Polypropylene structural foams: Measurements of the core, skin, and ...

Original article

Polypropylene structural foams: Measurements of the core, skin, and overall mechanical properties with evaluation of predictive models

Journal of Cellular Plastics 2017, Vol. 53(1) 25–44 ß The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0021955X16633643 cel.sagepub.com

Tarik Sadik1, Caroline Pillon1, Christian Carrot1, Jose´ A Reglero Ruiz2, Michel Vincent2 and Noe¨lle Billon2

Abstract Relationships for the prediction of various linear mechanical properties of polymeric sandwich foams obtained in injection processes were studied in comparison with shear, tensile, and flexural tests. The samples were obtained by a core-back foam injection molding process that enables one to obtain sandwich materials with dense skins and a foamed core as revealed by the morphological analysis. Tensile, shear, and flexural moduli were investigated for the skin, the core, and the overall foamed structure. In addition, the Poisson’s ratio of the skin was also determined. The core properties were specifically analyzed by machining the samples and removing the skins. Tensile and shear properties of the core can be well described by the Moore equation. The tensile modulus can be calculated by a linear mixing rule with the moduli of the skin and of the core in relation to the thickness of the layers. Shear and flexural moduli are described by a linear mixing rule on the rigidity in agreement with the mechanics of beams. Tensile modulus, out-of-plane shear modulus, and flexural modulus can finally be predicted by the knowledge of only very few data, namely the tensile modulus and

1

UMR 5223, Ingeńierie des Mate´riaux Polyme`res, Universite´ de Lyon, CNRS, Universite´ Jean Monnet, Saint Etienne, France 2 UMR 7635, Centre de Mise en Forme des Mate´riaux, MINES ParisTech, CNRS, Sophia Antipolis Cedex, France Corresponding author: Christian Carrot, UMR 5223 Ingeńierie des Mate´riaux Polyme`res, Universite´ Jean Monnet, CNRS, 23, Rue du Dr Paul Michelon, F-42023, Saint-Etienne Cedex 2, France. Email: [email protected]

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Journal of Cellular Plastics 53(1)

Poisson’s ratio of the matrix, the void fraction, and thickness of the core. The equations were proved to be physically meaningful and consistent with each other. Keywords Foaming, injection, polypropylene, elastic properties, sandwich foam, tensile, shear

Introduction The current environmental issues add pressure on the automakers to manufacture fuel-efficient cars with less CO2 emissions. Equipment manufacturers are focused on the research of a technology to develop lighter cars with the basic principle that a lighter vehicle requires less energy. A 100 kg weight reduction in a car means up to 5% of increase in fuel economy and 10 g/km emission reduction of CO2. The target is therefore to reduce the weight of the vehicles by 10–15% by 2015. European automakers have great experience in manufacturing fuel-efficient vehicles as they need to comply with the fuel economy standards and emission regulation for a long time. Weight reduction is sought by the use of lightweight materials; by the optimization of the design of different components; or by innovations in manufacturing process of course with balancing cost, weight, and performance. The weight fraction of plastics in cars is expected to grow since these materials provide reduction in interior and exterior parts including the use of composites for high-end luxury cars. In turn, the weight reduction in parts made with plastics also becomes a challenge.1 European governments funded project to develop processes that could lead to auto component weight reduction of up to 50%. For plastics parts, this may rely on an injection molding process that combines moving mold cores, a chemical blowing agent, and lightweight reinforcing fillers. The molding process also called core-back foam injection molding differs slightly from traditional injection molding. The polymer, containing a foaming gas dissolved in it, fills up the mold. The back portion of the mold is moved and the cavity volume quickly increases; therefore, the pressure exerted on the polymer in the cavity decreases. The sudden pressure drop promotes bubble nucleation and achieves expansion with a fine cell structure between two solid skins. This new in-mold foaming process may enable a weight reduction in the part of at least 30% compared to conventional solid molding. The use of new reinforcing fillers may reduce part weight by as much as 7% globally fulfilling the target. Injection molding of foamed materials based on flakes or fiber reinforced polypropylene (PP), which is a widely used polymer in automotive industry, and the resulting mechanical properties have been addressed in the case of glass fiber, cellulose fibers, nanosize calcium carbonate and talc with nitrogen or carbon dioxide as blowing agents.2–5 The present paper addresses the prediction of mechanical properties through analysis of foam structure, linear mechanical properties, and their modeling for PP reinforced with novel synthetic fibers and processed by core-back foam injection molding.

