Unit Cell Analysis Research Articles

Computationally-efficient homogenization and localization of unidirectional piezoelectric composites with partially cracked interface

The homogenized and localized responses of unidirectional piezoelectric composites with partially cracked interface are investigated in this contribution. To simulate the electro-elastic fully-anisotropic constitutive behavior, a micromechanical multiphysics finite-element model is developed through adopting higher-order internal trial functions. The generated results are validated against the independently-developed generalized Eshelby solution with piezoelectric effects, a comparable finite-volume (FV) technique, and ABAQUS simulations with excellent agreement. More importantly, the present work investigates thus-far little explored effects of several microstructural details including the fiber/matrix property ratio, an elliptical inclusion, multiple periodic inclusions, as well as the crack length and poling direction, on either effective properties or local stress/electric field distributions. It is also demonstrated that the present model is advantageous to commercial finite-element packages in two aspects. Firstly, the periodic boundary conditions of the nodal displacements and electric potential are enforced explicitly. Secondly, the ability to generate a complete set of homogenized moduli with two-dimensional architectures sets the present model apart from the readily available commercial codes that require a three-dimensional unit cell analysis.

Control of Spatial Wave Profiles in Finite Lattices of Repelling Magnets

Abstract We report on the control of the spatial wave profile of a chain of lumped magnets arranged in a repelling configuration. The spatial wave attributes are controlled by varying the spacing between the magnets, which in turn affects the equivalent stiffness of the lattice. The dispersion relation for an infinite lattice is first derived for linearized equations of motion to provide an insight into the effect of varying the lattice spacing on the wavelength and sonic speed. The unit cell analysis is verified using analytical spatial wave profiles for a finite magnetic lattice. We conclude the paper with an analytical derivation of a fuel/time optimal controller designed to control the spacing between the magnets, which is necessary for implementation aspects.

Deformation analysis of geosynthetic-encased stone column using cavity expansion models with emphasis on boundary condition

This paper presents a modified theoretical model to predict the deformation of geosynthetic-encased stone column (GESC) and surrounding soil, using cylindrical cavity expansion model (CEM). The model was distinguished for single GESC and GESC in groups with emphasis on the different boundary conditions. The displacement boundary of CEM was used for GESC in groups, and the stress boundary of CEM was adopted for single GESC. The plasticity development of the soil obeying the Mohr-Coulomb yielding criterion was considered. The stress and settlement of the GESC were analyzed by radial stress and vertical stress equilibrium. This method has been verified via comparison with test data and numerical simulation results. The influences of applied loading, geosynthetic encasement stiffness, and soil stiffness on the mechanical performance of the GESC and the surrounding soil have also been investigated. The proposed theoretical approaches are suitable for predicting the deformation of the GESC, and the surrounding soil. The proposed method in unit cell analysis was more reasonable for GESC in groups.

Characteristic Mode Analysis of Unit Cells of Metal-Only Infinite Arrays

Characteristic mode analysis of metal only unit cells of periodic structures is performed using Method of Moments based formulation. Ewald’s transformation is incorporated for a fast and cost efficient solution and the advantages over spatial Green’s function are discussed. The influence of the unit cell size on the characteristic modes is demonstrated. Various metal-only reflectarray elements are compared and their radiation characteristics are interpreted using the theory of characteristic modes. It is shown that characteristic modes of the unit cell can help us to understand the radiation and scattering behavior of the unit cell and this physical insight can be used in periodic array unit cell design.

Fast and Accurate Power Distribution Network Modeling of a Silicon Interposer for 2.5-D/3-D ICs With Multiarray TSVs

In this paper, we first propose power distribution network (PDN) model of the perforated power and ground (P/G) planes including substrate effects and multiarray through-silicon vias (TSVs) for a silicon interposer. Since it is almost impossible to simulate a high metal density perforated PDN structure, we first suggest a modeling methodology for P/G perforated planes to reduce simulation time significantly with a high accuracy. To obtain the PDN impedance of a silicon interposer faster, we convert the perforated planes to solid planes with a dielectric mixture. The capacitance (C) and conductance (G) component of perforated P/G planes are precisely estimated based on the conformal mapping method. From the estimated C and G, the physical dimension and material properties of the dielectric mixture of solid P/G planes are determined, respectively. Because silicon interposer PDN consists of a periodic structure, we design and analyze the unit cell of a PDN thoroughly. From the unit cell analysis, the electrical characteristic of an entire PDN for a silicon interposer is successfully estimated. The PDN impedance of the proposed solid and perforated P/G planes and simulation time to obtain each PDN impedance are compared and evaluated, respectively. The proposed methodology is validated by a full 3-D electromagnetic (EM) simulation in the frequency range from 0.01 to 20 GHz. We also proposed models of multiarray P/G TSVs that can be applied to the various P/G patterns for a silicon interposer. From the inductance matrix considering current directions, the self- and mutual inductances of TSVs are accurately calculated. Because quasi-transverse electromagnetic mode is propagated through TSVs and the substrate dielectric of a silicon interposer can be regarded as a uniform dielectric, the capacitance and conductance of TSVs can be calculated from the inductance matrix. The proposed model of multiarray TSVs is also analyzed and verified by EM simulations in the frequency range from 0.01 to 20 GHz with a high accuracy.

