Kirchhoff-Love Theory Research Articles

A transfer matrix model to predict acoustic attenuation of earplug filters made up of elastic film and air cavity

The sound attenuation of earplugs, commonly used for noise protection, can be improved by adding acoustic filters. However, the effect of acoustic filters featuring an elastic film and cylindrical air cavities on sound attenuation needs to be better understood. This study seeks to predict acoustic attenuation of an elastic film filter via the development of an analytical model. Its advantage, as compared to numerical simulations, is to offer quicker solutions and a more profound comprehension of each physical parameter's impact. The total acoustic transfer matrix of the filter has been established by coupling the air cavity with the film under assumption of plane wave. The transfer matrix of the vibrating elastic film is based on the Kirchhoff-Love theory and could take into account all its dynamic behaviors, including possible in-plane tension. The model is then validated by comparison with numerical simulations using the finite element method. It is found that the plane wave assumption is valid in almost practical geometries of the air cavity such that the sound attenuation obtained from the analytical model align consistently with those from simulations. Potential limitations are addressed and evaluated.

A four-node inverse curved shell element coupling MITC method for deformation reconstruction of plate and shell structures

In the field of structural deformation monitoring, the inverse finite element method (iFEM) has significant engineering value as a structural health monitoring technique that provides timely and reliable warnings for shell structures. However, existing inverse finite elements are mainly based on first-order shear deformation theory and kirchhoff-love theory, which are not suitable for deformation reconstruction in plate and shell structures of arbitrary thickness. This study integrates iFEM with the Mixed Interpolation of Tensorial Components (MITC) method to develop a novel four-node quadrilateral inverse curved shell element, named iMICS(inverse Mixed Interpolation Curved Shell)4, aimed at enhancing the accuracy and efficiency of deformation reconstruction in complex plate and shell structures. The method uses the MITC4 shell element as the kinematic framework and applies the least squares variational principle to achieve deformation reconstruction, effectively alleviating shear and membrane locking issues across structures of varying thickness. Numerical examples validate the superior performance of the iMICS4 element, demonstrating its promising application prospects.

Modeling and solution of eigenvalue problems of laminated cylindrical shells consisting of nanocomposite plies in thermal environments

This work is dedicated to the modeling and solution of eigenvalue problems within shear deformation theory (SDT) of laminated cylindrical shells containing nanocomposite plies subjected to axial compressive load in thermal environments. In this study, the shear deformation theory for homogeneous laminated shells is extended to laminated shells consisting of functionally graded (FG) nanocomposite layers. The nanocomposite plies of laminated cylindrical shells (LCSs) are arranged in a piecewise FG distribution along the thickness direction. Temperature-dependent material properties of FG-nanocomposite plies are estimated through a micromechanical model, and CNT efficiency parameters are calibrated based on polymer material properties obtained from molecular dynamics simulations. After mathematical modeling, second-order time-dependent and fourth-order coordinate-dependent partial differential equations are derived within SDT, and a closed-form solution for the dimensionless frequency parameter and critical axial load is obtained for first time. After the accuracy of the applied methodology is confirmed by numerical comparisons, the unique influences of ply models, the number and sequence of plies and the temperature on the critical axial load and vibration frequency parameter within SDT and Kirchhoff–Love theory (KLT) are presented with numerical examples.

Magneto-thermoelastic nonlinear dynamic modeling of a rotating functionally graded shell

Research is conducted on the dynamic modeling of a rotating functionally graded (FG) cylindrical shell subjected to magnetic and temperature fields. Based on the elasticity theory and generalized Hooke's law on the physical neutral surface, nonlinear geometric equations and thermoelastic constitutive relations are determined. According to the Kirchhoff-Love theory, variational formulas of strain energies for deformation, temperature, and centrifugal force are obtained. Considering the rotational effect, the kinetic energy and its variational formula are derived. The electromagnetic force model incorporating magnetization effect of the ferromagnetic FG shell is established by utilizing the electromagnetic theory. Subsequently, the magneto-thermoelastic dynamic model of the rotating FG shell is developed by adopting the Hamilton's principle. The model can reveal the coupling mechanisms of the interaction and superposition of multi-physical fields. Finally, taking the primary resonance as example, detailed numerical analyses are performed to investigate the effects of different parameters on vibration response and dynamical stability.

