Nonlinear Bending Behavior Research Articles

Piezoelectric Interaction Analysis and Load‐Deflection Prediction for Nonlinear Bending and Snap‐Through of Post‐Buckled FG Piezoelectric Beams

Revealing the interaction effect in the nonlinear bending of post‐buckled functionally graded (FG) piezoelectric beams promotes the intelligent development of structural design. The nonlinear bending behavior of post‐buckled piezoelectric beams is studied. Based on the displacement field constructed by the refined beam model, the post‐buckling configuration is considered. The nonlinear geometric relationship and the constitutive relationship in the multi‐physical field are modeled. In the framework of the principle of minimum potential energy, the nonlinear static models of post‐buckled FG piezoelectric beams are established. The post‐buckling configuration is obtained by the extended two‐step perturbation method. To obtain the approximate solution for the static problems of the post‐buckled FG piezoelectric beam, the perturbation scheme of electric potential and generalized displacement is proposed. The detailed parametric analysis indicates the interaction mechanism in the nonlinear bending of the post‐buckled FG piezoelectric beam. The snap‐through process and its sensitive interval are further captured. The back propagation neural network (BPNN) technique developed for the nonlinear bending problem of post‐buckled FG piezoelectric beams achieves accurate prediction.

A nonlinear elastic-plastic model describing the bending and fracture behavior of Caragana korshinskii Kom. branch wood

ABSTRACT Caragana korshinskii Kom. branch (CKB), a vital biomass source in northwest China, has not been thoroughly studied in terms of its bending fracture mechanics. This research aims to establish a nonlinear constitutive model that clarifies the fracture behavior of CKB when subjected to bending loads. The nonlinear bending fracture process of CKB was characterized through a three-point bending test, revealing stages of elastic deformation, elastic-plastic transition, and plastic fracture. A power-enhanced elastic-plastic model combined with continuous damage mechanics theory was used to develop an intrinsic bending fracture model, from which the equations governing bending damage evolution were derived. The model's accuracy was validated with simulations in Abaqus/Explicit, showing that CKB from various growth sites demonstrated distinct nonlinear bending behaviors. At the critical damage strain, cumulative damage led to a reduction in the effective elastic modulus, with the bending fracture curve exhibiting a power function trend. The parameters of the nonlinear elastic-plastic constitutive model were derived through curve fitting, yielding coefficients of determination (R²) exceeding 0.99. The consistency between the trends observed in actual tests and simulations, along with a peak bending force error of 7.7%, validated the finite element model's reliability and confirmed the intrinsic model's accuracy.

The stick-slip bending behavior of the multilevel helical structures: A 3D thin rod model with frictional contact

The multilevel helical structures in various engineering and natural fields offer excellent deformation flexibility and load bearing capabilities. Understanding the interplay between the local frictional contact and the geometric characteristics of the helical structure under complex external loads has attracted considerable interest. In this work, the effect of local frictional contact behaviors on the bending in multilevel helical structures is investigated by using a combination of theoretical modeling, finite element (FE) simulations, and experiments. In the case of pure bending, the kinematic parameters of the bent multi-stage helix are derived concisely by the idea of the kinematic analogy. The bending stiffness of the multi-stage helix is further obtained. In the case of the combined tension/torsion and bending, the 3D thin rod model incorporating Coulomb’s friction is established to describe the mechanical responses. It is found that the relationship between equivalent bending stiffness and the laying angle exhibits nonlinearity. A comparison with the classical Papailiou model reveals that, for helical structures at large laying angles, the influence of friction is primarily determined by the internal force in the tangential direction, which is the core assumption of the Papailiou model. However, in the case of small laying angles, the helical twisting characteristics and the contribution of the internal forces and moments in the other two directions (normal and binormal directions) to the friction cannot be ignored. Subsequently, a multilevel frictional contact transmission formulation is proposed according to the force action–reaction principle. Based on the above formulation, the non-simplified thin rod equations with Coulomb’s friction are extended to describe the multilevel stick-slip bending behaviors of the second stage cable (3*3). The dissipation capacity of helical structures is evaluated quantitatively under the hysteretic bending. Finally, the theoretical model is verified by FE simulations and experimental results. This work provides insights for unveiling the intrinsic relationship between the nonlinear bending and local frictional contact behaviors in the multilevel helical structures.

