Kirchhoff-Love Plate Theory Research Articles

Head-on collision between two hydroelastic solitary waves with Plotnikov-Toland's plate model

ABSTRACT Head-on collision between two hydroelastic solitary waves propagating at the surface of an incompressible and ideal fluid covered by a thin ice sheet is analytically studied by means of a singular perturbation method. The ice sheet is represented by the Plotnikov-Toland model with the help of the special Cosserat theory of hyperelastic shells and the Kirchhoff-Love plate theory, which yields the nonlinear and conservative expression for the bending forces. The shallow water assumption is taken for the fluid motion with the Boussinesq approximation. The resulting governing equations are solved asymptotically with the aid of the Poincare-Lighthill-Kuo method, and the solutions up to the third order are explicitly presented. It is observed that solitary waves after collision do not change their shapes and amplitudes. The wave profile is symmetric before collision, and it becomes, after collision, unsymmetric and titled backward in the direction of wave propagation. The wave profile significantly reduces due to greater impacts of elastic plate and surface tension. A graphical comparison is presented with published results, and the graphical comparison between linear and nonlinear elastic plate models is also shown as a special case of our study.

An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(\mathrm{div\,Div})$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(\mathrm{div\,Div})$. They are essential to construct basis functions for an approximation of $H(\mathrm{div\,Div})$. To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

Fully discrete DPG methods for the Kirchhoff–Love plate bending model

We extend the analysis and discretization of the Kirchhoff–Love plate bending problem from Führer et al. (in press) in two aspects. First, we present a well-posed formulation and quasi-optimal DPG discretization that include the gradient of the deflection. Second, we construct Fortin operators that prove the well-posedness and quasi-optimal convergence of lowest-order discrete schemes with approximated test functions for both formulations. Our results apply to the case of non-convex polygonal plates where shear forces can be less than L2-regular. Numerical results illustrate expected convergence orders.

Flexural wave band gaps in a multi-resonator elastic metamaterial plate using Kirchhoff-Love theory

Abstract We investigate theoretically the band structure of flexural waves propagating in an elastic metamaterial thin plate. Kirchhoff-Love thin plate theory is considered. We study the influence of periodic arrays of multiple degrees of freedom local resonators in square and triangular lattices. Plane wave expansion and extended plane wave expansion methods, also known as ω ( k ) and k ( ω ) , respectively, are used to solve the governing equation of motion for a thin plate. The locally resonant band gaps for square and triangular lattices present almost the same attenuation for all examples analysed. However, square lattice presents broader Bragg-type band gaps with higher attenuation than triangular lattice. An experimental analysis is conducted with a real elastic metamaterial thin plate with resonators in a square lattice. Modal analysis and forced response are computed by finite element method. Plane wave expansion, finite element and experimental results present good agreement.

A Simple High-Order Shear Deformation Triangular Plate Element with Incompatible Polynomial Approximation

The High-order Shear Deformation Theories (HSDTs) which can avoid the use of a shear correction factor and better predict the shear behavior of plates have gained extensive recognition and made quite great progress in recent years, but the general requirement of C1 continuity in approximation fields in HSDTs brings difficulties for the numerical implementation of the standard finite element method which is similar to that of the classic Kirchhoff-Love plate theory. As a strong complement to HSDTs, in this work, a series of simple High-order Shear Deformation Triangular Plate Elements (HSDTPEs) using incompatible polynomial approximation are developed for the analysis of isotropic thick-thin plates, cracked plates, and through-thickness functionally graded plates. The elements employ incompatible polynomials to define the element approximation functions u/v/w, and a fictitious thin layer to enforce the displacement continuity among the adjacent plate elements. The HSDTPEs are free from shear-locking, avoid the use of a shear correction factor, and provide stable solutions for thick and thin plates. A variety of numerical examples are solved to demonstrate the convergence, accuracy, and robustness of the present HSDTPEs.

