Modeling the Effect of Foam Density and Strain Rate on the Compressive Response of Polyurethane Foams.
Due to the high energy absorbing capacity combined with a low density, polymer foams are increasingly being employed as impact energy absorbers in automotive applications related to ensuring passive safety, such as bumpers, crash boxes, and side impact protection systems. Polyurethane (PU) foams can be used either as pre-fabricated inserts or by injecting PU components directly into cavities of structural parts where foaming occurs in situ ensuring an efficient mechanical interlocking. PU foams of density ranging from ca. 30 to 400 kg/[m.sup.3] are utilized in the automotive industry as reported in . The polyol component of the foams can be derived from natural sources such as, for example, vegetable oils, or from recycled petrochemical resources, thus further mitigating the environmental impact of production of automotive parts .
While energy-absorption and efficiency diagrams are instrumental in selection of foams for specified loading conditions , the design of structural parts with an impact mitigation functionality is generally performed using numerical simulation codes [4, 5]. The mechanical characteristics of foams are strongly affected by their density, therefore, the optimum foam density, providing the intended mechanical response at a minimum foam weight, can be determined by employing the FEM modeling  for, e.g., crash simulation. As input data for such simulations, compressive stress-strain diagrams of foams at different densities and loading rates are utilized. A model of rate- and density- dependent stress-strain response that could be calibrated against a limited set of test data would allow reducing the amount of actual foam tests needed at this design stage.
Several phenomenological models have been proposed for expressing the compressive stress a as a function of strain s in uniaxial compression at a reference strain rate and temperature, with effects of different strain rates [??] and temperatures T allowed for via a multiplicative correction factor g [6-14]:
[mathematical expression not reproducible] (1)
While separate expressions are employed for the elastic, plateau, and densification regimes of the stress-strain response in , most of the models use smooth continuous analytical functions f([epsilon]) containing several parameters characterizing the foams, which are evaluated by fitting f([epsilon]) to the experimental data [7-9, 11-13, 16-18]. Concerning loading rate effect, the relation for g ([??]) proposed by Nagy et al.  has been found applicable for PU foam response at strain rates ranging from [10.sup.-3] to ca. [10.sup.2] [s.sup.-1] [6, 9, 10, 13]. Variations of the Johnson-Cook relation for g([??]) have been introduced and validated for PU foams in [11, 12, 14]. To allow for the temperature effect on the stress-strain response of foams, temperature-dependent parameters of the Nagy relation have been employed in .
Explicit analytical relations for the parameters of f([epsilon]) = f([epsilon], [rho]) as functions of foam density [rho] have been considered in [17, 18]. Such an approach makes it possible to predict the compressive response of foams at densities different from those against which the model has been calibrated. However, the test results for all the different-density foams were used in evaluating the density dependence of model parameters in [17, 18], thus precluding a direct verification of the predictive potential of this approach. As concerns the effect of loading rate and/or temperature, characterized by factor g in Equation 1, it is typically evaluated separately for each foam density of interest [9-13] without considering possible analytical approximations of the effect of foam density on the parameters of g ([??],T).
In this study, a simple model for g([??]) as a function of foam density is introduced. Explicit empirical relations are used to express the dependence of the phenomenological model parameters on foam density for a selected functional form of Equation 1 previously demonstrated to provide a good description of PU foam response. By contrast to the models discussed above, such an approach enables predicting loading rate effect on foam response in compression at different foam densities without the need to recalibrate the model. For validation of the model, PU foams of a range of apparent densities, containing polyols derived from recycled polymers, have been produced and tested in quasi-static and dynamic compression up to a ca. 80% nominal strain. The model parameters have been evaluated using test results for foams of two different densities at low loading rates. The relative root mean square errors of the predicted response of foams of intermediate densities at different loading rates are calculated to quantify the accuracy of the prediction.
The constituents and procedure used to produce rigid PU foams containing polyols derived from recyclable resources have been described in detail in  for the specific case of foams of 200 kg/[m.sup.3] apparent density. Below, we briefly recapitulate the formulation of the foams and the method used for obtaining foams of different densities.
