Effect of strain rate and load orientation on cyclic response of anisotropic polyurethane foam

Abstract

Anisotropic cellular materials, such as polymeric foams, play an important role in structures subjected to cyclic loadings. The present paper provides an experimental investigation of the mechanical behavior of an anisotropic polyurethane foam subjected to cyclic compressive loadings under two perpendicular orientations: the rising and perpendicular directions. The foam samples are loaded under three different strain rates and various deformations. The experimental results are presented in terms of elasticity modulus, maximal compressive stress, effective energy absorption capacity, and residual strain. It is proved that the investigated polyurethane foam presents a macroscopic mechanical anisotropy caused by microscopic cell elongation in the foaming direction. Moreover, it is demonstrated that the mechanical behavior of the foam is fully influenced by both deformation rates and imposed strains. The experimental stress–strain curves are modelized using an empirical model considering an adjustable modulus of elasticity. The analytical results show a good agreement with the experiments.

Keywords

Polyurethane foam Cellular structure Compressive behavior Anisotropic behavior Empirical model

Introduction

Polymeric foams, such as Polyurethane Foams (PUF), are commonly used in everyday life, ranging from thermal and acoustic insulation to energy absorption and lightweight design.^1–4 Such a use is warranted for a low-cost manufacturing process and a range of relevant mechanical and thermal properties.^5–8 These unique characteristics are resulting from the complex cellular microstructure of the foams, which consists of cellular ribs.^9,10 For several years, great effort has been devoted to the study of the mechanical behavior of foams. Generally, polymeric foams exhibit a complex behavior.⁹ For compression tests, the stress–strain curve exhibits a primary elastic response explained by cell wall bending. Then, a plateau appears as the moment acting on the cell walls exceed its elastic resistance, so the material undergoes a continuous deformation without a significant reaction force until all the cells are totally collapsed. Finally, a densification zone appears where the stress increases exponentially because the opposite cell walls start to experience mutual contact.⁹ Garrido et al.¹¹ investigated the elastic response of a polyethylene terephthalate foam used as the core of an industrial sandwich panel. Tensile, compressive, and shear tests were carried out. The shear creep behavior of the foam was evaluated at different load levels. The experimental results contributed to obtaining the bending properties of the sandwich structure. Demirel et al.¹² studied the cyclic behavior of a polyurethane foam and showed that the indentation force deflection and the foam firmness depended significantly on the foam density. Furthermore, the experimental results obtained by Hwang et al.¹³ showed that the dynamic behavior of the PUF, analyzed via drop impact tests, was affected by the foam density and its temperature. Dynamic compression tests were carried out by Linul et al.¹⁴ to determine the variation in the compressive maximal stress and the energy absorption capacity of the PUF as a function of its density. The experimental results highlighted that these mechanical properties would increase with the foam density. In addition, they used the crushable foam model to predict the dynamic behavior of the studied foams. By using numerical simulation, Kim et al.¹⁵ described the stress–strain response of a PUF via different hyperelastic and hyperfoam constitutive models where the mechanical properties were obtained by uniaxial and volumetric compression tests. The numerical simulation demonstrated that the two studied models could describe the compressive behavior of the foam. Furthermore, the response of high-velocity impact tests on PUFs with different densities and thicknesses were given by Taherkhani et al.¹⁶ They found out that when the foam thickness was raised, the absorbed energy and the damage area would build up. He et al.⁵ developed a micro-macro model to predict the mechanical coupled-to-damage behavior of a PUF under complex thermal vibration conditions. They used a scanning electron microscope observation in order to establish the failure prediction model via a realistic representative volume element. Recently, the mechanical performance of a reinforced polyurethane foam has been studied by Kim et al.⁹ using quasi-static compression tests before and after drop impact tests. They found out that the plateau and the densification zones were not affected by different impact energies. However, there was a significant decrease in the elastic modulus of the foam after repetitive impact tests.