Parametric structural modelling of fish bone active camber morphing aerofoils

Camber morphing aerofoils have the potential to significantly improve the efficiency of fixed and rotary wing aircraft by providing significant lift control authority to a wing, at a lower drag penalty than traditional plain flaps. A rapid, mesh-independent and two-dimensional analytical model of the fish bone active camber concept is presented. Existing structural models of this concept are one-dimensional and isotropic and therefore unable to capture either material anisotropy or spanwise variations in loading/deformation. The proposed model addresses these shortcomings by being able to analyse composite laminates and solve for static two-dimensional displacement fields. Kirchhoff–Love plate theory, along with the Rayleigh–Ritz method, are used to capture the complex and variable stiffness nature of the fish bone active camber concept in a single system of linear equations. Results show errors between 0.5% and 8% for static deflections under representative uniform pressure loadings and applied actuation moments (except when transverse shear exists), compared to finite element method. The robustness, mesh-independence and analytical nature of this model, combined with a modular, parameter-driven geometry definition, facilitate a fast and automated analysis of a wide range of fish bone active camber concept configurations. This analytical model is therefore a powerful tool for use in trade studies, fluid–structure interaction and design optimisation.

A Hybrid High-Order method for Kirchhoff–Love plate bending problems

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in hk+3 is also derived for the L2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

Combined Semianalytical and Numerical Static Plate Analysis. Part 2: Resultant Multipoint Boundary Problem

The distinctive paper is devoted to solution of multipoint boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Coupled multilevel approximation model for extended domain and resultant multipoint boundary problem are constructed. Brief information about software systems and verification samples are presented as well.

Combined Semianalytical and Numerical Static Plate Analysis. Part 1: Formulation of the Problem and Approximation Models

The distinctive paper is devoted to solution of multipoint (particularly, two-point) boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains are under consideration.

Isogeometric collocation for Kirchhoff–Love plates and shells

With the emergence of isogeometric analysis (IGA), the Galerkin rotation-free discretization of Kirchhoff–Love shells is facilitated, enabling more efficient thin shell structural analysis. High-order shape functions used in IGA also allow the collocation of partial differential equations, avoiding the time-consuming numerical integration of the Galerkin technique. The goal of the present work is to apply this method to NURBS-based isogeometric Kirchhoff–Love plates and shells, under the assumption of small deformations.Since Kirchhoff–Love plate theory yields a fourth-order formulation, two boundary conditions are required at each location on the contour, generating some conflicts at the corners where there are more equations than needed. To remedy this overdetermination, we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions (i.e. the clamped, simply-supported, symmetric and free supports). Greville and alternative superconvergent points are used for NURBS basis of even and odd degrees, respectively. For square, circular, and annular flat plates, convergence orders are found to be in agreement with a-priori error estimates. The proposed isogeometric collocation method is then validated and benchmarked against a Galerkin implementation by studying a set of problems involving Kirchhoff–Love shells.

Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle

In this paper, dynamic behavior of double layered nanoplate systems (DLNPS) with respect to a moving nanoparticle is investigated. Both layers of DLNPS are assumed to be orthotropic and each layer is bearing a biaxial load while internal damping effects are also taken into account. Furthermore, coupling between layers are modeled using Kelvin-Voigt viscoelastic theory and moving nanoparticles path are assumed to be linear and circular with constant velocities. Governing equations of motion are derived by using D'Alembert's principle, Kirchhoff-Love plate and Eringen's nonlocal theory. Galerkin's and Laplace transform methods is used to solve the governing equations and analytical solution is presented for linear moving nanoparticle while close-form solution is obtained for circular moving nanoparticle. In order to clarify the influence of different parameters such as small scale effect, stiffness and damping in coupling, biaxial compression and tension of layers, path of the moving mass, etc. on dynamic behavior of each layer, parametric study is presented. Accordingly, with the brand new discussions in moving atoms, molecules, nanocars, nanotrims, point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding such kind of behaviors.

