Anisotropy of the stiffness and strength of rigid low-density closed-cell polyisocyanurate foams

Materials and Design 92 (2016) 836–845

Contents lists available at ScienceDirect

Materials and Design journal homepage: www.elsevier.com/locate/matdes

Anisotropy of the stiffness and strength of rigid low-density closed-cell polyisocyanurate foams J. Andersons a,b,⁎, M. Kirpluks a, L. Stiebra a, U. Cabulis a a b

Latvian State Institute of Wood Chemistry, Dzerbenes st. 27, LV-1006 Riga, Latvia Institute of Polymer Mechanics, University of Latvia, Aizkraukles st. 23, LV-1006 Riga, Latvia

a r t i c l e

i n f o

Article history: Received 7 July 2015 Received in revised form 19 December 2015 Accepted 21 December 2015 Available online 22 December 2015 Keywords: Polymer foams Anisotropy Stiffness Strength

a b s t r a c t The cells of polymer foams are usually extended in the foam rise direction, causing a geometrical anisotropy, the degree of which, characterized by the cell aspect ratio R, depends on foam density and production method. Such elongated cell shape translates into anisotropy of the mechanical properties of foams. Rigid low-density closedcell polyisocyanurate foams of apparent density ranging from ca. 30 to 75 kg/m3, containing polyols derived from renewable resources, have been produced and tested for the stiffness and strength in the foam rise and transverse directions in order to experimentally characterize their mechanical anisotropy. Analytical relations for foams with rectangular parallelepiped and tetrakaidecahedral (Kelvin) cells were considered for predicting the mechanical characteristics of the polyisocyanurate foams in terms of their apparent density, geometrical anisotropy, and characteristics of the base polymer. Open-cell models were found to produce conservative estimates of foam stiffness and strength, albeit very close for the former when a tapering strut geometry was allowed for in the Kelvin cell model. Extending the rectangular parallelepiped cell model to the closed-cell case allowed a reasonably good description of variations of both the stiffness and strength with foam density. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Rigid low-density closed-cell polyurethane (PU) and polyisocyanurate (PIR) foams are employed primarily as thermal insulation materials. In the automotive industry, such foams are incorporated in bumper and side impact protection systems to improve the passive safety of vehicles [1–3]. The polyols used in foam production can be derived from natural sources such as, e.g., vegetable oils [4–6], thus mitigating the environmental impact. Depending on production method, different relative extensions of cells along the foam rise direction are obtained, which are characterized by the aspect ratio (shape anisotropy ratio) R defined as the average ratio of cell size in the rise and transverse directions. The geometrical anisotropy of cells translates into anisotropy of the mechanical characteristics of foams, typically well described by the transverse isotropy, with the higher strength and stiffness exhibited in the rise direction. The shape anisotropy of cells generally decreases with increasing foam density [7]. Notably, the geometrical anisotropy factor of closed-cell rigid PU foams was reported to decrease from R = 2.5 for foams of relative density γ = 1.98% to R = 1.7 for the higher-density foams with γ = 2.94% [8]. This led to variation in the ratio of stiffness in the foam rise and transverse directions from ca. 17 to 4, while the anisotropy of ⁎ Corresponding author at: Latvian State Institute of Wood Chemistry, Dzerbenes st. 27, LV-1006, Riga, Latvia. E-mail address: [email protected] (J. Andersons).

http://dx.doi.org/10.1016/j.matdes.2015.12.122 0264-1275/© 2015 Elsevier Ltd. All rights reserved.

the tensile strength was less affected by R. For PU foams with a relative density in the range of 11 to 13%, the geometrical anisotropy R amounted to ca. 1.6 and that of the stiffness to 1.7 [9]. Such a pronounced anisotropy in the mechanical properties should be taken into account in a product design either via experimental characterization or modeling tools. Apparently, the most accurate prediction of the mechanical properties of foams can be achieved by a detailed analysis of their morphology by using, e.g., Computed Tomography (CT) with a subsequent numerical modeling of the response of a realistic microstructure by the Finite Element Method (FEM) [10,11]. Both linear elastic [12,13] and large-strain [14] responses of polymer foams have been modeled in such a manner. Another numerical, FEM-based approach involves the recreation of foam morphology based either on specific random tessellation procedures or regular cells, as in, e.g., [15]. Allowing for the realistic shape of foam struts, as well as variation of the cross-sectional area along their length, was shown to be essential for an accurate prediction of the elastic properties [16], compressive strength [17,18], and crushing [19] of foams. A FEM study of the effect of disorder in cell geometry on foam stiffness, based on the distributions of cell size and cell wall thickness derived from microscopy measurements of polymer foams, revealed that their Young's and shear moduli decreased with increasing dispersion of the geometrical variables mentioned [20]. Albeit accurate, the numerical models are demanding in terms of the computational resources needed for the characterization and digital reconstruction of foam morphology and a FEM analysis. A simpler

