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This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy, where the clamped boundary is partially debonded along the thickness direction. Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates, incorporating the Gurtin-Murdoch surface elasticity theory. For two typical cases of constant bending moment and uniform shear force in the debonded segment, the associated problems are reduced to two mixed boundary value problems. By solving the resulting mixed boundary value problems using the Fourier integral transform, a new type of singular integral equation with two Cauchy kernels is obtained for each case, and the exact solutions in terms of the fundamental functions are determined using the Poincare-Bertrand formula. Asymptotic elastic fields near the debonded tips including the bending moment, effective shear force, and bulk stress components exhibit the oscillatory singularity. The dependence relations among the singular fields, the material constants, and the plate's thickness are analyzed for partially debonded cantilever micro-plates. If surface energy is neglected, these results reduce the bending fracture of a macroscale partially debonded cantilever plate, which has not been previously reported. [ABSTRACT FROM AUTHOR]

AbstractThe present article aims at performing a global sensitivity analysis of modal properties of a frequency dependent viscoelastic sandwich plate. The latter are evaluated using a finite element model relying on Kirchhoff theory for the elastic faces and the Mindlin-Reissner approach for the viscoelastic layer. The plate’s characteristics are considered as random variables following log normal laws. Sparse polynomial chaos expansions are employed to evaluate Sobol’s indices. For high damping, the resonant frequencies and loss factors turn to be sensitive to the elastic face properties, while in the weak damping case they are mostly affected by the viscoelastic model parameters. [ABSTRACT FROM AUTHOR]

Recent studies have underscored the significance of the capillary fringe in hydrological and biochemical processes. Moreover, its role in shallow waters is expected to be considerable. Traditionally, the study of groundwater flow has centered on unsaturated-saturated zones, often overlooking the impact of the capillary fringe. In this study, we introduce a steady-state two-dimensional model that integrates the capillary fringe into a 2-D numerical solution. Our novel approach employs the potential form of the Richards equation, facilitating the determination of boundaries, pressures, and velocities across different ground surface zones. We utilized a two-dimensional Freefem++ finite element model to compute the stationary solution. The validation of the model was conducted using experimental data. We employed the OFAT (One_Factor-At-Time) method to identify the most sensitive soil parameters and understand how changes in these parameters may affect the behavior and water dynamics of the capillary fringe. The results emphasize the role of hydraulic conductivity as a key parameter influencing capillary fringe shape and dynamics. Velocity values within the capillary fringe suggest the prevalence of horizontal flow. By variation of the water table level and the incoming flow q0, we have shown the correlation between water table elevation and the upper limit of the capillary fringe. [ABSTRACT FROM AUTHOR]

• Flexoelectricity and surface effect on piezoelectric nanolaminates were reported. • Analytical solution of bending deformation and vibration were obtained. • Effects of residual surface stress and the flexoelectric coefficient were examined. • Effects of aspect ratio and plate thickness on bending behavior were examined. In this paper, a model of double-layer rectangular piezoelectric nanolaminates with different upper and lower surface properties is established. The bending and vibration behavior of the piezoelectric nanolaminates are evaluated and analyzed, taking into consideration the flexoelectric and surface effects. Utilizing the surface piezoelectric model, flexoelectric theory, and Kirchhoff plate theory, this study provides analytical solutions for the bending deflection and natural frequency of rectangular piezoelectric nanolamination materials under various boundary conditions. The results indicate that the effects of flexoelectric and surface influences on piezoelectric nanolaminates are associated with residual surface stress and flexoelectric coefficients. These effects can enhance the bending stiffness and decrease the natural frequency of the piezoelectric nanolaminates. Furthermore, altering the residual stress on the upper and lower surfaces of the laminates separately will result in different trends in bending deflection, while causing a consistent trend in natural frequency change. When the electric potential direction of piezoelectric nanolaminates is different, the bending deflection of piezoelectric nanolaminates will change accordingly. The natural frequency of piezoelectric nanolaminates is determined by the absolute value of the flexoelectric coefficients, and is independent of their positive or negative values. Both the surface effect and the flexoelectric effect are influenced by size-dependent behaviors. When the thickness of the laminates is sufficiently large, both the surface effect and flexoelectric effect can be disregarded. This paper's research methods and results provide theoretical models and analytical methods for the microstructure design, multi-physical field characterization, and bending deformation behavior of intelligent components containing rectangular piezoelectric nanolaminates. [ABSTRACT FROM AUTHOR]