Simulation and parameterization of nonlinear elastic behavior of cables

AbstractThis work contributes to the simulation, modeling, and characterization of nonlinear elastic bending behavior within the framework of geometrically nonlinear rod models. These models often assume a linear constitutive bending behavior, which is not sufficient for some complex flexible slender structures. In general, nonlinear elastic behavior often coexists with inelastic behavior. In this work, we incorporate the inelastic deformation into the rod model using reference curvatures. We present an algorithmic approach for simulating the nonlinear elastic bending behavior, which is based on the theory of Cosserat rods, where the static equilibrium is calculated by minimizing the linear elastic energy. For this algorithmic approach, in each iteration the static equilibrium is obtained by minimizing the potential energy with locally constant algorithmic bending stiffness values. These constants are updated according to the given nonlinear elastic constitutive law until the state of the rod converges. To determine the nonlinear elastic constitutive bending behavior of the flexible slender structures (such as cables) from the measured values, we formulate an inverse problem. By solving it we aim to determine a curvature-dependent bending stiffness characteristic and the reference curvatures using the given measured values. We first provide examples using virtual bending measurements, followed by the application of bending measurements on real cables. Solving the inverse problem yields physically plausible results.

Characterization of pressure-dependent nonlinear bending behavior of yarns and its application in modeling the compression of 2D woven fabrics

This paper investigates the pressure-dependent nonlinear bending behavior of yarns, which is essential for the application of the virtual fiber modeling (VFM) method in the mechanical analyses of fabrics. An experimental method, along with a theoretical model based on classical beam theory, is presented to characterize the varying bending stiffness of yarns under different pressures. A tailored beam user element is then developed, incorporating the nonlinear bending behavior and combined with a truss element to create a physics-based virtual fiber formulation. Utilizing this formulation, the original kinematic VFM method is extended for modeling the mechanical response of 2D woven fabrics under compression. The predicted results of the proposed model closely match the reported experiment, demonstrating the significance of introducing the nonlinear bending behavior of yarns. This method can be a valuable tool for the fabric compression process and generating realistic mesoscale geometries for textile composites.

Thermomechanical bending of functionally graded carbon nanotubes reinforced composite plate by meshless method

AbstractFunctionally graded (FG) carbon nanotubes (CNTs) reinforced composite possessing continuously varied material properties have received extensive concern in the area of aviation, automotive, military, and spaceflight. The nonlinear bending of FG CNT composite is performed by developing a novel meshless model on the base of Smoothed Particle Hydrodynamics (SPH) method. The equivalent temperature relative material properties of composite is established in the light of the extended rule of mixture model. Different gradient dispersions of CNTs are taken into account and modeled. The governing equations are explored using the Mindlin theory and discretized though a symmetric SPH method with high prediction ability. The developed model is verified by analyzed several examples and compared to the solution obtained in literature. Influences of CNT dispersion, content, plate aspect ratio, boundary conditions and lamination angle on the bending response are elaborated. The present study provides an alternative potential method to solve the mechanical performances of FG CNT composites based on the meshless SPH formulation.Highlights Thermo‐mechanics of FG‐CNTRC is firstly solved by SPH method. The model is verified by solving the benchmarks in literature. Nonlinear bending behaviors of FG‐CNTRC are demonstrated. The minimum deflection take place in FG‐X type composite. For laminate plate, the smallest deformation occurs in 0° CNTRC.

Nonlinear bending and buckling analysis of 3D-printed meta-sandwich curved beam with auxetic honeycomb core

The present study investigates the static responses of a 3D-printed polymeric meta-sandwich curved beam with an auxetic honeycomb core, including buckling and nonlinear bending behavior. After extracting the mechanical properties of the core material, the nonlinear governing equations for the curved beam under two types of loading, namely uniform transverse and axial mechanical loads, are derived based on the first-order shear deformation theory and von-Kármán nonlinearity. After examining the convergence of the static responses and validating them by using the Ritz method, the influence of various geometrical parameters of the 3D-printed meta-sandwich structure on the results of buckling and nonlinear bending behavior as well as their optimization using the response surface methodology (RSM) are studied. The results indicate that by increasing the thickness-to-length ratio of the inclined wall, the critical load of the auxetic sandwich beam is enhanced due to improved stiffness, and its lateral displacement declines. The amount of buckling load increase from m = 0.005 to m = 0.1 is 1.92% in the clamped-clamped boundary conditions, 1.56% in the clamped-simply supported conditions, and 1.2% in the simply supported-simply supported edge conditions. Therefore, by selecting appropriate geometrical parameters for the core cells, the occurrence of instability can be delayed, leading to enhanced buckling resistance of the structure. Furthermore, optimizing the results by using the RSM showed that the highest critical load value and the lowest transverse deflection value of the meta-sandwich curved beam are obtained when θc = 15, n = 1, m = 0.1, 2θ0= 30, b = 0.07, λ= 1.1, h/h2= 1.2, and the edge conditions are of the clamped-clamped type.

