Kirchhoff Plate Theory Research Articles

Analytical solution of dual-phase-lagging generalized thermoelastic damping in Levinson micro/nano rectangular plates

Analysis of dual-phase-lag (DPL) thermoelastic damping (TED) in Levinson micro/nano plate resonators is first carried out by using the complex frequency approach. Based on the Levinson plate theory and the DPL heat conduction model, two-way coupled equations of motion and heat conduction equation of the resonators are established, respectively. A hyperbolic shear stress shape function is selected to estimate the variation of the strain field along the plate thickness. Analytical solution of the temperature field with fluctuating characteristics is obtained from the quasi-1D heat conduction equation. The thermal bending moment is expressed in terms of the deflection in the structural vibration equation and an analytical solution of the complex frequency and the TED is derived for the simply supported rectangular micro/nano plates by using the complex frequency approach. Influence of the transverse shear deformation as well as the DPL thermal relaxation on the TED varying with the plate geometry, vibration modes and isothermal resonance frequency is examined quantitively in detail through plentiful numerical results. As a result, the existing theoretical study of DPL TED of micro/nanoplate resonators in the framework of the Kirchhoff plate theory was extended to the framework of the Levinson plate theory.

Theory of thin elastic plate: history and current state of the problem

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

A unified nonlinear dynamic model for bolted flange joint disk-drum structures under different interface states: Theory and experiment

Bolted flange joint disk-drum structures (BFJDDSs) are key components in aero-engines, however, bolt looseness inevitably occurs due to extreme service conditions. This can cause the bolted joint interface to exhibit complex contact behaviors, which significantly complicate the vibration characteristics of BFJDDSs. Existing dynamic models for BFJDDSs have not considered the effect of bolt looseness (slip and separation), so their vibration behaviors are not well understood. In this study, a unified nonlinear dynamic model for BFJDDSs considering different interface states (stick, slip, and separation) is proposed, which is effective under both bolt looseness and non-looseness (stick) conditions. The bi-linear hysteretic model combined with the piecewise linear model is used to simultaneously consider different interface states. The Kirchhoff plate theory, the Sanders’ shell theory, and the Euler-Bernoulli beam theory are used to derive the energy functions of the disk, drum, and flange, respectively. Then, the Lagrange equations are employed to derive the governing equations of BFJDDSs. Modal and nonlinear forced vibration experiments are carried out on a BFJDDS to prove the correctness of the proposed mathematical model. Research results show that the proposed mathematical model is able to achieve good predictions of nonlinear dynamic properties of BFJDDSs under both bolt looseness and non-looseness conditions. This model is also able to well predict the jumping phenomenon observed in the experiment.

Analysis of the electromechanical responses of sandwich circular nano-plate based on flexoelectric nano-ultrasonic transducer

Flexoelectric nano-ultrasonic transducer is an important development direction of ultra-high frequency ultrasonic transducers. The key to its development lies in addressing the electromechanical coupling dynamics of a flexoelectric nano-element, that is, sandwich circular nano-plate. In this paper, a novel dynamic theoretical model of sandwich circular nano-plate is developed based on the transversely isotropic flexoelectricity couple stress and Kirchhoff plate theories. Based on the novel model, this study demonstrates the influence of flexoelectricity effect and size effect on its electromechanical responses of the core component in the nano-ultrasonic transducer. Simultaneously, using finite element software, the paper presents the first six mode shapes of the structure and reveals the mode shape most conducive to the energy conversion of the transducer. The main objective of this study is to provide a theoretical foundation for the development of nano-ultrasonic transducers based on flexoelectric materials. The novel model can also be applied in various energy conversion devices, including flexoelectric actuator and flexoelectric energy harvesters.

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

Stability analysis of thin cylindrical shells under pure and three-point bending

Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler–Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.

