Exact axisymmetric solution of laminated transversely isotropic piezoelectric circular plates (Ⅱ)——Exact solution for elastic circular plates and numerical results

Emphasis is placed on purely elastic circular plates. Let the piezoelectric coefficients be equal to zero. Then two sets of uncoupled mechanical and electric equations are obtained and they can be solved independently. Two three-dimensional exact solutions of laminated transversely isotropic circular plate are derived under two boundary conditions, i.e. rigid slipping support and elastic simple support. For isotropic circular plates, the problem of multiple root is treated. At last, some numerical results of piezoelectric and purely elastic circular plates are presented and the applicability of classical plate theory is discussed.     

Fundamental solutions for plane problem of piezoelectric materials

Based on the basic equations of two-dimensional, transversely isotropic, piezoelectric elasticity, a group of general solutions for body force problem is obtained. And by utilizing this group of general solutions and employing the body potential theory and the integral method, the closed-form solutions of displacements and electric potential for an infinite piezoelectric plane loaded by point forces and point charge are acquired. Therefore, the fundamental solutions, which are very important and useful in the boundary element method (BEM), are presented.     

Three-dimensional elasticity solution of simple-supported rectangular plate on point supports,line supports and elastic foundation

This paper studies the bending of simple-supported rectangular plate on point supports, line supports and elastic foundation. On the basis of three-dimensional elasticity theory, the exact expressions of the displacement functions, which satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived. The reaction forces of the in- termediate supports are regarded as the unknown external forces acting on the lower surface of the ...     

Numerical simulation of circular jet impinging on hot steel plate

Flow structure and heat transfer characteristics of an axisymmetric circular jet impinging on a hot 1Crl8Ni9Ti medium plate have been simulated numerically using computational fluid dynamic (CFD) code. The relation between flow field of jet impingement and its heat transfer capability is analyzed, and the phenomenon that heat transfer at stagnation point is smaller than that of points directly around is discussed. The simulation result provides boundary conditions for thermal analysis of medium plate quenching.     

A theoretical study of asymptotic boundary conditions for the numerical solutions of open-boundary static electromagnetic-field problems

Based on the multipole expansion theory of the potential, a satisfactory interpretation is put forward of the exact nature of the approximations of asymptotic boundary condition (called the ABC) techniques for the numerical solutions of open-boundary static electromagnetic-field problems, and a definite physical meaning is bestowed on ABC, which provide a powerful theoretical background for laying down the operating rules and the key to the derivation of asymptotic boundary conditions. This paper is also intended to reveal the shortcomings of the conventional higher-order ABC, and at the same time to give the concept of a new type of higher-order ABC, and to present a somewhat different formulation of the new nth-order ABC. In order to test its feasibility, several simple problems of electrostatic potentials are analyzed. The results are found to be much better than those of conventional higher-order ABCs.     

Three-dimensional analysis of functionally graded piezoelectric plate with arbitrarily distributed material properties

An orthotropic functionally graded piezoelectric rectangular plate with arbitrarily distributed material properties was studied, which is simply supported and grounded（electrically） on its four lateral edges. The state equations of the functionally graded piezoelectric material were obtained using the state-space approach, and a Peano-Baker series solution was obtained for the coupled electroelastic fi elds of the functionally graded piezoelectric plate subjected to mechanical and electric loading on its upper and lower surfaces. The influence of different distributions of material properties on the structural response of the plate was studied using the obtained solutions.     

Heat transfer in a porous saturated wavy channel with asymmetric convective boundary conditions

The viscous flow in a wavy channel with convective boundary conditions is investigated. The channel is filled with a porous viscous fluid. Two cases of equal and different external convection coefficients on the walls are taken into account. Effect of viscous dissipation is also considered. The governing equations are derived employing long wavelength and low Reynolds number approximations. Exact closed form solutions are obtained for the simplified equations. Important physical features for peristaltic flow caused by the wavy wave are pumping, trapping and heat transfer rate at the channel walls. These are discussed one by one in depth and detail through graphical illustrations. Special attention has been given to the effects of convective boundary conditions. The results show that for Bi1≠Bi2, there exists a critical value of Brinkman number Brc at which the temperatures of both the walls become equal. And, for Bi1Bi2 and BrBrc, the temperature of the cold wall exceeds the temperature of hot wall.     

Interaction of oblique waves with an array of long horizontal circular cylinders

The scattering and radiations of linear oblique waves by multiple long horizontal circular cylinders submerged in water of finite depth are investigated using the multipole expansion method. Analytical expressions for the diffracted and radiated potentials are given as a linear combination of infinite multipoles. The unknown coefficients in the expressions are determined by using the addition theorem of the Bessel function and the cylinder boundary conditions. Also analytical expressions for wave forces, hydrodynamic coefficients and reflection and transmission coeffi-cients are derived. The present analytical solution is verified through the boundary element method and applied to investigate three different cases of the interaction of oblique waves with multiple submerged horizontal circular cylinders. The results show that the number of cylinders, the arrangement and spacing between cylinders play an important role in wave forces, hydrodynamic coefficients and reflection and transmission coefficients. Some interesting and important phenomena are ob-served in numerical experiments.     

Dynamic model building and simulation for mechanical main body of lunar lander

Focused on the dynamics problems of a lunar lander during landing process, the whole process was analysed in detail, and the linear elastic model of the moon soil was established by means of experiments-analogic method. Combining the way of elastic impact with the way of velocity replacement, the dynamics model of damping free vibration dynamics model with 3-degree of freedom(DOF) for lunar lander is obtained according to the vibration mechanics elementary theory. Based on Lagrange equations and the energy principle, the damping free vibration differential equations for the lunar lander with 3-DOF are derived and the equations are solved in simulation ways by means of ADAMS software. The conclusions obtained can be used for the design and manufacture of lunar lander.     

Periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance

The bifurcations and chaotic dynamics of a simply supported symmetric cross-ply composite lami- nated piezoelectric rectangular plate are studied for the first time, which are simultaneously forced by the transverse, in-plane excitations and the excitation loaded by piezoelectric layers. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by using the Hamilton’s principle. The Gal...     

Buckling of carbon fiber composite pyramidal truss core sandwich columns

This paper deals with the theoretical prediction of global buckling loads for carbon fiber composite pyramidal truss core sandwich columns. Different from thin plate structures, transverse shear effect can not be neglected for sandwich structures. In addition, the attributes of the laminated face sheets are considered in the present paper. A zig-zag displacement approximation is made. Based on the principle of minimum potential energy, equilibrium equations and boundary conditions are derived via the variational method. The critical buckling loads under various boundary conditions are presented. In order to validate the reasonableness of the equivalent-core method, the strain energies stored in the actual discrete truss members and the equivalent continuous homogenous core layer are calculated respectively and compared, and a good agreement is obtained. The proposed analytical method is verified by comparing with the published theoretical predictions and experimental results.     

Analytic solution for Reissner plate bending based on new symplectic approach

This paper presents the analytic solution for Reissner plate bending derived by the symplectic geometry approach.Firstly,the basic equations for Reissner plate are transferred into Hamilton canonical equations.And then the whole state variables are separated.Finally,the solution is obtained according to the method of eigenfunction expansion in the symplectic geometry.Only the basic elasticity equations of Reissner plate are used in the present study and the pre-selection of the deformation function is abandoned,which is requisite in classical solution methods.Therefore,the utilized approach is completely reasonable and theoretical.To verify the accuracy and validity of the formulations derived,the numerical results are presented to compare with those available in the open literatures.     

Minimum Energy Optimal Tracking

Based on the theory of generalized inverses and Bellman‘s dynamic programming approach, two forms of solutions to the optimal and minimum energy optimal tracking problems are preseated for discrete linear time-varying systems. In each ease, simple expressions for the minimum tracking error and minimum eontrol energy are derived.     

A numerical analysis of stress intensity factors for cracks emanating from an elliptical hole in a rectangular plate under biaxial loads

Due to the stress concentration effect around ahole, cracks are likely to initiate at the hole under theaction of fatigue loading. Consequently, a number ofpapers dealing with hole edge crack problems are avail-able. Bowie[1]gave solutions of a circular hole with asingle edge crack and a pair of symmetrical edge cracksin a plate under tension. Newman[2]provided a solu-tion by means of the boundary collocation method, andNisitani and Isida[3]by using the body force methodperformed analysis of …     

Behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading

The behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading was investigated. To make the analysis tractable, it was assumed that the material properties varied exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem could be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. The normalized stress and electrical displacement intensity factors were determined for different geometric and property parameters for permeable electric boundary conditions. Numerical examples were provided to show the effect of the geometry of the interacting cracks and the functionally graded material parameter upon the stress intensity factors of cracks.     

Three-dimensional consolidation deformation analysis of porous layered soft soils considering asymmetric effects

Long-term settlements for underground structures, such as tunnels and pipelines, are generally observed after the completion of construction in soft clay. The soil consolidation characteristic has great influences on the long-term deformation for underground structures. A three-dimensional consolidation analysis method under the asymmetric loads is developed for porous layered soil based on Biot’s classical theory. Time-displacement effects can be fully considered in this work and the analytical solutions are obtained by the state space approach in the Cartesian coordinate. The Laplace and double Fourier integral transform are applied to the state variables in order to reduce the partial differential equations into algebraic differential equations and easily obtain the state space solution. Starting from the governing equations of saturated porous soil, the basic relationship of state space variables is established between the ground surface and the arbitrary depth in the integral transform domain. Based on the continuity conditions and boundary conditions of the multi-layered pore soil model, the multi-layered pore half-space solutions are obtained by means of the transfer matrix method and the inverse integral transforms. The accuracy of proposed method is demonstrated with existing classical solutions. The results indicate that the porous homogenous soils as well as the porous non-homogenous layered soils can be considered in this proposed method. When the consolidation time factor is 0.01, the value of immediate consolidation settlement coefficient calculated by the weighted homogenous solution is 27.4% bigger than the one calculated by the non-homogeneity solution. When the consolidation time factor is 0.05, the value of excess pore water pressure for the weighted homogenous solution is 27.2% bigger than the one for the non-homogeneity solution. It is shown that the material non-homogeneity has a great influence on the long-term settlements and the dissipation process of excess pore water pressure.     

Application of generalized impedance boundary conditions to sommerfeld half-space problem

Generalized impedance boundary conditions are employed to simplify the solution of the Sommerfeld half-space problem. An analytical expression is derived for the Hertz potential of a vertical electric dipole over the earth’s surface, in which the earth is assumed to be a layered media or homogeneous dissipative half-space.A Sommerfeld type integral in the potential function is expressed as the sum of two parts: a zeroth order Hankel function and an absolutely convergent series of Bessel functions. In addition, two expressions in closed form are obtained as the far-field and near-field approximation of the present result.     

Free boundary value problems for a class of generalized diffusion equation

The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied.Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function,convection function,heat flux function,and power law parameters by utilizing the shooting technique.The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.     

Similarity solutions to a laminar boundary layer problem in power law fluids

A suitable similarity transformatiopn is introduced to reduce the laminar boundary layer equations of power law fluids to a class of singular nonlinear two-point boundary value problems.The skin friction and shear stress distributions for boundary layer flow over a moving flat plate are investigated by utilizing the shooting technique.Results indicate that for each fixed value of the power law exponent n or the velocity ratio parameter ξ,the skin friction and shear stress decrease with the increasing of n or ξ respectively.     

Interaction between subinterface cracks and interface in metal/piezoelectric ceramic bimaterials

A class of problems concerning subinterface cracks interacting with the interface in metal/ piezoelectric ceramic bimaterials are studied. The interaction problem is reduced to a system of integral equations with the aid of the pseudo-traction-electric-displacement method. The equations are solved numerically, and the stress intensity factor, the electric displacement intensity factor and the mechanical strain energy release rate are evaluated. Numerical results of a Cu/PZT-4 bimaterial are given and shown in figures, in which three kinds of remote loading conditions are considered to make comparisons . It is found that the electric loading at infinity plays a quite important role in the present interaction problem. In addition, a conservation law of the first component of the Jk-integral vector is found, which does lead to a consistency check to confirm the effectiveness of the PTEDM as well as the numerical results derived in this paper.