Dynamic behaviour of a thin laminated plate embedded with auxetic layers subject to in-plane excitation

The dynamic modelling of a simply-supported thin laminated plate subject to in-plane excitation is established based on the classic shear theory and von Kármán nonlinear theory. The method of multiple scales is used to determine an approximate solution for the system. According to solvability conditions, the nonlinear modulation equations arising from the principal parametric resonances are obtained and two possible nontrivial solutions are performed. To analyze the nonlinear dynamic response of the plate embedded with auxetic layers, 5-layered sandwich plate, in which two auxetic elastic layers are alternatively sandwiched between three positive Poisson’s ratio (PPR) elastic ones, is presented. The natural frequency of model (m, n) shows an increase with respect to the absolute value of Poisson’s ratio. Particularly, the amplitude-frequency responses of the laminated plate subject to principal parametric resonance are analyzed for different values of Poisson’s ratio. Moreover, it can be found that for model (m, n), there must be some certain value or interval of negative Poisson’s ratio (NPR), which, results in zero response effect, in other words, the in-plane excitation will be ineffective for this model when the Poisson’s ratio just lies at such a value or interval. Furthermore, it can also be observed that the certain interval of Poisson’s ratio becomes wider with the increase of damping.     

A theoretical computational model of a plate in hypersonic flow

A theoretical model of an elastic panel in hypersonic flow is derived to be used for design and analysis. The nonlinear von Kármán plate equations are coupled with 1st order Piston Theory and linearized at the nonlinear steady-state deformation due to static pressure differential and thermal loads. Eigenvalue analysis is applied to determine the system’s stability, natural frequencies and mode shapes. Numerically time marching the equations provides transient response prediction which can be used to estimate limit cycle oscillation amplitude, frequency and time to onset. The model’s predictive capability is assessed by comparison to an experiment conducted at a free stream flow of Mach 6. Good agreement is shown between the theoretical and experimental natural frequencies and mode shapes of the fluid–structure system. Stability analysis is performed using linear and nonlinear methods to plot stability, flutter and buckling zones on a free stream static pressure vs temperature differential plane.     

On the Role of In-Plane Compliance in Edge Wrinkling

Bifurcations of a thin circular elastic plate subjected to uniform normal pressure are investigated by taking into account the in-plane compliance of the edge restraint. This effect amounts to introducing a Hookean spring relating the radial components of the membrane stress tensor and the corresponding in-plane displacement fields. The addition of this new feature gives rise to an adaptive radial stretching of our configuration, which is intimately linked to the strength of the applied pressure. The Föppl-von Kármán nonlinear plate theory, in conjunction with singular perturbation arguments, help us to establish the nature of the localised wrinkling observed in numerical simulations. Asymptotic analysis of the problem provides some simple qualitative predictions for the dependence of the critical load on a number of key dimensionless parameters.     

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.     

Improvement of the acoustic black hole effect by using energy transfer due to geometric nonlinearity

Acoustic Black Hole effect (ABH) is a passive vibration damping technique without added mass based on flexural waves properties in thin structures with variable thickness. A common implementation is a plate edge where the thickness is locally reduced with a power law profile and covered with a viscoelastic layer. The plate displacement in the small thickness region is large and easily exceeds the plate thickness. This is the origin of geometric nonlinearity which can generate couplings between linear eigenmodes of the structure and induce energy transfer between low and high frequency regimes. This phenomenon may be used to increase the efficiency of the ABH treatment in the low frequency regime where it is usually inefficient. An experimental investigation evidenced that usual ABH implementation gives rise to measurable geometric nonlinearity and typical nonlinear phenomena. In particular, strongly nonlinear regime and wave turbulence are reported. The nonlinear ABH beam is then modeled as a von Kármán plate with variable thickness. The model is solved numerically by using a modal method combined with an energy-conserving time integration scheme. The effects of both the thickness profile and the damping layer are then investigated in order to improve the damping properties of an ABH beam. It is found that a compromise between the two effects can lead to an important gain of efficiency in the low frequency range.     

NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES

Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.     

Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory

The present study proposes a nonclassical Kirchhoff plate model for the axisymmetrically nonlinear bending analysis of circular microplates under uniformly distributed transverse loads. The governing differential equations are derived from the principle of minimum total potential energy based on the modified couple stress theory and von Kármán geometrically nonlinear theory in terms of the deflection and radial membrane force, with only one material length scale parameter to capture the size-dependent behavior. The governing equations are firstly discretized to a set of nonlinear algebraic equations by the orthogonal collocation point method, and then solved numerically by the Newton–Raphson iteration method to obtain the size-dependent solutions for deflections and radial membrane forces. The influences of length scale parameter on the bending behaviors of microplates are discussed in detail for immovable clamped and simply supported edge conditions. The numerical results indicate that the microplates modeled by the modified couple stress theory causes more stiffness than modeled by the classical continuum plate theory, such that for plates with small thickness to material length scale ratio, the difference between the results of these two theories is significantly large, but it becomes decreasing or even diminishing with increasing thickness to length scale ratio.     

