Von Karman Nonlinearity Research Articles

An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams

Abstract In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.

Analysis of post-buckling and delamination in laminated composite St. Venant–Kirchhoff beams using CZM and layer-wise TSNDT

A layer-wise third order shear and normal deformable plate/shell theory (TSNDT) incorporating a cohesive zone model (CZM) is used to study the initiation and growth of delamination in straight and curved laminated beams. Upon satisfaction of the delamination criteria at a point on the interface between two layers, displacements there of two abutting points on the interface between the two layers are made discontinuous. Delaminations under mode-I, mode-II and mixed-mode static and transient loadings have been studied. All geometric nonlinearities, including the von Karman nonlinearity, are considered. The material of each layer is assumed to be St. Venant–Kirchhoff for which the second Piola–Kirchhoff stress tensor is a linear function of the Green-St. Venant strain tensor. Example problems studied also include delamination growth during axial buckling of a three-layer beam. It is found that the consideration of inertia forces noticeably delays the buckling load and significantly affects the deformed shape of an axially compressed laminated beam.

Post-buckling behavior of composite and sandwich skew plates

Present analysis deals with the buckling and Post-buckling behavior of laminated composite and sandwich skew plates. Higher order shear deformation theory and von-Karman's nonlinearity is considered in the problem formulation. The physical domain is mapped into computational domain using linear mapping. Finite degree double Chebyshev series method is employed for spatial discretizations of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. Post-buckling response of the skew plates is investigated for different types of in-plane compressive loadings. Effects of skew angle, lamination scheme, core thickness and face to core material property ratio of the sandwich plate on the buckling and Post-buckling response are studied in detail.

Thermomechanical Elastic Post-Buckling of Functionally Graded Materials Plate with Random System Properties

This paper presents the stochastic post-buckling response of elastically supported FGM plate with random system properties subjected to uniform and nonuniform temperature change with temperature-dependent and -independent material properties. The FGMs plate is supported with two parameters of Pasternak foundation with Winkler cubic nonlinearity. The basic formulation is based on higher-order shear deformation theory (HSDT) with von-Karman nonlinearity using modified C0 continuity. A direct iterative-based nonlinear finite element method combined with first-order perturbation technique is used to compute the second-order statistics (mean and coefficient of variation) of post-buckling response of FGM plates.

Effect of Variable Fiber Spacing on Post-Buckling of Boron/Epoxy Fiber Reinforced Laminated Composite Plate

The primary objective of research is to determine the improved structural efficiency of plates can be with variable fiber spacing. The Post-buckling of laminated composite plate with variable fiber spacing is obtained numerically, using eight node isoparametric quadrilateral elements (serendipity element) with five degree of freedom per node. The mathematical formulation is based on first-order shear deformation theory and von-Karman non-linearity. The effect of the initial geometric imperfection, fiber spacing, orientation fiber, and direction of in-plane loading were considered. Numerical results for boron/Epoxy fiber reinforced laminates are presented for the different effects of the composite plate under in-plane loading. This study showed that the post buckling behavior of composite plate very sensitive for type of distribution fiber and the seventh distribution equation gives maximum buckling load and smallest deformation.

Stochastic Nonlinear Free Vibration Analysis of Piezolaminated Composite Conical Shell Panel Subjected to Thermoelectromechanical Loading With Random Material Properties

This paper presents the effect of randomness in material properties on piezolaminated composite geometrically nonlinear conical shell panel subjected to thermoelectromechanical loading acting simultaneously or individually. Material properties such as modulus ratio, Poisson’s ratio, and thermal expansion coefficients are modeled as independent random variables. The temperature field considered is assumed to be a uniform distribution over the shell panel surface and through the shell thickness and the electric field is assumed to be the transverse component Ez only. It is assumed that the mechanical properties do not depend on temperature and electric fields. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinearity. A C0 nonlinear finite element model based on direct iterative approach in conjunction with mean centered first order perturbation technique (FOPT) used by the present author for plate is now extended for conical shell panel to solve a random nonlinear generalized eigenvalue problem. Parametric studies are carried out to examine the effect of amplitude ratios, stacking sequences, cone angles, circumferential length to thickness ratios, piezoelectric layers, applied voltages, change in temperature, types of thermoelectromechanical loadings, and support boundary conditions on the dimensionless mean and coefficient of variance (COV) of laminated conical shell panels. The present outlined approach has been validated with those available results in literature and independent Monte Carlo simulation (MCS).

