Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests

Transcription

1 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Makram Elfarhani*, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Mechanics, Modeling and Production Laboratory (LA2MP), National Engineering School of Sfax (ENIS), B.P.W Sfax Tunisia Received: 14 August 2015, Accepted: 9 November 2015 Summary The flexible polyurethane foam exhibits a highly nonlinear and viscoelastic behavior under quasi-static and uni-axial compressive tests. The dependence of the displacement rate during the loading phase is significantly higher than that of the unloading phase. A new identification of the Force-Displacement curve based on experimental observations is proposed. The total foam response is modeled as a sum of a nonlinear elastic component and a viscoelastic component. The elastic force is modeled as a sum of orthogonal polynomials in displacement, while the viscoelastic force is modeled according to the hereditary approach in the loading half-cycle and the fractional derivative approach in the unloading halfcycle. The objective of this paper is to develop a model able to make predictions of the total foam force as well as simulations of its components. A parameter calibration procedure is established according to the difference force method between two unloading responses corresponding to two different displacement rates of the same foam sample. The validity of the model results is discussed by addressing three efficiency requirements: the accuracy of simulations to experimental measurements, the repeatability of results for both tests, and the accordance of predicted components of the total response with the phenomenological identification of the Force-Displacement curve. The proposed optimization methodology is found to simulate reasonably well the responses of * Corresponding author: makram.farhani@gmail.com Smithers Information Ltd Cellular Polymers, Vol. 35, No. 5,

2 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar three different types of soft foam. Some morphologic characteristics of these foams have clear influence on viscoelastic damping and residual effects. Keywords: Flexible polyurethane foam, Quasi-static behavior, Viscoelasticity, Compression rate, Force-displacement curve, fractional derivative approach, Hereditary approach Introduction Flexible polyurethane foam is an important cellular material with various engineering applications. The mechanical applications of this material are in progressing growth because of its reduced cost, low density, easy formatting and good energy absorption characteristics. Viscoelasticity and nonlinearity are the most interesting aspects of the mechanical behavior of foam. It is a blend between viscous liquid and elastic solid. Its response depends not only on the loading level and the present state but also on the rate of loading and the previous states. These two aspects complicate the understanding and modeling of the mechanical behavior of flexible polyurethane foam. Many investigators have studied foam behavior and followed several approaches to model its response. Gibson and Ashby (1997) followed the micromechanical approach to describe the mechanical properties of cellular materials by exploring the behavior of single cells. They modeled foam as an array of cubic cells. Warren and Kraynik (1991) considered open foam cells as a regular tetrahedral element. Zhu, et al. (1997) characterized high deformation of foams under compression tests by considering the Kelvin cell. However, the models of the micromechanical approach yielded good results for predicting the elastic behavior of foam; it does not heed the viscoelastic behavior of the material. The continuum mechanics approach is another commonly followed procedure to model the mechanical behavior of foams. In this approach, the size of the micro-structural units is neglected against the size of the material sample. The cellular solid is then assumed to be a continuum. The advantage of this procedure is that a single model allows representing various loading cases. Assuming flexible foam to be hyperelastic and deformation to be macroscopically homogeneous, energetic models (Ogden (1972), Blatz and Ko (1962), Yeoh (1993)) could serve to characterize foam response by the Finite Element Analysis. These models are primarily based on the determination of the analytical form of the deformation energy. A strain-energy function that depends on the deformation gradient is introduced in stress formulations. For the current investigation, a phenomenological approach based on experimental data and observation is adopted to model viscoelasticity and nonlinearity in foam behavior. 236 Cellular Polymers, Vol. 35, No. 5, 2016

