [45]. In evaluation of the

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Fig. 7. (a) Percentage of debris at each EP compact density. (b) Cumulative particle size distribution for entire compacts (EP particles and debris).

models, the value of Pcr is considered to lie within these bounds in order to preserve its physical interpretation. It should be noted that particle crushing and production of debris is not taken into account in any of the models' assumptions. Hence, it would be more appropriate to apply these models to the modified values. However, to investigate the effect of particle debris on the physical features of the models, the models were applied to moduli determined using both the raw compact densities (hereafter the experimental moduli) and moduli calculated using modified densities, hereafter the modified moduli. In the following, each model and its physical features is explained, and the applicability of these models to EP particle compaction data are discussed. 4.1. Phani models Phani and Niyogi [46] have derived a semi-empirical relation based on a simple model of applying a uniform state of stress on a porous body of constant cross sectional area and constant length. This model has been found to agree well with the data from many polycrystalline brittle solids over a wide range of porosity [46–49] and is expressed by: E ¼ E0 ð1−aPÞnE

ð9Þ

where E and E0 are Young's modulus at volume fraction porosity P and zero, respectively, and a and nE are packing geometry and pore structure dependent parameters, respectively. This model satisfies boundary conditions: E = E0 at P = 0, and E = 0 at P = Pcr = 1/a. Rice [50] theoretically derived the values of Pcr to be 0.785, 0.964 and 1.0 for cubic staking of cylindrical, spherical and hexagonal pores, respectively. Knudsen [51]

studied the contact area as a function of bulk density and calculated the value of Pcr for rhombohedral, orthorhombic and cubic packing of spherical particles to be 0.26, 0.397 and 0.476, respectively. From the theoretically derived values of Pcr and the existing relationship between Pcr and parameter a (i.e. Pcr = 1/a), the value of a lies in the range 1 ≤ a ≤ 3.85. For random packing with isolated spherical pores, constants a and nE were found to be about 1 and 2, respectively [46]. However, increase in nE corresponds to transition of pores from being spherical to be more interconnected. Phani and Sanyal [52] derived a relation between the shear modulus and Young's modulus of isotropic porous solid based on the Mori– Tanaka mean-field approach in which elastic properties are obtained by subjecting the inclusion (i.e. pores) to an effective stress or strain field. This relation is given by: G¼



  2 E 1−2ν 0 E n0 ð1 þ ν0 Þ þ G0 3 3 E0 E0

ð10Þ

where G is shear modulus, E is predicted from Eq. (9) and n0 is a constant for a given data set of a porous material related to pore morphology. The value of n0 can be evaluated using the experimentally measured values of E and G values at a single porosity value. This model satisfies boundary conditions: G = G0 when E = E0 at P = 0, and G = 0 when E = 0 at P = Pcr. Predicted values of Young's moduli from Eq. (9) along with the ones of shear moduli from Eq. (10) were used to evaluate Poisson's ratio by, ν¼

E −1 2G

ð11Þ

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Fig. 9. (a) Young's modulus (E) of packed beds of EP particles vs compact density. (b) Poisson's ratio of packed beds of EP particles vs compact density. In both graphs, the upper and right hand scales allow the Normalized moduli vs porosity to be read from the same graphs. In both graphs “Experimental moduli” refers to moduli determined based on raw compact density and “Modified moduli” refers to moduli determined based on modified density.

better agreement with the ones obtained based on the modified densities. It should be noted that Poisson's ratio is a small quantity dependent on differences of other elastic moduli and is hence very sensitive to error in them [19]. The considerable deviation from the experimental moduli of values predicted by the Phani model might be ascribed to the effect of the shape factor nE as a single parameter to reflect the change in porosity as compaction proceeds. This effect is discussed in more detail later. Fig. 8. (a) Particle density versus particle size, this graph also includes debris density versus debris size; (b) Inter-particle Porosity (excluding debris) versus compact density; (c) Inter-particle space filled by debris versus compact density.