J. Andersons et al. / Materials and Design 92 (2016) 836–845

alternative is provided by the models based on orientation distribution of foam struts [21,22], as well as models considering the mechanical response of a unit cell of foams [23–32]. The scaling relations of foam properties, derived from unit cell models, are useful for understanding the key aspects of the mechanical behavior of foams, as demonstrated in, e.g., [23]. Concerning anisotropic foams, models based on rectangular parallelepiped cell [23–25] and elongated tetrakaidecahedral or Kelvin cell geometry [26–29] have been developed, relating the stiffness and strength of foams to their density and properties of the solid cell wall material. The expressions for moduli and strength derived using a rectangular parallelepiped unit cell in [23] contain unspecified nondimensional constants of proportionality, so they have primarily been applied to relating the degree of mechanical anisotropy of foams, i.e. the ratios of mechanical characteristics along the principal axes of orthotropy, to the geometrical anisotropy [23,29]. In [23], the cross section of struts is assumed to be square and the cross-section area — the same for all the struts. By contrast, struts of a more general, rectangular, cross section are considered in [25]. Allowing for different dimensions of strut cross sections along the principal directions is shown to lead to anisotropy of the elastic properties of foams even for a cubic unit cell [25]. Most of the work on the anisotropic Kelvin cells is based on an apparently arbitrary simplifying assumption concerning their geometry that allows characterizing it by two independent parameters [26–29], which can be expressed via the foam anisotropy factor R and the relative density. A more general geometry of the Kelvin cell, allowing for three independent geometrical characteristics, one of which is determined via fitting the model prediction to the results of mechanical tests, has been proposed in [30–32]. Struts of uniform thickness and different cross-sectional shapes have been considered for modeling the elastic properties [26,30,31] and strength [26,28–30] of foams, while tapered struts have been considered in [27]. Although the models for anisotropic foams consider open-cell geometries, they should be applicable also to low-density closed-cell foams, as the effect of cell faces is expected to be minor when strut thickness exceeds considerably that of the faces, which is the case for lowdensity PU and PIR foams. In the current study, the dependence of strength and stiffness of PIR foams in the rise and transverse directions on relative density is determined experimentally, and the prediction accuracy of rectangular parallelepiped and tetrakaidecahedral cell models is evaluated. For that, the relations for the stiffness and plastic collapse stress of open-cell foams of the rectangular parallelepiped unit cell model [23] are reformulated in terms of the geometrical anisotropy and relative density of the foams and generalized to allow for the presence of cell walls; expressions for foam strength, based on the criterion of the appearance of plastic hinges in the struts, are derived for elongated Kelvin cells with a uniform strut geometry [30,32].

837

phosphate (TCPP), was supplied by Albemarle. A mix of water and cyclopentane was employed as a blowing agent. For the isocyanate component, a polymeric diphenylmethane diisocyanate — IsoPMDI 92140 supplied by BASF was used. The amount, in parts by weight (pbw), of the components in the formulation is presented in Table 1. The foam density was controlled by varying the relative amount of the blowing agent cyclopentane. The PIR foams were produced by mechanically mixing appropriate amounts of isocyanate and the polyol system, comprising polyols, surfactant, catalysts, and the blowing agent, for 10 to 15 s. The unreacted mixture was poured into a plastic mold of dimensions 20 × 30 × 10 cm for free foaming. The polymerization reaction took place at room temperature. The foam blocks were removed from the molds and conditioned at room temperature for at least 24 h. In the blocks produced, a denser surface layer had formed, which was cut off and discarded before making specimens for tests. To produce samples of a solid polymer, the mixture was prepared as described above, except that the blowing agent was not added. The mixture was cast in plastic ampoules of 80 mm length with an inner diameter of ca. 15 mm. To remove the air bubbles entrapped during mixing, the ampoules were centrifuged for 15 min at a speed of 5500 rpm in a centrifuge EBA20 (produced by Hettich Zentrifugen). 3. Experimental 3.1. Characterization of foam morphology A Tescan TS 5136 MM scanning electron microscope (SEM) with a secondary electron detector was used to take images of the surface of foam slices. For the SEM investigation, foam samples of dimensions 1 × 1 × 0.2 cm were cut and sputtered with gold by using an Emitech K550X sputter coater at a current of 25 mA and coating time of 2 min. The images obtained were processed with a Vega TC software. In order to assess the shape of cells, their average size, the cell size distribution, and anisotropy, images were taken parallel to the foam rise direction. The typical micrographs obtained are presented in Fig. 1. Cell dimensions in cross-sectional images were determined both in the rise and transverse directions. For each sample, more than 100 cells were measured. The geometrical anisotropy factor was calculated as.

R¼

n 1X h n i¼1 l

ð1Þ

where h and l denote cell dimensions in the rise and transverse directions, respectively, and n is the number of cells measured for a given sample.

2. Materials 3.2. Mechanical tests PIR foams and solid polymer specimens were manufactured as described in [33]. For the completeness and ease of reference, below we briefly recapitulate the description of the materials and production procedure employed. A rapeseed oil (RO) from the company Iecavnieks (Latvia) was used for the synthesis of polyol. The polyol was obtained by amidization of the RO with diethanolamine at a 140 °C temperature, employing zinc acetate as a catalyst. The molar ratio of RO to diethanolamine was 2.9/ 1.0. The synthesis process was controlled via the conversion degree of diethanolamine (acid value and NH value). The polyol component in this study contained both polyols from RO and a higher functional polyether polyol based on sorbitol Lupranol 3422, containing only secondary hydroxyl groups, OH Number 490 mg KOH/g, from BASF. Concerning additives, surfactant NIAX Silicone L6915, from Momentive Performance Materials, and catalyst Polycat 5 from Air Products were used. The flame retardant Tris-chloropropyl

Both tensile and compressive tests were performed in the direction transverse to that of foam rise. First, slices from the foam blocks were

Table 1 Formulation of PIR foams. Component Polyols Catalyst Surfactant Flame retardant Blowing agents Polyisocyanate

Amount, pbw RO-based Lupranol 3422 Polycat 5 L6915 TCPP Cyclopentane Water IsoPMDI 92140

70 30 1 1.5 30 12 1 164

838

J. Andersons et al. / Materials and Design 92 (2016) 836–845

Fig. 1. SEM micrographs of PIR foams with apparent densities of 31 (a) and 52 kg/m3 (b) in a plane parallel to the foam rise direction (the vertical direction in the pictures).

