Equivalent Single Layer Research Articles

Prediction and validation of flanking sound transmission in cross-laminated timber junctions with resilient interlayers

Propelled by its small carbon footprint, low weight and high stiffness, cross-laminated timber (CLT) has experienced a significant commercial growth in the last decade. However, these latter two features potentially contribute to poor acoustic performance. A critical factor is flanking sound, in which vibrational energy is transmitted between two elements connected through a common junction, driving the need to include vibration isolation solutions such as resilient interlayers in the junction design. In this work, the vibration reduction index across CLT junctions is predicted with an analytical wave approach for semi-infinite thin plates. Each CLT panel is modelled as an orthotropic or isotropic equivalent single layer (ESL). Multiple options to determine the ESL properties are compared, including the law of mixtures and the modified gamma method. The interlayer is considered to be a thick, flexible waveguide, whose out-of-plane motion is governed by shear. The prediction model is validated experimentally with on-site measurements for a vertical T-junction with a polyurethane foam interlayer. The predictions are moderately accurate in the entire frequency range, with deviations below 10 dB across all 1/3 octave bands. Depending on the layer configuration, the orthotropy of the ESL can significantly influence the predictions.

Estimation of random incidence sound transmission loss of panels based on normal incidence sound transmission loss measurements

The performance of building sound insulation is an important parameter for evaluating the quality of construction projects. So far, the sound insulation performance testing of building walls requires the installation of large-dimensioned wall samples in specialized sound insulation laboratories, which is time-consuming and costly. If small-dimensioned components of the wall panels can be applied and quickly tested at the project site, it could significantly improve efficiency and reduce costs. To address this issue, this paper investigates a method for estimating the random incidence sound transmission loss of wall panels based on the measurement of the normal incidence sound transmission loss of corresponding small samples. Initially, the sound transmission loss of single-layer and composite structures is theoretically analyzed based on the theories of Equivalent Single Layer method and Layer-Wise method. Subsequently, the random incidence sound transmission loss is estimated by the normal incidence sound transmission loss. The methodology for estimating the random incidence sound transmission loss of wall panels proposed in this study offers both a practical approach and theoretical support for the rapid assessment of the sound insulation performance of building walls. This study is of great significance for improving the quality of construction projects and inspection efficiency.

Novel High-Strength and High-Temperature Resistant Composite Material for In-Space Optical Mining Applications: Modeling, Design, and Simulation at the Polymer and Atomic/Molecular Levels.

This study explores the modeling, design, simulation, and testing of a new composite material designed for high-strength and high-temperature resistance in in-space optical mining, examining its properties at both the polymer and atomic/molecular levels. At the polymer level, the investigation includes mechanical and thermal performance analyses using COMSOL Multiphysics 6.1, employing layerwise theory, equivalent single layer (ESL) theory, and a multiple-model approach for mechanical modeling, alongside virtual thermal experiments simulating laser heating. Experimentally, porous Polyaniline (PANI) films are fabricated via electrochemical polymerization, with variations in voltage and deposition time, to study their morphology, optical performance, and electrochemical behavior. At the atomic and molecular levels, this study involves modeling the composite material, composed of Nomex, Kevlar, and Spirooxazine-Doped PANI, and simulating its behavior. The significance of this work lies in developing a novel composite material for in-space optical mining, integrating it into optical mining systems, and introducing innovative thermal management solutions, which contribute to future space exploration by improving resource efficiency and sustainability, while also enhancing the understanding of PANI film properties for in-space applications.

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

Efficient strategy for frequency design and bandwidth extension of curved piezoelectric ultrasonic micromachined transducers

Abstract This article proposes an efficient analytical model and strategy for designing curved piezoelectric micromachined ultrasonic transducers (curved PMUTs). The model is developed based on the Donnell-Mushtari-Vlasov (DMV) theory and the Equivalent Single Layer (ESL) method, and validated through Finite Element Analysis (FEA). Utilizing the model, we further analyze the diaphragm’s vibration modes and key design parameters. The proposed strategy is centered on 2 design equations, facilitating the rapid design of devices at any frequency through parametric sweeps. Furthermore, to minimize bandwidth loss, we employ the merging of adjacent vibration modes to broaden the bandwidth. Using the proposed method for modes merging, we have effortlessly designed devices with operating frequencies of 2.15 MHz, 6.3 MHz, 10.65 MHz, and 18.75 MHz in water. For comparison, we also designed planar PMUTs and general curved PMUTs operating around 6 MHz and 15 MHz. Compared to planar PMUTs, curved PMUTs show exceptional performance improvements in output pressure and sensitivity. Moreover, the proposed strategy for bandwidth extension results in 1.33× and 1.25× bandwidth improvements around 6 MHz and 15 MHz. The proposed design methodology is anticipated to assist engineers in designing high-performance PMUT arrays more efficiently and systematically.