Analysis of nonlinear bending behavior of nano-switches considering surface effects

Nano-switch structures are important control elements in nanoelectromechanical systems and have potential applications in future nanodevices. This paper analyzes the effects of surface effects, geometric nonlinearity, electrostatic forces, and intermolecular forces on the nonlinear bending behavior and adhesion stability of nano-switches. Based on the Von Karman geometric nonlinearity theory, four types of boundary conditions for the nano-switch structure were specifically calculated. The results show that surface effects have a significant impact on the nonlinear bending and adhesion stability of nano-switches. Surface effects increase the adhesion voltage of the nano-switch and decrease its adhesion displacement, and as the size of the nano-switch structure increases, the impact of surface effects decreases. A comparative analysis of the linear theory and the nonlinear theory results shows that the adhesion voltage predicted by the linear theory is smaller than that predicted by the nonlinear theory. The effect of geometric nonlinearity increases as the size of the nano-switch structure increases, as the distance between the electrodes increases, and as the aspect ratio of the nano-switch structure increases. These findings provide theoretical support and reference for the design and use of future nanodevices and nanoelectromechanical systems.

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

The nonlinear static behavior of hygrothermal magnetoelectroelastic(MEE) nanoplates considering the flexomagnetoelectric (FME) effect is researched by engaging the first-order shear deformation theory (FSDT). The constitutive equations of MEE nanoplates take into account the FME effect and hygrothermal effect. Leveraging the FSDT, von Karman's nonlinear equation and Hamilton's principle, the nonlinear control equation for hygrothermal MEE nanoplates can be derived by using the variational approach. The nonlocal strain gradient theory (NSGT) has application in the size effect of MEE nanoplates. The nonlocal nonlinear term in the control equation is handled by employing the Airy stress function, and then the non-linear mechanical model is solved by the Galerkin method. Thus, the nonlinear load-deflection curves (NLDC) of the MEE nanoplate are obtained via the introduction of material parameters, enabling an examination of the impact of various factors such as the FME effect, two small size parameters of NSGT and other parameters on the nonlinear bending behavior of MEE nanoplates. In conclusion, this study offers a theoretical foundation for taking into account the FME effect in the development of nanodevices, thereby contributing to advancements in this field.

Nonlinear bending behavior of functionally graded porous beams based on sinusoidal shear deformation theory

Nonlinear bending behavior of functionally graded porous beams based on sinusoidal shear deformation theory

Nonlinear bending characteristics of multidirectional functionally graded porous panels using higher-order finite element solution with parametric optimization

The current study aims to investigate the nonlinear bending behavior of unidirectional (1D) and multidirectional (2D/3D) functionally graded porous shell panels. Here, the material properties of the multidirectional functionally graded (FG)-panels are estimated by employing the representative volume element method and Voigt’s micromechanical homogenization scheme. The nonlinear mathematical model of the FG-panel is developed by incorporating Green-Lagrange type nonlinear strains in higher-order-midplane kinematics. The minimum potential energy principle is used to obtain the final form of the equilibrium equation for the multidirectional FG shell panel and further solved by using Picard’s iterative-based 2D-isoparametric finite element method via quadrilateral Lagrangian elements. The efficacy and robustness of the present model are checked by comparing its results with previously reported literature. Numerous examples are solved to analyze the impact of material parameters, such as volume fraction index and porosity index, and geometrical parameters, such as aspect ratio and support conditions, on the nonlinear deformation behavior of multidirectional FG shell panels. In addition, parametric optimization is carried out to minimize the flexural response by using the response surface methodology and, hence, determining the optimal values of multidirectional porous FG-panel. The results demonstrate that, within the defined ranges of FG-panel, lower values of porosity index and volume fraction index yield lesser deflection.