The effect of pore numbers in the cell walls of soybean oil polyurethane foam on sound absorption performance

Since environmental protection is of increasing importance, the sound absorption of polyurethane foams in which the normal petroleum content is partly replaced by soybean oil has been studied through simulations and experiments. In this study, the theoretical sound absorption coefficients of the foam with different microstructures were calculated by using Johnson-Champoux-Allard (JCA) method. The parameters required in the JCA method were calculated based on the specific finite element analysis of periodic unit cells (PUCs) under ideal conditions. The PUCs were modeled as tetrakaidecahedrons with different numbers of pores in the wall cells. Both the experimental results and simulation results show that the sound absorption performance of PU foam is influenced by pore numbers, and the more pores the better sound absorption performance. Simulations show that the number of pores influences the JCA parameters including air-flow resistivity, tortuosity and characteristic lengths.

An analytical framework for locally resonant piezoelectric metamaterial plates

We present an analytical modeling framework and its analysis for thin piezoelectric metamaterial plates to enable and predict low-frequency bandgap formation in finite structural configurations with specified boundary conditions. Using Hamilton’s extended principle and the assumptions of classical (Kirchhoff) plate theory, the governing equations and boundary conditions for the fully coupled two-dimensional electromechanical system are obtained. The two surfaces of the piezoelectric bimorph are segmented into non-overlapping opposing pairs of electrodes of arbitrary shape, and each pair of electrodes is shunted to an external circuit. This formulation can be used to study the effect of electrode shape on plate response for topology optimization and other vibration control applications. Using modal analysis, we show that for a sufficient number of electrodes distributed across the surface of the plate, the effective dynamic stiffness of the plate is determined by the shunt circuit admittance applied to each pair of electrodes and the system-level electromechanical coupling. This enables the creation of locally resonant bandgaps and broadband attenuation, among other effects, in analogy with our previous work for one-dimensional piezoelectric structures with synthetic impedance shunt circuits. It is also demonstrated that piezoelectric bimorph plates display significantly improved performance (i.e. electromechanical coupling) over bimorph beams, but require additional electrode segmentation to achieve metamaterial-type performance. The governing equations are also used for dispersion analysis using the plane wave expansion method, enabling the analytical dispersion analysis of unit cells with arbitrary electrode shapes. The modeling framework and approximate closed-form bandgap expressions are numerically validated using finite element analysis.

Hydrogen informed Gurson model for hydrogen embrittlement simulation

Hydrogen-microvoid interactions were studied via unit cell analyses with different hydrogen concentrations. The absolute failure strain decreases with hydrogen concentration, but the failure loci were found to follow the same trend dependent only on stress triaxiality, in other words, the effects of geometric constraint and hydrogen on failure are decoupled. Guided by the decoupling principle, a hydrogen informed Gurson model is proposed. This model is the first practical hydrogen embrittlement simulation tool based on the hydrogen enhanced localized plasticity (HELP) mechanism. It introduces only one additional hydrogen related parameter into the Gurson model and is able to capture hydrogen enhanced internal necking failure of microvoids with accuracy; its parameter calibration procedure is straightforward and cost efficient for engineering purpose.

Micro-structurally informed finite element analysis of carbon/carbon composites for effective thermal conductivity

A micro-structurally informed two scale unit cell analysis has been presented for the prediction of thermal conductivities of 3D hybrid and 4D in-plane carbon/carbon composites. Realistic micro-structural features such as bundle cross sections, alignments, and macro voids have been included directly in the unit cells geometry through reconstructed X-ray computed tomography images of the composite. Asymptotic homogenization technique has been used on both scales (bundle and composite) along with periodic boundary conditions to obtain effective thermal conductivities. The imperfectly bonded conditions at the fiber/matrix and fiber bundle/matrix interfaces have also been modeled by considering the contact and equivalent gap conductance between their surfaces. The model has been validated through the experiment conducted in the in-plane direction of 3D hybrid composite. Finally, the effective thermal conductivities of 3D hybrid composite in out-of-plane directions and of 4D in-plane composite in x, y, and z-directions have been predicted for full temperature range (300–1500 K).