Active vibration control effect of 3D printed cruciform honeycomb laminates based on fiber-reinforced shape memory polymer

Honeycomb structures offer the benefits of being lightweight and having high out-of-face compressive stiffness, making them a popular choice for aerostructures. The present study delineates the design and active vibration control effect for fiber-reinforced cruciform honeycomb laminates (CHLs) characterized by variable parameters. Employing a continuous fiber 3D printer, carbon fiber filament and shape memory polymers (SMPs) were utilized in the fabrication of the specimens. Differential equations of motion for CHLs were deduced from Kirchhoff plate theory. To ascertain their frequency response, these laminates were subjected to a sinusoidal swept frequency excitation. Subsequently, the classical proportional integral derivative (PID) program was employed to scrutinize the efficacy of closed-loop active vibration control on CHLs. The investigation extended to analyzing the impact of active vibration control across laminates with disparate parameters, including an examination of the convergence speed during vibration control procedures. A comparative analysis of simulated versus experimental data revealed a substantial consonance between the two, thereby corroborating the accuracy of the experimental findings. This study offers crucial insights and reference value for the utilization of CHLs in the aerospace industry.

Overcoming the cohesive zone limit in the modelling of composites delamination with TUBA cohesive elements

The wide adoption of composite structures in the aerospace industry requires reliable numerical methods to account for the effects of various damage mechanisms, including delamination. Cohesive elements are a versatile and physically representative way of modelling delamination. However, using their standard form which conforms to solid substrate elements, multiple elements are required in the narrow cohesive zone, thereby requiring an excessively fine mesh and hindering the applicability in practical scenarios. The present work focuses on the implementation and testing of triangular thin plate substrate elements and compatible cohesive elements, which satisfy C1-continuity in the domain. The improved regularity meets the continuity requirement coming from the Kirchhoff Plate Theory and the triangular shape allows for conformity to complex geometries. The overall model is validated for mode I delamination, the case with the smallest cohesive zone. Very accurate predictions of the limit load and crack propagation phase are achieved, using elements as large as 11 times the cohesive zone.

Flexoelectric and surface effects on bending deformation and vibration of piezoelectric nanolaminates: Analytical solutions

In this paper, a model of double-layer rectangular piezoelectric nanolaminates with different upper and lower surface properties is established. The bending and vibration behavior of the piezoelectric nanolaminates are evaluated and analyzed, taking into consideration the flexoelectric and surface effects. Utilizing the surface piezoelectric model, flexoelectric theory, and Kirchhoff plate theory, this study provides analytical solutions for the bending deflection and natural frequency of rectangular piezoelectric nanolamination materials under various boundary conditions. The results indicate that the effects of flexoelectric and surface influences on piezoelectric nanolaminates are associated with residual surface stress and flexoelectric coefficients. These effects can enhance the bending stiffness and decrease the natural frequency of the piezoelectric nanolaminates. Furthermore, altering the residual stress on the upper and lower surfaces of the laminates separately will result in different trends in bending deflection, while causing a consistent trend in natural frequency change. When the electric potential direction of piezoelectric nanolaminates is different, the bending deflection of piezoelectric nanolaminates will change accordingly. The natural frequency of piezoelectric nanolaminates is determined by the absolute value of the flexoelectric coefficients, and is independent of their positive or negative values. Both the surface effect and the flexoelectric effect are influenced by size-dependent behaviors. When the thickness of the laminates is sufficiently large, both the surface effect and flexoelectric effect can be disregarded. This paper's research methods and results provide theoretical models and analytical methods for the microstructure design, multi-physical field characterization, and bending deformation behavior of intelligent components containing rectangular piezoelectric nanolaminates.

Vibration of bolted composite cylindrical-cylindrical flanged shells considering contact characteristics

A semi-analytical model for bolted composite cylindrical-cylindrical flanged shells (BCCFS) considering the influence of the flange and the connection characteristics of the bolted flange joint is established. In the model, flanges are treated as annular plate structures and modeled using Kirchhoff plate theory. To simulate the connectivity performance of the bolted flange joint, a surface-spring model considering the pressure distribution characteristics is developed and applied to the model of the bolted structure. A method based on fractal contact theory is proposed to identify stiffness-related parameters of the surface-spring model, which avoids additional tensile experiments of the bolted joint. Sufficient modal tests are performed to validate the accuracy of the model. Furthermore, the effects of flange thickness, length ratio of two shells, and quantity of bolts on the free vibration characteristics of the structure are studied. The frequency veering phenomenon with the variation of size parameters is found and analyzed in depth.

