Nonlinear Bending Research Articles

The efficient data-driven solution to estimate the nonlinear bending of sandwich doubly curved panel subjected to transient loading

This paper investigates nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading which is extensively used in aerospace structures. Due to the lack of a dataset for the nonlinear transient bending of the sandwich doubly curved panel for the data-driven solution, mathematical modeling is used. The sandwich doubly curved panel is made of two graphene nanoplatelets (GPLs) reinforced face sheets, and a polymer core. In the mathematical modeling section, the effective Young's modulus is obtained via modified Halpin-Tsai methodology, and the role of the mixture is used to estimate other material properties of the composite face sheets. The presented structure is analyzed using Reddy's third-order shear deformation theory (RTSDT), which incorporates von Kármán nonlinear geometric assumptions and Hamilton's principle. The governing equations and associated boundary conditions are derived accurately by imposing shear stress-free conditions on the upper and lower surfaces of the panel. The meshfree radial point interpolation method (RPIM) with interpolation functions is utilized to determine the unknown degrees of freedom to numerically simulate the mid-plane of the sandwich doubly-curved panel. The Newton-Raphson approach is used to solve the nonlinear ordinary differential using the constant-average acceleration strategy from Newmark's time integration method. After obtaining the dataset, the data is collected, and using an efficient data-driven approach, the results are tested, validated, and trained. In future studies, the presented efficient data-driven approach can be used with low computational cost to simulate nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading for related aerospace industries.

Nonlocal nonlinear higher-order finite element model for static bending of curved nanobeams including graphene platelets reinforcement

The aim of this work is concerned with the nonlocal nonlinear bending characteristics of non-classical isotropic and composite curved beams based on graphene platelets reinforcement, viz., nano size/micro beams. The size-dependant effect prominent in micro/nanobeams is incorporated in the formulation based on the nonlocal theory using size-dependent continuum approach. The proposed structural model includes both moderately large displacement and rotations, respectively. The system governing equations are established in terms of displacement functions using the principle of virtual work statements. A three-noded shear deformable curved beam finite element using a higher-order sinusoidal function based beam theory is employed; the solutions are obtained by adopting direct iterative procedure. The efficacy of the model is tested against the analytical studies available for the nano-size beams. A detailed analysis is made to highlight the influence of various design variables like thickness ratio, curvature of beam, nonlocal parameter, and the support conditions on the nonlinear bending behavior of curved nano/micro-size beams.

Nonlinear bending examination of nanocomposite cylindrical shell exposed to mechanical loads using dynamic relaxation method

In the current context, the nonlinear bending examination of a carbon nanotube reinforced composite (CNTRC) shell exposed to mechanical loading using the first-order shear deformation shell model via the dynamic relaxation (DR) procedure, is perused. The composite shell is assumed to be reinforced in the longitudinal axis. The curved morphology distribution of CNTs is considered to be functionally graded (FG) or uniform along the shell thickness. Mechanical properties of the constituents are gathered based on the modified rule of the mixture. For verification, the present numeric outcomes are compared with available references and ABAQUS finite element package. Finally, the roles of effective items including carbon nanotubes (CNTs) distributions, thickness-to-radius and length-to-radius ratios, boundary conditions, and the volume fraction of CNTs are investigated on the maximum non-dimension deflection.

An Out-of-Plane Bending Fatigue Assessment Approach for Offshore Mooring Chains Considering the Real-Time Updating of Interlink Bending Stiffness

Fatigue failure caused by frequent tension and bending loads is a crucial safety concern for mooring chains used on floating structures in the oil and gas industry. The bending effect for a chain’s fatigue is usually not considered by existing fatigue analysis methods, and even if it is considered, existing commercial software can only calculate the constant interlink bending stiffness without consideration of the stiffness variations due to tension and friction. To address this issue, this study develops an in-plane bending (IPB)/out-of-plane bending (OPB) fatigue assessment approach for offshore mooring chains considering the time-varying nonlinear interlink bending stiffness. Initially, the mooring tension and IPB/OPB angles are calculated by an in-house code MeCAP (multi-element coupled analysis program), considering the time-varying interlink stiffness between the chain links. The fatigue life of mooring chains for pure tension–tension (TT) and OPB combined fatigue at different hotspot locations (A, B, and C) is calculated by MeCAP-fatigue, a newly-developed module in MeCAP. Based on the analysis model, a comparative study is implemented to investigate the effects of interlink stiffness on fatigue damage. The differences and advantages of the combined fatigue calculation method over the pure TT fatigue assessment method are discussed. The results illustrate that combined fatigue would be underestimated significantly if zero interlink stiffness (hinge joint assumption) is applied or even constant interlink stiffness (calculated by mooring pretension), while pure TT fatigue will be largely unaffected. Moreover, compared to pure TT fatigue, the OPB combined fatigue assessment method, considering the effects of OPB/IPB and the time-varying properties of the connection between chain links, could evaluate fatigue life more comprehensively.

