Classical Plate Research Articles

Reduced Order Modelling of Fully Coupled Electro‐Mechanical Systems Through Invariant Manifolds With Applications to Microstructures

ABSTRACTThis article presents the first application of the direct parametrisation method for invariant manifolds to a fully coupled multiphysics problem involving the nonlinear vibrations of deformable structures subjected to an electrostatic field. The formulation proposed is intended for model order reduction of electrostatically actuated resonating Micro‐Electro‐Mechanical Systems (MEMS). The continuous problem is first rewritten in a manner that can be directly handled by the parametrisation method, which relies upon automated asymptotic expansions. A new mixed fully Lagrangian formulation is thus proposed, which contains only explicit polynomial nonlinearities, which is then discretised in the framework of finite element procedures. Validation is performed on the classical parallel plate configuration, where different formulations using either the general framework or an approximation of the electrostatic field due to the geometric configuration selected are compared. Reduced‐order models along these formulations are also compared to full‐order simulations operated with a time integration approach. Numerical results show a remarkable performance both in terms of accuracy and the wealth of nonlinear effects that can be accounted for. In particular, the transition from hardening to softening behaviour of the primary resonance while increasing the constant voltage component of the electric actuation is recovered. Secondary resonances leading to superharmonic and parametric resonances are also investigated with the reduced‐order model.

Haar wavelet analysis of axisymmetric vibration of 2-D functionally graded porous thin annular plate resting on the Winkler foundation

Purpose In this paper, free axisymmetric vibration analysis of a two-directional functionally graded porous thin annular plate resting on the Winkler foundation is presented utilizing the classical plate theory (CPT). The mechanical properties are considered to be varying in the radial-thickness plane.Design/methodology/approach Based on the CPT, the governing differential equation of motion is derived. The highest-order derivative of displacement is approximated by Haar wavelets and successive lower-order derivatives are obtained by integration. The integration coefficients are calculated using boundary conditions. The fundamental frequency for clamped-clamped, clamped-simply supported, simply supported-clamped and simply supported-simply supported boundary conditions is obtained.Findings The effects of the porosity coefficient, the coefficient of radial variation, the exponent of power law, the foundation parameter, the aspect ratio and boundary conditions are investigated on fundamental frequency. A convergence study is conducted to validate the present analysis. The accuracy and reliability of the Haar wavelets are shown by comparing frequencies with those available in the literature. Three-dimensional mode shapes in the fundamental mode for all four boundary conditions are presented.Originality/value Based on the Haar wavelet method, a free axisymmetric vibration model of a porous thin annular plate is solved in which 2-D variation of mechanical properties is considered.

Free vibration analysis of a functionally graded porous nanoplate in a hygrothermal environment resting on an elastic foundation

This research investigates the free vibrational behavior of a functionally graded porous (FGP) nanoplate resting on an elastic Pasternak foundation in a hygrothermal environment. The nanoplate is modeled based on the nonlocal strain gradient theory (NSGT) and considering several plate theories including the CPT (classical plate theory), the FSDT (first-order shear deformation theory), and the TSDT (third-order shear deformation theory). Several patterns are investigated for the dispersion of pores, and the surface effects are incorporated to enhance the precision of the model. The governing equations and boundary conditions are derived via Hamilton's principle and an exact solution is provided via the Navier method. The impacts of several parameters on the natural frequencies are inspected such as length scale and nonlocal parameters, surface effects, porosity parameter, hygrothermal environment, and coefficients of the foundation. The results show that the impact of the porosity parameter on the natural frequencies of nanoplates is significantly dependent on the porosity distribution pattern. It is discovered that by increasing the porosity parameter from 0 to 0.6, the relative changes of natural frequencies vary from a decrease of 30 % to an increase of 6 %.

