Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

3D Finite Volume Scheme for Czochralski Single Crystal Growth

Czochralski (Cz) technique, which is used for growing single crystals, has dominated the production of single crystals for electronic applications. The Cz growth process involves multiple phases, moving interface and three-dimensional behavior. Much has been done to study these phenomena by means of numerical methods as well as experimental observations. A three-dimensional curvilinear finite volume based algorithm has been developed to model the Cz process. A body-fitted transformation based approach is adopted in conjunction with a multizone adaptive grid generation (MAGG) technique to accurately handle the three-dimensional problems of phase-change in irregular geometries with free and moving surfaces. The multizone adaptive model is used to perform a three-dimensional simulation of the Cz growth of silicon single crystals.Since the phase change interface are irregular in shape and they move in response to the solution, accurate treatment of these interfaces is important from numerical accuracy point of view. The multizone adaptive grid generation (MAGG) is the appropriate scheme for this purpose. Another challenge encountered is the moving and periodic boundary conditions, which is essential to the numerical solution of the governing equations. Special treatments are implemented to impose the periodic boundary condition in a particular direction and to determine the internal boundary position and shape varying with the combination of ambient physicochemical transport process and interfacial dynamics. As indicated above that the applications and processes characterized by multi-phase, moving interfaces and irregular shape render the associated physical phenomena three-dimensional and unsteady. Therefore a generalized 3D model rather than a 2D simulation, in which the governing equations are solved in a general non-orthogonal coordinate system, is constructed to describe and capture the features of the growth process. All this has been implemented and validated by using it to model the low pressure Cz growth of silicon. Accuracy of this scheme is demonstrated by agreement of simulation data with available experimental data. Using the quasi-steady state approximation, it is shown that the flow and temperature fields in the melt under certain operating conditions become asymmetric and unsteady even in the absence of extrinsic sources of asymmetry. Asymmetry in the flow and temperature fields, caused by high shear initiated phenomena, affects the interface shape in the azimuthal direction thus results in the thermal stress distribution in the vicinity, which has serious implications from crystal quality point of view.

Transverse mode formation in longitudinally pumped miniature slab lasers

The formation of transverse modes in longitudinally pumped miniature slab lasers is investigated theoretically and experimentally. The longitudinally non-uniform gain-guiding is studied by expanding the electric field into the Hermite-Gaussian functions that satisfy boundary conditions of the resonator. Non-Gaussian transversal beam profiles in the near field are found and the beam diameter is reduced when the pump spot becomes smaller. The experimental observation agrees with the theoretical calculation.

基于Berreman矩阵的双折射薄膜光谱响应特性计算

采用Berreman特征矩阵方法，通过数值计算研究了双折射薄膜的反射、透射等光谱响应特性。依据电磁场理论的电场分量、磁场分量的界面连续条件，推导了光波在各向异性双轴薄膜中的Berreman转移矩阵，用以分析含有各向异性介质层的复杂薄膜系统的光学性质。这些矩阵递推关系包含了界面处的多点反射，适用于一般的各向异性的多层膜系统，包括入射媒质或基底为各向异性的情况。在文中给出了各向同性入射媒质双轴各向异性膜层一各向同性基底薄膜系统的计算结果，验证了该计算方法的可行性，以此作为进一步研究各向异性薄膜和相关光学薄膜器

光学薄膜中界面层和表面吸附层对相位延迟的影响

多层介质反射镜在非正入射的时候，两个不同的偏振态之间会产生不同的相移．根据空气与膜层、膜层之间的实际情况，建立了界面层和表面吸附层模型，并运用它分析相位延迟产生误差的原因．通过优化设计，入射角为54°，在1285～1345nm之间p，s波获得了270±1°的相移，同时也使反射率在99．5％以上．用离子束溅射技术制备相位延迟膜，用分光光度计测试了光谱特性和用椭偏仪测试了相位特性，在相应波段获得了262．4±1．8°的相移，同时也使反射率在99．6％以上．误差的主要来源是离子源工作特性会产生不均匀的过渡层和最