For further realistic comprehension of the mechanical behavior of PUFs, recent investigations have been expanded to the study of the foam sensibility to the strain rate. Wang et al.¹⁰ and Koumlis et al.¹⁷ found out that the compressive elastic modulus of a PUF strongly depended on the strain rate. Furthermore, Koohbor et al.¹ developed a new method to quantify the inertia stress generated in a polymeric compressible foam during axial impact loading. They also discussed the effect of the strain rate on the impact response of the foam. Considering the different microstructure mechanisms that would occur during the uniaxial compression test of the PUF, Mane et al.¹⁸ showed that the yield strength went up with the increasing strain rate. They also observed that the densification zone took place much earlier along with dynamically deformed foams than with quasi-statically deformed foams. In addition, it was proved that the PUF had a higher energy absorption capacity when subjected to dynamic compression tests, compared to its deformation capacity in quasi-static compression tests. Li et al.¹⁹ investigated the strain rate effect on the PUF behavior by means of compression tests. They found out that the elastic stiffness, the yield stress, and the energy absorption capacity grew with the increasing strain rate. Moreover, Abdi et al.²⁰ compared the effect of different deformation rates on the flexural properties of both foamed-core and polymer-pin-reinforced-foamed-core sandwich panels. The obtained results showed that the strain rate effect was only related to the foam core properties. Furthermore, Zhang et al.²¹ investigated the polystyrene foam response to static and dynamic indentation tests. An analytical model was proposed and compared to the experimental and numerical results. They found out that the model could reproduce the indentation results of the polystyrene foam for various loading rates.

Generally, most of the above-mentioned work has considered isotropic behavior laws for foam materials. However, polymeric foams could exhibit anisotropic behavior caused by the manufacturing process. To further understand the response of the polyurethane foam under various load conditions, it is important to consider the anisotropy of foam materials. Several approaches have been suggested to explain the microscopic anisotropy of the polymeric foams, which consists of cell elongation through the rising direction.^9,19,22–24 One of the first examples of the experimental analysis of the PUF anisotropy was presented by Hubber et al.²⁵ They analyzed the foam microstructure via scanning electron micrographs and found out that the foam behavior was direction-dependent due to cell elongation in the rising direction. Afterward, the macroscopic mechanical properties of the foam were measured through the expansion direction and along two perpendicular directions. Lee et al.²² used the uniaxial tensile and compression tests through different directions in order to prove the anisotropic behavior of the polyurethane foam. They also put forward an anisotropic elasto-visco-plastic model to predict the PUF behavior coupled with damage. In the same vein, Marvi-Mashhadi et al.²³ studied the anisotropic behavior of closed-cell PUFs with various densities using compression tests, in the rising direction and in the perpendicular one. They also analyzed the elastic response in the representative volume element. They found out that the deformation along the rising direction was dominated by the axial deformation (buckling) of the struts, while bending would control the deformation perpendicular to the rising direction. A more interesting approach was suggested by Li et al.¹⁹ They performed both quasi-static and dynamic compression tests on polyurethane samples cut at a set of angles to the rising direction. They found out that the elastic stiffness, the yield stress, and the energy absorption capacity declined with the increasing cutting angle, which had a slight effect on the densification strain.

Most of the work cited previously mainly focused on analyzing the effect of the strain rate and the foam orientation on monotonic tests. The effect of the direction and the influence of the deformation rate on the cyclic behavior of these foams have been rarely reported in the literature. The purpose of this paper is to investigate the cyclic behavior of a PUF while considering together the foam anisotropy and the strain rate effect. To this end, the cellular microstructure of the foam is described. Then, cyclic compression tests under various conditions are experimentally analyzed. The effect of the displacement rate and level on the macroscopic response of the foam is discussed. Afterward, an empirical model is developed in order to reproduce the stress–strain response of the PUF.

Material and methods

Preparation of polyurethane foam

The studied foam is a rigid open-cell PUF used in sandwich panels with a density of 40 Kg/m³. It is produced under free expansion in one privileged orientation (the foaming direction also called rising direction) while it is blocked in the other two perpendicular directions, as shown in Figure 1. Samples are cut out into rectangular prisms according to two orientations: the foam rising direction (D1) and a transverse direction perpendicular to the rising one (D2). Ensured by the manufacturing process as in Figure 1, it can be assumed that the two directions (D2) and (D3) have nearly the same mechanical properties justified by the fact that the foam is blocked through these two directions. Furthermore, the thickness along (D2) is only 70 mm. Thus, the foaming process is assumed to be only along (D1). Normalized samples have a section of 50 × 50 mm² and a thickness of 50 mm. During all the tests, the ambient temperature and relative humidity are 25°C and 50%, respectively.