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

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

The Application of the Modified Lindstedt–Poincaré Method to Solve the Nonlinear Vibration Problem of Exponentially Graded Laminated Plates on Elastic Foundations

The solution of the nonlinear (NL) vibration problem of the interaction of laminated plates made of exponentially graded orthotropic layers (EGOLs) with elastic foundations within the Kirchhoff–Love theory (KLT) is developed using the modified Lindstedt–Poincaré method for the first time. Young’s modulus and the material density of the orthotropic layers of laminated plates are assumed to vary exponentially in the direction of thickness, and Poisson’s ratio is assumed to be constant. The governing equations are derived as equations of motion and compatibility using the stress–strain relationship within the framework of KLT and von Karman-type nonlinear theory. NL partial differential equations are reduced to NL ordinary differential equations by the Galerkin method and solved by using the modified Lindstedt–Poincaré method to obtain unique amplitude-dependent expressions for the NL frequency. The proposed solution is validated by comparing the results for laminated plates consisting of exponentially graded orthotropic layers with the results for laminated homogeneous orthotropic plates. Finally, a series of examples are presented to illustrate numerical results on the nonlinear frequency of rectangular plates composed of homogeneous and exponentially graded layers. The effects of the exponential change in the material gradient in the layers, the arrangement and number of the layers, the elastic foundations, the plate aspect ratio and the nonlinearity of the frequency are investigated.

Automated shape and thickness optimization for non-matching isogeometric shells using free-form deformation

AbstractIsogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff–Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

The effects of shear deformation and rotary inertia on the electrical analogs of beams and plates for multimodal piezoelectric damping

AbstractAnalogous electrical networks were previously derived from the Euler–Bernoulli and Kirchhoff–Love theories to represent beams and plates, respectively, for use in multimodal structural vibration damping. However, these networks do not account for shear deformations or rotary inertia, which can result in suboptimal vibration damping performance when used on moderately thick beams and plates. In this paper, we investigate the incorporation of shear deformation and rotary inertia using Timoshenko–Ehrenfest beam theory and Mindlin–Reissner plate theory to develop improved electrical networks that can more accurately represent thick beams and plates. Our findings suggest that the inclusion of shear deformation and rotary inertia can significantly improve the frequency coherence of the electrical networks and multimodal vibration damping for thicker structures. The electrical analogs presented here are of use for various applications, especially to conveniently design complex circuit topologies in fields spanning from vibration attenuation to energy harvesting.

Investigation of the vibration problem of sandwich heterogeneous orthotropic plates

In this study, the solution of the free vibration problem of sandwich plates consisting of heterogeneous orthotropic layers is presented. First, the Kirchhoff-Love theory (KLT) for homogeneous plates is extended to sandwich plates composed of heterogeneous orthotropic layers. Within the framework of Donnell theory, after establishing the basic relations of multilayer plates, governing equations are derived based on Airy stress and deflection functions. The solution of the governing equations is carried out by the Galerkin method and the analytical expression is found for the linear frequency of sandwich plates consisting of heterogeneous orthotropic layers. Finally, the effects of various factors such as heterogeneity, number of layers and their arrangement on the free vibration frequency of sandwich plates are examined.

Nonlinear geometric fluid-structure interaction model of multilayered sandwich plates in contact with unbounded or bounded fluid flow

