Nonlinear Deflection Research Articles

A Higher-Order Theory for Nonlinear Dynamic of an FG Porous Piezoelectric Microtube Exposed to a Periodic Load

This paper investigates the nonlinear dynamic deflection, natural frequency, and wave propagation in functionally graded (FG) porous piezoelectric microscale tubes under periodic load, hygrothermal conditions, and an external electric field. The piezoelectric material used to make the smart microtubes has pores that may be smoothly changed or uniformly distributed over the tube wall. Here, three types of porosity distribution are taken into consideration. The nonlinear motion equations are constructed using a novel shear deformation beam theory and the modified couple stress theory (MCST). The nonlinear motion equations are solved using the fourth-order Runge–Kutta technique and the Galerkin approach. The effects of various geometric parameters, porosity distribution type, porosity factor, periodic load amplitude and frequency, material length scale parameter, moisture, and temperature on the nonlinear dynamic deflection, natural frequency, and wave frequency of FG porous piezoelectric microtubes are explored through a number of parametric investigations.

A size-dependent nonlinear isogeometric approach of bidirectional functionally graded porous plates

Recent progress in materials science and fabrication technologies, such as additive manufacturing (AM) or 3D printing, has facilitated the design and production of intricate multi-directional functionally graded (FG) materials which are playing a pivotal role in interdisciplinary engineering applications. Developing mathematical modeling for such nanostructures, however, remains challenging. The primary objective of this research, therefore, is to present a mathematical model for studying the static bending characteristics considering the geometric nonlinearity of bidirectional FG nanoplates with internal pores. The mathematical formulations are established by utilizing the first-order shear deformation theory and the von-Kármán assumption within the framework of an isogeometric approach based on NURBS basis functions. To account for the influence of both strain gradient and nonlocal elasticity, we adopt the nonlocal strain gradient theory including two independent material length scale parameters to predict the geometric nonlinearity performance of structures at small-scale levels. The mechanical properties of multi-directional FG material models are assumed to be continuously graded in two directions based on power law, while internal pores can be dispersed into two typical distribution patterns. Several numerical investigations are performed to evaluate the substantial impact of certain parameters on the nonlinear deflections of bidirectional FG nanoplates with pores under various conditions. Compared to existing studies, the key contribution is to present a robust numerical model aimed at elucidating both the stiffness-softening and stiffness-hardening mechanisms of multi-directional FG nanostructures under static bending conditions with geometric nonlinearity. These findings enhance our insights into nonlinear bending performance and contribute valuable design guidelines for future small-scale engineering structures. These innovative designs have become popular and are now crucial in state-of-the-art engineering applications, such as aerospace, nuclear systems, or thermal resistance devices.

Hydrodynamic instability of shear imposed falling film over a uniformly heated inclined undulated substrate

Linear and weakly nonlinear stability analyses of an externally shear-imposed, gravity-driven falling film over a uniformly heated wavy substrate are studied. The longwave asymptotic expansion technique is utilized to formulate a single nonlinear free surface deflection equation. The linear stability criteria for the onset of instability are derived using the normal mode form in the linearized portion of the surface deformation equation. Linear stability theory reveals that the flow-directed sturdy external shear grows the surface wave instability by increasing the net driving force. On the contrary, the upstream-directed imposed shear may reduce the surface mode instability by restricting the gravity-driving force, which has the consequence of weakening the bulk velocity of the liquid film. However, the surface mode can be stabilized/destabilized by increasing the temperature-dependent density/surface-tension variation. Furthermore, the bottom steepness shows dual behavior on the surface instability depending upon the wavy wall's portion (uphill/downhill). At the downhill portion, the surface wave becomes more unstable than at the bottom substrate's uphill portion. Moreover, the multi-scale method is incorporated to obtain the complex Ginzburg–Landau equation in order to study the weakly nonlinear stability, confirming the existence of various flow regions of the liquid film. At any bottom portion (uphill/downhill), the flow-directed external shear expands the supercritical stable zones, which causes an amplification in the nonlinear wave amplitude, and the backflow-directed shear plays a counterproductive role. On the other hand, the supercritical stable region decreases or increases as long as the linear variation of density or surface tension increases with respect to the temperature, whereas the sub-critical unstable region exhibits an inverse trend.