cut, and then specimens were machined from them. The apparent density of each specimen was determined as the ratio of its weight and volume. For this purpose, the rectangular specimens used in compression tests were weighed and their dimensions measured. As concerns specimens for tensile tests, a rectangular piece of the specimen size was first cut from a foam slice and its density determined as described above, and then the wasted part of the dog-bone shape machined. For tension tests, dog-bone shape specimens with a rectangular test section of 85 mm length, 22 mm width, and 20 mm thickness were used. For compression tests, rectangular specimens of dimensions 80 × 30 × 30 mm where cut from the foam slices, with the larger dimension normal to the foam rise direction. In compression tests, the specimens were placed between aligned plates of the testing machine, and thin Teflon films were inserted between specimen ends and the plates to reduce friction. Both tension and compression tests were performed by a servo-hydraulic test machine with a 1 kN load cell in stroke control, at a displacement rate of 8 mm/min, which corresponds to a nominal strain rate of 10%/min in the gage section as stipulated by the respective standards ISO 1926 and ISO 844 for tensile and compressive tests of rigid cellular plastics. The strain in the loading direction was measured by an extensometer MTS 634.25F-24 with a 50 mm gage length. Only compression tests were performed in the foam rise direction, as the height of the foam blocks produced prevented machining of tension test specimens of the dimensions described above. Rectangular specimens of dimensions 40 × 20 × 20 mm where cut from the slices of foam blocks in the foam rise direction for compression tests. The tests were performed at a displacement rate of 4 mm/min (10%/min). The strain in the loading direction was measured by an extensometer MTS 632.26F-20 with an 8 mm gage length. To check for the effect of specimen slenderness, tests at a few selected densities of foams were performed also on specimens of dimensions 80 × 30 × 30 mm, for which a loading rate of 8 mm/min and an extensometer with a 50-mm gage length were applied. The tension tests of the solid polymer were performed using cylindrical dog-bone shape specimens manufactured by turning. The length of the test section of the specimens amounted to ca. 28 mm and the diameter to 5.5 mm. Specimen ends were glued into tubular aluminum tabs. The tests were carried out by stroke control with a displacement rate of 1 mm/min (3.57%/min).

[27] strut geometry are presented and expressions for foam strength, based on the appearance of plastic hinges in the struts, derived. Finally, the analysis of rectangular parallelepiped unit cell [23] is generalized to allow for the presence of cell walls by employing the approach applied to closed-cell isotropic foams in [23]. 4.1. Open-cell foams 4.1.1. Rectangular parallelepiped cell Relations for the stiffness and strength of anisotropic open-cell foams have been derived in [23] using a rectangular parallelepiped unit cell. Bending of its struts was considered as the principal mechanism of deformation, as shown schematically in Fig. 2 for the case of compression in the foam rise direction. Young's modulus of the foams in the rise direction, E3 , is expressed via the lengths l and h and the thickness te of square-section struts as [23]

E3 ¼ CEs

 4 te h l l

ð2Þ

4. Models of the mechanical properties of foams In the following, the relations derived in [23] for the stiffness and plastic collapse stress of open-cell foams by using a rectangular parallelepiped unit cell are recapitulated, expressing them in terms of the geometrical anisotropy and relative density of the foams and determining the proportionality constants left unspecified in [23], so that, at R = 1, the respective semi-empirical expressions for isotropic foams presented in [23] are recovered. Further, the expressions for Young's moduli obtained for elongated Kelvin cells with a uniform [30,32] and tapered

Fig. 2. Schematic of deformation of a rectangular parallelepiped cell in compression in the foam rise direction.

J. Andersons et al. / Materials and Design 92 (2016) 836–845

839

where Es designates the Young's modulus of cell wall material and C is a constant of proportionality that, according to the derivation [23], does not depend on cell geometry. The strut thickness to length ratio for the unit cell of given geometry is related to its geometrical anisotropy factor R ¼ h=l

ð3Þ

and the relative density of foams γ = ρ/ρs, given by the ratio of the apparent density of foams, ρ, to the density of the solid cell wall material, ρs, as [23]  2 te R ¼γ l Rþ2

ð4Þ

for l ≫ te. Substituting the ratios of cell geometry parameters in Eq. (2) by their expressions of Eqs. (3) and (4) and determining the value of C from the condition that Eq. (2) at R = 1 coincides with the semi-empirical expression for the Young's modulus of isotropic open-cell foams Ef = Esγ2 [23], we finally obtain  E3 ¼ E s

 3γ 2 3 R : Rþ2

Similarly, for the modulus in the transverse direction, E1, the expression derived in [23] is transformed into E1 ¼

    1 3γ 2 1 Rþ 2 : Es 2 Rþ2 R

ð6Þ

The strength of plastic foams is determined by the appearance of plastic hinges in their struts. For loading in the rise direction, the foam strength is expressed as [23] σ3 ¼ C

Mp l

3

;

ð7Þ

where Mp is the plastic moment of a foam strut. For struts with a square cross section, Mp is related to the yield strength σys of the cell wall material as 1 Mp ¼ σ ys t 3e : 4



3Rγ Rþ2

3=2 ;

ð9Þ

where the value of C in Eq. (7) has been determined from the condition that the expression for foam strength, Eq. (9), at R = 1 coincides with the semi-empirical expression for the plastic collapse stress of isotropic open-cell foams σf = 0.3σysγ3/2 [23]. In the same way, the expression derived in [23] for foam strength in the transverse direction is recast into  σ 1 ¼ 0:3σ ys

expressed as R¼

γ¼

4L sinθ pffiffiffi ; 2b

3Rγ Rþ2

3=2

Rþ1 2R2

:

ð10Þ

4.1.2. Elongated Kelvin cell The geometry of an elongated Kelvin cell, employed in [30–32] and characterized by the strut lengths b and L, and the angle θ, is shown in Fig. 3. The foam anisotropy factor R and its relative density γ are

ð11Þ

2L cosθ þ

4Að2L þ bÞ  pffiffiffi 2 ; L sinθ 2L cosθ þ 2b

ð12Þ

where A is the cross-sectional area of a strut. The foam moduli, derived in terms of cell geometry characteristics in [30] and expressed in [32] via the anisotropy factor R, relative density γ, b , are as follows: and an auxiliary non-dimensional parameter Q ¼ L cosθ

E3 ¼ E s

ð8Þ

Inserting the expression for plastic moment, Eq. (8), into Eq. (7) and taking into account Eq. (4), we arrive at σ 3 ¼ 0:3σ ys

Fig. 3. Schematic of an elongated Kelvin cell.

