Full-Field Experimental Study and Numerical Modeling of Soft Polyurethane Foam Subjected to Cyclic Loading ^†

Abstract

In this study, the responses of three soft open cell polyurethane foam samples (85, 63 and 46 kg/m³) subjected to four incremental cyclic compression load steps of 20%, 40%, 60%, and 80% strain are analyzed by digital image correlation. Facing large deformation, the foam behavior is investigated in terms of engineering and true strain curves. Poisson’s ratio is studied by a tangent Poisson function which is able to capture the instantaneous behavior of the foam. Experimental displacement maps, stress–strain, and axial vs. transverse strain curves are compared in terms of numerical results obtained by FEM analysis for ν = 0 and ν ≠ 0.

1. Introduction

Polyurethane foam (PUf) is a versatile material that is commercialized as being either rigid or soft. It is mainly used due to its good energy absorption capability and low weight. The exact structure and deformation mechanisms of this material are topics of great interest as, at large deformations, the material response becomes nonlinear. To analyze the mechanical behavior of soft polyurethane foam, full field, contactless, and non-destructive measurement techniques, like digital image correlation (DIC) [1], should be used. Several studies can be found in the literature that report the use of DIC for the mechanical characterization of foams, but most of them are related to rigid foams [2,3]. When soft and low-density materials are analyzed, events such as large deformations, severe illumination variations, or other phenomena can cause serious decorrelation effects in the deformed images and the measurement can fail [4,5]. In this work, the mechanical characteristics of three types of PUf (types A, B and C) with densities of 85, 63, and 46 kg/m³, respectively, are investigated by incremental DIC, exploiting the naturally speckled pattern of the specimens that is given by the presence of fireproof particles of expanded graphite [6,7]. Due to large deformation, the nonlinear response is analyzed, both experimentally and numerically (Abaqus 6.13), thereby plotting engineering and true strain curves. Stress–strain curves and the energy absorbing capability are evaluated here. An eventually auxetic behavior of the materials is investigated by a tangential Poisson’s function formulation which is more capable in terms of capturing the instantaneous behavior of the foam.

2. Materials and Methods

2.1. Materials and Specimens

The experiments were performed on three fire-retardant, soft, open cell polyurethane foams of different densities, manufactured by the same production process. These foams comply with the flammability requirements, so they are suitable for use in the aviation industry thanks to the presence of expanded graphite, which is visible in the form of small black dots that are dispersed throughout the material (Figure 1). Cubic specimens of foam types A, B, and C, with densities of 76 ÷ 94 kg/m³ (higher density), 60 ÷ 65 kg/m³ (average density) and 43 ÷ 49 kg/m³ (lower density), respectively, are represented in Figure 1a from left to right, while their characteristics are summarized in

Table 1. Designation and characteristics of the tested foams.

Designation	Type A	Type B	Type C
Foam type	Flexible polyurethane foam
Cells type	open
Density [kg/m3]	76 ÷ 94	60 ÷ 65	43 ÷ 49
Cell size [mm]	0.145 ± 0.018	0.210 ± 0.076	0.315 ± 0.064
Cell wall size [mm]	0.044 ± 0.008	0.043 ± 0.006	0.041 ± 0.011
Void ratio %	35 ± 0.024	53 ± 0.113	53 ± 0.113

. A comparison among the microstructures of the studied foam is presented by the stereoscopic images in Figure 1b–d.

2.2. Cyclic Compression Test

Cyclic compression tests were performed at room temperature in compliance with the UNI EN ISO 3386 standard [8]. Cubic specimens of 50 mm × 50 mm × 50 mm (see Figure 2a,b) were tested using the universal hydraulic test machine INSTRON 1342 (Instron, Norwood, MA, USA) equipped with a 10 kN load cell. The specimens were compressed between the flat plates in the displacement control mode at 50 mm/min during the loading phase and at 10 mm/min during the unloading phase. This arrangement ensured contact between the specimen and the plates. Four deformations levels were reached with 20%, 40%, 60%, and 80% strain, considering the initial heights of the specimens. Stress-strain relationships were obtained using the loading data acquired by the testing machine at a frequency of 20 Hz, and displacement data were obtained by the DIC technique at a frequency of 1 Hz. Before the test, the apparatus was calibrated and the pre-load conditions were evaluated to be as low as possible.

