Form Of Plates Research Articles

Use of Ni/CNT Particles as Additive Fillers in Ebonite Bipolar Plates for Proton-Exchange Membrane Fuel Cells.

Composite-based bipolar plates are a chance to substitute graphite and metal bipolar plates because of their corrosion resistance and better chemical and mechanical properties, but they still have various electrical conductivity properties. One of the problems is the loading of filler that depends on matrix types which will affect the electrical conductivity of bipolar plates; ebonite has potential as a matrix in composite bipolar plates because it is obtained from elastomer or rubber. In this work, nickel and carbon nanotubes (CNT) that have a high electrical conductivity will be investigated as additive fillers on graphite particles to enhance the electrical conductivity of ebonite bipolar plates. The formulation and characterization of ebonite bipolar plates with graphite and graphite Ni/CNT and their various contents are the main objectives of this research. Characterization by scanning electron microscopy (SEM) for identification and morphology of compounds and ebonite bipolar plates and Raman spectroscopy for identification of the type carbon in Ni/CNT was performed. Some tests such as bending/flexural tests, corrosion tests, and resistance testing for interfacial contact resistance were conducted to study the properties and optimum of the composite materials. In this research, Ni/CNT particles were added as additive fillers with graphite to enhance the electrical conductivity of fillers in ebonite bipolar plates of proton-exchange membrane fuel cells and their impacts were studied. By through-plane testing, graphite fillers were added in ebonite bipolar plates with 65-75% w/w content, achieving electrical conductivity values from 22.3 to 34 S/cm. This is still below the technical target set by the US DOE for composite bipolar plates. But by adding 30% Ni/CNT filler contents in ebonite bipolar plates at various filler contents from 65% to 75% w/w, one can achieve electrical conductivities from 104.35 to 165.52 S/cm. Only 65% w/w filler with 30% Ni/CNT can meet the technical targets such as a bending/flexural test value of 25.58 N/mm2, a corrosion test value of 0.894 μA/cm2 (I corr), and an interfacial contact resistance value of 3.09 mΩ cm2. Further improvements are needed based on fuel cell applications, as indicated by some additional data that did not meet technical targets.

Semi-analytical formulation to predict the vibroacoustic response of a fluid-loaded plate with ABH stiffeners

Stiffened structures are widely used in aeronautics, marine and rail industries. When stiffeners are integrated into host structures, so-called Bloch-Floquet waves are generated due to interactions between the host’s flexural waves and the stiffeners’ flexural and torsional waves. It is reported in the literature that these waves are often the source of undesirable noise and vibrations when the stiffened structure is excited by a force. To mitigate unwanted noise and vibrations from the stiffened structures, this study proposes to replace common rectangular stiffeners with acoustic black hole (ABH) stiffeners. To do this, a semi-analytical model is initially developed in the wavenumber domain to predict the forced vibroacoustic response of a 2D fluid-loaded infinite plate with stiffeners on one side. In the proposed model, the stiffeners are characterised by their translational and rotational dynamic stiffnesses which can be estimated by a finite element method (FEM). These dynamic stiffnesses are then coupled with the analytical formulation of the fluid-loaded plate to obtain the expressions of the spectral displacement and radiated pressure. Comparisons of the results in terms of the plate’s mean quadratic velocity and radiated sound power for the rectangular and ABH stiffeners show that by using the ABH stiffeners instead of the conventional stiffeners, one can significantly reduce the vibroacoustic response of light/heavy fluid-loaded plates.

Substructure-based topology optimization design method for passive constrained damping structures

Abstract This work presents a generalized substructure-based topology optimization method for passive constrained layer damping (PCLD) structures. Here, the model of PCLD structure is obtained by the Kirchhoff–Love thin plate formulation, and the whole structure is assumed to be composed of substructures with different yet connected scales and artificial lattice geometry features. Each substructure is condensed into a super-element to obtain the associated density-related matrices under the different geometry feature parameters, and the surrogate model for the stiffness and mass matrix of PCLD substructures with different densities has been particularly built. Using cubic spline interpolation, the derivatives of super-element matrices to the associated densities can be evaluated efficiently and accurately. The modal loss factor is defined as objective functions and topology optimization for the PCLD structures is formulated based on the model for PCLD plates that are described by combining the condensed substructures. Numerical examples under two lattice patterns of substructures and their corresponding physical tests show that the correctness and superiority of this substructure-based topology optimization approach for PCLD plates are verified.