Figure 1.

Schematic presentation of manufacturing process of foamed-core sandwich panels.

Microscopic observations

A transmitted light microscope (Figure 2) is used to observe the microstructure of a cross-section in plane (D1, D2) of the polyurethane foam. The sample is thinned to a transparent body and then placed under different magnifications to examine the anisotropy of the foam and approximately determine the thickness of the cell edges.

Figure 2.

Microscopic observation of thinned polyurethane foam.

Experimental tests

Experimental investigations are conducted according to ASTM D1621 for the compressive properties of rigid cellular plastics. In order to study the PUF behavior, cyclic compression tests are carried out under various strain rates and displacement levels for both considered directions. All the tests are performed using an Instron tensile machine presented in Figure 3. The rectangular prisms are placed between the two perfectly parallel platens, and the displacement is controlled. To ensure repeatability, three specimens are tested for each loading direction and strain rate. In order to study the effect of the strain rate and the loading direction on the polyurethane foam response, three displacement rates (5 mm/min, 50 mm/min, and 400 mm/min) are imposed along (D1) and (D2). Besides, for each strain rate and loading direction, three deformation levels have been reached (10%, 50%, and 80%) to study the effect of the loading level on the mechanical parameters of the foam. For all tests, the unloading phase follows the loading one.

Figure 3.

Compression tests using universal tensile machine: PUF samples are placed between two parallel platens, and displacement is controlled.

Modeling of loading-unloading compression test

Hardening model

By considering the experimental results, the behavior model of the foam is assumed to be elastoplastic and rate dependent, and the yield function is written as

f = \bar{σ} - (σ_{0} + H ({\bar{ε}}^{a})) {(\frac{\dot{\bar{ε}}}{{\dot{\bar{ε}}}_{r}})}^{m} \leq 0

(1)

where

\bar{σ}

is the equivalent stress,

{\dot{\bar{ε}}}_{r}

denotes a reference strain rate where the initial yielding stress is

σ_{0}, \dot{\bar{ε}}

is the strain rate through the (D1) direction, m is a mechanical parameter, and H is the hardening stress given as a function of the accumulated equivalent anelastic residual strain

{\bar{ε}}^{a}

described by the following Ludwick law

H ({\bar{ε}}^{a}) = k {({\bar{ε}}^{a})}^{n}

(2)

where k and n are hardening model parameters.

For uniaxial compression tests and for a reference strain rate of 0.017 s⁻¹ (corresponding to a displacement rate of 50 mm/min), the yield function, introduced in equation (1), is reduced to

f = σ - σ_{0} - H ({\bar{ε}}^{a})

(3)

In order to consider the densification aspect, the following hardening model is then proposed

H (ε^{a}) = k {(ε^{a})}^{n} \exp (α 〈 ε^{a} - ε_{s}^{a} 〉)

(4)

where α and

ε_{s}^{a}

are additional hardening model parameters, and

〈 〉

are the so-called Macauley brackets:

〈 x 〉 = (x + | x |) / 2

. Equation (4) clearly shows that no densification is supposed to occur for deformations below

ε_{s}^{a}

. The parameters corresponding to the two considered directions, (D1) and (D2), are identified in Table 1.

Table 1.

Hardening model parameters.

	σ ₀ (MPa)	k (MPa)	n	ε^a_s, %	α
Parallel	0.3	1	6	70	9
Perpendicular	0.2	0.97	2.5	59	3.1

The hardening model is calibrated via the uniaxial compression tests. Figure 4 presents the experimental results and the predictions of the proposed hardening model (equation (4)). There is a good agreement between these results.

Figure 4.

Presentation of experimental and analytical engineering stress-anelastic strain curves in compression test: (a) (D1) direction;(b) (D2) direction.

