Nonlinear Plate Research Articles

Geometrically nonlinear vibrations of plate embedding a large circular acoustic black hole

The concept of acoustic black hole (ABH) designates a decrease in bending stiffness and a localized added damping within a thin structure aiming to localize and efficiently dissipate vibrations. The small thickness results in large amplitude vibrations, which suggests to take into account geometrical nonlinearities. The subject has been previously studied on a beam with a one dimensional ABH. Previous research studied a geometrically nonlinear plate embedding a one dimensional ABH. This study aims at solving the Von Karman equations for a variable thickness plate, via a modal projection using the eigenmodes obtained with a finite element model. The use of a finite element model in this context allows to take into consideration more complex geometries. The methodology is then applied to a rectangular plate embedding a large circular ABH with added damping. The convergence of the nonlinear coefficients is studied and time domain computations show the relative importance of the geometrical nonlinearity.

Buckling of thin films: Lateral growth and kinetic evolution of telephone cords

We conduct a thorough investigation of the lateral growth and kinetic evolution of telephone cord buckles within thin films, employing both experimental and numerical techniques. Our exploration begins with in-situ experiments conducted on annealed silicon nitride films, aimed at capturing empirical data on the formation and evolution of these distinctive patterns. These experiments yield valuable insights into the morphological changes of telephone cords, such as wave flipping and merging, which lead to the enlargement of buckles at double wavelengths and widths. Subsequently, we employ a combined approach of geometrically nonlinear plate modeling and surface-based cohesive interface framework within a finite element numerical model to analyze the interplay between buckling-induced delamination and growth triggered by mode mixity-dependent interfacial toughness. Through this integrated approach, we effectively capture the mutual evolution of buckling and delamination occurrences, thus highlighting their inherently dynamic nature. Our numerical simulations show that the width and wavelength of telephone cords are doubled, consistent with experimental findings.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Frequency studies of nonlinear TSDT FGM plates in the linear shear corrections and thermal environment temperature

The effects of third-order shear deformation theory (TSDT) and varied linear shear correction coefficient on the vibration frequency of thick functionally graded material (FGM) plates with fully homogeneous equations under thermal environment and power law are investigated. The nonlinear coefficient c_1 term of displacement field of TSDT is included in the fully homogeneous equation under vibration of FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under vibration can be represented in the fully fifth-order polynomial equation, thus the natural frequency can be found. The natural frequency with/without the nonlinear c_1 term of displacement fields is investigated. It is a significant novelty with the consideration of the nonlinear c_1 term in the frequency computation.

A nested non-intrusive stochastic isogeometric method for nonlinear thermal vibration of FGM plates under random loading

This paper introduces an adaptive nested non-invasive dynamic SIGA method based on RBFNN for the nonlinear thermal vibration analysis of FGM plates under random loading. This method is robust and accurate in high-dimensional input spaces regardless of the sample sequence, addresses the low convergence rate of QMCS, and solves the stochastic dynamic response probability density function under random loading. Firstly, the multidimensional input space for random loading is constructed based on several mutually independent random variables via the stationary Gaussian stochastic process simulation technique. The nonlinear FGM plate model subjected to random loading in the thermal environment is modeled using HSDT with von Kármán strain-displacement relation within the IGA framework. Subsequently, a nested non-invasive dynamic response surface analysis method, SIGA-RBFNN, is proposed based on the RBFNN technique. Analysis of the FGM plate response under random loading demonstrates that SIGA-RBFNN, compared to tensor-product-based SIGA methods, is less affected by sample sequence types. It effectively addresses high-dimensional stochasticity and offers significant advantages in robustness, accuracy, and efficiency. Finally, SIGA-RBFNN compensates for the low convergence speed of QMCS and successfully solves the FGM plate displacement response statistics and probability density function under random loading.