耗散粒子动力学处理复杂固体壁面的一种有效方法

耗散粒子动力学(dissipative particle dynamics,DPD)作为一种介观尺度拉格朗日型粒子方法,已经成功地应用于微纳米流动和生化科技的研究中. 复杂固体壁面的处理和壁面边界条件的实施一直是DPD方法发展及应用的一个障碍. 提出了处理复杂固体壁面的一种新的方法. 复杂固体区域通过冻结随机分布并且达到平衡状态的DPD粒子代表;所冻结的DPD粒子位于临近流动区域的一个截距内;在靠近固体壁面的流动区域中设置流动反弹层,当流动DPD粒子进入此流动层后反弹回流动区域. 应用这种固体壁面处理方法.

Boundary Element Method (BEM) Analysis for Galvanic Corrosion of Hot Dip Galvanized Steel Immersed in Seawater

A numerical analysis of galvanic corrosion of hot-dip galvanized steel immersed in seawater was presented. The analysis was based on the boundary element methods (BEMs) coupled with Newton-Raphson iterative technique to treat the nonlinear boundary conditions, which were determined by the experimental polarization curves. Results showed that galvanic current density concentrates on the boundary of steel substrate and zinc coating, and the sacrificial protection of zinc coating to steel substrate results in overprotection of steel cathode. Not only oxygen reduction but also hydrogen reduction could occur as cathode reactions, which probably led up to the adsorption and absorption of hydrogen atoms. Flat galvanized steel tensile sample shows a brittle behavior similar to hydrogen embrittlement according to the SSRT (show strain rate test) in seawater.

二氧化碳在孔隙咸水层中多相流动数值模拟研究

There has been a growing concern about the use of fossil fuels and its adverse effects on the atmospheric greenhouse and ecological environment. A reduction in the release rate of CO2 into the atmosphere poses a major challenge to the land ecology of China. The most promising way of achieving CO2 reduction is to dispose of CO2 in deep saline aquifers. Deep aquifers have a large potential for CO2 sequestration in geological medium in terms of volume and duration. Through the numerical simulation of multiphase flow in a porous media, the transformation and motion of CO2 in saline aquifers has been implemented under various temperature and hydrostatic pressure conditions, which plays an important role to the assessment of the reliability and safety of CO2 geological storage. As expected, the calculated results can provide meaningful and scientific information for management purposes. The key problem to the numerical simulation of multiphase flow in a porous media is to accurately capture the mass interface and to deal with the geological heterogeneity. In this study, the updated CE/SE (Space and time conservation element and solution element) method has been proposed, and the Hybrid Particle Level Set method (HPLS) has extended for multiphase flows in porous medium, which can accurately trace the transformation of the mass interface. The benchmark problems have been applied to evaluate and validate the proposed method. In this study, the reliability of CO2 storage in saline aquifers in Daqingzi oil field in Sunlong basin has been discussed. The simulation code developed in this study takes into account the state for CO2 covering the triple point temperature and pressure to the supercritical region. The geological heterogeneity has been implemented, using the well known geostatistical model (GSLIB) on the base of the hard data. The 2D and 3D model have been set up to simulate the CO2 multiphase flow in the porous saline aquifer, applying the CE/SE method and the HPLS method .The main contents and results are summarized as followings. (1) The 2D CE/SE method with first and second –order accuracy has been extended to simulate the multiphase flow in porous medium, which takes into account the contribution of source and sink in the momentum equation. The 3D CE/SE method with the first accuracy has been deduced. The accuracy and efficiency of the proposed CE/SE method have been investigated, using the benchmark problems. (2) The hybrid particle level set method has been made appropriate and extended for capturing the mass interface of multiphase flows in porous media, and the numerical method for level set function calculated has been formulated. (3) The closed equations for multiphase flow in porous medium has been developed, adept to both the Darcy flow and non-Darcy flow, getting over the limitation of Reynolds number to the calculation. It is found that Darcy number has a decisive influence on pressure as well as velocity given the Darcy number. (4) The new Euler scheme for numerical simulations of multiphase flows in porous medium has been proposed, which is efficient and can accurately capture the mass interface. The artificial compressibility method has been used to couple the velocities and pressure. It is found that the Darcy number has determinant effects on the numerical convergence and stability. In terms of the different Darcy numbers, the coefficient of artificial compressibility and the time step have been obtained. (5) The time scale of the critical instability for critical CO2 in the saline aquifer has been found, which is comparable with that of completely CO2 dissolved saline aquifer. (6) The concept model for CO2 multiphase flows in the saline aquifer has been configured, based on the temperature, pressure, porosity as well as permeability of the field site .Numerical simulation of CO2 hydrodynamic trapping in saline aquifers has been performed, applying the proposed CE/SE method. The state for CO2 has been employed to take into account realistic reservoir conditions for CO2 geological sequestration. The geological heterogeneity has been sufficiently treated , using the geostatistical model. (7) It is found that the Rayleigh-Taylor instability phenomenon, which is associated with the penetration of saline fluid into CO2 fluid in the direction of gravity, has been observed in CO2 multiphase flows in the saline aquifer. Development of a mushroom-type spike is a strong indication of the formation of Kelvin-Helmholtz instability due to the developed short wavelength perturbations present along the interface and parallel to the bulk flow. Additional key findings: the geological heterogeneity can distort the flow convection. The ascending of CO2 can induce the persistent flow cycling effects. The results show that boundary conditions of the field site have determinant effects on the transformation and motion of CO2 in saline aquifers. It is confirmed that the proposed method and numerical model has the reliability to simulate the process of the hydrodynamic trapping, which is the controlling mechanism for the initial period of CO2 storage at time scale of 100 years.