Modeling Bloch Waves in Prestressed Phononic Crystal Plates

The study aims at investigating the effect of a generic state of prestress on the passbands and bandgaps of a phononic crystal plate. To this end, an Updated Lagrangian scheme is developed, which consists of two steps: first, a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load is conducted and then the Floquet-Bloch decomposition is applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration. In addition, a formula for the calculation the energy velocity is proposed. In the case of an epoxy plate with cylindrical steel inclusions, it is shown that, even in the case of prestress inducing full reversible deformation state, the band gap experiences a shift towards higher frequencies when the cell is subjected to a compressive prestress, whereas a frequency downshift is registered when the cell is subjected to traction. In particular, it is demonstrated that the frequency downshift of the bandgap for the phononic plate undergoing a tensile prestress is approximately 3.5\% with respect to the case of the phononic plate under compression. The results presented herein provide insights in the behavior of phononic crystal plates with tunable dispersive properties, and suggest new leverages for wave manipulation valuable in many application fields such as wave filters, waveguiding and beam splitting, sensing devices, and vibration shielding.

A high-efficient model for interface debonding analysis of carbon fiber-reinforced polymer composite

ABSTRACTDue to the difficulties in micro stress calculation of interface, most failure analysis models of macro composite structure mainly focus on the failure modes of fiber and matrix, which have not taken interface debonding into consideration. In this paper, a high-efficient failure analysis model is proposed to realize quick investigation of interface debonding as well as fiber and matrix failure in macro composite structure under complex stress condition. Stress amplification factors are adopted to realize the rapid calculation of micro stress at interface according to the macro stress of composite. Three-point bending tests with elaborate designed stacking sequences are carried out to extract the critical interfacial normal strength and shear strength used in this model. In the failure analysis of micro unit cell this model shows high accuracy in the prediction of interface debonding both in the failure modes and failure locations compared with cohesive zone model. To further inspect its capability in the analysis of macro composite structures, open-hole compression tests of quasi-isotropic laminates are carried out. Good agreements in the interface debonding mechanisms are obtained between numerical prediction and experimental results.

Application of damage model based on void growth analysis to ductile crack growth simulation

ABSTRACT The aim of this study is to confirm that ductile crack growth resistance of cracked components can be predicted by simulation model with the proposed damage model, which is derived by unit cell analyses with an initial void and reflects void growth behaviour until ductile crack initiation. Critical strain under constant stress triaxiality history, which is used in the damage model as ductility of the material, could be determined properly using tensile test results by inverse analyses. Tensile tests using notched round bar specimens, which simply represent stress triaxiality history similar to cracked component, were conducted for two different steels and the proposed damage model was found to be validated for evaluation of ductile crack initiation limit. Then, ductile crack growth tests of two types of bend specimens with a machined notch and a fatigue pre-crack were conducted for a mild steel. It was demonstrated that ductile crack growth resistance could be reproduced using simulation model with the proposed damage model with high accuracy and the influence of plastic constraint to ductile crack growth behaviour could be simulated.

Evaluation of metamaterial unit cell analysis techniques

The field of acoustic metamaterials has produced novel materials in a wide variety of applications. An important step in designing a metamaterial is unit cell analysis with subwavelength geometry. There are several techniques used for unit cell analysis when designing acoustic properties of interest for metamaterial applications. Such techniques include, but are not limited to, band diagrams, effective material properties, or half-space homogenization. This talk discusses the challenges and tradeoffs between analysis techniques and types of structures that lend to one method or another. Unit cell analysis methods will be evaluated to perform trade space exploration, including validation and parametric studies for metamaterial design.

An investigation of the effect of hydrogen on ductile fracture using a unit cell model

The effect of hydrogen on the ductility of metals is studied by incorporating the hydrogen diffusion process and the hydrogen enhanced localized plasticity (HELP) into a finite element program. A series of unit cell analyses are conducted under various stress states and the loading speed resulting in a steady state hydrogen distribution is determined. The evolution of the local stress and deformation states results in hydrogen redistribution in the material, which in turn changes the material's flow property due to the HELP effect. It is found that localized plastic deformation plays a major role in increasing the hydrogen concentration due to the newly generated trapping sites. The HELP effect promotes material failure by accelerating void growth, which is affected by the macroscopic stress state subjected by the material unit characterized by the stress triaxiality and the Lode parameter. For a constant Lode parameter, the effect of HELP on void growth and failure strain reduction increases with the stress triaxiality. For a constant stress triaxiality, the effect of HELP is highest when the Lode parameter is near 0. As the Lode parameter increases towards 1 or decreases towards −1, the HELP effect gradually diminishes.