This research investigates the free vibration of a rotating annular microplate under the flexoelectric effect. Initially, the Kirchhoff plate theory assumptions are used to express the displacement fields. After considering the displacement field, strains and their gradients are derived and substituted into the electric enthalpy and kinetic energy expressions. Subsequently, by applying Hamilton’s principle to the aforementioned equations, the electric and mechanical equations are computed. To derive the equations of motion, initially, the polarization vectors and their gradients are derived from the electric equations and associated boundary conditions. Subsequently, these are incorporated into the mechanical equations, which also encompass electric components. It is notable that by removing the time-dependent terms from the in-plane equations of motion, the static displacement due to rotation at each speed is obtained. After deriving the equations of motion and boundary conditions, these equations are non-dimensionalized using non-dimensionalizing relations. In the next step, Hamilton’s principle is used to discretize the equations and boundary conditions. Consequently, by applying the generalized differential quadrature method and extracting the stiffness and mass matrices resulting from the transverse equation of motion and boundary conditions, the natural transverse frequency of the rotating annular microplate under the flexoelectric effect is calculated. The results of this research are useful for promoting the use of rotating annular microplants under flexoelectric effect for microelectromechanical systems designers with high efficiency. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear free vibration of spinning pre-twisted functionally graded material (FGM) plates in thermal environment. Based on the Kirchhoff plate theory and von Kármán large-deformation theory, the nonlinear dynamic model is established via the Lagrange equation, where the warping effect is taken into account. The material properties are related to temperature and change according to a power-law exponent distribution. The theoretical solution of nonlinear-free vibration is obtained using the multiple scale method. The effects of power-law exponent, temperature difference, spinning speed, hub radius ratio, and twist angle on nonlinear-free vibration characteristics of the spinning pre-twisted cantilever FGM plates are discussed. The results show that the spinning pre-twisted cantilever FGM plate exhibits nonlinear hardening-spring vibration behavior, while the characteristics weaken with the increase of spinning speed. The twist angle and setting angle have coupling effects on the nonlinear vibration characteristics of spinning pre-twisted FGM plates. [ABSTRACT FROM AUTHOR]

This document is a foreword to a special issue of the journal "Mathematics & Mechanics of Solids" dedicated to Professor Alain Goriely. The foreword highlights Professor Goriely's achievements and contributions to the field of mathematics and mechanics, particularly in the areas of nonlinear elasticity and biological growth. It also mentions his involvement in scientific dissemination and his support for junior researchers and graduate students. The special issue includes articles on various topics related to Professor Goriely's research interests, such as theories of rods and shells, materials modeling, energy, biomechanics, and mathematical foundations of mechanics. The foreword expresses the hope that Professor Goriely will find the collection inspiring and wishes him many more successful years of research. [Extracted from the article]

A concise derivation of Kirchhoff's theory for naturally curved and twisted rods is presented as a prelude to the derivation of the theory for the elastic response of rods that have undergone prior plastic deformation. [ABSTRACT FROM AUTHOR]

Composite corrugated plates have a great potential in the application to morphing wings. However, it takes high computational cost to conduct flutter analysis with detailed 3D finite element models due to its structural complexity. In this study, an analytical method is proposed for flutter stability analysis of composite corrugated plates in supersonic flow. The trapezoidal and sinusoidal composite corrugated plate is homogenized as an equivalent anisotropic plate based on an energy approach. The flutter model for the composite corrugated plate is derived based on Kirchhoff plate theory and the equivalent stiffness properties. The unsteady aerodynamic pressure is evaluated by using the supersonic piston theory in which the corrugated section is taken into account. Hamilton's principle with the assumed mode method is applied to formulate the aeroelastic equation of the composite corrugated plate. The eigenvalue criterion is utilized to reveal the flutter mechanism and evaluate the stability of composite corrugated plates in supersonic flow. The accuracy and reliability of the present method are verified by comparing aeroelastic responses with those obtained from commercial software. Parametric studies concerning different parametric variables are also conducted. It is shown that the proposed method has sufficient accuracy and requires less computational effort, providing a theoretical basis for the utilization of trapezoidal and sinusoidal composite corrugated plates in morphing wings. [ABSTRACT FROM AUTHOR]

Currently, bending deformations of lithospheric plates are calculated based on Kirchhoff's theory of bending of thin plates, which was formulated about 170 years ago. This paper examines the possibility of refining these calculations based on the theory of bending of thick plates developed by Timoshenko and Reissner. A new unified derivation is presented of the Timoshenko equations for 2D bending of beams and the Reissner equations for bending of slabs by direct transformation of the general elasticity equations with a simple approximate replacement of power-cubic functions with effective linear ones. This allows a more detailed and clearer understanding of the differences between the equations and the sense of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the known test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are given. [ABSTRACT FROM AUTHOR]

On the 200 th birthday of Gustav Robert Kirchhoff: The plate theory was finalised 175 years ago Gustav Robert Kirchhoff (1824–1887) is remembered for five important achievements in 19th century physics: – Circuit laws (Kirchhoff's current and voltage law), 1845 – Theory of the thin, elastic plate, 1850 – Theory of the thin, elastic rod with large deformations, 1859 – Foundation of spectroscopy with Robert W. Bunsen (1811–1899), 1859 – Law of thermal radiation, 1859 There is also the Piola‐Kirchhoff stress tensor from continuum mechanics. This list of Kirchhoff's impact on physics could be continued. In addition to the biography, the following focuses on the genesis of the theory of the thin, elastic plate, which is important for reinforced concrete construction. [ABSTRACT FROM AUTHOR]