Experimental and numerical study of specific bending behavior of tufted multilayered reinforcements and effects of tufting density

The tufting technology enhances the delamination and impact resistance of composites. The presence of tufting yarns leads to a modification of the structure of the fabric, which affects the bending behavior of tufted reinforcements. Tufted reinforcements experience significant slippage during bending, challenging classical plate and shell theories. Therefore, the bending behavior of tufted reinforcement is investigated and a specific fibrous shell approach is proposed, assuming quasi-inextensibility of fibers and potential slippage between them. The influence of tufting yarns on the bending behavior is integrated into the constitutive model. The experimental results show that the bending stiffness of reinforcements can be increased by tufting density. The simulations exhibit a good agreement with experiments, demonstrating that the proposed approach can accurately capture not only the bending deflections of tufted reinforcements, but also the rotation of material direction indicators. Using the proposed numerical method, the prediction accuracy for deflection and rotation angle of tufted reinforcements reaches 90% in cantilever bending tests and 85% in three-point bending tests. These novel insights can deepen the understanding of the bending behavior of tufted reinforcements and be an asset of the developing numerical model for the forming simulation.

Plate buckling analysis in peridynamic framework

AbstractPeridynamics (PD) is a generalized continuum theory that is developed to account for long range internal force/moment interactions. PD equations of motion are of the integro‐differential type, and only several closed‐form analytical solutions are available for elementary structures. In this paper, governing equations for stability of Kirchhoff type (shear rigid) plates in the PD framework are derived. For several types of boundary conditions, closed form analytical solutions for the buckling loads are presented. The results are compared with the solutions according to the classical plate theory. A very good agreement between the non‐local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the developed approach.

Supra-pectineal quadrilateral buttress plating versus infra-pectineal plating in the management of quadrilateral plate fractures: A randomized controlled trial.

Management of quadrilateral plate fractures is technically demanding and requires specific fixation techniques. Infra-pectineal plating is the gold standard method of fixation. However, we recorded a high incidence of medial wall displacement and reoperations. Therefore, the aim of our study was to identify whether supra-pectineal quadrilateral buttress plating provides much more rigid fixation with a better functional and radiological outcome or not. The authors conducted this prospective, randomized control, single-blinded study at a level 1 single trauma centre. Between March 2022 and June 2023, 34 patients with quadrilateral plate fractures had anterior fixation, either via the anatomical QLP (17 cases) or infra-pectineal plating (17 cases) (Groups A and B, respectively). The radiological and clinical outcomes, as well as residual medial wall displacement, were the primary outcomes. The mean follow-up was 14.47months in group A and 15.24months in group B. In group A, the mean operative time (p = 0.02) was shorter, and the mean blood loss (p < 0.001) was significantly lower. However last follow-up showed no statistically significant differences as regards residual medial wall displacement (p = 1.0), final radiological (p = 0.86), and clinical outcomes (p = 1.0). Authors concluded that the anatomical QLP made it easier to reduce and fix acetabular fractures with a displaced medial wall. This was done by using multidirectional screws in the posterior column through its infra-pectineal extension and a strong screw purchase aimed at the posterior column through its supra-pectineal part. The two groups were similar in terms of final radiological and clinical outcomes, as well as residual medial wall displacement rates. However, the QLP had less morbidity than the classic infra-pectineal plating (shorter operation time and less blood loss).

Predicting initial microlayer thickness in nucleate boiling using Landau–Levich theory

A phenomenological model is proposed to estimate the initial thickness of the liquid microlayer forming beneath a vapour bubble growing on a solid surface upon nucleate boiling. The model employs an analogy between the microlayer formation and the classic plate withdrawal problem. It calculates the microlayer thickness by considering it as a Landau–Levich film, where the thickness is a function of the meniscus speed and radius of curvature. Given the nearly hemispherical shape of the bubble during the early growth stage when the microlayer is first deposited, we assume that the meniscus speed can be approximated by the bubble expansion rate, and estimate the meniscus curvature using the Rayleigh equations. Unlike previous theories that assume that the bubble radius growth is proportional to the square root of time, the proposed model does not rely on any specific law of growth for vapour bubbles. The model is validated for predicting the microlayer thickness in water and ethanol, showing good agreement with experimental measurements and empirical correlations. Subsequent analyses of the microlayer interface profile address inconsistent reports – some described a wedge-like shape, whereas others reported a slight outward curvature with decreasing thickness in the outer region. This discrepancy is attributed to a reduction in the expansion rate of the microlayer's outer edge, particularly when the bubble reaches its maximum width. Our model provides insights into microlayer dynamics, essential to boiling heat transfer, as the evaporative heat flux through the microlayer is very sensitive to its initial thickness.