Thermal vibration analysis of bi-directionally stepped porous functionally graded plates with segment-specific material property variation supported by Kerr foundation

This paper presents a comprehensive analysis of the nonlinear dynamic response of stepped rectangular plates made of functionally graded porous material (FGPM), supported by the Kerr foundation, in a thermal environment. A novel analytical framework is developed to explore geometric and material variations. The study investigates structural variations in plate thickness with abrupt changes in uni- or bi-directional orientations, examining both single and double stepped thickness profiles. Material properties vary with thickness, featuring distinct horizontal discontinuities across plate segments. Porosity distributions, both even and uneven, are addressed using modified mixture rules. Nonlinear kinematic relationships are established using Reddy’s third-order shear deformation plate theory and von Kármán’s nonlinear geometric assumptions, with equations of motion solved via Galerkin’s technique. This improved model effectively addresses non-continuous thickness variation through integral calculus, enhancing computational efficiency. Validation is achieved by comparing outcomes with published literature and Finite Element Analysis (FEA). The study investigates the influence of material properties, elastic foundation, boundary conditions, and geometric parameters on the free vibration and nonlinear behaviors of the plates. Some of the key findings include: increasing the thickness of the stepped segment significantly heightens the fundamental frequency while reducing vibrational amplitudes; optimizing the location of the stepped segment directly impacts the plate’s fundamental frequency and vibrational amplitudes; and as the load factor increases, the difference between linear and nonlinear deflection becomes evident. Therefore, accurate FGPM stepped plate design requires incorporating nonlinear terms in the strain–displacement relationships. Suggestions for future model modifications are also discussed, contributing to advancements in structural design and analysis.

Computational modelling and prediction of nonlinear deflection responses of bonded joint component with multiple damage (crack and delamination) – An experimental validation

The present research evaluates the presence of damage, including delamination and cracks, affects the deflection response of an adhesively bonded single lap joint (SLJ). In this study, the numerical analysis is performed for intact and influence of different damages in the adherend of SLJ in the commercial simulation software (ABAQUS). The circular delamination of radius ‘R’ 10 mm and crack having length ‘l’ 20 mm, the radius of curvature ‘Rc’ 0.2 mm are imposed into the laminate adherend using node-to-surface interaction technique. The stability of the simulation model is verified by comparing the deflection values from published results and experimental data. The confirmed simulation model is broadened to examine the impact of design factors (fiber orientation, loading position, delamination shape, crack position and orientation, and delamination position with and without crack) on the deflection response for various overlapping lengths (25, 30, 35, and 40 mm) for final design recommendation. Additionally, the sensitivity analysis was performed using a surrogate model (i.e., variance-based method) by constructing the second-order polynomial function to quantify the influence of damages under the input parameters. The detailed features and multivariate model functions have been analysed for the various damage cases of SLJ.

Nonlocal strain gradient-based nonlinear vibration analysis of nonlinear FG three-phase HJCNT/MWCNT/epoxy composite microbeam resonators using a modified micromechanical model

This paper aims to study nonlinear dynamic behavior of functionally graded (FG) three-phase composite microbeam resonators made of an epoxy matrix and two reinforcements namely multi-walled carbon nanotubes (MWCNTs) and hetero-junction carbon nanotubes (HJCNTs). The effective mechanical properties of the composite microbeam are obtained using the modified Halpin-Tsai micromechanical model. The microbeam surrounding medium is simulated using a two-parameter elastic foundation. The von-Karman’s geometric nonlinearity relations are incorporated and the equations of motion are derived based on the nonlocal strain gradient Euler–Bernoulli beam model. A new closed-form analytical solution is obtained using the homotopy perturbation method. The effects of vibration amplitude, nanofiber volume fraction, nanofiber distribution pattern, small-scale parameters and the foundation parameters on the nonlinear frequency and deflection of the FG three-phase composite microbeams are studied in detail. The findings of the paper are valuable for researchers in the field of microbeam resonators.