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The prediction of the mechanical properties of homogeneous and sandwich foams by models has been reviewed in a series of paper by Zhang et al.6–8 on polyethylene foams in various tests including shearing, tensile, and flexural modes. The idea of beam models equivalent cross-sections related to density distribution was initially proposed by Hobbs9 and reviewed by many authors.10,11 For homogeneous foams, various equations have been proposed to describe the tensile modulus (Ec) and the shear modulus (Gc) as a function of the void fraction fc and the matrix core modulus (Em or Gm). These are mainly the linear mixing rule Ec ¼ ð1 fc ÞEm

ð1Þ

Ec ¼ ð1 fc Þ2 Em

ð2aÞ

the empirical Moore equation12

and ad hoc derived power law with adjustable exponent Ec ¼ ð1 fc Þn Em

ð2bÞ

the Kerner equation,13 depending also on the Poisson’s ratio of the matrix polymer nm Ec ¼

1 fc Em m 1 þ 810v 75vm fc

ð3Þ

In all these equations the dependence of the tensile and shear moduli on void fraction is the same. In contrast, the equations proposed by Ramakrishnan14 are different for tensile and shear modulus Ec ¼

1þ

ð1 fc Þ2 ð15vm Þð3vm 1Þ fc 2ð75vm Þ

Em

ð4aÞ

and Gc ¼

ð1 fc Þ2 Gm m 1 þ 1119v 4ð1þvm Þ fc

ð4bÞ

For sandwich foams, the model must combine the properties of the core and skin. In this paper, the data are expressed in terms of modulus in the frame of Figure 1 with axis 1 and 2 in the plane of the sandwich and direction 3 being normal to this plane. In addition, direction 1 is the injection direction. Without orientation due to the process, the planar isotropy may be assumed, so that properties in directions 1 and 2 are equivalent.

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Figure 1. Reference axis for the mechanical data of sandwich structure.

The simple combination for the tensile modulus E (standing for E11 or E22) must be a linear mixing rule of the core and skin properties taking into account the relative thickness of both parts since the deformation of both layers is the same while the load is shared between them. Hence c c E ¼ Ec þ 1 ð5aÞ Em where c is the foamed core thickness and is the total sample thickness. For thin skins, an approximation could be written in the form E ð1 fÞ2 Em

ð5bÞ

For the shear modulus in the 1,2 direction G (standing for G12) according to Figure 1, linear mixing rule for the sandwich applies on rigidity rather than on tensile modulus.15 Therefore GI ¼ Gc Ic þ Gm Im with I, Ic, and Im being the inertia of the sandwich, core, and skin. This yields the mixing rule 3 3 ! c c G¼ Gc þ 1 Gm

ð6aÞ

ð6bÞ

Approximation for thin skins under the assumption that, in this mode, most of the load is supported by the skins could be expressed as

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ð6cÞ

Similarly, for the flexural modulus, the considerations on the bending of the test bar in small deformation can be expressed by a mixing rule on rigidity, which can be further simplified since flexural modulus and tensile modulus are the same for the isotropic materials, skin or core FI ¼ Fc Ic þ Fm Im ¼ Ec Ic þ Em Im where F is the flexural modulus of the sandwich foam. This yields a mixing rule in the form 3 3 ! c c F¼ Ec þ 1 Em