Application of suspended inter-array power cables in floating offshore wind farms

The aim of the present study is to propose an engineering approach of designing suspended inter-array power cable configuration for offshore wind farms. To achieve this, a spar type and a semi-submersible type floating offshore wind turbines (FOWTs) are modeled in the dynamic analysis software OrcaFlex. The accuracy of these models is validated against reference studies. Several different cable configurations connecting two spar FOWTs are assessed based on Fitness Factors using static and dynamic analyses. The non-linear bending behavior and the marine growth effects are considered in the analyses. The number of buoys is found to be the main parameter affecting the maximum effective tension and the minimum bending radii of the cables. The buoy spacing determines whether the cable touches the seabed, floats fully submerged, or floats at the sea surface. The influence of the cable length and the marine growth on the observed cable tensions and curvatures is small for the investigated configurations. The optimum cable configuration found for the two spar FOWTs is employed to connect two semi-submersible FOWTs. The influence of the floater type on the suspended power cable tension and curvature is found to be negligible. Analysis of power cable behavior in an offshore wind farm layout shows that a two-turbine setup with different environmental load directions can be used as a representative case for the suspended power cables design of the entire wind farm.

Development and Analysis of an Origami-Based Elastomeric Actuator and Soft Gripper Control with Machine Learning and EMG Sensors.

This study investigates the characteristics of a novel origami-based, elastomeric actuator and a soft gripper, which are controlled by hand gestures that are recognized through machine learning algorithms. The lightweight paper-elastomer structure employed in this research exhibits distinct actuation features in four key areas: (1) It requires approximately 20% less pressure for the same bending amplitude compared to pneumatic network actuators (Pneu-Net) of equivalent weight, and even less pressure compared to other actuators with non-linear bending behavior; (2) The control of the device is examined by validating the relationship between pressure and the bending angle, as well as the interaction force and pressure at a fixed bending angle; (3) A soft robotic gripper comprising three actuators is designed. Enveloping and pinch grasping experiments are conducted on various shapes, which demonstrate the gripper's potential in handling a wide range of objects for numerous applications; and (4) A gesture recognition algorithm is developed to control the gripper using electromyogram (EMG) signals from the user's muscles.

Large Deflection Geometrically Nonlinear Bending of Porous Nanocomposite Cylindrical Panels on Elastic Foundation

Large deflection nonlinear bending of functionally graded (FG) porous cylindrical panels reinforced with graphene platelets (GPLs) on a Pasternak-type elastic foundation is examined by developing a reliable and effective 2D meshfree-based nonlinear numerical method. The large displacement field is express by the first-order shear deformation theory (FSDT) and the von Kármán nonlinearity, and approximated by 2D natural element method (NEM) in conjunction with the stabilized MITC3+ shell concept and the shell surface–rectangular grid geometry transformation. The nonlinear simultaneous equations are solved by a load incremental Newton–Raphson scheme. The developed nonlinear numerical method is justified from by comparing with the reference solutions, and the load–deflection and bending moment of FG-GPLRC porous cylindrical panels on elastic foundation are scrutinizingly examined. Four different symmetric GPL distribution patters (except for FG-Λ) and three different symmetric porosity distributions are considered and their combined effects on the nonlinear bending behavior are investigated, as well as the effects of foundation stiffness and GPL amount. Also, the results are compared with those of FG CNT-reinforced porous cylindrical panels.

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.

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 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.

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.

Re-examination of nonlinear vibration, nonlinear bending and thermal postbuckling of porous sandwich beams reinforced by graphene platelets

This paper re-examines the nonlinear free vibration, nonlinear bending and thermal postbuckling behaviors of porous sandwich beams with graphene platelets (GPL) reinforcements rested on elastic foundations. We remove the equivalent isotropic model (EIM) and introduce an inhomogeneous model. The shear modulus together with the Young’s moduli for porous GPLRC core is determined by a generic Halpin–Tsai model with porosity. Mechanical properties of both porous GPLRC core and metal face sheets are temperature dependent. Based on the framework of higher order shear deformation beam theory, the motion equations of porous sandwich beams with GPL reinforcements are established. In the modeling, the von Kármán kinematic nonlinearity, beam-foundation interaction and thermal effect are considered. By employing the two-step perturbation method, the analytical solutions are obtained for the nonlinear vibration and nonlinear bending as well as thermal postbuckling problems. Numerical investigations are performed for comparing the results obtained from the EIM and the present model. The outcomes reveal that the EIM is invalid for linear free vibration, thermal buckling and postbuckling analyses of porous sandwich beams with GPL reinforcements, but the EIM is valid in some cases for nonlinear bending and nonlinear vibration analyses of the same beam.