Nonlinear eccentric low-velocity impact response of a polymer-carbon nanotube-fiber multiscale nanocomposite plate resting on elastic foundations in hygrothermal environments

In the present article, a nonlinear, eccentric, low-velocity impact response of a polymer-carbon nanotube-fiber multiscale nanocomposite plate on elastic foundations in hygrothermal conditions using the finite element method is performed. In this regard, the governing equations are derived based on higher-order shear deformation plate theory and von Kármán geometrical nonlinearity. Three types of distributions of the temperature field and moisture concentrations, namely, uniformly, linearly, or nonlinearly through the thickness direction of the plates are considered. The effective material properties of the multiphase nanocomposite are calculated using fiber micromechanics and Halpin–Tsai equations in hierarchy. The carbon nanotubes are assumed to be uniformly distributed and randomly oriented through the matrix. The contact force between the impactor and the plate is obtained with the aid of the modified nonlinear Hertzian contact law models. After examining the validity of the present work, the effects of the weight percentage of CNT, moisture concentration, temperature variations, distribution of temperature and moisture concentration, elastic foundation, and the eccentricity on the contact force, indentation, and central deflection of polymer-CNT-fiber multiscale nanocomposite plate are studied in details.     

Nonlinear Free Vibration of In-Plane Functionally Graded Rectangular Plates

Nonlinear free vibration of functionally graded (FG) plates with in-plane material inhomogeneity subjected to different boundary conditions is presented. The nonlinear equations of motion and the related boundary conditions are extracted based on the classical plate theory. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing equations in spatial domain. After validating the presented approach, parametric studies are performed to clarify the effects of different parameters on the nonlinear frequency parameters of the in-plane FG plates.     

Application of the boundary-domain element method to the pre/post-buckling problem of von Kármán plates

Based on the BEM formulations for the finite deflection problem of von-Kármán-type plates, this paper presents an incremental boundary-domain element method for the pre/post-buckling problem of thin elastic plates. As the governing equations involve the coupled in-plane and out-of-plane deformations as the nonlinear terms, the boundary integral equations are formulated in terms of the increment by using the fundamental solutions for the linear parts of the differential operators. Some of the innovations are made in order to improve the accuracy and accelerate the convergence of the solution procedure. The load-incrementation method and also the arc-length-incrementation method are employed for each incremental step. Numerical analysis is carried out and the results are compared with the available analytical solutions to demonstrate the effectiveness of the proposed method.     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

A simple finite element model for the geometrically nonlinear analysis of thin shells

A triangular flat finite element for the analysis of thin shells which undergo large displacements is proposed. It is based upon the geometrically nonlinear theory of von Kármán for thin plates and the total Lagrangian approach. It has a total of only twelve degrees of freedom, namely, three translations at each vertex and one rotation at each mid-side. The stiffness matrix and the tangent stiffness matrix are derived explicitly. The element is tested against nonlinear patch test solutions and its performance is evaluated by solving several standard problems. The directional derivatives of the potential energy function required for the stability analysis are also provided. Received 10 September 1997     

A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

The present paper concerns a semi-analytical procedure for moderately large deflections of an infinite non-uniform static beam resting on a nonlinear elastic foundation. To construct the procedure, we first derive a nonlinear differential equation of a Bernoulli–Euler–von Kármán “non-uniform” beam on a “nonlinear” elastic foundation, where geometrical nonlinearities due to moderately large deflection and beam non-uniformity are effectively taken into account. The nonlinear differential equation is transformed into an equivalent system of nonlinear integral equations by a canonical representation. Based on the equivalent system, we propose a method for the moderately large deflection analysis as a general approach to an infinite non-uniform beam having a variable flexural rigidity and a variable axial rigidity. The method proposed here is a functional iterative procedure, not only fairly simple but straightforward to apply. Here, a parameter, called a base point of the method, is also newly introduced, which controls its convergence rate. An illustrative example is presented to investigate the validity of the method, which shows that just a few iterations are only demanded for a reasonable solution.     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

BEM formulation for von Kármán plates

This work deals with nonlinear geometric plates in the context of von Kármán's theory. The formulation is written such that only the boundary in-plane displacement and deflection integral equations for boundary collocations are required. At internal points, only out-of-plane rotation, curvature and in-plane internal force representations are used. Thus, only integral representations of these values are derived. The nonlinear system of equations is derived by approximating all densities in the domain integrals as single values, which therefore reduces the computational effort needed to evaluate the domain value influences. Hyper-singular equations are avoided by approximating the domain values using only internal nodes. The solution is obtained using a Newton scheme for which a consistent tangent operator was derived.     