ð7aÞ

ð7bÞ

Mixing rule for the shear modulus in the plane of the sandwich (G13) was not studied in this paper. Nevertheless, an approximation could be easily derived from the assumption that most of the deformation is supported by the core and that the skins are not deformed G13 Gc

c

ð8aÞ

The above expression can be reduced for thin skins as G13 ð1 f Þ2 Gm

ð8bÞ

Materials and methods Materials The matrix polymer is provided by Sumika Company. It is made of PP compound with high fluidity containing 11% of mineral fillers as confirmed by thermogravimetric analysis (TGA). The differential scanning analysis (DSC) trace reveals the presence of a small amount of elastomeric olefinic copolymer in the PP (Figure 2). The clear separation of melting peaks and the occurrence of the melting peak of PP at the expected conventional temperature of 168 C indicate a simple blend of the olefinic polymers. The mineral fillers consist in a mix of talc with high performance and high modulus magnesium-based fillers that provide high performance in polyolefin compounds with reduced densities in comparison to traditional fillers. This filler is expected to allow stiffness with minimal loss of impact properties balance and reduced weight because the amount of filler required for given properties is reduced in comparison with low form factor fillers such as talc. Hybrid

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compounds with talc remain nevertheless necessary in order to keep reasonable economic viability in industrial applications. A foaming agent composed of sodium bicarbonate and citric acid (1/3 molar) dispersed at 65 wt% in polyethylene was used as a master batch. Two percent of this master batch was mixed with the PP for injection molding. In these proportions and for heating rates higher than 10 C/min, which is the case in the barrel of the molding machine, the decomposition takes place nearly in one step with maximum rate between 170 and 200 C (Figure 2) to yield water and carbon dioxide (1/1 molar) and sodium citrate as a by-product.

Processing Samples were injection molded into plaques of 165 mm 95 mm (and some smaller samples with dimensions 100 mm 100 mm). Material is injected into a mold provided with a moving back plate that is initially in the forward position. Once the skins of the part have solidified, the moving back plate retracts at a programmed position that gives the final thickness of molded parts. This lowers the pressure in the mold cavity. The chemical blowing agent that has previously reacted in the barrel of the injection molding machine and is dissolved in the melt, creates a cellular structure in the core. Samples were molded in various conditions varying the cooling time, temperature and pressure, opening time, and core-back displacement. This enables us to get samples with skin thicknesses 0.4 mm (and 0.65 mm for a few samples) and overall thicknesses varying between 1.7 and 3.1 mm. During the injection, the dissolved gas was expelled from the cooled skin or still remains for a small part, less than 0.3 g/kg,16–19 soluble in the polymer matrix at room temperature.

Figure 2. DSC trace of the polymer matrix and foaming agent during heating at 20 C/min.

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Thermal measurements DSC and TGA analysis were performed, respectively, in a TA DSC Q10 and a Mettler TGA/DSC 1 devices using nitrogen or air as the atmosphere. In particular, the amount of filler was determined by complete oxidation of the organic moieties.

SEM and density determination Morphological analysis, including skin () and core (c) thickness measurements, and void fraction (fc) of core of the sandwich foams were determined using an optical microscope and taking photos with a magnification of 5 on cryo-microtomed slices of the molded parts. Pictures were taken at various locations of the parts and analyzed using the ImageJ software. Contrast between skin (without bubbles) and core was sufficient to obtain satisfactory measurements of the thicknesses. Similarly, contrast between voids and bulk PP enables a very nice binarization of the images as shown in Figure 3. Void fraction (f) of the sandwich foam was calculated by f ¼ fc (c/). Void fraction f was also confirmed by density measurements.

Machining Parts were machined (removing 0.1 mm layers step by step) in order to remove skins and to get core samples for measurements of the mechanical properties. Skins in contact with the moving and fixed parts of the mold were also obtained by the same method. Figure 4 shows some of the machined samples.

Figure 3. Example of micrograph and the binarization to calculate the void fraction (core void fraction ¼ 0.53, total void fraction ¼ 0.36).

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Figure 4. Machined core of dumbbell shape testing bars.

