Kirchhoff Model Research Articles

Extension of paperboard modelling with Föppl–von Karman terms for improved stress state under suction pressure

A practical method for solving the bending stiffness of a thin orthotropic plate based on measured deflection and known loading was studied. The target application was paperboard manufacturing and related process control. Here, finite element (FE) method was used to create virtual paperboard and its deflection data. The accuracy of the linear Kirchhoff plate model used in the practical method was tested against the FE model with known load and material properties. It was found that in the case of significant deflections, the linear Kirchoff model was not accurate. The non-linear Föppl--von Kàrmàn model takes into account the occurring membrane stresses and extends on the Kirchhoff model, allowing for larger deflection. The non-linear Föppl--von Kàrmàn model was found to successfully describe the simulated situation and could be used to better solve for the paperboard's bending stiffness values over two axes.

Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia

This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of Dk−absorbing sets in the stochastic dynamical system Φk, and the asymptotic compactness of Φk is proved via decomposition. Finally, the family of Dk−random attractors is acquired for the stochastic dynamical system Φk in Vm+k(Ω)×Vm+kb0(Ω). These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.

An unsupervised learning-based guidewire shape registration for vascular intervention surgery robot

In a vascular interventional surgery robot(VISR), a high transparency master-slave system can aid physicians in the more precise manipulation of guidewires for navigation and operation within blood vessels. However, deformations arising from the movement of the guidewire can affect the accuracy of the registration, thus reducing the transparency of the master-slave system. In this study, the degree of the guidewire’s deformation is analyzed based on the Kirchhoff model. An unsupervised learning-based guidewire shape registration method(UL-GSR) is proposed to estimate geometric transformations by learning displacement field functions. It can effectively achieve precise registration of flexible bodies. This method not only demonstrates high registration accuracy but also performs robustly under different complexity degrees of guidewire shapes. The experiments have demonstrated that the UL-GSR method significantly improves the accuracy of shape point set registration between the master and slave sides, thus enhancing the transparency and operational reliability of the VISR system.

Longtime dynamics of solutions for higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff models with higher-order rotational inertia and nonlocal damping

The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff system with higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.

Bifurcation analysis of fractional Kirchhoff–Schrödinger–Poisson systems in R 3

In this paper, we investigate the bifurcation results of the fractional Kirchhoff–Schrödinger–Poisson system { M ( [ u ] s 2 ) ( − Δ ) s u + V ( x ) u + ϕ ( x ) u = λ g ( x ) | u | p − 1 u + | u | 2 s ∗ − 2 u i n R 3 , ( − Δ ) t ϕ ( x ) = u 2 i n R 3 , where s , t ∈ ( 0 , 1 ) with 2 t + 4 s > 3 and the potential function V is a continuous function. We show that the existence of components of (weak) solutions of the above equation bifurcates out from the first eigenvalue λ 1 of the problem ( − Δ ) s u + V ( x ) u = λ g ( x ) u in R 3 . The main feature of this paper is the inclusion of a potentially degenerate Kirchhoff model, combined with the critical nonlinearity.

Nonlinear oscillation of a microbeam due to an electric actuation—Comparison of approximate models

AbstractIn this paper, oscillations of the electrically actuated microbeam are considered, described by strongly nonlinear differential equations. One of the essentially nonlinear effects is the pull‐in phenomenon, that is, the transition of the oscillatory regime to the attraction regime. This effect is considered in the paper on the basis of various approximate models. In particular, the error of replacing the original nonlinearity by its expansion in the Maclaurin series is estimated. It is shown that such an approximation can only be used for voltage values that are far from critical. Estimates are also given for initial displacements and velocities, which guarantee an oscillating character of solutions. The electrically activated oscillations of the microbeam are considered taking into account the geometric nonlinearity within the framework of the Kirchhoff model. The dependence of pull‐in value on geometric nonlinearity is investigated.