We consider the following Kirchhoff problem - (a+b ∫R³ |▽u|²) Δu + (1+εV(x))u = |u|p-2u, where a,b > 0, and 2 < p < 6. Under suitable assumptions on V, by using the Lyapunov-Schmidt reduction method, we obtain the existence of multi-bump solutions. [ABSTRACT FROM AUTHOR]

• An effective mathematical model for bolted flange joints is proposed. • The proposed model is applicable for both bolt looseness and no-looseness conditions. • Experimental results validate the proposed model. • The soft nonlinearity characteristics of bolted flange joint multi-plate structure under base excitation are revealed. Bolted flange joints are widely used in various engineering structures to combine two or more substructures together. However, the understanding of mechanical mechanism of bolted flange joints is still insufficient due to the lack of efficient mathematical models. This paper proposes an effective mathematical model for bolted flange joints, which is then employed to conduct vibration analysis on bolted flange joint multiple-plate structures. During the modeling process, the flange is modeled by the Kirchhoff plate theory, and the bolted joints are modeled by the two-dimensional face constraint in bolted joint affected region (BJAR). The artificial spring technique is adopted to simulate the face constraint. The friction contact model and equivalent linearization technique are adopted to analyze the stick and slip states of bolted joints. By adopting the Chebyshev orthogonal polynomials as admissible functions, governing equations are derived. Then, the Rayleigh-Ritz method is applied to study the free vibration characteristics, and the Newmark-beta method is employed to study the vibration response under base excitation. The experimental studies are performed on the bolted flange joint two-plate structure (BFJTPS) to illustrate the effectiveness of the proposed theoretical model. The results indicate that the proposed mathematical model can well predict the vibration characteristics of multiple-plate structures under both bolt looseness and no-looseness conditions. The soft nonlinearity phenomenon under bolt looseness condition can be successfully revealed by the model. [ABSTRACT FROM AUTHOR]

In the framework of the work the description of sphere wave filed in the far observation region is discussed. The consideration is based on the decomposition of the wave field over the longitudinal and transverse spatial parameters of the problem in the direction of observation. It is shown that the approximation of a sphere wave by the flat field is correct only if the consideration is conducted in the limit of solid angle. The magnitude of the solid angle, which includes the area of flatness of the spherical wave, is determined using the so-called wave parameters corresponding to the observation area and the illuminated point. The wave field created by a small flat area is defined as a limit case of a superposition field generated by a system of point sources. It is shown that in an observation point the wave field of a small flat area can be characterized by the angle between the area normal and the vector indicting from the given area the observation point. The problem of description of a wave filed in the framework of a small flat aria is also discussed. A new method for deriving of the well-known obliquity factor of the Kirchhoff diffraction theory is suggested. [ABSTRACT FROM AUTHOR]

Flexoelectricity, referring to the electrical polarization generated by strain gradients, is an electromechanical coupling phenomenon in all dielectric materials. Based on the Kirchhoff plate theory, this paper investigates the influence of flexoelectricity on the electromechanical coupling response of piezoelectric circular nanoplates with different electric boundary conditions. Using the variational principle and Gibbs free energy, the governing equations and boundary conditions of piezoelectric nanoplates are derived. The analytical solutions of the deflection, polarization, and induced electric potential are obtained. The results show that the flexoelectric effect is size-dependent and has a more significant influence on the electrostatic responses than the piezoelectric effect at the nanoscale. The results also show that the induced electric potential due to the flexoelectric effect may be helpful for sensing or energy harvesting designs. [ABSTRACT FROM AUTHOR]

• The symplectic superposition method for buckling of thick plates is developed. • New benchmark buckling solutions of TSDT-based non-Lévy-type plates are obtained. • The effects of boundary conditions and geometric dimensions are revealed. • A quantitative comparison study between TSDT and CPT/FSDT is conducted. The third-order shear deformation theories (TSDTs) are frequently employed to enhance the accuracy of mechanical behavior characterization for thick plates. However, the analytic buckling solutions based on the TSDTs are rare due to the major challenge in mathematical treatments. This study introduces an analytic symplectic superposition method to derive new benchmark buckling solutions of TSDT-based rectangular thick plates with non-Lévy-type boundary conditions (BCs), which are inaccessible using conventional analytic approaches. The governing equations are formulated in the Hamiltonian system. The symplectic framework-based mathematical techniques are manipulated, which shapes the characteristic of the method that a rational derivation is realized without any predetermined solution forms. The comprehensive buckling load factors and mode shapes of fully clamped plates and their variations are given, all of which are verified through comparison with the finite element method as well as literature results. Utilizing the obtained solutions, a comprehensive parametric analysis is conducted to elucidate the effects of BC, thickness-to-width ratio, and aspect ratio on critical buckling load factors. A comparison study is made between the results by the TSDT and those by the Kirchhoff theory and the first-order shear deformation theory, which demonstrates the superior accuracy of the former. The proposed rigorous framework offers a rational route to obtaining more benchmark buckling solutions. [ABSTRACT FROM AUTHOR]