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

Hamiltonian formalism for bistable-multilayered plates under non-mechanical stimuli

Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding symplectic dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an exact analytical solution in the membrane problem. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including pitchfork bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of boundary conditions across various equilibria, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, serving as an efficient approach for analyzing adaptive and morphing structures.

A0 mode Lamb wave propagation in a nonlinear medium and enhancement by topologically designed metasurfaces for material degradation monitoring

This paper intends to provide an application example of using metamaterials for elastic wave manipulation inside a nonlinear waveguide. The concept of phase-gradient metasurfaces, in the form of artificially architectured structures/materials, is adopted in nonlinear-guided-wave-based structural health monitoring (SHM) systems. Specifically, the second harmonic lowest-order antisymmetric Lamb waves (2nd A0 waves), generated by the mutual interaction between primary symmetric (S) mode and antisymmetric (A) mode waves, show great promise for local incipient damage monitoring. However, the mixing strength is adversely affected by the wave beam divergence, which compromises the 2nd A0 wave generation, especially in the far field. To tackle this problem, a metasurface is designed to tactically enhance the 2nd A0 waves through manipulating the phases and amplitudes of both primary waves simultaneously. After theoretically revealing the features of the 2nd A0 wave generation in a weakly nonlinear plate, an inverse-design strategy based on topology optimization is employed to tailor-make the phase gradient while ensuring the high transmission of the primary waves, thus converting the diverging cylindrical waves into quasi-plane waves. The efficacy of the design is tested in a 2nd-A0-wave-based SHM system for material degradation monitoring. Results confirm that the manipulated S and A mode waves can propagate in a quasi-planar waveform after passing the surface-mounted metasurface. Changes in material properties inside a local region of the host plate can be sensitively captured through examining the variation of the 2nd A0 wave amplitude. The concept presented here not only showcases the potential of metamaterial-enhanced 2nd A0 waves for material degradation monitoring, but also illuminates the promising direction of metamaterial-aided SHM applications in nonlinear waveguides.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

Nonlinear dynamic behavior of single-layered black phosphorus with an attached mass

The investigation of black phosphorus-based (BP)-based mass sensors provides theoretical support for the development of mass-detection devices. This study examines the non-linear dynamic behavior of a rectangular single-layered BP (SLBP) with an attached mass through the utilization of molecular dynamics (MD) simulations and a nonlinear orthotropic plate model (NOPM) with a concentrated mass. The results indicate that significant deformation of an SLBP with an attached mass necessitates consideration of geometric nonlinearity, although the attached mass does not affect the deformation. Additionally, this paper discusses the impact of the attachment mass and the amplitude of harmonic force on the non-linear forced vibration of the SLBP. It is observed that as the attachment mass increases, the nonlinear vibration resonance frequency decreases, while the peak amplitude increases. Furthermore, the thermal nonlinear vibration of SLBP with an attached mass has been investigated, revealing that an increase in the attached mass leads to a decrease in the nonlinear vibration frequency but an increase in the amplitude of the nonlinear vibration of SLBP with an attached mass. Overall, comparison with MD simulation results, this investigation suggests that NOPM with a concentrated mass effectively describes the nonlinear vibration behavior of an SLBP with an attached mass, providing theoretical support for designing such devices to detect attached masses.

Bending stiffness of ionically bonded mica multilayers told by its bubbles

Revealing the bending stiffness of layered materials is crucial for guiding their applications with notable out-of-plane deformation, such as in flexible electronics. To this end, dedicated methods have been developed, but usually involving precise manipulation of atomically thin flakes or cross-section characterization with atomic resolution, hindering their widespread adoption. Here, we utilize mica as a case study to demonstrate that bubbles spontaneously formed during mechanical exfoliation provide a facile but reliable approach for investigating its bending mechanics. Through topographical analysis of bubbles with widely distributed sizes, a bending stiffness is extracted following a nonlinear plate theory. The less bending stiffness than the ideal non-linear plate solution indicates a moderate interlayer slip, as confirmed by molecular dynamics simulations. The interlayer shear coefficient for mica is higher than that for multilayer graphene, which is attributed to its strong interfacial shear strength inheriting from its interlayer ionic bonding.