Thermal buckling of variable stiffness composite laminates using high order plate finite elements

This paper proposes a study on the thermal buckling of Variable Angle Tow (VAT) composite plates using high-order theories. Here, the governing equations are derived via the principle of virtual work. Under the assumption of linear pre-buckling, the stability problem is reduced to a linear eigenvalue analysis considering proportional geometric stiffness. In contrast, a constant thermal load is assumed to be known along the plate thickness, and the uncoupled thermo-mechanical formulation is used, where the thermal effects are described as external loads. The plate is discretized using the Finite Element Method (FEM) and high-order theories are developed using the Carrera Unified Formulation (CUF). Using the CUF, the equations are expressed as an invariant of the plate theory approximation order. Therefore, Equivalent Single Layer (ESL) and Layer-Wise (LW) models can be easily implemented. Several geometries and lamination cases are considered for verification purposes, including different side-to-thickness ratios and fiber orientations, which result in various anisotropy effects. In addition, the effect of changing constraints and materials is evaluated. Particular attention is paid to the effect of the structural theory approximation on the evaluation of the thermal buckling load. It is shown that the correct evaluation is highly dependent on the edge-to-thickness ratio and on the anisotropy given by both the fiber orientation and the material properties. As a final remark, sensitivity analysis and best fiber angle solutions are discussed to highlight the importance of the LW modeling approach.

Exact solutions for clamped spherical and cylindrical panels via a unified formulation and boundary discontinuous method

Numerous works in both recent and historical literature have concentrated on formulating theories to perform static analysis on simply-supported shell structures. However, it is worth noting that obtaining analytical solutions for clamped boundary conditions presents a strong challenge. In this paper, closed-form solutions for clamped cross-ply laminated and sandwich shells are achieved by employing a robust and hybrid methodology not previously reported in the literature. The high versatility of the Carrera Unified Formulation (CUF), based on the Equivalent-Single-Layer (ESL) description, is utilized to implement several refined shell theories. The Principle of Virtual Displacements (PVD) is utilized to derive the strong form of the governing equations in terms of displacement variables. As the main novelty, these equations are solved by the Boundary Discontinuous Fourier-based method (BDM) which provides highly accurate analytical solutions. The validity and robustness of the proposed methodology are assessed through a detailed comparison with references available in the open literature, as well as with FEM 3D results obtained with commercial software. Furthermore, the stress recovery technique is exploited to fulfill zero-stress and interlaminar continuity (IC) conditions. The findings might be useful in training artificial intelligence (AI) models, which, for instance, could facilitate the development of digital twin structures.

Non-polynomial zig-zag unified shear deformation formulation solved via boundary discontinuous method for laminated sandwich plates

Analytical solutions of mechanical problems with refined theories and challenge boundary conditions are still a topic of research due to intrinsic complexity. In this paper, a generalized refined theory is formulated to obtain analytical solutions for the static problem of cross-ply laminated and sandwich plates with several boundary conditions. These theories are implemented in the framework of the Carrera unified formulation (CUF) based on the equivalent-single-layer (ESL) approach. The displacement field of ESL-based models are expanded over the thickness direction by using non-polynomial terms, e.g. trigonometric, hyperbolic and the so-called zig-zag (ZZ) Murakami’s functions. The principle of virtual displacements (PVD) statement is used to derive the strong form of the governing equations in terms of displacements variables. These equations are solved by the boundary-discontinuous double Fourier series approach which provides highly accurate analytical solutions. This work gives a detailed comparison of results obtained using the presented ZZ models highlighting the differences with 3D FEM results. Furthermore, the influence of non-polynomial and ZZ terms in each ESL model is investigated. The superior robustness of the presented methodology compared to past works is demonstrated, making the proposed results suitable as a benchmark for validating new theories and finite elements. The findings can be also useful to train artificial intelligence (AI) models that can allow to the development of faster digital twin structure technology.

Equivalent single‐layer Mindlin theory of laminated piezoelectric plates and application

AbstractBased on the Mindlin first‐order shear deformation theory, this paper proposes an equivalent single layer (ESL) plate theory to analyze the electro‐mechanical coupling problem of laminated piezoelectric plates (LPPs). The main features of the proposed approach are: (i) It assumes that the electric potential across the thickness is a polynomial function, ensuring its continuity at the interface. (ii) The electric displacements are continuous at the interface, in line with the interface continuity condition between laminated plates. The theoretical solutions for the deformation and electric potential of LPPs are obtained. The validity and accuracy of the theoretical solutions are confirmed through comparison with results of two‐ and four‐layer LPPs obtained from the three‐dimensional finite element method (FEM). The numerical results discuss the influence of different series expansions and emphasize the necessity of high‐order expansion. Meanwhile, the range of application of three‐dimensional FEM is discussed. It is expected that such a new analytical method can be instructive to the optimal design of piezoelectric device.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