Forced transverse vibrations of an elastically connected nonlocal orthotropic double-nanoplate system subjected to an in-plane magnetic field

Nonlocal forced transversal vibrations of an orthotropic double-nanoplate system are considered in this paper. Both nanoplates are rectangular, simply supported, and coupled by a Winkler elastic medium. In order to take into account small-scale effects, we use Eringen’s nonlocal continuum theory. Based on the nonlocal constitutive relation and Kirchhoff–Love plate theory, the system of two coupled nonhomogeneous partial differential equations of motion is derived, where the effects of the Lorentz magnetic force are obtained via a Maxwell relation. Analytical methods are employed to consider the dynamic responses of the orthotropic double-nanoplate system for three different cases of external transversal load. Also, closed-form solutions of the response amplitudes for forced vibration are derived and then validated with the existing results. The nonlocal parameter, magnetic fields, and exciting loads are discussed in detail for all three cases.

Application of the generalized Hooke's law for viscoelastic materials (GHVMs) in nonlocal free damped vibration analysis of viscoelastic orthotropic nanoplates

Reviewing the literature reveals that in all previous research works related to the damped vibration analysis of nanoplates, the material damping of nanoplates has been represented by Kelvin-Voigt model without any reasonable justification.Recently a refined 3-D theory of linear viscoelasticity termed the generalized Hooke's law for viscoelastic materials (GHVMs), is developed by Rajabi and Hosseini-Hashemi [1]. In that theory new 3-D linear viscoelastic constitutive equations are derived which bridge the differential form of linear viscoelasticity and the integral form of linear viscoelasticity.In the present paper vibration characteristics of simply supported orthotropic nanoplates are studied using the nonlocal Kirchhoff-Love plate theory in conjunction with GHVMs. Using GHVMs the material damping of the nanoplates is represented by other rheological models namely the Maxwell model and the standard linear solid model for illustration purposes. Results have revealed that for the same physical and geometrical properties and also for the same contents of damping, the characteristic of various rheological models differ considerably from each other. This indicates that for correct and accurate modeling of 2-D and 3-D viscoelasticity problems in nanoscale it is crucial to perform the material characterization experimentally or by MD simulation.

Analytical solutions for sandwich plates considering permeation effect by 3-D elasticity theory

In this paper, an exact analytical solution for simply supported sandwich plate which considers the permeation effect of adhesives is presented. The permeation layer is described as functionally graded material (FGM), the elastic modulus of which is assumed to be graded along the thickness following the exponential law. Based on the exact three-dimensional (3-D) elasticity theory, the solution of stresses and displacements for each layer is derived. By means of the recursive matrix method, the solution can be efficiently obtained for plates with many layers. The present solution obtained can be used as a benchmark to access other simplified solutions. The comparison study indicates that the finite element (FE) solution is close to the present one when the FGM layer in the FE model is divided into a series of homogeneous layers. However, the present method is more efficient than the FE method, with which the mesh division and computation are time-consuming. Moreover, the solution based on Kirchhoff-Love plate theory is greatly different from the present solution for thick plates. The influence of the thickness of the permeation layer on the stress and displacement fields of the sandwich plate is discussed in detail. It is indicated that the permeation layer can effectively relieve the discontinuity stress at the interface.

Study on coupled analysis for compact acoustic systems using finite-difference time-domain method based on Kirchhoff-Love plate theory and elastic wave

In this paper, we propose a coupled analysis method for compact acoustic systems (e.g., micro speakers, electret condenser microphones) using finite-difference time-domain (FDTD) method based on the Kirchhoff-Love plate theory and elastic wave. Acoustic systems are usually analyzed through any numerical analysis technique based on wave propagation. However, an exact frequency response is rarely obtained for compact acoustic systems. This is because the standard FDTD method cannot treat the change of the frequency response due to the air viscosity and cannot also treat the wave propagation in a solid. Thus, we cannot obtain exact frequency responses for compact acoustic systems through the analysis of coupled fluid-solid model based on standard FDTD method. In order to solve these problems, we utilize a coupled analysis method based on Kirchhoff-Love plate theory and elastic wave FDTD method considering air viscosity. The analysis method models diaphragm and voice coil as a two-dimensional plate model base...