as a function of porosity. The required data for pore-free solid phase (i.e. E0, G0 and ν0) were obtained from Table 2. Non-linear regression coefficients to fit the model to the experimental and modified moduli are presented in Table 3. The regression value for nE in both cases (i.e. based on the experimental and modified moduli) was indicative of the compacts' pore structure being non-spherical and partially interconnected. The regression value of constant a (i.e. Pcr = 1/a) gives Pcr of 1.0. For packed beds of EP particles, the range Ptap b Pcr b Pgreen was experimentally determined to be 0.96 b Pcr b 0.98. Thus, the estimated value of Pcr by Eq. (9) is slightly higher than the expected range. Application of the Phani model to both experimental and modified moduli is illustrated in Fig. 11a–d. In both cases, Young's modulus values predicted by Eq. (9) show a considerable deviation (Fig. 11a and c). Though, predicted Poisson's ratios (Fig. 11b and d) do not follow the same trend as experimentally measured values, they show slightly

Fig. 10. Constrained modulus of packed EP particle beds measured via elastic wave speed (this work) and mechanical tests [23]. Data using the raw densities and the densities modified to exclude debris are shown.

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Table 3 Physical parameters of mathematical models applied to the experimental and modified moduli. The physical parameters of the modified moduli are given in a parenthesis. Model Name

Physical Parameters

Phani Nielsen Rice [1] Wang Gibson-Ashby

a = 1 (1); nE = 2.8 (3.095); n0 = 1.104 (1.053); Pcr = 1 (1) β = 0.24 (0.12); Pcr = 0.965 (0.979) b'E = 0.036 (0.02); b'G = 0.0327 (0.02); Pcr = 0.976 (0.978) b = 3 (3.63); c = 3.1(3.08); d = 0.95 (1); Pcr = 1 (1) C1 = 0.1 (0.068); C1′ = 0.1 (0.072); C2 = 0.037 (0.026); C2′ = 0.039 (0.0276); Pcr = 1 (1)

1. The subscript E in b'E refers to Young's modulus and subscript G in b'G refers to shear modulus.

4.2. Nielson model Nielsen [53] proposed a model for predicting elastic moduli of porous materials based on the composite sphere assemblage (CSA) approach. CSA considers a porous body to be constituted of congruent composite elements consisting of a spherical pore embedded in a concentric spherical shell of matrix material. It is assumed that the composite spheres are available in an infinite range of sizes and they are distributed in a way that smaller composite spheres fill all the interstices between larger spheres. This model was originally introduced by Hashin and assumed the applied stress on the assembled body is uniformly distributed (hydrostatic) around each inclusion so that strains can be calculated to obtain elastic moduli for the body. Nielsen set forth to predict the elastic moduli for porous materials by assuming two types of composite elements: one as described above and the other one made of spherical matrix embedded in a concentric spherical shell of pore. These two types of CSAs defined two bounds of continuous solid phase (isolated pores) and continuous pore phase (isolated solid phase) where transition between them (different mixtures of them)

results in porous material with a wide range of porosity of any geometry. The Nielsen model for Young's modulus and Poisson's ratio are expressed as: E¼

2βE0 ð1−P Þð5ν 0 −7ÞðP cr −P Þ 2β ð5ν 0 −7ÞðP cr −P Þ þ P cr P ðν 0 þ 1Þð15ν0 −13Þ

ð12Þ

ν¼

2βν0 ð5ν0 −7ÞðP cr −P Þ þ P cr P ðν 0 þ 1Þð5ν 0 −3Þ   2β 5ν 0 −7 ðP cr −P Þ þ P cr P ðν 0 þ 1Þð15ν 0 −13Þ