New Accomplishments on the Equivalence of the First-Order Displacement-Based Zigzag Theories through a Unified Formulation

The paper presents a critical review and new accomplishments on the equivalence of the first-order displacement-based zigzag theories for laminated composite and sandwich structures. Zigzag theories (ZZTs) have widely spread among researchers over the last few decades thanks to their accuracy in predicting the response of multilayered composite and sandwich structures while retaining the simplicity of their underlying equivalent single-layer (ESL) theory. The displacement field consists of two main contributions: the global one, able to describe the overall structural behaviour, and the local layer-wise one that considers the transverse shear continuity at the layer interfaces that describe the “zigzag” displacement pattern typical of multilayered structures. In the framework of displacement-based linear ZZTs, various assumptions have been made on the local contribution, and different theories have been deduced. This paper aims to provide a unified formulation for first-order ZZTs, highlighting some common aspects and underlying equivalencies with existing formulations. The mathematical demonstrations and the numerical examples prove the equivalence of the approaches to characterising local zigzag enrichment. Finally, it is demonstrated that the kinematic assumptions are the discriminants of the ZZTs’ accuracy.

Equivalent single-layer model for hierarchical diamond honeycomb sandwich panels using variational asymptotic method

Sandwich panels with a hierarchical core facilitate the optimal utilization of limited materials to enhance bearing capacity in strategic locations. Nevertheless, the complex sub-structural cells have hindered extensive investigation into their static and modal characteristics. The objective of this study is to obtain the effective plate properties by homogenizing the representative unit cell and subsequently applying them to a two-dimensional equivalent single layer model (2D-ESL) based on the variational asymptotic method (VAM). Furthermore, three-point bending test of 3D-printed specimen and 3D detailed finite element results are employed to confirm the predictions of 2D-ESL under various load and boundary conditions. The effects of material factors (such as fiber volume fraction and layup configuration) and structural parameters (including aspect ratio, hierarchy ratio, and slenderness ratios of ribs and ligaments) on the equivalent stiffness and modal characteristics are also examined. Moreover, sandwich panels with different hierarchical configurations (including 2, 3 and infinite segments within the inclined ribs) are investigated to provide valuable insights for selecting appropriate hierarchies based on specific conditions. Notably, the transition from polyline to arc-shaped segments has been proven to effectively mitigate local stress concentration, leading to a more uniform distribution of local displacement.

Aeroelasticity analysis of anisogrid lattice sandwich cylindrical shells: Extended potential and piston theories

Based on the extended potential and piston aerodynamics theories, buckling and vibration analysis of sandwich cylindrical shells subjected to external airflow in both subsonic and supersonic regimes is presented. Sandwich cylindrical shells are considered to be made of same homogenous face sheets and an anisogrid composite lattice core with hexagonal cells. An equivalent single layer (ESL) theory in the framework of classical Love’s shell theory is used to formulate the global mechanical behavior of such shells. According to the extended potential and piston aerodynamic theories, aerodynamic pressure distributions are determined for steady, inviscid, irrotational compressible, and external airflow. The closed–form solutions of critical upstream pressure and natural frequency are developed for different boundary conditions using the Galerkin method. Finally, case studies are presented to compare between results of extended potential and piston theories and investigate the effects of upstream speed, geometric ratios, boundary conditions, and lattice specifications on various response quantities. Findings indicate that the differences in results of extended potential and piston theories are less than 3% in the supersonic regime with 2.3 ≤ M ∞ ≤ 2.7 for intermediate-length shells with a wide range of R . The piston theory is inadequate for analyzing aeroelasticity behavior of very long thin sandwich cylindrical shells.

A unified framework for equivalent single layer theories of composite beams, plates, and shells: a review

This article presents a comprehensive review of the unified framework for the equivalent single layer theories of composite beams, plates and shells. First, the approach to unified displacement fields has two main categories: one is to start from the assumption of linear displacement field variables. And the another is to start from the assumption of transverse strains and transverse displacement variable. In terms of unified shape functions, the shape function is defined as the function that exist in the transverse strain rather than the in-plane displacement variables. Moreover, the basic requirements for the choose of shape functions are also given and analysed in this paper. The unified shape functions are introduced in terms of the several functions and the unified polynomial model. In addition, a construction method for new shear strain shape functions is described in this review. Finally, the current research progress and future prospects are also given.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

Effect of porosity on the modal response of doubly-curved laminated shell structures made of functionally graded materials employing higher order theories

