Kirchhoff-Love Plate Theory Research Articles

FREE VIBRATION ANALYSIS OF CRACKED KIRCHHOFF-LOVE PLATE USING THE EXTENDED RADIAL POINT INTERPOLATION METHOD

The Kirchhoff-Love plate theory is appropriate for analysing thin plate structures. In a simple form, only one degree of freedom (per node) is needed to describe the behaviour of the plate, thus saving the computational cost. Besides, the analysis of cracked structures is important because it is related to the lifetime of the structures. Therefore, this paper uses the extended radial point interpolation method (XRPIM) to investigate the free vibration of the Kirchhoff-Love plate. The XRPIM is based on RPIM so the requirement for calculating the second-order derivative in Kirchhoff-Love theory is easily done. The numerical results from this study are compared with other researchers to verify the accuracy of the method.

Multi-directional Falling Weight Deflectometer (FWD) testing and quantification of the effective modulus of subgrade reaction for concrete roads

ABSTRACT Falling Weight Deflectometry (FWD) tests are performed around the centres of two rectangular concrete plates, with geophones measuring vertical deflections in eight directions. Experimental results allow for quantifying asymmetries regarding the structural behaviour. Significant asymmetries are found for a 22-year-old plate scheduled for replacement. A new plate, tested a few weeks after production, is found to behave in a virtually double-symmetric fashion. Structural analysis of the new plate is based on Kirchhoff–Love plate theory, using free-edge boundary conditions. The support of the plate is provided by a Winkler foundation. Performing a static analysis, the uniform modulus of subgrade reaction is optimised to reproduce the measured deflections. The result is not convincing. The model is extended towards consideration of a second optimisation variable: a uniform auxiliary surface load. This allows for reproducing the measured deflections. The auxiliary load is superimposed with the pressure resulting from the Winkler foundation. This yields a realistic distribution of subgrade pressure. Dividing it by the deflections results in the distribution of the effective modulus of subgrade reaction. Finally, the analysis is extended towards the consideration of inertia forces. They increase the effective moduli of subgrade reaction determined by means of static analysis by less than 3.5%.

Stable and accurate numerical methods for generalized Kirchhoff–Love plates

Kirchhoff–Love plate theory is widely used in structural engineering. In this paper, efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff–Love plate model subject to three common physical boundary conditions: (i) clamped; (ii) simply supported; and (iii) free. The generalization stems from the inclusion of additional physics to the classical Kirchhoff–Love model that accounts for bending only. We solve the model equation by discretizing the spatial derivatives using second-order finite-difference schemes, and then advancing the semi-discrete problem in time with either an explicit predictor–corrector or an implicit Newmark-Beta time-stepping algorithm. Stability analysis is conducted for the schemes, and the results are used to determine stable time steps in practice. A series of carefully chosen test problems are solved to demonstrate the properties and applications of our numerical approaches. The numerical results confirm the stability and 2nd-order accuracy of the algorithms and are also comparable with experiments for similar thin plates. As an application, we illustrate a strategy to identify the natural frequencies of a plate using our numerical methods in conjunction with a fast Fourier transformation power spectrum analysis of the computed data. Then we take advantage of one of the computed natural frequencies to simulate the interesting physical phenomena known as resonance and beat for a generalized Kirchhoff–Love plate.

Recovery-based a Posteriori Error Analysis for Plate Bending Problems

We present two new recovery-based a posteriori error estimates for the Hellan–Herrmann–Johnson method in Kirchhoff–Love plate theory. The first error estimator uses a postprocessed deflection and controls the \(L^2\) moment error and the discrete \(H^2\) deflection error. The second one controls the \(L^2\times H^1\) total error and utilizes superconvergent postprocessed moment field and deflection. The effectiveness of the theoretical results is numerically validated in several experiments.