Closed-form effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates

This study derives closed-form expressions for the effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates. The derivation utilizes the homogenization method based on equivalent strain energy and the Kirchhoff plate theory. Rigorous accuracy assessments of the closed forms are conducted through comparisons with the existing literature and direct finite element analysis. A detailed parametric study is also conducted using the obtained closed forms to provide insights into the relationships between the effective elastic properties and the unit-cell shape. Lastly, general forms for some effective properties applicable to out-of-plane and in-plane situations are presented for hexagonal lattices.

Theoretical investigation on scattering of ultrasonic Lamb waves at damaged area in quasi-isotropic CFRP laminates based on Mindlin plate theory

ABSTRACT With the wide application of carbon fibre reinforced plastic (CFRP) structure, a structural health monitoring (SHM) system that can easily detect damages in CFRP structures is urgently needed. Since it is possible to detect damages effectively in a large area in CFRP by using the scattering of Lamb waves, in this research, we investigated the scattering of Lamb waves at an impact damaged area in quasi-isotropic (QI) CFRP laminate. By modelling the impact damage as a cylinder area with a uniform stiffness degradation, the scattering waves are calculated based on Mindlin plate theory, which is more precise than Kirchhoff plate theory. It is shown that in the low-frequency region, the calculation results based on Kirchhoff and Mindlin plate theory was similar, but in a higher frequency range, the results differed. The calculation results were compared with the simulation results, suggesting the validity of Mindlin plate theory within a practical frequency range. Moreover, we validated the analytical and simulation results experimentally using a QI CFRP laminate with a hole. We also conducted some parameter study based on the analytical calculation to clarify influential stiffness components. Since scattering behaviour of Lamb waves can be calculated in a fast and easy way using this method, this work is useful for the successful development of a scattering-based SHM system.

Nonlinear deformations of size-dependent porous functionally graded plates in a temperature field

In this paper, the Variational Iteration Method (VIM) or Extended Kantorovich Method (EKM) is formulated for the first time for flexible porous functionally graded (PFGM) plates with different boundary conditions and geometric nonlinearity according to the theory of Theodore von Karman. Its accuracy and efficiency are demonstrated. The plate is subjected to a uniform transversal load and temperature field. The displacement field of the plate is approximated based on the classical plate theory (CTP) or Kirchhoff's plate theory. The governing equations are derived using Hamilton's principle. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. The material properties vary with thickness and are temperature dependent. Four porosity distribution patterns are considered in this study. Several examples are solved to demonstrate the proposed algorithm. The results obtained are compared with solutions obtained by the Bubnov-Galerkin Method (BGM) in higher approximations, the Finite Difference Method (FDM) of second order accuracy, as well as with results obtained by the Finite Element Method (FEM) of other authors. The results include an analysis of the effect of size dependent parameters, porosity type pattern, porosity index, functionally graded index, temperature field and different types of boundary conditions on the stress–strain state and bending deflection of plates.

Micromachined piezoelectric sensor with radial polarization for enhancing underwater acoustic measurement