Introduction/purpose: This article presents the bending examination of advanced-generation composite structures with specific properties exposed to different loads. Methods: This paper thus proposes and introduces a new generalized fivevariable shear strain theory for calculating the static response of functionally graded rectangular plates made of ceramic and metal. Notably, our theory eliminates the need for a shear correction factor and ensures zero-shear stress conditions on both the upper and lower surfaces. Numerical investigations are introduced to interpret the influences of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the bending behavior of FGPs. These analyzes are then compared to the results available in the literature. Results: Preliminary results include a comparative analysis with standard higher-order shear deformation theories (PSDPT, ESDPT, SSDPT), as well as Mindlin and Kirchhoff theories (FSDPT and CPT). Conclusion: Our theory contributes alongside established theories in the field, providing valuable insights into the static thermomechanical response of functionally graded rectangular plates. This encompasses the influence of volume fraction exponent values on nondimensional displacements and stresses, the impact of aspect ratios on deflection, and the effects of the thermal field on deflection and stresses. Numerical examples of the bending examination of advanced-generation composite structures with specific properties exposed to different loads demonstrate the accuracy of the present theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a class of frictional antiplane contact problems of p(x)-Kirchhoff type on a bounded domain Ω ⊆ R². Using an abstract Lagrange multiplier technique and the Schauder fixed point theorem we establish the existence of weak solutions. Imposing some suitable monotonicity conditions on the datum f1 the uniqueness of the solution is obtained. [ABSTRACT FROM AUTHOR]

Analytical techniques used for estimating thermoelastic damping by incorporating both mechanical and thermal interactions between surfaces and the rest of the bulk are intricate and challenging due to the limited understanding of the damping mechanisms in extra-thin films subjected to forced vibrations. This paper proposes a modified model to analytically calculate the thermoelastic damping of ultrathin elastic films due to surface effects and analyzes the thermoelastic damping variation with different factors through numerical experiments on two materials. The model considers surface stresses derived from the elastic surface theory using Kirchhoff's kinetic hypothesis and determines thermoelastic damping by considering thermal dissipation and elastic potential energy. The results show that surface effects significantly influence the thermoelastic damping of the film, and the specific behavior of a thin film's thermoelastic damping with respect to film thickness is impacted by various factors, including material property, the variation range of film thickness, and the forced vibration frequency. This study provides insights into the thermoelastic damping behavior of thin films and has important implications for the development of nanoscale oscillators in MEMS or NEMS systems. [ABSTRACT FROM AUTHOR]

The ground surface deformation induced by shield tunnels passing through enclosure structures of existing tunnels is a particular underground construction scenario that has been encountered in Wuhan Metro Line 12 engineering cases in China. Timely ground deformation prediction is important to keep shield tunneling safe. However, the classic ground deformation theory is difficult to accurately predict for this ground deformation. This paper develops a semi-analytical method to predict ground heave considering the space effect in this engineering condition. Based on the improved ground deformation theory, a novel deformation prediction method for the ground and enclosure structure is derived and combined with Kirchhoff plate theory. Comparing with field deformation measurements, the maximum difference between the measured and calculated deformation is 14.6%, which demonstrates that the proposed method can be used to predict the ground heave induced by shield tunnels passing through the enclosure structure of existing tunnels. The parameters of the underground diaphragm wall used in Wuhan Metro Line 12 are further studied in detail. The results show that the ground heaves have a positive correlation with the embedded ratio of the diaphragm wall, but a negative correlation with its elastic modulus and thickness. However, the thickness and embedded ratio have a limited effect on ground heaves. This study provides a technical reference for optimizing the setting of enclosure structures in order to protect existing buildings. [ABSTRACT FROM AUTHOR]

Laminated plates with holes are often used in industrial applications such as aeronautics, automobiles, and marine. It is necessary to present a study of the combined effect of deformation theories and thickness variation for the laminated plate with a hole. This manuscript considers analysis theories named: Kirchhoff, layer-wise, and Reissner-Mindlin, to study their validity for different thickness aspect ratios under transverse compression loading. Studies are conducted on CFRP laminate (symmetric cross-ply and quasi-isotropic), and the performed numerical (FEM) analysis processes are validated through existing literature. Transverse shear, circumferential, radial, and radial-hoop stress variations with stress concentration factors are presented along the thickness and around the hole configurations (18 cases are covered) of the plate. Also, the resultant effects are discussed on in-plane, and out-of-plane stresses for laminates to specify the selection of design conditions (theory, thickness, model, and laminates). This research may provide engineers and researchers with various assessments and design insight for laminate structures with a hole. [ABSTRACT FROM AUTHOR]