Modelling and Parametric Study for Panel Flutter Problem using Functionally Graded Materials

Abstract In this paper we will demonstrate the possibility of weight optimization of panels under aero-thermal loading in hypersonic flow using functionally graded materials (FGM). The in-plane volume fraction of two constituents (Aluminium and Nickel) is modelled using polynomial distributions. Different material grading layouts are investigated, including cases with Nickel concentrated at corners, sides, midpoints and center. The solution of the problem utilized a higher order element with C1 continuity. The study covers the linear boundaries of the panel flutter problem as well as the non-linear post-buckling deflections. The results indicated Nickel placement strategies are shown to enhance dynamic pressure and vibration performance for a given mass reduction through optimal center and edge localization. Overall, the integrated modelling approach demonstrates the potential to systematically optimize stability, weight and integrity in hypersonic flow to optimize the weight of panels subject to aero-thermal loads.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.

Application of pseudo-parameter iteration method to nonlinear deflection analysis of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation

In this study, the nonlinear deflection of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation was analyzed using the pseudo-parameter iteration method (PIM), which is a novel iterative semi-analytic method for solving ordinary/partial differential equations. To do this, we set six types of infinite beams with concave and convex shapes under static loading conditions. To calculate the nonlinear deflection of the infinite beam with variable cross-sections, the Bernoulli-Euler beam equation (fourth-order ordinary differential equation) considering changing beam flexural rigidity was introduced, and the PIM was adopted to this equation. Through the numerical experiment, it was confirmed that the nonlinear deflections calculated via the PIM are quite close to the exact solution within a few iterations. In addition, the graph of error quickly reaches the steady state error for all cases as the number of iterations increases.

A mechanics-based model for predicting flexible needle bending with large curvature in soft tissue

Percutaneous insertion is one of the most common minimally invasive procedures. Compared with traditional straight rigid needles, bevel-tipped flexible needle can generate curved trajectories to avoid obstacles and sensitive organs. However, the nonlinear large deflection problem challenges the bending prediction of the needle, which dramatically influences the surgical success rate. This paper analyzed the mechanism of needle-tissue interaction, and established a mechanics-based model of the needle bending during an insertion. And then, a discretization of the bending model was adopted to accurately predict the large bending of the needle in soft tissue. Insertion experiments were conducted to validate the bending prediction model. The results showed that the large needle bending was predicted with the mean/RMSE/maximumu error of 0.42 mm / 0.26 mm / 1.08 mm, which was clinically acceptable. This proved the rationality and accuracy of the proposed model.

Nonlinear Deflection Characteristics of Weakly Bonded Curved Composite Structure Under Hygro-Thermo-Mechanical Loadings

A customized MATLAB algorithm is developed for internally separated laminated composite panels experiencing large geometric deformations. The algorithm is designed to calculate nonlinear deflection responses under the effect of combined hygro-thermo-mechanical (HTM) loading. The hygrothermal (HT) load on the panel is in-plane, whereas the mechanical load acts upon the structure transversely. The analysis has adopted various kinematic theories and finite element (FE) techniques to determine the deformations computationally. The deflection behavior of the composite is characterized through a macro mechanical model considering the nonlinearity in geometry with and without accounting for the stretching effects across the panel thickness. Additionally, the changes in composite properties due to the environment and/or loadings are adopted to achieve a realistic response, preserving continuity assumptions between the individual layers of the weakly bonded structure. Moreover, various numerical examples are examined through different models to illustrate the influences of environmental factors and design-specific parameters on the flexural strength of weakly bonded structures. The findings strongly emphasize the necessity of employing diverse kinematic models when examining laminated structures, both with and without HT loading, while also acknowledging the potential for debonding.