软土亚塑性模型与一维非线性固结理论研究

With the great development of Tianjing New Coastal District economy, people need more land to build and live. Land subsidence, which is caused by its special engineering geological conditions, has restricted the further development in the district. Soft soil consolidation is main factor of land subsidence ;thus , on the basis of consolidation theory, the paper make further study on soft soils one-dimension nonlinear consolidation which contains two parts:(1) the nonlinear consolidation of permeability coefficient and compressibility coefficient changing with time and depth, which means real one-dimension nonlinear consolidation;(2) the non-homogeneous consolidation of permeability coefficient and compressibility coefficient only changing with depth. Firstly, nonlinear characteristics of soft soils are elaborated. Hypoplastic theory is introduced to establish a modified soft soils nonlinear constitutive model; the nonlinear governing equation of compressibility coefficient is built, and the nonlinear characteristics of compressibility coefficient are analyzed. Secondly, Considering Load Fluctuation and soil thickness changing ,the consolidation characteristics of single layer is discussed in the paper; meanwhile, on the basis of the Davis and Raymond’s hypothesis and single layer nonlinear consolidation equation, the doubled-layer one-dimension nonlinear consolidation equation is also derived. The solution of the equation is obtained by analytical method, and the consolidation characteristics of doubled-layer soft soil nonlinear theory is also analyzed. Finally, based on assumption that permeability coefficient and compressibility coefficient is varying along depth, single layer soil one-dimension non-homogeneous consolidation differential equation is derived; and the approximate solution is obtained. Furthermore, the single layer non-homogeneous consolidation is extended to double layer non-homogeneous consolidation theory. By using parabolic differential scheme, the matrix equation is established; and the solution of the matrix equation is obtained by chase method. Consolidation characteristics of soil soft single (double) layer non-homogeneous consolidation theory and Terzaghi’s theory are also discussed.