Featured Application: The drive and deployment unit of the variable diameter internal drive device described in this paper is primarily intended for use in space deployment structures. This innovative mechanism is expected to serve as a potential deployment mechanism for large-aperture space telescopes, providing the capability to explore distant deep space. The limitations associated with traditional screw-based and cable-based deployment mechanisms for space deployable structures, such as deployment stiffness, accuracy, and distance, are effectively overcome by introducing the concept of a variable diameter internal drive device. To enhance stability during the discontinuous surface transition phase inside the tubular structure, a thin-walled flexible beam structure is adopted for the driving deployment unit. The analysis employs the spatial absolute nodal coordinate formulation, integrating the pose states of each node into the global coordinate system. The three-dimensional displacement field and rotational angle changes at different time intervals are obtained using the unit shape function matrix and Kirchhoff theory. Subsequently, a dynamic model of the corresponding spatial nodes is established using the virtual work principle. This significant improvement enhances the dynamic characteristics of the coupled rigid-flexible deformation of the driving deployment unit under radial and axial external forces, surpassing the previously used flexible beam structure. The dynamic simulation analysis is performed using the finite element method and validated through experimental tests. The experimental results confirm the driving deployment unit's stability and successful achievement of the desired functionalities, as demonstrated by the endpoint displacement, three-dimensional centroid displacement, and trajectory rotation angle. [ABSTRACT FROM AUTHOR]

We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel 'Blake-Zisserman-Kirchhoff' functional on a natural space of 'flat and fractured' two-dimensional isometric immersions that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids. [ABSTRACT FROM AUTHOR]

• Bending edge wave is studied in a thin transversely isotropic viscoelastic plate supported by a viscoelastic foundation. • The nonlocal elasticity theory is used to extract the micro-scale effect on the travelling edge wave. • The Kirchhoff plate theory and the sinusoidal waveform are considered to obtain the closed-form solution. • It is observed that due to the present of viscosity in the foundation the speed of the wave is reduced. The analysis consists of the dynamic behaviour of a thin semi-infinite functionally graded plate under bending edge wave propagation. The plate is made of transversely isotropic materials supported by an elastic foundation. A quadratic variation of the properties of the materials is used to examine the behaviour of the FGM (Functionally Graded Materials) plate in the bending edge wave propagation. The Eringen differential form is employed to transition from non-local to local elasticity. The parameter entailed in the differential form is used to extrapolate the microstructure's impact on the bending edge wave propagation in the functionally graded plate. Additionally, the surface elasticity theory is included to investigate the surface effects on the dispersion of edge wave. The viscosity effect of the plate and the foundation are taken into consideration while formulating the problem. To develop the kinematics of FGM plates, the Kirchhoff plate theory is considered. The motion of the plate is taken along the transverse direction of the plate. The viscoelastic theory proposed by Kelvin-Voigt is used to formulate the mathematical theory of the proposed model. A coupled equation corresponding to the moments and shear forces has been obtained while formulating the plate equation of motion. The rotatory inertia effect is neglected due to the small deflection of the plate. The numerical simulation presented in the paper shows the influence of the viscosity of the elastic foundation, surface effects, non-local elasticity, and the density of the materials on the natural frequency of bending edge wave propagation. [ABSTRACT FROM AUTHOR]

The main focus of this paper lies in investigating the existence of infinitely many positive weak solutions for the following elliptic-Kirchhoff equation with Dirichlet boundary condition The methodology adopted revolves around the technical approach utilizing the direct variational method within the framework of anisotropic variable exponent Sobolev spaces. [ABSTRACT FROM AUTHOR]

Quantifying the intrinsic mechanical properties of one-dimensional nanostructures such as nanotubes and nanowires is technically challenging due to their extremely small sizes and large aspect ratios. In comparison with direct tensile measurements, displacement responses of an end-clamped rod under transverse loads are more significant and more feasible for experimental characterization. However, the displacement–force relationship could be non-trivial due to the presence of geometrical nonlinearity and contributions from both stretching and bending. Choosing a simple but reliable model to extract the mechanical parameters from experimental data is thus important for the design of tests. Starting from the fully nonlinear, extensible Kirchhoff rod theory, we explore the application scope of several simplifications by referring to recent experimental studies on carbon nanotubes. The horizontal displacement is shown to be crucial information for strain analysis in the stretching-dominated regime, and the constant-tension assumption fails at large loading amplitudes. The capability of several simplified models is assessed through the Euclidean distance between deflection curves, as well as the error in estimating the strain distribution. Practical issues such as boundary slippage and dynamical effects are also discussed. This study offers a theoretical groundwork to understand the mechanical responses of one-dimensional nanostructures in typical experimental setups and provides a standard or guideline for the experimental design. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear energy sink (NES) and a giant magnetostrictive-piezoelectric (GMP) energy harvester (NES-GMP) are investigated to suppress the nonlinear aeroelastic responses and to absorb the mechanical energy of an embedded plate interacting with external subsonic airflow. The analytical model of the embedded plate attaching the NES-GMP is established by using the Hamilton's principle based on the Kirchhoff plate theory and incompressible subsonic aerodynamic model. The natural frequencies of the plate with the NES-GMP and with the NES are analyzed by solving the generalized eigenvalue problems. The global amplitude-frequency responses of the pure plate, the plate with NES, and the plate with NES-GMP are compared to show the vibration suppression effects of the proposed method. Based on the energy analysis, the energy transfer mechanisms of the plate with the NES-GMP are studied. The results reveal that the input energy of the NES-GMP converts into the strain energy by the Terfenol-D layer and then transforms into the electric energy by the piezoelectric layer. Furthermore, numerical simulations also indicate that the maximum harvested energy is obtained when the airflow velocity approaches the critical divergence flow velocity of the plate, and the most effective harvesting installation position is in a specific annular region near the edge of the plate. [ABSTRACT FROM AUTHOR]