Plane wave expansion and extended plane wave expansion formulations for Mindlin-Reissner elastic metamaterial thick plates

The plane wave expansion (PWE) and the extended plane wave expansion (EPWE) formulations are derived in order to obtain the complex dispersion relation of flexural waves in a metamaterial Mindlin-Reissner thick plate with multiple periodic resonators. These new formulations are useful to obtain more accurate results at higher frequencies than those based on Kirchhoff-Love thin plate theory. Multiple periodic resonators with multiple degrees of freedom (M-DOF) attached to metamaterial thick plates broaden the applications for wave attenuation in high frequencies.

A robust DPG method for large domains

We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincaré-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff–Love plate bending model, and also include fully discrete variants where optimal test functions are approximated. Numerical experiments for both model problems, including an-isotropic domains and mixed boundary conditions, confirm our findings.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

A generalized methodology for thermoelastic damping in axisymmetric vibration of circular plate resonators covered by multiple partial coatings

Microresonators used for microelectromechanical systems (MEMS) are becoming more and more complex in terms of structural geometry and mode shapes. Accompanying this increase in complexity is the challenge to develop an accurate analytical method to evaluate the thermoelastic damping (TED). This paper firstly presents a generalized methodology for TED in axisymmetric vibration of circular plate microresonators covered by multiple partial coatings. The frequency and mode shape function of plates are solved by the Kirchhoff–Love plate theory. The thermoelastic temperature fields are acquired by solving the one-way coupled heat conduction along the thickness direction with Green’s function method, accordingly obtaining the effects of partial coating on the microplate. The present model can reduce to that of the fully covered bilayer plates, and is validated by finite element model (FEM). This methodology produces an explicit model for TED in the form of one infinite series with rapid converge, and a simple model is given by retaining the first term. The accuracy and practicality of the simple model are investigated. The results suggest that SiO2 coating with thermally perfect interfaces can reduce the TED at the fundamental mode. However, SiO2 coating significantly increases TED values with thermally imperfect interfaces. Three distinctive peaks of TED spectra in Si/DLC/SiO2 plates are observed. This work can be utilized to optimize the TED in the composite circular microplates and gain more insights into the intrinsic dissipation.

Shape-shifting panel from 3D printed undulated ribbon lattice

Materials that change their shape in response to external stimuli open up new prospects for efficient and versatile design and shaping of three-dimensional objects. Here, we present a novel class of micro-structures exhibiting an extension-bending coupling (EBC) effect, that can be harnessed as an elementary building block for shape-shifting panels. They are built with a single material as a network of undulated ribbons. The deformations mechanisms of both single and connected undulated ribbons are analysed using the finite element method to explain the main features of the EBC mechanism. For a particular micro-structure of the proposed class, the elastic response is investigated both under small strain assumption combining two-scale homogenization with Kirchhoff–Love plate theory, and at finite strains relying on numerical analysis. The range of achievable EBC ratio is then assessed with respect to the geometric parameters of the unit cell. Patterned specimens are manufactured using a commercial FFF Ultimaker 3D printer and are mechanically tested at finite strain up to 20%. The displacement measured by point tracking match the predictions from the finite element simulations and indicate that the structure maintain its properties at finite strain. Moreover, a tensile test load with point-like boundary is proposed to highlight exceptional out of plane displacement. The proposed ribbon based architectures can be combined with active materials for the actuation of shape shifting structures, like soft robots, control systems and power devices.

Thermoelastic Damping and Frequency Shift in Kirchhoff Plate Resonators Based on Modified Couple Stress Theory With Dual-Phase-Lag Model

The present investigation deals with study of thermoelastic damping and frequency shift of Kirchhoff plate resonators by using generalized thermoelasticity theory of dual-phase-lag model. The basic equations of motion and heat conduction equation are written with the help of Kirchhoff-Love plate theory and dual phase lag model. The analytical expressions for thermoelastic damping and frequency shift of modified couple stress dual-phase-lag thermoelastic plate have been obtained. A computer algorithm has been constructed to obtain the numerical results. Influences of modified couple stress dual-phase-lag thermoelastic plate, dual- phase-lag thermoelastic plate and Lord-Shulman (L-S, 1967) thermoelastic plate with few vibration modes on the thermoelastic damping and frequency shift are examined. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically for clamped-clamped and simply-supported boundary conditions. It is observed from the results that the damping factor and frequency shift have noticed larger value in the presence of couple stress for varying values of length but opposite effect are shown for varying values of thickness in case of both vibration modes and boundary conditions.

