Investigation of the Variants of Independent Elastic Constants of Rigid Polyurethane Foams with Symmetry Elements

Abstract

Rigid PU foams have wide practical applications, and their mathematical modelling would benefit from deeper knowledge about the variants of independent elastic constants of symmetric PU foams. Therefore, in this study, various symmetry elements of rigid PU foams were analysed in relation to the characteristics of production moulds and technologies. The generalised Hooke’s law was considered together with additional relationships valid for certain types of symmetry. Variants of independent elastic constants were determined for orthotropic, orthotropic with a rotational symmetry, and isotropic PU foams. For transtropic PU foams, nine variants of independent elastic constants were identified and corresponding equations for the components of response strain tensor were derived. Then, in order to investigate the results provided by the 9 variants, 12 elastic constants were determined experimentally in compression and shear for free-rise, rigid, and quasi-transtropic PU foams with average densities of 34 kg/m³, 55 kg/m³, and 75 kg/m³. Based on the analysis of (a) measurement uncertainties and (b) satisfying of the transtropy equations, an assessment was made of the correspondence of the experimentally determined elastic constants to the constants of a perfectly transtropic material. This made it possible to identify variants of independent constants that ensure the best correspondence between the calculated strains and the set of average strains.

1. Introduction

Rigid polyurethane (PU) foams are polymer–gas composites with outstanding thermal insulation properties and acceptable load bearing capacity, which are applied in various engineering solutions [1,2]. The foams are used in the construction sector in sandwich structures and panels; as thermal insulation of mortars for masonry; in the transport sector in cooling vans and vehicle insulation; for both heat and cold saving insulation in stores, warehouse buildings, and constructions; for the thermal insulation of cryogenic tanks in space vehicles; as emergency shock absorbers; as electric insulation in electronics, equipment housing, and components; in radomes (protective structures for out-door antennas, locators and telescopes); for providing a radio-frequency transparent layer along dimensional stability, etc. [3,4]. Practical applications and mathematical modelling require knowledge on the symmetry of PU foams’ structure and physical/mechanical properties.

Anisotropic materials are usually classified by their symmetry properties with respect to orthogonal transformations of the coordinate system [5,6]. For PU foams, produced by foaming of the liquid chemical formulation in a mould, the symmetry of the moulded blocks is determined mainly by the characteristics of the mould: its absolute and relative dimensions, shape (rectangular, circular cylindrical, conical), open or sealed [3,4]. A study of cellular structure of rigid PU foams in moulded free-rise rectangular blocks (dimensions 200 cm × 100 cm × 30 cm) by cutting cubical samples (size 5 cm × 5 cm × 5 cm) from the block is described in [3]. The foams were found to have nearly spherical cells in the centre of the blocks (quasi-isotropic) and elongated cells, orientated mainly in rise direction, near the walls of the mould.

The effect of processing temperature and mould size (aluminium cylinders with diameters of 29 mm, 41 mm, and 51 mm) on the average density and density gradients (radial and vertical) for free-rise, water blown, and rigid polyurethane foam systems was studied in [7]. Both average density and radial density gradients decreased with increasing processing temperature and larger mould sizes. In the study [8], rigid transtropic PU foams were produced in free-rise process, in relatively high, circular cylindrical aluminium moulds with an inner diameter of ≈29 mm and height (a) ≈140 mm and (b) ≈457 mm, as well as in conical moulds. Microscopy revealed transtropy in the central part of the cylindrical PU foams’ samples, while general anisotropy was identified near the wall of the metallic mould.

Symmetry elements of rigid PU foams blocks, produced in delimited foaming in a rectangular, sealed mould [9], are discussed in [10]. The blocks were shaped as a truncated pyramid, with top dimensions of 15 cm × 15 cm, bottom of 14 cm × 14 cm, and a height of 5 cm. The fourth-order rotational symmetry around the rise direction was identified for blocks produced in the mould with square cross-section, which facilitates forecasting of locations with similar foaming conditions in the block.

Neumann’s principle can be extended to symmetric PU foams; when the structure of a material is invariant with respect to certain symmetry elements, any of its physical properties must be invariant with respect to at least the same symmetry elements [11,12]. The general equations of linear elasticity and independent elastic constants are discussed in [12,13] for the main kinds of symmetry. Since components of stiffness and compliance tensors have to be invariant at symmetry transformations of a material, reduction in the number of independent elastic constants due to symmetry is analysed in [14,15].

Representation theory [16,17] deals with the effect of symmetries on solutions of equations and behaviour of objects. When symmetry elements of a certain medium form a finite group, the number of independent components of compliance tensor s_ijkl can be determined by means of linear representation theory of groups [18]. In [18], the independent components of the elasticity tensors of fourth- to twelfth-order are derived for several kinds of anisotropic media. One of the variants of five independent elastic constants is considered for rigid transtropic (montropic) PU foams in [19]. In [20], calculation of the dependent constants of transtropic plastics foams with pronounced strut-like structure is considered for the same variant of independent constants. In [21], equations are derived expressing a variant of five independent moduli of transtropic light-weight foams, but the relationships remain in too general a form, which complicates practical application. In [22], an advanced mechanical characterisation of one of the variants of elastic constants of orthotropic PU foams is carried out. In [23], the numerical values of the same variant of stiffness tensor are calculated numerically for orthotropic foams using finite element simulations and ANN, but other variants are not considered.

An analysis of scientific information sources revealed a limited number of theoretical studies relevant to determining variants of independent elastic constants of PU foams with symmetry elements. The methodology for deriving the variants is not clearly outlined. A shortage of experimental data for various variants of constants was identified as well.