Multiscale analysis of fatigue crack initiation life for unidirectional composite laminates

Three dimensional (3D) finite element analysis along with a damage model based on kinetic theory of fracture (KTF) are used to determine the onset of fatigue crack initiation in unidirectional carbon fiber reinforced composites. In addition, the Multicontinuum theory (MCT) and micromechanics are employed separately to enable the assessment of the stress state distribution at the length scale level of the constituents. While the MCT is implemented in the macroscale model formulation, micromechanical analysis of a representative unit cell has been also developed to capture the stress field in the micro length scale level. An ABAQUS™ user material (UMAT) subroutine has been developed and used for implementing the formulation. Numerical examples of various tests including tension, three point bending, four point bending, and short beam shear test of a unidirectional (UD) laminate are considered and the results are compared against actual test data. A detailed discussion, challenges and future plans on the applicability of the KTF are concluding this work.

Analysis and design of an ultra-thin metamaterial absorber and its application for in-band RCS reduction of antenna

ABSTRACTThis paper presents the analysis and design of an ultra-thin metamaterial absorber (MA) for the in-band Radar Cross Section (RCS) reduction of a patch antenna. The proposed MA consists of a periodic array of double square loop structures printed on an FR4 substrate with a thickness of 0.022 λo. A detailed theoretical analysis of the MA unit cell is conducted to investigate the absorbing characteristics of the MA. The simulated and measured results show that MA significantly reduces the RCS of the proposed antenna within its impedance bandwidth and a maximum RCS reduction of 15 dB is achieved at 6.75 GHz. The impedance bandwidth of the proposed antenna is 6.6–7 GHz, which is also improved by the loading of the MA. The results also show that after loading the MA on the patch antenna, the RCS of the proposed antenna is significantly reduced and there is no degradation observed in its radiation properties.

Prediction of temperature-dependent properties of short fiber reinforced thermoplastic by unit cell analysis

Prediction of temperature-dependent properties of short fiber reinforced thermoplastic by unit cell analysis

Efficient Simulation of Substrate-Integrated Waveguide Antennas Using a Hybrid Boundary Element Method

This paper presents a hybrid boundary element method for the efficient simulation of substrate-integrated waveguide (SIW) horn antennas. It is applicable with good accuracy to relatively thin structures with conventional circular ground vias. In the multiscale simulations, a 2-D contour integral method is used for the modeling of the fields inside the structure with numerous vias and a method of moments is used for the radiated fields outside. The contour integral method is extended in this paper by a new waveguide port of finite size based on the unit cell analysis of an SIW segment. Several SIW horn antennas are studied to validate the proposed method in terms of the input impedance, the field distribution on the aperture, and the radiation diagram. The proposed method shows good to reasonable accuracy and has a numerical efficiency which is about 2–3 orders higher than FEM-based full-wave simulations. It is therefore well suited for fast optimizations.

Realisation of a locally resonant metamaterial on the automobile panel structure to reduce noise radiation

Abstract This study aims to demonstrate the reduction in noise and vibration of an automobile dash panel structure by applying a locally resonant metamaterial (LRM). LRMs are implemented by adding a subwavelength array of local resonators to a host structure. Here, the minimum repeated structure is called a unit cell. Owing to dynamic absorption in a unit cell, LRM has a stop band (or also called a band gap), which is a frequency band in which waves cannot propagate. Taking advantage of the stop band, various studies have been conducted on mechanical noise and vibration problems. However, no case has yet been published regarding the application of LRMs to industrial structures such as automobile dash panels because of their complex shape. To overcome the difficulties of LRM realisation in dash panels, an attachable local resonator (ALR) is proposed in this study. The proposed ALR consists of a local resonator and a permanent magnet to be attached on the dash panel surface. To predict the stop band formation, a finite element-based unit cell analysis method is developed, and several ALRs are designed based on the stop band. The designed ALRs are applied to an actual automobile dash panel structure and the acoustic and vibration responses are measured. The measurement results demonstrate that the vibration and noise radiation from the dash panel structure are greatly reduced in the designed stop bands.

Dynamic homogenization of resonant elastic metamaterials with space/time modulation

We present a variational multiscale strategy in the spirit of FE $$^2$$ method for simulating the real time evolution of resonant elastic metamaterials with space/time modulation. The implicit time discretization used guarantees the stability of the numerical solution, while the accuracy is quantified by direct comparison with the response obtained via direct numerical simulation. The results showcase a remarkable accuracy of the method, even when the dynamic material properties drastically change over length scales that are equal to the macroscopic mesh size and time scales that are of the same order of magnitude than the external excitations. We further propose a strategy to identify the band gap structure of these metamaterials based on their real time response. The methodology is verified for spatially periodic time invariant systems via classical unit cell analyses in the frequency domain.