Unloading elastic modulus model

In order to describe the unloading elastic modulus, the widely used exponential formula of Yoshida et al.²⁶ is considered as follows

E_{u} = E_{0} - (E_{0} - E_{a}) (1 - \exp (- ξ ε_{0}^{a}))

(5)

where E_u is the unloading elastic modulus, E₀ is the initial elastic modulus, E_a is the elastic modulus obtained for an infinitely large anelastic prestrain

ε_{0}^{a}

, and ξ is a mechanical parameter that determines the decrease rate of E_u.

In order to consider a non-linear unloading, the unloading elastic modulus should depend on the normalized stress point σ₁/σ_0, as suggested by Chatti et al.²⁷ for metallic materials.

As illustrated in Figure 5, E_u is equal to the initial elastic modulus for the normalized stress point σ₁/σ₀=1 (at the beginning of the unloading stage) and it decreases as the stress point goes down. Accordingly, the following relation is put forward²⁷

E_{u} = [E_{0} - (E_{0} - E_{a}) (1 - \exp (- ξ ε_{0}^{a}))] σ_{1} / σ_{0}

(6)

Figure 5.

Definition of unloading elastic modulus: The unloading elastic modulus depends on the imposed strain level $ε_{0}^{a}$ , reached just before unloading stage and on the normalized stress point σ₁/σ₀.

However, for foam materials, it is observed that the unloading decrease is more important than for metallic materials. Thus, equation (6) is modified as

E_{u} = [E_{0} - (E_{0} - E_{a}) (1 - \exp (- ξ ε_{0}^{a}))] δ σ_{1} / σ_{0}

(7)

where δ is a stiffness reduction coefficient.

Furthermore, to consider the densification aspect, equation (7) is adjusted as

E_{u} = [E_{0} - (E_{0} - E_{a}) (1 - \exp (- ξ ε_{0}^{a}))] δ \frac{σ_{1}}{σ_{0}} (1 + α {〈 ε_{0}^{a} - ε_{s}^{a} 〉}^{n})

(8)

where α,

ε_{s}^{a}

, and n are the parameters of the unloading elastic modulus model depending on the (D1) and (D2) directions.

Calibration of the unloading model

To determine the PUF elastic modulus for every strain level, nine progressive loading-unloading cycles for strains ranging from 10% to 90% are imposed. Specimens are compressed under a displacement rate of 50 mm/min in both considered directions. The experimental results are plotted in Figure 6. This figure is considered to determine the experimental elastic modulus in the unloading stage given as a function of the anelastic strain. Figure 7 gives the unloading elastic modulus for directions (D1) and (D2) by experiments and predictions with identified parameters reported in Table 2.

Figure 6.

Stress–strain response in cyclic compression tests: (a) parallel; (b) perpendicular orientations: Nine progressive loading-unloading cycles are applied with increasing strain levels for each cycle from 10% to 90%.

Figure 7.

Unloading elastic modulus vs. anelastic strain: (a)l direction (D1) and (b) direction (D2) for different stress points σ₁/σ₀.

Table 2.

Identification of the elastic unloading modulus parameters.

	E ₀ (MPa)	E _a (MPa)	ξ	n	α	ε_a^s(%)	δ
Parallel (D1)	4.86	0.95	15	4	4.8	29	15
Perpendicular (D2)	2.62	0.82	20	1.5	21	25	4

Results and discussions

Microstructure of polyurethane foam

Figure 8 presents the microscopic observations of the investigated polyurethane foam. The graph shows that the foam presents a cellular microstructure with a pore size in the order of few hundred microns. The cell edges are about 10 μm thick. Besides, the polyurethane foam cells are mostly oriented in one direction, which is assumed to be the foaming direction. This figure clearly shows that the foam is anisotropic and highly polydisperse. In addition, the geometrical degree of anisotropy, DA, which is the ratio of the cell size in D1 to that in D2, is equal to 2.35.

Figure 8.

Microscopic observation of polyurethane foam obtained by transmitted light microscope in plane (D1, D2).

