Unified 2D continuous and reduced order modeling of thermomechanically coupled laminated plate for nonlinear vibrations

A unified formulation of the thermomechanical problem of laminated plates with von Karman nonlinearities, undergoing finite amplitude vibrations, is presented. It integrates mechanical and thermal aspects, by addressing them in parallel via the introduction of generalized 2D variables and equations also for the latter. The formulation virtually embeds a multitude of possible models, resulting from different assumptions about the plate mechanical and thermal configurations. The obtained continuous model is then subjected to a minimum reduction via Galerkin procedure. Some analyses of free and forced nonlinear vibrations under variable mechanical and/or thermal excitations are also carried out, to get some hints on the importance of different thermal aspects associated with membrane and bending dynamics, and on the possibility to catch them via variably simplified models.     

Micromechanical analysis of the fully coupled finite thermoelastic response of rubber-like matrix composites

A micromechanical analysis for the prediction of the coupled thermoelastic response of multiphase composites that include rubber-like phases is presented. Rubber-like solids are highly nonlinear thermoelastic materials that exhibit anomalous behavior referred to as the thermoelastic inversion effect. Results are presented which show that the derived micromechanical model is capable of predicting this effect in nylon/rubber composites subjected to appropriate thermal loadings assuming one-way coupling. For full thermomechanical coupling, the nonlinear response and induced temperatures under several types of mechanical loading are investigated.     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

Sensitivity analysis of the thermomechanical response of welded joints

A computational procedure is presented for evaluating the sensitivity coefficients of the thermomechanical response of welded structures. Uncoupled thermomechanical analysis, with transient thermal analysis and quasi-static mechanical analysis, is performed. A rate independent, small deformation thermo-elasto-plastic material model with temperature-dependent material properties is adopted in the study. The temperature field is assumed to be independent of the stresses and strains. The heat transfer equations emanating from a finite element semi-discretization are integrated using an implicit backward difference scheme to generate the time history of the temperatures. The mechanical response during welding is then calculated by solving a generalized plane strain problem. First- and second-order sensitivity coefficients of the thermal and mechanical response quantities (derivatives with respect to various thermomechanical parameters) are evaluated using a direct differentiation approach in conjunction with an automatic differentiation software facility. Numerical results are presented for a double fillet conventional welding of a stiffener and a base plate made of stainless steel AL-6XN material. Time histories of the response and sensitivity coefficients, and their spatial distributions at selected times are presented.     

A decoupled modal analysis for nonlinear dynamics of an orthotropic thin laminate in a simply supported boundary condition subject to thermal mechanical loading

In this paper, we analyze the nonlinear dynamic response of an orthotropic laminate in a simply supported boundary condition subject to thermal and mechanical loading. The equation of motion for the laminate’s deflection is obtained in a decoupled Duffing equation by means of a Galerkin-type method without Berger’s approximations. The Duffing equation incorporates an arbitrary thermal field, with both the in-plane and transverse temperature variations in a steady-state and a transient state. The formulation indicates that the transverse temperature variation produces an additional pressure load, while the in-plane temperature variation affects the system frequency. The equation allows for characterization of the laminate behaviors in nonlinear thermal buckling, thermal vibration and thermal mechanical response.     

Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate

Global bifurcations and multi-pulse chaotic dynamics are studied for a four-edge simply supported composite laminated piezoelectric rectangular plate under combined in-plane, transverse, and dynamic electrical excitations. Based on the von Karman type equations for the geometric nonlinearity and Reddy’s third-order shear deformation theory, the governing equations of motion for a composite laminated piezoelectric rectangular plate are derived. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional non-autonomous nonlinear system is simplified to a three-order standard form by using the method of normal form. The extended Melnikov method is improved to investigate the six-dimensional non-autonomous nonlinear dynamical system in mixed coordinate. The global bifurcations and multi-pulse chaotic dynamics of the composite laminated piezoelectric rectangular plate are studied by using the improved extended Melnikov method. The multi-pulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.     

Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow

Turbo-machineries, as key components, have wide applications in civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Kármán nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton's principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.     

Large deflection of a rectangular magnetoelectroelastic thin plate

Based on the von Karman plate theory of large deflection, we derive the nonlinear partial differential equation for a rectangular magnetoelectroelastic thin plate under the action of a transverse static mechanical load. By employing the Bubnov-Galerkin method, the nonlinear partial differential equation is transformed to a third-order nonlinear algebraic equation for the maximum deflection where a coupling factor is introduced for determining the coupling effect on the deflection. Numerical results are carried out for the thin plate made of piezoelectric BaTiO₃ and piezomagnetic CoFe₂O₄ materials. Some interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.     