Based on the Kirchhoff-Love theory and von Kármán's geometric nonlinearity, the nonlinear vibration analysis of multilayered sandwich plates under ideal flow is presented. Rectangular plates are considered to be laminated of orthotropic composite layers as upper and lower face sheets as well as a central lightweight core made of functionally graded (FG), anisogrid lattice or metal foam materials. Three coupled equations of motion are reformulated into one decoupled governing equation in terms of transverse deflection (w) using the method of inverse differential operator. For the first time, an impermeability condition is developed that considers the moderately large deflection of the contact surface in order to correct the pressure distribution of fluid flow. Based on the potential flow theory, closed-form expressions of hydrodynamic pressure are obtained for different fluid boundary conditions including infinite depth of fluid, flow bounded by a rigid wall and flow between two identical elastic plates. Then, the Galerkin and multiple-scale perturbation methods are used to obtain closed-form solution of nonlinear natural frequency in terms of maximum vibration amplitude. Finally, case studies are presented to investigate the effects of geometric ratios, boundary condition, material distribution, layup, characteristics of flow and geometric nonlinearity on the response quantities. Findings indicate that the large deformation term in the developed impermeability condition has significant effect on the pressure distribution and its effect is more pronounced for thicker plates with larger mode number in the flow direction. Therefore, the novel nonlinear fluid-structure interaction model should be considered to study flow-induced dynamic behavior of sandwich plates with different boundary conditions. In addition, the snap-through behavior is seen in the nonlinear frequency-upstream speed curves and the incipient points of dynamic instability can vary remarkably with the fluid boundary conditions.

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Vibration analysis of radial tire using the 3D rotating hyperelastic composite REF based on ANCF

In order to analyze the influence of tread structure and rotating angular speed on the vibration characteristics of radial tire, a 3D rotating hyperelastic composite REF (3D-RHCREF) model is proposed. Different form uniform cylindrical tread in other REF models, the tread is simplified as a hyperelastic composite revolving shell, whose discretization and parameterization are achieved by using three-order segmented Bézier curve and complete-gradient ANCF space curved trapezoid element. In the proposed 3D-RHCREF, the generalized inertial force with the centrifugal force and Coriolis force are obtained by introducing a rotation matrix into the framework of ANCF. Based on Kirchhoff-Love theory and hyperelastic constitutive relation, the generalized hyperelastic force of composite tread is derived and the compensation of initial generalized hyperelastic force is proposed. According to the D'Alembert principle, the nonlinear dynamic differential equations are established, then the modal equations are derived on the basis of the equilibrium position. By comparing with the FEM model of radial tire, the validity of the proposed model and calculation method is verified. The 3D-RHCREF is used to analyze the influences of internal pressure, tread pressure sharing ratio, belt structure and rotating angle speed on the vibration characteristics of radial tire. The vibration frequency will increase with the increase of internal pressure and the decrease of tread pressure sharing ratio; different belt structures of tread will cause changes in vibration frequency; the radial vibration frequency will be split into two traveling wave frequencies in two directions with the increase of rotating angular speed, and the circumferential/lateral vibration frequencies will almost only increase slightly with the increase of rotating angular speed.

Buckling behavior of multilayer cylindrical shells composed of functionally graded nanocomposite layers under lateral pressure in thermal environments

In this study, the stability behavior of multilayer cylindrical shells made of functionally graded nanocomposite layers (FG-NCLs) subjected to the lateral pressure in thermal environments is investigated. It is postulated that nanocomposite layers forming layered cylindrical shells are made of single-walled carbon nanotube (SWCNT)-reinforced polymers that have four types of profiles based on the uniform and linear distributions of mechanical properties. The material properties of SWCNTs are assumed to be dependent on location as well as temperature and are obtained from molecular dynamics simulations. The governing equations are derived as partial differential equations within shear deformation theory (SDT) and solved in a closed form, using the Galerkin procedure, to determine the lateral critical pressure (LCP) in thermal environments. The numerical representations relate to the buckling behavior of multilayer cylindrical shells made of functionally graded nanocomposite layers under the uniform lateral pressure for different CNT patterns and temperatures within SDT and Kirchhoff-Love theory (KLT).

A new combined asymptotic-tolerance model of thermoelasticity problems for thin uniperiodic cylindrical shells

AbstractThe objects of consideration are thin linearly thermoelastic Kirchhoff–Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selecteddynamic thermoelasticity problemsfor the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a newprocedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

Shape sensing for thin-shell spaceborne antennas with adaptive isogeometric analysis and inverse finite element method