Mechanical measurements All mechanical characterizations were achieved in the linear domain. Tensile modulus (E11) was measured at 22 C in a Shimadzu AGS-X Series tensile machine using a 10 kN cell and a noncontact digital video extensometer TRViewX. Crosshead speeds of 1, 10, and 50 mm/min were used. Because of the small strain at break, the original section was used in the calculation. Dumbbell shape testing bars with dimensions of the ISO 1/2 standard (gauge length width thickness: 50 mm 10 mm 4 mm) were injected for resin characterization. For some of the samples, tensile modulus of plain plates was also obtained in the transverse direction (E22), normal to injection direction. Tensile characterization of modulus (E11) of the core was performed on machined dumbbell-shaped testing bars with smaller dimensions (gauge length width thickness: 36 mm 4.5 mm 1.55 mm) extracted from the injected plates. Shear measurements were obtained by torsion experiments in simple shear. Small oscillating strains were applied on rectangular specimens, using either the parallelepipedic part of the injection-molded dumbbellshaped samples or alternatively, samples cut in the plain or foamed plates. An ARES TA instrument was used in torsion mode in the linear domain with 0.3– 0.5% strain during temperature ramps at 1 rad/s from 70 until 70 C. Isothermal tests were also carried out during frequency sweeps between 0.1 and 100 rad/s at room temperature (22 C). The shear modulus measured in torsion was G12*. Flexural modulus was measured according to NF EN ISO 178 standards at 22 C on rectangular specimens cut in plates and at 22 C. The width (w) was 25 mm and the distance between support (l) was variable according to the thickness (from 27 mm for 1.7 mm thickness until 42 mm for 2.6 mm). The velocity is of the

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crosshead displacement at the center of the beam was 1 mm/min. Strain and stress were calculated from the deflection (d) using the following equations 6de l2

ð9aÞ

3F1 2We2

ð9bÞ

"¼ ¼

where w, e, l are, respectively, the width, thickness, and distance between support of the sample, F is the force and d is the deflection, s and e are the stress end strain.

Results Morphology of the foamed plates Figure 5 shows the photographs of some sections across the thickness of the plates for various injection molding conditions and mold opening between 0.5 and 1.3 mm. In all cases, the pictures show a well-defined core–skin structure. SEM pictures of the skins in their section do not give evidence of bubbles. In these zones, the quick cooling of the polymer under pressure at the contact of the cool mold walls rapidly freezes the skin layer. The thickness of the solid skin depends on the cooling conditions, with values between 0.4 and 0.65 mm. Most samples presented in this paper have skin thicknesses of 0.4 mm. During the filling and packing processes, the gas is dissolved in the polymer melt due to the high-pressure values. In the mold opening, foaming occurs and most of the gas is released to create the bubble structure, although a small amount of the gas may remain in the solid skin. The situation of the core is different since the opening of the mold before the complete solidification of the polymer enables expansion of the gas in the molten polymer and the foaming of the core before its solidification. The SEM pictures of the core (Figure 6) show a regular foamed structure though some smaller bubbles are generally observed in the transition zone within the core. Bubble size is decreasing when the core-back displacement is larger. The average results of the measurements on the pictures of Figure 6 are given in Table 1. Measurements of the density of the foamed plates confirm the reliability of the image analysis and the homogeneity of the structure throughout the samples since whereas image analysis is a very local sampling, the density is a macroscopic measure. The calculation was performed using the value of 0.98 cm3/g for the density of the PP matrix. The reduction of density of the plates is between 25 and 35%. The variation of the void fraction fc with thickness can be well described using the following equation, which is only valid in this range of thicknesses fc ¼ 0:3 0:25 with in mm.

ð10Þ

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Figure 5. Photographs of sandwich foams at various overall thicknesses.

Tensile modulus Figure 7 shows a typical strain–stress curve obtained in tensile experiments for the skin, core, and sandwich foam. The experiment on the skin was obtained on a sample that was injection molded with no core-back displacement. The thickness of the sample was the nominal gap in the mold. Indeed attempts to measure the tensile modulus on machined skins were unsuccessful or gave inconsistent results. Sometimes, results were also found to be very different if the machined skins were taken on one side or the other of the molded plates though no cooling difference was used during the process. This is probably because the skins are so thin that the accuracy on the measurement is poor. Experiments on the core were performed after machining the samples. The curves show an expected trend for the various parts with the lowest modulus for the core and the highest one for the skin whereas values of the sandwich foam remain between the previous results. Table 2 gives the averaged values of the tensile modulus on five samples determined from the tangent at the origin on the strain–stress curve. In this table, experiments at various crosshead speeds are reported in some cases showing the viscoelastic effect on the data of the skins obtained on samples without wall retraction during the process. The viscoelastic behavior of the polymer induces a strong dependence of the modulus on the tensile speed. The value of 1 mm/min was selected to be comparable with the data in the flexural tests. Also, the data on the skin at 50 mm/min can be compared with data of the polymer injected in plain dumbbell test bars with a standard injection mold showing a lower modulus of the