济阳-昌潍坳陷火成岩油气藏体系形成机理和分布规律研究

Jiyang & Changwei depressions are two neighboring depressions in Bahai Bay Basin, the famous oil rich basin in East China. The exploration activities in the past 40 years has proved that, within the basins, there exists not only plentiful sandstone hydrocarbon reservoirs (conventional), but also abundant special reservoirs as igneous rock, mudstone and conglomerate ones which have been knowing as the unconventional in the past, and with the prospecting activity is getting more and more detailed, the unconventional reservoirs are also getting more and more important for further resources, among which, the igneous lithological reservoir be of significance as a new research and exploration area. The purpose of this paper is, with the historical researches and data as base, the System Theory, Practice Theory and Modern Comprehensive Petroleum Geology Theory as guide, the theoretical and practice break through as the goal, and the existing problems in the past as the break through direction, to explore and establish a valid reservoir formation and distribution models for igneous strata in the profile of the eastern faulted basins. After investigating the distribution of the igneous rocks and review the history of the igneous rocks reservoirs in basins, the author focused on the following issues and correspondingly the following progresses have been made: 1.Come to a new basin evolution and structure model named "Combined-Basin-bodies Model" for Jiyang even Eastern faulted basins based on the study on the origin and evolution of Jiyang & Changwei basins, depending on this model, every faulted basin in the Bo-hai Bay Basin is consisted of three Basin-Bodies including the Lower (Mesozoic), Middle (Early Tertiary) and the Upper (Late Tertiary) Bodies, each evolved in different geo-stress setting and with different basin trend, shape and igneous-sedimentary buildings system, and from this one to next one, the basin experienced a kind of process named "shape changing" and "Style changing". 2. Supposed a serious of new realizations as follows (1) There were "multi-level magma sources" including Upper mantel and the Lower, Middle and even the Upper Shell magma Chambers in the historical Magma Processes in the basins; (2) There were "multi-magma accessing or pass" from the first level (Mantel faults) to the second, third and fourth levels (that is the different levels of fault in the basin sediment strata) worked in the geo-historical and magma processes; (3) Three tectonic magma cycles and more periods have been recognized those are matched with the "Basin -body-Model" and (4)The geo-historical magma processes were non-homogeneous in time and space scale and so the magma rocks distributed in "zones" or "belts". 3. The study of magma process's effect on basin petroleum conditions have been made and the following new conclusions were reached: (1) the eruptive rocks were tend to be matched with the "caped source rock", and the magma process were favorable to the maturing of the source rocks. (2) The magma process were fruitful to the accumulation of the non-hydrocarbon reservoirs however a over magma process may damage the grade of resource rock; (3) Eruptive activity provided a fruitful environment for the formation of such new reservoir rocks as "co-eruptive turbidity sandstones" and "thermal water carbonate rocks" and the intrusive process can lead to the origin of "metamorphism rock reservoir"; (4) even if the intrusive process may cause the cap rock broken, the late Tertiary intrusive rocks may indeed provide the lateral seal and act as the cap rock locally even regionally. All above progresses are valuable for reconstructing the magma-sedimentary process history and enriching the theory system of modem petroleum geology. 4. A systematic classification system has been provided and the dominating factors for the origin and distribution of igneous rock reservoirs have been worked out based on the systematic case studies, which are as follows: (1) The classification is given based on multi-factors as the origin type, litho-phase, type of reservoir pore, reservoir ability etc., (2) Each type of reservoir was characterized in a detailed way; (3) There are 7 factors dominated the intrusive reservoir's characteristics including depth of intrusion, litho-facies of surrounding rocks, thickness of intrusive rock, intrusive facies, frequency and size of the working faults, shape and tectonic deformation of rock, erosion strength of the rock and the time of the intrusion ect., in the contrast, 4 factors are for eruptive rocks as volcanic facies, frequency and size of the working faults, strength of erosion and the thermal water processing. 5. Several new concept including "reservoir litho-facies", "composite-volcanic facies" and "reservoir system" ect. Were suggested, based on which the following models were established: (1) A seven reservoir belts model for a intrusive unit profile and further more, (2) a three layers cubic model consisted of three layer as "metamorphic roe layer", "marginal layer" and "the core"; (3) A five zones vertical reservoir sequence model consisted of five litho-facies named A, B, C, D and E for a original lava unit and furthermore three models respectively for a erosion, subsidence and faulted lava unit; (4) A composite volcanic face model for a lava cone or a composite cone that is consisted of three facies as "crater and nearby face", "middle slope" and "far slope", among which, the middle slope face is the most potential reservoir area and producible for oil & gas. 6. The concept of "igneous reservoir" was redefined as the igneous, and then a new concept of "igneous reservoir system" was supposed which means the reservoir system consisted of igneous and associated non-igneous reservoirs, with non-hydrocarbon reservoir included. 7. The origin and distribution of igneous reservoir system were probed and generalized for the exploration applications, and origin models of the main reservoir sub-systems have been established including those of igneous, related non-igneous and non-hydrocarbon. For intrusive rocks, two reservoir formation models have been suggested, one is called "Original or Primary Model", and the another one is "Secondary Model"; Similarly, the eruptive rock reservoirs were divided in three types including "Highly Produced", "Moderately Produced" and "Lowly Produced" and accordingly their formation models were given off; the related non-igneous reservoir system was considered combination of eight reservoirs, among which some ones like the Above Anticline Trap are highly produced; Also, the non-hydrocarbon. Trap system including five kinds of traps was discussed. 8. The concept models for four reservoir systems were suggested, which include the intrusive system consisted of 7 kinds of traps, the land eruptive system with 6 traps, the under water eruptive system including 6 kinds of traps and the non-hydrocarbon system combined by 5 kinds of traps. In this part, the techniques for exploration of igneous reservoir system were also generalized and probed, and based on which and the geological progresses of this paper, the potential resources and distributions of every reservoir system was evaluated and about 186 millions of reserves and eight most potential non-hydrocarbon areas were predicted and outlined. The author believe that the igneous reservoir system is a very important exploration area and its study is only in its early stage, the framework of this paper should be filled with more detailed studies, and only along way, the exploration of igneous reservoir system can go into it's really effective stage.