Nonlinear deflection and traveling wave solution for FG-GPLs reinforced microtubes embedded in Kerr foundation and conveying magnetic fluid

The nonlinear dynamic deformation analysis and traveling wave solution of microtubes reinforced with functionally graded (FG) graphene platelets (GPLs) embedded in Kerr foundation with three-parameter are studied. It is assumed that the composite microtubes convey a fluid and are exposed to a magnetic field. The displacement field of the microtubes is modeled using a higher-order shear deformation beam theory. Furthermore, to capture the tiny size impact, the modified couple stress theory with a single material parameter is applied. The nonlinear equations of motion of the fluid-conveying microtube are derived using Hamilton’s variational principle considering von-Karman’s strain–displacement relations and the fluid velocity potential. The material properties of the microtubes are calculated using the modified Halpin–Tsai model and the mixture rule. The GPL volume fraction is continuously varied across the thickness based on a refined cosine rule, taking into account different patterns of GPLs distribution. The partial differential equations are transformed into ordinary equations with different boundary conditions using the Galerkin technique. These equations are solved employing the fourth order Runge–Kutta method. While, as a special case, the nonlinear motion equation of Euler–Bernoulli microtube, that is surrounded by a shear foundation and conveying microfluid, is solved analytically based on Jacobi elliptic functions. Finally, a variety of numerical examples are presented to illustrate the impacts of magnetic field, elastic foundation parameters, material length scale parameter, flow velocity, Knudsen number, length-to-radius ratio, and length-to-thickness ratio on the nonlinear dynamic deflection and axial displacement waves in the FG-GPLs microtubes.

Cooperative correction method for distortion and dispersion of deflected field of view in Risley-prism bionic-human-eye imaging systems.

The Risley-prism imaging system (RPIS) is a powerful way to achieve bionic human eye imaging with great advantages on large field of view (FOV) and variable resolution imaging owing to the autonomous controlled deflection of light. But the imaging dispersion originating from nonlinear and uneven light deflection results in limited imaging wavelength that seriously hinders its application. The existing solutions for imaging dispersion mainly rely on the hardware, which generally has bulky structure and limited improvement on image. Besides, the existing image evaluation methods for dispersion are not suitable for RPIS due to inhomogeneous dispersion. Herein, this paper systematically analyzes the mechanism and characteristics of dispersion in the RPIS, and proposes a cooperative correction method for image distortion and dispersion of multiple-color imaging, achieving the elimination of distortion and dispersion simultaneously without changing the optical structure. A dispersion evaluation index based on Pearson's correlation coefficient (PCC) is also established, and the objectivity and validity of the index are proved by experiments. Furthermore, a kind of compact RPIS based on an RGB camera is built, and both indoor and outdoor experiments are conducted. The experimental results demonstrate that proposed algorithm has strong universality and robustness for various scenes and targets.

Distributed finite‐time adaptive fault‐tolerant consensus control of second‐order multi‐agent systems under deception attacks

SummaryThis study addresses the issue of distributed fault‐tolerant consensus control for second‐order multi‐agent systems subject to simultaneous actuator bias faults in the physical layer and deception attacks in the cyber layer. Cyber‐physical threats (malicious state‐coupled nonlinear attacks and physical deflection faults), unknown control gains, external disturbances and uncertainties force the failure of the existing graph theory‐based consensus control schemes, leading to disruptions in the cooperation and coordination of multi‐agent systems. Then, the power integrator‐based virtual control is incorporated in the distributed fault‐tolerant consensus to achieve unknown parameter estimations with the adaptive technique. The consensus‐based robustness to lumped uncertainties, resilience to attacks, compensation to faults, and novel finite‐time convergence of the neighborhood errors and velocity errors are also realized within a prescribed finite‐time settling bound. The simulation is conducted to verify the effectiveness of the distributed finite‐time adaptive fault‐tolerant consensus algorithm.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation

<abstract><p>We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the $ L^r $-integral input-to-state stability estimate for the solution whenever $ r\in [2, +\infty] $ by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.</p></abstract>