Nonlinear free vibration analysis of rotating composite Timoshenko beams

Linear and nonlinear free vibrations of rotating composite Timoshenko beams are investigated in this paper. The formulation is based on the assumptions of Timoshenko beam theory and the nonlinear von-Karman strain–displacement relationships. The updated equations of motion about the prestressed configuration are obtained from the three-coupled partial differential equations of motion. The differential transform method is implemented to find the prestressed configuration due to centrifugal forces. The Galerkin discretization approach is applied to the linearized updated equations of motion to determine the linear normal modes and the associated natural frequencies of the rotating composite Timoshenko beam. The rotation speed and number of layers variation effects on the flapping natural frequencies are studied and the necessity of employing the Timoshenko beam model for the composite beams is illustrated. The direct multiple scales method is applied to the three-coupled nonlinear updated partial differential equations for the nonlinear vibration study especially on the flapping backbone curves. The number of layers variation effects is investigated on the three lowest flapping backbone curves. The present results are validated with the previous literature results.

Convergence of solutions of von Karman evolution equations to equilibria

We consider von Karman evolution equations with nonlinear interior dissipation and with clamped boundary conditions. Under some conditions we prove that every energy solution converges to a stationary solution and establish a rate of convergence. Earlier this result was known in the case when the set of equilibria was finite and hyperbolic. In our argument we use the fact that the von Karman nonlinearity is analytic on an appropriate space and apply the Lojasiewicz–Simon method in the form suggested by A. Haraux and M. Jendoubi.

STOCHASTIC THERMAL POST-BUCKLING RESPONSE OF LAMINATED COMPOSITE CYLINDRICAL SHELL PANEL WITH SYSTEM RANDOMNESS

The effect of random system properties on thermal post-buckling temperature of laminated composite cylindrical shell panel with temperature independent (TID) and dependent (TD) material properties subjected to uniform temperature distribution is examined in this study. System properties such as material properties, thermal expansion coefficients and lamina plate thickness are modeled as independent basic random variables. The basic formulation is based on higher-order shear deformation (HSDT) theory with von-Karman nonlinearity using modified C0 continuty. A direct iterative-based C0 nonlinear finite element method (FEM) combined with Taylor series-based mean-centered first-order perturbation technique (FOPT) developed by the authors for composite plate is extended for shell panel with reasonable accuracy to compute second-order statistics of post-buckling temperature of cylindrical shell panel. Typical numerical results for second order statistics (mean and coefficient of variance) of thermal post-buckling temperature of laminated cylindrical shell panel are obtained through numerical examples for various support conditions, amplitude ratios, shell thickness ratios, aspect ratios, lamination lay-up sequences, curvature to length ratios, types of material properties with the effect of random system parameters. The performance of outlined approach has been validated with those results available in the literatures and independent MCS.

Electro-dynamic behavior of an electrically actuated micro-beam: Effects of initial curvature and nonlinear deformation

This paper presents a curved beam model for the nonlinear electro-dynamic analysis of micro-beams with initial curvature, taking into account the nonlinear electric force and nonlinear deformation. The governing equations of motion and boundary conditions are derived in an arc coordinate system. Unlike previous studies based on von Karman nonlinearity or considering mid-plane stretching only, the present study does not involve any assumptions on nonlinear deformation. Differential quadrature method and Petzold–Gear BDF method are employed to obtain the nonlinear electro-dynamic behavior of curved micro-beams. The effects of nonlinear deformation, initial gap and initial rise on the nonlinear electro-dynamic characteristics are studied.