This study aimed to determine variants of independent elastic constants for rigid-moulded PU foams with symmetry elements. The generalised Hooke’s law is considered in a Cartesian coordinate system, together with additional relationships valid for different types of symmetry. Nine variants of independent elastic constants are identified for the wide-spread transtropic PU foams and corresponding equations are derived for the response strain components. The elastic constants are determined experimentally for quasi-transtropic PU foams with average densities of 34 kg/m³, 55 kg/m³, and 75 kg/m³ and the variants of independent constants, which provide the best correspondence with the constants of perfectly transtropic PU foams, are outlined.

2. Theoretical Part

2.1. Orthotropic PU Foams

Anisotropic rigid PU foams can be obtained by spray-on technology [3,4,24,25]. For anisotropic foams, the generalised Hooke’s law in an ortho-normal coordinate system X_i, i = 1, 2, 3 is written as ε_ij = s_ijklσ_kl, where ε_ij is the strain tensor, σ_ij is the stress tensor, and s_ijkl is the fourth-rank tensor of elastic compliance; i, j, k, l = 1, 2, 3 [13,14]. An elastic material with stress and strain symmetry has 21 independent elastic constants, since s_ijkl = s_jikl = s_ijlk = s_jilk.

Homogeneous blocks of rigid PU foams with certain symmetry elements are produced in rectangular or circular, cylindrical moulds, in free or delimited foaming [3,8,9]. Furthermore, it is assumed that the transversal dimensions of a mould are large compared to the dimensions of the foams’ structural elements. Only that part of the moulded blocks is considered which is sufficiently far from the walls of a block, where contact with the mould leads to un-adiabatic processes and inhomogeneous structure.

Orthotropic PU foams are produced in free-rise, in a high, long, and narrow rectangular mould, whose height and one transversal dimension significantly exceed the other transversal one: L₂, L₃ >> L₁ and L₃ > L₂ (Figure 1).

Orthotropic PU foams have three mutually orthogonal symmetry planes: X₁OX₂, X₂OX₃, and X₁OX₃. Directing the coordinate axis OX₁, OX₂, and OX₃ perpendicular to each plane, the generalised Hooke’s law in tensor notations is written as [12,13,14]

ε₁₁ = s₁₁₁₁σ₁₁ + s₁₁₂₂σ₂₂ + s₁₁₃₃σ₃₃,  
ε₂₂ = s₂₂₁₁σ₁₁ + s₂₂₂₂σ₂₂ + s₂₂₃₃σ₃₃,
ε₃₃ = s₃₃₁₁σ₁₁ + s₃₃₂₂σ₂₂ + s₃₃₃₃σ₃₃,
ε₂₃ = 2s₂₃₂₃σ₂₃,         
ε₁₃ = 2s₁₃₁₃σ_13, and       
ε₁₂ = 2s₁₂₁₂σ₁₂.         

(1)

Using the engineering constants—the elastic moduli E_i, shear moduli G_ij, and Poisson’s ratios ν_ij, i and j = 1, 2, and 3—Equation (1) becomes

$$\begin{array}{l}{\mathsf{\epsilon }}_{11}={\displaystyle \frac{1}{{\mathrm{E}}_1}}{\mathsf{\sigma }}_{11}-{\displaystyle \frac{{\nu }_{21}}{{\mathrm{E}}_2}}{\mathsf{\sigma }}_{22}-{\displaystyle \frac{{\nu }_{31}}{{\mathrm{E}}_3}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{22}=-{\displaystyle \frac{{\nu }_{12}}{{\mathrm{E}}_1}}{\mathsf{\sigma }}_{11}+{\displaystyle \frac{1}{{\mathrm{E}}_2}}{\mathsf{\sigma }}_{22}-{\displaystyle \frac{{\nu }_{32}}{{\mathrm{E}}_3}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{33}=-{\displaystyle \frac{{\nu }_{13}}{{\mathrm{E}}_1}}{\mathsf{\sigma }}_{11}-{\displaystyle \frac{{\nu }_{23}}{{\mathrm{E}}_2}}{\mathsf{\sigma }}_{22}+{\displaystyle \frac{1}{{\mathrm{E}}_3}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{23}={\displaystyle \frac{1}{{2\mathrm{G}}_{23}}}{\mathsf{\tau }}_{23},\\ {\mathsf{\epsilon }}_{13}={\displaystyle \frac{1}{{2\mathrm{G}}_{13}}}{\mathsf{\tau }}_{13}, \mathrm{and}\\ {\mathsf{\epsilon }}_{12}={\displaystyle \frac{1}{{2\mathrm{G}}_{12}}}{\mathsf{\tau }}_{12}.\end{array}$$

(2)

In Poisson’s ratios, ${\nu }_{\mathrm{i}\mathrm{j}}=-{\displaystyle \frac{{\mathsf{\epsilon }}_{\mathrm{j}\mathrm{j}}}{{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{i}}}}$, the first index corresponds to the direction of applied stress and the second to that of response strain; ν_ij ≠ ν_ji. In denotations of tangential stresses τ_ij and shear moduli G_ij, the first index corresponds to the plane of action of the applied stress and the second to its directions; G_ji = G_ij. Equations (2) comprise 12 constants: 3 moduli, E₁, E₂, E₃; 6 Poisson’s ratios, ν₂₁, ν₁₂, ν₃₁, ν₁₃, ν₃₂, ν₂₃; and 3 shear moduli, G₂₃, G₁₃, G₁₂. Owing the symmetry of compliance coefficients s_iijj = s_jjii in Equation (1), we obtain s₁₁₂₂ = s₂₂₁₁, s₁₁₃₃ = s₃₃₁₁, and s₂₂₃₃ = s₃₃₂₂. Then, it follows from (2)