Incompatible-mode geometrically non-linear finite element for micropolar elasticity

In this work a new three-dimensional geometrically non-linear hexahedral micropolar finite element enhanced with incompatible modes is presented. The analytical model is expressed in terms of Biot-like stress and couple-stress tensors and corresponding Biot-like strain and curvature tensors, with a linear, elastic and isotropic constitutive law. The numerical model is derived based on the principle of virtual work, and the residual derivation together with the linearisation and static condensation procedure is given in detail. The newly developed finite element is tested against the analytical solution of the geometrically non-linear micropolar pure bending problem and the element accuracy and robustness is compared against hexahedral Lagrangian finite elements of first and second order on several numerical examples. It is shown that the newly presented element is fast convergent, more robust and more accurate than the available Lagrangian elements. Moreover, the operator split and static condensation provide for a significantly lower computational cost than standard elements.

Geometrically nonlinear bending of stiffened composite skewed cylindrical shells under transverse pressure

The literature review on relative performances of laminated composite skewed shells confirms that research reports on bending performances of moderately thin, stiffened, laminated composite skewed cylindrical panels, using the geometrically nonlinear approach, are not available. This paper aims to fill that deficiency and proposes a finite element code combining eight-noded, doubly curved elements with modified Sanders’ first approximation theory for thin shells and von Kármán-type nonlinear strains. Correctness of the proposed geometrically nonlinear bending formulation for skewed shells are verified through solutions of benchmark problems. The deflections, force, and moment resultants are reported for different skew angles, laminations, stacking sequences, radius of curvature, plan dimension ratios, and stiffener properties like orientations, numbers, and eccentric positions. The results are discussed critically which reveals that shells having curved edges free and straight edges clamped fabricated using 0°/90°/0° laminate offer the best performances. The biaxial stiffeners, nx = 7, ny = 7, show the minimum deflections and stress resultants. The skewed shells offer greater deflections and hence, must be avoided in industrial practices.

Solitary wave solution for the non-linear bending wave equation based on He’s variational method

A beam vibration originating in the beam porous structure or on a non-smooth boundary might make its vibrating energy concentrated on a single wave, leading to a solitary wave. This paper applies the variational approach to analysis of the soliton basic property, and the effect of the fractal dimensions on the solitary wave is elucidated. This paper is to draw attention the beam soliton property be-yond its widely known resonance and periodic and chaotic properties.

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.

Bayesian eikonal tomography using Gaussian processes


 Eikonal tomography has become a popular methodology for deriving phase velocity maps from surface wave phase delay measurements. Its high efficiency makes it popular for handling datasets deriving from large-N arrays, in particular in the ambient-noise tomography setting. However, the results of eikonal tomography are crucially dependent on the way in which phase delay measurements are predicted from data, a point which has not been thoroughly investigated. In this work, I provide a rigorous formulation for eikonal tomography using Gaussian processes (GPs) to smooth observed phase delay measurements, including uncertainties. GPs allow the posterior phase delay gradient to be analytically derived. From the phase delay gradient, an excellent approximate solution for phase velocities can be obtained using the saddlepoint method. The result is a fully Bayesian result for phase velocities of surface waves, incorporating the nonlinear wavefront bending inherent in eikonal tomography, with no sampling required. The results of this analysis imply that the uncertainties reported for eikonal tomography are often underestimated.