Large deformation of hyperelastic modified Timoshenko–Ehrenfest beams under different types of loads

In this paper, a nonlinear finite element formulation for a hyperelastic, modified Timoshenko–Ehrenfest beam with geometrical and material nonlinearities is developed for the first time. A new five-parameter beam element is introduced. The parameters contain displacement, values of difference vector and a through-the-thickness scalar value. Moreover, a new procedure is employed to apply the moment to the beam. The constitutive formulation is derived for Saint Venant–Kirchhoff (SVK) as well as the compressible neo-Hookean (n-H) hyperelastic model. The beam is subjected to different loads such as dead load, point and distributed loads, and follower pressure. To demonstrate the applicability of the formulations, several examples are solved. The results reveal that this formulation can capture the previous results reported in the literature. Furthermore, a comparative study is done between two hyperelastic models. It is demonstrated that in the case of large rotation of the beam, both Saint Venant–Kirchhoff and neo-Hookean models show the same behavior. However, in case of large deformation with large strains of the beam, the Saint Venant–Kirchhoff model behaves stiffer than the neo-Hookean hyperelastic model.

Symmetric and asymmetric separated variables. II. A second integrable case of the complex Kirchhoff model

In the present paper, we find a second new integrable case of the complex Kirchhoff model on e*(3). We show that the obtained integrable model possesses two types of separation of variables (SoV): symmetric and asymmetric ones. The asymmetric SoV—similar to that of the Clebsch model (F. Magri and T. Skrypnyk, arXiv:1512.04872) and of the first integrable case of the complex Kirchhoff model (T. Skrypnyk, J. Geom. Phys. 172, 104418 (2022)]—is characterized by two different curves of separation. In the case of symmetric SoV, both curves of separation are the same. This case—similar to the integrable case of the complex Kirchhoff model of Skrypnyk, J. Geom. Phys. 172, 104418 (2022)—yields a direct analog of the famous Weber–Neumann separated coordinates. The obtained separation curves and Abel-type equations for the constructed model can be viewed as trigonometric degenerations of the corresponding separation curves and Abel-type equations of the Clebsch model [F. Magri and T. Skrypnyk, arXiv:1512.04872 and T. Skrypnyk, J. Geom. Phys. 135, 204–218 (2019)], while separation curves and Abel-type equations of Skrypnyk, J. Geom. Phys. 172, 104418 (2022) can be viewed as their rational degenerations.

Physical nonlinearity in porous functionally graded kirchhoff nano-plates: Modeling and numerical experiment

In this paper, mathematical models of porous functionally graded inhomogeneous plates were constructed. The following hypotheses were used as the foundation: the kinematic Kirchhoff model was used; nano effects were considered by the modified couple stress theory; material properties depend on coordinates (x,y,z), and for metals, they also hinge on the stress-strain state at a point in the plate volume, i.e. there is a dependence σi(ei). This dependence can be any analytical function derived from experimentation, which means that it is possible to take into account elastic-plastic deformations according to the deformation theory of plasticity. The ceramic material is resilient. The required variational and differential equations and the boundary conditions were derived from the Lagrange functional. The plate material's porosity was characterised by six types. For numerical solutions, the methods of reduction of the partial differential equation to the ordinary differential equation were applied, as well as iterative methods nested one into another, based on the following techniques: variational iterations method (VIM), Agranovskii-Baglai-Smirnov method (ABSM) and Birger's method of variable parameters of elasticity. For each of these methods, there are theorems proving their convergence. Each problem that describes the stress-strain state of Kirchhoff nano-plates was solved using the following analytical methods: Exact Navier's solution (ENS), Bubnov-Galerkin method (BGM), Levy solution (LS), numerically analytical Kantorovich-Vlasov (KVM) methods, variational iterations method (VIM), Vaindiner method (VaM) and numerical finite difference method (FDM). The convergence of these methods was investigated and the optimal method for calculating such structures, both in terms of accuracy and speed, was determined for the elliptic 2D PDEs of arbitrary area Ω. Numerical results for both elastic nano-plates and elastoplastic nano-plates have been presented.

Exponential attractor for Kirchhoff model with time delay and thermal effect

The Kirchhoff model is derived from the vibration problem of stretchable strings. In this paper, we focus on the long-time dynamics of the Kirchhoff model with time delay and thermal effect. We first proved the existence and uniqueness of the solution through the Faedo–Galerkin method and prior estimates. After that, it is further proved that the dynamical system has a bounded absorption set, and it is also verified that the system is quasi-stable on the bounded absorbing set. Finally, the existence of exponential attractor in dynamical system is proved. This innovatively considers the long-time dynamical behavior of Kirchhoff model under the simultaneous action of variable coefficient, thermal effect and time delay comprehensively. It also promotes the relevant conclusions of the Kirchhoff model. The findings of this paper provide a theoretical basis for subsequent application and research.