Rigid PU foams are produced by mixing two, a polyol and an isocyanate, components. As the basis of polyol formulation, aromatic polyester polyols, derived from the side-stream substance of polyethylene terephthalate production, were used (at ca. 65% of the polyol component). A higher functional polyether polyol (~15%) was added to increase the cross-linkage density of the polymeric matrix. An additive surfactant (~2%) was used to obtain closed-cell PU foams. The reactive delayed action time amine-based catalysts (~2%) were employed. Tris(chloropropyl)phosphate was utilized as a flame retardant (~15%). As a chemical blowing agent for foaming PU, distilled water was applied. To obtain rigid PU foams with different densities, the water fraction was varied; for PU foams with a density of 150 kg/[m.sup.3] and lower, in addition to water, a Solkane 365/227 physical blowing agent was used. As the isocyanate component, polymeric diphenylmethane diisocyanate was utilized (NCO = 31.5 wt.%). The ratio of the polyol and isocyanate components was 1:1.46 pbw.
In order to obtain isotropic foams of a selected apparent density, the foaming was performed in a closed metallic mold as follows. The isocyanate and polyol components were weighted and stirred for 15 s by a mechanical stirrer at 2000 rpm. The reacting mixture was poured into a stainless steel mold, preheated to 50 [degrees]C and placed on balance, and the mold was closed. Air was allowed to escape from the mold through an opening in the lid, which was subsequently sealed. The foams were post-cured via placing the mold in an oven at 50 [degrees]C for 2 h. Upon removing the mold from the oven and cooling, the foam block was extracted and conditioned at room temperature and 50% relative humidity for 24 h before being cut into specimens for mechanical testing. The apparent density of each specimen was determined according to ISO 845. As an example, SEM image of foams of ca. 150 kg/[m.sup.3] density is shown in Figure 1 demonstrating isotropic morphology.
For quasi-static tests, specimens of dimensions 40 x 40 x 40 mm were cut out of the conditioned foam blocks. Uniaxial compression tests with stroke control were performed on a Zwick/Roell Z100 test set-up equipped with a 100 kN load cell at ambient laboratory conditions of ca. 23 [degrees]C temperature and 40% relative humidity. The engineering strain was evaluated based on the cross-head displacement according to ISO 844. The tests were performed at constant strain rates of 0.00167, 0.167, and 0.5 [s.sup.-1] up to a ca. 80% strain. Four or five specimens at each foam density and loading rate were tested.
For high-speed tests, cylindrical foam specimens with a 20 mm diameter and ca. 22 mm height were machined. An Instron Dynatup 9250HV instrumented drop tower impact testing system with a 16 kN load cell was used for dynamic compression tests. Foam specimens were attached to the striking head by an oil-based adhesive and crushed against the horizontal surface of a rigidly fixed steel plate at a speed of 4 m/s, which translates into engineering strain rate, experienced by the specimen upon impact, of ca. 180 [s.sup.-1]. The variation of impact force, striker speed, and displacement during the contact of the foam specimen with the steel plate were recorded. The impact absorption brakes were put up so that the striker was stopped by them at a displacement corresponding to about an 80% compressive engineering strain of the foam specimen. The engineering strain was evaluated based on the striker displacement since the onset of contact between the specimen and the steel plate.
In order to apply Equation 1 to modeling of foam response in compression, the functions f([epsilon]) and g([??], T), as well as the dependence of their parameters on foam density, need to be specified. As the analytical form of f([epsilon]), representing in Equation 1 the engineering stress as a function of strain in uniaxial compression of foams at a reference strain rate [??], we select the expression proposed in , which reads as follows:
[mathematical expression not reproducible] (2)
Equation 2 has been shown to provide a close fit to the test results of PU foams [12, 18]. Whereas the exponents m and n in Equation 2 are density-independent material characteristics, the parameters A, B, and E are functions of foam density .
The parameters A and E can be interpreted as the plateau stress and the Young's modulus of foams, respectively, as demonstrated in . For PU foams, a power-law dependence of both, modulus and plateau stress (or compressive strength), on foam density has been reported, see, e.g. [20-22], therefore we assume that
[mathematical expression not reproducible] (3)
[mathematical expression not reproducible] (4)
where [C.sub.E] and [C.sub.A] are pre-factors with the dimension of stress, [k.sub.E] and [k.sub.A] designate the respective exponents, and [rho.sub.s] is density of the monolithic PU. The latter is introduced as a normalizing factor to render the argument of the power functions in Equations 3 and 4 dimensionless. For the parameter B in Equation 2, describing the densification behavior of foams, we also select a power-law dependence on foam density, as suggested in :
[mathematical expression not reproducible] (5)
Regarding the strain rate effect, different analytical expressions for g = g ([??]) have been applied to describe the PU foam response with an apparently commensurable accuracy, such as that proposed by Nagy et al. [6, 9, 10, 13], Cowper-Symonds rule in , and variations of the Johnson-Cook relation in [11, 12]. We select the form
[mathematical expression not reproducible] (6)
proposed in , because its applicability to PU has been confirmed experimentally in [6, 9] for strain rates [??] ranging from [10.sup.-3] to ca. [10.sup.2] [s.sup.-1]. The results presented in  for PU foams suggest a relatively modest variation of the parameters a and b of Equation 6 with foam density. As a first approximation, we assume them to be linear functions of density:
[mathematical expression not reproducible] (7)
[mathematical expression not reproducible] (8)
Inserting the relations given by Equations 2 and 6 into Equation 1, we obtain the following expression for engineering stress as a function of strain, strain rate, and foam density (the latter via the density-dependent parameters of Equation 9) in uniaxial compression:
[mathematical expression not reproducible] (9)
Equation 9 comprises seven parameters - five for the response at a reference strain rate, Equation 2, and two for describing the strain rate effect, Equation 6. Since each of the density-dependent parameters is represented as a function (either power-law, Equations 3-5, or linear, Equations 7 and 8) of foam density containing two material constants, the minimum set of experimental data needed for calibration of the model Equation 9 comprises compression tests of foams of two different densities, to be performed at two different strain rates (i.e. the reference strain rate and another one).
Results and Discussion
It has been argued in  that, for Equation 2 to hold over a range of foam densities with the same values of parameters of Equations 3-5, the deformation and crushing mechanisms of foams in this density range have to be similar. The latter mechanisms are determined primarily by the morphology of foams. PU foams are known to exhibit three distinct types of morphology, depending on the apparent density [rho], see, e.g. . High-density foams ([rho] exceeding ca. 300 kg/[m.sup.3]) have the structure of a solid PU with isolated spherical voids, low-density foams ([rho] less than ca. 70 kg/[m.sup.3]) consist of a network of struts forming Kelvin cells with thin membrane faces, while medium-density foams possess rounded polyhedral cells.
The present study being focused on medium-density foams, we selected foams with apparent densities of [rho.sub.1] = 113 and [rho.sub.2]0. = 311 kg/[m.sup.3] for evaluation of Equation 2 parameters. A quasi-static loading rate of 10%/min, stipulated by the ISO 844 standard, was chosen as the reference rate, i.e. [??] = 0.00167[s.sup.-1]. The values of parameters were determined as those minimizing the sum [DELTA] of relative squared errors of the stresses according to Equation 2 with respect to the experimental values for all the specimens tested at both the foam densities mentioned:
[mathematical expression not reproducible] (10)
where [mathematical expression not reproducible] denotes the experimentally determined value of stress at a strain s of foams with density [rho.sub.i]i, and [mathematical expression not reproducible] is the respective value of stress given by Equation 2. The density-independent exponents of Equation 2 amounted to n = 1.85 and m = -25.16, while those of the parameters of Equations 3-5 expressing A, E, and B as functions of foam density are presented in Table 1 (density of the monolithic PU here and in the following is assumed to be [rho.sub.s] = 1210 kg/[m.sup.3], based on ).
The respective best-fit stress-strain diagrams according to Equation 2 are plotted in Figure 2 by dashed lines with round markers. A good accuracy of approximation is observed - the diagrams given by Equation 2, with the parameter values listed above, are within the experimental scatter among the four experimental diagrams for each of the 113 and 311 kg/[m.sup.3] density foams. Notably, the accuracy of prediction by Equation 2 of the response of foams with intermediate densities, not used in estimation of its parameters, is also good, as seen in Figure 2, where the predicted stress-strain diagrams of foams with densities in the range of 150 to 253 kg/[m.sup.3] are plotted by dashed lines with square markers.
Having thus established the description of foam deformation at the reference strain rate, we proceed with evaluation of the rate effect according to Equation 6. For this purpose, we use the test results at a strain rate exceeding the reference one by two orders of magnitude, i.e. [??] = 0.167 [s.sup.-1], for foams of the same densities p1 = 113 and [rho.sub.2] = 311 kg/[m.sup.3] as before. First, the values of a([rho.sub.i]) and b([rho.sub.i]) minimizing the sum [DELTA.sub.i] of relative squared errors of stress
[mathematical expression not reproducible] (11)
for i = 1 and 2 are determined. In Equation 11, [mathematical expression not reproducible] denotes the experimentally determined value of stress at a strain [epsilon.sub.j] of foams with density [epsilon.sub.i], and [mathematical expression not reproducible] is the respective value of stress given by Equation 9. Then, the parameters [C.sub.a] and [k.sub.a] of Equation 7 are calculated using the estimates of a([rho.sub.1]) and a([rho.sub.2]) obtained, and the parameters of Equation 8 are evaluated using the respective estimates of b([rho.sub.i]). The results are presented in Table 2. This completes calibration of the model presented by Equation 9.
