The dynamic relations for highly porous fibrous materials, having analytical expressions for dynamic viscous drag forces and oscillatory solid-to-fluid heat transfer, are now extended towards open-cell foam materials where the struts of the foam are considered to be primarily cylindrical except in the region of the joints. By also including analytical expressions for the stiffness of the foam cell, an entirely analytically-based model is presented for the acoustics of highly-porous, open-celled foam materials. This approach is extremely efficient, requiring only the mean cell size, mean strut diameter, and constitutive properties of the solid foam material and the surrounding viscous fluid as input. The acoustic performance prediction of not only isotropic foam cell designs, but also anisotropic ones may be performed rapidly and virtually, without the need for the determination of poroelastic material properties from existing material samples. The steps required for the development of the analytical foam-cell model are presented, along with the acoustic performance prediction of a typical Melamine foam cell, yielding very promising results in comparison against measurements. In order to understand the suitability of the cylindrical foam strut assumption, a viscous drag force comparison with foam struts having square and triangular cross-sectional profiles is also presented.

A simplified analytical model based on the Kelvin cell micro geometry has been developed for estimating the dynamic drag impedance of a periodic open cell material based on a Kelvincell based micro structure. The calculated dynamic drag impedance estimates similar properties as the static flow resistivity, but is based on dynamic micro scale estimates of the viscous losses neglecting interactions between struts. The Kelvin cell model can have a controlled degree ofanisotropy. Implementing the micro model in a state space transfer matrix method allows for absorption calculations of anisotropic micro material including stiffness to be calculated with limited input information. In addition to the solid constituent material parameters only cell height,strut thickness and twist angle, determining the degree of anisotropy, are required. The modelling allows for fast computations making optimization feasible, which is demonstrated by optimizing a two-layer porous material with the degree of anistropy as a design variable. An optimized design with enhanced absorption at lower frequencies, where the two layers have different anisotropic cell geoemetry, is discussed.

Multilayer panels consisting of a load carrying structure, a porous material for thermal and acoustic insulation and an interior trim panel is a very common type of design for vehicles. Weight as well as total build height are usually limiting constraints on the design. The idea of using an anisotropic porous material instead of an isotropic one to improve the sound transmission loss without adding a lot of weight or thickness is explored in the paper. By using a state space formulation of the transfer matrix method transmission loss it is possible to include anisotropic material properties in the calculation. The anisotropic material is modelled by a combination of a simplified analytical model for the acoustic losses and inverse estimation of the 21 independent elastic constants of the Hooke’s tensor. The porous material, which has typical dimensions possible to 3D print, is based on a Kelvin cell micro model that has a controlled degree of anisotropy. 

Porous foam materials with high porosity are used as part of multi-layer panels for sound insulation andabsorption in transportation vehicles. The acoustic properties of the foam are highly dependant on the microgeometryof the foam cells. In this work, simplified analytical models for calculating the viscous dynamicdrag forces, and oscillatory heat transfer within fibrous materials have been adapted towards foam materialshaving micro-cell geometries composed primarily of cylindrical struts. The analytical results are comparedto full visco-thermal numerical models of an isotropic unit foam cell, For both high porosity foam and lowporosity foam examples with typical dimensions for 3D printed materials, the agreement between analyticaland thermoviscous numerical models is shown to be very good. This approach model can easily be extendedto anisotropic materials as well.