ð5Þ

~ 5 R3 γ2 =ð2Q þ T Þ2 Q  ; ~ 5 R3 γ= 2QT þ T 2 4C 0 T þ Q

ð13Þ

5

E1 ¼ E s

C0



~ Rγ 2 =ð2Q þ T Þ2 8Q   ; ~ 2 R2 T þ 8Q ~ 3 Rγð8 þ QT Þ= 2QT þ T 2 32Q 3 þ Q

ð14Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ~ ¼ 2 þ 2Q, T ¼ 16 þ Q ~ 2 R2 , and we have denoted by C0 the where Q ratio of parameters C1 and C2 introduced in [32]. For struts with a Plateau border (i.e., three touching circular arcs) cross section, C 0 pffiffiffi pffiffiffi 2 ¼ ð 3−π=2Þ =ð10 3−11π=2Þ. The parameter Q has to be determined separately, e.g. via fitting the predicted ratio of elastic moduli in the principal material directions, Eqs. (13) and (14), to the experimentally determined value, as in [32]. Relations for the tensile strength of foams have been derived in [30] assuming a brittle failure of struts, i.e. by applying the criterion of maximum stress for their fracture. Based on the expressions for maximum bending moments in the struts of a unit cell presented in [30], the analysis can be straightforwardly extended to the yielding of struts. Assuming that the foams fail at an applied stress causing the development of a fully plastic moment in the struts, which is given by Eq. (8) for struts with a square cross section, and expressing the cross-sectional area A from Eq. (12), after elementary transformations we find, that the

840

J. Andersons et al. / Materials and Design 92 (2016) 836–845

strength of plastic foams in their rise direction is ~ Q 4

σ 3 ¼ σ ys

~ γRQ 2Q þ T

!3=2 :

pffiffiffi 3−π=2, CI = 0.1338, β = 1.24, and

where C A ¼ ð15Þ

1=2 Z

1=2 Z

2

ξ

C1 ¼ −1=2

C2 ¼

dξ; 2 f ðξÞ

−1=2

dξ : f ðξÞ

ð23Þ

For loading in the transverse direction, either struts of length L or b can fail. The respective critical stress is

σ L1

¼ σ ys

1 R2

~ γRQ 2Q þ T

4.2. Closed-cell foams

!3=2 ð16Þ

for the struts of length L and ~ ~ γRQ Q σ b1 ¼ σ ys pffiffiffi 2Q þ T 2 2QR

!3=2 ð17Þ

for the struts of length b to yield. The foam strength, being determined by the first mode of failure to occur, is given by the smallest of σL1 and σb1. Their ratio depends only on the foam anisotropy and cell geometry parameters R and Q: σ b1 1 ¼ R σ L1 2

! pffiffiffi 2 þ1 : Q

ð18Þ

The traditional approach to defining the geometry of an elongated Kelpffiffiffi vin cell corresponds to the fixed value of the parameter Q ¼ 2. In such b L a case, it follows from Eq. (18) that σ1 N σ1 for anisotropic foams, hence the transverse strength is given by Eq. (16). The geometry of the elongated Kelvin cell considered in [27] implicpffiffiffi itly assumes that Q ¼ 2. Then, Eq. (11) reduces to R ¼ tanθ:

ð19Þ

Although the cells are treated as regular in [27], the strut morphology is analyzed in detail and allowed for in the model. The cross-sectional area A of the struts was found to vary along their length, increasing from the smallest value A0, reached in the mid-span, towards nodes as A(ξ) = A0f(ξ), where ξ is the coordinate measured from the mid-span and normalized by strut length. Selecting an even forth-order polynomial as the symmetric function f(ξ) provided good approximation of measurements of the area A [27]. The cross sections of struts were considered as having Plateau borders consisting of three touching circular arcs. In order to accurately represent the volume of the struts, the material in a node was partitioned between the struts joined in the node, which resulted in the following relation for the relative density of foams as a function of r, the radius of the Plateau border at the mid-span normalized by strut length: γ ¼ kr n

ð20Þ

with the parameters k and n depending on foam anisotropy [27]. Treating the struts as beams and allowing for their axial and shear deformations, the following expressions were derived in [27] for Young's moduli in the foam rise, E3, and transverse, E1, directions: E3 ¼ Es

C I C 2A sinθ r 4

h i ; C 1 þ 2C I C A C 2 sin2 θ þ 2βð1 þ νs Þ cos2 θ r 2

Although only open-cell cell foams are considered in [23], the effect of cell walls on the mechanical properties of anisotropic foams with a rectangular parallelepiped geometry can be modeled by using the same approach as that applied to closed-cell isotropic foams in [23]. First, we express the geometrical characteristics of closed-cell foams via their relative density. Taking into account the fact that each strut belongs to four neighboring cells and each face is shared by two adjacent cells, the volume of solid contained in the struts of an anisotropic unit cell is Ve = t2e (h + 2l) and that in the faces is Vf = tf(2lh + l2), where tf is the cell face thickness and l ≫ te. The volume fraction of solid in cell struts φ then is φ = Ve/(Ve + Vf) and the relative density of foamsγ = (Ve + Vf)/hl2; solving these relations for the relative thicknesses of cell struts and faces, the following expressions are obtained:  2 te R ¼ φγ ; l Rþ2 tf R ¼ ð1−φÞγ : l 2R þ 1

ð24Þ

When considering foam modulus, the strain energy in cell faces due to an axial loading is estimated in [23] based on the axial strain taken to be proportional to the ratio of the displacement of struts in bending, δ, to the respective cell dimension. Then, the work of external forces applied to the midpoints of cell edges is equal to the sum of the elastic energy of struts in bending and the deformation energy of faces. For loading along the foam rise direction, the elastic energy of the bent struts is proportional to

Es Iδ23 3

l

, while that of cell faces – to the product

of the strain energy Esε23/2 and their volume 4lhtf, which leads to F 3 δ3 ¼ a

Es Iδ23 l

3

þ bEs

 2 δ3 lht f h

ð25Þ

where I = t4e /12, and a and b are numerical prefactors. Since E3 = σ3/ε3, σ3 ~ F3/l2, and ε3 ~ δ3/h, it follows from Eq. (25) that the stiffness of closed-cell foams E3 in their rise direction is E 3 ¼ a0 E s