Nonlinear vibration analysis of thin plate with periodically distributed tunable-stiffness nonlinear oscillators in subsonic flow

This paper provides a new passive control method for nonlinear vibration of subsonic thin plates. The control method uses periodically distributed tunable-stiffness nonlinear oscillators (NLO). By using von Karman’s nonlinear plate theory and subsonic aerodynamic model, the nonlinear dynamic model of thin plate is established based on Hamilton’s principle. Through analyzing the system’s nonlinear dynamics bifurcation, the impact of periodic distribution on the stability is discussed. The amplitude-frequency responses are discussed under different NLO installation modes and design parameters by using Matcont software. The results show that the periodically distributed tunable-stiffness nonlinear oscillators can improve the stability of the system and have a remarkable suppression effect. The optimal control scheme for vibration suppression by periodically distributed tunable-stiffness nonlinear oscillators is obtained. The results obtained from this research can offer a valuable understanding of nonlinear vibration control, thereby enabling the identification of various control approaches that can be implemented in practical applications during future design endeavors.

Multiple scattering of local nonlinear resonators on a thin plate

Developing nonlinear resonant metamaterial plates requires an effective wave simulation method to analyze the wave interactions between multiple nonlinear scatterers. The multiple scattering method has the advantages of high computational efficiency and closed-form displacement solutions in modeling the finitely periodic scatterer array. However, the multiple scattering method of nonlinear scatterers is absent in the available studies of plate structures. In this paper, a nonlinear multiple scattering approach is formulated and analyzed for thin plates with nonlinear resonant scatterers. The resonant scatterer consists of an inner plate with a local resonator modeled by the nonlinear mass-spring system. The motion equation of resonant scatterers is solved using the perturbation technology and wave expansion method. The T-matrix of the resonant scatterer of each order is then derived at the arbitrary harmonic frequency from continuous interfacial conditions of resonant scatterers. Finally, the multiple scattering is formulated for a finitely periodic array of resonant scatterers on a thin plate. Numerical simulations capture the scattering characteristics and stop-band formations of fundamental and second-harmonic waves. Furthermore, the finitely periodic array of nonlinear and linear resonant scatterers achieves the nonreciprocal transmission of flexural waves. These numerical results demonstrate the potential applications of the proposed method in designing metamaterial-based plates with complex transmission properties.

Influence of modelling approach for reinforced concrete underground structures, with application to the CMS cavern at CERN

Representative modelling of reinforced concrete (RC) components in underground structures is essential for accurate assessment of structural performance (deformations and internal forces) within numerical simulations. This paper examines the implications of selecting different structural modelling approaches within the seismic (dynamic) finite element analysis of a buried structure of complex shape, using the CMS (Compact Muon Solenoid) Detector Cavern of the Large Hadron Collider in Geneva, Switzerland, as a case study. Two alternate modelling approaches were employed to model the cavern lining: (i) a composite continuum approach, with the concrete and embedded reinforcement being explicitly modelled; and (ii) the use of a nonlinear elasto-plastic plate element. The pre-earthquake ground initial conditions were determined through simulation of the construction and detector installation operations consistent with field measurements from extensometers and internal survey of floor deformations. The results demonstrate the importance of adopting a non-linear continuum modelling approach in representing the RC lining under strong shaking events to avoid under-prediction of seismic actions at locations of potential seismically induced damage. Such an approach will be essential in 3D problems where multi-axial dynamically varying stresses are applied on the RC section. Finally, it offers a realistic approach in representing structures of complex shape and that contains volume and thick elements.

Homogenization of nonlinear Cosserat plate including growth theory when the thickness of the plate and the size of the in-plane heterogeneities are of the same order of magnitude

In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.