Direction effect and polyurethane foam anisotropy

In this section, the effect of the cell orientation on the quasi-static response of the foam is discussed. Figure 9 presents stress–strain curves for different strain rates in compression with respect to the two directions (D1) and (D2). It is clearly shown that the foam behavior presents a significant difference between directions (D1) and (D2), regardless of the imposed strain rate. In fact, for the (D1) direction, there is a primary linear zone, followed by a plateau, where stress remains almost constant. Then a densification zone appears, and the stress rises speedily. However, for the (D2) direction, the stress increases continuously after the first linear zone until densification is reached. This difference is expected since the PUF cells are elongated through the rising direction, causing microscopic anisotropy.²³ The most likely explanation of the stress–strain trends is that compression along the (D1) direction is dominated by axial deformation via cell edge buckling, which leads to a linear trend.¹⁹ Then, a progressive collapse of the cells takes place, and the stress remains almost constant, generating a plateau. Finally, the cell edge crush leads to the cell collapse and therefore to the interaction of opposite cell edges responsible for the consequent steep stress increase. On the other hand, the deformation is controlled by cell bending along (D2),¹⁹ and the stress builds up continuously until reaching the stress densification zone when the cell edges are collapsed.

Figure 9.

Stress–strain curves in directions (D1) and (D2) at displacement rates of: (a) 5 mm/min, (b) 50 mm/min, and (c) 400 mm/min.

Strain rate effect

Figure 10 illustrates the strain rate dependence of the PUF for the rising and perpendicular directions in compression tests. For all displacement rates, the stress–strain curves exhibit similar trends when a specific orientation is considered. Table 3 summarizes the mechanical PUF parameters for different strain rates and loading directions. It can be seen that the yield strength (10% deformation stress) and the compressive strength (stress corresponding to strain of 80%) together grow with the increasing displacement rate, and that for all loading directions. In fact, the number of interlocked chains for polymeric materials increases as the strain rate goes up, so the stress builds up. However, the elastic modulus is almost the same for all strain rates through direction (D1), whereas it goes up with the strain rate through direction (D2). This result is expected since the linear elastic zone along (D1) is generated by the buckling of the cell edges.¹⁹ As the buckling mechanism is not affected by the strain rate variation, the elastic modulus along (D1) is almost constant.

Figure 10.

Stress–strain curves at different strain rates for directions (a) (D1) and (b) (D2).

Table 3.

Mechanical properties of PUF for different displacement rates and loading directions.

Displacement rate		5 mm/min	50 mm/min	400 mm/min
Elastic modulus E (MPa)	D1	4.91 ± 0.91	4.86 ± 1.10	5.01 ± 0.75
—	D2	1.71 ± 0.28	2.62 ± 0.16	2.91 ± 0.12
Yield strength σ₀ (MPa)	D1	0.25 ± 0.01	0.30 ± 0.01	0.32 ± 0.01
—	D2	0.14 ± 0.01	0.17 ± 0.01	0.19 ± 0.01
Maximal strength σ_max (MPa)	D1	0.45 ± 0.02	0.48 ± 0.01	0.54 ± 0.03
—	D2	0.75 ± 0.04	0.80 ± 0.02	1.01 ± 0.06
Effective energy absorption capacity W(KPa)	D1	200 ± 3	229 ± 7	241 ± 3
—	D2	142 ± 9	156 ± 7	194 ± 3

Furthermore, during quasi-static compression tests, the effective energy absorption capacity (the energy absorbed per unit volume and corresponding to the area limited by the loading and the unloading stress–strain curve) is more important for the (D1) direction rather than the (D2) direction, and it builds up with the increasing strain rate. In fact, since the cells are elongated in (D1), they have more capacity to deform along this direction. Therefore, the elementary displacement δu, defined in Figure 11 of the point of application of the applied force F, along the direction (D1), is greater than that of the applied force under the other space directions. Consequently, the elementary work of the force applied in the (D1) direction is greater than that of the force applied in the (D2) direction. Thus, the energy absorption capacity along (D1) is greater than the one along (D2).

Figure 11.

Schematic illustration of applied force on elementary cell in (D1) and (D2 ) .

Strain level effect

Figure 12 reports the stress–strain curves at different loading levels with respect to the three distinct zones: (i) the linear zone at the strain level of 10%, (ii) the plateau zone at the strain level of 50%, and (iii) the densification zone at the strain level of 80%. The above-mentioned figure and Table 4 shows that the mechanical parameters of the foam depend on the imposed strain level. In particular, the PUF capacity to absorb energy is more important in direction (D1) than that in direction (D2), and it increases significantly with the strain level. Moreover, the residual strain builds up considerably with the rising deformation level.