Thermal effect on transverse vibrations of a nonhomogeneous rectangular thin plate subjected to a known temperature distribution

In this work, a model of thermoelasticity based upon the Kirchhoff–Love plate theory is constructed for studying the thermoelastic vibration of an arbitrary functionally graded rectangular thin plate subjected to a temperature distribution. The problem is solved in the context of the theory of dual-phase-lag of thermoelasticity. The plate is taken to be clamped on two opposite edges; one of those edges is subjected to a given temperature distribution, while the other is thermally insulated. The normal mode analysis is employed to find exact expressions for temperature, deflection, thermal stresses, and bending moments. As an illustrative example, the results were presented graphically for a plate made of a silicon material to show the consistency of the results.

Vibrations and buckling of orthotropic small‐scale plates with complex shape based on modified couple stress theory

AbstractFree vibrations of microplates with non‐classical shape are considered in the paper. Governing equations are based on the modified couple stress theory and Kirchhoff–Love plate theory. It is assumed that the plate is isotropic or orthotropic and satisfy various boundary conditions. An analysis is performed by the variational‐structural method which is based on R‐functions theory and Ritz method. The testing of the proposed method is carried out by comparison with known in literature results and analytical solutions. The developed method is applied to the study of the influence of the material length scale parameter, boundary conditions, shape parameters, material characteristics on vibration frequencies. Also, size‐dependent analysis for buckling of the isotropic and orthotropic microplate subjected to uniaxial load is performed by the proposed approach. Critical loads for a plate with complex shape, different types of boundary conditions and material are calculated in order to study their effect on buckling.

Variational approach to modeling soft and stiff interfaces in the Kirchhoff-Love theory of plates

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems.

Trace operators of the bi-Laplacian and applications

Abstract We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a nonclosed space. This can be fixed by switching to the Kirchhoff–Love traces from Führer et al. (2019, An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation. Math. Comp., 88, 1587–1619). Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the discontinuous Petrov–Galerkin method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the nonclosedness of a trace space) our analysis applies to any space dimension larger than or equal to two.

Free Vibration Investigation of Submerged Thin Circular Plate

Free vibration of coupled system including clamped-free thin circular plate with hole submerged in three-dimensional cylindrical container filled with inviscid, irrotational and compressible fluid is investigated in this work. Numerical approach based on the finite element method (FEM) is performed using the Comsol Multiphysics software, in order to analyze qualitatively the vibration characteristics of the plate. Plate modeling is based on Kirchhoff–Love plate theory. Velocity potential is deployed to describe the fluid motion since the small oscillations induced by the plate vibration is considered. Bernoulli’s equation together with potential theory is applied to get the fluid pressure on the free surface of the plate. To prove the reliability of the present numerical solution, a comparison is made with the results in the literature, which shows a very good agreement. Then, different parameters effect including fluid density, fluid height, free surface wave, hole radius and hole eccentricity on the natural frequencies of the coupled system is discussed in detail. Some three-dimensional mode shapes of the submerged plate are illustrated. Furthermore, the obtained results can serve as benchmark solutions for the vibration control, parameter identification and damage detection of plate.

FGM micro/nano-plates within modified couple stress elasticity

A unified formulation for plate bending problems is developed within the modified couple stress theory of elasticity with incorporating the deformation assumptions of three plate bending theories, such as the Kirchhoff-Love theory, 1st and 3rd order shear deformation plate theories. The microscopic structure of material is reflected in this higher-grade continuum theory via one material coefficient called the micro-length scale parameter. Furthermore the material can be composed of two micro-constituents what is included in the employed continuum model by functional gradation of the Young modulus through the plate thickness with assuming power-law dependence of volume fractions of micro-constituents on the transversal coordinate. The boundary restrictions on the bottom and top surfaces of the plate are discussed in details together with derivation of governing equations and physical boundary conditions on the plate edges. For numerical solution a novel method is developed with using the Moving Finite Element approximation for field variables. Several numerical simulations are devoted to study the influence of micro-length scale parameter as well as the level of gradation of Young’s modulus on coupled bending and in-plane deformation response modes.