$$\begin{gathered} {\displaystyle \frac{{\nu }_{\mathrm{i}\mathrm{j}}}{{\mathrm{E}}_{\mathrm{i}}}}={\displaystyle \frac{{\nu }_{\mathrm{j}\mathrm{i}}}{{\mathrm{E}}_{\mathrm{j}}}}; \mathrm{i}, \mathrm{j}=1, 2 \mathrm{and} 3; \mathrm{i}\ne \mathrm{j}; \\ {\displaystyle \frac{{\nu }_{21}}{{\mathrm{E}}_2}}={\displaystyle \frac{{\nu }_{12}}{{\mathrm{E}}_1}},{\displaystyle \frac{{\nu }_{31}}{{\mathrm{E}}_3}}={\displaystyle \frac{{\nu }_{13}}{{\mathrm{E}}_1}} \mathrm{and} {\displaystyle \frac{{\nu }_{32}}{{\mathrm{E}}_3}}={\displaystyle \frac{{\nu }_{23}}{{\mathrm{E}}_2}}. \end{gathered}$$

(3)

and there remain 12 − 3 = 9 independent constants. The moduli G₂₃, G₁₃, G₁₂, E₁, E₂, and E₃ cannot be expressed by the other constants in any way (each modulus E₁, E₂, and E₃ is comprised in two equations of Equations (3)). Considering all possible combinations of constants and taking into account Equations (3), eight variants of nine independent constants were identified (

Table 1. Variants of independent constants of orthotropic PU foams; n is the ordinal number of a variant.

n	Variant	n	Variant
1	G23, G13, G12, E1, E2, E3, ν21, ν31, ν32	5	G23, G13, G12, E1, E2, E3, ν12, ν31, ν32
2	G23, G13, G12, E1, E2, E3, ν21, ν31, ν23	6	G23, G13, G12, E1, E2, E3, ν12, ν31, ν23
3	G23, G13, G12, E1, E2, E3, ν21, ν13, ν32	7	G23, G13, G12, E1, E2, E3, ν12, ν13, ν32
4	G23, G13, G12, E1, E2, E3, ν21, ν13, ν23	8	G23, G13, G12, E1, E2, E3, ν12, ν13, ν23

).

2.2. Symmetry to the Rotation Angle

When PU foams are produced in the free-rise, in a high mould of equal transversal dimensions of the rectangular cross-section L₃ >> L₁, L₂ and L₁ = L₂, then, strictly speaking, the foams in the moulded block cannot be considered as perfectly transtropic, since the diagonal of the cross-section of mould is longer than the side: d = $\sqrt{2}$L₂ (Figure 2).

Only in the central part of the mould can a transtropic structure of foam be expected. The size and shape of the transtropic part has to be estimated individually for each mould and PU foam’s formulation. At the same time, an additional element of structural and elastic symmetry appears; namely, fourth-order rotational symmetry with respect to the rotation angle α = 360°/4 about the OX₃ axis (Figure 2). Then, the additional relationships E₁ = E₂ and G₂₃ = G₁₃ are valid. From Equations (3), we obtain that

$$\begin{gathered} {\displaystyle \frac{{\nu }_{21}}{\mathrm{E}}}={\displaystyle \frac{{\nu }_{12}}{\mathrm{E}}},\mathrm{i}.\mathrm{e}.,{\nu }_{21}={\nu }_{12}; \\ {\nu }_{31}={\nu }_{32}, \mathrm{and} {\nu }_{13}={\nu }_{23}. \end{gathered}$$

(4)

Taking into account Equations (4), the following notations are further adopted for simplicity: E₁ = E₂ = E, E₃ = E′; ν₂₁ = ν₁₂ = ν, ν₃₁ = ν₃₂ = ν′, ν₁₃ = ν₂₃ = ν″; G₂₃ = G₃₂ = G₃₁ = G₁₃ = G′; and G₁₂ = G₂₁ = G. Here, E and E′ are Young’s moduli under uniaxial compression/tension in the direction of the plane of isotropy and perpendicular to it, ν is Poisson’s ratio characterising transverse compression/tension in the plane of isotropy under tension/compression in the same plane, ν′ is Poisson’s ratio characterising transverse compression/tension in the plane of isotropy under tension/compression in direction perpendicular to the plane of isotropy, and ν″ is Poisson’s ratio characterising transverse compression/tension in the plane perpendicular to the plane of isotropy under tension/compression in the plane of isotropy [12,13,14]. Then, implementing the new notations, Equations (2) takes the form

$$\begin{array}{l}{\mathsf{\epsilon }}_{11}={\displaystyle \frac{1}{\mathrm{E}}}{(\mathsf{\sigma }}_{11}-\nu {\mathsf{\sigma }}_{22})-{\displaystyle \frac{{\nu }^{\prime }}{{\mathrm{E}}^{\prime }}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{22}={\displaystyle \frac{1}{\mathrm{E}}}(-\nu {\mathsf{\sigma }}_{11}+{\mathsf{\sigma }}_{22})-{\displaystyle \frac{{\nu }^{\prime }}{{\mathrm{E}}^{\prime }}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{33}={\displaystyle \frac{{\nu }^{″}}{\mathrm{E}}}\left(-{\mathsf{\sigma }}_{11}-{\mathsf{\sigma }}_{22}\right)+{\displaystyle \frac{1}{{\mathrm{E}}^{\prime }}}{\mathsf{\sigma }}_{33},\\ {\mathsf{\epsilon }}_{23}={\displaystyle \frac{1}{2{\mathrm{G}}^{\prime }}}{\mathsf{\tau }}_{23},\\ {\mathsf{\epsilon }}_{13}={\displaystyle \frac{1}{2{\mathrm{G}}^{\prime }}}{\mathsf{\tau }}_{13},\mathrm{and}\\ {\mathsf{\epsilon }}_{12}={\displaystyle \frac{1}{2\mathrm{G}}}{\mathsf{\tau }}_{12}.\end{array}$$