Propagation of bending waves along the edge of a point-loaded piezoelectric plate on elastic foundation

An investigation on the propagation of bending waves along the edge of a piezoelectric thin plate resting on an elastic foundation (cf. Pasternak foundation) and under the action of a point-load is carried out in this paper. The displacement field of the plate is based on the classical plate theory. The Kelvin-Voigt type viscoelastic model is used to examine the viscous effect on the characteristics of the bending wave. The Moore-Gibson-Thompson (MGT) thermoelastic equation is solved for an external time variant harmonic heat source at the upper surface of the plate. The piezoelectric potential is obtained for an electrode-covered upper surface and an electrically shorted lower surface. The displacements of the propagating bending wave along the edge are examined under a time-harmonic point-load acting near the vicinity of the free edge of the plate. The well-known Green function technique and Fourier transform method are implemented to generate an analytical solution for the non-homogeneous boundary value problem. The examination of the derived explicit form of the dispersion relationship and the displacements of the bending edge wave under external loading are discussed and presented both analytically and graphically, and the results are summarized.

Dynamic Locked Plating Using Near-Cortical Over-Drilling versus Standard Locked Plating for Fixation of Lower Femur Fractures

Abstract Background Fracture Healing in distal femoral fractures is greatly influenced by fixation method. Classic compression plate requires creation of a rigid plate/bone construct which involves soft-tissue stripping and devitalization of the underlying bone, this may lead to inhibited callus formation. Hence, the concept of “Dynamic” Locked Plates has been proposed to decrease construct stiffness and improve callus formation. Aim of the Work To compare functional and radiological outcomes in lower femur fractures treated by dynamic locked plating (using near cortical over drilling) to that treated by standard locked plating. Patients and Methods Prospective study was conducted starting from May 2020 to January 2023 including 30 patients aged from 18 to 65 years old with distal femur fracture treated by dynamic locked plating with near cortex over drilling (15 patients) or standard locked plating (15 patients). We excluded patients with open fractures, periprosthetic fractures, floating knee injuries and poly-traumatized patients. The follow up period was 1 year. Results The current study demonstrated no statistically significant difference between both groups regarding time of union, knee range of motion, nonunion rate, or other specific complications. However, the DLP group was noted to have statistically significant higher callus scores at six weeks postoperative (p value = 0.015) indicating increased callus formation. Conclusion Dynamic locked plating seems to result in improved callus formation and bone healing without sacrificing fixation strength compared with traditional locked plating when used for fractures of the distal femur.

Influence of Transverse Normal Strain on Buckling and Vibration of Sandwich Plates

This study presents the buckling and free vibration behavior of simply supported square and rectangular sandwich plates. A third-order shear deformation theory considering both the transverse shear and normal deformations is used. The condition of zero transverse shear stresses on the top and bottom surfaces of the plate is satisfied. Hence, the present theory does not require the shear correction factor generally associated with the first-order shear deformation theory. Governing equations and boundary conditions are obtained using the principle of minimum potential energy. The closed-form solutions of simply supported sandwich plates are obtained. The effect of the side-to-thickness ratio and plate aspect ratio on buckling and free vibration responses of simply supported square and rectangular sandwich plates is examined. Parametric effects of face sheet thickness ratios and load factor upon the critical buckling load for uniaxial and biaxial buckling analysis of square and rectangular sandwich plates for different staking sequences are studied. All numerical solutions are obtained using MATLAB® programs. The results obtained from the present theory are compared with those from classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, and the exact three-dimensional elasticity solutions as applicable.