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.     

Dynamic buckling of thin-walled composite plates with varying widthwise material properties

The paper deals with dynamic response of a thin-walled rectangular plate subjected to in-plane pulse loading. The plate is made of orthotropic (fibre composite) material in which the principal directions of orthotropy are parallel to the plate edges. The plate is characterised by a widthwise varying fibre volume fraction. The structures are assumed to be simply supported at the loaded ends and at non-loaded ends with five different boundary conditions (both simply supported, both fixed, simply supported fixed, simply supported free edge, fixed free edge). In order to obtain the equations of motion the non-linear theory of orthotropic thin-walled plates has been modified in such a way that it additionally accounts for all components of inertial forces. The differential equations of motion have been obtained from Hamilton’s Principle. The problem of nonlinear static stability was solved with the second order of the Koiter’s asymptotic stability theory of conservative systems. The results obtained from analytical–numerical method were compared with the results from finite element method (FEM).     

ON THE REFINED FIRST-ORDER SHEAR DEFORMATION PLATE THEORY OF KARMAN TYPE

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｕｓｅｏｆｌａｍｉｎａｔｅｄｃｏｍｐｏｓｉｔｅｓｉｎｔｈｉｎ＿ｗａｌｌｅｄｓｔｒｕｃｔｕｒｅｓｉｎｃｒｅａｓｅｓｓｏｔｈａｔｅｆｆｅｃｔｓｏｆｔｒａｎｓｖｅｒｓｅｓｈｅａｒｄｅｆｏｒｍａｔｉｏｎｓｃａｎｎｏｔｂｅｎｅｇｌｅｃｔｅｄａｎｄｉｎｖｏｋｅｑｕｉｔｅｃｏｍｐｌｅｘｅｓｉｎｎｏｎｌｉｎｅａｒａｎａｌｙｓｉｓ.Ｉｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｎｏｎｌｉｎｅａｒａｎａｌｙｓｉｓｏｆｌａｍｉｎａｔｅｄｐｌａｔｅｓａｎｄｓｈｅｌｌｓｃｏｕｎｔｉｎｇｆｏｒｔｒ…     

Nonlinear vibration analysis of functionally graded beams considering the influences of the rotary inertia of the cross section and neutral surface position

In the paper work, the nonlinear vibration response of functionally graded (FG) Euler–Bernoulli beam resting on elastic foundation is studied. Based on von Kármán’s geometric nonlinearity, the partial differential governing equations describing the nonlinear vibration of FG Euler–Bernoulli beam are derived from Hamilton’s principle and are reduced to an ordinary nonlinear differential equation with quadratic and cubic nonlinear terms via Galerkin’s procedure. Due to unsymmetrical material variation along the thickness of FG beam, the neutral surface concept is proposed to remove the stretching and bending coupling effect and the rotary inertia of the cross section is incorporated to obtain an analytical solution. Numerical results are presented to show the effects of the nonlocal parameters and vibration amplitude on the frequency responses. This results may be useful in design and engineering applications.     

Analysis on nonlinear dynamics of a deploying composite laminated cantilever plate

This paper presents the analysis on the nonlinear dynamics of a deploying orthotropic composite laminated cantilever rectangular plate subjected to the aerodynamic pressures and the in-plane harmonic excitation. The third-order nonlinear piston theory is employed to model the transverse air pressures. Based on Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the nonlinear governing equations of motion are derived for the deploying composite laminated cantilever rectangular plate. The Galerkin method is utilized to discretize the partial differential governing equations to a two-degree-of-freedom nonlinear system. The two-degree-of-freedom nonlinear system is numerically studied to analyze the stability and nonlinear vibrations of the deploying composite laminated cantilever rectangular plate with the change of the realistic parameters. The influences of different parameters on the stability of the deploying composite laminated cantilever rectangular plate are analyzed. The numerical results show that the deploying velocity and damping coefficient have great effects on the amplitudes of the nonlinear vibrations, which may lead to the jumping phenomenon of the amplitudes for first-order and second-order modes. The increase of the damping coefficient can suppress the increase of the amplitudes of the nonlinear vibration.     