MACROMECHANICAL ANALYSIS OF SPHERICALLY VOIDED BIAXIAL CONCRETE SLABS

The use of spherically voided biaxial concrete slab (SVBS) system, which uses hollow plastic balls as infill material, has increased widely because of its reduced weight-to-strength and weight-to-stiffness ratios when compared to solid concrete slabs. However, SVBS is a heterogeneous composite structure in which building a representative continuum model poses a significant challenge. To mitigate this challenge, the feasibility of determining the macromechanical structural behavior of spherically voided biaxial concrete slabs is studied using plate theories, aided by mechanical properties that were determined from a homogenization process of the representative volume element (RVE). This paper presents numerical analysis results of SVBS using both Mindlin-Reissner (thick) and Kirchhoff-Love (thin) plate theories. The results from both theories predicted the slab behavior reasonably well and they were within 10% of each other with the exception of the prediction of the twisting moment. Possible explanation of this deviation is provided in the paper.

Vibration of thin elastic FGM plates with multi-gradation effects

In this paper we investigate the vibration response of thin elastic FGM plates with combination of transversal and/or in-plane gradation of various material parameters subjected to transient tension loading. The equations of motion and initial-boundary conditions for transient problems are derived within Kirchhoff-Love plate theory. For numerical modelling of plates with dynamic multi-gradation coupling effects, it is developed the strong formulation with using the meshless approximation of field variables by the Moving Least Square (MLS) approximation scheme. Several numerical examples are presented for illustration of the multi-gradation coupling effects in vibration response of elastic FGM plates.

Stability of parametric vibrations of an isolated symmetric cross-ply laminated plate

The stochastic stability of parametric vibrations of an isolated symmetric cross-ply laminated plate system is investigated in this paper. The mechanical model of the plate system is given as a symmetric cross-ply laminated plate viscoelastically connected with two additional thin plates in order to satisfy necessary isolation (mechanical, thermal, acoustical, etc.) or structural requirements. Several important applications of the model are given in the introduction of the study. The investigated model is new and the adapted procedure is based on the analytical determination of the variable stability conditions as a consequence of the plate system being under the influences of the in-plane stochastic forces. The dynamic equilibrium equations are derived on the basis of the first-order shear deformation theory for laminated plates and Kirchhoff-Love plate theory for thin external plate-isolators. The stochastic stability of the coupled system, under parametric excitation of white noise, is investigated analytically through the moment Lyapunov exponents and Lyapunov exponents. The system of stochastic differential equations of motion is decoupled by using the contact transformation method. Then an intensive analytical procedure is continued through the perturbation method which is used in order to obtain the explicit asymptotic expressions for these exponents for the class of problems with three-degrees-of-freedom stochastic systems. The moment, almost-sure stability boundaries and critical excitation are presented and discussed for varying system parameters. The influences of the laminated plate parameters, damping coefficients and spectral density are taken into consideration in numerical examples. It is shown that thin plate-isolators could improve stochastic stability performances of composite laminated plates.

Passive vibration control using a metaconcrete thin plate

We investigated the band structure of flexural waves propagating in a metaconcrete thin plate, consisting of a concrete matrix reinforced by steel inclusions coated by rubber in square and triangular lattices. We considered the classical Kirchhoff-Love thin plate theory. We also studied the influence of inclusion geometry - circular, square and rotated square with a 45° angle of rotation with respect to the x, y axes. Improved plane wave expansion method was used to solve the wave equation considering flexural wave propagation in the xy plane. Flat bands and locally resonant band gaps were observed for all types of inclusion studied regarding square and triangular lattices. The locally resonant band gaps for square and rotated square inclusions in a triangular lattice were slightly shifted to higher frequencies than that given by circular inclusion. We suggest that the metaconcrete thin plate studied should be feasible for flexural vibration control in low frequencies.