ð13Þ

where β is a shape factor characterizing the ability of the material to transfer stress and providing information on the shape of the pores and their inter-connectivity. The Shape factor β is in the range 0 b β b 1. The lower bound, β = 0, is when the pore phase totally surrounds the solid phase (i.e. particle) while the upper bound, β = 1, corresponds to when solid phase completely surrounds the pore phase (i.e. isolated pores). Hence, the lower value of β correlates with highly interconnected pores and smaller contact areas between particles, while higher values of β indicates that pores are becoming increasingly more isolated. The Nielsen model satisfies boundary conditions at zero porosity and correctly predicts both solid phase elastic moduli (i.e. E = E0, ν = ν0). It also predicts the presence of the critical porosity at which Young's mod5ν0 −3 . Given that ulus goes to zero and Poisson's ratio is given by ν Cr ¼ 15ν 0 −13

Poison's ratio for isotropic elastic material lies in the physically realistic range 0 ≤ ν0 ≤ 0.5, this model predicts critical Poisson's ratio to lie in the range 0.09 ≤ νcr ≤ 0.23, which does not cover the expected range 0 ≤ νcr ≤ 0.25. This limited range may have a significant effect on the quality of predicted values for Poisson's ratio. Non-linear regression analysis was conducted to fit the Nielson model to the experimental and modified moduli and results are shown in Table 3. The predicted values of Pcr for both the raw compact

Fig. 11. (a) Normalized Young's modulus vs porosity (based on experimental moduli); (b) Normalized Poisson's ratio vs porosity (based on experimental moduli); (c) Normalized Young's modulus vs porosity (based on modified moduli); (d) Normalized Poisson's ratio vs porosity (based on modified moduli).

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and modified densities were within the specified range 0.96 b Pcr b 0.98 and the low value of β is representative of a high-volume fraction of pores compared to the solid phase. Fig. 11a and b show that the agreement between the Neilson model and experimental moduli is not very good. In case of Poisson's ratio, the discrepancy can be attributed to the low value of νcr (i.e. 0.09 b νcr b 0.23) and the shape factor β. Similar to nE in the Phani model, the shape factor β in the Nielson model is used as a single value to accommodate microstructural change during the compaction for compacts of different densities with a wide range of pore shapes and geometry. This problem might be alleviated by solving Eqs. (9) and (12) point by point. The results of this calculation for both shape factors (nE and β) as a function of porosity are presented in Fig. 12a. As can be seen, both shape factors change as a function of porosity and the results show the opposite of what was expected. At lower porosity β approaches zero while at successively higher porosity where the structure is open and interconnected it approaches 0.5 for experimental moduli and exceeded the upper bound of unity for modified moduli. Similar to β, the evolution of shape factor nE with porosity does not follow the expected behaviour defined by Phani and Niyogi. To accommodate these differences, shape factors were expressed as a function of porosity P, given by, SðP Þ ¼ η1 þ η2 P þ η3 P 2 þ η4 P 3 þ ::::: þ ηn P n−1

ð14Þ

where S is a shape factor, either nE or β,and ηn (n = 1.2,3,..) are empirical constants. Applying the modified form of shape factor in Eq. (14) to the experimental and modified moduli up to n = 4, both Nielson's and Phani models are replotted and presented in Fig. 12b-c. It can be seen that both Nielson's and Phani models for Young's modulus show close agreement with the experimental moduli when the shape factor was

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defined as a function of porosity. In the case of Poisson's ratio, the modified form of shape factor improved the results of the Phani model but not those from the Neilson model (Fig. 12c). Therefore, as the Nielson model does not exhibit good agreement with both Young's modulus and Poisson's ratio, it is not considered to be a good model for the prediction of elastic properties of EP particle compacts. 4.3. Minimum solid area (MSA) models The minimum solid area (MSA) model assumes the macroscopic elastic response is related to the load-bearing area of the solid phase material. The key assumption in MSA models is that normalized elastic moduli scales according to the minimum area of solid phase per unit area in a plane perpendicular to the direction of applied load, i.e. EE0 ¼ MSA. In this context, a common approach is to assume an idealized microstructure, including regular stacking of solid particles in a void matrix, and regular stacking of pores (spheres, cylinders, etc.) in a solid matrix. The initial and major development of such models was Knudsen's model [51]. Knudsen investigated the ideal microstructure of three particle stackings (i.e. simple cubic, orthorhombic, rhombohedral) and found the stacking of particles and hence the pores is significant. In addition, plotting of resultant models as a function of porosity on semi-log plots for low to intermediate porosity was approximated by, E ¼ E0 e−bP

