Periodic Porous Materials Research Articles

3D Printing of Periodic Porous Metamaterials for Tunable Electromagnetic Shielding Across Broad Frequencies

HighlightsThe stable periodic porous shielding materials were prepared by combining the strategies of 3D printing and metamaterial design.The relationship between porous material structure and electromagnetic interference shielding efficiency (EMI SE) effectiveness was deeply explored, revealing the important structural parameters for realizing tunable EMI SE property.The optimized design of the periodic porous shielding box achieves effective EMI shielding in a wide wavelength range (over 2.4 GHz).

Nonlocal dynamic response of FGP sandwich microbeam with 2D PSH network incorporating adatoms-surface interactions energy under magnetic field

The present work models and studies the resonance frequency change due to adsorption phenomena in a simply-supported adatom-microbeam system under an axial magnetic environment. The microbeam structure is a functionally graded (FG) sandwich that includes a rectangular configuration and a ceramic perforated core with periodic square holes network and FG porous material faces. To evaluate the adsorption-induced energies produced by van der Waals interaction (vdW), the Morse and Lennard-Jones (6–12) potentials are included in conjunction with nonlocal material elasticity to adopt the small-scale impact. Moreover, the explicit formulas for the inertia moments and shear force are developed by the Timoshenko beam equations, considering the residual stress influence as an additional axial load. The Navier-type solution and the differential quadratic method (DQM) are implemented to compute the solution of the governing equations. It is established that the resonance frequency change for the O/Si (100) and H/Si (100) is dependent on the geometrical, perforation, and porosity parameters, adsorption density, mode number, and the magnetic field intensity as well. Therefore, these numerical results have been discussed in detail to properly examine dynamic vibration behavior and a suitable mass detection design of M/NEMS structures.

Microstructural and transport characteristics of triply periodic bicontinuous materials

Three-dimensional (3D) bicontinuous two-phase materials are increasingly gaining interest because of their unique multifunctional characteristics and advancements in techniques to fabricate them. Because of their complex topological and structural properties, it still has been nontrivial to develop explicit microstructure-dependent formulas to predict accurately their physical properties. A primary goal of the present paper is to ascertain various microstructural and transport characteristics of five different models of triply periodic bicontinuous porous materials at a porosity ϕ1=1/2: those in which the two-phase interfaces are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two different pore-channel structures. We ascertain their spectral densities, pore-size distribution functions, local volume-fraction variances, and hyperuniformity order metrics and then use this information to estimate certain effective steady-state as well as time-dependent transport properties via closed-form microstructure–property formulas. Specifically, the recently introduced time-dependent diffusion spreadability is determined exactly from the spectral density. Moreover, we accurately estimate the fluid permeability of such porous materials from a closed-form formula that depends on the second moment of the pore-size function and the formation factor, a measure of the tortuosity of the pore space, which is exactly obtained for the three minimal-surface structures. We also rigorously bound the permeability from above using the spectral density. For the five models with identical cubic unit cells, we find that the permeability, inverse of the specific surface, hyperuniformity order metric, pore-size second moment and long-time spreadability behavior are all positively correlated and rank order the structures in exactly the same way. We also conjecture what structures maximize the fluid permeability for arbitrary porosities and show that this conjecture must be true in the extreme porosity limits by identifying the corresponding optimal structures.

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

Longitudinal wave propagation analysis in Entangled metallic wire materials using an improved wave and finite element method

This paper proposes a novel concept and methodology to study longitudinal wave propagation in entangled metallic wire material (EMWM), a porous structural network consisting of spatial helix wires. The numerical model of EMWM is constructed using representative volume elements (RVE) obtained through X-ray tomography and skeletonization of a standard specimen. By increasing the approximate periodic boundary, the improved wave and finite ele-ment method (WFE) is employed to model EMWM as approximately periodic structures, allowing for numerical simulations of wave propagation characteristics. Frequency bands and resonance modes of EMWM are calculated to analyze the characteristics of one-dimensional wave propagation. The numerical results show that EMWM exhibits unique frequency band characteristics compared to typical periodic porous materials. In the frequency range of 15 to n n 35 kHz, the first pass band corresponds to the tensile-compressive motion of the entire specimen, while the higher pass bands correspond to localized vibrations within the spatial wires. Additionally, the frequency bands can be adjusted across a wide range based on the relative density of EMWM specimens. The improved WFE method and obtained numerical data can facilitate the application of EMWM on high frequency vibration isolation.

