GREEN＇S FUNCTION AND EFFECTIVE ELASTIC STIFFNESS TENSOR FOR ARBITRARY AGGREGATES OF CUBIC CRYSTALS

A closed but approximate formula of Green‘s function for an arbitrary aggregate of cubic crystallites is given to derive the effective elastic stiffness tensor of the polycrystal. This formula, which includes three elastic constants of single cubic crystal and five texture coefficients,accounts for the effects of the orientation distribution function (ODF) up to terms linear in the texture coefficients. Thus it is expected that our formula would be applicable to arbitrary aggregates with weak texture or to materials such as aluminum whose single crystal has weak anisotropy.Three examples are presented to compare predictions from our formula with those from Nishioka and Lothe‘s formula and Synge‘s contour integral through numerical integration. As an application of Green‘s function, we briefly describe the procedure of deriving the effective elastic stiffness tensor for an orthorhombic aggregate of cubic crystallites. The comparison of the computational results given by the finite element method and our effective elastic stiffness tensor is made by an example.     

Thermal-stress induced phenomena in two-component material:Part Ⅱ

The paper deals with analytical models of the elastic energy gradient Wsq representing an energy barrier. The energy barrier is a surface integral of the elastic energy density Wq. The elastic energy density is induced by thermal stresses acting in an isotropic spherical particle （q = p） with the radius R and in a cubic cell of an isotropic matrix （q = m）. The spherical particle and the matrix are components of a multi-particle-matrix system representing a model system applicable to a real two-component material of a precipitation-matrix type. The multi-particle-matrix system thus consists of periodically distributed isotropic spherical particles and an isotropic infinite matrix. The infinite matrix is imaginarily divided into identical cubic cells with a central spherical particle in each of the cubic cells. The dimension d of the cubic cell then corresponds to an inter-particle distance. The parameters R, d along with the particle volume fraction v = v（R, d） as a function of R, d represent micro- structural characteristics of a real two-component material. The thermal stresses are investigated within the cubic cell, and accordingly are functions of the microstructural charac- teristics. The thermal stresses originate during a cooling pro- cess as a consequence of the difference am - ap in thermal expansion coefficients between the matrix and the particle, am and ap, respectively. The energy barrier Wsq is used for the determination of the thermal-stress induced strengthening aq. The strengthening represents resistance against com- pressive or tensile mechanical loading for am - ap 〉 0 or am - ap 〈 0. respectively.     

Bifurcation and chaos of the circular plates on the nonlinear elastic foundation

According to the large amplitude equation of the circular plate on nonlinear elastic foundation , elastic resisting force has linear item , cubic nonlinear item and resisting bend elastic item. A nonlinear vibration equation is obtained with the method of Galerkin under the condition of fixed boundary. Floquet exponent at equilibrium point is obtained without external excitation. Its stability and condition of possible bifurcation is analysed. Possible chaotic vibration is analysed and studied with the method of Melnikov with external excitation . The critical curves of the chaotic region and phase figure under some foundation parameters are obtained with the method of digital artificial.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Dynamic quasi-continuum model for plate-type nano-materials and analysis of fundamental frequency

A dynamic quasi-continuum model is presented to analyze free vibration of plate-type cubic crystal nano-materials.According to the Hamilton principle,fundamental governing equations in terms of displacement components and angles of rotations are given.As an application of the model,the cylindrical bending deformation of the structure fixed at two ends is analyzed,and a theoretical formula evaluating the fundamental frequency is obtained by using Galerkin's method.Meanwhile,the solution for the classical continuous plate model is also derived,and the size-dependent elastic modulus and Poisson's ratio are taken in computation.The frequencies corresponding to different atomic layers are numerically presented for the plate-type NaC l nano-materials.Furthermore,a molecular dynamics(MD)simulation is conducted with the code LAMMPS.The comparison shows that the present quasi-continuum model is valid,and it may be used as an alternative model,which reflects scale effects in analyzing dynamic behaviors of such plate-type nano-materials.     

BOUNDARY INTEGRAL FORMULAS FOR ELASTIC PLANE PROBLEM OF EXTERIOR CIRCULAR DOMAIN

After the stress function and the normal derivative on the boundary for the plane problem of exterior circular domain are expanded into Laurent series, comparing them with the Laurent series of the complex stress function and making use of some formulas in Fourier series and the convolutions, the boundary integral formula of the stress function is derived further. Then the stress function can be obtained directly by the integration of the stress function and its normal derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function is convenient to be used for solving the elastic plane problem of exterior circular domain.     

Response of loose bonding on reflection and transmission of elastic waves at interface between elastic solid and micropolar porous cubic crystal

The problem of reflection and transmission of plane periodic waves incident on the interface between the loosely bonded elastic solid and micropolar porous cubic crystal half spaces is investigated. This is done by assuming that the interface behaves like a dislocation, which preserves the continuity of traction while allowing a finite amount of slip. Amplitude ratios of various reflected and transmitted waves have been depicted graphically. Some special cases of interest have been deduced from the present investigation.     

CALCULATION ANALYSIS OF SHEARING SLIP FOR STEELCONCRETE COMPOSITE BEAM UNDER CONCENTRATED LOAD

The strain difference of steel and concrete under vertical concentrated load was analyzed on the basis of elastic theory, and was compared with ideal solution of steel and concrete under vertical uniform load. The results indicate that the computing formula concluded from the paper is believable. The practical structure usually bears concentrated load, so it can be used in the practical engineering.     

Thermal-stress induced phenomena in two-component material： part I

The paper deals with analytical fracture mechanics to consider elastic thermal stresses acting in an isotropic multi-particle-matrix system. The multi-particle-matrix system consists of periodically distributed spherical particles in an infinite matrix. The thermal stresses originate during a cooling process as a consequence of the difference αm - αp in thermal expansion coefficients between the matrix and the particle, αm and αp, respectively. The multi-particle-matrix system thus represents a model system applicable to a real two-component material of a precipitation-matrix type. The infinite matrix is imaginarily divided into identical cubic cells. Each of the cubic cells with the dimension d contains a central spherical particle with the radius R, where d thus corresponds to inter-particle distance. The parameters R, d along with the particle volume fraction v = v（R, d） as a function of R, d represent microstructural characteristics of a twocomponent material. The thermal stresses are investigated within the cubic cell, and accordingly are functions of the microstructural characteristics. The analytical fracture mechanics includes an analytical analysis of the crack initiation and consequently the crack propagation both considered for the spherical particle （q = p） and the cell matrix （q = m）. The analytical analysis is based on the determination of the curve integral Wcq of the thermal-stress induced elastic energy density Wq. The crack initiation is represented by the determination of the critical particle radius Rqc = Rqc（V）. Formulae for Rqc are valid for any two-component mate- rial of a precipitate-matrix type. The crack propagation for R 〉 Rqc is represented by the determination of the function fq describing a shape of the crack in a plane perpendicular     

The impact torsional buckling of elastic cylindrical shells with arbitrary form imperfection

A perturbation analysis for the impact torsional buckling of imperfective elastic cylindrical shells subjected to a step torque is given..The imperfection is supposed to be small and has arbitrary form.It is shown that only the imperfection which has the shape of static torsional buckling mode could influence the critical step torque.Finally a formula is presented for the critical step torque.     

The singular perturbation method applied to the nonlinear stability problem of a shallow spherical shell (II)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author’s previous paper[11].     

THE SINGULAR PERTURBATION METHOD APPLIED TO THE NONLINEAR STABILITY PROBLEM OF A SHALLOW SPHERICAL SHELL(Ⅱ)

In this paper we consider the nonlinear stability of a thin elastic circular shallowspherical shell under the action of uniform normal pressure with a clamped edge.When thegeometrical parameter k is large,the uniformly valid asymptotic solutions are obtained bymeans of the singular perturbation method.In addition,we give the analytic formula fordetermining the centre deflection and the critical load,and the stability curve is also derived.This paper is a continuation of the author’s previous paper[11].     

APPROXIMATE SOLUTION FOR BENDING OF RECTANGULAR PLATES—Kantorovich-Galerkin’s Method

This paper derives the cubic spline beam function from the generalized beamdifferential equation and obtains the solution of the discontinuous polynomial underconcentrated loads,concentrated moment and uniform distributed by using delta function.By means of Kantorovich method of the partial differential equation of elastic plates whichis transformed by the generalized function(δ function and σ function),whetherconcentrated load,concentratedmoment,uniform distributed load or small-square load canbe shown as the discontinuous polynomial deformed curve in the x-direction and the y-direction.We change the partial diffkerential equation into the ordinary equation by usingKantorovich method and then obtain a good approximate solution by using Glerkin’smethod.In this paper there are more calculation examples involving elastic plates withvarious bounndary-conditions,various loads and various section plates,and the classicaldifferential problems such as cantilever plates are shown.     

On crack-tip strain energy release rate in non-principal directions of elasticity for simple layer plate of composite materials

In this paper,the fracture problem in non-principal directions of elasticity for a simple layer plate of linear-elastic orthotropic composite materials is studied.The formulae of transformation between characteristic roots,coefficients of elastic compliances in non-principal directions of elasticity and corresponding parameters in principal directions of elasticity are derived.Then,the computing formulae of strain energy release rate under skew-symmetric loading in terms of engineering parameters for principal directions of elasticity are obtained by substituting crack-tip stresses and displacements into the basic formula of the strain energy release rate.     

Parametric resonances of convection belt system

Based on the Coriolis acceleration and the Lagrangian strain formula, a gen- eralized equation for the transverse vibration system of convection belts is derived using Newton＇s second law. The method of multiple scales is directly applied to the govern- ing equations, and an approximate solution of the primary parameter resonance of the system is obtained. The detuning parameter, cross-section area, elastic and viscoelastic parameters, and axial moving speed have a significant influences on the amplitudes of steady-state response and their existence boundaries. Some new dynamical phenomena are revealed.     

APPROXIMATE SOLUTION FOR BENDING OF RECTANGULAR PLATES Kantorovich-Galerkin’s Method

This paper derives the cubic spline beam function from the generalized beam differential equation and obtains the solution of the discontinuous polynomial under concentrated loads, concentrated moment and uniform distributed by using delta function. By means of Kantorovich method of the partial differential equation of elastic plates which is transformed by the generalized function (δ function and σ function), whether concentrated load, concentrated moment, uniform distributed load or small-square load can be shown as the discontinuous polynomial deformed curve in the x-direction and the y-direction. We change the partial differential equation into the ordinary equation by using Kantorovich method and then obtain a good approximate solution by using Glerkin’s method. In this paper there ’are more calculation examples involving elastic plates with various boundary-conditions, various loads and various section plates, and the classical differential problems such as cantilever plates are shown.     

Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory

The deviation from the classical elastic characteristics induced by the free surface energy can be considerable for nanostructures due to the high surface to volume ratio. Consequently, this type of size dependency should be accounted for in the mechanical behaviors of nanoscale structures. In the current investigation, the influence of free surface energy on the nonlinear primary resonance of silicon nanoshells under soft harmonic external excitation is studied. In order to obtain more accurate results,the interaction between the first, third, and fifth symmetric vibration modes with the main oscillation mode is taken into consideration. Through the implementation of the Gurtin-Murdoch theory of elasticity into the classical shell theory, a size-dependent shell model is developed incorporating the effect of surface free energy. With the aid of the variational approach, the governing differential equations of motion including both of the cubic and quadratic nonlinearities are derived. Thereafter, the multi-time-scale method is used to achieve an analytical solution for the nonlinear size-dependent problem. The frequency-response and amplitude-response of the soft harmonic excited nanoshells are presented corresponding to different values of shell thickness and surface elastic constants as well as various vibration mode interactions. It is depicted that through consideration of the interaction between the higher symmetric vibration modes and the main oscillation mode, the hardening response of nanoshell changes to the softening one. This pattern is observed corresponding to both of the positive and negative values of the surface elastic constants and the surface residual stress.     

Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform was employed in the analysis of the semi-infinite elastic foundation with vertical load.And an integral representations for the displacements of the semi-infinite elastic foundation was presented.The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semi- infinite elastic foundation.Some computational results and the analysis on the influence of parameters were presented.     

THE CONVERGENT CONDITION AND UNITED FORMULA OF STEP REDUCTION METHOD

The step reduction method was first suggested by Prof. Yeh Kai-yuan. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time. its ealculuting time is very short and convergent speed very fast. In this paper, the convergent condition and nited formula of step reduction method are given by mathemutical method. It is proved that the solution of displacement and stress resultants obtained by this method can eonverge to exact solution uniformly, when the convergent condition is sutisfied. By united formula, the analytic solution solution can be expressed as matrix form, and therefore the former complicated expression can be avoMed. Two numerical examples are given at the end of this paper which indicate that. by the theory in this paper, a right model can be obtained for step reduction method.     

Binary discrete method of topology optimization

The numerical non-stability of a discrete algorithm of topology optimization can result from the inaccurate evaluation of element sensitivities. Especially, when material is added to elements, the estimation of element sensitivities is very inaccurate, even their signs are also estimated wrong. In order to overcome the problem, a new incremental sensitivity analysis formula is constructed based on the perturbation analysis of the elastic equilibrium increment equation, which can provide us a good estimate of the change of the objective function whether material is removed from or added to elements, meanwhile it can also be considered as the conventional sensitivity formula modified by a non-local element stiffness matrix. As a consequence, a binary discrete method of topology optimization is established, in which each element is assigned either a stiffness value of solid material or a small value indicating no material, and the optimization process can remove material from elements or add material to elements so as to make the objective function decrease. And a main advantage of the method is simple and no need of much mathematics, particularly interesting in engineering application.