The vibrational power flow and damage identification of functionally graded material (FGM) plates with a through-width edge crack are studied in this paper based on the Kirchhoff plate theory. The distribution of material properties obeys the power law function with respect to the thickness. The linear spring model is employed to simulate the crack in the FGM plate. The reference surface of the FGM plate is the physical neutral surface, and the governing equations are derived based on Hamilton's principle. The wave propagation approach is employed to compute the input and the transmitted power flows of intact and cracked FGM plates under the harmonic distributed transverse load. The input and transmitted power flows are sensitive to the crack location, crack depth, and gradient index. For the cracked FGM plate, a novel damage index is defined via the transmitted power flow distribution and continuous wavelet transform. The crack location in the FGM plate can be detected by finding the peak of the damage index. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the (p, q)-Kirchhoff type problem - 1 + a ∫ R N | ∇ u | p d x Δ p u - 1 + b ∫ R N | ∇ u | q d x Δ q u + V ε (x) | u | p - 2 u + | u | q - 2 u = f (u) in R N , where ε > 0 is a small parameter, a , b > 0 , 1 < p < q < N < 2 q , Δ s u = div (| ∇ u | s - 2 ∇ u) , with s ∈ { p , q } , is the s-Laplacian, V : R N → R is a continuous potential, V ε (x) = V (ε x) and f : R → R is a continuous non-linearity without satisfying usual Ambrosetti–Rabinowitz condition. By using Nehari manifold techniques and perturbation methods, we establish the existence of positive ground state solution, and by those methods combined with Ljusternik–Schnirelmann theory, we investigate the multiplicity and concentration of solutions. In particular, our results extend and improve the works of He and Zou (J Differ Equ 252:1813–1834, 2012). [ABSTRACT FROM AUTHOR]

Depending on the load level, structures can experience a material nonlinearity known as elastoplasticity, which has an important role in the behaviour of structures. In order to avoid the elastoplastic behaviour, it is necessary to find the optimal thickness distribution, which corresponds to the minimum mass that provides an elastic behaviour for a certain load level. The elastoplasticity analysis of functionally graded axisymmetric shells under axisymmetric mechanical loading, and the subsequent optimization, was performed by using a simple conical frustum finite element model with two nodal circles; three degrees of freedom per node, which was based on Kirchhoff's theory allowing for shear deformation; and using a reduced numerical integration procedure that is essential for its success when applied to thin shells. The formulation accounts for the calculation of the displacements and through-thickness stress distribution, including the effective stress. In this work, the thickness was the design variable in the optimization procedure and the mass was the objective function that needed to be minimized subject to a constraint imposed on the effective stress. The optimization solutions were obtained by using a feasible arc interior point gradient-based algorithm. Some illustrative examples were performed, and the corresponding results are presented and discussed. [ABSTRACT FROM AUTHOR]

In the past, to model fibre stiffness of finite-radius fibres, previous finite-strain (nonlinear) models were mainly based on the theory of non-linear strain-gradient (second-gradient) theory or Kirchhoff rod theory. We note that these models characterize the mechanical behaviour of polar transversely isotropic solids with infinitely many purely flexible fibres with zero radius. To introduce the effect of fibre bending stiffness on purely flexible fibres with zero radius, these models assumed the existence of couple stresses (contact torques) and non-symmetric Cauchy stresses. However, these stresses are not present on deformations of actual non-polar elastic solids reinforced by finite-radius fibres. In addition to this, the implementation of boundary conditions for second gradient models is not straightforward and discussion on the effectiveness of strain gradient elasticity models to mechanically describe continuum solids is still ongoing. In this paper, we develop a constitutive equation for a non-linear non-polar elastic solid, reinforced by embedded fibers, in which elastic resistance of the fibers to bending is modelled via the classical branches of continuum mechanics, where the development of the theory of stresses is based on non-polar materials; that is, without using the second gradient theory, which is associated with couple stresses and non-symmetric Cauchy stresses. In view of this, the proposed model is simple and somewhat more realistic compared to previous second gradient models. [ABSTRACT FROM AUTHOR]