This work presents an underwater acoustic sensor featuring a radial polarized piezoelectric diaphragm, which aims to work in d33 mode to enhance its performance. For the sake of accomplishing the in-plane polarization, interdigital electrodes based on semi-circular ring patterns are designed on both sides and aligned along the thickness direction of the diaphragm. Effects of electrode parameters on the polarization orientation are numerically analyzed by the conversion relationship between different coordinates. Using the Kirchhoff plate theory and Rayleigh-Ritz method, a mathematical model is developed to investigate the dynamic properties of the sensor. The results indicate that the output voltages increase with the increase of the electrode inner radius, width, and spacing but decrease with the increase of the diaphragm thickness and radius. Sensor prototypes are fabricated by the microelectromechanical system (MEMS) technology in a clean room and characterized by impedance spectrums. An underwater testing system is established to conduct the experimental verification, whose results are in good accord with those of mathematical modeling. The sensor output voltage has fine linearity with the underwater acoustic pressure and achieves a flat frequency response. Moreover, the sensor has a sensitivity of −172.7 dB (Ref. 1 V/μPa) superior to the reported devices in the same categories. This work provides significant guidance for modeling radial field piezoelectric diaphragms and designing more efficient piezoelectric microsensors, which have a fine application prospect in underwater acoustic measurement.

Free vibrations of axially loaded thin-walled shaft-disk rotors subjected to non-uniform temperature field

Research on vibrations of shaft-disk rotors in non-uniform thermal field is essential for the design of rotating machinery. In this paper, the dynamic model of axially loaded thin-walled shaft-disk rotors in axial non-uniform temperature field is developed, considering three-dimensional displacement fields of the shaft and disk. This model introduces the temperature-dependent properties of materials, and gyroscopic and centrifugal effects. Based on the Euler-Bernoulli beam theory and the Kirchhoff plate theory, Lagrange's equation is used to derive the differential equations of motion. By comparing with the results of the existing reference and finite element results, the accuracy of the present model is verified. The effects of the axial displacement of shaft, temperature-dependent material property, temperature difference, temperature distribution, disk position and axial preload on the free vibration of shaft-disk rotors are discussed. Results show that the present model can effectively predict free vibrations of shaft-disk rotors in axial non-uniform temperature field. The effect of thermal field on vibrations of shaft-disk rotors is affected by the rotational speed and disk location as well.

Free vibration analysis of corrugated plate based on improved Fourier series method

Corrugated plates are widely used in ships, underwater vehicles and other engineering fields due to their good bending stiffness and strength characteristics. The properties of corrugated plates are determined by their corrugated shape, wavelength, and wave height, which affect each direction differently. In order to model these properties quickly and accurately, this paper employs an improved Fourier series method. The method is based on the Kirchhoff plate theory and equivalent stiffness method, expanding the displacement of the corrugated plate using Fourier series. By applying the energy principle, the free vibration characteristics of the corrugated plate can be determined. This approach allows for efficient and comprehensive multi-parameter control modeling of corrugated plates. The influence of structural form, material parameters and convergence on the vibration characteristics of corrugated plate is analyzed. This paper lays a foundation for the subsequent dynamic study of sandwich corrugated plate structure.

Mistuning identification and model updating of a blisk with piezoelectric excitation components

Integrally bladed disks (blisks) are extensively used in military and commercial aircraft engines. There are inevitable deviations between blades called mistuning, leading to the vibration localization of blisks, which potentially results in high cycle fatigue (HCF). Furthermore, the dynamic performance is sensitive to mistuning patterns. Hence, to predict the vibration response of mistuned blisks accurately, the mistuning pattern needs to be identified experimentally and verified under traveling wave excitation (TWE). Piezoelectric TWE is a promising testing technique for blisk dynamics in a non-rotating state, offering advantages such as high excitation force, wide bandwidth, and a simple setup. However, piezoelectric materials are generally arranged on blades forming an electromechanically coupled structure, which challenges the existing mistuning identification methods. In this paper, an integral mistuning identification and model updating method with traveling wave excitation verification is developed. The detuning strategy is used to split natural frequencies of blades, whereupon the response is measured blade by blade. Measured response functions of isolated blades can be retrieved by data reconstruction. Then, the mistuning patterns of elastic modulus and damping ratio are identified by Kirchhoff-plate theory and half-power bandwidth method, respectively. Experimental studies are carried out to validate the method. A dummy blisk with piezoelectric patches is designed and the dynamic properties are calculated. Blade-by-blade measurement and TWE test are conducted. Results show that natural frequencies of the updated model are in excellent agreement with the measured ones. Under TWE, the deviations of natural frequencies for all blades are less than 0.25%. The response deviation of the updated model is decreased by 89.4% on average compared with the original model and the average response deviation of the updated model is 7.4%.