3 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Depending on the displacement rate and the frequency of solicitation, there are different types of compressive tests which can be classified into static, quasi-static and dynamic. Following the phenomenological approach, many different constitutive laws have been developed to characterize the mechanical behavior of viscoelastic materials under quasi-static loading. For linear viscoelasticity and unidirectional excitation, rheological models like Burger and Zener models are very often used to approximate the behavior of continuum viscoelastic materials. These models reflect the response of systems comprised by different combinations of viscous dampers and elastic springs. The generalization of the stress-strain relation in rheological representations leads to a differential form, in which the constitutive law is given in terms of a differential equation of integer orders (Ferry, 1970). The application of the Boltzmann s principle of superposition conduces to the convolution form that introduces the integral term with a relaxation kernel. An alternative class derived from the integral formulation is the hereditary approach (Christensen, 1982) in which it is supposed that the stress in the present time explicitly depends on the integral in the past time of the deformation relatively to the appropriate relaxation. Muravyov and Hutton (1997) assumed the kernel to be a sum of exponentials. White (2000), Singh (2000a) and Ippili et al. (2003a) used this formulation to model the viscoelastic behavior of polyurethane foam and identified the model parameters from quasi-static and dynamic measurements. Jmal et al. (2014) obtained accurate results using the hereditary model for different types of polyurethane foam. The most important advantage of the hereditary models is their mathematical simplicity that does not pose problems in the process of parameters calibration with the experimental data. However, the major disadvantage of the hereditary model is that it cannot completely predict the real viscoelastic behavior of certain materials, because it does not allow it to exhibit two important viscoelastic phenomena in soft foams: the Mullins effect (1948) and the residual deformation after loading. Bagley and Torvik (1983) proposed another approach based on the fractional derivative calculus, and provided the theoretical basis for the use of this formulation to characterize the viscoelasticity behavior. The use of this approach has theoretical reasons, since the relationship between stress and strain in the molecular theory of Rouse (1953) is given with a formulation of fractional derivatives. In this approach, it is assumed that the constitutive law contains derivations in time with non-integer orders. Deng et al. (2006), proposed a five-parameter fractional derivative and they obtained good predictions to experimental data. In the investigation described in this paper, hereditary and fractional derivative formulations are considered to model the mechanical behavior of three types of flexible polyurethane foam subjected to quasistatic uni-axial compressive tests. Each formulation is used according to a new identification of the Force-Displacement curve based on experimental Cellular Polymers, Vol. 35, No. 5,

4 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar observations. This model is used to simulate the response of three different types of flexible polyurethane foam to assess its ability to cover a wide range of soft foam and to investigate the effects of some morphologic characteristics in model predictions. Foam Force-Displacement Curve Experimental Procedure Polyurethane Foam Specimens We consider three types of open cell flexible polyurethane foams designated as: foam type A, foam type B, and foam type C. The foam specimens were obtained from a local manufacturer. All samples were of cubic form (75 mm) and they were cut from the center into free rise foam blocks (1900 mm x 1400 mm x 980 mm) of different densities. These foams were manufactured by reacting toluylene diisocyanate (TDI 80) with polyether polyols. The density of each type of foam was adjusted via the type and the proportion of Polyol as well as the content of the physical blowing agent. So the designation of the considered foams was based on the concentration and reactivity of the used Polyols. The morphological characteristics and principal industrial applications of each type of foam are presented in Table 1. Table 1. Morphological characteristics and industrial applications of considered foams Designation Foam A Foam B Foam C Density (kg /m 3 ) Pore size (μm) Porosity 95 % 94% 97% Industrial Applications Automotive seat backs Automotive seat headrest Furniture backs Furniture arms Automotive seat cushions Furniture cushions Mattress cores Mattress padding Mattress cores Bedding Furniture solid cores Carpet Experimental Setup Experiments were conducted in a universal testing machine. The foam blocks were placed between two cylindrical plates. The bottom plate was 238 Cellular Polymers, Vol. 35, No. 5, 2016

5 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests maintained fix, it held the foam specimens, while the top plate of the machine was programmed to move down, at a constant rate, up to 80% of the foam height, and return to initial position at the same rate. The sampling rate was set at 16 samples per second. The experimental setup is shown in Figure 1. Figure 1. Experimental setup of uni-axial compressive tests We should note that during all experiments, the samples were loaded in the rise direction since in most comfort application, soft foams are oriented so as to support compressive forces only in that orientation. The material response and the compression displacement were measured by using, respectively, a load cell that had a 500 N capacity, and a displacement sensor. The testing machine provided a displacement resolution of mm, and an accuracy of the testing force less than 0.5 % of indicative value, and it had a speed range between to 500 mm/min. Obtained Force-Displacement Curves In this study, we assume a Poisson s ratio of zero for the material samples (Mills and Gilchrist, 2000). As known, the force-displacement curve for soft foam includes both hysteretic and nonlinear behavior. In Figure 2, we show the typical quasi-static force-displacement curve for three types of soft polyurethane foam loaded and then unloaded. In cellular materials literature, this complex behavior of polyurethane foam has been attributed to the viscoelastic nature of solid polymer and to the cell structure of the materiel. Following the micro-mechanical approach, Ashby Cellular Polymers, Vol. 35, No. 5,

6 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 2. Typical compressive force-displacement curves for three types of soft polyurethane foams in quasi-static regime 240 Cellular Polymers, Vol. 35, No. 5, 2016