 4 tf te h 0 þ b Es : l l l

ð26Þ

Inserting the expressions of geometrical parameters according to Eq. (24) into Eq. (26) and determining the values of the numerical factors so that the stiffness of closed-cell isotropic foams, given by Ef = Es[(φγ)2 + (1− φ)γ] [23], is regained at R = 1, we obtain " E3 ¼ E s

3φγ Rþ2

2

# 3R R þ ð1−φÞγ : 2R þ 1 3

ð27Þ

ð21Þ When estimating the modulus in the transverse direction, equal displacement δ1 of the struts of length l and h is assumed [23]; hence, the

2C I C 2A cosθ r 4  hpffiffiffi  i o ; E1 ¼ Es  n  pffiffiffi pffiffiffi R C 1 R2 þ 2 cosθ þ C I C A C 2 2 cosθ þ 2 cos2 θ þ 2βð1 þ ν s Þ 2 sin2 θ þ 2 cosθ r2

ð22Þ

elastic energy of the bent struts is proportional to Es Iδ21 ð 13 þ 13 Þ and l

h

that of cell faces - to the product of Esε21/2 and their volume 2l(l + h)tf. Noting that E1 = σ1/ε1, σ1 ~ F1/lh, and ε1 ~ δ1/l, and repeating the

J. Andersons et al. / Materials and Design 92 (2016) 836–845

841

derivation procedure described above, one finally obtains E1 ¼

Es 2

"

#    3φγ 2 1 Rþ1 R þ 2 þ 3ð1−φÞγ : Rþ2 2R þ 1 R

ð28Þ

To take into account the effect of cell walls on the plastic collapse stress, scaling relations for cubic cells were obtained in [23] by a similar procedure, considering the plastic work spent in bending the respective struts and deformation of cell faces. Repeating the derivation for the rectangular parallelepiped cell geometry [23] and determining the constants of proportionality so that the semi-empirical expression of strength for isotropic plastic foams σf = σys[0.3(φγ)3/2 + 0.4(1 − φ)γ] is regained at R = 1, one obtains for strength in the foam rise, σ3, and transverse, σ1, directions: σ 3 ¼ σ ys

σ 1 ¼ σ ys

!   3Rφγ 3=2 3R 0:3 þ 0:4ð1−φÞγ Rþ2 2R þ 1 !   3Rφγ 3=2 R þ 1 Rþ1 0:3 þ 0:6ð1−φÞγ : Rþ2 2R þ 1 2R2

Fig. 5. Variation of the geometrical anisotropy R with the apparent density of foams.

ð29Þ

ð30Þ

It is seen that the relations for the stiffness and strength of closedcell anisotropic foams, Eqs. (27) to (30), are reduced to those presented in Section 4.1.1 for open-cell anisotropic foams at φ = 1, and to the semi-empirical relations for isotropic closed-cell foams of [23] at R = 1, as they should. 5. Results and discussion 5.1. Foam morphology The elongation in the foam rise direction and the marked variability in the size of cells was observed at all the foam densities considered, as exemplified by the micrographs in Fig. 1 for foams with apparent densities of 31 and 52 kg/m3. The histograms of cell dimensions in the principal directions are presented in Fig. 4 for the same foam densities. With increasing density, the cell size decreased, while the coefficient of variation of cell dimensions appeared unaffected by foam density and amounted to ca. 19 and 16% in the foam rise and transverse directions, respectively. The geometrical anisotropy R, determined using Eq. (1), was found to slightly decrease with growing density, as seen in Fig. 5. Notably, the scatter of R among cells, characterized by the coefficient of variation of ca. 10%, was markedly, up to 1.9 times, smaller than that of cell dimensions. The content of closed cells in the foams amounted to ca. 91% [34]. The density of the core of foam blocks, obtained upon cutting off the surface layer of ca. 1-cm thickness, as described in Section 2, appeared to be uniform. This is confirmed by the relatively low scatter in the

apparent density of specimens, measured as described in Section 3.2 — the coefficient of variation for the density of specimens cut from the same foam block did not exceed 5%. 5.2. Mechanical properties Typical stress–strain diagrams in tension transverse to the foam rise direction are shown in Fig. 6. As expected, an increase in foam stiffness and strength, accompanied by reduction in the failure strain, is seen with increasing foam density. Macroscopically brittle fracture was observed, with a crack typically originating in the gage section of specimens. The fracture surface was visually examined for the presence of entrapped air bubbles; if such were observed, the data were discarded. Young's modulus was determined from the initial, linear part of the stress–strain diagram. The foam modulus and strength as functions of the apparent density are presented in Fig. 7a and 7b, respectively. Compressive tests of foams were performed both in the rise and transverse directions. Foams failed by developing a kink band spanning the whole width of the specimens. Typically, this led to an abrupt reduction in the stress in stroke-controlled tests in the foam rise direction, while in the transverse direction, the variation in the stress was much more gradual as exemplified in Fig. 8 for the foams of density 34 kg/ m3. The maximum stress reached during the test was taken as the foam strength. The experimentally determined variation of the stiffness and strength with foam density is shown in Fig. 9 for compression in the rise direction and in Fig. 7c and 7d - in the transverse direction. It is seen in Fig. 9 that the tests of specimens with aspect ratios of 2 and 2.7 produced close values of mechanical characteristics. The anisotropy of stiffness, characterized by the ratio of moduli in the rise and transverse direction E3/E1 of ca. 3, is more pronounced than that of strength, σ3/σ 1, amounting to ca. 1.4, cf. Figs. 7 and 9. As discussed in Section 4, the stiffness of low-density foams is governed

Fig. 4. Histograms of cell dimensions in the rise and transverse directions of foams with apparent densities of 31 (a) and 52 (b) kg/m3.