Polyamide infiltration phenomenon on microfiltration support reduced by covalent organic frameworks net for faster desalination

Covalent organic frameworks (COFs) are a new periodic porous crystalline material, that has nano pore sizes. Herein, a novel COFs MPDTp was introduced into nylon microfiltration substrate to reduce polyamide (PA) infiltration condition for highly permselective nanofiltration (NF). The resultant PA-COF/nylon membrane was achieved after twice interfacial polymerization (IP) processes. The first was the introduction of COFs, and the second was the preparation of PA selected layer. The optimal membrane showed outstanding pure water permeance (PWP) of 27.3 L m−2 h−1 bar−1, which was about three times larger than the PA/nylon membrane, and rejection of 98.4 % for Na2SO4. The excellent performance was benefit from the existence of COFs, on the one hand, the COFs on the surface and inside could be seem as a net which prevented PA infiltration and blockage. On the other hand, a thinner PA separation layer was generated due to the slower piperazine (PIP) diffusion rate, which was also certified by molecular dynamics (MD) simulation. Therefore, this work not only demonstrates the high application value for desalination of the PA-COF/nylon membrane, but also provides a new strategy to adjust the relationship between the microporous substrate and the PA layer.

Application of Lévêque Analogy to Open Cellular Porous Materials

Open-porous materials are used for thermal management applications, catalyst support, heat exchangers, and others owing to their enhanced heat transfer. Designing porous material for a specific heat transfer application requires quantifying effective thermal conductivity, pressure drop, and heat transfer as computed data or experimental correlations. This work extends the Lévêque analogy proposed for calculating heat transfer performance from pressure drop data for tube bundles and cross-rod matrices to porous materials. The current work shows the applicability of this analogy for periodic open-cellular porous materials. Seven unit cell geometries representing periodic cellular structures with 75% - 95% porosity are considered. Periodic flow and heat transfer simulation is performed in ANSYS Fluent for Reynolds numbers between 1 and 100 to provide pressure drop and heat transfer results. Permeability and inertial coefficient are estimated, and permeability is correlated successfully with the porosity and fiber diameter of the unit cell geometries. This correlation is validated using experimental data available in the literature, which allows for predicting pressure drops directly from the geometry. Lévêque’s analogy in the original form is explored using the pressure drop coefficient and Nusselt number, which provides a 44% frictional contribution to the total pressure drop. The correlation for the current geometries is further improved using a modified version of the Lévêque analogy, where the exponent is estimated using best fit instead of fixing at 1/3. The modified correlation is validated using experimental data from the literature, and a guideline to predict pressure drop and heat transfer from geometry is provided for periodic open-cellular porous materials.

Acoustic modelling of 3D-printed hybrid materials: a preliminary study

Analytical and finite element (FE) models are developed to predict the sound absorption coefficient of hybrid materials obtained through assembly of folded quarter-length resonators and periodic porous material. The analytical model uses the sum of the acoustic admittances. The acoustic admittance of resonators is simulated with Stinson's model. Johnson-Champoux-Allard-Lafarge (JCAL) model and transfer matrix method (TMM) are used to calculate the acoustic admittance of 1, 2 and 4-layers of porous materials with different geometric parameters. The FE model is implemented using the COMSOL Multiphysics. The helical tubes are simulated using the visco-thermal acoustic module while the porous layers are simulated with JCAL poro-acoustic module. 30-mm thick samples of hybrid materials are produced via additive manufacturing (AM). Normal incident sound absorption coefficient of the hybrid materials is measured using an impedance tube. The experimental and simulated sound absorption coefficients are compared. The impact of the structure parameters (resonator diameter and length as well as number of porous layers) on the sound absorption is assessed. The hybrid materials present low frequency and broadband sound absorption. Higher broadband sound absorption is obtained with 4-layers of porous material. Low frequency absorption ( 1000 Hz) is achieved with long folded resonators (L 100 mm).

