Kirchhoff-Love Plate Research Articles

Plate and beam moment-field formulation

Governing equations of the linear elastic Euler–Bernoulli beam and pure bending Kirchhoff–Love plate are developed in a pure stress form with the evolution of the bending moments fields. In the case of elastodynamics, the governing equations are used to find mode restrictions of vibration within both structures. These vibration restrictions are then compared with the solutions found through the displacement formulation finding both forms to give the same solution under the same boundary conditions. The waves are assumed to be harmonic with only spatial contributions being examined, validating the stress approach. More advanced cases involving anisotropic and non-classical boundary conditions are also explored to demonstrate further implications of following the stress evolution. Elastostatic plates are examined under the compatibility conditions to be met for moment fields to possess uniqueness and to show the CLM (Cherkaev–Lurie–Milton) invariance of the inhomogeneous plane-stress problem.

Exploring the Multiplication of Resonant Modes in Off-Center-Driven Chladni Plates from Maximum Entropy States

In this study, the resonant characteristics of the off-center-driven Chladni plates were systematically investigated for the square and equilateral triangle shapes. Experimental results reveal that the number of the resonant modes is considerably increased for the plates under the off-center-driving in comparison to the on-center-driving. The Green’s functions derived from the nonhomogeneous Helmholtz equation are exploited to numerically analyze the information entropy distribution and the resonant nodal-line patterns. The experimental resonant modes are clearly confirmed to be in good agreement with the maximum entropy states in the Green’s functions. Furthermore, the information entropy distribution of the Green’s functions can be used to reveal that more eigenmodes can be triggered in the plate under the off-center-driving than the on-center-driving. By using the multiplication of the resonant modes in the off-center-driving, the dispersion relation between the experimental frequency and the theoretical wave number can be deduced with more accuracy. It is found that the deduced dispersion relations agree quite well with the Kirchhoff–Love plate theory.

Nonlinear Control System for Flat Plate Structures Considering Interference Based on Operator Theory and Optimization Method

In recent years, vibration control utilizing smart materials has garnered considerable attention. In this paper, we aim to achieve vibration suppression of a plate structure with a tail-fin shape by employing piezoelectric actuators—one of the smart materials. The plate structure is rigorously modeled based on the Kirchhoff–Love plate theory, while the piezoelectric actuators are formulated in accordance with the Prandtl–Ishlinskii model. This research proposed a control system that addresses the interference effects arising during vibration control by dividing multiple piezoelectric elements into two groups and implementing MIMO control. The efficacy of the proposed control method was validated through simulations and experiments.

Spatial and frequency identification of the dynamic properties of thin plates with the Frequency-Adapted Virtual Fields Method

Vibroacoustic inverse methods use the measured response of a vibrating structure to identify a structural parameter or a dynamic load. Two inverse methods are considered, the Force Analysis Technique (FAT) and the Virtual Fields Method (VFM). The Corrected Force Analysis Technique (CFAT) is a variant of FAT that corrects its singularity. This correction allows the method to be applied in the high-frequency domain, when the number of measurement points per flexural wavelength becomes small. In this study, the proposed novelty is the development of a Frequency-Adapted VFM (FA VFM) to the case of a Love–Kirchhoff plate. Thanks to this method, the VFM can now be applied to identify the equivalent bending stiffness and structural damping of a thin plate when the number of measurement points per wavelength is small. The method has previously been developed for an Euler–Bernoulli beam. An experimental identification of the complex bending stiffness of an locally damped aluminium plate using Laser Doppler Velocimetry (LDV) data and the developed method is performed. The experimental study shows for the first time that the FA VFM can be used to map the equivalent bending stiffness and structural damping as a function of position on a plate and identify these parameters as a function of frequency over a large frequency band. The results of the Frequency-Adapted VFM are compared with those of CFAT and the classical VFM approach. FA VFM results are more accurate than those of classical VFM and similar to those of CFAT.