(5)

Due to the symmetry of compliance coefficients s₁₁₃₃ = s₃₃₁₁, from Equations (5), we obtain

$${\displaystyle \frac{{\nu }^{″}}{\mathrm{E}}}={\displaystyle \frac{{\nu }^{\prime }}{{\mathrm{E}}^{\prime }}}.$$

(6)

Altogether, Equations (5) comprises seven constants: E, E′, ν, ν′, v’’, and G, G′. The constants G′ and ν, G are independent, since they cannot be expressed by the other ones, but E, E; ν′, and v’’ can be expressed from (6) in four variants:

$${\mathrm{E}}^{\prime }=\mathrm{E}{\displaystyle \frac{{\nu }^{\prime }}{{\nu }^{″}}}, \mathrm{E}={\mathrm{E}}^{\prime }{\displaystyle \frac{{\nu }^{″}}{{\nu }^{\prime }}}, {\nu }^{\prime }={\mathrm{E}}^{\prime }{\displaystyle \frac{{\nu }^{″}}{\mathrm{E}}}, \mathrm{and} {\nu }^{″}=\mathrm{E}{\displaystyle \frac{{\nu }^{\prime }}{{\mathrm{E}}^{\prime }}}.$$

(7)

Thus, four variants of six independent constants have been determined for PU foams with a fourth-order rotational symmetry axis OX₃: (1) G′, G, E, ν, ν′, ν″; (2) G′, G, E′, E, ν, ν″; (3) G′, G, E′, ν, ν″, ν′; and (4) G′, G, E, E′, ν, ν′.

2.3. Transtropic PU Foams

If PU foams are symmetrical to an arbitrary rotation angle α about the OX₃ axis, the order of rotational symmetry is infinite and the foams are transtropic with the plane of isotropy X₁OX₂ and monotropy axis OX₃. Then, the following relationship is valid:

$${\mathrm{G}=\mathrm{G}}_{12}={\displaystyle \frac{{\mathrm{E}}_1}{2(1+{\nu }_{12})}}.$$

(8)

Equation (8), together with Equations (7), permits the reduction of the number of independent constants to five in Equations (5). Perfectly transtropic PU foams can be produced free-rise, in a comparatively high, cylindrical mould with constant transversal dimensions: L₃ >> L₁, L₂ and L₁ = L₂ = D, where D is the diameter of cross-section of the mould [7,8] (Figure 3).

Let us estimate how many variants of the five independent elastic constants can exist for transtropic PU foams. With this aim, we consider the constants E, E′, ν, ν′, ν″ and G, G′ of Equations (5). Obviously, the shear modulus G′ has to be present in all variants, since it cannot be expressed by other constants. E, G, and ν can be expressed by other ones in three variants from Equation (8):

$$\mathrm{G}={\displaystyle \frac{\mathrm{E}}{2(1+\nu )}}, \mathrm{E} = 2\mathrm{G}(1 + \nu ) \mathrm{and} \nu ={\displaystyle \frac{\mathrm{E}}{2\mathrm{G}}}-1.$$

(9)

E′, E″, ν′, and ν″ can be expressed via each other in four variants, given in Equations (7). Variants of constants are analysed in Table S1 of the Supplementary Materials, where the sub-variants from columns 1, 2, and 3 are summarised in column 4. When there were five constants in a variant, it was written in engineering constants with indices in the column 5 and an ordinal number “n” was assigned to it in the column 6. If there were six constants in the variant, constants from column 3 were considered one by one for expression via Equations (7). When the resulting variant comprised five constants, it was compared with the numbered variants, which were identified already previously. When the variant was new, it was written in engineering constants with indices and a number was assigned to it. When the variant coincided with an already identified one, it was omitted. If the resulting variant still had six elements, it was omitted as well. Altogether, nine different variants were identified (

Table 2. Variants of independent elastic constants of transtropic PU foams; n is the ordinal number of the variants.

n	Variant	n	Variant	n	Variant
1	G13, E1, E3, ν12, ν31	4	G13, G12, E3, ν12, ν31	7	G13, G12, E1, E3, ν31
2	G13, E1, E3, ν12, ν13	5	G13, G12, E3, ν12, ν13	8	G13, G12, E1, E3, ν13
3	G13, E1, ν12, ν31, ν13	6	G13, G12, ν12, ν31, ν13	9	G13, G12, E1, ν31, ν13

).

Strain components ε_ij of the system of Equations (5) were expressed via the independent constants of each variant Equations (S1)–(S9) of the Supplementary Materials). The strain tensors (S1)–(SS9) fully describe the stress–strain state of transtropic PU foams in the principal coordinate system.

2.4. Isotropic PU Foams

When all directions in a PU foam’s material are elastically equivalent and principal, the foams are isotropic. Then, E′ = E, ν′ = ν″ = ν; G′ = G = E/[2 (1 + ν)], and E, ν, and G can be expressed via each other in three variants, which provides three different variants of independent constants: (1) E, ν, (2) G, ν, and (3) E, G.

The free-rise technology causes some elongation of PU foams’ cells parallel to the rise direction and the foams produced are transtropic to some degree [3,7,8,9]. Quasi-isotropic PU foams can be produced in the free-rise, in a low, cylindrical mould of dimensions L₁ = L₂ = R and L₃ ≤ L₁, L₂; Figure 4. Perfectly isotropic PU foams are obtained in foaming under an overpressure, in a sealable mouldmold, Figure 5 [10].