7 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests and Gibson (1997) proposed an identification of the Force-Displacement curve based on the deformation mechanisms within the material cells. Micro-Mechanical Identification of the Force-Displacement Curve According to Ashby and Gibson (1997), three compression regions can be identified in the curve: linear, buckling, and densification. Each region corresponds to dissimilar compression mechanism. The linear region, is given by the bending of the cell walls, and it corresponds to first deformations of the material struts under loading. The buckling region (plateau region) corresponds to the collapsing of the foam cells by elastic buckling. In this phase, the struts bend with little increase in force. Consequently, the foam stiffness decreases since the material becomes effectively softer. The densification region corresponds to completely collapsed cells, opposite cell walls touch each other and further strain compresses the solid itself. The stress then increases rapidly with increased compression, and the foam becomes stiffer. Phenomenological Identification of the Force-Displacement Curve Viscoelastic materials display both «viscous» and «elastic» properties (Ward, 1983). The viscous properties are caused by the motion of the air contained within the foam cells. The elastic properties are due to the elasticity and the structure of the polymer matrix. Several researchers including Singh et al. (2000b) have adopted that the response of foam under loading than unloading compressive tests, is an additive sum of viscoelastic component, and elastic component. Many of the behavioral features of the elastic and viscoelastic components are not fully understood. Therefore, the first goal of the present investigation was to find a new identification of the compressive Force- Displacement curve to specify the regions of domination of each component. This may help to get adequate mathematical models that predict and quantify the behavioral features of the elastic and viscoelastic components. Based on experimental observation, it is evident that the viscoelastic component is affected by the displacement rate of compression tests, contrary to the elastic component that does not depend on the loading rate, or the number of cycles. Thus, comparing the shapes of the compressive Force-displacement curve, under various loading conditions, and for different types of flexible foam is a good starting point to identify the regions of domination of each Cellular Polymers, Vol. 35, No. 5,

8 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar component. In the research reported in this paper, two types of quasi-static testing configurations were used: a single cycle compressive tests for different time durations, and a multi-cycle test with a constant displacement rate. The results of compressive uni-axial tests performed with different time durations (on each type of considered soft polyurethane foams) are illustrated in Figure 3. For each type of soft foam, it is noticeable that despite all curves have the same shape, the dependence of the loading rate is significantly higher in parts of the curve, and completely or partially absent in others. Thus, according to this criterion, four regions can be identified in the loading and unloading curves: (1) linear elastic region, (2) viscoelastic damping region, (3) elastic spring back region, and (4) recovery region. The linear elastic region is absolutely independent of the strain rate. Therefore, soft foam response is entirely elastic. The material deforms as elastic solid. Then, the viscoelastic component may not be mentioned in this part of the curve. This can be explained by the fact that the air contained within the cells remains motionless. The viscoelastic damping region depends dramatically on the loading rate. The higher the displacement rate, the more elevated the magnitude of soft foam response. Hence, in addition to the elastic component, the viscoelastic component exists in this region. Here, the dependence on the loading rate is due to the damping character of the foam caused by the air exhaust through the material pores. The value of the viscoelastic component, during this phase of the loading half-cycle, is relative to the amount of air trapped into the polymer matrix, and also to its velocity during its exhaust. The elastic spring back region depends very slightly on the displacement rate. The elastic return of each type of foam is nearly the same whatever the loading rate. Here, the weak dependence is due to the residual strain caused during the loading process. This phenomenon is a manifestation of matrix polyurethane viscoelasticity. Clearly, at the end of the unloading process (at 4% compression level of foam A, 8% of foam B, and 5% of foam C), the materials responses are quite nonexistent, on account of the contact loss between the foam samples and the upper plate of the machine. In this region, soft foam does not recover instantaneously, but takes several hours or even days to completely recuperate its initial dimensions. Hence, contrary to the first region, the recovery region includes only a purely viscoelastic component, since the response is not instantaneous and the deviation keeps decreasing with time. In Figure 4, we show the results of four consecutive compression tests performed on each type of considered flexible polyurethane foams. For multi-cycle tests, the linear elastic region fades at the second cycle. In elasticity literature, this effect is often attributed to the Mullins effects (1948). It can be explained by the fact that the foam is not yet fully recovered. Furthermore, the foam response decreases, in the viscoelastic damping region, 242 Cellular Polymers, Vol. 35, No. 5, 2016

9 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Figure 3. Compressive force-displacement curves performed with different loading rates on three types of flexible polyurethane foam Cellular Polymers, Vol. 35, No. 5,

10 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 4. Typical responses of flexible polyurethane foams subjected to four cycles of uni-axial compression loading and unloading 244 Cellular Polymers, Vol. 35, No. 5, 2016

11 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests with the number of cyclic tests. This is, at least, due to the attenuation of the viscoelastic component since the material has not yet reached its ultimate settling point. Regarding the elastic spring back region, it is almost not affected by the number of cycles. This confirms that the foam response in this region is dominated by a nonlinear elastic component. Concerning the recovery region, it is completely absent in the second and third tests. To sum up, the phenomenological approach led to a new identification of the Force-Displacement curve for uni-axial compressive test of polyurethane foam in the quasi-static regime. As shown in Figures 3 and 4, the four regions can be clearly identified in the curve, when soft foam is subject to a monocycle, while the linear elastic region and the recovery region appear only in the first cycle and fade in the subsequent cycles when the material is subjected to a multi-cycle test. The next goal in this work is to find the model that refers to the new curve identification to describe the viscoelastic behavior of flexible polyurethane foams. The Elastic-Viscoelastic Model As reported by many investigators, foam displays a highly nonlinear behavior. This is due to the elastic component (White et al., 2000). According to the curve identification described above; the elastic component is present in both the damping and the elastic spring back regions. In addition to non linearity, foam is also highly viscoelastic. Except the linear elastic region, viscoelasticity is manifested by various aspects and origins in the three other regions. In the damping region, it is, principally, caused by the flow of air through the foam matrix, and it is manifested by the high dependence on the loading rate. In the elastic spring back and recovery regions, it occurs due to the viscoelastic nature of the foam matrix, and it is displayed by the delay in recovery and the memory of the loading history. Eventually, this second viscoelasticity manifestation affects the damping region starting from the second cycle. Hence, the viscoelastic components during loading and unloading must be modeled using different viscoelastic approaches, and then different mathematical formulations. Therefore, the total force exerted by soft foam samples during the damping region (loading half-cycle according to Figure 6), is assumed to be: Where E L (t) is the elastic component during the loading phase, and V D (t) is the Damping component. In the elastic spring back region, the viscoelastic component displays the residual force behavior instead of pneumatic damping. (1) Cellular Polymers, Vol. 35, No. 5,

12 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Thus, the total force exerted by soft foam during the unloading-half cycle, is assumed to be: Where E U (t) is the elastic component during the unloading phase, and V R (t) is the Residual force component. (2) The Displacement Expressions For a quasi-static compression test with a constant loading rate, the displacement x(t) is a linear function of time in each half cycle. Figure 5 shows the imposed displacement of the top plate testing machine during loading and unloading of foam blocks. According to Figure 5, the displacement expression can be deduced as: Where =2A/T is the displacement rate; A the maximum displacement, and T the monocycle duration. (3) Figure 5. Imposed displacement of compressive uni-axial tests 246 Cellular Polymers, Vol. 35, No. 5, 2016

13 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests The Viscoelastic Damping Component In linear viscoelasticity literature, the most widely used viscoelastic models are referred to the hereditary approach whose formulations are drawn from the general differential consecutive law that relies on certain combinations of springs and dashpots (Ottosen and Ristinmaa, 2005). Thus, the hereditary approach is adopted to describe the pneumatic damping of foam during the loading half cycle. In this modeling formulation, the force, at given time, is explicitly dependent on the integrated effect of previous displacement states, appropriately weighing by the relaxation kernel (Christensen, 1982). Muravyov and Hutton (1997) assumed the kernel to be of exponential terms. Hence, the expression of the viscoelastic damping component is given by: Where a i and α i are the viscoelastic parameters and N is the order of the viscoelastic kernel. The developed damping component formulation is generated by a combination of Equation (3) (the case of loading half-cycle) and Equation (4). Then, its expression in displacement can be deduced as follows: (4) Since this formulation is based on a generalization of the classical rheological models, it does not allow it to exhibit residual deformation after loading. Therefore, the hereditary approach is inadequate for the modeling of the viscoelastic residual force component during the unload process. (5) The Viscoelastic Residual Force Component In linear viscoelasticity literature, another popular approach to describe the viscoelastic behavior of cellular solids is based on the fractional derivative models. This approach is particularly promising, because it has clear experimental and mathematical interpretations. Using the simplified form of the empirical fractional model (Bagley and Torvik, 1983), the expression of the viscoelastic residual force is assumed here to be: (6) Cellular Polymers, Vol. 35, No. 5,

14 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Where Γ(x) is the complete gamma function; R is a constant of the model and β is the fractional derivative order. It can be noticed, that the fractional operator expression (in Equation (6)), may characterize the memory effects of previous loading, since its mathematical structure contains three tools needed to describe viscoelastic behavior: (1) the time derivative, that allows to describe the changes of material response over time, (2) the convolution integral, which presents a distinct class of viscoelastic models in itself; this formula implies that the influence of each previous disturbance depends on its location in the time scale, and (3) the fractional derivative order which gives the flexibility of representing the status between purely elastic and purely viscous properties. The developed residual force component formulation is generated by a combination of Equation (3), (the case of unloading half-cycle) and Equation (6). Thus, its expression in displacement can be deduced as follows: (7) The Elastic Component In elasticity literature, nonlinearity in foam behavior has often been described by assuming that the material is hyperelastic. In this case, constitutive models are based on the strain-energy function which depends on a deformation gradient. Several popular hyperelastic models have been used to describe nonlinear elasticity in foams: including the Ogden model (1972), the Mooney-Rivlin model (1948), the Yeoh model (1993), and many others. However, polynomials models have a clear physical interpretation, and give good results in term of experimental data fitting accuracy. White et al. (2000) and Singh et al. (2003) introduced a fifth order polynomial to model the large deformation dynamic behavior of foam. Ippili (2003b) showed that the elastic force can be better written in terms of orthogonal polynomials. The reason is that polynomial terms (x, x 2, x 3,, x n ) are not orthogonal over the considered range of compression [0,A]. This non-orthogonality disturbs the understanding of the participation of each term apart in the model. In this study, the elastic component (E (t)), is modeled as a sum of orthogonal polynomials in displacement x(t). The orthogonality condition between two polynomial terms over the range of compression [0, A] is given by: (8) 248 Cellular Polymers, Vol. 35, No. 5, 2016

15 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests The orthogonal polynomial terms up to the seventh order are expressed as follows: Thus, the nonlinear component force is expressed as: (9) Where k j, j=1..7, are the stiffness parameters. (10) Identification Process of Parameters In this investigation and for each type of considered flexible polyurethane foams, the identification method is mainly based on two assumptions of the nonlinear elastic force behavior: a) The independence of displacement rate, b) The independence of the phase of loading (the elastic component of the loading half-cycle is identical to that of the unloading half-cycle). In Figure 6, we show the responses of each type of foam subjected to two uni-axial compression tests with different loading rates: 10 mm/min (Test 1) and 25 mm/min (Test 2). The first assumption allows eliminating the elastic component E from the spring back region, by establishing the difference force between two unloading responses which correspond to two different loading rates of the same foam block (Figure 6). This helps to determine firstly the viscoelastic residual force component V R, and then, the nonlinear elastic force. Once the elastic component is known, the second assumption allows eliminating the Cellular Polymers, Vol. 35, No. 5,

16 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 6. Uni-axial force-displacement curves for three types of flexible polyurethane foam with two compression rates (10 mm/min, and 25 mm/min) 250 Cellular Polymers, Vol. 35, No. 5, 2016

17 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests elastic effects from the pneumatic damping region, and finally determining the viscoelastic damping component V D. The final step consists of validating the analytical model by a simulation of the total force during the loading and unloading phases. The methodology of the identification parameters process is described in the flowchart of Figure 7. Figure 7. Flowchart of the identification parameters methodology Cellular Polymers, Vol. 35, No. 5,

18 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar In each step, the system optimization of parameters is based on the fitting of the analytical model with experimental data. The numeric calculations consist of minimizing an objective function which is the least mean square error between experimental measurements and the mathematical formulation of the model. Numeric calculations were performed in Matlab, by using the optimization toolbox. The lsqnonlin algorithm was exploited to calculate the estimates of the model parameters. The curve fitting toolbox was also used to determine the stiffness parameters of the nonlinear elastic force. Estimation of the Viscoelastic Residual Force Component In the stage of estimating the viscoelastic residual force component, removing elastic effects from the unloading response allows to generate a formulation that depends only on the viscoelastic residual parameters. The experimental data are obtained by subtracting two unloading responses of a same foam block, and at a same imposed compression level, but with two different loading rates. The testing machine should be settled to have the same sample rate of the displacement signal for both compression tests. Singh (2000) found that only a two day restoration period was enough to ensure repeatability. For this reason, the foam block was maintained at reset, between the two compression tests, for a sufficient time (16 days) to completely recover. This ensures that the material sample had effectively reached its final settling points. Equations (2) and (7) were used to generate the analytical formulation. Hence, the viscoelastic residual force difference is expressed in x(t) as follows: Here x(t) is in the range of [0,A], and t is in the range of [T/2,T]. The parameters of Equation (11) are estimated by minimizing the objective function Q I given by: (11) Where S is the number of samples (A=S.dx). The experimental data and the prediction of the residual force difference of each type of foam are shown (12) 252 Cellular Polymers, Vol. 35, No. 5, 2016

19 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests in Figure 8. The recovery region is eliminated from the experimental data, since the foam response is null in this part (according to phenomenological identification of force-displacement curve previously discussed). Figure 8. Experimental data and predicted simulations of the difference between viscoelastic residual forces for each type of foam Cellular Polymers, Vol. 35, No. 5,