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.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

Non-coercive problems for elastic plates with thin junction

We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

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.

Homogenization of non-rigid origami metamaterials as Kirchhoff–Love plates

Origami metamaterials have gained considerable attention for their ability to control mechanical properties through folding. Consequently, there is a need to develop systematic methods for determining their effective elastic properties. This study presents an energy-based homogenization framework for non-rigid origami metamaterials, effectively linking their mechanical treatment with that of traditional materials. To account for the unique mechanics of origami systems, our framework incorporates out-of-plane curvature fields alongside the usual in-plane strain fields used for homogenizing planar lattice structures. This approach leads to a couple-stress continuum, resembling a Kirchhoff–Love plate model, to represent the homogeneous response of these lattices. We use the bar-and-hinge method to assess lattice stiffness, and validate our framework through analytical results and numerical simulations of finite lattices. Initially, we apply the framework to homogenize the well-known Miura-ori pattern. The results demonstrate the framework’s ability to capture the unconventional relationship between stretching and bending Poisson’s ratios in origami metamaterials. Subsequently, we extend the framework to origami lattices lacking centrosymmetry, revealing two distinct neutral surfaces corresponding to bending along two lattice directions, unlike in the Miura-ori pattern. Our framework enables the inverse design of metamaterials that can mimic the unique mechanics of origami tessellations using techniques like topology optimization.

Nonlinear combination resonance analysis of parametric-forced excitation for an axially moving piezoelectric rectangular thin plate in thermal- electromechanical coupling field

In this paper, the combination resonance of parametric-forced excitation characteristics for an axially moving rectangular piezoelectric plate under a thermal-electromechanical field is studied. Based on Kirchhoff-Love plate theory and Von Karman theory, the transverse vibration governing equations are derived from Hamilton's principle. The equations are discretized to ordinary differential equations by the Galerkin method. Then, the multiple scales method is applied to solve the system combination resonance equation, two different resonance states and corresponding amplitude-frequency response equations are obtained by eliminating the secular term, respectively. Additionally, the stability of the steady-state responses is analyzed by Lyapunov stability. Based on the numerical analysis, the influence of axial velocity, external voltage, central temperature difference, structural damping, and other parameters on nonlinear combination resonance response are investigated. The effect of parameter variation on period-doubling bifurcation and the chaotic motion of the system are also discussed by the system global bifurcation diagram.

Wave interaction with multiple adjacent floating solar panels with arbitrary constraints

The problem of wave interaction with multiple adjacent floating solar panels with arbitrary types and numbers of constraints is considered. All the solar panels are assumed to be homogeneous, with the same physical properties, as well as modeled by using the Kirchhoff-Love plate theory. The motion of the fluid is described by the linear velocity potential theory. The domain decomposition method is employed to obtain the solutions. In particular, the entire fluid domain is divided into two types, the one below the free surface, and the other below elastic plates. The velocity potential in the free surface domain is expressed into a series of eigenfunctions. By contrast, the boundary integral equation and the Green function are employed to construct the velocity potential of fluid beneath the entire elastic cover, with unknowns distributed along two interfaces and jumps of physical parameters of the plates. All these unknowns are solved from the system of linear equations, which is established from the matching conditions of velocity potentials and edge conditions. This approach is confirmed with much higher computational efficiency compared with the one only involving eigenfunction expansion for the fluid beneath each plate. Extensive results and discussions are provided for the reflection and transmission coefficients of water waves, maximum deflection, and principal strain of the elastic plates; especially, the influence of different types and numbers of edge constraints are investigated in detail.