The manuscript investigates the modal response of laminated anisotropic doubly-curved shell structures of variable thickness made of Functionally Graded Materials (FGM), according to an efficient equivalent single layer strategy where the displacement field is described employing a condensed unified formulation, accounting for a higher order through-the-thickness expansion. A generalized power law distribution is adopted for the assessment of the FGM layers, whereas the presence of voids is considered starting from different homogenization assumptions. Unlike previous works which presented a variation of the homogenized material properties within the shell solid, in this paper the problem of material porosity is addressed, therefore a general distribution of the volume fraction of the constituent materials is considered, along with the presence of voids. To this end, both linear and trigonometric through-the-thickness distributions are here assumed. The fundamental governing equations are derived starting from a proper set of curvilinear principal coordinates, while a generalized set of blending functions accounts for arbitrarily shaped structures. Non-conventional boundary conditions are, here, modelled with a distribution of linear springs along the shell edges. The numerical implementation of the problem is performed with the generalized differential quadrature method. A systematic set of validating examples is presented, whose results are compared to predictions based on refined models, as well as some experimental evidence. After a validation step, an extensive parametric analysis points out the sensitivity of the porosity parameters on the modal response of structures with different curvatures and external constraints, with valid findings for engineering design purposes.

Higher-order modal parameter estimation and verification of cross-laminated timber plates for structural-acoustic analyses

Cross-Laminated Timber (CLT) is a wood composite that is popular due to its favourable stiffness-to-mass-density ratio and environmental benefits, among other positive attributes. This paper presents the estimation of higher-order modal parameters and their use in the verification and validation of an Equivalent Single-Layer (ESL) CLT model. The eigenvectors, eigenfrequencies, and modal damping ratios of 20 out-of-plane vibration modes were experimentally determined. The experimentally determined eigenfrequencies and eigenmodes were correlated and compared to those derived from a numerical model. The modal damping ratios were compared to damping loss factors derived from the power injection method. A broadband frequency view is also considered, with the comparison of experimental and numerical forced response models. As part of the verification and validation process, a framework of key indices and respective criteria is suggested and discussed in this contribution. Over the course of the paper, an ESL CLT model which derives its properties from a layerwise basis is validated in detail against experimental measurements. The results are intended to be of relevance to both structural and acoustic domains.

Equivalent single layer approach for ultimate strength analysis of box girder under bending load

The objective of this study is to assess the ultimate strength of box girder using equivalent single layer (ESL) approach. Box girders, composed of stiffened panels, are subjected to a four-point bending load for the ultimate strength analysis. The results of ESL were validated using FEM and experiments. Modeling of ESL includes a single plate with the same stiffness as the actual stiffened panel topology, as opposed to traditional FEM modeling and test specimen, where the stiffener is explicitly modeled. For ESL approach, VUGENS subroutine in Abaqus software is used to define the non-linear stiffness properties. Material and geometric non-linearities were considered during simulations. The ESL stiffness matrix was derived from unit cell simulations under six loading conditions. In this case, a unit cell is a periodic structure that represents the entire stiffened panel model. Contact interaction is utilized to simulate the controlled movement of a hollow cylinder contacting a box girder, a precise modeling approach essential for comprehensive analysis. The explicit dynamic analysis is used to allow the large deformation and non-linear material behavior on stiffened panels. The effect of initial imperfection, residual stress, and stiffener slenderness on the ultimate strength was studied. The ultimate strength of a box girder is determined by its maximum bending moment. According to the results, the ESL model achieves results similar to the traditional FEM model and experiment. The accuracy of ESL to FEM in the ultimate strength assessment varies depending on the magnitude of residual stress and initial imperfection applied to the stiffened panel.

Mathematical modeling and dynamic analysis of carbon fiber reinforced plastic shaft-disk-bearing system considering internal damping effect

The internal damping has significant influence on the dynamic characteristics of carbon fiber reinforced plastic drive shaft. In this paper, a dynamic model of the single-span multi-disk carbon fiber reinforced plastic shaft with elastic support is established based on layer-wise theory, damping specific capacity model, and Euler–Bernoulli beam theory. The model takes into account physical factors such as rigid body displacement caused by elastic support deformation, gyroscopic effect of concentrated mass disk, viscous internal damping of the carbon fiber reinforced plastic shaft, and lamination parameters. The model is used to analyze the natural angular frequency and critical instability speed of carbon fiber reinforced plastic shaft-disk-bearing system. The analysis results are compared with those obtained from the equivalent modulus beam theory, equivalent single layer theory, simplified homogenized beam theory, and quadrature element method. All the results demonstrate the accuracy of the proposed model, thereby validating its effectiveness. Furthermore, the model is utilized to investigate the influence of lamination parameters and mass ratio on the natural angular frequency and critical instability speed. This study provides valuable insights for the dynamics and stability design of the carbon fiber reinforced plastic shaft-disk-bearing system.