Figure 12.

Stress–strain curves for different loading levels along (D1) at displacement rate of: (a) 5 mm/min, (c) 50 mm/min, (e) 400 mm/min; and along (D2) at displacement rate of (b) 5 mm/min, (d) 50 mm/min, (f) 400 mm/min.

Table 4.

Mechanical properties of PUF with different displacement rates, loading directions, and strain levels.

Parameter	Strain	5 mm/min		50 mm/min		400 mm/min
—	—	D1	D2	D1	D2	D1	D2
Maximal strength (MPa)	10%	0.25 ± 0.01	0.14 ± 0.01	0.30 ± 0.02	0.17 ± 0.02	0.32 ± 0.01	0.19 ± 0.01
—	50%	0.28 ± 0.01	0.26 ± 0.02	0.31 ± 0.01	0.3 ± 0.02	0.32 ± 0.01	0.3 ± 0.02
—	80%	0.45 ± 0.02	0.75 ± 0.04	0.48 ± 0.01	0.80 ± 0.02	0.54 ± 0.03	1.01 ± 0.06
Effective energy absorption capacity (KPa)	10%	8.5 ± 2.7	2.9 ± 0.5	16.0 ± 0.8	6.2 ± 0.4	18.3 ± 1.1	7.4 ± 0.8
—	50%	114.0 ± 3.5	66.9 ± 3.1	127.0 ± 6.5	72.4 ± 1.2	131.7 ± 4.6	75.1 ± 7.9
—	80%	200 ± 3	142 ± 9	229 ± 7	156 ± 7	241 ± 3	194 ± 3
Residual deformation (%)	10%	2.5 ± 0.1	0.00 ± 0.01	1.7 ± 0.2	0.00 ± 0.01	1.6 ± 0.1	0 ± 0
—	50%	24.2 ± 1.7	13.1 ± 1.4	22.4 ± 1.2	9.6 ± 1.1	21.8 ± 2.3	4.4 ± 2.3
—	80%	36.3 ± 2.6	13.2 ± 0.7	32.7 ± 0.7	12.4 ± 0.3	29.5 ± 3.1	11.4 ± 0.2

In addition, the residual strain is greatly higher in direction (D1) than in direction (D2). In fact, for direction (D2), there is no residual deformation after the unloading stage for a strain level of 10%; that is, the deformation is totally recoverable. Thus, the first zone of the stress–strain curve in direction (D2) reveals a linear elastic behavior explained by the bending of cell edges. When the flexural stress exceeds the plastic moment, particularly for 50% of the imposed strain, the deformation is both elastic and plastic, so it is partially recoverable. Then, as the applied deformation increases, the foam gets more plastic and the residual strain grows. However, considering direction (D1), the deformation occurs via buckling of the cell edges. Since buckling is unstable, it results in a deformation localization mechanism.^19,28 Consequently, for 10%, only the cell edges in contact with the moving platter deform elastically and then plastically. Hence, the deformation is partially recoverable, but the residual deformation is small. By applying 50% of the strain, the above-mentioned zone is plastically deformed, and the adjacent zones are also deformed. Thus, there is an increase in the residual strain. When densification is reached, the cells begin to crush, and the edges come into mutual contact. Thus, the deformation is non-recoverable regardless of the strain rate and the direction. As a result, there are higher residual strains. Furthermore, the residual strain decreases with the increasing deformation rate: If the foam is loaded with a low velocity, it takes a long time to return to its initial shape, and an increase in the deformation rate allows the foam to recover almost fully, explained by the viscosity aspect of the cell edges.

Model validation

The previous tests and the analytical study clearly prove that the polyurethane foam mechanical parameters are described by an anisotropic behavior. The proposed model is compared to the experimental response in compression tests. Figure 13 shows that the use of the model described by Eq. (4) and Eq. (8) is in good agreement with the experiments. To conclude, the suggested analytical model considering an adjustable modulus of elasticity can accurately predict the polyurethane foam behavior in cyclic loading-unloading compression tests.