三维油藏数值模拟正反演算法研究

The dynamic prediction of complex reservoir development is one of the important research contents of dynamic analysis of oil and gas development. With the increase development of time, the permeabilities and porosities of reservoirs and the permeability of block reservoir at its boundaries are dynamically changing. How to track the dynamic change of permeability and porosity and make certain the permeability of block reservoir at its boundary is an important practical problem. To study developing dynamic prediction of complex reservoir, the key problem of research of dynamic prediction of complex reservoir development is realizing inversion of permeability and porosity. To realize the inversion, first of all, the fast forward and inverse method of 3-dimension reservoir simulation must be studied. Although the inversion has been widely applied to exploration and logging, it has not been applied to3-dimension reservoir simulation. Therefore, the study of fast forward and inverse method of 3-dimension reservoir simulation is a cutting-edge problem, takes on important realistic signification and application value. In this dissertation, 2-dimension and 3-dimension fluid equations in porous media are discretized by finite difference, obtaining finite difference equations to meet the inner boundary conditions by Peaceman's equations, giving successive over relaxation iteration of 3-dimension fluid equations in porous media and the dimensional analysis. Several equation-solving methods are compared in common use, analyzing its convergence and convergence rate. The alternating direction implicit procedure of 2-dimension has been turned into successive over relaxation iteration of alternating direction implicit procedure of 3-dimension fluid equations in porous media, which possesses the virtues of fast computing speed, needing small memory of computer, good adaptability for heterogeneous media and fast convergence rate. The geological model of channel-sandy reservoir has been generated with the help of stochastic simulation technique, whose cross sections of channel-sandy reservoir are parabolic shapes. This method makes the hard data commendably meet, very suit for geological modeling of containing complex boundary surface reservoir. To verify reliability of the method, theoretical solution and numerical solution are compared by simplifying model of 3-dimension fluid equations in porous media, whose results show that the only difference of the two pressure curves is that the numerical solution is lower than theoretical at the wellbore in the same space. It proves that using finite difference to solve fluid equations in porous media is reliable. As numerical examples of 3-dimension heterogeneous reservoir of the single-well and multi-well, the pressure distributions have been computed respectively, which show the pressure distributions there are clearly difference as difference of the permeabilities is greater than one order of magnitude, otherwise there are no clearly difference. As application, the pressure distribution of the channel-sandy reservoir have been computed, which indicates that the space distribution of pressure strongly relies on the direction of permeability, and is sensitive for space distributions of permeability. In this dissertation, the Peaceman's equations have been modified into solving vertical well problem and horizontal well problem simultaneously. In porous media, a 3D layer reservoir in which contain vertical wells and horizontal wells has been calculated with iteration. For channel-sandy reservoir in which there are also vertical wells and horizontal wells, a 3D transient heterogeneous fluid equation has been discretized. As an example, the space distribution of pressure has been calculated with iteration. The results of examples are accord with the fact, which shows the modification of Peaceman's equation is correct. The problem has been solved in the space where there are vertical and horizontal wells. In the dissertation, the nonuniform grid permeability integration equation upscaling method, the nonuniform grid 2D flow rate upscaling method and the nonuniform grid 3D flow rate upscaling method have been studied respectively. In those methods, they enhance computing speed