Nonlinear dynamic response of FG-GPLRC beams induced by two successive moving loads

In this study, the nonlinear dynamic analysis of functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) beams is examined. The material properties can be estimated by using the modified Halpin-Tsai model and rule of mixture. Based on the third-order shear deformation theory and the von Kármán assumption, the equations of motion is derived and solved by using Jacobi-Ritz method incorporating with iteration procedure. Several effects such as number of layers, weight fraction of graphene nanoplatelets, material distributions, beam geometry, velocity of moving loads, distance between the loads are considered. For nonlinear forced vibration, the results indicate a considerable dissimilarity in the nonlinear dynamic deflection of the beams under moving loads when compared to linear analysis. The incorporation of additional GPLs into the beams yields a significant improvement in the strength of the beam, resulting in lower deflection. The new findings on nonlinear response of the beams excited by one and two moving loads are presented.

Nonlinear bending of unsymmetric double-layer lattice truss core sandwich beams

This paper studied the nonlinear bending of unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). Five unsymmetric cases are considered by varying different materials and geometric thicknesses in two layers. To establish a theoretical model for the unsymmetric double-layer LTCSBs, Allen’s model and von Kármán nonlinear theory are employed. The present theoretical model is established by considering the midplane of the mid-sheet as the reference plane. Ritz method is applied to determine the deflection for the nonlinear bending, and verified by the finite element method (FEM). Finally, the effects of the loading direction, unsymmetric factors, and boundary conditions on the nonlinear bending behaviors of LTCSBs are studied. It is found that the nonlinear deflections of unsymmetric LTCSBs may exhibit softening-spring nonlinearity in one direction.

Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory

This article investigates a comprehensive analysis of the bending problem pertaining to magneto-electro-elastic (MEE) nanobeams on the Winkler-Pasternak foundation. Taken into account the influence of size effects of nano structures, a non-classical mechanical model is established to describe the bending behavior for MEE nanobeams. In this research, the nonlocal modified couple stress theory is utilized to accurately capture both the softening and hardening effects resulting from the size effect. The theory of third-order shear deformation and von Karman geometric nonlinear theory are employed in conjunction with the introduction of Maxwell's equation to develop the model of MEE nanobeam, the control equation of the nonlinear bending problem of MEE nanobeams is obtained by Minimum potential energy principle, and it is solved by Galerkin method. Finally, this study provides a detailed discussion on the influences of the ratio of nonlocal parameters and length-scale parameters, Winkler-Pasternak coefficient, external magnetic potential, external voltage and span-thickness ratio on nonlinear deflection of nanobeam.

Numerical modeling of geometrically nonlinear responses of smart magneto-electro-elastic functionally graded double curved shallow shells based on improved FSDT

The use of functionally graded magneto-electro-elastic (FG-MEE) shells in smart composite structures like sensors, transducers, vibration monitoring, aeronautics, shape tracking, medical and sonar applications has a lot of promise. The accurate modeling approach for situations with several physics linked is quite difficult. In this paper, an improved first-order shear deformation hypothesis is used to inspect the magneto-electro-elastic coupling performance of thin-walled smart structures with laminate design presenting FGM composite as inactive material and using the magneto-piezoelectric patches as combined sensor and actuator features. The govern equations of motion, electric and magnetic distributions through the thickness direction of shell element are derived from the virtual of Hamilton principle. The simulations are carried out using a self-coded FE software. A numerical analysis of the nonlinear deflection of FG-MEE shells is performed under multiphysics loading. To ensure that the current MEE formulation is accurate, the findings were compared to existing solutions from the literature, and remarkable agreement was identified. A parametric analysis is conducted to demonstrate how the material composition affects the deflection throughout the thickness. Further, new numerical results, physical insights and finding have been provided in the present manuscript taking into account influences of several important parameters such as material graduation index and various boundary conditions. The novel MEE FE model suggested in this study is highly useful for the accurate development and design of FG-MEE structures.