Nonlinear viscoelastic analysis of orthotropic beams using a general third-order theory

The displacement based finite element model of a general third-order beam theory is developed to study the quasi-static behavior of viscoelastic rectangular orthotropic beams. The mechanical properties are considered to be linear viscoelastic in nature with a scope to undergo von Kármán nonlinear geometric deformations. A differential constitutive law is developed for an orthotropic linear viscoelastic beam under the assumptions of plane-stress. The fully discretized finite element equations are obtained by approximating the convolution integrals using a trapezoidal rule. A two-point recurrence scheme is developed that necessitates storage of data from the previous time step only, and not from the entire deformation history. Full integration is used to evaluate all the stiffness terms using spectral/hp lagrange polynomials. The Newton iterative scheme is employed to enhance the rate of convergence of the nonlinear finite element equations. Numerical examples are presented to study the viscoelastic phenomena like creep, cyclic creep and recovery for thick and thin beams using classical mechanical analogues like generalized n-parameter Kelvin-Voigt solids and Maxwell solids.     

Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress

A microstructure-dependent nonlinear theory for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, is developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, temperature-dependent properties, and the von Kármán geometric nonlinearity. Classical and first-order shear deformation theories are considered in the study. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material plate. The theories presented herein can be used to develop analytical solutions of bending, buckling, and free vibration for the linear case and finite-element models for the nonlinear case to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on linear and nonlinear response of axisymmetric analysis of circular plates.     

Nonlinear Deflection Control of Laminated Plates Using Third-Order Shear Deformation Theory

In this paper, the third-order shear deformation theory is employed to study static and dynamic deflection control of laminated composite plates. The effects of shear deformation and geometric nonlinearity (in the von Kármán sense) on the bending and transient response are investigated using the finite element method. Magnetostrictive material, Terfenol-D, layers are used to actively control the deflection via simple negative velocity feedback control in a closed loop. The effects of the lamination scheme, types of load, and boundary conditions on the deflection are investigated.     

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.     

A consistent deduction of von Kármán-type plate theories from three-dimensional nonlinear continuum mechanics

Summary Classical and refined plate theories derived from linear continuum mechanics lead to correct results only if the transverse deflection of the plate is small compared to its thickness. In the case of large deflections, geometrical nonlinearities have to be incorporated. For the classical Kirchhoff plate theory, a suitable extension for moderate rotations has been presented by von Kármán in 1911.Starting from the three-dimensional equations of nonlinear continuum mechanics, a family of von Kármán-type plate theories is deduced. For the derivation, the kinematical variables are replaced by a series representation and the principle of virtual displacements is used. It can be shown that most plate theories can be obtained from this type of theory and that the kinematical assumptions must fulfill certain conditions to obtain a solvable system of equations.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part I: Theory and solutions

Buckling and postbuckling behavior are presented for fiber reinforced composite (FRC) laminated cylindrical shells subjected to axial compression or a uniform external pressure in thermal environments. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic non-linearity and including the extension-twist, extension-flexural and flexural-twist couplings. The thermal effects are also included, and the material properties of FRC laminated cylindrical shells are estimated through a micromechanical model and are assumed to be temperature dependent. The non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths of FRC laminated cylindrical shells.     

Non-linear bending analysis of laminated sector plates using Generalized Differential Quadrature

Non-linear static analysis of laminated sector plates with any combination of clamped, simply supported and free edges is presented using Generalized Differential Quadrature (GDQ) method. Particular interest of this study is large deformation of asymmetric sector plates with free edges. Based on the first-order shear deformation theory and von Kármán type non-linearity, the governing system of equations include a system of 13 partial differential equations (PDEs) in terms of unknown displacements, forces and moments. Successive application of the GDQ technique to the governing equations resulted in a system of non-linear algebraic equations. The Newton–Raphson iterative scheme is then employed to solve the system of non-linear equations. Illustrative examples are presented to demonstrate accuracy and rapid convergences of the method with small number of grid points. Predictions of the presented method show very good agreement with other numerical studies available in the literature. Further results for asymmetric laminated sector plates with free edges are also presented for future references.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching. The formulation is general in the sense that it can be used to compare the influence of different structural theories, through static and dynamic analyses of curved nanobeams. The governing equations derived here are solved introducing a 3-nodes beam element. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loadings, boundary conditions, and nonlocal scale parameter are brought out on the static bending behaviors of curved nanobeams.     