Fast analysis and prediction approach for geometrically nonlinear bending analysis of plates and shells using artificial neural networks

For the first time, this paper introduces a Fast Analysis and Prediction (FAP) approach for geometrically nonlinear bending analysis of plates and shells. The proposed approach consists of two phases: fast analysis and prediction using artificial neural networks (ANNs) combined with the Bayesian regularization algorithm. In the first phase (fast analysis), geometrically nonlinear bending analysis of a structure is performed using only several load steps. This enables the efficient computation within a short period of time. In the second phase (prediction using ANNs), the deflections obtained in the first phase are utilized as the data for training the ANNs. The trained networks are then used to quickly predict deflections at any load step. The calculation process in the second phase is also very fast. For the analysis package in the first phase, the formulation of this paper is originated from the literature for geometrically nonlinear analysis of composite shells. High accuracy and efficiency of the FAP approach were demonstrated via solving nonlinear bending problems of plates and shells including an isotropic square plate, a composite cylinder subjected to internal pressure, a FG-CNTRC panel and a pinched cylinder with end diaphragms. The proposed approach offers significantly computational savings (up to 70%) compared to the conventional one. ANNs combined with the Bayesian regularization algorithm, which are proposed for the FAP approach in this work, possess the outstanding advantages compared to many existing neural networks. The FAP approach can be applied to simulations, observations, or experiments where nonlinear responses are present or data is limited. Applying the FAP approach to such areas can significantly save labor, time, and money.

State-Spatial Viewpoint of Stress in Thermally Postbuckled and Nonlinearly Bent FG Saturated Poroelastic Circular Plates

A dimensionless discrete state-space mathematical model is proposed for stress analysis of an axially-symmetric saturated poroelastic circular plate being loaded by temperature rise of one of the flat faces. Computation of stress by means of this model is free from a need for differentiation. The curves of thermal postbuckling, thermal nonlinear bending, radial stress, circumferential stress and resultant radial force have been achieved by means of six state-space variables and differential quadrature procedure. The effects of loading temperature, boundary conditions, porosity, pore distribution and fluid pore pressure on the thermal post buckling and nonlinear bending phenomena have been investigated. The critical postbuckling thermal load has been determined by means of a numerical algorithm that does not converge to trivial solution because it starts from a relatively great numerical thermal load. Contrary to the governing equation in usual form, the governing equations in state space together with the state equations equalize the number of the governing assembled difference equations and that of the unknowns, and eventually do not require to apply the so-called auxiliary extra boundary conditions at the nodes adjacent to the boundary nodes. As another advantage, in state-spatial viewpoint, there is not any necessity for computing the coefficients of derivatives with order higher than unity.

An improved FSDT with an efficient mesh-free approach for nonlinear static analysis of FG-GOPRC beams

The aim of this paper is to enhance the First Shear Deformation Theory (FSDT) beam and apply it to investigate the linear and nonlinear static bending behavior of beams containing functionally graded graphene oxide powder-reinforced composite (FG-GOPRC). In this study, we propose an improvement to the FSDT by utilizing a high-order mesh-free approach. Unlike some existing FSDT methods, our proposed study uses a nonlinear shear deformation term coupling, which is crucial to illustrate the effect of nonlinear cases of various boundary conditions under a strong formulation, even without selective or reduced integration, resulting in lower computation costs. Using the principle of minimum potential energy, the governing equations are rewritten in a matrix–vector strong form and discretized using the Radial Point Interpolation Method with variable shape parameter (VS-RPIM) and linearized using a High Order Path Following Techniques (HOPFT). To account for the behavior of GOPRC, we utilized the modified Halpin–Tsai model to estimate the effective Young’s modulus, while the rule of mixture was employed to determine the effective Poisson’s ratio. Furthermore, the weight fractions of the graphene oxide powder (GOP) vary continuously through the thickness direction of the composite beams. To validate the proposed numerical model, we provide various comparison studies to demonstrate its robustness and accuracy. Additionally, we conduct parametric studies to examine the effects of different parameters, such as length-to-thickness ratio, GOP distribution patterns, weight fraction, and size of GOP, types of loads, and boundary conditions on the nonlinear bending behavior of FG-GOPRC beams. We also investigate the neglected term in some literature, which is of interest in our study. In summary, this paper proposes an enhanced FSDT to investigate the nonlinear static behavior of FG-GOPRC beams. The proposed model is validated through various comparison studies and parametric analyses, demonstrating its robustness and accuracy. Our study provides useful insights into the nonlinear bending behavior of FG-GOPRC beams and highlights the importance of nonlinear shear deformation term coupling.

