Nonlinear Bending Research Articles

Accurate non-linear bending analytical solutions setting new benchmarks for rectangular thin plates with four clamped edges

Accurate non-linear bending analytical solutions setting new benchmarks for rectangular thin plates with four clamped edges

Nonlinear bending and vibration analysis of a variable-width piezoelectric nanoplate with flexoelectric effects

Nonlinear bending and vibration analysis of a variable-width piezoelectric nanoplate with flexoelectric effects

Analyzing flexoelectric polarization of suspended membrane by nonlinear bending theory of plate

Strain gradient breaks inversion symmetry and induces flexoelectric polarization as well as electromechanical coupling in all material systems, though the effect is usually only significant at the nanoscale. Two-dimensional (2D) materials and thin membranes thus provide an ideal platform to explore flexoelectricity, which has been widely pursued, yet quantitative theoretical analysis is needed to guide the rapid experimental developments. In this work, we develop 2D flexoelectric model for suspended membrane based on von Kármán plate theory, and implement it into finite element computation using conforming BCIZ element. Numerical results and discussions on flexoelectric polarization in suspended membrane under uniform pressure or concentrated load are presented, which are validated by piezoresponse force microscopy (PFM) experiments under a range of membrane thicknesses and loading forces showing good agreement with computations. Since large strain gradient often exists in samples with small size in one or two dimensions, the method we develop provides a powerful tool to study a wide range of low-dimensional materials and structures with flexoelectric effect.

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.

On the nonlinear mechanics of hybrid skew aerospace structures

This study proposes a functionally graded (FG) nanocomposite hybrid skew plate element which is fabricated from a polymeric matrix and is reinforced with two well-known nanoscale reinforcements, namely, carbon nanotubes (CNTs) and graphene platelets (GPLs). The Mori–Tanaka approach, the Halpin-Tsai scheme and standard rule of mixtures are employed to estimate the effective mechanical properties of the hybrid plates. To model this structural element, Reddy's third-order shear deformation theory (TSDT) is used. By considering the von-Kármán assumptions and using the Hamilton's principle, the nonlinear static bending and free vibration governing equations are determined. Then the isogeometric analysis (IGA) is employed to establish the structural stiffness and mass matrices of the skew hybrid plates. After validating the correctness of the presented solution procedure with the previously published results, the effects of various variables such as number of layers, distribution pattern of reinforcements (CNTs and GPLs) and their volume fractions, skew angle and width-to-thickness ratio are investigated. It is found that with considering nonlinear von-Kármán relationship in the strain field, the distributions of σxx and σyy about the mid-plane are no longer symmetric.

Non-linear bending analysis and control of graphene-platelets-reinforced porous sandwich plates with piezoelectric layer subjected to electromechanical loading

Piezoelectric materials as the controlling element have been widely utilized to produce intelligent engineering structures, while these smart structures may fail to realize effective control of composite structures with large deformations. However, investigations on such issues are less reported in published literature, as an accurate and efficient model is required to well forecast the geometrically nonlinear behaviors of smart sandwich structures. As a result, a novel sinusoidal Legendre global-local higher-order shear deformation plate theory (SLHSDT) has been developed to accurately capture geometrically nonlinear behaviors of piezoelectric sandwich plates. The proposed model can fulfill the compatible conditions of transverse shear stresses and contain transverse normal strain, which can ensure precision in predicting electromechanical behaviors. The multi-patch isogeometric analysis (IGA) method for sandwich plates partially bonded with piezoelectric layers is proposed to overcome C1-continuity between patches for the first time. Moreover, the Newmark-β method and Newton-Raphson technique are attempted to solve the nonlinear equations. The present model has been utilized to investigate electromechanical behaviors of laminated structures with piezoelectric layers, which has been compared with the published results. In addition, experiments on macro fiber composite (MFC) integrated sandwich plates have been also carried out in the present work, which can effectively verify the performance of proposed model. Subsequently, the proposed model is employed to study electromechanical behaviors of the five-layer piezoelectric sandwich plates containing internal pores and graphene platelets. Then, influences of the porosity coefficient and GPLs weight fraction on the nonlinear electromechanical behaviors of sandwich plates are investigated. Eventually, the active control on nonlinear behaviors of piezoelectric porous sandwich plates with GPLs reinforcement is studied by using a closed-loop control system, and an effective approach slowing down large deformation has been proposed by selecting an appropriate distribution of GPLs along the thickness direction.