The main novelty of the paper is that analytical, experimental, and numerical analyses are used to investigate the free vibration problem of a sandwich structure in which Nanocomposites skins (SiO2/epoxy and Al2O3/epoxy) at different densities are used as the face sheet. The volume fraction's of nanoparticle addition varies (0% to 2.5%). The present free vibration was derived based on Kirchhoff's theory and aspiration to obtain the natural frequency. The results show that in structures with SiO2 nanoparticles with a density of 1180 kg/m3, the optimum increase (VF = 2.5%) is 50% in Young's modulus and 22% in natural frequency, while at a density of 1210 kg/m3 is 56 % in Young's modulus and 24.5% in natural frequency. Furthermore, the same structures reinforced with Al2O3 Nano-particles show that at the density of 1180 kg/m3, the optimum (VF=2.5%) parentage increase in Young's modulus is 41% and 19% in natural frequency, while at the density of 1210 kg/m3 is 46% in Young's modulus and 21% in natural frequency. A numerical investigation was used to validate the obtained results of the analytical solution. The findings also show an acceptable convergence between analytical and numerical techniques with a maximum discrepancy not exceeding 3%. [ABSTRACT FROM AUTHOR]

The stall aerodynamic model based on Kirchhoff's theory of flow separation is widely used in the identification and modeling of stall aerodynamic parameters. However, it has two defects. First, its model structure is significantly different from the pre-stall model used for a small attack angle, meaning the identification results cannot be combined with the pre-stall model to form the full flight envelope model. Second, the pitching moment model, which is used in conjunction with the Kirchhoff lift model, cannot accurately describe the aircraft stall pitching moment characteristics. To ensure the compatibility of the two models, this paper proposes a method to determine some unknown parameters in the stall model. The mechanism for the pitching moment generation of the aircraft stall is analyzed, and two high-order correction terms are added to the pitching moment model to better describe the longitudinal stall aerodynamic characteristics. Based on the identification of aerodynamic parameters, a longitudinal stall aerodynamic modeling method used for the aircraft stall process is developed. The identification and simulation validation results based on the quasi-steady stall flight data of a civil aircraft show that the improved stall model can accurately describe the quasi-steady stall pitching moment. The established stall aerodynamic model can accurately characterize the longitudinal quasi-steady stall aerodynamic characteristics of aircraft under different stall degrees. [ABSTRACT FROM AUTHOR]

As a first endeavor, thermo-electro-mechanical analysis of quadrilateral and triangular piezoelectric nanoplates are investigated based on the nonlocal theory and the Kirchhoff plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage, and a uniform temperature rise. Hamilton's principle is employed to derive the governing equations. The B3-spline finite strip method used to determine the natural frequencies, buckling loads, and corresponding mode shapes of displacement and the electric potential of quadrilateral and triangular piezoelectric nanoplates, for the first time. The comprehensive parametric study is conducted to explore the effect of the nonlocal parameter, geometrical shape, thermo-electro-mechanical loadings, boundary conditions, aspect ratio, and side length. It is shown that small-scale effect plays a considerable role in the buckling and vibration behavior of quadrilateral and triangular piezoelectric nanoplates. [ABSTRACT FROM AUTHOR]

In this paper, we study variation-inequality initial-boundary value problems with fouth order p(x)-Kirchhoff operators. First, an operator is constructed based on the Leray Schauder principle, and the existence of solutions is obtained. Secondly, the stability and uniqueness of the solution are analyzed after the conditions are appropriately relaxed on the Kirchhoff operators. [ABSTRACT FROM AUTHOR]

We consider the following singularly perturbed Kirchhoff-type equations - ε 2 M ε 2 - N ∫ R N | ∇ u | 2 d x Δ u + V (x) u = | u | p - 2 u in R N , u ∈ H 1 (R N) , N ≥ 1 , where M ∈ C ([ 0 , ∞)) and V ∈ C (R N) are given functions. Under very mild assumptions on M, we prove the existence of single-peak or multi-peak solution u ε for above problem, concentrating around topologically stable critical points of V, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. (Arch Ration Mech Anal 213(3):931–979, 2014) [ABSTRACT FROM AUTHOR]

Based on Kirchhoff plate theory and the Rayleigh-Ritz method, the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method (ICCM). The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate. The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method. From the continuity condition of the vibration displacement function at the cutout, the transition matrix between the two coordinate systems is constructed, and the mass and stiffness matrices are completely obtained. As a result, the calculation is simplified and the computational efficiency of the solution is improved. In this paper, numerical examples and modal experiments are presenteed to validate the effectiveness of the modeling methods, and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics. [ABSTRACT FROM AUTHOR]