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Figure 6. (a) Morphology of the core for various thickness (2.2, 2.4, and 2.6 mm from left to right). Horizontal direction corresponds to axis 3, perpendicular to injection direction. (b) Details of the transition zone (2) between the skin (1) and the center zone (3), core is made of (2) and (3).

Table 1. Typical values of thicknesses, void fractions, and densities of the samples of Figure 6. Thickness (mm)

2.2

2.4

2.6

Skin thickness (mm) Core void fraction fc Void fraction f Density (measured) d (g/cm3) Core density dc (g/cm3)

0.41 0.41 0.26 0.73 0.58

0.42 0.47 0.31 0.67 0.50

0.42 0.53 0.36 0.65 0.49

skin. This may be due to the orientation of the magnesium fibers that are less oriented in the plates than in the tensile dumbbells. Also the crystallization conditions may be significantly different. Data of two sandwich foams with various thicknesses are also reported showing the evident effect of a larger core and consequently a larger void fraction. The Poisson’s ratio was also measured using an

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Figure 7. Typical strain–stress curve during tensile experiments on skin, core, and sandwich foam (test speed: 50 mm/min).

Table 2. Tensile modulus of skin, core, and structural foams. Sample

E (22 C, in MPa)

Crosshead speed (mm/min)

Matrix (bulk)

2600 60 2200 60

50 1

Skin (no opening)

2300 60 1900 60

50 1

Core (machined, fc ¼ 0.53) Structural foam (2.2 mm) Structural foam (2.4 mm) Structural foam (2.6 mm)

390 20 1560 60 1300 60 1160 60

10 50 50 50

optical camera and marks on the testing bars, with an average value after five tests of 0.43 0.06 on a strain domain between 1.9 and 3% of strain.

Shear modulus (1, 2 axis) The shear modulus was measured using oscillatory rectangular torsion during isochronal temperature sweeps and isothermal frequency sweeps. Figure 8 shows

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Figure 8. Complex shear modulus (1,2 component) (a) and tangent of the loss angle (b) of skin, core, and sandwich foam in temperature ramp at 1 rad/s and in frequency sweep at 22 C (c).

the results obtained in both modes. Figure 8(a) shows the variation of the complex modulus for the matrix polymer, skin, core, and sandwich foam. This shows that the skin shear modulus is close to that of the neat PP. Also the moduli rank as expected, the core being the most compliant material. Figure 8(b) shows the tangent of the loss angle. The absorption capacity is generally also higher for foamed material, either core or sandwich, than for solid material and skin. The loss peaks also appear at temperatures similar to those of the matrix indicating no new polymer relaxation connected to the structure. The relaxation peak near 0 C is the glass transition of amorphous PP while that at 50 C reveals the presence of a small amount of elastomer, presumably an ethylene–propylene copolymer as already mentioned in the ‘‘Materials’’ section. The frequency dependence of the moduli at 22 C (Figure 8(c)) indicates a viscoelastic effect that is the same as that observed on the tensile modulus. A 15% decrease on tensile modulus was reported for the tensile modulus for crosshead speeds between 50 and 1 mm/min. The same decrease can be noticed for frequencies with a ratio of 50, for example between 50 and 1 rad/s. The values of G*12 at 22 C and 1 rad/s are reported in Table 3. However, it is important to say that crosshead speeds in tensile experiments and frequencies in oscillatory shear cannot be simply compared because there is no direct and easily demonstrable relation between them. Nevertheless, it is also remarkable that the shear modulus that can be calculated from the tensile value

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Table 3. Shear modulus G (¼G*12) of skin, core, and structural foam. Sample

G (1 rad/s, 22 C, in MPa)

Matrix (bulk) Skin (no opening) Core (machined) Structural foam (2.6 mm)

690 630 100 170

at 1 mm/min of Table 2 with the Poisson’s ratio of 0.43 (660 MPa) is close to the experimental value of Table 3.