2.3. DIC Set-Up and Measurements

Digital image correlation (DIC) is a full-field, non-contact optical technique that allows one to determine maps of displacement and strain over an entire surface under examination. It is based on the comparison of the differences between a reference image, generally acquired at zero load, and an image of the same surface that is captured after deformation of the sample [9]. In this study, since large deformations were achieved, an incremental DIC approach was adopted [10] to overcome issues such as reflections and, therefore, losses of correlation. Basically, the incremental DIC was applied by comparing the image n to the previously recorded one, (n-1). The main components of the DIC measurement setup are summarized in Figure 2b. The experiments were performed using a single camera, therefore in 2D, because, due to the symmetry of the specimen, it is sufficient to analyze only the deformations in the X–Y plane. The acquisition system used the ISTRA 4D software package (Dantec Dynamics A/S, Copenhagen, Denmark), which received data from a GigE CCD Manta camera (1628 × 1436 pixels) AVT (Allied Vision Technologies GmbH, Stadtroda, Germany), imaging through a XENOPLAN 1.4/23-0902 lens. All the experiments were carried out under optimized lighting conditions, obtained by arranging a LED light to graze the surface of the specimen to avoid unwanted reflections and light peaks. Before gathering measurements, the whole system was calibrated and some preliminary analyses were performed to evaluate the measurement noise and the suitability of the speckle pattern. Indeed, the surfaces of the specimens were not treated, so the natural texture of the foam, given by the morphology of the cell walls and the expanded graphite particles contained in the material, was analyzed considering the Shannon entropy values (SE) [11], the black/white ratio, and the sizes of speckles. For the data analysis, it was assumed that the Y-axis coincides with the direction of the load and the X-axis coincides with the transverse one. The stress–strain curves were calculated by placing two virtual reference points, P₁ and P₂, along the load direction on the X–Y plane, near the contact with the plates. For more details regarding the measurements and the DIC parameter settings, see the work of Casavola et al. [6]. Since large deformations are involved, the real (or Hencky) deformation of the specimens, ε_i, was considered, given by the ratio between the extension and the instantaneous length of the specimen during the testing time t:

$${\epsilon }_i=\frac{L_i-L_{i-1}}{L_i}=\ln \left(1+e_i\right) 0\le i\le t$$

(1)

From Equation (1), a simple way to calculate the instantaneous component of the strain is the logarithm of the engineering strain e. Being the Poisson’s ratio, ν, a constant that governs the transverse deformation of an isotropic material in the linear elastic strain regime, it is defined as the negative ratio between the transverse engineering strain, e_x, and the axial engineering strain, e_y:

$$\nu =-\frac{e_x}{e_y}$$

(2)

Equation (2) must satisfy –1 ≤ ν ≤ 0.5 [12]. For linear elastic materials subject to small deformations, there is a rigorous definition of the Poisson’s ratio, where this does not occur in case of non-linear deformations. Indeed, hyperelastic foams are characterized by a Poisson ratio that varies strongly with the deformation level [13,14]. So, for an hyperelastic foam, the expression in Equation (2) is the Poisson’s secant ratio, which can be obtained for every small load increment as a function of the global axial strain [15]. In this regard, some authors [14,15] have proposed a tangential Poisson function formulation, defined as:

$${\nu }_{tan}=-\frac{d{e}_x}{d{e}_y}$$

(3)

This formula is able to highlight the “instant” value of the Poisson ratio. The expression in Equation (3) is given by the tangent to the transverse strain curve versus the global axial deformation for each load increase. For more details on Poisson’s function, see Ref. [14].

2.4. Numerical Modelling

The finite element model, built in Abaqus 6.13 (Dassault System, Vélizy-Villacoublay, France), reproduced the compression tests while considering both ν = 0 and ν ≠ 0. The condition ν ≠ 0 is obtained while including, point by point, the experimental nominal transverse strain of each tested foam. The models were discretized using the CPS4 four-node flat element and 2601 nodes. The Hyperfoam material model [16] was used to reproduce the mechanical behavior of the hyperelastic foams. This elastic model considers characteristics such as an isotropic material, non-linear behavior, very large volumetric variations, elastic deformations up to 90%, and energy dissipation. The Hyperfoam material modes are based on a strain energy function, U, originally proposed by Ogden [17]:

$$U={\displaystyle \sum }_{i=1}^N\frac{2{\mu }_i}{{\alpha }_i{}^2}\left[{\widehat{\lambda }}_1^{{\alpha }_i}+{\widehat{\lambda }}_2^{{\alpha }_i}+{\widehat{\lambda }}_3^{{\alpha }_i}-3+\frac{1}{{\beta }_i}\left({\left(J^{el}\right)}^{-{\alpha }_i{\beta }_i}-1\right)\right]$$