Practically, the central part of any PU foam’s block, produced in the free-rise, in sufficiently low and large mould of equal transversal dimensions can be considered as quasi-isotropic.

3. Materials and Experimental Methods

Three rectangular blocks of rigid, free-rise, core PU foams of standard petrochemical formulation, with dimensions of 500 mm × 600 mm × 200 mm and average densities of 34 kg/m³, 55 kg/m³, and 75 kg/m³ were acquired from Bayer AG (Leverkusen, Germany). The geometric centre “O” of each block was determined as the crossing point of the spatial diagonals (Figure S1 of Supplementary Materials). Two parallelepipeds, located on either side of the geometric centre and the plane X₁OX₃, were cut from each block: N1 (200 mm × 150 mm × 100 mm) for compression samples and N2 (100 mm × 122 mm × 100 mm) for shear samples (Figures S2 and S3 of Supplementary Materials). Compression and shear samples were made with a band saw, with seam allowances of 1–2 mm; the samples were then sanded to the final dimensions.

To ensure a homogeneous stress field in the measurement zone, the uniaxial compression samples were made as rectangular prisms of dimensions 50 mm × 50 mm × 100 mm [3,26] (Figure S2 of Supplementary Materials). The experimental setup and measurements were carried out according to methodology given in [26]. Deformation parallel to the loading direction was measured with a clip-on extensometer (MTS Model 632.11C-20 type; MTS Systems Corporation, Eden Prairie, MN, USA) attached in the middle of sample’s height, on a measurement base of 10 mm, and with a strain rate 10%/min. The deformations perpendicular to the loading direction were measured with two similar extensometers on the sides of sample’s cross-section [26]. The elastic constants E₁, ν₂₁, ν₃₁, E₂, ν₁₂, ν₃₂, E₃, ν₁₃, and ν₂₃ were determined from the stress–strain curves according to ISO 844:2021.

The shear samples were made as rectangular prisms with dimensions of 14 mm × 22 mm × 100 mm (Figure S3 of Supplementary Materials). Deformation was measured in the centre of a sample’s side, on a 6 mm × 38 mm measurement base, at a strain rate of 3.4 mm/min (9%/min) [3,27]. The experimental setup and measurements were carried out according to methodology given in [27]. Constants G₁₂, G₁₃, and G₂₃ were determined from corresponding stress–strain curves according to the standard ASTM C273/C273M.

The density of all samples was determined according to ISO 845:2006. Both in compression and in shear, four samples were tested under reproducibility conditions and at ambient temperatures of 20 °C ≤ T ≤ 22 °C and relative humidity of 49% ≤ RH ≤ 53%; no conditioning was made for the samples. All samples were tested on the electromechanical testing machine 2166P-5 which used the automated data-gathering system Spider v1.22. Tangents were drawn to the uniaxial compression curves “σ₁₁–ε₁₁”, “σ₂₂–ε₂₂”, and “σ₃₃–ε₃₃”, as well as to the shear curves “τ₁₂–ε₁₂”, “τ₁₃–ε₁₃”, and “τ₂₃–ε₂₃” in the elastic region, in order to determine the elastic constants (Figures S4 and S5 of the Supplementary Materials). The limit stresses σ₁₁^lim, σ₂₂^lim, and σ₃₃^lim, as well as τ₁₂^lim, τ₁₃^lim, and τ₂₃^lim, were calculated as the stresses at which the curves deviated from tangents.

4. Numerical Calculations

4.1. The Elastic Constants

The strains ε_ij, corresponding to the nine variants of independent elastic constants of transtropic PU foams, were calculated numerically by a PC code. Based on the superposition principle for linear systems, virtual complex loading of three PU foams’ cubes with densities of 34 kg/m³, 55 kg/m³, and 75 kg/m³ was considered. (1) Hydrostatic pressure σ_HP along with (2) shear were applied to the cubes’ faces parallel to the X₁OX₂, X₁OX₃, and X₂OX₃ planes:

σ₁₁ = σ₂₂ = σ₃₃ = σ_HP;
τ₁₂ = τ₂₁ = τ, and τ₁₃ = τ₃₁ = τ₂₃ = τ₃₂ = τ′.

(10)

Input data for the elastic constants and the stresses were provided to the PC code.

The experimental data identified some orthotropy of foams in the rectangular parallelepiped-shaped blocks: E₂ > E₁, ν₁₂ > ν₂₁, ν₁₃ > ν₂₃ (Tables S2–S5 of the Supplementary Materials). Therefore, constants linked to OX₁ and OX₂ directions were averaged:

E = ½(E₁ + E₂), ν = ½(ν₁₂ + ν₂₁),
ν′ = ½(ν₃₁ + ν₃₂), ν″ = ½(ν₁₃ + ν₂₃),
G′ = ½(G₁₃ + G₂₃), but
G = G₁₂, and E′ = E_3.

(11)

The modulus G was assumed to be equal to G₁₂ (due to lacking experimental data for G₂₁). The constants, calculated from Equations (11), are given in

Table 3. The averaged experimental data of PU foams.

Average Density ρav; kg/m3	E; MPa	ν	E′; MPa	ν′	ν″	G; MPa	G′; MPa	A
34	4.3	0.29	10.4	0.48	0.23	1.8	2.6	2.4
55	11.3	0.36	19.4	0.51	0.22	3.9	5.0	1.7
75	19.7	0.31	28.7	0.41	0.24	7.1	9.0	1.5

. The average anisotropy degree of the foams in blocks was estimated as A = E′/E [7] (Table 3).