On the refined first-order shear deformation plate theory of Kármán type

A new refined first-order shear-deformation plate theory of the Kármán type is presented for engineering applications and a new version of the generalized Kármán large deflection equations with deflection and stress functions as two unknown variables is formulated for nonlinear analysis of shear-deformable plates of composite material and construction, based on the Mindlin/Reissner theory. In this refined plate theory two rotations that are constrained out in the formulation are imposed upon overall displacements of the plates in an implicit role. Linear and nonlinear investigations may be made by the engineering theory to a class of shear-deformation plates such as moderately thick composite plates, orthotropic sandwich plates, densely stiffened plates, and laminated shear-deformable plates. Reduced forms of the generalized Kármán equations are derived consequently, which are found identical to those existe in the literature. Foundation item: the National Natural Science Foundation of China (59675027) Biography: Zhang Jianwu (1954-)     

宽翼缘梁非线性温差自应力级数解

为了客观反映宽翼缘梁在非线性温差作用下的自应力分布特点,克服按梁理论计算温度自应力时的不足,本文从弹性力学的分析方法入手,按平面应力状态分析翼缘板和腹板的温度应力。通过翼缘板与腹板连接处的变形协调条件及平衡条件建立补充方程,求解艾瑞应力函数中的积分常数,推导了翼缘板与腹板的纵、横向正应力、剪应力及位移分量解析式。导出的翼缘板纵向正应力公式可自动考虑宽翼缘梁沿横截面宽度自应力不均匀分布的特点。对一宽翼缘T梁的计算表明,当翼缘板相对于腹板发生温差变化时,沿其宽度的纵向温度自应力分布很不均匀,在翼缘板根部自应力较大而在悬臂端则显著减小。按通常基于梁理论的温度自应力计算方法,无法反映这种应力分布规律。     

ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

The effect of initial geometric imperfection on the nonlinear resonance of functionally graded carbon nanotube-reinforced composite rectangular plates

The purpose of the present study is to examine the impact of initial geometric imperfection on the nonlinear dynamical characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular plates under a harmonic excitation transverse load. The considered plate is assumed to be made of matrix and single-walled carbon nanotubes (SWCNTs). The rule of mixture is employed to calculate the effective material properties of the plate. Within the framework of the parabolic shear deformation plate theory with taking the influence of transverse shear deformation and rotary inertia into account, Hamilton’s principle is utilized to derive the geometrically nonlinear mathematical formulation including the governing equations and corresponding boundary conditions of initially imperfect FG-CNTRC plates. Afterwards, with the aid of an efficient multistep numerical solution methodology, the frequency-amplitude and forcing-amplitude curves of initially imperfect FG-CNTRC rectangular plates with various edge conditions are provided, demonstrating the influence of initial imperfection, geometrical parameters, and edge conditions. It is displayed that an increase in the initial geometric imperfection intensifies the softening-type behavior of system, while no softening behavior can be found in the frequency-amplitude curve of a perfect plate.     

The nonlinear vibrations of functionally graded cylindrical shells surrounded by an elastic foundation

This paper reports the result of an investigation on the nonlinear vibrations of functionally graded cylindrical shell surrounded by an elastic foundation, based on Hamilton’s principle, von Kármán nonlinear theory, and the first-order shear deformation theory. Material properties are assumed to be temperature dependent. The surrounding elastic medium is modeled as Winkler foundation model, Pasternak foundation model, and nonlinear foundation model. Galerkin’s method is utilized to convert the governing partial differential equations to nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Considering the primary resonance case, the method of multiple scales is used to study the frequency response of nonlinear vibrations and the softening/hardening behavior. Parametric effects on the nonlinear vibrations are investigated.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

Bending of a highly stretched plate containing an eccentrically plate-reinforced circular hole

In this paper, we consider the problem of finding the stress distribution in a highly stretched plate containing a circular hole that is eccentrically reinforced by thickening the plate, on one side only, in an annular region concentric with the hole. A solution of the nonlinear Kármán plate equations is obtained that is asymptotically valid for large membrane stresses. We show that, except for a narrow bending boundary layer in the neighbourhood of the boundary between the reinforced area and the rest of the plate, a state of plane stress prevails and the reinforced area undergoes a transverse deflection that brings its middle surface into the plane of the middle surface of the plate.     

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads in thermal environments

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads is investigated. A new micro-macro-mechanical model of unit cells is suggested. In this model, a 3D braided composite may be considered as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate. The material properties of the epoxy are expressed as a linear function of temperature. Based on Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, analytical solutions for nonlinear bending behavior of simply supported 3D braided rectangular plates are obtained using mixed Galerkin-perturbation method. The numerical examples concern effects of geometric parameters, of fiber volume fraction, braiding angle and load boundary condition.