Vibration analysis of functionally graded epoxy/graphene composite plates using the Boundary Element Method and new micromechanical model

This paper studies the free vibration and harmonic response of Functionally Graded Graphene/Epoxy composite plates, considering the First Order Shear Deformation Plate Theory. The weight fraction of graphene variation along the thickness direction with graphene homogeneous dispersed in a polymer matrix. The effective Young Modulus of the plate is predicted by a new micro-mechanical model. The model includes the aspect ratio and weight fraction of nanoplatelets embedded into the matrix. The modal and harmonic of Graphene/Epoxy plates are obtained by the Boundary Element Method formulation for composite plates. The Young modulus calculated from the proposed micro-mechanical model highly agrees with those obtained from experimental results reported in the literature. Results demonstrate a high correlation of the micromechanical model with experimental results and other analytical models. Graphene weight fraction and ratio aspect increase the effective Young modulus of the composite. Boundary Element results show high agreement with Finite Element solutions. Numerical results for modal and harmonic analysis show concentrating graphene near the top and bottom surfaces of the plate is the most effective way to reinforce the plate for increased natural frequencies. The results demonstrate the BEM formulation and micromechanical model can be used as reliable engineering tools for the vibrational analysis of functionally grade graphene composite plates.

An SPH formulation for general plate and shell structures with finite deformation and large rotation

In this paper, we propose a reduced-dimensional smoothed particle hydrodynamics (SPH) formulation for quasi-static and dynamic analyses of plate and shell structures undergoing finite deformation and large rotation. By exploiting Uflyand–Mindlin plate theory, the present surface-particle formulation is able to resolve the thin structures by using only one layer of particles at the mid-surface. To resolve the geometric non-linearity and capture finite deformation and large rotation, two reduced-dimensional linear-reproducing correction matrices are introduced, and weighted non-singularity conversions between the rotation angle and pseudo normal are formulated. A new non-isotropic Kelvin-Voigt damping is proposed especially for the both thin and moderately thick plate and shell structures to increase the numerical stability. In addition, a shear-scaled momentum-conserving hourglass control algorithm with an adaptive limiter is introduced to suppress the mismatches between the particle position and pseudo normal and those estimated with the deformation gradient. A comprehensive set of test problems, for which the analytical or numerical results from literature or those of the volume-particle SPH model are available for quantitative and qualitative comparison, are examined to demonstrate the accuracy and stability of the present method.

Effects of a nonlocal microstructure on peeling of thin films

In this work, starting from an approach previously proposed by the Authors, we put forward an extension to the large deformation regime of the dimensionally-reduced formulation for peridynamic thin plates, including both hyperelasticity and fracture. In particular, the model, validated against numerical simulations, addresses the problem of the peeling in nonlocal thin films, which when attached to a soft substrate highlights how nonlocality of the peeled-off layer might greatly influence the whole structural response and induce some unforeseen mechanical behaviours that could be useful for engineering applications. Through a key benchmark example, we in fact demonstrate that de-localization of damage and less destructive failure modes take place, these effects suggesting the possibility of ad hoc conceiving specific networks of nonlocal interactions between material particles, corresponding to lattice-equivalent structure of the nonlocal model treated, of interest in designing new material systems and interfaces with enhanced toughness and adhesive properties.

A T-splines-oriented isogeometric topology optimization for plate and shell structures with arbitrary geometries using Bézier extraction

Recently, the non-uniform rational B-splines (NURBS) have been considerably employed in modeling plate and shell structures, or developing the related size, shape or topology optimization methods for their design. However, the NURBS with the tensor product feature strongly hinders the effectiveness of the optimization on complex structures. The primary intention of the current research is to propose a new T-splines-oriented Isogeometric Topology Optimization (T-ITO) method and then address the critical design problem of plate and shell structures with arbitrary shapes in practical applications. Firstly, the Bézier extraction is utilized in the T-splines to map the geometrical model into a family of Bézier elements, each of which is presented by a local Density Distribution Function (DDF) for elementary topology. A global DDF is constructed by an assembly of all local DDFs with a natural connection to the present structural topology. Secondly, the T-splines-based IsoGeometric Analysis (T-IGA) formulation with the Bézier extraction for complex plate and shell structures is developed using the Kirchhoff-Love theory, where a universal numerical implementation framework is constructed, including the Rhino, MATLAB, export modules with all geometric information and import modules for the analysis. Thirdly, a mathematical formulation for plate and shell structures with arbitrary geometries is developed using the T-ITO method to improve structural loading-capability, and the sensitivity analysis with respect to design variables is rigorously derived in detail. Finally, several numerical examples of plate and shell structures with both regular and elaborate geometries are tested to demonstrate the validity, effectiveness, superiorities and indispensability of the T-ITO method.