842

J. Andersons et al. / Materials and Design 92 (2016) 836–845

Fig. 6. Stress–strain diagrams in tension in the transverse direction of foams with different densities.

mainly by the elastic bending of foam struts, while the foam strength is related to the formation of plastic hinges in the struts. This leads to different dependences of the stiffness and strength of foams on the geometrical anisotropy R, resulting in different degree of anisotropy of these mechanical characteristics. Notably, not only the transverse stiffness determined from tensile and compressive tests agree within the experimental scatter, but the values of tensile and compressive strengths are also close as seen in Fig. 7. The latter indirectly corroborates the assumption that the foam strength is determined by the onset of plastic bending of struts. Tensile tests of the monolithic polymer of the same composition as that used in foam production revealed an elastic–plastic response with a failure strain typically exceeding 6%, as exemplified in Fig. 10. A small fraction (ca. 14%) of specimens failed at strains lower than 3.5%, before reaching the inflection point on the engineering stress–strain

Fig. 8. Stress–strain diagrams in compression in the rise and transverse direction of foams with an apparent density of 34 kg/m3.

curve, apparently due to the surface flaws introduced during their manufacture. Such data were discarded from a further strength analysis. Young's modulus Es of 2300 ± 180 MPa was determined from the initial, linear part of stress–strain diagrams. Since the ultimate load-bearing capacity of the polymer rather than the onset of plastic deformation was of interest, the yield stress σys of each specimen was defined as the maximum stress attained during the test. The average σys amounted to 61.5 ± 3.7 MPa. The density ρs of the monolithic polymer was determined to be ca. 1210 kg/m3 [33]. 5.3. Rectangular parallelepiped cell models Assuming that the mechanical characteristics of the cell wall material are the same as those determined by testing the base polymer, the foam stiffness (Eqs. (5) and (6)) and strength (Eqs. (9) and (10)) for

Fig. 7. Young's modulus and strength in the transverse direction of foams in tension (a, b) and compression (c, d) as functions of foam density.

J. Andersons et al. / Materials and Design 92 (2016) 836–845

843

Fig. 9. Young's modulus (a) and strength (b) in compression in the rise direction of foams as functions of foam density; specimen aspect ratios 2 (○) [33] and 2.7 (●).

the rectangular parallelepiped open-cell geometry were evaluated. It is seen in Figs. 7 and 9 that the model provides conservative estimates of the mechanical characteristics, and the discrepancy grows with foam density. Notably, the strain rate in foam tests exceeded that in tests of the monolithic polymer, therefore the experimental foam characteristics, measured at a greater strain rate, would be expected to exceed the predicted ones, based on polymer tests at a lower strain rate. However, the rate effect is moderate for PU foams as demonstrated experimentally in, e.g., [35–37], with variation in the loading rate by an order of magnitude causing variation in the foam stiffness and strength by a few percent. Specifically, if follows from the data presented in [35] for PU foams of 60 kg/m3 density that the reduction of their loading rate from 10%/min to 3.57%/min (corresponding to the rates in foam and monolithic polymer tests as reported in Section 3.2) would lead to a decrease in their stiffness and strength by ca. 2.4%. Such a variation is within the scatter range of the test results presented in Figs. 7 and 9, so the contribution of the loading rate effect in the discrepancy of the experimental and predicted foam characteristics is likely to be negligible. A slightly closer agreement between the predicted and experimentally determined rise-direction properties, Fig. 9, than between those in the transverse direction, Fig. 7, is seen. This appears consistent with neglecting the contribution of cell faces on foam response since the stiffness and strength in the foam rise direction should be less affected by the presence of cell walls. In view of the lack of direct experimental measurements, the fraction of solid in cell struts, φ, was assumed to vary linearly with the foam density φ ¼ a1 þ a2 ρ

ð31Þ

and the parameters of Eq. (31) were determined by fitting the theoretical relation for the transverse stiffness of closed-cell forms, Eq. (28), to test results. A close agreement of Eq. (28) with measured modulus variation with foam density could be achieved this way, as seen in Figs. 7a and 7c. Using the expression Eq. (31) for φ with the same, previously obtained values of a1 and a2, the predicted rise-direction stiffness, Eq. (27), and strength, Eq. (29), were found to be in reasonably good agreement with test results, as seen in Fig. 9, while a close but conservative estimate of the transverse strength, given by Eq. (30), was obtained, Figs. 7b and 7d. According to Eq. (31), the volume fraction of solid in cell struts varies from 0.98 to 0.94 with foam density growing from 35 to 75 kg/m3, which appears relatively low compared with the values of φ reported for PU foams in the literature. An analysis of five PU foams by using the SEM, reported in [38], suggested that 10 to 20% of the polymer was contained in cell walls, hence φ was in the range of 0.8 to 0.9. In [39,40], φ was estimated by fitting the semi-empirical relation for the modulus of isotropic cubic foams [23] to test results. The value of φ = 0.9 provided good agreement between the predicted and measured, in quasi-static tests, stiffness data for the whole foam density range of 120 to 600 kg/m3 considered in [39]. The same value of φ also yielded reasonable agreement of the theoretical elastic collapse stress [23] and experimental compressive strength of foams. Similarly, φ = 0.89 was found to ensure the fit of predicted stiffness to the experimental storage modulus for high-density (γ in the range of 0.33 to 0.85) foams, but failed to describe the compressive strength of the foams [40]. Although it appears plausible that the fraction of polymer in the struts of lowdensity PU foams may exceed that in high-density ones, direct experimental measurements of φ are lacking. However, the closed-cell model results shown in Figs. 7 and 9 suggest that φ can be used at least as a fitting parameter for anisotropic foams too. 5.4. Elongated Kelvin cell models

Fig. 10. Typical stress–strain diagram of a monolithic polymer in tension.