Hybrid acoustic materials through assembly of tubes and microchannels: design and experimental investigation

PurposeThe purpose of this paper is the design and experimental investigation of compact hybrid sound-absorbing materials presenting low-frequency and broadband sound absorption.Design/methodology/approachThe hybrid materials combine microchannels and helical tubes. Microchannels provide broadband sound absorption in the middle frequency range. Helical tubes provide low-frequency absorption. Optimal configurations of microchannels are used and analytical equations are developed to guide the design of the helical tubes. Nine hybrid materials with 30 mm thickness are produced via additive manufacturing. They are combinations of one-, two- and four-layer microchannels and helical tubes with 110, 151 and 250 mm length. The sound absorption coefficient of the hybrid materials is measured using an impedance tube.FindingsThe type of microchannels (i.e. one, two or four layers), the number of rotations and the number of tubes are key parameters affecting the acoustic performance. For instance, in the 500 Hz octave band (α500), sound absorption of a 30 mm thick hybrid material can reach 0.52 which is 5.7 times higher than the α500 of a typical periodic porous material with the same thickness. Moreover, the broadband sound absorption for mid-frequencies is reasonably high with and α1000 > 0.7. The ratio of first absorption peak wavelength to structure thickness λ/T can reach 17, which is characteristic of deep-subwavelength behaviour.Originality/valueThe concept and experimental validation of a compact hybrid material combining a periodic porous structure such as microchannels and long helical tubes are original. The ability to increase low-frequency sound absorption at constant depth is an asset for applications where volume and weight are constraints.

An FFT‐based Galerkin method for the effective permeability of porous material

AbstractA numerical scheme which is a combination of FFT and Galerkin methods is proposed for calculating the effective permeability of periodic porous material. First, a periodic stress field whose divergence is equal to the subtraction of body force from prescribed pressure gradient is introduced to reformulate the governing equations. Then, an auxiliary problem is established to replace the prescribed pressure gradient with the resulting initial prescribed stress. An FFT‐based Galerkin method of stress control type, the associated projection operator for zero divergence condition and the treatment of composite pixels/voxels are presented for solving the viscous flow in periodic porous material. Finally, numerical examples of viscous flows in 2D parallel fissured medium, 2D medium with regularly packed squares, 2D medium with regularly packed and rotated squares, 3D medium with regularly packed ellipsoids, and 3D medium with randomly packed ellipsoids are implemented. Numerical results calculated by FFT‐based Galerkin method agree well with analytical solutions and finite element results. The proposed FFT‐based Galerkin method is effective for predicting the macro permeability of porous material with complex meso structures.

Phase behaviour of a simple fluid confined in a periodic porous material

Using density functional theory for a simple fluid of hard spheres with a square-well attraction, we study the phase behaviour of the fluid confined in triply periodic porous material, where the pore space is bounded by a bicontinuous minimal surface. We combine our numerical results with the morphometric thermodynamics, an ansatz that expresses thermodynamic quantities as a linear combination of four additive geometrical terms, and analyse the dependency of the transition point between a liquid and a gas on the geometrical properties of the confining space. By demonstrating that the morphometric approach can be employed in order to account for the thermodynamic behaviour of a simple fluid in a complex porous material, we make the first step towards a better understanding of experimental observations that can help to characterise the pore space.