Flexural modulus Flexural modulus was measured on polymer matrix, solid skin, and on sandwich foams. As previously, data on solid skins were obtained on injection-molded samples with conditions of no core-back displacement. No data could be obtained on the foamed core material with satisfactory reproducibility. The strain–stress curves are presented in Figure 9 and the modulus is given in Table 4. Data on the bulk matrix were obtained at 10 mm/min on injection-molded bars. As in the previous results concerning the tensile modulus and shear modulus, Table 4 evidences a reduction of the flexural modulus of 15% from bulk to skin samples. This is mainly due to the combination of lower orientation of fibers in plates and of viscoelastic effect as the crosshead speed was 10 times higher for the bulk injected bars. As expected, the sandwich foam has also a lower modulus than the solid polymer or skin due to the multilayer structure and in particular the skin and core structure. Nevertheless, the bending resistance in flexion is not given by a simple linear mixing law on the flexural moduli. Whereas the flexural modulus of the skin part is 15% lower than that of the bulk, it still remains 25% higher for sandwich foams in comparison to a simple blending law with overall thicknesses of 2.4 and 2.6 mm and skin thickness of 0.4 mm (see Table 1). This is due to the strong effect of the skins and because they have roughly the same thickness of 0.4 mm, the flexural modulus shows only a small dependence to the overall thickness.

Models and discussion Model for the core structure Table 5 shows the relative errors (%) with respect to average value, on the prediction by the various models using equations (1) to (4). The empirical Moore model presents a good agreement within 10% error, both for the tensile and shear modulus in agreement with the findings of Zhang et al.20 The Ramakrishnan model gives

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Figure 9. Typical strain–stress curve during flexural experiments on skin and sandwich foam (speed test: 1 mm/min).

Table 4. Flexural modulus of skin and structural foam. Sample

Flexural modulus (1 mm/min, 22 C, in MPa)

Matrix (bulk) Skin (no opening) Structural foam (2.4 mm) Structural foam (2.6 mm)

2300(a) 1920 1470 1460

a

Test speed: 10 mm/min.

similar predictions and relative error values because the correction term introduced by the Poisson’s ratio is negligible with respect to the experimental value for the polymer matrix. The Kerner model, being mostly a linear mixing rule with a similar correction, is inadequate. Therefore, the Moore model provides a good representation of the modulus data of homogeneous and isotropic foams for tensile modulus and shear properties. Though the theoretical foundation of the model has not been given in the literature, simple considerations can be used to explain the dependence of mechanical properties with the void fraction considering the stored energy in a unit volume of material in linear conditions. This quantity is proportional to the product of the stress and the strain. In bulk materials, under a given macroscopic linear strain go, the

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Table 5. Errors on prediction of the various models for tensile and shear moduli of the foamed core. Model

Error on E (%)

Error on G* (%)

Linear mixing rule (equation (1)) Moore (equation 2(a)) Kerner (equation (3)) Ramakrishnan (equation (4))

180 9 90 12

160 10 60 13

stress s in a polymer of modulus Em is given by the conventional elastic relation s ¼ Em go, leading to a stored energy per unit volume Wo ffi Em go2. In a homogeneous foam, most of the strain is stored by the cellular structure (strain amplification), the strain in the polymer is reduced by a factor (1 fc), and the elastic stress in the polymer is reduced by the same factor leading to a stored energy per unit volume Wf ffi Em (1 fc)2 go2. Therefore, the modulus of the foam (Ec) is given by Ec ¼ Em (1 fc)2.