Error analysis of the element-free Galerkin method for a nonlinear plate problem

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results.

Frequency sweep study on the generation of dual-mode second harmonics (DMSH) on an isotropic nonlinear elastic cylindrical rod by T(0,1) mode

Nonlinear ultrasonic (NLU) guided wave (GW) measurements are more sensitive to microscale defects than linear measurements. This nonlinear phenomenon is helpful for early-stage damage detection in material for nondestructive evaluation (NDE) and structural health monitoring (SHM) applications. The amplitude of the higher harmonics generated in the material serves as the basis for the NLU measurement. Recent studies have reported the presence of dual-mode second harmonic (DMSH) on an isotropic nonlinear elastic plate and cylindrical structures. The fundamental non-dispersive mode shear horizontal SH0 and torsional T(0,1) modes can generate simultaneously propagating dual-mode second harmonic(DMSH) on plate and cylindrical guided media respectively. For the case of cylinders, at phase matching condition two dominant second harmonic fundamental longitudinal (l(0,1)) and orthogonal torsional (t(0,1)┴) mode are generated with significant amplitude. The particle vibration of the t(0,1)┴ mode was present in both orthogonal directions compared to conventional T(0,1) mode and hence called orthogonal torsional mode. The present work aims to study the behaviour of DMSH at different frequencies through a frequency sweep study on a weakly nonlinear elastic cylinder of circular cross-section via numerical simulations and validated experimentally for some selected cases. The wave propagation of the T(0,1) mode at several selected frequencies from 300 kHz to 2.2 MHz is chosen to explore the second harmonic mode for better understanding. At frequencies below 1.25 MHz, l(0,1) mode is faster than t(0,1)┴ mode; above 1.25 MHz, it was the other way around. The behaviour of harmonics was observed to be in accordance with the group and phase velocities at the corresponding frequencies in the dispersion curves. The evidence for the presence of DMSH was validated experimentally for some selected frequency cases at different propagation distances. The significance of DMSH is explored via numerical simulations and found that t(0,1)┴ mode is sensitive towards material degradation, while all other generated wave modes were not affected. Research shows that enough energy is available in the generated DMSH to be noticed experimentally and in numerical simulations. This improved understanding of the generation of DMSH across different frequencies would be helpful in new advanced NLU-based NDE and SHM applications.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

Modeling Geometrically Nonlinear FG Plates: A Fast and Accurate Alternative to IGA Method Based on Deep Learning

Modeling Geometrically Nonlinear FG Plates: A Fast and Accurate Alternative to IGA Method Based on Deep Learning

The Adjustable Bandgap of Novel Local Resonance Sandwich-Like Metamaterial Plates with Buckling Beam Oscillators

Sandwich structures possess a high stiffness-to-mass ratio, which facilitates flexural wave propagation at lower frequencies and enhances their vibration isolation and sound insulation capabilities, therefore, sandwich-like metamaterial plates with the nonlinear local oscillator is constructed in this paper to obtain adjustable bandgap in low-frequency. The buckling beam oscillator consists of a small mass, linear support spring, and buckling elastic elements for nonlinear stiffness. First, governing equations of flexural vibrations of plates are deduced based on the Hamilton principle. By utilizing the harmonic balance method (HBM), the dispersion relation of a sandwich-like plate can be determined. Theoretical analysis reveals the presence of structural bandgap ranges at lower frequencies, which are subject to the excitation amplitudes. And in order to validate the theoretical findings, the finite element method (FEM) is employed. Moreover, the influence of structural parameters, including oscillator mass, linear spring stiffness, buckling beam angle, and Young’s modulus, are conducted to enhance the vibration control in the low-frequency. Interestingly, it is observed that the effectiveness of low-frequency suppression in the nonlinear metamaterial plate varies significantly across different boundary conditions. Therefore, the sandwich structures with nonlinear local oscillators can provide an effective way to suppress low-frequency vibration and maintain dynamic stability.