greatly, but the computing speed of 3D flow rate upscaling method is faster than that of 2D flow rate upscaling method, and the precision of 3D flow rate upscaling method is better than that of 2D flow rate upscaling method. The results also show that the solutions of upscaling method are very approximating to that of fine grid blocks. In this paper, 4 methods of fast adaptive nonuniform grid upscaling method of 3D fluid equations in porous media have been put forward, and applied to calculate 3D heterogeneous reservoir and channel-sandy reservoir, whose computing results show that the solutions of nonuniform adaptive upscaling method of 3D heterogeneous fluid equations in porous media are very approximating to that of fine grid blocks in the regions the permeability or porosity being abnormity and very approximating to that of coarsen grid blocks in the other region, however, the computing speed of adaptive upscaling method is 100 times faster than that of fine grid block method. The formula of sensitivity coefficients are derived from initial boundary value problems of fluid equations in porous media by Green's reciprocity principle. The sensitivity coefficients of wellbore pressure to permeability parameters are given by Peaceman's equation and calculated by means of numerical calculation method of 3D transient anisotropic fluid equation in porous media and verified by direct method. The computing results are in excellent agreement with those obtained by the direct method, which shows feasibility of the method. In the dissertation, the calculating examples are also given for 3D reservoir, channel-sandy reservoir and 3D multi-well reservoir, whose numerical results indicate: around the well hole, the value of the sensitivity coefficients of permeability is very large, the value of the sensitivity coefficients of porosity is very large too, but the sensitivity coefficients of porosity is much less than the sensitivity coefficients of permeability, so that the effect of the sensitivity coefficients of permeability for inversion of reservoir parameters is much greater than that of the sensitivity coefficients of porosity. Because computing the sensitivity coefficients needs to call twice the program of reservoir simulation in one iteration, realizing inversion of reservoir parameters must be sustained by the fast forward method. Using the sensitivity coefficients of permeability and porosity, conditioned on observed valley erosion thickness in wells (hard data), the inversion of the permeabilities and porosities in the homogeneous reservoir, homogeneous reservoir only along the certain direction and block reservoir are implemented by Gauss-Newton method or conjugate gradient method respectively. The results of our examples are very approximating to the real data of permeability and porosity, but the convergence rate of conjugate gradient method is much faster than that of Gauss-Newton method.

Electro-Osmotic Flow and Mixing in Heterogeneous Microchannels

Analytical and numerical studies of secondary electro-osmotic flow EOF and its mixing in microchannels with heterogeneous zeta potentials are carried out in the present work. The secondary EOFs are analyzed by solving the Stokes equation with heterogeneous slip velocity boundary conditions. The analytical results obtained are compared with the direct numerical simulation of the Navier-Stokes equations. The secondary EOFs could transport scalar in larger areas and increase the scalar gradients, which significantly improve the mixing rate of scalars. It is shown that the heterogeneous zeta potentials could generate complex flow patterns and be used to enhance scalar mixing.

Intensity Distribution Design for Laser-Induced Thermal Loading Based on Numerical Simulation

To accomplish laser-induced thermal loading simulation tests for pistons,the Gaussian beam was modulated into multi-circular beam with specific intensity distribution.A reverse method was proposed to design the intensity distribution for the laser-induced thermal loading based on finite element(FE) analysis.Firstly,the FE model is improved by alternating parameters of boundary conditions and thermal-physical properties of piston material in a reasonable range,therefore it can simulate the experimental resul...