Since the anisotropy degree A of foams in the three blocks is different, as seen in Table 3, the same-name moduli E_i and G_ij of the three blocks cannot be directly compared and the trendlines are not given in Figures S6 and S8 of the Supplementary Materials. It was shown in [10] that Poisson’s ratio depends on structural anisotropy and does not directly depend on density; therefore, the dependence of ν₁₂, ν₁₃, ν₂₁, ν₂₃, ν₃₁, and ν₃₂ on anisotropy degree is given in Figure S7 of the Supplementary Materials.

For an idealised, perfectly transtropic material, the density and degree of transtropy [10] are the same at all points of the block, so Equations (6) and (8) have to be satisfied exactly. In practice, the density and degree of transtropy vary from point to point in moulded blocks [3,4,7,8,10]. Therefore, it was necessary to evaluate the correspondence of the experimentally obtained elastic constants of PU foams (Tables S2–S5 of Supplementary Materials) to constants of a perfectly transtropic material. With this aim, the values of ratios f₁ = ν″/E and f₂ = ν′/E′ were calculated from Equations (6) and (11) with the experimental data from Tables S2–S4 of Supplementary Materials as the input. The standard uncertainties were estimated (in Supplementary Materials, section “Analysis of uncertainties”, points “1. Ratio f₁” and “2. Ratio f₂”) [28,29,30,31], and the ranges of f₁ and f₂ values were calculated. When necessary, expanded uncertainties U were estimated using the effective degrees of freedom and the coverage factor (Tables S7 and S8 of Supplementary Materials). The obtained ranges of f₁ and f₂ values were compared with each other. Then the values of modulus G were calculated from Equations (8) and (11) with the experimental data from Tables S2 and S3 of Supplementary Materials as the input. The standard uncertainties were estimated and the ranges of G values were calculated in Supplementary Materials, section “Analysis of uncertainties”, point “3. Modulus G”. The obtained ranges of G values were compared with the ranges of G values from direct experiments (Table S5 of Supplementary Materials).

The numerical results show that in the limits of the combined standard uncertainties u_c(f₁) and u_c(f₂), associated with the f₁ and f₂ estimates, there is no overlapping of the ranges of values of f₁ and f₂ (

Table 4. The ranges of values of the functions f₁ and f₂ at the combined standard uncertainties u_c.

Average Density ρav; kg/m3	f1	f2	uc(f1)	uc(f2)	Range of f1 Values	Range of f2 Values
34	0.053	0.046	0.0022	0.0026	0.051 ≤ f1 ≤ 0.055	0.044 ≤ f2 ≤ 0.049
55	0.019	0.026	0.0005	0.0009	0.019 ≤ f1 ≤ 0.020	0.025 ≤ f2 ≤ 0.027
75	0.012	0.014	0.0006	0.0006	0.011 ≤ f1 ≤ 0.013	0.014 ≤ f2 ≤ 0.015

). Overlapping of the ranges of the values of f₁ and f₂ means that there exist such values of f₁ and f₂, for which Equation (6) is satisfied exactly: f₁ = f₂ = ${\displaystyle \frac{{\nu }^{\prime }}{\mathrm{E}}}={\displaystyle \frac{{\nu }^{″}}{{\mathrm{E}}^{\prime }}}$. The wider the overlap, the better the agreement of the experimental data set and that of a perfectly trantropic material.

In the limits of the expanded uncertainties U(f₁) and U(f₂), the range of f₁ values overlaps with the range of f₂ values for PU foams’ of average densities 34 kg/m³ and 75 kg/m³. At the average density 55 kg/m³, there is still no overlapping (

Table 5. The ranges of values of the functions f₁ and f₂ at the expanded uncertainties U.

Average Density ρav; kg/m3	f1	f2	U(f1)	U(f2)	Range of f1 Values	Range of f2 Values
34	0.053	0.046	0.0052	0.0069	0.048 ≤ f1 ≤ 0.058	0.039 ≤ f2 ≤ 0.053
55	0.019	0.026	0.0012	0.0025	0.018 ≤ f1 ≤ 0.021	0.024 ≤ f2 ≤ 0.028
75	0.012	0.014	0.0014	0.0017	0.011 ≤ f1 ≤ 0.013	0.013 ≤ f2 ≤ 0.016

). The two other variants of Equation (6) ${\displaystyle \frac{\mathrm{E}}{{\nu }^{″}}}={\displaystyle \frac{{\mathrm{E}}^{\prime }}{{\nu }^{\prime }}}$, and ${\displaystyle \frac{{\nu }^{″}}{{\nu }^{\prime }}}={\displaystyle \frac{\mathrm{E}}{{\mathrm{E}}^{\prime }}}$ provide similar results. It is concluded that Equation (6) is satisfied for the experimental data of the PU foam samples of average densities of 34 kg/m³ and 75 kg/m³ and is not satisfied for the data of foams with the average density of 55 kg/m³.

The numerical results show (

Table 6. The range of values of modulus G at the combined standard uncertainty u_c.

Average Density ρav; kg/m3	Equations (8) and (11)			Direct Experimental Data
	G; MPa	uc(G)	Range of G Values	G; MPa	uc(G)	Range of G Values
34	1.7	0.05	1.6 ≤ G ≤ 1.7	1.8	0.25	1.55 ≤ G ≤ 2.05
55	4.2	0.09	4.1 ≤ G ≤ 4.3	3.9	0.35	3.55 ≤ G ≤ 4.25
75	7.5	0.26	7.3 ≤ G ≤ 7.8	7.1	0.30	6.80 ≤ G ≤ 7.40

) that already in the limits of standard uncertainties, the range of G values estimated from Equations (8) and (11) overlaps with the range of G values, determined from direct experiments (Table S5 of Supplementary Materials) for all the three considered average densities (34 kg/m³, 55 kg/m³, and 75 kg/m³). Therefore, there was no necessity to determine the range of G values corresponding to the expanded uncertainty U(G). It can be concluded that Equation (8) is satisfied for the experimental data of the PU foam samples of the average density 34 kg/m³, 55 kg/m³, and 75 kg/m³.