ð15Þ

where b is related to particle stacking and hence a function of pore shape, geometry and alignment with respect to the stress axis. This equation has been criticised [34] for not satisfying the boundary

Fig. 12. (a) Shape factor vs compact porosity for Phani and Neilson models applied to both experimental moduli and modified moduli; (b) Phani and Nielson models with modified shape factors for Young's Moduli as a function of porosity; (c Phani and Nielson models with modified shape factors for Poisson's ratio as a function of porosity.

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condition that E = 0 at P = 1. However, such criticism is only valid when critical porosity occurs, at about unity. Anderson [54] analytically proved the existence of such exponential dependence on porosity from strain analysis of isolated ellipsoidal pores. He stated that Eq. (15) becomes invalid at very high pore fractions where constant b does not reflect pore interactions occurring as the concentration of pores increases. Rice [50] discussed that this relationship (Eq. (15)) provides a good approximation for effective properties up to 1/3–1/2 of P/Pcr and holds true for all elastic moduli. In attempts to estimate minimum solid area for higher porosity, Rice and Wang proposed two equations. Rice [55] suggested that the role of pores and solid phase material can be exchanged (i.e. stacking of pores rather than particles) leading to the theoretical equation:  0 M ¼ M0 1−e−b ð1−P=Pcr Þ

ð16Þ

where M is Young's, shear or bulk modulus and b′ is related to pore stacking. Rice discussed that b′ decreases as b increases. For instance, for spherical pores in cubic stacking, the value of b is 2.7 while the value of b′ is 0.5. For other pore shapes and stackings, values of b and b′ can be obtained from Fig. 2 in Ref [55]. This model predicts zero stiffness as porosity goes to unity but it does not satisfy the zero porosity boundary condition since (1 − exp (− b′(1 − P/Pcr)) does not go to unity as P goes to zero. Rice explained this by expressing that Eq. (16) is a continuation of Eq. (15) for higher porosity values and no extrapolation between them is required. Wang [29] used the ideal model of spherical particles in a simple cubic stacking developed by Knudsen [51] and modified it for real microstructures. The modifications account for misalignment of uniaxial applied stress which induces shear and hinge effects at the neck (small contact area between two particles). Wang derived a complex relation between porosity and Young's modulus and proposed an approximate solution to his exact solution which could satisfactorily cover a wide range of porosity, given by, h  i 3 E ¼ ES exp − bP þ cP2 þ dP þ :::

ð17Þ

where b, c and d are empirical constants. Brown et al. [56] analytically derived the value of b for different idealized pore geometry and orientations and proposed a similar equation to Wang's model, but in the context of strength, given by ln ðσ=σ 0 Þ ¼ − ∑ bi P i − i

2

3

j 1 1 1 ∑ bi P i − ∑ bi P i − ∑ bi P i 2! i 3! i j! i ð18Þ

where bi is related to pore geometry and orientation of the ith kind of pore and Pi is the contribution made by the pores of ith kind to the total porosity. Therefore, the constants of Eq. (17) can be expressed in terms of constants of Eq. (18). Wang has shown the proposed equation with quadratic exponent can predict the Young's modulus of a porous material up to a porosity of 38% [29] and additional higher order terms can be included for higher porosity. The Wang model with a cubic exponent was used for prediction of elastic moduli of EP particle compacts. Non-linear regression analysis was conducted to fit the Wang and Rice models to the experimental and modified moduli. Results are presented in Table 3. The critical porosity value predicted by the Wang model is slightly higher than the specified range 0.96 b Pcr b 0.98 while the value predicted by the Rice model (for both shear and Young's moduli) is within the range. As shown in Fig. 11a and c, the Rice model shows very close agreement with experimental moduli, however, it did not show such agreement with modified moduli. Similarly, the Wang model shows better agreement with experimental moduli than the modified Young's modulus values (Fig. 11a and c). The regression parameters in Wang's model