Analysis and measurement of vibration characteristics of a hollowing defect based on a laser self-mixing interferometer

Abstract To solve the problems with the existing methods for detecting hollowing defects, such as inconvenient operation, low efficiency and intense subjectivity, and to improve the efficiency of the acoustic-optic fusion method for detecting hollowing defects, in this paper the vibration characteristics of hollowing defects are measured and analyzed using a laser self-mixing interferometer. The ceramic tile above the hollowing defect is equivalent to a thin circular plate with peripheral fixed support. According to Kirchhoff’s classical circular plate theory and the circular plate displacement function based on the improved Fourier series, a theoretical model of a circular plate is established. By solving the characteristic equation, the theoretical modal parameters of hollowing defects are obtained. Subsequently, an experimental system based on a laser self-mixing interferometer is built, and modal experiments are carried out using the hammering method. The experimental modal parameters are obtained with a professional modal analysis software. Through comparative analysis between the theoretical and experimental modal parameters, the error of the natural frequency results is found to be tiny and the mode shapes are consistent. These results provide theoretical guidance for a practical non-destructive acoustic-optic fusion method for detecting hollowing defects.

Integrated analytical, experimental, and numerical study of the effect of hydro-static loading on the vibrational response of clamped circular plates

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between a water cavity of varying fluid cavity height and the vibration of a thin circular plate subjected to its hydro-static pressure. We extend the application of the strong modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solution of a thin clamped circular plate with uniform radial tension. Using a set of geometric and physical properties for the system, the resonance frequencies, response functions and non-dimensionalised added virtual mass incremental (NAVMI) factors are calculated and investigated as a function of hydro-static pressure and fluid cavity height, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation and to uncover experimental insights into the physical problem. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, resonance frequencies and NAVMI factors and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated, along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal compositions of the coupled system modes as a function of cavity height and hydro-pressure, the deformation of the coupled system mode shapes and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies and NAVMI factors. In totality, this study provides detailed modelling and prediction of the frequency response, resonance frequencies, added mass factors, modal contributions, and deformation of the coupled mode shapes, offering comprehensive insights with wide applicability.

Free vibration and transient response of initially compressed CNT-reinforced composite plates

This paper investigates the linear free vibration and nonlinear transient response of carbon nanotube (CNT)-reinforced composite plates subjected to pre-existent compressive load in thermal environments. CNTs are reinforced into matrix according to uniform and functionally graded distributions. The properties of constitutive materials are assumed to be temperature-dependent while the effective properties of nanocomposite are determined using an extended rule of mixture. Governing equations are established within the framework of classical plate theory incorporating von Kármán nonlinearity and initial geometrical imperfection. Analytical solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to obtain a time differential equation including both quadratic and cubic nonlinear terms. Fourth-order Runge–Kutta numerical integration scheme is employed to determine dynamical deflection-time response of nanocomposite plates. Numerical analyses are presented to consider the influences of CNT volume fraction, CNT distribution, initial compressive load, geometrical imperfection, elevated temperature and plate geometry on the natural frequencies and nonlinear dynamical response of nanocomposite plates. The study reveals that the natural frequency and dynamical deflection are strongly decreased and increased when initial compressive load is increased, respectively. Numerical results also find that the natural frequencies are enhanced and dynamical deflection is dropped as a result of increase in volume percentage of CNTs, respectively.

Dynamic analysis of laminated plate and ballastless track slab on Pasternak foundation under moving load based on different shear deformation theory

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Dynamic response of sandwich functionally graded nanoplate under thermal environments and elastic foundations using dynamic stiffness method

The present paper introduces the development of dynamic stiffness method for analyzing small-scale sandwich functionally graded nanoplates resting on elastic foundation in thermal environments. The mathematical formulation is based on classical plate theory in conjunction with nonlocal elasticity theory. The governing equation is derived using Hamilton’s principle. The dynamic stiffness matrix is obtained through the application of the Levy displacement approach and assembled to form the global stiffness matrix. The final matrix is solved for natural frequency of the plates using the Wittrick–Williams algorithm. The proposed methodology is validated against existing literature, demonstrating a strong agreement. Various parametric studies explore the effects of thermal environments, volume fraction index, sandwich configurations, elastic foundation characteristics, nonlocal parameter and boundary conditions. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of sandwich functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Nonlinear free vibration analysis of the rectangular conductive elastic plate in magnetic field based on homotopy perturbation method

AbstractThis article tries to investigate the nonlinear free vibrations of the rectangular conductive elastic plate in uniform magnetic fields under the classic plate theory considering nonlinear strain‐displacement. The formulation of the governing equations integrates the electromagnetic forces arising from the motion of the plate. The nonlinear motion equations are dimensionless, which takes the effect of in‐plane inertia into account. The equations are solved by the Galerkin method and homotopy perturbation method (HPM). The effectiveness of the solution is verified. According to the solutions by HPM, the effects of the initial conditions, length‐to‐thickness ratio, and magnetic induction intensity on the nonlinear free vibrations behavior of conductive elastic plates are discussed.