20 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar In the optimization procedure, various algorithms such as Levenberg Marquardt and trust region can be used to minimize the objective function of Equation (12). Getting the global minimum depends mostly on the starting values of parameters. The curve fitting toolbox of Matlab was exploited to get random start points. Optimum values of the model can be distinguished from local minima by the quality of prediction statistics as well as the significance of the reconstructed residual force components. The results of optimization parameters are presented in Table 2. The residual force component curves, for the three types of flexible polyurethane foam, are plotted in Figure 9 using the estimated parameters and Equation (7). Calculation of the Nonlinear Elastic Component The calculation is performed by minimizing the objective function Q II given by: E is the analytical expression of the elastic component, and E data is its reference data obtained by subtracting the viscoelastic residual force V R from the total unload response of foam. In Figure 10, we show the plotting of the total foam response during the unloading phase and the obtained elastic component of each type of foam. The elastic data curve keeps almost the same shape with a small shift, since the magnitude of the viscoelastic residual force is relatively very low. It appears that the elastic component data for the two tests coincide completely. Therefore, the identification procedure was performed for a unique data curve which generates common optimum stiffness parameters for both tests. The optimization system was executed by using the nonlinear curve fitting tool lsqcurvefit of Matlab. At this stage, we considered different orders of the orthogonal polynomials given by Equation (9). As mentioned before (in subsection 3.4), it is found that the 7 th order is the smallest order which gives accepted simulation. In Figure 11, we represent the fitted curves corresponding to the 5 th, 6 th, and 7 th orthogonal polynomials order of foam A, and the calibrated curves with only 7 th orthogonal polynomials order of foam B and foam C. The estimated stiffness parameters of the three types of foam are grouped in Table 2. (13) Estimation of the Viscoelastic Damping Component The viscoelastic damping component data is calculated by removing the elastic effects from total foam response during the loading phase. According 254 Cellular Polymers, Vol. 35, No. 5, 2016

21 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests to Equation (1), we obtain: (14) Figure 9. Viscoelastic residual force plotting of each type of foam for Test 1 and Test 2 Cellular Polymers, Vol. 35, No. 5,

22 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 10. Total unload foam responses and elastic component data for each type of foam 256 Cellular Polymers, Vol. 35, No. 5, 2016

23 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Figure 11. Elastic component data and fitted curves based on 5th, 6th, and 7th orthogonal polynomials order for foam A and only 7th orthogonal polynomials order for foam B and foam C Cellular Polymers, Vol. 35, No. 5,

24 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar The analytical formulation is given by Equation (5). The damping component parameters are calibrated by minimizing the objective function Q III expressed as: The predicted and the calculated data of viscoelastic damping component are shown in Figure 12. It is clear that the three orders hereditary model provides a good simulation; however the two orders model did not lead to acceptable fitting results. In fact, the accuracy of the model prediction is obtained by considering the parameters: a 1, a 2, and a 3 to be real form, while the parameters: a 1, a 2, and a 3, are chosen to be complex form. The predicted parameters are presented in Table 2. (15) Results and Discussion The following table contains the values of estimated parameters. Table 2. Optimized parameters values Parameters Foam A Foam B Foam C Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 β R (N.s β.m -1 ) k 1 (N. m -1 ) k 2 (N.m -2 ) k 3 (N.m -3 ) k 4 (N.m -4 ) k 5 (N. m -5 ) k 6 (N. m -6 ) k 7 (N. m -7 ) a 1 (N.m.s -1 ) a 2 (N.m.s -1 ) a 3 (N.m.s -1 ) Re(a 1 ) (s -1 ) Im(a 1 ) (s -1 ) Re(a 2 ) (s -1 ) Im(a 2 ) (s -1 ) Re(a 3 ) (s -1 ) Im(a 3 ) (s -1 ) Cellular Polymers, Vol. 35, No. 5, 2016

25 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Figure 12. Viscoelastic Damping component data and predictions based on two and three order hereditary model for foam A and only three order hereditary model for foam B and foam C Cellular Polymers, Vol. 35, No. 5,

26 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Model Validation In this paragraph, the validity of the model results is discussed by addressing three efficiency requirements: the simulation accuracy with experimental measurements, the repeatability of results, and the accordance of predicted components of the total response with the phenomenological identification of the curve. Accuracy of the Model Fit In Figure 13, we show the experimental data and the predicted total response of each type of foam. It is clear that the optimization system gives accurate simulation results with a small bias. The fit statistics relative to each identification process step are presented in Table 3. As shown in this table, we note that for each type of foam the R-square values of all identification procedure steps are very near to 1. This indicates that the major proportions of variance of each of the measurement data are considered by the model. In addition, the adjusted R-square values confirm that prediction results have good quality, since fits in step 4 explain at least 96% of the total variations in experimental data about the average values. Table 3. Fit statistics Foams Identification Process R-Square Adj. R-Square RMSE Foam A Foam B Foam C Step (V R /F U ) Step (E/F L ) (E/F U ) Step (V D /F L ) Step Step (V R /F U ) Step (E/F L ) (E/F U ) Step (V D /F L ) Step Step (V R /F U ) Step (E/F L ) (E/F U ) Step (V D /F L ) Step Cellular Polymers, Vol. 35, No. 5, 2016