The effect of anisotropic damage evolution on the behavior of ductile and brittle matrix composites

Anisotropic damage evolution laws for ductile and brittle materials have been coupled to a micromechanical model for the prediction of the behavior of composite materials. As a result, it is possible to investigate the effect of anisotropic progressive damage on the macroscopic (global) response and the local spatial field distributions of ductile and brittle matrix composites. Two types of thermoinelastic micromechanics analyses have been employed. In the first one, a one-way thermomechanical coupling in the constituents is considered according to which the thermal field affects the mechanical deformations. In the second one, a full thermomechanical coupling exists such that there is a mutual interaction between the mechanical and thermal fields via the energy equations of the constituents. Results are presented that illustrate the effect of anisotropic progressive damage in the ductile and brittle matrix phases on the composite’s behavior by comparisons with the corresponding isotropic damage law and/or by tracking the components of the damage tensor.     

含高温度梯度及接触热阻非线性热力耦合问题的谱元法

建立了含高温度梯度及接触热阻的非线性热力耦合问题的谱元法格式, 考虑了温度相关的热导率、弹性模量、泊松比和热膨胀系数, 以及界面应力相关的接触热阻的影响. 谱元法的插值函数基于非等距分布的Lobatto结点集或第二类Chebyshev结点集, 兼具谱方法的高精度和有限元法的灵活性. 数值算例表明, 建立的谱元法计算格式可以高效高精度地求解域内高温度梯度以及含接触热阻的非线性热力耦合问题, 不仅收敛速度快于传统有限元法, 而且用较少的自由度和较短的计算时间即可得到比传统有限元法更高精度的计算结果, 在工程实际热力耦合问题中具有广阔的应用前景.      

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.     

Nonlinear stability and vibration of pre/post-buckled microstructure-dependent FGPM actuators

The present study deals with the study of the nonlinear stability and small free vibration of microstructure-dependent functionally graded piezoelectric material (FGPM) beams in pre/post-buckling regimes. The Timoshenko beam theory with various inplane and out-of-plane boundary conditions are considered under different types of mechanical and thermal loads. The beam is assumed to be under inplane mechanical, thermal, and electrical excitations. Each thermo-electro-mechanical property of the beam is graded across the thickness (i.e., height) of the beam, based on a power law model. The von Kármán type geometric nonlinearity is included to account for the large deflection behavior of the beam under inplane loads. The modified couple stress theory is included to account for the size effects. A weak-form, displacement-based, finite element formulation is developed to discretize the equations of motion. The resulting system of nonlinear algebraic equations is solved using Newton’s iterative method. The numerical results of frequencies and lateral deflections as a function of load parameters reveal the existence of bifurcation or critical states in some cases. The effects of load type, microstructural dependency, boundary conditions, beam geometry, composition rule of the constituents, and actuator voltage are discussed through the various parametric studies.     

Dynamic response of a deepwater riser subjected to combined axial and transverse excitation by the nonlinear coupled model

In offshore engineering long slender risers are simultaneously subjected to both axial and transverse excitations. The axial load is the fluctuating top tension which is induced by the floater’s heave motion, while the transverse excitation comes from environmental loads such as waves. As the time-varying axial load may trigger classical parametric resonance, dynamic analysis of a deepwater riser with combined axial and transverse excitations becomes more complex. In this study, to fully capture the coupling effect between the planar axial and transverse vibrations, the nonlinear coupled equations of a riser’s dynamic motion are formulated and then solved by the central difference method in the time domain. For comparison, numerical simulations are carried out for both linear and nonlinear models. The results show that the transverse displacements predicted by both models are similar to each other when only the random transverse excitation is applied. However, when the combined axial dynamic tension and transverse wave forces are both considered, the linear model underestimates the response because it ignores the coupling effect. Thus the coupled model is more appropriate for deep water. It is also found that the axial excitation can significantly increase the riser’s transverse response and hence the bending stress, especially for cases when the time-varying tension is located at the classical parametric resonance region. Such time-varying effects should be taken into account in fatigue safety assessment.     