Static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates

The static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates (MGEKLPs), which incorporate two length-scale parameters related to strain gradient and rotation gradient effects, are comprehensively analyzed under various load forms and boundary conditions (BCs). The study of static stability employs static balance method and an improved energy method by introducing higher-order deformation gradients and corresponding energy terms. Utilizing the variational method, a sixth-order fundamental buckling differential equation for MGEKLPs under both transverse and in-plane loads is derived, serving as the foundation for the static balance method. The static stability analysis of MGEKLPs examines the combined effects of strain and rotation gradients on size-dependent critical buckling loads. Building on generalized strain energy with higher-order deformation energy, the energy method of classical elastic thin plate model is enhanced and applied to the static stability analysis of MGEKLPs. This approach enables the investigation of static stability without being constrained by the need to solve complex differential equations, making it applicable to various BCs and load scenarios. While static stability provides a description of stable state of an elastic system, dynamic stability offers a more scientific and rigorous analysis. The dynamic stability of simplified gradient elastic Kirchhoff–Love plates (SGEKLPs) with curved edges and different BCs is further investigated by combining the generalized strain energy with Lyapunov’s second stability method, presenting the dynamic stability criterion in the form of norms. A strict description of the dynamic stability of a SGEKLP over the entire time domain is provided for different supporting conditions, including case where all edges are supported and case with free edges. The analysis of size-dependent static and dynamic stabilities offers theoretical guidance for designing elastic thin plates with microstructures.

Evaluation of cutout factors with small and narrow fields using various dosimetry detectors in electron beam keloid radiotherapy.

The inherent problems in the existence of electron equilibrium and steep dose fall-off pose difficulties for small- and narrow-field dosimetry. To investigate the cutout factors for keloid electron radiotherapy using various dosimetry detectors for small and narrow fields. The measurements were performed in a solid water phantom with nine different cutout shapes. Five dosimetry detectors were used in the study: pinpoint 3D ionization chamber, Farmer chamber, semiflex chamber, Classic Markus parallel plate chamber, and EBT3 film. The results demonstrated good agreement between the semiflex and pinpoint chambers. Furthermore, there was no difference between the Farmer and pinpoint chambers for large cutouts. For the EBT3 film, half of the cases had differences greater than 1%, and the maximum discrepancy compared with the reference chamber was greater than 2% for the narrow field. The parallel plate, semiflex chamber and EBT3 film are suitable dosimeters that are comparable with pinpoint 3D chambers in small and narrow electron fields. Notably, a semiflex chamber could be an alternative option to a pinpoint 3D chamber for cutout widths≥3 cm. It is very important to perform patient-specific cutout factor calibration with an appropriate dosimeter for keloid radiotherapy.

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Integrated analytical, experimental, and numerical study of vibrational response in clamped circular plates with pressurised air cavities

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between the sound field produced within a pressurised air-filled back cavity and the vibration of a thin clamped circular plate subjected to its pressure. Previous investigations into the interaction of sound fields with vibrating boundaries have primarily been restricted to non-pressurised cavities. We extend the application of the weak modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solutions for a pressurised cylindrical sound field and a thin clamped circular plate with uniform radial tension. We describe the vibration response of the coupled system in terms of the uncoupled cavity and plate modes. Using a set of geometric and physical properties for the system, the resonance frequencies and response functions are calculated and the coupling between the sound field and the plate is investigated as a function of cavity pressure, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, mode shapes, resonance frequencies and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal contributions as a function of cavity pressure and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies.