Rapid heating vibrations of FGM annular sector plates

In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.

     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

Some computational aspects of large deformation,rate-dependent plasticity problems

The governing equations for a finite element formulation of boundary value problems for large deformation metal forming processes are derived using a principle of virtual work formulated in a Lagrangian reference system. The updated Lagrangian method is used to simplify the equations. The resulting nonlinear equilibrium equations are solved using Newton's method.A constitutive model for large deformation, rate-dependent plasticity is developed. The model incorporates a one parameter, implicit integration operator for stability and accuracy. The stressrate/strain-rate relation is written in terms of the Jaumann rate of stress.Numerous example problems are solved to demonstrate the effectiveness of the numerical algorithms.     

Steady Homann flow and heat transfer of an electrically conducting second grade fluid

The steady axisymmetric flow and heat transfer of an incompressible, electrically conducting non-Newtonian second grade fluid impinging on a flat plate is investigated. An external uniform, transverse magnetic field is applied at the surface of the plate. Similarity transformation is used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. An effective numerical scheme has been adopted to solve the nonlinear ordinary differential equations. The effects of non-Newtonian flow parameters and the magnetic field on the momentum and thermal boundary layers are discussed in detail and shown graphically. It is interesting to find that the non-Newtonian parameter and the magnetic parameter have opposite effects on the momentum and thermal boundary layers. The skin friction coefficient decreases exponentially with an increase in the non-Newtonian viscoelastic parameter and increases linearly with an increase in the magnetic parameter.     

Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory

In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.

     

The effect of nonlinear elasticity on the large amplitude free vibration behavior of elastic plates at small scale

The effect of nonlinear elasticity on the free vibration behavior of elastic plates has been evaluated by employing continuum mechanics model. The second-order non-linear stress–strain relationship has been considered and the Kirchhoff’s hypothesis has been applied on the elastic plate. The large deformation during vibration has also been considered. By using the Hamilton principle, the governing equations of the free vibration of the plate under different boundary condition have been obtained. In order to get the explicit solutions of the governing equations, the Galerkin’s method and the harmonic balance method have been utilized. The relationship between the vibration frequency and the vibration amplitude has been discussed and the vibration frequencies of different shaped plate have been compared. It is perceived that the nonlinear elasticity has a distinct effect on the free vibration of the plate.

     

悬臂式蜂窝夹层板的非线性动力学建模及分析

蜂窝夹层结构因其良好的力学特性,在众多工程领域具有非常广泛的应用.本文建立了悬臂边界条件下,蜂窝夹层板的动力学模型并研究其非线性动力学行为.选取文献中更加接近实体有限元解的等效弹性参数公式对蜂窝芯层进行等效简化,得到六角形蜂窝芯的等效弹性参数.基于Reddy高阶剪切变形理论,应用Hamilton原理建立悬臂式蜂窝夹层板在受到面内激励和横向激励联合作用下的偏微分运动方程.然后利用Galerkin方法得到两自由度非自治常微分形式运动方程.在此基础上,通过对悬臂式蜂窝夹层板进行数值模拟分析系统的非线性动力学.结果表明面内激励和横向激励对系统的动力学特性有着重要影响,在不同激励作用下系统会出现周期运动、概周期运动以及混沌运动等复杂的非线性动力学响应.     

Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method

A general finite strip method of analysis is presented for the geometrically nonlinear analysis of laterally loaded, rectangular, isotropic plates. The analysis is based on the use of Mindlin plate theory and therefore includes the effects of transverse shear deformation. The nonlinearity is introduced via the strain-displacement equations and correspondingly the analysis pertains to problems involving moderate displacements but small rotations. The principle of minimum potential energy is used in the development of the strip and the complete plate stiffness equations and the latter equations are solved using the Newton-Raphson method. In numerical applications a particular type of finite strip is used in which all five reference quantities (three displacements and two rotations) are represented by cubic polynomial interpolation across the strip whilst the ends of the strip are simply supported for bending/shearing behaviour and immovable for membrane behaviour. These applications are concerned with uniformly loaded plates of both thin and moderately-thick geometry and detailed presentation is given of both displacement- and force-type quantities.     

薄板的几何非线性求积元分析

针对薄板非线性迭代计算量很大的问题,依据von Kárman薄板非线性理论构造能量泛函,并用数值积分和数值微分进行离散,得到非线性方程组,从而利用求积元法(Quadrature Element Method,QEM)求解薄板的中等挠度的弯曲和非线性屈曲问题,得到可信的结果.算例表明:在处理薄板几何非线性问题上,QEM计算效率很高,应用潜力很大.     

Numerical analysis of single-layered graphene sheets by a mesh-free approach

The paper presents a numerical analysis of the behavior of single-layered graphene sheet (SLGS) using a mesh-free approach based on a nonlocal continuum plate model (NCPM). The adopted NCPM constructed by incorporating the nonlocal elasticity theory into the first order shear deformation elastic plate theory is able to capture small length scale effects. Through the NCPM, the SLGS is modeled as a continuous orthotropic nanoplate. the obtained nonlocal nonlinear partial differential equations completed by boundary conditions are solved numerically by a mesh-free approach associating the asymptotic numerical method with the mesh-free collocation method based on moving least square approximation. The effects of the small-scale parameter and aspect ratio on the nonlinear bending and post-buckling behaviors of SLGS are considered. Good agreement has been established between the obtained results and those of the literature.

     

Axisymmetric bending of circular and annular sandwich plates with nonlinear elastic core material

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagrange's formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.     

The performance of Mindlin plate finite strips in geometrically nonlinear analysis

The finite strip method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates within the context of Mindlin plate theory wherein the effects of transverse shear deformation are included. The analysis is of the Lagrangian type with the nonlinearity introduced by the inclusion of certain nonlinear terms in the strain-displacement equations. Following on from a related earlier investigation which dealt with a particular finite strip model, the performance of a range of different models is investigated. Linear, quadratic, cubic, and quartic polynomial interpolation is used in the different models in representing the variation of the five relevant displacement type quantities across a strip: also, both analytical (exact) and numerical (reduced, selective) schemes of integration are used in the crosswise direction in evaluating the stiffness properties of the various models. The ends of the finite strips are simply supported for out-of-plane behaviour and immovable for inplane behaviour. Detailed results are presented of the application of seven types of finite strip model to a range of plate problems, all involving uniformly loaded, square plates but with thin or moderately thick geometry and with simply supported or clamped longitudinal edges.     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Three-dimensional fluid-structure coupling in transient analysis

The numerical simulation of nonlinear, transient fluid-structure interactions (FSI) is a current area of concern by researchers in various fields, including the field of nuclear reactor safety. This paper primarily discusses the formulation used in an algorithm that couples three-dimensional hydrodynamic and structural domains. Here, both the fluid and structure are discretized using finite elements. The semi-discretized equations of motion are solved using an explicit temporal integrator.Coupling is accomplished by satisfying interface mechanics. The structure imposes kinematic constraints to the moving fluid boundary, and the fluid in turn provides an external loading on the structure. At each interface node, normals are computed from the nodal basis functions of only the hydrodynamic nodes. By defining the interface normal in this manner, it becomes independent of the type of structural boundary (i.e. shell, plate, continuum, etc.) and thus makes this aspect of the coupling independent of the structure type. A penalty type gap-impact element is developed to model the impact region between the fluid and structure.Results for several problems are presented and these include a comparison between analytical results for a FSI problem and numerical predictions.     

几何非线性损伤粘弹性中厚板的动力学行为分析

根据Timoshenko几何变形假设和Boltzmann叠加原理,推导出控制损伤粘弹性Timoshenko中厚板的非线性动力方程以及简化的Galerkin截断方程组;然后利用非线性动力系统中的数值方法求解了简化方程组．通过分析可知,板在谐载荷的作用下,具有非常丰富的动力学特性．同时研究了板的几何参数、材料参数及载荷参数对损伤粘弹性中厚板动力学行为的影响．     

The optimal solution for the flow of a fourth-grade fluid with partial slip

The steady flow of a non-Newtonian fluid when slippage between the plate and the fluid occurs is considered. The constitutive equations of the fluid are modeled for a fourth-grade non-Newtonian fluid with partial slip; they give rise to nonlinear boundary value problems. Analytical solutions are obtained using powerful analytic techniques for solving nonlinear problems, homotopy perturbation and optimal homotopy asymptotic methods. The results obtained are compared with the numerical results and it is shown that solutions exist for all values of the non-Newtonian parameters. The solutions valid for the no-slip condition for all values of the non-Newtonian parameters can be derived as special cases of the present analysis. Finally the solutions are discussed using a graphical approach.     

Numerical analysis on snapping induced by electromechanical interaction of shuffling actuator with nonlinear plate

When a voltage is applied between two electrodes consisting of a rigid ground and a deformable rectangular plate that has two parallel free edges, and one movable edge and one fixed edge, the plate bends under the electrostatic force generated in it. Accompanying to the bending deformation, the movable edge of the plate results in some displacement along the movable direction, which is the main clue of the MEMS actuator of shuffling movement. As the control voltage is applied up to a critical value, the phenomenon of snapping occurs at the plate electrode under the interaction of electromechanical coupling such that the shuffling displacement is obtained as possibly as large. Based on the nonlinear theory of plates with the von Karman’s type deformation and the electrostatic theory, a theoretical analysis for the snapping behavior is quantitatively displayed in this paper. For this purpose, a numerical code is proposed by associating the increment finite element methods for deformation with the moment method for electrostatic fields as well as the arc-length control approach in the quantitative calculations. The numerical results for some case studies show that the path of bending deformation from stability into instability passing through the snapping criterion can be tracked well by this numerical code, while the characteristic curves of the maximum deflection in the plate and the shuffling displacement versus the control voltage, which are mainly concerned in the design of the MEMS shuffling actuator, are obtained additionally.     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.