Carrera Unified Formulation (CUF) for the composite plates and shells of revolution. Layer-wise models

Here, higher order layer-wise models of elastic composite multilayer plates and shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the thickness coordinates. Following the Unified Carrera Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order rectangular plate and cylindrical shell supported on the edges under sinusoidal loading, are considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric conical, spherical, elliptical and catenoidal shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order composite laminate shells.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Investigations on adapted interpolation orders for a mixed isogeometric plate formulation

AbstractIn order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non‐matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner‐Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non‐uniform rational B‐splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero‐energy modes.

Effect of grading pattern on free vibration analysis of FGM plates carrying concentrated and distributed mass

The present article aims to develop a finite element formulation for FGM plates carrying concentrated and distributed mass. First-order shear deformation theory with consideration of physical neutral surface has been employed. Nine noded isoparametric element with five degrees of freedom at each node is used for this finite element analysis. To understand the effect of grading pattern under overhead mass carrying conditions, three types of FGMs have been analyzed. The effect of non-dimensional parameters such as aspect ratio, thickness ratio, power index, the intensity of mass etc. are extensively studied. It will bring new insights to the field of FGM structures.

Extended plane wave expansion formulation for viscoelastic phononic thin plates

The extended plane wave expansion (EPWE) formulation is derived to obtain the complex band structure of flexural waves in viscoelastic thin phononic crystal plates considering the Kirchhoff–Love plate theory. The presented formulation yields the evanescent behavior of flexural waves in periodic thin plates considering viscoelastic effects. The viscosity is modeled by the standard linear solid model (SLSM), typically used to closely model the behavior of polymers. It is observed that the viscoelasticity influences significantly both the propagating and evanescent Bloch modes. The highest wave attenuation of the viscoelastic phononic thin plate is found around a unit cell filling fraction of 0.37 for higher frequencies considering the least attenuated wave mode. This EPWE formulation broadens the suitable methods to handle evanescent flexural waves in 2-D thin periodic plate systems considering the effects of viscoelasticity on wave attenuation.

Simulation of linear elastic structural elements using the Petrov–Galerkin finite element method

AbstractIn this contribution, it is demonstrated that the mesh sensitivity of linear elastic Reissner–Mindlin finite‐element plate formulations can be significantly reduced by using a Petrov–Galerkin‐based approach. In contrast to the usual Bubnov–Galerkin method, Petrov–Galerkin methods are generally characterized by the fact that the test function and the trial function are approximated using different shape functions. To provide an overview, established Petrov–Galerkin methods for 2D solid elements, which have been shown to reduce mesh sensitivity, are reviewed first. It is then investigated whether a suitable Petrov– Galerkin plate formulation can be developed. In this context, it is demonstrated that a full Petrov–Galerkin method leads to problems in the treatment of transverse shear locking. However, the proposed partial Petrov–Galerkin method shows the desired mesh‐insensitive behavior.

FFT-based multiscale scheme for homogenisation of heterogeneous plates including damage and failure

This paper introduces a new approach that combines FFT-based multiscale homogenisation with the Lippman–Schwinger equation to efficiently and accurately analyse plate structures with periodic micro-structures. For the first time, the plate periodic Lippman–Schwinger equation is derived to address the auxiliary cell problem. The proposed method comprises two key components: (i) a 3D plate formulation using a novel finite prism approach, and (ii) solving the Lippman–Schwinger equations by means of Green’s functions. Solutions for both the classical and first-order plate theories are derived and implemented for linear and nonlinear problems, completed with an open-source code provided.The efficiency and accuracy of the proposed method are assessed through several case studies, including complex woven composites. It is demonstrated that proposed method achieves a comparable level of accuracy to 3D solid problems while significantly reducing the required computing effort (i.e., milliseconds in computing time as opposed to hours required for 3D solid models). Moreover, nonlinear progressive damage problem involving an actual plate woven composite model is investigated. The results obtained show good agreement with experimental measurements for the progressive damage in plain woven composite, further highlighting the effectiveness of the proposed method.

Embedded boundary conditions for shear-deformable plate bending

An efficient procedure for embedding kinematic boundary conditions of shear-deformable plate bending is based on stabilized variational formulations, obtained by Nitsche’s approach for enforcing surface constraints. Effective use of the plate thickness as scaling in the rotational constraint terms leads to a uniform problem statement with a single stabilization parameter. An alternative form of the plate kinematics precludes the vulnerability of computation based on the standard Reissner–Mindlin formulation to shear locking, yet raises other challenges in the form of higher-order regularity and unconventional boundary conditions. The embedded boundary Nitsche approach provides a framework that easily accommodates these features. The accuracy of the approach for both plate formulations is verified by representative computations with bicubic B-splines, exhibiting optimal rates of convergence, and robust performance with respect to values of the parameters.

Shear deformable plate by SGBEM: Indirect “progenitor matrix” approach

The Mindlin-Reissner's formulation of shear deformable plate is addressed through the Symmetric Galerkin Boundary Element Method (SGBEM). The thick plate's elastic problem is resolved and fundamental solutions matrix is obtained. The application of Betti's theorem in the unlimited domain allows obtaining the Somigliana's Identities (S.I.s). The solving system is obtained by an indirect approach, i.e., through the “progenitor matrix”. The “progenitor matrix” allows simulating all possible plate's load conditions and is the starting point for an automatic calculation code. The coefficients are calculated through the theory of distributions, using double integrals without resorting to regularization techniques but by exploiting the expansion in power series of the Bessel functions present in the fundamental solutions.Particular attention is paid to the presence of domain loads, whose domain integral is transformed into a boundary one using the Radial Integral Method technique (RIM).These strategies lead to a robust procedure that allows obtaining good results even in the presence of boundary sparse discretization. This is demonstrated by the results of the examples carried out which, compared with the analytical solutions present in the literature, show a very good convergence.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

Deep-Learning-Based Acoustic Metamaterial Design for Attenuating Structure-Borne Noise in Auditory Frequency Bands

In engineering acoustics, the propagation of elastic flexural waves in plate and shell structures is a common transmission path of vibrations and structure-borne noises. Phononic metamaterials with a frequency band gap can effectively block elastic waves in certain frequency ranges, but often require a tedious trial-and-error design process. In recent years, deep neural networks (DNNs) have shown competence in solving various inverse problems. This study proposes a deep-learning-based workflow for phononic plate metamaterial design. The Mindlin plate formulation was used to expedite the forward calculations, and the neural network was trained for inverse design. We showed that, with only 360 sets of data for training and testing, the neural network attained a 2% error in achieving the target band gap, by optimizing five design parameters. The designed metamaterial plate showed a -1 dB/mm omnidirectional attenuation for flexural waves around 3 kHz.

Higher-order models for the passive damping analysis of variable-angle-tow composite plates

In this paper, the damped free-vibration and frequency response analysis of variable-angle-tow composite plates embedding viscoelastic layers with frequency-dependent properties are performed. The governing equations are derived from the Principle of Virtual Displacements and higher-order Layer-Wise models are used for the unknown variables description in the thickness direction, furthermore a nine-node finite plate element is employed to solve them. The plate formulation presented in this paper is more efficient with respect to three-dimensional approaches, typically used for this kind of structures, and it is more accurate than existing reduced order methods thanks to the use of the mixed interpolation of tensorial components technique, employed to solve the problem of the shear and membrane locking phenomena. Different multilayered structures have been considered, laminated with unidirectional cross-ply layers or with curvilinear fibers path ones. The use of viscoelastic soft sheets permits to have a passive damping of the structural vibration. The frequency dependent properties of the viscoelastic materials are described through the use of a Kelvin-Voigt model. Numerical solutions are presented for the analysis of variable-angle-tow composites with different boundary-conditions and various lamination schemes.

Vibrations and bending of thin laminated square plates with holes in gradient elasticity: A finite element solution

Nonlocal elasticity is becoming more and more popular due to the introduction of advanced constituents and the need of a peculiar approach able to capture size-dependent effects. Strain gradient theory falls within this aim, and it is included into the thin plate formulation in the current paper. Following the procedure developed in Bacciocchi et al. (2020, 2021), a finite element model based on higher-order Hermite interpolating functions is used here to evaluate the mechanical behavior of laminated composite square plates with holes. These peculiar configurations cannot be solved analytically.