Modelling of a visco-hyperelastic polymeric foam with a continuous to discrete relaxation spectrum approach

Highlights

• Mechanical behaviour of a polymeric foam at large strain and in a wide range of strain rates represented by a visco-hyperelastic model.

• Straightforward parameter identification procedure consisting in combining viscoelastic tests at small strain from DMA with cyclic tests at large strain from UTM.

• Formulation making use of the continuous relaxation time spectrum and nonlinear setting of large strains.

• Identification procedure able to take into account variability of experimental DMA tests.

Abstract

The prediction of compressive properties of foams at large strains and in a wide range of strain rates is still an open issue. In this work we propose a visco-hyperelastic formulation, suitable for large finite strain applications, for the prediction of the compressive response of foams that takes into account the viscoelasticity of the polymer, nonlinear damping, nonlinear behaviour of the cellular structure and effect of gas permeability through the pores at high strain rates. A mathematical expression of the continuous relaxation spectrum is proposed to model the viscoelastic behaviour of the polymer. The relaxation spectrum is then discretized with the desired accuracy required for the subsequent numerical simulations. The model parameters are identified by coupling dynamic measurements at small strain with static ones at large strain. The results are validated by comparing numerical predictions with experimental data from compressive tests up to 50% strain performed at strain rates spanning over 6 degrees of magnitude.

Introduction

The peculiar mechanical properties of polymeric foams (Gibson and Ashby, 2014), which are due to their cellular structure and to the polymer properties, offer several advantages in applications where low density, capability of large deformations and cushion properties are needed such as in lightweight structures, impact shock absorbers (Sun and Li, 2018), stimuli responsive porous devices (D’Auria et al., 2016), and so on. Properties like Young’s modulus, yield stress and Poisson’s ratio strictly depend on the viscoelastic behaviour of the polymer, on the actual porosity and on the cellular morphology. The mechanical response of a polymeric foam to a time dependent loading depends not only on the viscoelasticity of the polymer itself but also on the actual cellular morphology and on the interaction between air and porosity during the change in volume and shape with the deformation. For this reason a predictive model has to take into account all these effects

A possible approach to foam modelling consists in taking into account the microstructure of the foam, and several nonlinear micromechanics models have been developed to incorporate the effects of the polymer and of the cell parameters (shape, size, and cell size distribution) (Danielsson, Parks, Boyce, 2004, Demiray, Becker, Hohe, 2006, Hohe, Becker, 2003, Mihai, Goriely, 2015, Segerstad, Toll, 2008). The cell type (e. g., open- or closed-cells) has a significant influence on the compression behaviour (Gibson and Ashby, 2014). In general, the complex foam structure is the cause of the highly non-linear mechanical behaviour at large strains. For example, the buckling and collapsing of cells can result in negligible lateral expansion in compression  (Iba et al., 2016), unlike in conventional solid materials (Tang et al., 2012).

The use of an appropriate strain energy density function is one of the most employed approach to take into account the nonlinearity at large strain. In the case of a purely elastic foam, for which all the strain variations are instantaneously recovered as soon as the load is released, a suitable hyperelastic potential can be employed. On the other hand, in the case of a viscoelastic material an extra stress component due to viscosity must be taken into account, and visco-hyperelastic models must be used. Typically, only uniaxial data are used to find an appropriate expression of the potential, because of the technical issues in carrying out multiaxial tests on foams, as evidenced by H. Iba et al in Iba et al. (2016).

A complete characterization of the viscoelasticity may require in principle a very large number of parameters. One of the most adopted approaches to identify the relaxation spectrum is through periodic loading tests in small strain regime. In the literature, various functional expressions of the latter for different types of viscoelastic materials have been proposed (Lee, Bae, Cho, 2017, Luo, Lv, Liu, 2018, Malkin, 2006b, Stankiewicz, 2012). Ferry Ferry (1980) showed that relaxation spectra can be expressed in discrete or continuous forms, constituted by infinitesimal contributions of an infinite number of relaxation times. Fuoss and Kirkwood (Fuoss, Kirkwood, 1941, Lee, Bae, Cho, 2017) derived an analytical equation that relates a continuous relaxation time spectrum to the loss modulus. They demonstrated the uniqueness of continuous relaxation spectrum by managing the expression in terms of the Fourier transform.

From a practical point of view, discrete spectra save computational time and allow for a simpler mathematical formulation. For this reason they are largely employed in numerical methods to account for the viscoelastic properties of the material. A discrete relaxation spectrum leads to practical difficulties because it is an ill-posed inverse mathematical problem (Elster, Honerkamp, Weese, 1992, Stankiewicz, 2012). In fact, being one of the possible approximations of the continuous solution it is not unique  (Lee, Bae, Cho, 2017, Malkin, Masalova, 2001).

A further promising approach to characterize the viscoelasticity is the use of the fractional derivative method. As reported by Bagley Bagley and Torvik (1983) and Nutting (1921), the relevance of the fractional calculus applied to mechanical properties of viscoelastic materials can be effective since the stress relaxation phenomenon appears to be proportional to time raised to fractional powers (Appendix A.4). Later observations by Gemant Gemant, 1936, Gemant, 1938 showed that, similarly, also the frequency dependence of mechanical properties varies with fractional powers of frequency itself, suggesting therefore to use differentials with fractional order to model materials (Bagley and Torvik, 1983). Even if promising, the fractional calculus is not implemented in commercial codes yet, which makes it less useful for numerical calculations and has not been considered here.

Foams may be classified with respect to the cellular morphology, open-cell or closed-cell being the main types. In closed-cell foams, the internal volumes of cells are not interconnected, and the gas entrapped in cells results in a stress contribution directly dependent on their volume reduction. In open-cell foams, cells are interconnected and the gas can flow through interconnections during deformation. In particular, at high strain rates the mechanical contribution depends mainly on the foam permeability, which non-linearly depends on the strain and changes with the cell geometry during deformation. It can typically be modelled by means of the Darcy law (Dawson, Germaine, Gibson, 2007, Dawson, McKinley, Gibson, 2009, Hilyard, Collier, 1987). Hilyard and Collier (1987) analytically correlating the variation of the permeability k and the inertial flow coefficient B with the applied compressive strain ε. The consequence is that at low strain rates air can freely move through the cells and with negligible effects on the foam stiffness (Gibson, Ashby, Iba, Nishikawa, Urayama, 2016), while in the short timescale of high strain rates, the airflow can significantly increase the stiffness of the foam.

In the present work the mechanical response of the foam has been identified by coupling simple experimental evaluations with a model based on three mechanical contributions. The first is the polymer viscoelasticity, which is modelled through the identification of the polymer viscoelastic parameters by using an approach that exploits the typical shape features of the mechanical response in the frequency domain, namely two stress plateau regions and a transition zone, where the stress-strain response monotonically increases with frequency. The master curve from the experimental identification step is fitted by using a summation, representing the overall relaxation time spectrum, of skewed Gaussian curves, each characterized by a set of three parameters only. The relaxation spectrum was then discretized through the choice of a finite number of equispaced relaxation times obtained after setting the desired accuracy. The effects of the porous structure under large deformations, in terms of elastic and damping parameters, and of airflow, which is modelled by using the Darcy law, were added to the viscoelastic contribution.