(4)

where N is the order of the strain energy potential (N ≤ 6), μ_i is related to the initial shear modulus, α_i specifies the shape of the stress–strain curve, while β_i determines the degree of compressibility and is directly related to the Poisson’s ratio, ν_i. For further details see the software guide [16].

3. Results

3.1. Natural Pattern Preliminary Evaluation and Measurements

The results of the assessment of the speckle pattern quality, obtained through a Python routine and a MATLAB^® routine, for the Shannon entropy and the black/white ratio values, respectively, are shown in

Table 2. Comparison between the Shannon entropy values and the black/white ratio of the analyzed foams, containing flame retardant graphite particles, and of a generic soft PU foam.

	Type A	Type B	Type C	Generic
SE	7.07	7.21	7.18	6.22
B/W ratio	58.6%	59.8%	63.6%	19.1%

. In this first analysis, the natural texture of the three tested foams, given by the presence of small black particles of expanded graphite is compared with a generic soft PUf one.

The analysis of the speckle size, performed in MATLAB^®, gave values mainly distributed between 2 and 4 pixels. The accuracy of the measurement was assessed, so the average error, as a percentage, was calculated considering all the values recorded during a single test, for each material. For the displacements along the Y and X directions, the average maximum errors were 1.65% and 2.91%, respectively.

3.2. Cyclic Compression Test

Experimental and numerical stress–strain curves obtained for compression until 80% strain are shown in Figure 3a along the positive axis for ease of reading. Also, a magnification related to the first phase of compression up to 10% of strain is given in Figure 3a. An eventually auxetic behavior of the foams is investigated by plotting the transverse vs. axial strain curves, as shown in Figure 3b. Facing large deformations, curves were reported while considering both the engineering (Exp_ENG) and the true (Exp_TRUE) strain data for all the three foams. These experimental curves were also proposed while considering the numerical model output (Eng_ν, True_ν). Obviously, the influence of the transverse strain is missed when ν = 0, as can be seen in the corresponding horizontal curve in Figure 3b. The auxetic behavior of the foams is explored during the first and the last steps of deformation, namely 20% and 80% strain, through a tangential Poisson’s function formulation, as shown in Figure 3c. The magnification in Figure 3c, until 10% strain, highlights the differences of the tangential Poisson values in the first instant of loading between the first and the fourth cycle. Foams are mainly used to dissipate energy through the deformation of their cellular structure by keeping the stress within a given threshold, so the energy absorbing characteristics are relevant. The area subtended by the stress–strain curve represents the energy density per unit volume, W, dissipated by the material, expressed in kJ/m³:

$$W={{\displaystyle \int }}_0^{ϵ}\sigma dϵ$$

(5)

In Figure 3d, the energy absorbing capability of each foam is provided, together with magnification until 10% strain is reached for the first and last instances of deformation (i.e., 20% and 80% of strain).

In

Table 3. Plateau stress σ_pl and modulus of elasticity E_5% up to 5% of deformation and the plateau modulus E_pl for the first and fourth compression cycles (20% and 80% of deformation).

Specimen	σpl 20% [kPa]	E5% 20% [kPa]	Epl 20% [kPa]	σpl 80% [kPa]	E5% 80% [kPa]	Epl 80% [kPa]
Type A	10.3	159.5	23.7	7	110.7	24.1
Type B	7.2	125.7	14.3	4.4	87	16.4
Type C	3.4	73.7	6	2.3	38.7	5.8

, the elastic modules of the foams are calculated until the linear elastic limit of 5% of strain [18]. Also, the plateau modules, E_pl, and the collapse stress plateau values, σ_pl, are reported for 20% and 80% strain. The plateau stress is given by the intersection between the linear approximation of the initial elastic curve up to 5% of strain and the linear approximation of the plateau region, whose slope is properly represented by the plateau modulus. The experimental results, in the form of displacement maps obtained by DIC, are reported in the top row of Figure 4 along the Y (a, b) and X (c, d) axes, while the numerical results obtained through Abaqus CAE are reported in the bottom row of Figure 4 for the same compression steps along the Y (e, f) and X (g, h) axes. Considering the real experiment, the 2D square specimens were constrained to the upper end, preventing displacement along the Y-axis while, at the lower end of the specimen, a compression of up to 80% has been imposed. More information about the analysis for each cyclic compression step and the Hyperfoam material constant has been reported by Casavola et al. [6,7].