In this paper, the following fractional Kirchhoff-type problem...where a,b>0 are constants,s∈ (3/4,1),2*s = 6/3-2s,V:R³-R is a continuous function, and f:R³×R-R is a continuous function, is considered. It is demonstrated that the fractional Kirchhoff-type equation has a radial sign-changing solution ub and a radial solution ub when f does not satisfy the subcritical growth condition and the usual Nehari-type monotonicity condition. The main tools are the constraint variational method and some analysis techniques. [ABSTRACT FROM AUTHOR]

Analytical models to describe the mechanical behaviour of an adaptive sun protection facade system (Adaptex) are presented. First, a one-dimensional (1D) rigid link model is developed. Stretching and bending deformation of textile bands are represented with the aid of extensional and rotational springs respectively. The model is described by two generalized coordinates only. The second analytical model employs a continuous description based on the Kirchhoff plate theory. The Rayleigh–Ritz method is applied to analyse the deformation behaviour of the textile bands. In comparison with the experimental results, it can be found that the continuous model simulates the real deformation of the facade system, thus its morphological behaviour, very effectively. The results of a parametric study indicate that the stiffness of the textile bands, the geometric dimensions and the position of load application have significant effects on the deformation behaviour. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices. The Kirchhoff index of G is defined as Kf (G) = n Pn1 i=1 > n = 0 are the Laplacian eigenvalues of G. Some bounds on Kf (G) in terms of graph parameters such as the number of vertices, the number of edges, first Zagreb index, forgotten topological index, etc., are presented. These bounds improve some previously known bounds in the literature. [ABSTRACT FROM AUTHOR]

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Analytical models to describe the mechanical behaviour of an adaptive sun protection facade system (Adaptex) are presented. First, a one-dimensional (1D) rigid link model is developed. Stretching and bending deformation of textile bands are represented with the aid of extensional and rotational springs respectively. The model is described by two generalized coordinates only. The second analytical model employs a continuous description based on the Kirchhoff plate theory. The Rayleigh--Ritz method is applied to analyse the deformation behaviour of the textile bands. In comparison with the experimental results, it can be found that the continuous model simulates the real deformation of the facade system, thus its morphological behaviour, very effectively. The results of a parametric study indicate that the stiffness of the textile bands, the geometric dimensions and the position of load application have significant effects on the deformation behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the Kirchhoff-Choquard problem with upper critical exponent and general nonlinearity. Some appropriate restrictions are imposed on parameters. Further-more, when the nonlinearity satisfies subcritical growth conditions, the existence of global minimizers and mountain pass type solutions of the problem are established by using the variational properties and fibbing maps. [ABSTRACT FROM AUTHOR]

This paper presents an investigation on vibration characteristics of graphene nanoplatelet (GPL) reinforced disk-shaft rotor with eccentric mass resting on elastic supports. The shaft is made of common metal material while the disk is made of graphene nanoplatelet reinforced material. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary smoothly and continuously along the thickness direction of the disk, which leads to a functionally gradient (FG) structure. The effective properties of the disk are determined via the Halpin-Tsai micromechanics model and the rule of mixture. In accordance with the Timoshenko beam theory and Kirchhoff plate theory, the governing equations which take into account gyroscopic effect due to rotation are derived by using the Lagrange equation. The substructure modal synthesis method and the assumed modes method are utilized to obtain the natural frequencies, and the Runge-Kutta method is adopted to solve the forced vibration response of the rotor system, where the centrifugal force and gravity of an eccentric mass in the shaft are considered as generalized forces. The present analysis is validated through comparison with both experiment and finite element (FE) results. A detailed parametric study is conducted to examine the effects of the rotating speed, GPL weight fraction, GPL distribution pattern, length-to-thickness ratio and length-to-width ratio of GPLs, disk position, disk dimension, and support stiffness on vibration behaviors of the nanocomposite disk-shaft rotor system. [ABSTRACT FROM AUTHOR]

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations. [ABSTRACT FROM AUTHOR]

The design of structures that can yield efficient sound insulation performance is a recurring topic in the acoustic engineering field. Special attention is given to panels, which can be designed using several approaches to achieve considerable sound attenuation. Previously, we have presented the concept of thickness-varying periodic plates with optimized profiles to inhibit flexural wave energy propagation. In this work, motivated by biological structures that present multiple locally resonant elements able to cause acoustic cloaking, we extend our shape optimization approach to design panels that achieve improved acoustic insulation performance using either thickness-varying profiles or locally resonant attachments. The optimization is performed using numerical models that combine the Kirchhoff plate theory and the plane wave expansion method. Our results indicate that panels based on locally resonant mechanisms have the advantage of being robust against variation in the incidence angle of acoustic excitation and, therefore, are preferred for single-leaf applications. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'. [ABSTRACT FROM AUTHOR]