Figure 13.

Comparison of analytic and experimental results for two strains 50% and 80% for: (a) rising direction; (b) perpendicular direction.

Conclusion

The effects of the strain rate, the strain level, and the foam anisotropy on the cyclic compressive behavior of a rigid open-cell polyurethane foam have been investigated. It has been found that the PUF presents a cellular microstructure providing a privileged direction: the rising direction. The experimental results have proved that this microscopic anisotropy influences the cyclic mechanical behavior of the foam. The elastic modulus, the maximal compressive stress, the residual strain, and the effective energy absorption capacity have been discussed by means of cyclic compression under various conditions. The effect of the displacement level on the PUF response has been also investigated, and it has been found that the foam behavior is highly dependent on the deformation level, particularly the residual strain. Furthermore, based on an analytical study, a new approach has been developed to describe the loading-unloading compression cycle of the polyurethane foam. In fact, the suggested model uses an adjustable modulus of elasticity depending on the strain level.

The present study can be extended to a morphological characterization of the PUF microstructure in order to develop a relation between the geometrical degree of anisotropy and the ratio of the investigated mechanical properties in directions (D1) and (D2). Afterward, the microstructure representations can be used to simulate a whole sandwich structure subjected to cyclic loading where the used isotropic elastoplastic model fails to predict accurately the real behavior of foam materials.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Dorra Ben Abdeljelil

References

Koohbor

Singh

Kidane

Radial and axial inertia stresses in high strain rate deformation of polymer foams. Int J Mech Sci 2020; 181: 105679. DOI: 10.1016/j.ijmecsci.2020.105679.

Liu

Subhash

Gao

XL.

A parametric study on crushability of open-cell structural polymeric foams. J Porous Mater 2005; 12(3): 233–248. DOI: 10.1007/s10934-005-1652-1.

Sun

QM.

Dynamic compressive behaviour of cellular materials: A review of phenomenon, mechanism and modelling. Int J Impact Eng 2018; 112: 74–115. DOI: 10.1016/j.ijimpeng.2017.10.006.

Ousji

, et al. Air-blast response of sacrificial cladding using low density foams: Experimental and analytical approach. Int J Mech Sci 2017; 128–129: 459–474. DOI: 10.1016/j.ijmecsci.2017.05.024.

Qiu

, et al. Construction of 3-D realistic representative volume element failure prediction model of high density rigid polyurethane foam treated under complex thermal-vibration conditions. Int J Mech Sci 2021; 193: 106164. DOI: 10.1016/j.ijmecsci.2020.106164.

Atiqah

Jawaid

Sapuan

, et al. Physical and thermal properties of treated sugar palm/glass fibre reinforced thermoplastic polyurethane hybrid composites. J Mater Res Technol 2019; 8(5): 3726–3732. DOI: 10.1016/j.jmrt.2019.06.032.

Kairytė

Vaitkus

Kremensas

, et al. Moisture-mechanical performance improvement of thermal insulating polyurethane using paper production waste particles grafted with different coupling agents. Constr Build Mater 2019; 208: 525–534. DOI: 10.1016/j.conbuildmat.2019.03.048.

Zhu

ai Xu

Synthesis and properties of rigid polyurethane foams synthesized from modified urea-formaldehyde resin. Constr Build Mater 2019; 202: 718–726. DOI: 10.1016/j.conbuildmat.2019.01.035.

Kim

, et al. Mechanical Performance Degradation of Glass Fiber-reinforced Polyurethane Foam Subjected to Repetitive Low-energy Impact. Int J Mech Sci 2021; 194: 106188. DOI: 10.1016/j.ijmecsci.2020.106188.

10.

Wang

Huang

YH.

Viscoelastic properties of foam under hydrostatic pressure and uniaxial compression. Proced Eng 2013; 67: 397–403. DOI: 10.1016/j.proeng.2013.12.039.

11.

Garrido

Correia

JR.

Elastic and viscoelastic behaviour of sandwich panels with glass-fibre reinforced polymer faces and polyethylene terephthalate foam core. J Sandw Struct Mater 2018; 20(4): 399–424. DOI: 10.1177/1099636216657388.