Nonlinear Thermomechanical Post-Buckling Analysis of Thin Functionally Graded Annular Plates Based on Von-Karman's Plate Theory

In this article, a nonlinear bending and axisymmetric thermal buckling analysis of thin functionally graded (FG) annular plates is presented based on classical nonlinear Von-Karman's plate theory. Using a shooting method the nonlinear ordinary differential equations are solved numerically. The presented solution is validated by comparing the obtained results with those of an isotropic plate. The emphasis is placed on investigating the effect of varying the gradient index of the FG plate and thermal boundary conditions on the nonlinear bending and post buckling behavior of the FG plate in detail.

Non-linear analysis of functionally graded circular plates under asymmetric transverse loading

Based on the first-order shear deformation plate theory with von Karman non-linearity, the non-linear axisymmetric and asymmetric behavior of functionally graded circular plates under transverse mechanical loading are investigated. Introducing a stress function and a potential function, the governing equations are uncoupled to form equations describing the interior and edge-zone problems of FG plates. This uncoupling is then used to conveniently present an analytical solution for the non-linear asymmetric deformation of an FG circular plate. A perturbation technique, in conjunction with Fourier series method to model the problem asymmetries, is used to obtain the solution for various clamped and simply supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified by comparison with the existing results in the literature. The effects of non-linearity, material properties, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. It is found that linear analysis is inadequate for analysis of simply supported FG plates which are immovable in radial direction even in the small deflection range. Furthermore, the responses of FG materials under a positive load and a negative load of identical magnitude are not the same. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effect in clamped FG plates being stronger than that in simply supported plates.

Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates.

We study the influence of von Karman nonlinearity, van der Waals force, and a athermal stresses on pull-in instability and small vibrations of electrostatically actuated mi-croplates. We use the Galerkin method to develop a tractable reduced-order model for elec-trostatically actuated clamped rectangular microplates in the presence of van der Waals forcesand thermal stresses. More specifically, we reduce the governing two-dimensional nonlineartransient boundary-value problem to a single nonlinear ordinary differential equation. For thestatic problem, the pull-in voltage and the pull-in displacement are determined by solving apair of nonlinear algebraic equations. The fundamental vibration frequency corresponding toa deflected configuration of the microplate is determined by solving a linear algebraic equa-tion. The proposed reduced-order model allows for accurately estimating the combined effectsof van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflectionprofile with an extremely limited computational effort.

POSTBUCKLING RESPONSE OF FUNCTIONALLY GRADED RECTANGULAR PLATES SUBJECTED TO THERMO-MECHANICAL LOADING

The present work deals with the stability analysis of shear deformable functionally graded rectangular plates subjected to thermo-mechanical loads. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a simple power law distribution of the volume fraction of the constituents. An analytical approach based on fast converging Chebyshev polynomials is presented. The formulation is based on first-order shear deformation plate theory and von-Karman nonlinearity. The temperature-dependent thermal and mechanical properties of FGM plate are considered. The variations in the buckling load/temperature due to various parameters are studied. The postbuckling response of the clamped FGM plate, subjected to in-plane edge compressive mechanical loading (uniaxial, biaxial, shear), thermal (uniform in-plane temperature, transverse temperature gradient) and thermo-mechanical loading are presented. It is observed that volume fraction index greatly affects the buckling load/temperature of the plate.

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature of composite laminates

Design and control optimization is presented to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The total elastic energy of the laminates is taken as a measure of the dynamic response. The optimization control problem is solved under constraints on the laminate thickness and the control energy produced by a transverse dynamic load distributed over the upper surface of the laminate. The constrained control objective is expressed as the sum of the total elastic energy and penalty term involving the control force, which may be considered as a measure of the control energy. The thickness of layers and the fibers orientation angles are taken as optimization design variables. The design and control objectives are formulated based on shear deformation theory accounting for the von-Karman nonlinearity. The displacements are chosen as the sum of time-independent displacements due to the static thermal load and time-dependent displacements due to the initial disturbances and the applied control force. Liapunov–Bellman theory is used to obtain the optimal control force, buckled deflections and controlled elastic energy. Numerical examples are presented for angle-ply antisymmetric laminates with simply supported edges. Graphical studies are carried out to show the advantages of the present design and control procedures.