An effective model for bolted flange joints and its application in vibrations of bolted flange joint multiple-plate structures: Theory with experiment verification

Bolted flange joints are widely used in various engineering structures to combine two or more substructures together. However, the understanding of mechanical mechanism of bolted flange joints is still insufficient due to the lack of efficient mathematical models. This paper proposes an effective mathematical model for bolted flange joints, which is then employed to conduct vibration analysis on bolted flange joint multiple-plate structures. During the modeling process, the flange is modeled by the Kirchhoff plate theory, and the bolted joints are modeled by the two-dimensional face constraint in bolted joint affected region (BJAR). The artificial spring technique is adopted to simulate the face constraint. The friction contact model and equivalent linearization technique are adopted to analyze the stick and slip states of bolted joints. By adopting the Chebyshev orthogonal polynomials as admissible functions, governing equations are derived. Then, the Rayleigh-Ritz method is applied to study the free vibration characteristics, and the Newmark-beta method is employed to study the vibration response under base excitation. The experimental studies are performed on the bolted flange joint two-plate structure (BFJTPS) to illustrate the effectiveness of the proposed theoretical model. The results indicate that the proposed mathematical model can well predict the vibration characteristics of multiple-plate structures under both bolt looseness and no-looseness conditions. The soft nonlinearity phenomenon under bolt looseness condition can be successfully revealed by the model.

Semi-Analytical Prediction of Ground Surface Heave Induced by Shield Tunneling Considering Three-Dimensional Space Effect

The ground surface deformation induced by shield tunnels passing through enclosure structures of existing tunnels is a particular underground construction scenario that has been encountered in Wuhan Metro Line 12 engineering cases in China. Timely ground deformation prediction is important to keep shield tunneling safe. However, the classic ground deformation theory is difficult to accurately predict for this ground deformation. This paper develops a semi-analytical method to predict ground heave considering the space effect in this engineering condition. Based on the improved ground deformation theory, a novel deformation prediction method for the ground and enclosure structure is derived and combined with Kirchhoff plate theory. Comparing with field deformation measurements, the maximum difference between the measured and calculated deformation is 14.6%, which demonstrates that the proposed method can be used to predict the ground heave induced by shield tunnels passing through the enclosure structure of existing tunnels. The parameters of the underground diaphragm wall used in Wuhan Metro Line 12 are further studied in detail. The results show that the ground heaves have a positive correlation with the embedded ratio of the diaphragm wall, but a negative correlation with its elastic modulus and thickness. However, the thickness and embedded ratio have a limited effect on ground heaves. This study provides a technical reference for optimizing the setting of enclosure structures in order to protect existing buildings.

Nonlinear Free Vibration of Spinning Pre-Twisted Functionally Graded Material Plates in Thermal Environment

This paper investigates the nonlinear free vibration of spinning pre-twisted functionally graded material (FGM) plates in thermal environment. Based on the Kirchhoff plate theory and von Kármán large-deformation theory, the nonlinear dynamic model is established via the Lagrange equation, where the warping effect is taken into account. The material properties are related to temperature and change according to a power-law exponent distribution. The theoretical solution of nonlinear-free vibration is obtained using the multiple scale method. The effects of power-law exponent, temperature difference, spinning speed, hub radius ratio, and twist angle on nonlinear-free vibration characteristics of the spinning pre-twisted cantilever FGM plates are discussed. The results show that the spinning pre-twisted cantilever FGM plate exhibits nonlinear hardening-spring vibration behavior, while the characteristics weaken with the increase of spinning speed. The twist angle and setting angle have coupling effects on the nonlinear vibration characteristics of spinning pre-twisted FGM plates.