Kirchhoff-Love Plate Theory Research Articles

Dynamics and stability of an axially moving plate interacting with surrounding airflow

The dynamics and stability of an axially moving plate interacting with surrounding axial airflow are investigated. The dynamic model of the axially moving plate is established based on the Kirchhoff–Love plate theory and the linear potential flow model. The natural frequencies of the plate under different moving velocities and airflow velocities are calculated by solving the generalized eigenvalue problems of the dynamic system, from which the stability of the plate is analyzed. The effects of the surrounding airflow and the length to width ratio of the plate on the critical divergence velocity and the flutter velocity are discussed. From the simulation, it can be seen that with the flow velocity increasing, the natural frequencies of the plate decrease and then couple, resulting in the plate losing stability of the divergence and flutter types. With the length to width ratio of the plate increasing, the critical divergence velocity and the flutter velocity of the plate decrease. The critical divergence velocity and flutter velocity of the plate in surrounding airflow are lower than those in vacuum, which indicates that it is necessary to consider the effects of the surrounding airflow when analyzing the stability of the axially moving plate.

Asymptotic near-field analysis of wedges in anisotropic composite plates using first-order shear deformation theory

In the present work, a complex potential approach is proposed in order to study the singular solution behaviour at wedges or sharp notches in fibre-reinforced composite plates using first-order shear deformation plate theory. The singularity exponent γ as a measure of the strength of the singularity is calculated for different notch opening angles and boundary conditions along the notch faces. Furthermore, the influence of the fibre orientation on the singularity exponent is discussed in detail. Asymptotic solutions of the governing system of partial differential equations are derived employing a Lekhnitskii-like formalism using three holomorphic potentials. Choosing the complex potentials according to prescribed boundary conditions finally leads to an eigenvalue problem where the singularity exponents appear as roots of the corresponding characteristic equation. In contrast to the classical Kirchhoff-Love plate theory, it is shown that the present approach allows for a distinction between singularities associated to transverse shear forces and to bending moments and that the fibre orientation significantly affects the singularity exponent. The obtained asymptotic near fields are compared to finite element data. The findings are in very good agreement with numerical results and results from literature available for the limit case of isotropic material behaviour.

Powell–Sabin B‐splines and unstructured standard T‐splines for the solution of the Kirchhoff–Love plate theory exploiting Bézier extraction

SummaryThe equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B‐splines and unstructured standard T‐splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B‐splines result in ‐continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non‐uniform rational B‐splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B‐splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T‐splines can be modified such that they are ‐continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T‐splines, Powell–Sabin B‐splines and NURBS‐to‐NURPS (non‐uniform rational Powell–Sabin B‐splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plate. Copyright © 2015 John Wiley & Sons, Ltd.

Damping in thin circular viscothermoelastic plate resonators

The governing equations of transverse motion and heat conduction of a homogenous, isotropic, thermally conducting, Kelvin–Voigt-type medium, based on Kirchhoff–Love plate theory, are established for out-of-plane vibrations of a generalized viscothermoelastic circular thin plate. The analytical expressions for thermoelastic damping of vibration and frequency shift are obtained for generalized and coupled viscothermoelastic plates. It is noticed that the damping of vibrations significantly depends on mechanical relaxation times and thermal relaxation time in addition to thermomechanical coupling in a circular plate under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The numerical results may also be illustrated in the case of a circular plate and an axisymmetric circular plate for clamped and simply supported boundary conditions for fixed aspect ratio, fixed radius, and fixed thickness, respectively.

Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models

• Extension to stress gradient elasticity of two variational principles of class elasticity. • Extra boundary conditions as congruence conditions at the boundary. • Extension to stress gradient elasticity of Timoshenko beam and Kirchoff plate. • Gradient-induced boundary conditions for beams and plates in stress gradient elasticity. • Comparison of stress gradient elasticity theories by Forest and Sab and by Polizzotto. The principle of minimum total potential energy and the primary principle of virtual power for stress gradient elasticity are presented as kinematic type constructs dual of analogous static type principles from the literature (Polizzotto, 2014; Polizzotto, 2015a). The extra gradient-induced boundary conditions are formulated as “boundary congruence conditions” on the microstructure’s deformation relative to the continuum, which ultimately require that the normal derivative of the stresses must vanish at the boundary surface. Two forms of the governing PDEs for the relevant boundary-value problem are presented and their computational aspects are discussed. The Timoshenko beam and the Kirchhoff–Love plate theories are extended to stress gradient elasticity under the assumption that stress gradient effects do not propagate transversally. It is shown that for beam models no extra gradient-induced boundary conditions are required, whereas for plate models such conditions must be enforced at the contour surface of the plate, where the normal derivative of the stress resultants are required to vanish. Appendix A is devoted to some basic aspects of the mechanics of the microstructure; Appendix B to a comparison between the present theory and an analogous theory from the literature (Forest and Sab, 2012).

Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory

This study deals with forced vibration analysis of a microplate subjected to a moving load. The formulation is developed based on the modified couple stress theory in conjunction with Kirchhoff–Love plate theory. The equations of motion of the problem are derived using Lagrange’s equations. In order to obtain the response of the microplate, the trial function for the dynamic deflection is expressed in the polynomial form. The equations of motion are solved by using the implicit time integration Newmark-β method, and then displacements, velocities and accelerations of the microplate at the considered point and time are determined. Five different sets of boundary condition are considered. For this purpose, boundary conditions are satisfied by adding some auxiliary functions to the trial functions. A parametric study is conducted to study the effects of the material length scale parameter, plate aspect ratio, boundary conditions and the moving load velocity on the dynamic response of the microplate. Also, in order to validate the present formulation and solution method, some comparisons with those available in the literature are performed. Good agreement is found. The results show that the dynamic deflections are significantly affected by the scale parameter and the load velocity.

Temperature effects on the vibration and stability behavior of multi-layered graphene sheets embedded in an elastic medium

This communication presents the thermal vibration and stability analysis of the multi-layered graphene sheets (MLGS) modeled as multi-nanoplate system (MNPS) embedded in an elastic medium. Nonlocal elasticity and Kirchhoff–Love plate theories are used for nanoplates, which are considered as orthotropic. Elastic medium is represented by the Winkler’s spring model, which couples adjacent nanoplates in MNPS. Governing equations are derived and exact closed form solutions for nonlocal frequencies, critical buckling loads and critical buckling temperature are obtained by using the Navier’s and trigonometric methods. The analytical results are validated with the corresponding results in the literature, results obtained via molecular dynamics simulation and numerical method of solution of the system of algebraic equations. To consider the thermal effects on the vibration and stability behavior of MNPS we adopted the coefficient of thermal expansion (CTE) obtained from the experimental results found in the literature. In the parametric study, effects of temperature change, nonlocal parameter, number of nanoplates and change of mediums stiffness on dimensionless nonlocal frequencies, critical buckling load and critical buckling temperature are investigated.

Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers.

The Chladni nodal line patterns and resonant frequencies for a thin plate excited by an electronically controlled mechanical oscillator are experimentally measured. Experimental results reveal that the resonant frequencies can be fairly obtained by means of probing the variation of the effective impedance of the exciter with and without the thin plate. The influence of the extra mass from the central exciter is confirmed to be insignificant in measuring the resonant frequencies of the present system. In the theoretical aspect, the inhomogeneous Helmholtz equation is exploited to derive the response function as a function of the driving wave number for reconstructing experimental Chladni patterns. The resonant wave numbers are theoretically identified with the maximum coupling efficiency as well as the maximum entropy principle. Substituting the theoretical resonant wave numbers into the derived response function, all experimental Chladni patterns can be excellently reconstructed. More importantly, the dispersion relationship for the flexural wave of the vibrating plate can be determined with the experimental resonant frequencies and the theoretical resonant wave numbers. The determined dispersion relationship is confirmed to agree very well with the formula of the Kirchhoff-Love plate theory.

Nonlocal mass-nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field

Nano-materials such as graphene sheets have a great opportunity to be applied in development of a new generation of nanomechanical sensors and devices due to their unique physical properties. Based on the nonlocal continuum theory and vibration analysis, the single-layered graphene sheet with attached nanoparticles affected by in-plane magnetic field is proposed as a new type of the mass-nanosensor. The nonlocal Kirchhoff-Love plate theory is adopted to describe mechanical behavior of single-layered graphene sheet as an orthotropic nanoplate. The equation of motion of a simply supported orthotropic nanoplate is derived, where the influence of Lorentz magnetic force is introduced through classical Maxwell׳s equations. Complex natural frequencies, damped frequency shifts and relative shift of damping ratio for nanoplate with attached nanoparticles are obtained in the explicit form. The influences of the nonlocal and magnetic field parameter, different mass weights and positions of attached nanoparticles and damping coefficients on the relative damped frequency shift and relative shift of damping ratio are examined. The presented results can be useful in the analysis and design of nanosensors applied in the presence of strong magnetic field. Our results show that magnetic field could be successfully used to improve sensibility performances of nanomechanical sensors.