A tangent finite-volume direct averaging micromechanics framework for elastoplastic porous materials: Theory and validation

In this communication, we present a reformulated finite-volume direct averaging micromechanics (FVDAM) framework for periodic multiphase materials with elastic-plastic phases. The original elastoplastic version is formulated using the secant stiffness matrix approach, which requires a computationally intensive solution procedure of the governing differential equations. For the first time, the reformulation makes use of the tangent plasticity matrix approach that incorporates linearities to the unit cell boundary value problem. The tangential FVDAM theory is implemented using a quasi-Newton-Raphson strategy that quantifies errors in the evaluation of surface-averaged stresses due to the linearization, hence allowing large load increments. The tangential formulation is vigorously and fully assessed vis-à-vis the secant approach for porous unit cells on several aspects, including the convergence of the algorithmic implementation with mesh and step sizes, solution accuracy, and computation times. The reformulation simplifies the solution to the governing differential equations relative to its predecessor, albeit at the cost of more computational efforts for the reforming and reinverting of the global tangent stiffness matrix (or the so-called tangent operator). Additionally, advantages of the FVDAM theory relative to the classical finite-element homogenization are highlighted. The tangent FVDAM is employed to demonstrate the plasticity-triggered architectural effects in the response of periodic porous materials under different loading modes. The common mechanisms responsible for differences in the elastic-plastic response are attributed to the effective and hydrostatic stress alteration in the matrix phase. The current work paves the ground for the incorporation of FVDAM into readily-available commercial computational tools through the user material subroutines that are based on tangent stiffness approaches, which will produce a paradigm shift for FVDAM to solve the challenging multiscale structural problem.

Generalized tessellations of superellipitcal voids in low porosity architected materials for stress mitigation

Stress concentration is a crucial source of mechanical failure in structural elements, especially those embedding voids. This paper examines periodic porous materials with porosity lower than 5%. We investigate their stress distribution under planar multiaxial loading, and presents a family of geometrically optimized void shapes for stress mitigation. We adopt a generalized description for both void geometry and planar tessellation patterns that can handle single and multiple voids of arbitrary void shape at a generic angle. The role of void shape evolution from diamond to rectellipse on the stress-distribution is captured at the edge of voids in a representative volume element (RVE) made of non-equal length periodic vectors. Theoretical derivations, numerical simulations along with experimental validation of the strain field in thermoplastic polymer samples fabricated by laser cutting unveil the role of geometric parameters, e.g. superellipse order, aspect ratio and rotation angle, that minimize stress peak and ameliorate stress distribution around voids. This work extends and complements classical theory by providing fundamental insights into the role that tessellation, void shape and inclination play in the stress distribution of low-porosity architected materials, thus introducing essential guidelines of broad application for stress-minimization and failure mitigation in diverse sectors.

Impact of entry–exit loss on the measurement of flow resistivity of porous materials

This study experimentally investigates the impact of low pore density and thickness on the flow resistivity, hence permeability, of periodic open porous materials. An experimental setup is developed to measure the flow resistivity of porous samples adhering to the guidelines provided by the International Standards Organization 9053 and American Society for Testing and Materials C522 standards. The flow resistivity of three different porous microlattices is considered. Two of the three lattices are obtained from a monospherical void subtraction method, while the third is the octet-truss lattice. The monospherical void geometries are body-centered-cubic and A15 lattices. The pore density, measured in terms of pores-per-inch (PPI), was varied from 1 PPI to 12 PPI for all three microlattices considered. At low PPIs, the pore sizes are on the order of macroscale thickness. Hence, the volume-averaging or homogenized equivalent fluid model may not be valid. The thickness variation for a fixed PPI is also considered over limits where traditional volume-averaging methods break down. It is shown that at extremely low PPIs and thicknesses, entrance and exit losses become dominant, and the flow resistivity measured as per standards ceases to be a constant with PPI or thickness. Furthermore, existing two-dimensional modeling approaches for losses due to flow in pipes are adopted to predict the entry and exit pressure loss coefficients for the considered microlattices with appropriate modifications.