Dispersion and Energy Characteristics of Bending Waves in a Plate Lying on a Two-Parameter Elastic Foundation

The propagation of bending waves in a plate resting on a two-parameter elastic foundation is considered. In contrast to the classical Kirchhoff model, the mathematical model used here takes into account not only the kinetic and potential energies of bending vibrations, but also the kinetic energy due to the inertia of rotation of the plate elements during bending. The dispersion equation, phase velocity, energy transfer velocity, and energy characteristics of waves propagating in the plate are analyzed depending on the ratio of the coefficients determining the shear and compression stiffness of the elastic foundation. Conditions are found under which waves with phase and group velocities having opposite directions (frequently called “backward” waves), can exist in the plate. It is demonstrated that such waves significantly change the behavior of the energy flux. In addition, relations are found that relate the kinematic and average energy characteristics of the waves.

Dispersion and Energy Characteristics of Bending Waves in a Plate Lying on a Two-Parameter Elastic Foundation

The propagation of bending waves in a plate resting on a two-parameter elastic foundation is considered. In contrast to the classical Kirchhoff model, the mathematical model used here takes into account not only the kinetic and potential energies of bending vibrations, but also the kinetic energy due to the inertia of rotation of the plate elements during bending. The dispersion equation, phase velocity, energy transfer velocity, and energy characteristics of waves propagating in the plate are analyzed depending on the ratio of the coefficients determining the shear and compression stiffness of the elastic foundation. Conditions are found under which waves with phase and group velocities having opposite directions (frequently called “backward” waves), can exist in the plate. It is demonstrated that such waves significantly change the behavior of the energy flux. In addition, relations are found that relate the kinematic and average energy characteristics of the waves.

The asymptotic behaviors of solutions for higher-order (m 1, m 2)-coupled Kirchhoff models with nonlinear strong damping

Abstract The Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.

Well-posedness and global attractor of Kirchhoff equation with memory term and thermal effect

The Kirchhoff model is derived from the vibration problem of stretchable strings. In this paper, the Kirchhoff equation with thermal effects and memory terms is studied. First, the existence and uniqueness of the solution to the equation are obtained using the Faedo–Galerkin method. Then, the existence of a global attractor is established by proving the existence of a bounded absorbing set and the asymptotic smoothness of the semigroup. This paper innovatively considers the long-term dynamic behavior of the Kirchhoff model under the simultaneous action of variable coefficients, memory, and thermal effects comprehensively. It also promotes the relevant conclusions of the Kirchhoff model. The findings of this paper provide a theoretical basis for subsequent application and research.

Atomistically-informed continuum modeling and isogeometric analysis of 2D materials over holey substrates

This work develops, discretizes, and validates a continuum model of a molybdenum disulfide (MoS2) monolayer interacting with a periodic holey silicon nitride (Si3N4) substrate via van der Waals (vdW) forces. The MoS2 layer is modeled as a geometrically nonlinear Kirchhoff–Love shell, and vdW forces are modeled by a Lennard-Jones (LJ) potential, simplified using approximations for a smooth substrate topography. Both the shell model and LJ interactions include novel extensions informed by close comparison with fully-atomistic calculations. The material parameters of the shell model are calibrated by comparing small-strain tensile and bending tests with atomistic simulations. This model is efficiently discretized using isogeometric analysis (IGA) for the shell structure and a pseudo-time continuation method for energy minimization. The IGA shell model is validated against fully-atomistic calculations for several benchmark problems with different substrate geometries. Agreement with atomistic results depends on geometric nonlinearity in some cases, but a simple isotropic St.Venant–Kirchhoff model is found to be sufficient to represent material behavior. We find that the IGA discretization of the continuum model has a much lower computational cost than atomistic simulations, and expect that it will enable efficient design space exploration in strain engineering applications. This is demonstrated by studying the dependence of strain and curvature in MoS2 over a holey substrate as a function of the hole spacing on scales inaccessible to atomistic calculations. The results show an unexpected qualitative change in the deformation pattern below a critical hole separation.