From the numerical results above, it was concluded that in the limits of expanded uncertainties ± U (≈95% of all random points, [28,29,30]), the experimentally obtained sets of elastic constants of PU foams correspond with the constants of a transtropic material for foams from the blocks of average densities 34 kg/m³ and 75 kg/m³. The sets of experimental elastic constants of PU foams from the block of average density 55 kg/m³ do not correspond with the constants of a transtropic material. These conclusions are valid only for the particular PU foam blocks produced under certain technological conditions, e.g., foams in a block with an average density of 55 kg/m³ produced under other technological conditions may well correspond to a transtropic material.

4.2. The Stresses of Complex Loading

The hydrostatic pressure has to ensure that response strains do not go beyond the elastic region of the uniaxial stress–strain curves. Therefore, the limit stresses σ_11lim, σ_22lim, and σ_33lim of the elastic region of experimental curves “σ₁₁–ε₁₁”, “σ₂₂–ε₂₂”, and “σ₃₃–ε₃₃” were determined and compared. Since σ_33lim > σ_11lim, σ_22lim for the considered foams’ densities, σ_11lim and σ_22lim were selected for numerical calculations and the hydrostatic pressure was determined as σ_HP = ½(σ_11lim + σ_22lim).

The shear stresses also have to ensure that the response strains do not go beyond the elastic region of shear stress–strain curves. The shear stress τ in the isotropy plane X₁OX₂ was determined as the limit stress τ_12lim of the elastic region of experimental curves “τ₁₂–ε₁₂”: τ = τ_12lim. The shear stress τ′ in the X₁OX₃ and X₂OX₃ planes was determined as τ′ = ½(τ_13lim + τ_23lim), where τ_13lim and τ_23lim are the limit stresses of the elastic region of experimental curves “τ₁₃–ε₁₃” and “τ₂₃–ε₂₃”.

The virtual stresses calculated for different densities of PU foams, are given in Table S6 of the Supplementary Materials. In order to stay in the elastic region at all the considered densities, the stresses σ_HP and τ, τ′, determined for the most light-weight PU foam’s (block No. 1, average density 34 kg/m³), were used in complex loading:

σ_HP = σ_HP¹ = ½(σ_11lim¹ + σ_22lim¹),
τ = τ_12lim¹, and τ′ = ½(τ_13lim¹ + τ_23lim¹).

(12)

The stresses, used in numerical calculations of response strains ε_ij, are given in

Table 7. The stresses of virtual complex loading.

Average Density ρav; kg/m3	Hydrostatic Pressure σHP; MPa	Shear Stress
		τ′; MPa	τ; MPa
34, 55 and 75	−0.045	0.042	0.030

.

Then the strains ε_ij were calculated for the nine variants of independent constants at each average density of PU foams. The statistical characteristics (the average value, standard deviation, and coefficient of variation) were calculated for the set of nine values of each same-name strain ε_ij:

$$\begin{gathered} {\mathsf{\epsilon }}_{\mathrm{ijav}} = {\displaystyle \frac{1}{9}}{\sum }_{\mathrm{n}=1}^9{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\left(\mathrm{n}\right)}, {\mathrm{s}}_{\mathrm{ij}} = \sqrt{{\displaystyle \frac{\sum _{n=1}^9{[{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\left(\mathrm{n}\right)}-{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\mathrm{a}\mathrm{v}}]}^2}{8}}} \mathrm{and} \\ {\mathrm{v}}_{\mathrm{ij}} = {\displaystyle \frac{{\mathrm{s}}_{\mathrm{i}\mathrm{j}}}{{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\mathrm{a}\mathrm{v}}}}; \end{gathered}$$

(13)

where i, j = 1, 2, 3 and the ordinal number “n” of a variant is a non-tensorial index n = 1, 2, …, 9. The relative difference between the average strain ε_ijav and the same-name strain ε_ij(n) for the n-th variant of independent constants was estimated as

$${\mathrm{R}}_{\mathrm{ij}\left(\mathrm{n}\right)}={\displaystyle \frac{{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\mathrm{a}\mathrm{v}}-{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\left(\mathrm{n}\right)}}{{\mathsf{\epsilon }}_{\mathrm{i}\mathrm{j}\mathrm{a}\mathrm{v}}}}, \mathrm{where} \mathrm{n}=1, 2, \dots , 9.$$

(14)

The set of average values of strains ε_11av, ε_22av, ε_33av, ε_23av, ε_31av, and ε_12av characterises, in a certain sense, the set of strains of perfectly transtropic PU foams. Therefore, the value of summary relative difference between each of the six average strains ε_kav and each of the six same-name strains ε_k(n) was calculated for the nine variants of independent constants:

$${\mathrm{R}}_{\left(\mathrm{n}\right)} = {\sum }_{\mathrm{k}=1}^6\mid {\mathrm{R}}_{\mathrm{k}\left(\mathrm{n}\right)}\mid ={\sum }_{\mathrm{k}=1}^6{\displaystyle \frac{\mid ({\mathsf{\epsilon }}_{\mathrm{k}\mathrm{a}\mathrm{v}}-{\mathsf{\epsilon }}_{\mathrm{k}\left(\mathrm{n}\right)})\mid }{{\mathsf{\epsilon }}_{\mathrm{k}\mathrm{a}\mathrm{v}.}}}$$

(15)

where n = 1, 2, …, 9 and k = 1 at ij = 11, k = 2 at ij = 22, k = 3 at ij = 33, k = 4 at ij = 23, k = 5 at ij = 31, and k = 6 at ij = 12. The variant of independent constants, which provided the smallest summary relative difference R_(n), was considered to have the best conformity with the constants of perfectly transtropic PU foams.