As the preventive maintenance paradigm transfers to condition-based maintenance, deformation monitoring has become a fundamental system capacity in aerospace engineering. In this study, a novel shape sensing method is proposed for accurate and efficient reconstruction of full-field deformation of thin shell structures from discrete strain measurements. Firstly, a flexible isogeometric approach based on the geometry-independent field approximation is developed for characterizing the geometric and physical domains, which fully unlocks the potential of local refinement while preserving the original exact geometry without re-parameterization. On this basis, a posteriori error estimation algorithm is put forward to automatically drive the adaptive refinement procedure, reducing the discretization error with a fast convergence rate. Subsequently, according to the Kirchhoff-Love theory and the least-squares variational principle, an isogeometric inverse-shell element is created to integrate the inherent advantages of adaptive isogeometric analysis with excellent shape-sensing capabilities of the inverse finite element method. Moreover, a smoothing technique is applied to replenish strain data into each inverse shell element, by which the compatibility between the interpolated and measured strain components is also enforced. Finally, the excellent accuracy and efficiency of the proposed deformation reconstruction framework are verified using both experimental and numerical strain data for two thin-shell spaceborne antennas.

Multi-scale procedure for the mechanical analysis of composite laminate structures considering mixed boundary conditions

This paper presents a multi-scale procedure for the study of flat composite structures with discontinuities. In this procedure, the structure is solved using shell elements while the laminate performance and the structural discontinuities (e.g. connections or change in the laminate thickness) are analysed with a subscale model made with solid 3D elements. The kinematics of both models are coupled following the Kirchhoff–Love theory. This coupling is used during the homogenization procedure where the characteristic behaviour of the different micro-models is obtained. Periodical boundary conditions are used for the laminates whereas a combination between periodical and linear boundary conditions are used for the discontinuities. The proposed procedure allows to reproduce accurately the structure elastic behaviour, as well as the stress and strain states in regions with discontinuities, which until now could only be accurately simulated by means of expensive numerical models using volumetric solid elements.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in the Reissner Approximation

Elastic bending stresses and strains in the lithosphere are typically calculated based on the Kirchhoff–Love (thin) plate theory. The criterion for its applicability is the smallness of plate thickness to length ratio. In oceanic plates, due to the buoyant force of the mantle, the main deformations are not uniformly distributed along the plate but are concentrated in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of the actual length, and the criterion of a thin plate is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are much more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate bending function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

Vibration of Conjugated Shell Systems Under Combined Static Loads

We propose a mathematical model of vibration of elastic systems formed by conjugated shells of revolution with different geometry in the field of combined static axisymmetric loads. The model is based on the concepts of geometrically nonlinear theory of mean bending and realized within the framework of the classical Kirchhoff–Love theory with the use of contemporary methods of applied mathematics and numerical analysis. The spectral picture of a shell structure with elements of positive, zero, and negative Gaussian curvature is constructed. This picture enables us to detect resonance situations under specific dynamic actions and determine dangerous combinations of static loads in the analysis of stability of the equilibrium states of the structure.

Asymptotic Study of Plate Bending for a Strongly Orthotropic Materia

The technique of asymptotic averaging has been developed for three-dimensional partial differential equations with rapidly oscillating coefficients. For example, for the equations of elasticity theory. Then it was modified and applied to thin bodies in the form of plates (homogeneous or inhomogeneous, with even front surfaces or not), described by the three-dimensional theory of elasticity. In these cases, asymptotic solutions were constructed with respect to one small parameter, which is usually the ratio of the plate thickness to the characteristic plan size. The averaging technique in this case also reduces the dimension of the problem, i.e. reduces the three-dimensional boundary value problem to some two-dimensional one. In this article, we substantiate the application of the method to a problem with two small parameters in the case of a homogeneous thin, highly orthotropic plate bent by a surface load without taking into account body forces. The second small parameter is the ratio of the transverse elastic moduli to the moduli in the plan of the plate. It is shown that strong orthotropy is equivalent to an increase in the thickness of the equivalent plate. A procedure for obtaining the stress distribution over the plate thickness is described for three approximations. The first approximation gives the classical Kirchhoff theory, also called the Kirchhoff-Love theory, and the third approximation coincides with the Ambartsumian theory and allows one to find transverse shear and normal stresses. The consideration of cylindrical bending makes it possible to find solutions in the framework of classical plate theories in the form of formulas, as well as three approximations of the asymptotic theory, which simplifies comparison. Examples are considered when the averaged orthotropic moduli are taken for a single-layer fibrous composite.