The elongated Kelvin cell model developed by Sullivan et al. [30–32] encompasses a cell geometry parameter Q, which is evaluated using the experimentally determined stiffness anisotropy of foams in [32]. In our case, the best fit of the predicted ratio of elastic moduli in the principal material directions, Eqs. (13) and (14), and the experimental one was obtained at Q = 1.35. Using this value of Q, foam stiffness was evaluated by Eqs. (13) and (14) and presented in Figs. 7 and 9. It is seen that the predicted stiffness estimate is conservative and slightly lower than that of the rectangular open-cell model. Note that the relations for foam strength, Eqs. (15) to (17), although derived for the same elongated Kelvin cell geometry, differ from those in [30,32] in that fully plastic bending of the struts is used as the failure criterion and a square cross section of the struts is assumed. It follows from Eq. (18) at the given value of Q that the L-struts determine the

844

J. Andersons et al. / Materials and Design 92 (2016) 836–845

foam strength in the transverse direction, hence Eqs. (15) and (16) are used for estimating the strength. Notably, the predicted plastic collapse stress, both in the rise, Fig. 9, and transverse directions, Figs. 7b and 7d, appeared rather close to that provided by the rectangular open-cell model despite the marked difference in cell geometry. The principal distinctive features of the Kelvin cell model used by Gong et al. [27] consist in an accurate consideration of strut geometry and partitioning of the material in a node among the struts joined in the node, which necessitates a detailed study of foam morphology. Such a morphological analysis has been performed in [16,27] for polyester urethane foams of relative density in the range of 2.2–2.8% and geometrical anisotropy of 1.2 to 1.4. Since the PU foam characteristics mentioned are close to those of the foams in the current study, we assume that the strut geometry is also the same. Variation of the normalized area of strut cross section along a strut of PU foams was approximated in [16] by the polynomial 4

2

f ðξÞ ¼ 96ξ þ ξ þ 1

ð32Þ

with the normalized coordinate ξ ranging from − 0.5 to 0.5. Based on Eq. (32), the auxiliary parameters of the model given by Eq. (23) were evaluated at C1 = 0.0156947, C2 = 0.641421. The parameters of Eq. (20) relating the relative density of foams to strut dimensions for the geometrical asymmetry R = 1.4 were determined as n = 1.6839 and k = 0.1453 in [16]. Eqs. (21) and (22) with such parameters provide very close, albeit slightly conservative, estimates of foam stiffness, as seen in Figs. 7 and 9. Thus, the stiffness model of open-cell anisotropic Kelvin foams allowing for a realistic strut geometry [27] is also applicable to low density closed-cell foams. Analytical relations for the strength of anisotropic Kelvin foams have been derived only for the case of open cells with struts of uniform thickness [26,28–30]. Therefore, it appears of interest to extend the strength analysis of foams with doubly tapered, variable thickness struts (as performed, e.g., in [41,42]) to the case of elongated Kelvin cells, allowing for the experimentally determined strut geometry. 6. Conclusions Low-density closed-cell PIR foams have been produced using a polyol system partially derived from rapeseed oil. Free-rise foams of density in the range of ca. 30 to 75 kg/m3, with a geometrical anisotropy factor R of 1.43 to 1.56, were obtained. The anisotropy of foam stiffness, characterized by the ratio of Young's moduli along the foam rise and transverse directions, amounted to ca. 3, while that of the strength was found to be lower, at approximately 1.4. Models assuming a regular foam structure with a rectangular parallelepiped or elongated Kelvin cells were applied for the prediction of foam stiffness and strength by using, for the Young's modulus and yield strength of the cell wall material, the corresponding mechanical characteristics obtained by tensile tests of the specimens of base polymer. The open Kelvin cell model allowing for the doubly tapered strut morphology was found to produce close conservative estimates of foam stiffness. Considering a rectangular parallelepiped as the foam unit cell, the stiffness was underestimated by up to 50% and the strength by up to 30% when the open-cell model was applied. However, extending the model to closed-cell foams and treating the fraction of material in cell struts as a fitting parameter allowed a reasonably accurate description of foam stiffness and closer, although still conservative, estimation of foam strength along the principal axes. Acknowledgments The research has been funded, in part, by the EU Commission through FP7 Project EVOLUTION-314744 and the ERDF via project 2010/0290/2DP/2.1.1.1.0/10/APIA/VIAA/053.