The calibration procedure described above can be summarized as follows:
* model parameters n, m and parameters of the Equations 3, 4, and 5 are determined by fitting Equation 2 (with E, A, B substituted by the expressions given by Equations 3, 4, and 5, respectively) to the experimental stress-strain data of foams of two different densities [rho.sub.1] and [rho.sub.2], tested at a strain rate [??], so that the sum A of relative squared errors of the stresses given by Equation 10 is minimized;
* parameters of the rate-dependence relation (6) for foams of density [rho.sub.1], designated as a([rho.sub.1]), b([rho.sub.1]), are determined by fitting Equation 9 to the experimental stress-strain data of foams tested at a strain rate [mathematical expression not reproducible], so that the sum [DELTA.sub.1] of relative squared errors of the stresses given by Equation 11 is minimized. Similarly, parameters a([rho.sub.2]), b([rho.sub.2]) for foams of density [rho.sub.2] are determined by fitting Equation 9 to the experimental stress-strain data of foams tested at the same rate [??], so that the sum [DELTA.sub.2] of relative squared errors of the stresses given by Equation 11 is minimized;
* parameters [C.sub.a], [k.sub.a] of Equation 7 are evaluated using the obtained a([rho.sub.1]), a([rho.sub.2]) values as [C.sub.a] = (a([rho.sub.2])[rho.sub.1] - a([rho.sub.2])[rho.sub.2])/([rho.sub.1] - [rho.sub.2]) and [k.sub.a] = (a([rho.sub.1]) - a([rho.sub.2]))[rho.sub.s]/([rho.sub.1] - [rho.sub.2]). Parameters [C.sub.b], [k.sub.b] of Equation 8 are obtained from the above relations upon substituting a by b in them.
A comparison of the experimental diagrams in Figure 3 with those according to Equation 9 reveals both a close approximation (shown by dashed lines with round markers) of the compressive response of 113 and 311 kg/[m.sup.3] foams by the theoretical relation and a reasonably accurate prediction (plotted by dashed lines with square markers) of foam deformation for intermediate densities at the strain rate of 0.167 [s.sup.-1].
For quantitative characterization of accuracy of approximation and prediction by Equation 9 of the stress-strain response of foams in the 100 to 300 kg/[m.sup.2] density range, the relative root mean square error is calculated for each foam density and loading rate combination as follows
[mathematical expression not reproducible] (12)
where [mathematical expression not reproducible] denotes the experimentally determined value of stress at a strain [epsilon.sub.j] for foams of the given density and loading rate, [mathematical expression not reproducible] is the respective value of stress according to Equation 9, and N is the number of the stress-strain data points acquired during tests. The results are presented in Table 3.
The error of approximation of the experimental stress-strain diagrams used in calibration of the model (underlined in Table 3) ranges from 3.9 to 7.5%, while the error of prediction for foam densities of ca. 150-250 kg/[m.sup.3] at the loading rates used in model calibration varies from 5.7 to 8.5%. Such a small difference between error ranges for approximation and prediction suggests that using the minimum dataset for model calibration may be an acceptable option in the trade-off between the amount of testing needed for calibration and its accuracy.
Figure 4 presents foam test results at a three times higher strain rate, [??] = 0.5 [s.sup.-1], together with the predicted response. The latter is seen to agree well with test results, being mostly within the scatter band of experimental diagrams for each foam density; the prediction errors of 6.1-12.5% are also comparable to those at lower strain rates as seen in Table 3.
The drop tower test system allows reaching strain rates of practical interest for automotive applications of foams in, e.g., bumpers and crash boxes, that by two orders of magnitude exceed the highest rate of quasi-static tests employed in the model calibration described above. It should be noted that, during a drop tower compression test, the speed of the striker, and hence the strain rate the foam specimen experiences, gradually decreases. However, for the practically important plateau regime of deformation, during which most of its energy is absorbed, this reduction can be minimized , e.g., by a judicious selection of striker weight for a given foam density and impact speed. The stress-strain diagrams of 113 and 253 kg/[m.sup.3] foams at an initial strain rate of [??] = 180 [s.sup.-1] are shown in Figure 5. The speed of striker at the instant when the specimens were compressed to a 60% engineering strain, which roughly corresponds to the end of the plateau regime of deformation, had dropped by less than 5% for the lower-density and by 15% for the higher-density foams. Considering the modest strain rate effect exhibited by PU foams, such a reduction in loading speed should have a negligible effect on foam deformation. The reduction in speed is, however, more pronounced in the foam densification regime, when the crushing stress increases very rapidly with strain.