Nonlinear Bending of FG-CNTR Curved Nanobeams in Thermal Environments

Nonlinear Bending of FG-CNTR Curved Nanobeams in Thermal Environments

Prediction of sectional collapse of thin-walled structure under pure bending by nonlinear composite beam theory

Brazier (1927) found that when one dimension of the beam cross-section was relatively smaller than the others, large in-plane displacements over the cross-section might occur, even though the strains could remain very small. Under this circumstance, the so-called Brazier effect refers to the cross-sectional ovalization, which leads to nonlinear bending buckling and collapses. This paper studies the Brazier effect by the nonlinear Variational Asymptotic Beam Sectional Analysis (VABS) theory which considers finite cross-sectional deformations. Nonlinear VABS reduces three-dimensional (3D) continuum to a one-dimensional (1D) beam analysis and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities without unnecessary kinematic assumptions. The present theory is implemented using the finite element method (FEM) in the VABS code, a general-purpose beam cross-sectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model using the Euler–Bernoulli beam theory. The deformation gradient tensor is directly used to deal with finite deformation, various strain definitions, and several types of material laws. Numerical examples demonstrate the capabilities of VABS to predict the sectional collapse of thin-walled structures under pure bending.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

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.

Nonlinear thermal-mechanical coupled isogeometric analysis for GPLs reinforced functionally graded porous plates

This work presents an isogeometric model for thermal-mechanical coupled analysis of geometrically nonlinear responses of graphene platelets reinforced functionally graded porous (GPLs-FGP) plates. The closed-cell Gaussian random field (GRF) scheme, Halpin-Tsai micromechanics model and rule of mixture are utilized to determine the material properties of GPLs-FGP plates with three types of porosity distribution and GPLs dispersion pattern. Unlike previous studies, the influences of GPLs and porosity on the thermal conductivity of GPLs-FGP plates are considered simultaneously. A geometrically nonlinear isogeometric analysis (IGA) model is developed base on a four-variable refined plate theory (RPT), using the von Kármán nonlinear theory, Hamilton’s principle and Total Lagrangian (TL) incremental scheme. After ensuring the accuracy and reliability of the present model through comparison with several examples, the thermal conductivity, temperature distribution, geometrically nonlinear static bending and transient responses of the plate, taking into account the temperature effects of porosity and graphene under thermal-mechanical loads are investigated. Numerical results show that porosity exerts influences on the thermal conductivity and temperature distribution of GPLs-FGP plates, which subsequently impact their geometrically nonlinear thermal-mechanical coupled responses. The results of this study are helpful for designing and manufacturing GPLs-FGP structures with improved performance.

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.

A new approach to the calculation of variable tangent bending stiffness for helical strands

As the cross-section of the helical strand increases, the effect of its nonlinear bending characteristics becomes significant. To accurately simulate the response of helical strands under such conditions using the finite element method, determining the tangent bending stiffness of helical strands is indispensable. However, the approach of calculating the tangent bending stiffness of helical strands based on the stick-slip state of wires is not universally applicable across all strand models. In this study, we propose an approach for calculating the tangent bending stiffness of strands based on two mechanical models of helical strands. This proposed approach uses the ratio of the “actual increment” to the “theoretical maximum increment” of the nonlinear component of the wire axial force to calculate the contribution of one wire to the tangent bending stiffness of the strand. Significantly, this approach can be applied to different strand models, each of which may adopt different slip criteria. By comparing the results with the numerical solutions, we validate the effectiveness of the proposed approach for calculating the tangent bending stiffness of helical strands on different strand models. Furthermore, it is pointed out that the analytical results of wire sliding may vary significantly depending on the adopted slip criteria.