Static Analysis of a Nano Plate by Using Generalized Differential Quadrature Method

In this paper, static analysis of a nano rectangular plate subjected to uniform distributed load is studied based on modified couple stress theory (MSCT) by using Generalized Differential Quadrature (GDQ) Method. The inclusion of an additional material parameter enables the new plate model to capture the size effect. The new non-classical plate model reduces to the classical plate model when the length scale parameter is set to zero. In deriving of the governing equations, the minimum total potential energy principle is used. In the solution of the governing equations, the DQM method is used for simply-supported nano plate within the Kirchhoff-Love plate theory. In the numerical results, the influences the material length scale parameter and the dimension parameters of the plate on the static deflection of the nano plate is presented. Also, the difference between the classical theory (CT) and MSCT is investigated for static responses of nano plate.

Axisymmetric free vibration of closed thin spherical nanoshells with bending effects

Nanoscale spheres have led to a growing interest in their potential applications in a wide range of technological fields. Hollow nanospheres can be modelled as closed spherical shells in the analysis of their mechanical behaviour, the ratio of thickness to radius being used to ascertain the pertinence of considering only membrane forces, or both membrane forces and bending moments. Nonlocal elasticity theory has been widely used to analyze the mechanical behaviour of nanostructures. This paper investigates the free axisymmetric vibration of nanoscale spherical shells accounting for both types of internal force, by extending the Kirchhoff–Love plate theory to Eringen nonlocal elasticity theory. The influence of coupled size and bending effects on the frequencies and modal shapes is studied, revealing specific features that cannot be observed in an uncoupled analysis. This study could be useful in biomedical and bioengineering applications as well as in other fields related to nanotechnology.

Transient dynamics of an elastic Hele-Shaw cell due to external forces with application to impact mitigation

We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure field. We model the flow field via the lubrication approximation and the plate deformation by the Kirchhoff–Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed-form expressions. We examine a distributed external pressure, spatially uniform and linearly increasing with time, acting on the elastic plate over a finite region and during a finite time period, similar to the viscous–elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by an order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure. The results thus suggest that elastic Hele-Shaw configurations may be used to achieve significant impact mitigation.

Fast analytical design of MEMS capacitive pressure sensors with sealed cavities

This paper presents a fast design strategy for MEMS capacitive pressure sensors with sealed cavities. Based on circular Kirchhoff-Love plate theory and the perfect gas law, the underlying analytical model allows for a rapid and accurate evaluation of the sensitivity of the sensors, crucial for improving their design in function of the reference pressure applied in the sealed cavities. The accuracy of the new model is demonstrated by comparing its predictions with more computationally expensive simulation techniques (high-order parametric element and three-dimensional finite element models) and with experimental measurements performed on a 300µm diameter membrane fabricated using the Poly-SiGe platform developed at imec.

Validation of a Novel Experimental and Computational Methodology to Measure Intercellular Forces during Tissue Morphogenesis

During three-dimensional tissue development, cell-cell confinement, cell-cell traction forces, and the resulting intracellular tension play a key role in regulating cell proliferation differentiation, migration, and apoptosis. Yet, the molecular mechanisms that underlie the biochemical response of living cells to these mechanical forces are largely elusive, in part due to the lack of suitable methods to measure accurately the intercellular forces in three-dimensional, developing living tissues with high spatial and temporal resolution. We have developed a novel experimental and computational methodology to measure the spatial and temporal evolution of the intercellular forces during tissue development while simultaneously monitoring the dynamics of the resulting proliferation, differentiation, and migration of the cells within the tissue using a small, slender, elastic PDMS microrod. The microfabricated rod is functionalized and embedded in the tissue as we track its shape deformations over time. As tissue forms around the rod, the cells in the developing tissue exert stresses producing bending deformations along the length of the microrod. By tracking these deformations we can backtrack a solution to the primary forces exerted by the cells. Preliminary experimental results on wild type HaCat cells show that the method has enough sensitivity to measure single cell-to-cell forces and to evaluate the dynamics of wound healing processes. A validation strategy for this method is presented and tested in 1D and 2D scenarios. Traction forces are obtained solving the classical solution of the elastostatic equation from the deformations obtained from fluorescent beads on polyacrylamide substrates. Direct measurements of the intercellular stresses are then compared with calculations of the monolayer stresses obtained solving the Kirchhoff-Love thin plate theory equations.