27 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Figure 13. Experimental and predicted total response of each type of foam for Test 1 and Test 2 Cellular Polymers, Vol. 35, No. 5,

28 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 13. Cont'd Cellular Polymers, Vol. 35, No. 5, 2016

29 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests In fact, the fit quality in step 4 is due to good prediction in steps 2 and 3, rather than step 1. Because the contribution of viscoelastic residual force (estimated in step 1) in the total unloading response of foam (calculated in step 4) does not exceed 2% for foam A and foam C and 6% for foam B. However, the contribution of the nonlinear elastic force (predicted in step 2) and the viscoelastic damping force (predicted in step 3) in the total loading force are respectively over 30 % and 50 %. From here it appears that the major benefit of the identification process, presented in the flowchart of Figure 7, is that it allows to extract the experimental data of each component force apart, and then to calibrate its parameters separately. This advantage allows avoiding the admixture problem that occurs often in the difference force method between two different components. To sum up, the model simulations of each type of foam display good accuracy in all steps of the identification process. Repeatability of Results Concerning the repeatability of results, Figure 14 shows that the shape of each component force of the Test 1 has a good similarity with that of Test 2. Moreover, according to Figure 10, the elastic component is completely identical for both tests. This maintains perfect similarity between viscoelastic damping components of both experiments. Hence, any difference between shapes and values of these components is entirely attributed to the difference of the displacement rate between the two tests. Regarding the residual force components, it can be seen in Figure 9 that, for each type of foam, both residual force curves have the same shape and almost the same magnitude. Hence, the model results are quite repeatable for Test 1 and Test 2. Accordance with the New Identification of the Force-Displacement Curve In this investigation, the new identification of the force-displacement curve presents the theoretical base of model formulations. Thus, conformity of the predicted results with the phenomenological hypothesis is a strong validity requirement of the model. According to model simulations, it is clear that the components of the total response of soft foam have the following remarkable characteristics: The nonlinear elastic force does not depend on the displacement rate. As seen in Figures 10 and 14, for each type of foam, the elastic component of Test 1 is perfectly conform to that of Test 2. The viscoelastic damping force displays a high dependence of the loading rate. However, the two damping forces of Tests 1 and 2 keep Cellular Polymers, Vol. 35, No. 5,

30 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar practically the same shape with a significant shift due, primarily, to the variation of displacement rate. The viscoelastic residual force has a relatively low magnitude compared to other components. This approbates the dominance of elasticity in the unloading response of soft foams, as these materials restore instantaneously up to 95% of their initial dimensions; however the residual strain disappears progressively if enough time is allowed. The residual force component has negative values (especially for Test 2). This indicates that the residual force resists the elastic effect of foam. Indeed, opposition between the two unloading forces can be explained by residual strain as well. From here, it can be deduced that the obtained results are in good accordance with the basic hypothesis and they have logical physical meaning. Effects of Morphological Foam Characteristics in Model Simulations Effect of Pore Size and Density in the Viscoelastic Damping Force As seen in Figure 12, the shape of the damping curve can be likewise identified in three regions: first pick, plateau region, and second pick. For the first and second picks, the viscoelasticity is manifested only through the response of polymer matrix structure as the air is still motionless in the first pick and it is completely absent in the second pick. However, in the plateau region, the damping force is governed by the rate of the air flow through the polymer matrix and the structure of the polyurethane foam, especially porosity and pore size. From results of Table 3, it appears that the more the pore size of soft foam is important, the more the contribution of the damping force in the total loading response of the material is elevated. For example, foams A and C which have respectively mean pore sizes of 225 and 155 microns produce responses that contain respectively 58% and 53% of viscoelastic damping effects. Yet, foam B which has a mean pore size of 370 microns gives a response that contains 67% of damping effects. Inversely the nonlinear elastic force of the loading half-cycle decreases with the foam pore size, since it is dominated by the viscoelastic damping component in this phase. As for the density effects, we can clearly observe from Figure 12 that the magnitude of the plateau region of the viscoelastic damping force is significantly influenced by foam density. The lower the density (the quantity of air trapped within the foam matrix is respectively high) the higher magnitude of damping force. 264 Cellular Polymers, Vol. 35, No. 5, 2016