A geometrically exact formulation for thin multi-layered laminated composite plates: Theory and experiment

A geometrically exact approach is employed to formulate the equations of motion of thin multi-layered isotropic and laminated composite plates subject to excitations that cause large strains, displacements, and rotations. The linearization of the obtained semi-intrinsic theory leads to the Mindlin–Reissner theory while an ad hoc truncated kinematic approximation delivers, as a by-product, the Föppl–von Kármán theory of plates. An experimental validation is sought for fully clamped plates which are either of the isotropic single-layered type or of the multi-layered laminated composite type. To this end, nonlinear equilibrium paths are constructed both theoretically and experimentally when the plates are subject to a quasi-statically increasing central point load. The comparisons between the experimentally obtained results and those furnished by the geometrically exact theory as well as by the Föppl–von Kármán (FVK) theory show the high accuracy of the proposed nonlinear theory while the FVK theory becomes increasingly inaccurate at deflection amplitudes of the order of the plates thickness.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

Nonlinear elastic mechanics of the ball-loaded blister test

The nonlinear elastic mechanics of spherically capped shaft or ball-loaded blister tests is presented. In the test model, a thin film is attached to a substrate with a circular hole running through the thickness of the substrate. A central load is applied to the film through the hole by a spherically capped shaft or a ball with a finite radius. The deformed blister is divided into two parts: a circular region in contact with the sphere of the cap or ball and an outer noncontact annulus. The Reissner’s plate theory is employed to describe the deformation of the contact part and the von Kármán plate theory for the noncontact annulus. A constitutive equation of coupled linear springs is obtained to quantify the effect of the substrate deformation on the blister deflection. For small deflection, the analytical solution of load-deflection is derived. For large deflection, an iteration approach is adopted to predict numerically the load-deflection curve. Finite-element analysis is conducted to verify the analytical and numerical solutions. The influence of the substrate deformation, residual stress, radius of the spherical cap or ball and the friction between the film and ball on the load-deflection relation is investigated.     

Coupled deflection analysis of thin-walled Timoshenko laminated composite beams

For the deflection analyses of thin-walled Timoshenko laminated composite beams with the mono- symmetric I-, channel-, and L-shaped sections, the stiffness matrices are derived based on the solutions of the simultaneous ordinary differential equations. A general thin-walled composite beam theory considering shear deformation effect is developed by introducing Vlasov’s assumptions. The shear stiffnesses of thin-walled composite beams are explicitly derived from the energy equivalence. The equilibrium equations and force-deformation relations are derived from energy principles. By introducing 14 displacement parameters, a generalized eigenvalue problem that has complex eigenvalues and multiple zero eigenvalues is formulated. Polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to the equilibrium equations. Then the displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to nonzero and zero eigenvalues, respectively. Finally, the stiffness matrices are evaluated by applying the member force-displacement relations to the displacement functions. In addition, the finite beam element formulation based on the classical Lagrangian interpolation polynomial is presented. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the finite element results using the isoparametric beam elements and the detailed three-dimensional analysis results using the shell elements of ABAQUS. Particularly the effects of shear deformations on the deflection of thin-walled composite beams with the mono-symmetric I-, channel-, and L-shaped sections with various lamination schemes are investigated.     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

This article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.     

Nonlocal theories for bending,buckling and vibration of beams

Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.     

Torsional buckling and postbuckling of double-walled carbon nanotubes by nonlocal shear deformable shell model

This paper presents an investigation on the buckling and postbuckling of double-walled carbon nanotubes (CNTs) subjected to torsion in thermal environments. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects and van der Waals interaction forces. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent and are obtained from molecular dynamics (MD) simulations. The small scale parameter _e0a$e_{0}a$ is estimated by matching the buckling torque of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. The results show that buckling torque and postbuckling behavior of CNTs are very sensitive to the small scale parameter _e0a$e_{0}a$. The results reveal that the size-dependent and temperature-dependent material properties have a significant effect on the torsional buckling and postbuckling behavior of both single-walled and double-walled CNTs.     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Non-linear buckling and postbuckling of shear deformable anisotropic laminated cylindrical shell subjected to varying external pressure loads

Non-linear buckling and postbuckling of a moderately thick anisotropic laminated cylindrical shell of finite length subjected to lateral pressure, hydrostatic pressure and external liquid pressure has been presented in the paper. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic non-linearity and including the extension/twist, extension/flexural and flexural/twist couplings. The non-linear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a circumferential stress along with an associate shear stress when the shell is subjected to external pressure.