Post-buckling of cross-ply laminated rectangular plates containing short random fibers

Post-buckling of cross-ply laminated composite plate containing randomly oriented short spatial fibers in each layer is obtained analytically, using fast converging double Chebyshev series. The mathematical formulation is based on first-order shear deformation theory and von-Karman non-linearity. The effective elastic properties of the composites are expressed analytically in terms of phase properties, orientation angles, volume fraction, and fiber shape. The effects of fiber orientation in the composites, fiber volume fraction, fiber aspect ratio, and plate span to thickness ratio on the buckling and post-buckling strength are studied. Numerical results for E-glass/Epoxy fiber reinforced laminates are presented for the different boundary conditions and the number of layers of the composite. The results indicate that complete random distribution of the fibers in the composites gives higher buckling and post-buckling strength.

Efficient Higher-Order Shell Theory for Laminated Composites with Multiple Delaminations

A higher-order zigzag theory has been developed for laminated composite shells with multiple delaminations. General tensor-based formulation is developed for arbitrary curved shell with exact geometric description. A laminated shell theory with multiple delaminations for general lamination configurations is obtained by superposing a cubic varying displacement on a zigzag linearly varying displacement. The von Karman nonlinearity is included in the formulation for the potentialities of addressing problems requiring geometric nonlinearity such as large deflection and postbuckling problems. When top and bottom surface transverse shear stress free conditions and interface transverse shear continuity conditions including delamination interfaces are imposed the displacement of the minimal degrees of freedom is obtained. The proposed displacement field can systematically handle the number, shape, size, and locations of the delaminations. Through the variational principle, equilibrium equations and variationally consistent boundary conditions are obtained. To assess the accuracy and efficiency of the present theory, the linear buckling problem of cylindrical shell with multiple delaminations has been analyzed. The higher-order zigzag theory should work as an efficient tool for analyzing the behavior of composite laminated shells with multiple delaminations.

NON-LINEAR TRANSIENT ANALYSIS OF MODERATELY THICK LAMINATED COMPOSITE PLATES

The non-linear transient analysis of the shear deformable laminated composite plates, subjected to step, ramp and sinusoidal loading is presented. The clamped, simply supported, free and their combinations (non-Levy type) of boundary conditions are considered. The formulation is based on the Mindlin first order shear deformation theory and von-Karman non-linearity. The methodology of the solution is based on the Chebyshev series technique. Houbolt time marching scheme and quadratic extrapolation technique are used for the temporal discretization and linearization respectively. The effects of magnitude and duration of loading, rotary inertia, in-plane, inertia, b/a, a/h, lamination scheme and boundary conditions on the central displacement response are studied. Typical results are presented in dimensionless graphical forms for different parameters and loading conditions.

Non-linear bending analysis of composite laminated plates using a refined first-order theory

A refined single-layer model of geometrically non-linear composite laminates is presented using a mixed variational approach. The present model accounts for Reissner–Mindlin’s assumptions on displacement and continuous stress distributions at the layer interfaces. Therefore, the present first-order plate theory recovers the actual interlaminar stress state without losing its simplicity and leads to a consistency with the theory of elasticity. Furthermore, the stresses are consistent with the surface conditions. The rationale for the shear correction factor used in other first-order theories is obviated. The governing equations including the von Karman non-linearity are deduced with the required boundary conditions. A wide variety of linear and non-linear results for cross-ply symmetric and antisymmetric laminates are presented. A bending analysis is made to illustrate the influence of the geometric non-linear effect on the transverse deflections and the stresses. Some of the present linear results are compared with their counterparts in the literature. The present results are in good agreement with the results of others.

Thermal Postbuckling of Imperfect Shear-Deformable Laminated Plates on Two-Parameter Elastic Foundations

Thermal postbuckling analysis is presented for a simply supported, shear-deformable composite laminated plate subjected to uniform or nonuniform parabolic temperature loading and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Reddy's third-order shear-deformation plate theory with von Karman nonlinearity is used. The governing equations also include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, plate aspect ratio, fiber orientation, thermal load ratio, and initial geometric imperfections, is studied. Typical results are presented in dimensionless graphical form.