The response predicted by Equation 9 is in good agreement with experimental diagrams in the elastic and plateau deformation regimes of foams, while the stresses in the densification regime are slightly overestimated, see Figure 5. Notably, the accuracy of prediction appears to be comparable for foams of 113 kg/[m.sup.3] density (relative error of 28%), i.e. the foams used for model calibration at lower strain rates, and for 253 kg/[m.sup.3] foams (relative error of 34%), for which the density-dependent parameters of Equation 9 are obtained by interpolation according to Equations 3-5, 7 and 8, calibrated using tests of 113 and 311 kg/[m.sup.3] foams. However, the accuracy of Equation 9 in extrapolation to a foam density outside of this range is somewhat inferior, as seen in Figure 6 for 83 kg/[m.sup.3] foams. It is likely to be related to different morphology and crushing mechanisms of lower density foams. The results reported suggest that the density- and rate-dependent behavior of medium density PU foams in compression can be predicted with a reasonable accuracy (see Table 3) by using the phenomenological model of deformation Equation 9 with density- dependent parameters, thus considerably reducing the amount of actual foam tests needed for a preliminary design of foam-filled impact-absorbing components.
The phenomenological models used to describe the effect of density on the stress-strain diagrams of foams in compression at a fixed loading rate and the effect of loading rate at a fixed foam density have been combined in the present study allowing for the dependence of the loading rate effect on foam density in compression. The minimum experimental data set for calibration of the model consists of compression test results at two different loading rates of foams with two different densities. The model has been verified by applying it to description of the compressive response of medium-density rigid closed-cell polyurethane foams. The model parameters were evaluated using test results at two low loading rates, differing by two orders of magnitude, of ca. 100 and 300 kg/[m.sup.3] density foams. Accuracy of the predicted response was quantified by the relative root mean square error of stress prediction, which ranged from ca. 6 to 12% at low strain rates and reached up to 34% at the higher strain rate. By means of such an interpolative model, the amount of actual foam tests needed for enabling preliminary design and optimization of foam-filled components subjected to dynamic loads can be considerably reduced.
The financial support by the EU Commission through FP7 Project EVOLUTION-314744 and by the National research program IMATECH is gratefully acknowledged.
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Mikelis Kirpluks and Ugis Cabulis, Latvian State Institute of Wood Chemistry
Janis Andersons, University of Latvia
Guntis Japins and Kaspars Kalnins, Riga Technical University
Received: 08 Dec 2017
Accepted: 10 Jan 2018
e-Available: 08 May 2018
TABLE 1 The values of parameters for power-law dependences, given by Equations 3-5, of the characteristics E, A, and B on foam density, determined based on tests of foams with 113 and 311 kg/[m.sup.3] density at 0.00167 [s.sup.-1] strain rate. Parameters E, Equation 3 A, Equation 4 B, Equation 5 Pre-factor C, MPa 1735 70.8 72.4 Exponent k 1.74 1.84 2.31 TABLE 2 The values of parameters for linear dependences, given by Equations 7 and 8, of the rate effect characteristics a and b on foam density, determined using tests of foams with 113 and 311 kg/[m.sup.3] density at 0.167 [s.sup.-1] strain rate. Parameters a, Equation 7 b, Equation 8 C 0.0438 0.0307 k -0.0575 -0.144 TABLE 3 The relative root mean square error D of approximation (underlined) and prediction of foam stress-strain response by Equation 9 as a function of foam density and loading rate. Foam Loading rate, 1/s density, kg/[m.sup.3] 0.00167 0.167 0.5 180 110 3.9 5.2 6.1 28.3 150 7.8 5.7 6.5 - 210 8.5 7.5 8.7 - 250 6.1 6.2 8.3 33.6 310 5.4 7.5 12.5 -
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|Author:||Kirpluks, Mikelis; Cabulis, Ugis; Andersons, Janis; Japins, Guntis; Kalnins, Kaspars|
|Publication:||SAE International Journal of Materials and Manufacturing|
|Date:||Jul 1, 2018|
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