A Study of Composite Beam with Shape Memory Alloy Arbitrarily Embedded under Thermal and Mechanical Loadings

The constitutive relations and kinematic assumptions on the composite beam with shape memory alloy (SMA) arbitrarily embedded are discussed and the results related to the different kinematic assumptions are compared. As the approach of mechanics of materials is to study the composite beam with the SMA layer embedded, the kinematic assumption is vital. In this paper, we systematically study the kinematic assumptions influence on the composite beam deflection and vibration characteristics. Based on the different kinematic assumptions, the equations of equilibrium/motion are different. Here three widely used kinematic assumptions are presented and the equations of equilibrium/motion are derived accordingly. As the three kinematic assumptions change from the simple to the complex one, the governing equations evolve from the linear to the nonlinear ones. For the nonlinear equations of equilibrium, the numerical solution is obtained by using Galerkin discretization method and Newton-Rhapson iteration method. The analysis on the numerical difficulty of using Galerkin method on the post-buckling analysis is presented. For the post-buckling analysis, finite element method is applied to avoid the difficulty due to the singularity occurred in Galerkin method. The natural frequencies of the composite beam with the nonlinear governing equation, which are obtained by directly linearizing the equations and locally linearizing the equations around each equilibrium, are compared. The influences of the SMA layer thickness and the shift from neutral axis on the deflection, buckling and post-buckling are also investigated. This paper presents a very general way to treat thermo-mechanical properties of the composite beam with SMA arbitrarily embedded. The governing equations for each kinematic assumption consist of a third order and a fourth order differential equation with a total of seven boundary conditions. Some previous studies on the SMA layer either ignore the thermal constraint effect or implicitly assume that the SMA is symmetrically embedded. The composite beam with the SMA layer asymmetrically embedded is studied here, in which symmetric embedding is a special case. Based on the different kinematic assumptions, the results are different depending on the deflection magnitude because of the nonlinear hardening effect due to the (large) deflection. And this difference is systematically compared for both the deflection and the natural frequencies. For simple kinematic assumption, the governing equations are linear and analytical solution is available. But as the deflection increases to the large magnitude, the simple kinematic assumption does not really reflect the structural deflection and the complex one must be used. During the systematic comparison of computational results due to the different kinematic assumptions, the application range of the simple kinematic assumption is also evaluated. Besides the equilibrium study of the composite laminate with SMA embedded, the buckling, post-buckling, free and forced vibrations of the composite beam with the different configurations are also studied and compared.

Strain gradient theory with couple stress for crystalline solids

A new phenomenological strain gradient theory for crystalline solid is proposed. It fits within the framework of general couple stress theory and involves a single material length scale Ics. In the present theory three rotational degrees of freedom omega (i) are introduced, which denote part of the material angular displacement theta (i) and are induced accompanying the plastic deformation. omega (i) has no direct dependence upon u(i) while theta = (1 /2) curl u. The strain energy density omega is assumed to consist of two parts: one is a function of the strain tensor epsilon (ij) and the curvature tensor chi (ij), where chi (ij) = omega (i,j); the other is a function of the relative rotation tensor alpha (ij). alpha (ij) = e(ijk) (omega (k) - theta (k)) plays the role of elastic rotation reason The anti-symmetric part of Cauchy stress tau (ij) is only the function of alpha (ij) and alpha (ij) has no effect on the symmetric part of Cauchy stress sigma (ij) and the couple stress m(ij). A minimum potential principle is developed for the strain gradient deformation theory. In the limit of vanishing l(cs), it reduces to the conventional counterparts: J(2) deformation theory. Equilibrium equations, constitutive relations and boundary conditions are given in detail. For simplicity, the elastic relation between the anti-symmetric part of Cauchy stress tau (ij), and alpha (ij) is established and only one elastic constant exists between the two tensors. Combining the same hardening law as that used in previously by other groups, the present theory is used to investigate two typical examples, i.e., thin metallic wire torsion and ultra-thin metallic beam bend, the analytical results agree well with the experiment results. While considering the, stretching gradient, a new hardening law is presented and used to analyze the two typical problems. The flow theory version of the present theory is also given.