for experimental Young's moduli in terms of the constants in Eq. (18), correspond to; b - cylindrical pores aligned perpendicular to the loading direction; c - oblate pores (with ratio of about 0.8) in a cubic stacking; and d - a combination of about 70% cubical pores in 〈100〉 orientation with 30% cylindrical pores parallel to the loading direction [57–59], respectively. The interpretation of the regression parameters in Wang's model for modified Young's moduli is similar to those for experimental moduli, except for constant b (3.63) which corresponds to a combination of about 77% cubical pores in 〈110〉 direction with 23% cubical pores in 〈111〉 direction. This approximation seems reasonable with respect to the cell shape in EP particles (Fig. 1b) and the fact that pores can have different directions in a real packing of particles. The predicted Young's and shear moduli based on both experimental and modified moduli were found to lie within simple cubic and rhombohedral packing of pores. For comparison, a similar range of values can be acquired from Fig. 2 in Ref [55]. Rice discussed that porous bodies are better modelled by combining two or more idealized pore structures, as opposed to just one. Consequently, the values of b′ (Table 3) can be ascribed to different combinations of these two pore packing geometries (e.g. simple cubic and rhombohedral) and their interaction as porosity evolves. Poisson's ratio values were calculated using predicted Young's and shear modulus values from Eq. (16) and combining them using Eq. (11). Poisson's ratio values predicted by the Rice model based on raw compact density show close agreement with experimentally measured Poisson's ratio values (Fig. 11b). However, the deviation of Young's and shear moduli values predicted by the Rice model based on modified moduli resulted in poor predictive results for Poisson's ratio values (Fig. 11d). As a whole, the Rice model was shown to be successful in predicting elastic properties of EP particle compacts only when the raw compact density is considered. 4.4. Gibson and Ashby model Gibson and Ashby [7] derived a semi-imperial model for the relationship between elastic moduli and porosity via dimensional arguments (using standard beam theory) for a cellular solid. For simplicity, they assumed cell struts and walls of uniform dimensions, while in reality these will usually be tapered toward and thinner in the centre (i.e. due to the effect of surface tension during the foaming process). The correction for this tapering was implemented by arranging the cubic cells in a staggered stacking, so that their members meet at their midpoint. This model has been shown by many authors (e.g., [60–65]) to successfully predict elastic moduli of closed cell foams and are given by the following expression: E ¼ C 1 ϕ2 ð1−P Þ2 þ C 01 ð1−ϕÞð1−P Þ E0

ð19Þ

G ¼ C 2 ϕ2 ð1−P Þ2 þ C 02 ð1−ϕÞð1−P Þ E0

ð20Þ

where C and C′ are constants of proportionality, ϕ is the volume fraction of solid in the cell- struts, which can be obtained from the relative area of struts and face on a plane section as explained in Ref. [66], and the remaining fraction (1-ϕ) is the solid contained in the cell faces. The quadratic term describes the contribution of the cell struts bending to the modulus and is the same for open cell foams. The linear term corresponds to the cell walls' lateral stretching. Gibson and Ashby suggested C1 and C1′ should be about unity and C2 and C2′ should be about 3/8 for porous materials with ν0 = 0.33, where theses satisfy boundary conditions at P = 0 and P = Pcr = 1. However, the values of C1 and C1′ will change based on volume fraction of solid in the cell struts, the variable geometry of the foam and uncertainty in the value of the solid Young's modulus, E0 [7]. As a result, the values of these constants depend on the type of a foam and vary from one foam to another.