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.

Stiffness and stability of bamboo stem- A optimal design perspective

During the evolution process, a bamboo stem achieves a significant height (up to 20 m) to fulfil its phototropic requirements. While on land, the stem is mostly subjected to bending load which makes it liable to fail by uprooting. However, this failure is prohibited by smart structure of bamboo stem which includes graded arrangement of fibre bundles in the cross-section and a tapered cantilever form of the stem. This paper attempts to understand the optimal design of bamboo stem through the relationship between the stellar arrangement of stiff fibre bundles in the cross-section and the tapered form. In this work, a comparison between two types of stellar arrangement, namely uniform and graded, is presented in view of non-linear bending analysis through elastica theory and fracture-induced delamination, both numerically. It is observed from the results that a bamboo stem prefers to evolve with graded stellar arrangement which provides gradation of stiffness and toughness over the cross-section; the trend in toughness being opposite to that of stiffness. Moreover, interplay of stellar arrangement and gradation of stiffness-toughness thereof is found to be the governing mechanism for ensuring its mechanical integrity and stability in view of an optimal design perspective. The smart structure of bamboo is recommended for bio-mimicking.

Thermally Induced Nonlinear Bending and Stability of Nanocomposite Curved Pipes Reinforced by CNTs

This research is concerned with the thermal bending and stability of temperature-dependent nanocomposite curved pipes strengthened with carbon nanotubes (CNTs) subjected to uniform temperature rise. Thermo-mechanical characteristics of the polymer composite pipe are assumed to vary entirely in the thickness by a non-uniform function of the radius. Five different patterns are selected to model the propagation profile of CNTs amongst the pipe thickness. Based on the shear deformation and von-Karman kinematic hypothesis, nonlinear balance equations of the polymer curved pipe are determined via varying the total potential energy of the system. Governing equations as a set of coupled nonlinear differential equations are analytically solved using a perturbation-based technique. Closed-form solutions are derived to estimate large-amplitude deflection of nanocomposite curved pipes with pinned and clamped boundaries under uniform thermal loading. The obtained results show the influences of important parameters such as material/geometrical characteristics and foundation stiffness on the thermally induced nonlinear response of polymer nanocomposite curved pipes.

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.

Nonlinear wave propagation in homogenized strain gradient 1D and 2D lattice materials: Applications to hexagonal and triangular networks

AbstractThis work aims to analyze the propagation of fully nonlinear waves, encompassing shear, extension, and bending deformation modes, within homogenized periodic nonlinear hexagonal and triangular networks, successively considering 1D and 2D situations. The wave analysis is conducted from the expression of the effective strain energy density of periodic hexagonal and triangular lattices in the nonlinear regime by a continualization of the discrete lattice equations, considering all forms of energy. We incorporate strain gradient effects into the continuous model to account for the wave‐dispersive nature. The resulting second‐gradient nonlinear continuum exhibits subsonic and supersonic propagation modes. We first examine in a 1D situation the dynamical response of the hexagonal and triangular lattices, considering varying levels of nonlinearity quantified by a single scalar valued parameter. We further evaluate the impact of a fully nonlinear analysis compared to an analysis solely based on the shear energy, regarding both supersonic and subsonic modes. The nonlinear wave propagation analysis is then extended to a 2D situation for the same two lattices. It is shown that the longitudinal mode exhibits a higher frequency at a low degree of nonlinearity; however, as the degree of nonlinearity increases, the shear mode surpasses the longitudinal mode in terms of frequency. As the wavenumber increases, the nonlinearity has a lesser impact on the frequency behavior, and the phase velocity is more influenced by other factors, such as the second gradient contributions of the effective constitutive law. Such a behavior indicates a transition from a highly nonlinear behavior at lower wave numbers to a more linear behavior at higher wave numbers.

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.