4. Discussion

Speckle pattern quality assessment is an important aspect to consider in order to obtain good texture correlation and reliable measurements [6]. Some authors [19,20] have shown that the greater the Shannon entropy entity values are, the lower the measurement error associated with the image. The results reported in Table 2 are consistent with those obtained by Belda et al. [19]. Furthermore, the Shannon entropy values obtained by analyzing foams containing fireproof graphite are always greater than those obtained when analyzing generic foam. Good results were also obtained for the black/white ratio and the average speckle size; in fact, respective values of 50% and a range of 2 ÷ 4 pixels are suggested in different studies [21], which is consistent with those reported in Table 2 and those in the results section. So, it can be concluded that the graphite fireproof particles improve the contrast and lead to better results when compared to a surface in which only the empty/full contrast of the cells is exploited. The measurement errors, as also stated by other authors [5], can be neglected with respect to the large deformations that were reached (until 40 mm). The cyclic compression and compression behavior of soft polyurethane foam is of great interest for the design of energy absorbing devices. In general, the behavior of hyperelastic cellular materials is described by three typical regions, distinguishable in the graphs in Figure 3a, where the first elastic region is due to the bending of the cell walls at low stresses. Furthermore, as the load increases, the cell walls collapse due to the recoverable elastic buckling, the curve has a plateau approximately parallel to the X-axis (high level of deformation in face of a minor increase in stress). Finally, at higher stress values, the cells collapse and the opposite walls touch, giving rise to the last region in which the stress suddenly increases without an appreciable increase in deformation (densification) [18]. The results obtained from the stress–strain curves in Table 3 show that, from the first to the fourth load cycle, the elastic modules until 5% strain decrease by 30% for the type A and type B foams and by 47% for the type C foams. Additionally, as the cycles increased, the plateau stress values decreased by 32% for type A and C and by 38% for type B, while the plateau modules differed slightly for all the foam types (Table 3). The numerical stress–strain curves in Figure 3a, obtained using the Hyperfoam material model with a strain potential order N = 2, show that the experimental curves were well reproduced while considering both ν = 0 and ν ≠ 0. Figure 3b show that foams expanded until 2% of transverse strain and then, at about 20% of axial compressive strain, there was an inversion of the trend, which was also well reproduced by the numerical simulations. The presence of trend inversion means that these foams behave auxetically. More specifically, at the beginning of the load phase, the lateral strain increases as the foam is compressed. Once the transition point is reached, the transverse strain decreases with approximately the same slope of the initial part, which indicates that the specimen contracts with a further increase in compression. In contrast with the experimental results, the numerical curves present two different slopes between the portion of the curve that precedes and follows the transition point. The auxetic behavior of the foams was also deepened by the tangential Poisson function formulation (Figure 3c), which highlights the instantaneous tendency of a material to distort and change in volume. The comparison between the first and fourth cycle shows that the tangential Poisson’s function reached negative values close to 20% of strain and then approached zero when close to 80% strain, i.e., where the foam is totally compressed without the possibility to undergo further transversal deformation. The specific strain energy absorbed by the foams increased with the amount of deformation and with the foam density. Experimental displacement maps obtained by DIC along the Y-axis (Figure 4a,b) show horizontal bands due to progressive cell collapse, while those obtained along the X-axis (Figure 4c,d) confirm the tendency of the material to expand until 20% strain and then shrink at 80% strain. The experiments were well reproduced by the Hyperfoam material model (Figure 4e–h), but, at high strain, local effects were missed along the X-axis.

5. Conclusions

Large deformations have been successfully captured by DIC on unpainted soft polyurethane foams and the results have been reproduced well by a Hyperfoam material model. The auxetic behavior of these conventional foams was studied by transverse vs. axial strain analyses and has been confirmed by a tangential Poisson function formulation.