Nonlinear Static Bending Research Articles

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.

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.

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.

Nonlinear Static Bending and Forced Vibrations of Single-Layer MoS2 with Thermal Stress.

Single-layer molybdenum disulfide (MoS2) has been a research focus in recent years owing to its extensive potential applications. However, how to model the mechanical properties of MoS2 is an open question. In this study, we investigate the nonlinear static bending and forced vibrations of MoS2, subjected to boundary axial and thermal stresses using modified plate theory with independent in-plane and out-of-plane stiffnesses. First, two nonlinear ordinary differential equations are obtained using the Galerkin method to represent the nonlinear vibrations of the first two symmetrical modes. Second, we analyze nonlinear static bending by neglecting the inertial and damping terms of the two equations. Finally, we explore nonlinear forced vibrations using the method of multiple scales for the first- and third-order modes, and their 1:3 internal resonance. The main results are as follows: (1) The thermal stress and the axial compressive stress reduce the MoS2 stiffness significantly. (2) The bifurcation points of the load at the low-frequency primary resonance are much smaller than those at high frequency under single-mode vibrations. (3) Temperature has a more remarkable influence on the higher-order mode than the lower-order mode under the 1:3 internal resonance.

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.

Geometrically nonlinear bending mesh-free analysis of functionally graded porous sandwich beam

In this work, geometrically nonlinear static bending analysis of sandwich beams with a porous core and two skins made of functionally graded materials using a mesh-free method is presented. The material types of the core and face sheets of the beam are chosen so that the material continuity between the layers is guaranteed. The nonlinear governing equation including geometric nonlinearity is established via the principle of virtual work. This equation is discretized into a system of algebraic equations by the mesh-free approach, which is based on the C1 point interpolation method and polynomial basis functions, and then solved by the direct iterative method. The convergence of the mesh-free method is tested to determine the sufficient mesh level for the analysis. Comparative and comprehensive studies are performed to investigate both the correctness and the influences of several important parameters and boundary conditions on the linear and nonlinear deflections of the beam.

Nonlinear static analysis of functionally graded porous sandwich plates resting on Kerr foundation

The main purpose of this work is to explore the nonlinear static bending analysis of functionally graded porous sandwich plates with a ceramic core (FSCC) subjected to a transverse uniform load resting on Kerr foundation (KF) using the mixed interpolation of tensorial components for the four-node rectangular element (called MITC4). Porosity appearing in two skin layers is assumed according to the uneven porosity distribution with a porosity factor Ω . The advantage of this element is fast convergence, high accuracy and cancels the shear-locking phenomenon. The present study considers the geometrical nonlinearity resulting from mid-plane stretching based on the von Kármán geometrical nonlinearity. The first-order shear deformation theory (FSDT) is employed to approximate the displacement field of the plate due to its simplicity and efficiency. The current formulation’s authenticity is evaluated by comparing it with previously published works. For the first time, the nonlinear bending of FSCC subjected to a transverse uniform load resting on the Kerr foundation is investigated. Furthermore, the influence of input parameters on the nonlinear bending behavior of FSCC is fully provided.

STATIC BENDING ANALYSIS OF VARIABLE THICKNESS MICROPLATES USING THE FINITE ELEMENT METHOD AND MODIFIED COUPLE STRESS THEORY

This paper presents the nonlinear static bending analysis of variable thickness microplates by using the finite element method and modified couple stress. The present theory and mathematical model are confirmed by comparing the numerical data with those of open literatures. A parameter study is carried out to investigate the mechanical behavior of the structure, especially, the effect of nonlinearlity. The computed data can be used as a good reference in the use and design of these types of structures in engineering practice.

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

New finite modelling of the nonlinear static bending analysis of piezoelectric FG sandwich plates resting on nonlinear elastic foundations

This paper presents a new finite modelling of the nonlinear static bending analysis of piezoelectric functionally graded (FG) sandwich plates resting on nonlinear elastic foundations. Finite element formulations are derived by using the first-order shear deformation theory (FSDT) of Mindlin and the finite element method. The proposed theory and mathematical model of this work are verified by comparing the results with those of other methods, and they are in good agreement. A parameter study is conducted to investigate the effects of geometrical and physical properties such as nonlinear factors, volume fraction index, nonlinear foundation parameters, applied voltages, boundary conditions, etc., on the nonlinear mechanical behaviors of piezoelectric FG sandwich plates. The novel numerical results of this work are very important, and can be used as a good reference to examine related structures as well as of use in engineering practice.

