A Biphasic Continuum Approach for Viscoelastic High-Porosity Foams: Comprehensive Theory, Numerics, and Application

Abstract

This work aims to present a valuable and predictive model for the description of polymeric soft foams which are widely exploited as packaging, padding, or cushioning materials. Belonging to the class of cellular solids, polymeric foams and synthetic sponges can be regarded as permeable or impermeable fluid-saturated porous materials depending on whether their microstructure is open-celled or closed-celled. The pore space formed by the cellular polymer skeleton is filled with a pore fluid (e.g., liquid or gas) which can significantly affect the transient compressive and impact response making particularly air-filled foamed plastics a light-weight and shock-absorbing material par excellence.In view of their practical application, it is the intention to model the macroscopic bulk response of high-porosity foams without getting lost in the description of the deformation habits of a distinct cell on the microstructure. It is rather the goal to phenomenologically capture the coupled dissipative phenomena stemming from the viscoelasticity of the polymer matrix and the usually underestimated or even neglected influence of the viscous pore-fluid motion which essentially constitute the favorable characteristics of these materials. Therefore, a general framework is derived by merging the advances in porous media theories and the state of the art in finite solid viscoelasticity. Against a thermodynamically consistent background, a constitutive setting is presented where, based on the internal variable concept, an extended Ogden-type viscoelasticity law is embedded into the macroscopic Theory of Porous Media (TPM). By regarding foamed polymers as immiscible binary solid-fluid B. Markert ( ) Institute aggregates, essential nonlinearities are modularly included in the formulation. In detail, the developed biphasic continuum approach accounts for the relevant physical properties emanating from the cellular microstructure, such as the densification behavior under finite compression, the trapped or potentially moving and interacting viscous pore fluid (compressible or incompressible), and the superimposed intrinsic viscoelasticity of the polymeric skeleton itself.The numerical treatment of this strongly coupled solidfluid problem is carried out through the Finite Element Method (FEM) where the governing equations are formulated in the primary variables solid displacement and porefluid pressure. The two-field variational problem is then efficiently solved by recourse to stable mixed finite element formulations and an appropriate implicit time-stepping scheme in combination with a two-stage solution strategy for the discrete nonlinear equation system.Finally, in order to demonstrate its capabilities, the presented model is calibrated to the energy-absorbing response of an open-celled polyurethane (PUR) foam as it shows all conceivable nonlinearities under absolute finite viscoelastic deformations. The FE simulations of a real impact experiment on fragmented PUR foam blocks and the fast compressive loading of a whole car seat cushion...

Comparison of monolithic and splitting solution schemes for dynamic porous media problems

“…Basically, the additional bubble functions enrich the corresponding trial space thereby guaranteeing stability [29,31]. For a comparative study of the MINI element applied to porous media problems, see [50].…”

Variationally consistent derivation of the stress partitioning law in saturated porous media

“…This issue has received much attention both from a purely theoretical point of view, as a general problem of continuum mechanics, (Bedford and Drumheller, 1983;Bennethum, 2006;Bennethum and Cushman, 1996;Bowen, 1982;Coussy et al, 1998;De Boer, 1996;Ehlers, 2002;Gajo, 2010;Gray et al, 2009;Truesdell, 1957), and in more applicative contexts, such as soil mechanics (Skempton, 1960a;dell'Isola and Hutter, 1998), biomechanics (Ateshian et al, 1997;Lai and Mow, 1980;Mow et al, 1980) and impact engineering (Markert, 2008), dealing with specific classes of multiphase media.…”

The influence of viscosity on the response of open-cell polymeric foams in uniaxial compression: experiments and theoretical model

“…The system has many solutions. Indeed, for any fixed, sufficiently small δε i , for each j there are three elongations ε A + δε j which satisfy equation (20), one on each of the two ascending branches and one on the descending branch of the curve f τ .…”

“…While these aspects have been studied in the context of filler-reinforced rubbers and polymers, we are not aware of any specific study for foams, except the study of rate dependence made in [20]. The stress softening of filled rubbers was investigated by Mullins [22].…”