The tuning of the low frequency sound absorption of open-cell anisotropic porous materials is studied in the form of an optimisation problem. Using modelling based on micro-structural representations of the anisotropic elasticity, the dynamic viscous drag forces and cell porosity, a physically meaningful dependence between these quantities is ensured. The micro-geometry used here is Kelvin Cell based, which may be distorted in a controlled way, creating a degree of anisotropy in the dynamic viscous drag forces and the elastic properties. Here, the distortions are implemented through twisting the square faces of the cells, resulting in a controllable interaction between shear and compression in the elastic deformation. Low frequency sound absorption is maximised using a gradient-based optimisation approach. Importantly, the design parameters in this approach are purely geometrical: the twist angles, strut radii and the respective layer thicknesses. A multi-layer arrangement undergoing plane wave acoustic excitation is considered as an application example.

Constitutive models for the vibroacoustics of porous materials have previously been defined for lattice cell, foam and fibrous materials in terms of dynamic viscous drag forces and oscillatory solid to fluid heat transfer effects. Where the microgeometries are cylindrical in nature, analytical expressions have been derived to efficiently represent the viscous and thermal effects. A logical extension of this work is towards porous materials consisting of spherical shapes, which may also be defined using analytical relations. In this work, the analytical dynamic viscous drag force and oscillatory thermal impedance expressions of a sphere undergoing rectilinear oscillations in a viscous fluid are derived. A transfer matrix model of acoustic wave propagation is then used to predict the sound absorption performance of an array of packed spheres, using only the porosity and mean diameters of the spheres, and the constitutive properties of the solid spheres and the surrounding viscous fluid as modelling inputs. The results compare very well with published measurements.

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.

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. 

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.

Vehicle structures such as train floors or car roofs are usually built as multi-layer panels, where a foam is placed between a load-carrying structure and an interior panel. The foam adds acoustical and thermal performance, but very little weight. In most contributions introducing foams for acoustic treatment, these have been considered isotropic, with acoustic losses mainly dependingon properties in the thickness direction. Another mechanism investigated here is the possibilityfor the acoustic flow in the foam to change from acting only in the thickness direction but rather to be re-directed to also travel in-plane, where dimensions are substantially larger than in the thickness direction, permitting more losses as the wave travels through the material. That kind of effect would result in higher acoustic losses without increasing the thickness of the vehicle panel and better use of the allowable space to achieve acoustic and functional requirements, i.e. a better functional density. A first step is to investigate how the absorption properties of an anisotropic foam differs from an isotropic foam. The chosen approach is to use an analytical micro-modelto calculate the dynamic drag impedance (flow resistivity on micro-scale) for an anisotropic opencell foam material. Based on a simple micro-scale geometry of Kelvin cells, it has been shown that  simple cell alterations to the micro-geometry, such as stretching, twisting and tilting results in an anisotropic foam structure. The anisotropic flow resistivity tensor is not diagonal and uniform, but different directions can have different magnitudes and it can display off-diagonal coupling terms. The influence of such micro-scale distortions on the flow resistivity, and on the resulting sound absorption is investigated with the purpose of improving the acoustic performance without adding volume. Future steps include to modify the functional density and tailor the sound transmission loss to a specific application.

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.

The dynamic equations of a transversely isotropic fibrous, highly porous material are presented in terms of microstructure-derived analytical expressions for viscous dissipation, and analytical expressions for the oscillatory heat transfer between the thermal fields of the solid cylindrical glassfibres and the surrounding viscous fluid. This represents the non-equilibrium thermal expansion of the fluid, occurring when waves propagate in the porous material, and results in a frequency-dependent scaling of the fluid dilatation term. A state-space transfer matrix solution of the governing equations has been introduced, allowing the numerical acoustical performance of the fibrous material to be investigated, including the acoustical effects of heat transfer. In order to understand the dissipation mechanisms of the viscous and thermal boundary layers on the surface of the fibres and the validity of the assumptions made in the current model, a thermoviscous acoustic fluid finite element procedure has also been introduced. The results from these simulations illustrate the frequency-dependent interaction of the boundary layers between neighbouring fibres in the porous material.

This paper focusses on the modelling and prediction of two of the main mechanisms of acoustic attenuation, the dynamic drag and thermal heat transfer, of a bundle of fibres typical of lightweight fibrous porous materials. The methodology uses geometrical properties derived from microscopy, and is based on the assumption that the interaction between the shear stress fields as well as the thermal fields of neighbouring fibres, may be neglected in the predicted dynamic impedances of an individual fibre. Analytical procedures are discussed which provide an estimation of the impedances acting on infinite longitudinal and transversely orientated cylinders. The frequency-dependent viscous and thermal contributions may be scaled in terms of statistical fibre diameter distributions and orientation angles. Using these analytical models, a three-dimensional poroelastic model of coupled acoustic and structural wave propagation through a transversely isotropic, fibrous, highly porous thermal insulation material is presented.