Bending edge wave on a functionally graded fluid‐saturated porous magneto‐electro‐elastic plate

AbstractThis study investigates the localized wave propagation at the free edge of a thin semi‐infinite Magneto‐Electro‐Elastic (MEE) plate. The plate is supported by a two‐parameters elastic foundation. Biot's porosity theory is considered to formulate the porosity in the model, and the pore structure is saturated with an incompressible fluid. The Kirchhoff–Love plate theory determines the plate displacement field. A non‐homogeneity in the material parameters is considered, which varies with the spatial coordinate along the thickness direction of the plate. In order to illustrate the impacts of the surface elements on bending wave propagation, the model incorporates the surface elasticity theory. The short circuit boundary conditions are implemented to derive the general form of the potential function connected with the electric and magnetic fields, respectively. The dispersion relation for bending edge wave on a MEE fluid‐saturated porous plate is obtained by considering a sinusoidal waveform, which propagates along the free edge of the plate. A numerical method named Finite Difference (FD) scheme is employed to analyze the stability of the numerical scheme and also extract the expression of the velocity profiles of the bending edge wave in the absence of the pore structure. The impacts of the various parameters included in the considered model are examined using a numerical example, from which conclusions regarding their sensitivity are derived.

Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates

Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates

Physics-informed neural networks for structural health monitoring: a case study for Kirchhoff–Love plates

Physics-informed neural networks for structural health monitoring: a case study for Kirchhoff–Love plates

Vibroacoustics Analysis of Plates with Bioinspired Surface Modifications

This study presents a simplified semi-analytical model to analyze the dynamic behavior of a plate with shark skin biomimetic surface modifications. The base plate is considered to be thin isotropic homogeneous and modeled using a Kirchhoff–Love plate theory, whereas the dermal denticles are modeled as a distributed array of point mass and replicated using a Dirac delta function. A semi-analytical formulation, which takes into account closed-form beam mode shapes, with the Galerkin technique to minimize errors is utilized to study the free and forced vibration response of modified plates. Harmonic analysis is carried out to understand the dynamic behavior of the modified plate both quantitatively as well as qualitatively. A parametric study varying the array spacing and the boundary conditions is carried out to understand the effect of surface modification on power radiation characteristics. It appears that adding dermal denticles reduces the peak velocity amplitude and peak radiated power at all resonance frequencies. A separate study by varying the base plate material reveals that a higher reduction in the dynamic response and the radiated power can be achieved by increasing the mass ratio. Hence, surface modification using such a shark skin layer may prove beneficial to reduce acoustic radiation and dynamic response and to optimize the design.

An ANN-BCMO Approach for Material Distribution Optimization of Bi-Directional Functionally Graded Nanocomposite Plates with Geometrically Nonlinear Behaviors

This study commences with the application of an efficient artificial neural network (ANN)-balancing composite motion optimization (BCMO) approach for finding the optimal material distribution of bi-directional functional graded nanocomposite (FGN) thin plates considering geometrically nonlinear behaviors. To this regard, an ANN-based surrogate model of the high-fidelity isogeometric analysis (IGA) of the geometrically nonlinear Kirchhoff–Love plates based on the Von Kármán nonlinearity theory is first constructed and then employed to predict the values of objectives and constraints required in the BCMO framework. The multi-mesh design approach is utilized to form two separate nonuniform rational B-spline meshes for optimal material distribution and analysis meshes. The unknown control point values of the design mesh are herein selected as the continuous design variables. This allows a possibility to fully explore the complex distribution of optimal material profiles without requiring a significant number of variables. Selected numerical examples with different plate geometries and loading conditions are presented to illustrate the merit features of the proposed approach. Obtained results reveal not only its high accuracy and significant efficiency compared with the conventional approach coupling the BCMO with IGA direct analysis but also the stable ability of the BCMO in finding global optimum material distributions in comparison with two commonly used metaheuristics algorithms, i.e., particle swarm optimization and genetic algorithm.

Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method.