Small size-effect isogeometric analysis for linear and nonlinear responses of porous metal foam microplate

The main objective of this paper is to explore the linear and nonlinear responses of porous metal foam micro-plate using a modified couple stress theory coupled with isogeometric analysis. The material properties of metal foam micro-plate are derived from three porosity variation models, namely uniform, symmetric and asymmetric variation across the thickness of the microplate. The stress-strain and the modified couple stress constitutive relations are obtained using a seventh order distributed function based on the general shear deformation theory. Furthermore, the Hamilton’s principle is utilized to establish the size-effect equilibrium equations for metal foam micro-plate. The presented numerical solution is able to capture the small size effect of microplate by considering the material length scale parameter in the classical continuum theory. The reported results demonstrate the effect of small size-effect, porosity variation, porosity coefficient, slenderness ratio on free vibration, linear and nonlinear static bending of porous metal foam micro-plate. The role of pattern porosity variation in the nonlinear responses becomes more important by increasing the porosity coefficient. The pattern of symmetric porosity presents the best performance of linear and nonlinear responses.

Modified couple stress-based nonlinear static bending and transient responses of size-dependent sandwich microplates with graphene nanocomposite and porous layers

This paper investigates the nonlinear static bending and dynamic transient responses of the rectangular and circular sandwich microplates composed of functionally graded graphene nanocomposite (GNC) face sheets and a porous polymeric core layer based on the modified couple stress theory (MCST) within the isogeometric analysis (IGA) framework. The four-variable higher-order shear deformation refined plate theory (RPT) and the von Kármán large deflection assumption are synthetically employed to describe the kinematics of the microplate. The nonlinear governing equations of motion are derived from the Hamilton’s principle. The nonlinear bending problem is solved by the Newton–Raphson technique under a displacement criterion strategy, while the dynamic responses under transient loads are obtained by the Newmark integration scheme in conjunction with Newton–Raphson iterative procedure. Comparisons are carried out to test the accuracy of iteration procedure in present formulation and the reliability in capturing small scale effect of present model. Obtained outcomes reveal that raising the material length scale (MLS) parameter and volume fraction index of graphene nanofillers both leads to reductions in nonlinear static deflections, dynamic amplitudes of deflection and periods of motion. The examples for porosity coefficient show that embedding pores in core layer leads to slight changes in terms of static and dynamic responses and an improvement in weight reduction of the sandwich microplate.

Nonlinear electromechanical analysis of axisymmetric thin circular plate based on flexoelectric theory

Flexoelectricity will dominate the electromechanical coupling of intelligent components in MEMS/NEMS due to its size-dependency. This paper focuses on investigating the flexoelectric responses of intelligent components of the circular plate type, which are commonly used in MEMS/NEMS. Utilizing Hamilton’s principle, the nonlinear flexoelectric circular plate model is presented by combining von Kármán plate theory and flexoelectric theory. The equilibrium equations and all boundary conditions are obtained and then discretized. The nonlinear static bending of the simply supported axisymmetric flexoelectric circular plate is investigated by combining DQM and iteration method. The distributions of dimensionless bending deflection and electric potential are analyzed under different loads. Moreover, the nonlinear free vibration behaviors are also investigated by combining the Galerkin method and Lindstedt–Poincaré Method. The flexoelectric effect and stiffening effect of strain gradient are revealed. This paper will be helpful to promote the application of flexoelectric intelligent components of the circular plate type, which are encountered commonly in engineering.

Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.

Linear and nonlinear static bending of sandwich beams with functionally graded porous core under different distributed loads

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Kármán. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique. The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio, loading types, porous distributions of the core, etc., are presented in graphical and tabular forms, serving as a benchmark solution for future studies.