References [1] F. Billotto, M. Mirdamadi, B. Pearson, Design, application development, and launch of polyurethane foam systems in vehicle structures, SAE Technical Paper 2003-010333, 2003. [2] C. Fremgen, L. Mkrtchyan, U. Huber, M. Maier, Modeling and testing of energy absorbing lightweight materials and structures for automotive applications, Sci. Technol. Adv. Mater. 6 (2005) 883–888. [3] Y.Y. Tay, C.S. Lim, H.M. Lankarani, A finite element analysis of high-energy absorption cellular materials in enhancing passive safety of road vehicles in side-impact accidents, Int. J. Crashworth. 19 (2014) 288–300. [4] O. Faruk, M. Sain, R. Farnood, Y. Pan, H. Xiao, Development of lignin and nanocellulose enhanced bio PU foams for automotive parts, J. Polym. Environ. 22 (2014) 279–288. [5] U. Stirna, U. Cabulis, I. Beverte, Water-blown polyisocyanurate foams from vegetable oil polyols, J. Cell. Plast. 44 (2008) 139–160. [6] Z.S. Petrović, Polyurethanes from vegetable oils, Polym. Rev. 48 (2008) 109–155. [7] J.R. Dawson, J.B. Shortall, The microstructure of rigid polyurethane foams, J. Mater. Sci. 17 (1982) 220–224. [8] M. Ridha, V.P.W. Shim, Microstructure and tensile mechanical properties of anisotropic rigid polyurethane foam, Exp. Mech. 48 (2008) 763–776. [9] A.R. Hamilton, O.T. Thomsen, L.A.O. Madaleno, L.R. Jensen, J.C.M. Rauhe, R. Pyrz, Evaluation of the anisotropic mechanical properties of reinforced polyurethane foams, Compos. Sci. Technol. 87 (2013) 210–217. [10] C. Petit, S. Meille, E. Maire, Cellular solids studied by X-ray tomography and finite element modeling — a review, J. Mater. Res. 28 (2013) 2191–2201. [11] V. Srivastava, R. Srivastava, On the polymeric foams: modeling and properties, J. Mater. Sci. 49 (2014) 2681–2692. [12] F. Fischer, G.T. Lim, U.A. Handge, V. Altstadt, Numerical simulation of mechanical properties of cellular materials using computed tomography analysis, J. Cell. Plast. 45 (2009) 441–460. [13] E. Sadek, N.A. Fouad, Finite element modeling of compression behavior of extruded polystyrene foam using X-ray tomography, J. Cell. Plast. 49 (2013) 161–191. [14] J.G.F. Wismans, L.E. Govaert, J.A.W. Van Dommelen, X-ray computed tomographybased modeling of polymeric foams: the effect of finite element model size on the large strain response, J. Polym. Sci. B Polym. Phys. 48 (2010) 1526–1534. [15] N.J. Mills, Modeling the dynamic crushing of closed-cell polyethylene and polystyrene foams, J. Cell. Plast. 47 (2011) 173–197. [16] W.-Y. Jang, A.M. Kraynik, S. Kyriakides, On the microstructure of open-cell foams and its effect on elastic properties, Int. J. Solids Struct. 45 (2008) 1845–1875. [17] Y. Takahashi, D. Okumura, N. Ohno, Yield and buckling behavior of Kelvin open-cell foams subjected to uniaxial compression, Int. J. Mech. Sci. 52 (2010) 377–385. [18] W.-Y. Jang, S. Kyriakides, A.M. Kraynik, On the compressive strength of open-cell metal foams with Kelvin and random cell structures, Int. J. Solids Struct. 47 (2010) 2872–2883. [19] W.-Y. Jang, S. Kyriakides, On the crushing of aluminum open-cell foams: part II analysis, Int. J. Solids Struct. 46 (2009) 635–650. [20] Y. Chen, R. Das, M. Battley, Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams, Int. J. Solids Struct. 52 (2015) 150–164. [21] E. Kontou, G. Spathis, V. Kefalas, Statistical model for the compressive response of anisotropic polymeric and metallic foams, J. Mater. Sci. 47 (2012) 5326–5332. [22] A. Lagzdins, A. Zilaucs, I. Beverte, J. Andersons, Calculating the elastic constants of a highly porous cellular plastic with an oriented structure, Mech. Compos. Mater. 49 (2013) 121–128. [23] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, 1997. [24] A.T. Huber, L.J. Gibson, Anisotropy of foams, J. Mater. Sci. 23 (1988) 3031–3040. [25] T. Zhang, A general constitutive relation for linear elastic foams, Int. J. Mech. Sci. 50 (2008) 1123–1132. [26] A.G. Dement'ev, O.G. Tarakanov, Model analysis of the cellular structure of plastic foams of the polyurethane type, Polym. Mech. 6 (1970) 744–749. [27] L. Gong, S. Kyriakides, W.-Y. Jang, Compressive response of open-cell foams. Part I: morphology and elastic properties, Int. J. Solids Struct. 42 (2005) 1355–1379. [28] M. Ridha, V.P.W. Shim, L.M. Yang, An elongated tetrakaidecahedral cell model for fracture in rigid polyurethane foam, Key Eng. Mater. 306–308 (2006) 43–48. [29] Z.X. Lu, J.X. Huang, Z.S. Yuan, Effects of microstructure on uniaxial strength asymmetry of open-cell foams, Appl. Math. Mech. — Engl. Ed. 36 (2015) 37–46. [30] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, A general tetrakaidecahedron model for opencelled foams, Int. J. Solids Struct. 45 (2008) 1754–1765. [31] R.M. Sullivan, L.J. Ghosn, Shear moduli for non-isotropic, open cell foams using a general elongated Kelvin foam model, Int. J. Eng. Sci. 47 (2009) 990–1001. [32] R.M. Sullivan, L.J. Ghosn, B.A. Lerch, Application of an elongated Kelvin model to space shuttle foams, J. Spacecr. Rocket. 46 (2009) 411–418. [33] U. Cabulis, I. Sevastyanova, J. Andersons, I. Beverte, Rapeseed oil-based rigid polyisocyanurate foams modified with nanoparticles of various type, Polimery 59 (2014) 207–212. [34] M. Kirpluks, U. Cābulis, M. Kurańska, A. Prociak, Three different approaches for polyol synthesis from rapeseed oil, Key Eng. Mater. 559 (2013) 69–74. [35] L. Peroni, M. Avalle, M. Peroni, The mechanical behaviour of polyurethane foam: multiaxial and dynamic behaviour, Int. J. Mater. Eng. Innov. 1 (2009) 154–174. [36] K.Y. Jeong, S.S. Cheon, M.B. Munshi, A constitutive model for polyurethane foam with strain rate sensitivity, J. Mech. Sci. Technol. 26 (2012) 2033–2038. [37] D.A. Apostol, D.M. Constantinescu, Temperature and speed of testing influence on the densification and recovery of polyurethane foams, Mech. Time-Depend. Mater. 17 (2013) 111–136.

J. Andersons et al. / Materials and Design 92 (2016) 836–845 [38] D.W. Reitz, M.A. Schuetz, L.R. Glicksman, A basic study of aging of foam insulation, J. Cell. Plast. 20 (1984) 104–113. [39] S.H. Goods, C.L. Neuschwanger, C.C. Henderson, A.D.M. Skala, Mechanical properties of CRETE, a polyurethane foam, J. Appl. Polym. Sci. 68 (1998) 1045–1055. [40] F. Saint-Michel, L. Chazeau, J.-Y. Cavaillé, E. Chabert, Mechanical properties of high density polyurethane foams: I. Effect of the density, Compos. Sci. Technol. 66 (2006) 2700–2708.

845

[41] H.S. Kim, S.T.S. Al-Hassani, Plastic collapse of cellular structures comprised of doubly tapered struts, Int. J. Mech. Sci. 43 (2001) 2453–2478. [42] H.S. Kim, S.T.S. Al-Hassani, The effect of doubly tapered strut morphology on the plastic yield surface of cellular materials, Int. J. Mech. Sci. 44 (2002) 1559–1581.