Reproducibility of sound-absorbing periodic porous materials using additive manufacturing technologies: Round robin study

The purpose of this work is to check if additive manufacturing technologies are suitable for reproducing porous samples designed for sound absorption. The work is an inter-laboratory test, in which the production of samples and their acoustic measurements are carried out independently by different laboratories, sharing only the same geometry codes describing agreed periodic cellular designs. Different additive manufacturing technologies and equipment are used to make samples. Although most of the results obtained from measurements performed on samples with the same cellular design are very close, it is shown that some discrepancies are due to shape and surface imperfections, or microporosity, induced by the manufacturing process. The proposed periodic cellular designs can be easily reproduced and are suitable for further benchmarking of additive manufacturing techniques for rapid prototyping of acoustic materials and metamaterials.

Computation of dispersion diagrams for periodic porous materials modeled as equivalent fluids

The application of shift cell technique is presented and discussed for periodic porous media described with equivalent fluid models: as it can be found in literature, it consists in a reformulation of classical Floquet-Bloch (F-B) conditions, in which the phase shift of the boundary conditions, related to wave propagation, is integrated into the partial derivative operator. Consequently, the periodicity is included in the overall behavior of the structure, while continuity conditions are imposed at the edges of the unit cell. Its major advantage stands in allowing the introduction of a generic frequency dependence of porous material behavior, through the resolution a quadratic eigenvalue problem, providing an efficient way to compute the dispersion curves of a porous material modeled as an equivalent fluid. A validation and a computational cost comparison are performed between the shift cell technique and the classical F-B approach, pointing out that the first can provide, among its other advantages, a sensible computational time reduction for this kind of analyses. The derivation of the equivalent acoustic properties of the unit cell from its dispersion characteristics is also investigated. To this aim, group velocity matrix formulation and a branch-tracking algorithm are described. Some test cases are used for validating the proposed methodology.

Numerical calculation of permeability of periodic porous materials: Application to periodic arrays of spheres and 3D scaffold microstructures

SummaryIn this paper, an efficient numerical method is proposed to calculate the anisotropic permeability in porous materials characterized by a periodic microstructure. This method is based on pore‐scale fluid dynamic simulations using a static volume of fluid method. Unlike standard solution procedures for this type of problem, we here solve an average constitutive equation over both fluid and solid domain by use of a subgrid model to accurately capture momentum transfer from the fluid to solid interface regions. Using numerical simulations on periodic arrays of spheres, we first demonstrate that, by using the subgrid interface model, more accurate results can be produced, for the velocity and pressure fields, than via more conventional approaches. We then apply numerical upscaling over the unit cell to calculate the full anisotropic permeability from the pore‐scale numerical results. The obtained permeability values for a variety of periodic arrays of spheres in different arrangements and packing orders are in good agreement with semianalytical results reported in literature. This validation allows for the permeability assessment of more complex structures such as isotropic gyroid structures, or anisotropic cases, here modeled in their simplest form, the ellipsoidal inclusion.

Multiscale computational method for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains

In this paper, a novel multiscale computational method is presented for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains. In these porous materials, heat transfer at microscale has an important impact on the macroscopic temperature field. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on asymptotic homogenization method. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method (FEM) and finite difference method (FDM). Finally, some numerical examples are shown, which demonstrate the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains.

Multiscale analysis and computation for coupled conduction, convection and radiation heat transfer problem in porous materials

This paper discusses the multiscale analysis and numerical algorithms for coupled conduction, convection and radiation heat transfer problem in periodic porous materials. First, the multiscale asymptotic expansion of the solution for the coupled problem is presented, and high-order correctors are constructed. Then, error estimates and their proofs will be given on some regularity hypothesis. Finally, the corresponding finite element algorithms based on multiscale method are introduced and some numerical results are given in detail. The numerical tests demonstrate that the developed method is feasible and valid for predicting the heat transfer performance of periodic porous materials, and support the approximate convergence results proposed in this paper.

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

A novel second‐order two‐scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non‐periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non‐periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.