The present work's main objective is to investigate the natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM). The behavior of the supports is described by the fractional order derivatives of the Riemann-Liouville type. The subspace iteration method, in conjunction with the continuation method, is used as a tool to solve the non-linear eigenproblem. A deterministic core for solving structural eigenvibrations is the Finite Element Method. The probabilistic analysis includes the Monte-Carlo simulation and the semi-analytical approach, as well as the iterative generalized stochastic perturbation method. Probabilistic structural response in the form of up to the second-order characteristics is investigated numerically in addition to the input uncertainty level. Finally, the probabilistic relative entropy and the safety measure are estimated. The presented investigations can be applied to the dynamics of foundation plates resting on viscoelastic soil.

Stable Discrete Bending by Analytic Eigensystem and Adaptive Orthotropic Geometric Stiffness

In this paper, we address two limitations of dihedral angle based discrete bending (DAB) models, i.e. the indefiniteness of their energy Hessian and their vulnerability to geometry degeneracies. To tackle the indefiniteness issue, we present novel analytic expressions for the eigensystem of a DAB energy Hessian. Our expressions reveal that DAB models typically have positive, negative, and zero eigenvalues, with four of each, respectively. By using these expressions, we can efficiently project an indefinite DAB energy Hessian as positive semi-definite analytically. To enhance the stability of DAB models at degenerate geometries, we propose rectifying their indefinite geometric stiffness matrix by using orthotropic geometric stiffness matrices with adaptive parameters calculated from our analytic eigensystem. Among the twelve motion modes of a dihedral element, our resulting Hessian for DAB models retains only the desirable bending modes, compared to the undesirable altitude-changing modes of the exact Hessian with original geometric stiffness, all modes of the Gauss-Newton approximation without geometric stiffness, and no modes of the projected Hessians with inappropriate geometric stiffness. Additionally, we suggest adjusting the compression stiffness according to the Kirchhoff-Love thin plate theory to avoid over-compression. Our method not only ensures the positive semidefiniteness but also avoids instability caused by large bending forces at degenerate geometries. To demonstrate the benefit of our approaches, we show comparisons against existing methods on the simulation of cloth and thin plates in challenging examples.

Optimal design of piezoelectric energy harvesters for bridge infrastructure: Effects of location and traffic intensity on energy production

Piezoelectric energy harvesters (PEHs) can be used as an additional power supply for a Structural Health Monitoring (SHM) system. Its design can be optimised for the best performance, however, the optimal design depends on the input vibration and locations. In this work, we extend an optimisation framework from our previous studies to include the effect of the PEH’s location, which uses the cantilevered PEH Kirchhoff–Love plate model discretised by IsoGeometric Analysis (IGA) and coupled with Particle Swarm Optimisation (PSO) algorithm to find the designs with maximum energy outputs for a large number of input acceleration histories, extracted from the recorded dynamic response data of a real cable-stayed bridge in Australia. Then, clustering and evaluation under 24-h excitation are performed to find several best PEHs for the entire bridge. By comparing the best PEH at all locations with a typical benchmark device, the substantial performance improvement brought by the optimisation framework of this work is verified. The comparison results show a significant energy harvesting enhancement of 1.6 times at the best locations, 2.4 times at A2 and A3, and 3.6 times at the edge girders, respectively. The results also reveal the variability of the best design at different locations of the same structure to maximise the energy harvesting of the entire bridge, and demonstrate the design rule that the fundamental frequency of the device can be tuned within a certain frequency range to improve the robustness of PEHs design. Additional studies are performed to explore the effect of location and traffic intensity on energy harvesting by analysing the optimal PEH design and location throughout the bridge structure. The results indicate that the key factors of maximising energy harvesting efficiency are related to the input excitation and the mode of vibration being excited. The position of maximum displacement in the vibration mode corresponds to the best location for energy harvesting. Also, the best device has a fundamental frequency close to the frequency of the corresponding vibration mode. In addition, the change of traffic intensity affects the amount of convertible mechanical energy and also directs the tuning of the fundamental frequency of the PEH to achieve the highest energy conversion.