抛物面型激光推力器的热力冲击响应

通过实验、机理分析和数值模拟系统地分析了大气模式激光推进中抛物面型推力器的热力冲击问题。在分析激光推进中存在的4种热载荷（入射、辐射、透射和运流）的基础上,建立了相应的热力耦合动态计算方法。多脉冲推进的计算温升与实验结果吻合。计算表明,入射吸收和高温辐射是造成抛物面型激光推力器温升的主要原因,并预测推力器在熔化前首先发生拉伸破坏,揭示了激光推进中热力冲击破坏的机理和严重性。     

Nonlinear characterization of concurrent energy harvesting from galloping and base excitations

An energy harvester is proposed to concurrently harness energy from base and galloping excitations. This harvester consists of a triangular cross-sectional tip mass attached to a multilayered piezoelectric cantilever beam and placed in an incompressible flow and subjected to a harmonic base excitation in the cross-flow direction. A coupled nonlinear-distributed-parameter model is developed representing the dynamics of the transverse degree of freedom and the generated voltage. The galloping force and moment are modeled by using a nonlinear quasi-steady approximation. Under combined loadings and when the excitation frequency is away from the global natural frequency of the harvester, the response of the harvester mainly contains these two harmonic frequencies. Thus, the harvester’s response is generally aperiodic and is either periodic with large period (i.e., period- \(n\) ), or quasi-periodic, or chaotic. To characterize the harvester’s response under a combination of vibratory base excitations and aerodynamic loading, we use modern methods of nonlinear dynamics, such as phase portraits, power spectra, and Poincaré sections. A further analysis is then performed to determine the effects of the wind speed, frequency excitation, base acceleration, and electrical load resistance on the performance of the harvester under separate loadings.     

Hygroelasticity analysis of an elastically restrained functionally graded porous metamaterial circular plate resting on an auxetic material circular plate

The main objective of this research is to investigate the hygroelastic behavior of a non-homogeneous circular plate made up of porous metamaterial resting on an auxetic material plate. The mechanical properties of the main plate, as well as moisture concentration, vary as an exponential function in the transverse direction. Poisson's ratio is constant. The elastic supporting medium is developed by considering the structurestructure coupling. Based on the linear hygroelasticity theory, the governing state equations in terms of displacements and moisture concentration are acquired. At first, the Fickian equation is solved to compute the nonlinear distribution of moisture through the plate thickness, and then the state equations are semi-analytically solved using the statespace(SS) method and the differential quadrature(DQ) rule to predict the elastic field quantities. A comprehensive parametric analysis is accomplished to elucidate the effects of key parameters on the steady-state response of the plate under the mechanical and hygral loads.     

NONLINEAR BUCKLING BEHAVIOR OF DAMAGED COMPOSITE SANDWICH PLATES CONSIDERING THE EFFECT OF TEMPERATURE-DEPENDENT THERMAL AND MECHANICAL PROPERTIES

On the basis of the first-order shear deformation plate theory and the zig-zag deformation assumption, an incremental finite element formulation for nonlinear buckling analysis of the composite sandwich plate is deduced and the temperature-dependent thermal and mechanical properties of composite is considered. A finite element method for thermal or thermo-mechanical coupling nonlinear buckling analysis of the composite sandwich plate with an interfacial crack damage between face and core is also developed. Numerical results and discussions concerning some typical examples show that the effects of the variation of the thermal and mechanical properties with temperature, external compressive loading, size of the damage zone and ply angle of the faces on the thermal buckling behavior are significant. Project supported by the National Natural Science Foundation of China (No. 59975013).     

Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory

In this paper, an analysis on the nonlinear dynamics and chaos of a simply supported orthotropic functionally graded material (FGM) rectangular plate in thermal environment and subjected to parametric and external excitations is presented. Heat conduction and temperature-dependent material properties are both taken into account. The material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion for the orthotropic FGM rectangular plate are derived by using the Hamilton’s principle. The Galerkin procedure is applied to the partial differential governing equations of motion to obtain a three-degree-of-freedom nonlinear system. The resonant case considered here is 1:2:4 internal resonance, principal parametric resonance-subharmonic resonance of order 1/2. Based on the averaged equation obtained by the method of multiple scales, the phase portrait, waveform and Poincare map are used to analyze the periodic and chaotic motions of the orthotropic FGM rectangular plate. It is found that the motions of the orthotropic FGM plate are chaotic under certain conditions.     

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.