Mathematical modeling of nonlinear thermodynamics of nanoplates

The theory of nonlinear statics and dynamics of flexible plates, taking into account the modified couple stress theory and temperature field, is developed. The theory is based on the classical Kirchhoff model for an isotropic elastic body. Geometric nonlinearity is taken into account according to the von Kármán model. Variational differential equations are yielded by the Hamilton principle, and partial differential equation (PDE's) related to displacements of the middle surface and deflection are derived. The hypotheses of the modified couple stress theory implied an increase of the order of the system of partial differential equations due to the fact that moments of higher order appeared. The resolved PDE's are reduced to the Cauchy problem by the method of finite differences of the second order accuracy, which is solved by methods of the Runge-Kutta type (from the fourth to the eighth order of accuracy) and the Newmark method. No restrictions are imposed on the temperature field, and it determined from the solution of the three-dimensional heat equation using the finite element method (FEM). The convergence of the proposed algorithms is investigated depending on the number of finite elements and the time step. Static problems are solved by the dynamic approach. Nonlinear dynamics is analyzed based on (Fourier spectrum, wavelets of different types and phase portraits). The analysis of Lyapunov exponents obtained by different methods (Kantz, Wolf, Rosenstein, Sano-Sawada and the authors method) is carried out, which validated occurrence of chaotic vibrations. Hyperchaotic vibrations have been detected and studied. It is also illustrated how increase of the temperature field influences localization of the chaotic zones for size-dependent plates.

Thermoelastic Damping Limited Quality Factor Enhancement and Energy Dissipation Analysis of Rectangular Plate Resonators Using Nonclassical Elasticity Theory

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and needs to be alleviated in vibrating resonators to mitigate parameters by improving the thermoelastic damping limited quality factor, QTED. The maximum energy dissipation is also interrelated with the critical dimension h c of the plates, and by optimizing the dimensions, the peaking of energy dissipation can be diminished. As the size of the devices is scaled down, classical continuum theories become incompetent to explain the size-effect related mechanical nature at the micron and submicron levels, and, as a result, nonclassical continuum theories have been pioneered with the inception of internal length scale parameters. In this work, an analysis of isotropic rectangular microplates based on the Kirchhoff model and a higher order theory like Modified Couple Stress Theory is utilized to study size-dependent thermoelastic damping and its impact on the quality factor and critical dimensions. The Hamilton principle is adapted to derive the governing equations of motion, and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the plates. Five different structural materials (PolySi, diamond, Si, GaAs, and SiC) are used for optimizing QTED and hc, which depends on two material performance index parameters: the thermoelastic damping index (TDI) and the material thermal diffusion length, l T . According to this work, the maximum QTED is attained for PolySi with the lowest TDI, and hcmax is obtained for Si with the maximum l T . The impacts of the dimensionless length-scale parameters (l/h), vibration modes, and boundary conditions (clamped-clamped and simply supported) on QTED and hc are also investigated. From the current analysis, QTED can be further enhanced by selecting higher vibration modes and clamped-clamped boundary conditions. QTED can be maximized by fixing the internal length scale parameter (l) and making the thickness of the beam equal to l. The analytical study is numerically simulated by using MATLAB 2015 software. Prior knowledge of QTED and hc will help designers to produce high-performance and low-loss resonators for the futuristic technological applications.

Well-posedness and stability for Kirchhoff equation with non-porous acoustic boundary conditions

In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the L2(Ω) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates.

Quasi-analytical and rigorous modeling of interference microscopy

We present an extended vectorial Kirchhoff model of coherence scanning interferometry including several vector rotations occurring in the imagining and scattering process as well as polarization dependent reflection coefficients. For validation simulated results are compared to those of the conventional scalar Kirchhoff model and a rigorous finite element modeling.

Long‐Term Dynamic Behavior of a Higher‐Order Coupled Kirchhoff Model with Nonlinear Strong Damping

In this paper, we study the long‐time dynamics problem of a class of higher‐order Kirchhoff coupled systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and Faedo–Galerkin method; secondly, the family of global attractors of these problems is proved by using the compactness theorem. The results of the Kirchhoff coupled group are promoted through research.