Model for the overall sandwich foam The models for the sandwich foam were compared with the experimental results for various core back displacements. The evolution of the tensile modulus with sample thickness is plotted in Figure 10. Data were measured for total sample thicknesses between 1.7 and 3.1 mm with a remarkable agreement with the sandwich model (equation 5(a)) employing the Moore equation (equation 2(a)) for the core. This is in consistency with the data obtained for the core solely and confirms that the Moore equation, although it is a simple model, is also a reliable and physically meaningful equation to describe the tensile behavior of foamed sandwich. Data in the transverse direction (perpendicular to the flow) are slightly under the prediction curve, showing a less degree of orientation of the magnesium fibers. Indeed, it is important to notice that the data for the polymer matrix were measured in the injection direction or on dumbbell test bars injected on purpose, therefore with probably a significant level of orientation. Figure 11 shows the comparison of the experimental shear modulus values (measured in direction 1,2) with the mixing law models on rigidity (equation 6(b)) or on modulus (similar to equation 5(a) but employing the shear modulus). The third model analyzed is the skin approximation (equation 6(c)) that neglects the core contribution to the shear modulus. The two mixing laws use the Moore equation (equation 2(a)) for the core and the tensile modulus (1900 MPa) and the Poisson’s ratio of the polymer matrix (0.43) for the calculation of the shear modulus of the polymer matrix. The skin approximation gives good results for large thicknesses which sounds logical because in this situation, the skin contribution becomes dominant while that of the core is negligible due to the small shear modulus of the foam.

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Figure 10. Comparison of experimental data of tensile modulus with the sandwich model and various equations for the core modulus.

Figure 11. Comparison of experimental data of shear modulus with theoretical models.

Finally, Figure 12 shows the comparison of the experimental flexural modulus with theoretical models, both based on mixing rules. The prediction of the mixing rule based on rigidity is correct. On the other hand, the mixing rule based on modulus gives too much importance to the core part, leading to wrong predictions,

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Figure 12. Comparison of experimental data of flexural modulus with the mixing rules models.

whereas in experimental flexural tests, the outer skin provides the main contribution to stress.

Conclusions In this paper, the reliability of relationships for the prediction of various linear mechanical properties of polymeric sandwich foams obtained in injection molding processes was studied. Samples were obtained by a novel injection molding process that enables one to obtain sandwich with dense skins and a foamed core during the injection molding by a core-back opening of the mold after the filling stage. The foaming of the core is therefore obtained by expansion of the foaming agent that remains dissolved in the molten polymer until releasing the pressure during the core-back expansion. Samples with various thicknesses were studied. Morphology analysis reveals a sharp core–skin transition and therefore a welldefined sandwich structure. For similar cooling conditions, the skin thickness remains constant. For this reason, the modeling of such structure is simple and models for conventional sandwich composites may be applied. Tensile, shear, and flexural moduli were investigated for the skin, the core, and the overall foamed structure whenever it is possible. In addition, the Poisson’s ratio of the skin was determined and found in good agreement with tensile and shear moduli in corresponding viscoelastic conditions (temperature, frequency, or strain rate). The core properties were specifically analyzed by machining the samples and removing the skins. Tensile and shear properties of the core can be well described

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by the Moore equation. A tentative explanation of this empirical relation was proposed using the arguments that the deformation is mainly supported by the voids while the stress is mainly supported by the polymer. The tensile modulus is calculated by a linear mixing rule with the moduli of the skin and of the core in relation to the thickness of the layers. Shear and flexural moduli are described by a linear mixing rule on the rigidity in agreement with the mechanics of beams. Finally, tensile modulus, out-of-plane shear modulus, and flexural modulus can be correctly predicted from only very few data: the tensile modulus and Poisson’s ratio of the matrix, the void fraction, and thickness of the core. Moreover, a relation between the overall thickness and the void fraction could be determined. In our work, an empirical relation in well-defined cooling conditions was given but a meaningful relation could be found after a deeper study of the process analyzing the overall thickness, amount of foaming agent, process temperature, and cooling time. Also, additional experiments could be carried out to study the in-plane shear modulus (which can be mainly defined by the core properties) and also the compression modulus. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is part of the project PLUME funded by the French government in the frame of the FUI grantings by BPI France and thanks to Mecaplast, Sumika Polymer Compounds and Cero companies.

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