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The Gibson–Ashby relation provides a good approximation for the cellular structure of EP particles shown in Fig. 1b. For a given porosity and solid fraction of cell struts (ϕ = 0.78), the regression analysis yielded the values C1 and C1′ in Eq. (19) for both experimental and modified Young's modulus values (see Table 3). Application of the Gibson and Ashby model to the experimental and modified Young's modulus values is presented in Fig. 11a and c. This model shows good agreement with the experimental Young's modulus values. In the case of the modified Young's moduli, the Gibson and Ashby model demonstrates reasonable average agreement with the moduli for porosity higher than 0.87, while for lower porosity a considerable deviation is observed. Gibson and Ashby suggested that if C 1 and C 1 ′ are about unity and GE ≈ 38 , which holds true for polycrystalline metallic materials [67], Poisson's ratio of the foam is about 0.33. Since constants C1 and C1′ are not unity (see Table 2), this relation does not hold and Poisson's ratio should be calculated using two elastic moduli. For this purpose, Young's moduli from Eq. (19) along with the shear moduli predicted from Eq. (20) were used to calculate Poisson ratios by Eq. (11). For a given porosity and solid fraction of cell struts (ϕ = 0.78), the regression analysis yielded the values of C2 and C2′ in Eq. (20) for both experimental and modified shear modulus values (see Table 3). It is noteworthy that although C1 and C1′ are not unity, the ratio of shear to Young's moduli is almost 3/8, which gives implicit satisfaction of the boundary condition (explained above). The results for Poisson's ratio are presented in Fig. 11b and d. As can be seen, Poisson's ratio values predicted by the Gibson and Ashby model show good agreement with both the experimental and modified Poisson's ratio. Overall, the Gibson and Ashby model gives a satisfactory means of predicting the elastic properties of EP particle compacts based on experimental moduli and modified moduli at the upper end of the porosity range.

5. Conclusions Elastic wave velocity measurements were used to characterise the elastic properties of packed beds of EP particles in terms of two isotropic elastic moduli (Young's modulus and Poisson's ratio). Particle distributions within EP particle compacts after compaction showed that as compact density increases, the percentage of larger particles decreases due to breakage into smaller particles and the production of debris. Despite this process, it was found that the inter-particle porosity remains relatively constant with increasing compact density. The mass% of debris rises to quite high levels however, due to the high density of debris, the volume of debris never exceeded 20% of the available inter EP-particle space. Due to the formation of debris, the experimental compact densities were modified by deducting the debris mass from the total mass of EP particles and new densities were calculated. Young's modulus calculated based on the experimental and modified density values showed a decrease with porosity. However, Poisson's ratio was relatively independent of porosity. The properties of solid perlite were investigated by sintering the powdered perlite into low porosity solid samples and the application of elasticity theories. The elastic properties of solid perlite were found to be very close to those for solid obsidian reported by Manghnani and co-workers [20]. Using these properties of solid perlite, porosityelastic moduli relations were investigated. Four analytical models predicting the elastic moduli of packed beds of EP particles from the properties of the parent material were investigated. Although these models were not developed to explicitly handle debris, the Wang, Rice, and Gibson and Ashby models showed reasonable agreement with experimental moduli. However, none of the original models predicted the modified moduli well although the Wang and Gibson and Ashby models gave a reasonable average trend. It was shown that modifying the Phani model shape factor and expressing it as a function of porosity can provide a satisfactory means of predicting the elastic

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properties of EP particle compacts based on both the experimental and the modified densities. Acknowledgements The authors gratefully acknowledge assistance from staff in the electron microscope and X-ray unit at the University of Newcastle, NSW, Australia. H. H. Allameh Haery acknowledges receipt of University of Newcastle IPRS and Postgraduate Research scholarships. We are also grateful to one of the reviewers for suggestions concerning the role of debris in the measurements. References [1] P.A. Ciullo, Industrial Minerals and their Uses: A Handbook and Formulary, Noyes Publications, Westwood, N.J, 1996. [2] H. Allameh-Haery, E. Kisi, T. Fiedler, Novel cellular perlite–epoxy foams: effect of density on mechanical properties, J. Cell. Plast. (2016). [3] G. Straffelini, V. Fontanari, A. Molinari, True and apparent Young's modulus in ferrous porous alloys, Mater. Sci. Eng. A 260 (1999) 197–202. [4] J.R. Moon, Elastic moduli of powder metallurgy steels, Powder Metall. 32 (1989) 132–139. 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