12.

Demirel

Ergun Tuna

Evaluation of the cyclic fatigue performance of polyurethane foam in different density and category. Polym Test 2019; 76: 146–153. doi: 10.1016/j.polymertesting.2019.03.019.

13.

Hwang

Kim

, et al. Dynamic compressive behavior of rigid polyurethane foam with various densities under different temperatures. Int J Mech Sci 2020; 180(March): 105657. DOI: 10.1016/j.ijmecsci.2020.105657.

14.

Linul

Şerban

Marsavina

, et al. Assessment of collapse diagrams of rigid polyurethane foams under dynamic loading conditions. Arch Civ Mech Eng 2017; 17(3): 457–466. DOI: 10.1016/j.acme.2016.12.009.

15.

Kim

Shin

Rhim

, et al. Calibration of hyperelastic and hyperfoam constitutive models for an indentation event of rigid polyurethane foam. Compos B Eng 2019; 163(2018), 297–302. DOI: 10.1016/j.compositesb.2018.11.045.

16.

Taherkhani

Sadighi

Vanini

, et al. An experimental study of high-velocity impact on elastic-plastic crushable polyurethane foams. Aerosp Sci Technol 2016; 50: 245–255. DOI: 10.1016/j.ast.2015.11.013.

17.

Koumlis

Lamberson

Strain Rate Dependent Compressive Response of Open Cell Polyurethane Foam. Exp Mech 2019; 59(7): 1087–1103. DOI: 10.1007/s11340-019-00521-3.

18.

Mane

, et al. Mechanical Property Evaluation of Polyurethane Foam under Quasi-static and Dynamic Strain Rates- An Experimental Study. Proced Eng 2017; 173: 726–731. DOI: 10.1016/j.proeng.2016.12.160.

19.

Guo

Zhou

, et al. Response of anisotropic polyurethane foam to compression at different loading angles and strain rates. Int J Impact Eng 2019; 127: 154–168. DOI: 10.1016/j.ijimpeng.2018.12.009.

20.

Abdi

Azwan

Abdullah

, et al. Flatwise compression and flexural behavior of foam core and polymer pin-reinforced foam core composite sandwich panels. Int J Mech Sci 2014; 88: 138–144. DOI: 10.1016/j.ijmecsci.2014.08.004.

21.

Zhang

, et al. Indentation of expanded polystyrene foams with a ball. Int J Mech Sci 2019; 161–162: 105030. DOI: 10.1016/j.ijmecsci.2019.105030.

22.

Lee

JM.

Failure analysis of reinforced polyurethane foam-based LNG insulation structure using damage-coupled finite element analysis. Compos Struct 2014; 107(1): 231–245. DOI: 10.1016/j.compstruct.2013.07.044.

23.

Marvi-Mashhadi

Lopes

LLorca

Effect of anisotropy on the mechanical properties of polyurethane foams: An experimental and numerical study. Mech Mater 2018; 124: 143–154. DOI: 10.1016/j.mechmat.2018.06.006.

24.

Guo

Shim

VPW.

A constitutive model for transversely isotropic material with anisotropic hardening. Int J Sol Struct 2018; 138: 40–49. DOI: 10.1016/j.ijsolstr.2017.12.026.

25.

Huber

Gibson

LJ.

Anisotropy of foams. J Mater Sci 1988; 23(8): 3031–3040. DOI: 10.1007/BF00547486.

26.

Yoshida

Uemori

Fujiwara

Elastic-plastic behavior of steel sheets under in-plane cyclic tension-compression at large strain. Int J Plast 2002; 18(5–6): 633–659. DOI: 10.1016/S0749-6419(01)00049-3.

27.

Chatti

Hermi

The effect of non-linear recovery on springback prediction. Comput Struct 2011; 89(13–14): 1367–1377. DOI: 10.1016/j.compstruc.2011.03.010.

28.

Shim

VPW

Lim

CT.

Plastic deformation modes in rigid polyurethane foam under static loading. Int J Sol Struct 2001; 38(50–51): 9267–9279. DOI: 10.1016/S0020-7683(01)00213-X.