Geometrically nonlinear isogeometric analysis of smart piezoelectric FG plates considering thermal effects of piezoelectric stress and dielectric constants

In this paper, the isogeometric analysis (IGA) integrated with a simple first-order shear deformation theory (S-FSDT) with only four variables per node is extended to study geometrically nonlinear natural frequency, nonlinear static bending and transient dynamic response of two types of smart piezoelectric functionally graded plates (PFGPs) in thermal environment. The neutral surface is introduced to avoid the stretching-bending coupling effect. The nonlinear strain–displacement relation of the plates is obtained by combining the von Karman strains with the S-FSDT. To investigate the nonlinear mechanical behavior of PFGPs in thermal environment, closer to the real situation, an augmented piezoelectric constitutive equation considering the thermal effects of piezoelectric stress and dielectric constants is used. The accuracy and effectiveness of developed approach are demonstrated through numerical experiment with comparison. We additionally investigate the variation of temperature distribution on geometrically nonlinear response of PFGPs suffering thermo-electro-mechanical load.

Nonlinear Static Bending and Free Vibration Analysis of Bidirectional Functionally Graded Material Plates

Bidirectional functionally graded material (2D-FGM) plates have mechanical properties that vary continuously in both the thickness and one-edge directions; these plates are more and more widely used in design and engineering applications. When these structures are subjected to strong loads, they can be largely deformed; therefore, nonlinear calculations, in this case, are necessary. In this paper, nonlinear static bending and nonlinear free vibration behaviors of 2D-FGM plates are studied by using the finite element method based on the third-order shear deformation theory; the Newton-Raphson method is used to solve this problem. The accuracy of this approach is confirmed by comparing the results with respect to other papers. The effects of some numerical aspect ratios such as volume fraction index and thickness-to-length ratio on nonlinear static bending and free vibration of the plates are explored. This study shows that there is a big difference between the numerical results obtained from the nonlinear problem and those from the linear one.

Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT

The present paper investigates the nonlinear static bending behavior of infinite length cylindrical panels made of functionally graded porous (FGP) materials. The shell with clamped edges is subjected to uniform temperature rise and transverse pressure loading. Thermomechanical properties of the shell with even distributed porosities are temperature-dependent and are graded along the thickness. The principle of virtual displacement and nonlocal strain gradient theory (NSGT) are employed to derive the equilibrium equations of the shallow shell resting on nonlinear elastic foundation. The concept of physical neutral surface (PNS) and higher-order shear deformation theory are also included into the formulation. Using the uncoupled thermoelasticity and Donnell kinematic assumptions, three differential equations of the shell under thermomechanical loading are established. The nonlinear system of governing equations is solved for the shell with infinite length which is clamped on both straight edges and free at other curved edges. The two-step perturbation technique and Galerkin procedure are employed to derive analytical solutions for the thermal, mechanical, and thermomechanical responses of cylindrical panels. The important parameters governing the bending behavior of shells under thermal-mechanical coupling load are identified and discussed. Parametric studies are given to show the influences of nonlocal/length scale parameter, porosity coefficient, foundation stiffness, geometrical parameter and functionally graded pattern. It is shown that the geometrical parameters have an important role on the bending behavior of cylindrical panels. Also, the temperature dependence of material properties results in higher deflection of the heated shells.

Nonlinear stress-driven nonlocal formulation of Timoshenko beams made of FGMs

Motivated by the paradoxical results obtained from the differential nonlocal elasticity theory in some cases (e.g., bending and vibration problems of cantilevers), several attempts have been recently made to develop nonlocal beam models based on the integral (original) formulation of Eringen’s nonlocal theory. These models can be classified into two main groups including strain- and stress-driven ones which have the capability of capturing the softening and hardening behaviors of material caused by nanoscale (nonlocal) effects, respectively. In the present paper, a novel stress-driven nonlocal formulation is developed for the nonlinear analysis of Timoshenko beams made of functionally graded materials. To this end, the governing equations are first derived in the context of integral form of stress-driven nonlocal model. The proposed model can be used for arbitrary kernel functions, and the paradox related to cantilever is resolved by it. The governing equations of stress-driven model in differential form together with corresponding constitutive boundary conditions are also derived. The Timoshenko beam under various end conditions is considered as the problem under study whose nonlinear static bending is analyzed. Furthermore, the generalized differential quadrature method is employed in the solution procedure. The effects of nonlocal parameter, FG index, length-to-thickness ratio and nonlinearity on the deflection of fully clamped, fully simply supported, clamped–simply supported and clamped–free beams are investigated. The presented formulation and results may be helpful in understanding nonlocal phenomena in nano-electro-mechanical systems.