A boundary element formulation for analysis of elastoplastic plates with geometrical nonlinearity

In this paper a new boundary element method formulation for elastoplastic analysis of plates with geometrical nonlinearities is presented. The von Mises criterion with linear isotropic hardening is considered to evaluate the plastic zone. Large deflections are assumed but within the context of small strain. To derive the boundary integral equations the von Kármán’s hypothesis is taken into account. An initial stress field is applied to correct the true stresses according to the adopted criterion. Isoparametric linear elements are used to approximate the boundary unknown values while triangular internal cells with linear shape function are adopted to evaluate the domain value influences. The nonlinear system of equations is solved by using an implicit scheme together with the consistent tangent operator derived along the paper. Numerical examples are presented to demonstrate the accuracy and the validity of the proposed formulation.     

A perturbation solution of von-Kármán circular plates with different moduli in tension and compression under concentrated force

By modifying classical von-Kármán equations, we established bimodular von-Kármán equations of thin plates with different moduli in tension and compression. Adopting central deflection as a perturbation parameter, we used a perturbation method to solve the equations under various boundary conditions, including rigidly clamped, loosely clamped, simply hinged, and simply supported. The relation of load versus central deflection and stress formulas were derived via the perturbation solution obtained. The numerical simulation also shows that the perturbation solution based on central deflection is overall valid. The results indicate that when the compressive modulus of materials is greater than the tensile one, the bearing capacity of the plate will be further strengthened, which should be considered in the analysis and design of plate-like structures with obvious bimodular effect. Moreover, by comparing with the case under uniformly distributed load, the plate-membrane transition under centrally concentrated force presents discontinuity to some extent.     

Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading

The nonlinear dynamic behavior of simply supported tapered sandwich plates subjected to air blast loading is investigated theoretically and numerically. The plate is supposed to have both tapered core and tapered laminated face sheets and be subjected to uniform air blast load. The theory is based on a sandwich plate theory, which includes von Kármán large deformation effects, in-plane stiffnesses, inertias and shear deformations. The sandwich plate theory for plates with constant thickness which have one-layered face sheets found in the literature is developed to analyze the tapered sandwich plates with multi-layered face sheets. The equations of motion are derived by the use of the virtual work principle. Approximate solution functions are assumed for the space domain and substituted into the equations. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The tapered sandwich plate subjected to air blast load is also modelled by using the finite element method. The displacement–time and strain–time histories are obtained. The theoretical results are compared with finite element results and are found to be in an agreement.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.     

Hybrid control of vibrations of a smart von Kármán plate

The suppression of large vibrations of a smart thin elastic rectangular von Kármán’s plate is considered. The plate is subjected to external disturbances and generalized control forces produced by electromechanical feedback. The considered nonlinear initial-boundary value problem is spatially discretized by means of the time spectral method. The implicit Newmark-β iterative method is employed for the time integration of the obtained system of nonlinear equations of motion. Nonlinear controllers are designed, based on a fuzzy inference system. Two numerical algorithms involving a general control of displacement/velocity and a direct control of the Fourier coefficients are proposed. The techniques have been implemented within MATLAB environment with the use of the fuzzy logic toolbox. Numerical examples are presented.     

Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green–Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green–Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain–displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences

Surface stresses can significantly affect the mechanical behavior of structures when they are scaled down to deep submicron dimensions. The Gurtin–Murdoch surface elasticity theory has the capability to capture the size-dependent behavior of nanostructures due to the surface stress effect in a continuum manner. The present work is concerned with the application of Gurtin–Murdoch theory to the nonlinear free vibration analysis of circular cylindrical nanoshells with considering surface stress and shear deformation effects. The nonlinear governing equations of motion together with the corresponding boundary conditions are firstly derived using Hamilton’s principle, the first-order shear deformation shell theory and von Kármán’s assumption. An analytical approach is then presented to solve the nonlinear free vibration problem. Selected numerical results are given to illustrate the effects of surface energy on the nonlinear free vibration behavior of shear deformable nanoshells with different material and geometrical parameters. It is shown that there is a large difference between the results of Gurtin–Murdoch theory and those of its classical counterpart for very thin nanoshells.     

Semi-implicit numerical simulations of geometrically nonlinear beam,plate, and shell dynamical systems

The following article serves three purposes: (i) it presents a simple semi-implicit numerical formulation for nonlinear structural dynamics problems, which is computationally inexpensive and simple to use in nonlinear dynamics and chaos simulations; (ii) it serves as an introduction to numerical studies of nonlinear structural dynamics for engineering students; and (iii) it formulates a nonlinear structural dynamical system for studies of nonlinear dynamics and chaos. Numerical formulations along with results are presented for nonlinear oscillators, beams, Föppl–von Kármán plates, and thin shallow shells.     

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 analysis of axially-loaded functionally graded cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to compressive axial loads and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of axially-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials and under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied.     

Error bounds in nonlinear elastostatics

The boundary value problem of place and traction in nonlinear hyper-elastostatics is considered. As a consequence of convexity of the strain energy function in some neighborhood of a nondegenerate critical point in a quotient space the constitutive equations are invertible. Complementary functionals and a generalized interaction energy lead to estimates for the error energy and error norm without use of the orthogonality conditions. Introduction of extended singular Green states leads formally to pointwise estimates for some field quantities. Numerical results for the von Kármán plate are presented.