Nonlinear static bending and dynamic behaviors of graphene platelets reinforced dielectric porous arches

This paper comprehensively investigates the electro-induced nonlinear static bending and vibration behaviors of the graphene platelets reinforced dielectric porous (GPLRDP) arches. The effective dielectric permittivity and Young's modulus of GPLRDP composites are determined based on the effective medium theory (EMT). The nonlinear governing equations for the GPLRDP arch under the electrical voltage are derived by using the Hamilton principle and are numerically solved by utilizing the differential quadrature method (DQM) together with an iterative scheme to obtain its electro-induced nonlinear static bending and dynamic responses. The influences of the porosity, GPL concentration, boundary condition, central angle, and DC and AC voltage on the nonlinear behaviors of the GPLRDP arch are discussed in detail. It is found that the addition of GPLs has significant effects on the electro-induced mechanical behaviors of the GPLRDC arch when the GPLs weight fraction exceeds the percolation threshold. Numerical analysis proves that the electro-induced static and dynamic behaviors of the GPLRDC arch can be designed and actively tuned through adjusting material and structural parameters, providing valuable insights for the development of smart GPLRDP arches.

Nonlinear bending and vibration analysis of sandwich plate with graphene reinforced micro composite facing sheets and auxetic honeycomb core in thermal environment

Nonlinear bending and vibration analysis of sandwich plate with graphene reinforced micro composite facing sheets and auxetic honeycomb core in thermal environment

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

In this paper, the nonlinear electromechanical bending of a bi-modular piezoelectric laminated beam is studied based on the principle of minimum potential energy and the Adomian decomposition method. The different tensile-compressive Young’s modulus of the core and piezoelectric layers, and the different tensile-compressive piezoelectric coefficients are considered. The electromechanical governing equations and related boundary conditions are obtained by using the principle of minimum potential energy. The deflection, neutral layer and interlaminar stresses of the beam are solved by the Adomian decomposition method and the iterative method, and verified by the finite element model and Galerkin's method. Results show that the applied voltage and the bi-modular characteristics affect the position of the neutral layer and the interlaminar stresses. Compared with the bi-modular properties of the core layer, the influences of the bi-modular properties of the piezoelectric layer on the neutral layer are relatively unobvious. In addition, the interlaminar stresses between the piezoelectric layer and core layer can be increased or decreased, depending on the relative magnitude of the applied voltage ratio and the bi-modular ratio. The results obtained are helpful for the analysis of the electromechanical coupling mechanism and design of piezoelectric composites and structures with bi-modular characteristics.

Analyzing the bending deformation of van der Waals-layered materials by a semi-discrete layer model

Van der Waals (vdW)-layered materials, such as graphite, exhibit unique mechanical properties owing to their structural and mechanical anisotropies. This study reports the development of a mechanical model that reproduces the characteristics of the nonlinear and reversible bending deformation of vdW-layered materials, while taking into account the microscopic mechanism of the discrete interlayer slips. The vdW-layered material was modeled as a stack of interacting discrete deformable layers (semi-discrete layer model), and the interlayer interaction was modeled using a cohesive zone model that reproduced the localized interlayer slip. Using the finite-element method, out-of-plane bending deformation analyses were performed on the cantilevers of the highly oriented pyrolytic graphite (HOPG) and MoTe2, and the validity of the model was verified by comparing it with the experimental results. The model accurately reproduced the loading and unloading behaviors in the experiments for the submicron HOPG cantilevers or the large nonlinear and reversible deformation with a hysteresis loop. Furthermore, the model reproduced well the characteristics of the bending experiments for the micro-MoTe2 cantilevers, or the intermittent decrease in stiffness during the loading process and deformation restoration during the unloading process. These results demonstrated that the designed semi-discrete layer model can be universally applied to reproduce the bending deformation characteristics of a variety of vdW-layered materials and can be employed to effectively elucidate the underlying deformation mechanisms.

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.

Method of Solving Geometrically Nonlinear Bending Problems of Functionally Graded Shallow Shells of Complex Shape

Method of Solving Geometrically Nonlinear Bending Problems of Functionally Graded Shallow Shells of Complex Shape

Towards realistic nonlinear elastic bending behavior for cable simulation

AbstractThis contribution aims to characterize and model the nonlinear elastic behavior of cables for reliable simulations. To simulate the nonlinear elastic behavior of cables, we use an iterative method, where at each step, an algorithmic local bending stiffness constant is used and updated according to the current cable state. We also formulate an inverse problem to determine the properties of real cables. By solving the inverse problem, the nonlinear elastic behavior for given measurement data is identified, yielding a curvature‐dependent bending stiffness characteristic. In addition, we propose an alternative method based on the balance equations for rods in static equilibrium to identify the bending stiffness characteristic. We apply both methods to experimental data, and the results are compared and discussed.

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.