31 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Figure 14. Simulations of viscoelastic damping, viscoelastic recovery, and nonlinear elastic components of total foam response of each type of foam Cellular Polymers, Vol. 35, No. 5,

32 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar Figure 14. Cont'd Cellular Polymers, Vol. 35, No. 5, 2016

33 Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane Foam Viscoelastic Response Under Quasi-Static Compressive Tests Effect of Porosity in the Viscoelastic Residual Force and the Elastic Force Concerning the residual force component, it is clear that this force is influenced by the foam porosity. The lower the porosity, the more important the residual effects. For example, foams A and C which have respectively a porosity of 95% and 97% produce responses that contain 2 % of viscoelastic residual effects, while foam B, which has a porosity of 94%, gives a response that contains 6% of residual effects. Graphically, this observation has strong confirmation in Figure 6, since the recovery region that reflects residual strain in the foam is more extended in foam B rather than in foam A and than in foam C. Inversely the nonlinear elastic force of the unloading half-cycle decreases with the foam porosity, since it dominates the viscoelastic residual component in this phase. Conclusions In this work, the modeling formulations are based on the hereditary and the derivative fractional approaches. The hereditary approach is used to describe the viscoelastic damping component while the derivative fractional approach is used to characterize the viscoelastic residual force component. The purely elastic component is modeled as seventh order orthogonal polynomials. The basic equations of this model are in agreement with the phenomenological identification of the Force-Displacement curve described in the first part. The parameter calibration procedure is based on the difference of the foam response between two unloading paths. The quasi-static experiments were conducted on cubic samples of three types of flexible polyurethane foam with two loading rates: 10 mm/min and 25 mm/min. the model results verify three validity requirements: prediction accuracy, repeatability, and accordance with theoretical assumptions. The accuracy of the model simulations of three industrial soft polyurethane foams with different morphological characteristics is a strong indication that the proposed model is a general representation of a wide range of flexible polyurethane foam in the quasi-static regime. This enhances the model efficiency as it can be used to examine each effect (elastic, damping, or residual) apart in the total foam response, and to select with specified criteria the suitable type of foam for specific industrial applications. Moreover, this model may help seat car and furniture manufacturers to obtain flexible polyurethane foam with appropriate mechanical behavior based on morphological characteristics. Meanwhile, future works would address the ability of the model to predict the response of soft foam subjected to multi-cycle tests. In this case, the Cellular Polymers, Vol. 35, No. 5,

34 Makram Elfarhani, Abdessalem Jarraya, Said Abid, and Mohamed Haddar difference force method should be applied between two successive loading phases for the same displacement rate to estimate viscoelastic damping parameters of each cycle. References 1. Bagley, R.L., Torvik, P.J., A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 (3) Blatz, P.J., Ko W.L., Application of Finite Elastic Theory to the Deformation of Rubbery Materials. Transactions of the society of Rheology, 6, pp Christensen, R., Theory of Viscoelasticity. Academic Press, New York. 4. Deng, R., Davies, P., Bajaj, A.K., A nonlinear fractional derivative model for large uni-axial deformation behavior of polyurethane foam. Signal Processing Ferry, J.D., Viscoelastic Properties of Polymers. Wiley, New Jersey. 6. Gibson, L.J. and Ashby, M.F., Cellular Solids: Structure and Properties.2 nd edition, Cambridge University Press, Cambridge (UK). 7. Ippili, R.K., Widdle, R.D, Davies, P., Bajaj, A.K., Modeling and identification of polyurethane foam in uniaxial compression: combined elastic and viscoelastic response. Proceedings of the 2003 ASME Design Engineering Technical Conferences, DETC2003/VIB-48485, Chicago, Illinois, September Ippili, R.K., System identification of quasi-static foam behavior and its application in the prediction of static equilibrium position of a car seat occupant. Master s thesis, Perdue University, west Lafayette, IN Jmal, H., Dupuis, R., Aubry, E., Quasi-static behavior identification of polyurethane foam using a memory integer model and the difference-forces method. Journal of Cellular Plastics, SAGE Publications (UK and US), 2011, pp Mills, N.J., Gilchrist, A., Modelling the indentation of low density polymer foams. Cellular Polymers. 19 (6) Mullins, L., Effect of stretching on the properties of rubber. Rubber Chemistry and Technology 21, Muravyov, A., and Hutton, S.G., Closed-form solutions and the eigenvalue problem for vibration of discrete viscoelastic systems. J. of App. Mechanics Ogden, R.W., Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber like solids. Proceedings of the 268 Cellular Polymers, Vol. 35, No. 5, 2016