Coupled extensional-flexural vibration behaviour of a system of elastically connected functionally graded micro-scale panels

This study is concerned with the free vibration behaviour of a system of elastically connected functionally graded micro-scale panels. The mechanical properties of the micro-panel are assumed to have a through-thickness variation and governed by a power-law relation in terms of the constituents’ volume fractions. The biharmonic equations governing the motion of each micro-panel are formulated through the adoption of the energy method along with the postulates of the Kirchhoff–Love plate theory. Concentrating on the asynchronous motion of the connected micro-panels, the study investigates the shift of the natural frequencies of the system as a result of variation in the: aspect ratio of the micro-panel, span-to-thickness ratio of the micro-panel; gradient index; small-scale effect; and the ratio of the Young’s modulus. Estimates of the natural frequencies, under the assumption of simply-supported edges of the micro-panels, are provided by the Navier’s solution method. The qualitative assessment of the mode...

Feasibility of using PZT actuators to study the dynamic behavior of a rotating disk due to rotor-stator interaction.

In this paper, PZT actuators are used to study the dynamic behavior of a rotating disk structure due to rotor-stator interaction excitation. The disk is studied with two different surrounding fluids—air and water. The study has been performed analytically and validated experimentally. For the theoretical analysis, the natural frequencies and the associated mode shapes of the rotating disk in air and water are obtained with the Kirchhoff-Love thin plate theory coupled with the interaction with the surrounding fluid. A model for the Rotor Stator Interaction that occurs in many rotating disk-like parts of turbomachinery such as compressors, hydraulic runners or alternators is presented. The dynamic behavior of the rotating disk due to this excitation is deduced. For the experimental analysis a test rig has been developed. It consists of a stainless steel disk (r = 198 mm and h = 8 mm) connected to a variable speed motor. Excitation and response are measured from the rotating system. For the rotating excitation four piezoelectric patches have been used. Calibrating the piezoelectric patches in amplitude and phase, different rotating excitation patterns are applied on the rotating disk in air and in water. Results show the feasibility of using PZT to control the response of the disk due to a rotor-stator interaction.

Modelling the mechanical behaviour of pit membranes in bordered pits with respect to cavitation resistance in angiosperms.

Various correlations have been identified between anatomical features of bordered pits in angiosperm xylem and vulnerability to cavitation, suggesting that the mechanical behaviour of the pits may play a role. Theoretical modelling of the membrane behaviour has been undertaken, but it requires input of parameters at the nanoscale level. However, to date, no experimental data have indicated clearly that pit membranes experience strain at high levels during cavitation events. Transmission electron microscopy (TEM) was used in order to quantify the pit micromorphology of four tree species that show contrasting differences in vulnerability to cavitation, namely Sorbus aria, Carpinus betulus, Fagus sylvatica and Populus tremula. This allowed anatomical characters to be included in a mechanical model that was based on the Kirchhoff-Love thin plate theory. A mechanistic model was developed that included the geometric features of the pits that could be measured, with the purpose of evaluating the pit membrane strain that results from a pressure difference being applied across the membrane. This approach allowed an assessment to be made of the impact of the geometry of a pit on its mechanical behaviour, and provided an estimate of the impact on air-seeding resistance. The TEM observations showed evidence of residual strains on the pit membranes, thus demonstrating that this membrane may experience a large degree of strain during cavitation. The mechanical modelling revealed the interspecific variability of the strains experienced by the pit membrane, which varied according to the pit geometry and the pressure experienced. The modelling output combined with the TEM observations suggests that cavitation occurs after the pit membrane has been deflected against the pit border. Interspecific variability of the strains experienced was correlated with vulnerability to cavitation. Assuming that air-seeding occurs at a given pit membrane strain, the pressure predicted by the model to achieve this mechanical state corresponds to experimental values of cavitation sensitivity (P50). The results provide a functional understanding of the importance of pit geometry and pit membrane structure in air-seeding, and thus in vulnerability to cavitation.

Elastic Properties and Stability of Physisorbed Graphene

Graphene is an ultimate membrane that mixes both flexibility and mechanical strength, together with many other remarkable properties. A good knowledge of the elastic properties of graphene is prerequisite to any practical application of it in nanoscopic devices. Although this two-dimensional material is only one atom thick, continuous-medium elasticity can be applied as long as the deformations vary slowly on the atomic scale and provided suitable parameters are used. The present paper aims to be a critical review on this topic that does not assume a specific pre-knowledge of graphene physics. The basis for the paper is the classical Kirchhoff-Love plate theory. It demands a few parameters that can be addressed from many points of view and fitted to independent experimental data. The parameters can also be estimated by electronic structure calculations. Although coming from diverse backgrounds, most of the available data provide a rather coherent picture that gives a good degree of confidence in the classical description of graphene elasticity. The theory can than be used to estimate, e.g., the buckling limit of graphene bound to a substrate. It can also predict the size above which a scrolled graphene sheet will never spontaneously unroll in free space.

Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium

Modeling of nano-structure systems using nonlocal theories has received a great attention in recent years. However, there are not so many papers giving the exact solution for the free vibration problem of multiple nano-structure systems especially when damping properties of the system are considered. Here, we analyzed the free transverse vibration of a viscoelastic multi-nanoplate system (MNPS) embedded in a viscoelastic medium taking into account small-scale effects by using the nonlocal theory of Eringen. The modified Kelvin–Voigt viscoelastic constitutive relation was used to model material and nano-scale characteristics of nanoplates. Based on the Kirchhoff–Love plate theory, D’ Alembert’s principle and viscoelastic constitutive relation, we derived the homogeneous system of m partial differential equations of motion coupled through the viscoelastic interaction. The closed form solutions of undamped natural frequencies and modal damping factor were obtained utilizing the Navier’s method and trigonometric method for the case of “Cantilever-Chain” system. The proposed analytical results are verified with the results available in the literature and excellent agreement is achieved. In addition, we investigated the influences of a nonlocal and viscoelastic parameter, plate aspect ratio and coefficients of a viscoelastic medium on the complex eigenvalues through numerical simulations.

Finite-difference time-domain analysis of structure-borne sound using a plate model based on the Kirchhoff-Love plate theory

Finite-difference time-domain analysis of structure-borne sound using a plate model based on the Kirchhoff-Love plate theory

Low-complexity computation of plate eigenmodes with Vekua approximations and the method of particular solutions

This paper extends the method of particular solutions (MPS) to the computation of eigenfrequencies and eigenmodes of thin plates, in the framework of the Kirchhoff-Love plate theory. Specific approximation schemes are developed, with plane waves (MPS-PW) or Fourier-Bessel functions (MPS-FB). This framework also requires a suitable formulation of the boundary conditions. Numerical tests, on two plates with various boundary conditions, demonstrate that the proposed approach provides competitive results with standard numerical schemes such as the finite element method, at reduced complexity, and with large flexibility in the implementation choices.

An asymptotic method for deriving the equations of the Reissner-Mindlin plate theory

The equations of the Kirchhoff-Love and Reissner-Mindlin plate theories are derived with the use of the asymptotic homogenization method.

Buckling analysis of Levy-type orthotropic stiffened plate and shell based on different strain-displacement models

In the present work a model able to predict the buckling behavior of thin, orthotropic, stiffened plates and shells subject to axial compression is proposed. In the context of the Kirchhoff-Love plate theory and making use of different strain-displacement models – namely the von Kármán model, the Koiter–Sanders shell model, an enhanced von Kármán model and a spurious model commonly adopted in literature – the equilibrium equations have been solved by the Levy-type approach. The results obtained highlight the influence of each non-linear strain-displacement term and show that the von Kármán model can noticeably overestimate the buckling load when the critical mode involves significant in-plane displacements.

Characterisation of the bending stiffness components of MDF panels from full-field slope measurements

This paper deals with the characterisation of bending stiffness components of medium density fibreboard (MDF) by carrying out a single plate bending test. The approach couples full-field slope measurements with an inverse identification method. MDF panels manufactured with different fractions of Eucalyptus fibres and sugarcane bagasse particles were used. The slope maps generated across the plate surface were measured by the deflectometry technique. The curvature fields of the deformed plate were reconstructed by numerical differentiation afterwards. The virtual fields method was then implemented for material parameter identification under the framework of Kirchhoff–Love plate bending theory. The elastic properties obtained from the proposed data reduction (i.e. simultaneous identification of modulus of elasticity, Poisson’s ratio and shear modulus) were compared with values determined from classical three-point bending tests and reported in the literature. The set of properties were found in relatively good agreement.

Dynamic analysis of an axially translating plate with time-variant length

The linear dynamic analysis of a translating cantilever plate model characterized by time-variant length and axial velocity is investigated. A length-dependent governing partial differential equation (PDE) of motion is formulated by the extended Hamilton’s principle based on Kirchhoff–Love plate theory. The tension in the system arising from the longitudinal accelerations and in-plane stresses are incorporated. Further, the extended Galerkin method along with the Newmark direct time integration scheme is employed to simulate the response of the system. Stability and vibration characteristics are studied according to the quadratic eigenvalue problem from the governing PDE, which demonstrates that the coupling effects between the axial translation motion and the flexural deformation stabilizes the system during the extension and destabilizes it in the retraction. The computation results show that the translating velocity and the aspect ratio affect the natural frequencies and stability for the out-of-plane vibration of the moving plate. A domain for the traveling velocity associated with the stable system is given, and the critical failure velocity is also predicted based on numerical simulations.