5. Results and Discussion

The numerical results for response strains ε_ij at the nine variants of independent constants are given in Table S9 of the Supplementary Materials. It can be seen that the highest strains correspond to the PU foams of the lowest average density 34 kg/m³. At higher average densities of 55 kg/m³ and 75 kg/m³, the strains are smaller both in hydrostatic pressure and shear due to higher elastic moduli. All strains remain in the linear regions of the stress–strain diagrams. The same-name strains are different for the nine variants of independent constants; the coefficient of variations for ε₁₁, ε₂₂, ε₂₃, ε₃₁, and ε₁₂ is within a range 0% ≤ v ≤ 21%. The high values of v for the strain ε₃₃ (111%, 171%, and 33%) are caused by the comparatively small absolute values of ε₃₃ at different variants when any deviation of an input constant from the perfect value causes a comparatively high deviation from the average in the value of ε₃₃.

The average strains in the plane of isotropy X₁OX₂ are equal and considerably higher than those along the rise direction OX₃: ε_11av = ε_22av and |ε_11av|, |ε_22av| >> |ε_33av| (Tables S9 and S10 of the Supplementary Materials). The fifth variant of constants (G₁₃, G₁₂, E₃, ν₁₂, and ν₁₃) provides the best correspondence with constants of a transtropic material for foams of average density 34 kg/m³ and the fourth variant (G₁₃, G₁₂, E₃, ν₁₂, and ν₃₁) for foams with an average density of 75 kg/m³ because these variants ensure the smallest summary relative differences R₍₅₎ = 2.640 and R₍₄₎ = 0.273 (Table S10 of the Supplementary Materials). The average strains of the foams in the block of average density 55 kg/m³ (which does not correspond to a perfectly monotropic material), have the biggest coefficients of variations, as shown in

Table 8. The average strains of rigid PU foams; v is the coefficient of variations.

Average Strain	ρ = 34 kg/m3		ρ = 55 kg/m3		ρ = 75 kg/m3
	εijav	v; %	εijav	v; %	εijav	v; %
ε11av	−0.0054	14	−0.0015	21	−0.0010	13
ε22av	−0.0054	13	−0.0015	17	−0.0010	11
ε33av	−0.0001	111	−0.0002	171	−0.0003	33
ε23av	0.0082	0	0.0043	0	0.0024	0
ε31av	0.0082	0	0.0043	0	0.0024	0
ε12av	0.0086	4	0.0038	4	0.0021	3

.

When a material is perfectly transtropic, in the limits of uncertainties, all variants of the experimentally determined independent elastic constants have to provide equal numerical results for same-name strain components ε_ij of the system of Equations (5). However, in practice, the distribution of density and anisotropy degree in the moulded rectangular and cylindrical blocks of free-rise PU foams is non-uniform, as indicated by the density gradients and convex surface of rise [3], e.g., a density gradient dρ/dx = −10 kg/m³/5 cm from edge to centre of a rectangular free-rise block of average density 74 kg/m³ is reported in [27]. Then the testing samples have different densities and degrees of transtropy depending on location in the block (Tables S2–S5 of the Supplementary Materials). As a result, the set of the measured elastic constants does not fully conform to the set of constants of transtropic material and the strain components are different at different variants.

The non-uniform distribution of density and anisotropy degree throughout a PU foam’s block is mainly due to technological conditions (kind and rate of foaming, size and temperature of the mould, ambient environment, etc.) [2,3,4,10] and, as such, it is present in PU foam blocks of any density. The conformity of the experimental constants to those of a perfectly transtropic material can be improved by achieving a sufficient number of large blocks produced under similar technological conditions to ensure making of the testing samples from compact volumes of similar locations in the blocks. The distribution of density and anisotropy degree in the blocks should be studied prior to any other measurements.

6. Conclusions

(1)

Symmetry elements of rigid PU foams were considered in connection with characteristics of production moulds and technologies. The variants of independent elastic constants were determined for orthotropic, orthotropic with a rotational symmetry, and isotropic PU foams.

(2)

Nine variants of independent elastic constants were identified for transtropic PU foams and corresponding equations of the generalised Hooke’s law were derived for the components of response strain.

(3)

Correspondence of the experimentally determined elastic constants of rigid PU foams with constants of perfectly transtropic material was assessed, based on analysis of satisfying of the transtropy equations and of measurement uncertainties.

(4)

The variants of independent constants were outlined, providing the best conformity with the set of average strains, which characterises in certain meaning the set of strains of perfectly transtropic PU foams. The non-uniform distribution of density and anisotropy degree in the PU foam blocks are suggested as the main reason for the experimental constants not fully corresponding to those of a transtropic material.

(5)

The variant of independent elastic constants, which is the most appropriate for experimental determination or mathematical modelling, has to be selected in practice. All nine variants comprise the shear modulus G₁₃, therefore the height of the moulded PU foam blocks has to be sufficient for such a length of shear samples, which ensures prevalence of shear over bending.

(6)

No limitations specific to PU foams were implemented; therefore, the variants of independent elastic constants are valid for other materials with symmetry elements like other plastic foams, fibreglass–plastic composites, veneer, wood, etc. Further research may be directed towards determining variants of independent elastic constants using the elastic potential.