NONLINEAR SATURATION OF BAROCLINIC INSTABILITY IN THE GENERALIZED PHILLIPS MODEL(Ⅱ)-THE LOWER BOUND ON THE DISTURBANCE ENERGY AND POTENTIAL ENSTROPHY TO THE NONLINEARLY UNSTABLE BASIC FLOW

On the basis of the nonlinear stability theorem in the context of Arnol’d’s second theorem for the generalized Phillips model,nonlinear saturation of baroclinic instability in the generalized Phillips model is investigatedThe lower bound on the disturbance energy and potential enstrophy to the nonlinearly unstable basic flow in the generalized Phillips model is presented,which indicates that there may exist an allocation between a nonlinearly unstable basic flow and a growing disturbance     

STUDY OF THE EQUIVALENT THEOREM OF GENERALIZED VARIATIONAL PRINCIPLES IN ELASTICITY

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.     

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of planeproblems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with fouredges built-in as the basic system and then find displacement expressions of the actualsystem by using the reciprocal theorem between the basic system and actual system withvarious edge conditions.When only displacement edge conditions exist,obtaining displacement expressions bymeans of the method of reciprocal theorem is actual.But in other conditions,when staticforce edge conditions or mixed ones exist,the obtained displacements are admissible.Inorder to find actual displacement,the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple,convenient andgeneral one for the solution of plane problems of elasticity of the rectangular plates withvarious edge conditions.Evidently,it is a new method.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

ON THE LAW OF ENERGY CONSERVATION IN THE CLASSICAL ELECTRODYNAMICS OF DEFORMABLE MEDIA——A GENERALIZATION OF THE POYNTING THEOREM

This paper establishes the generalized Poynting theorem for the electrodynamics of deformable media with a view to shedding some light on the detailed mechanism of energy transfer between the electromagnetic field and the deformable media. Global field equations are chosen as the starting point and specialized forms of the theorem are derived based on the special postulates for the electromagnetic body force.     

ON THE EMBEDDING AND COMPACT PROPERTIES OF FINITE ELEMENT SPACES

In this paper, the generalized Sobolev embedding theorem and the generalized Rellich-Kondrachov compact theorem for finite element spaces with multiple sets of functions are established. Specially, they are true for nonconforming, hybrid and quasi-conforming element spaces with certain conditions.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied.This study is a forward development of the study of thestability in the Liapunov sense;furthermore,it is of significance in practice sinceperturbations are often not single in the time domain.Malk in proved a general theoremabout thesubject.To apply the theorem,however,the user has to construct a Liapunovfunction which satisfies specified conditions and it is difficult to find such a function fornonlinear nonautonomous systems.In the light of the principle of Liapunov’s indirectmethod,which is an effective method to decide the stability of nonlinear systems in theLiapunov sense,the authors have achieved several important conclusions expressed in theform of theorems to determine the stability of nonlinear nonautonomous systems under thefrequently-acting perturbation.     

ON THE COMPACTNESS OF QUASI-CONFORMING ELEMENT SPACES AND THE CONVERGENCE OF QUASI-CONFORMING ELEMENT METHOD

In this paper, the compactness of quasi-conforming element spaces and theconvergence of quasi-conforming element method are discussed The well-known Rellichcompactness theorem is generalized to the sequences of quasi-conforming element spaceswith certain properties. and the generalized Poincare inequality.The generalized Friedrichsinequality and the generalized inequality of Poincare-Friedrichs are proved true for them.The error estimates are also given. It is shown that the quasi-conforming element method isconvergent if the quasi-conforming element spaces have the approximability and the strongcontinuity, and satisfy the rank condition of element and pass the test IPT As practicalexamples, 6-paramenter. 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and21-paramenter quasi-conforming elements are shown to be convergent, and theirL~(2‘2)(Ω)-errors are O(h_τ)、 O(h_τ)、O(h_τ~2)、O(h_τ~2), O(h_τ~3), and O(h_τ~4) respectively.     

The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model

The nonlinear thermoelastic responses of an elastic medium exposed to laser generated shortpulse heating are investigated in this article. The thermal wave propagation of generalized thermoelastic medium under the impact of thermal loading with energy dissipation is the focus of this research. To model the thermal boundary condition(in the form of thermal conduction),generalized Cattaneo model(GCM) is employed. In the reference configuration, a nonlinear coupled Lord-Shulman-type generalized thermoelasticity formulation using finite strain theory(FST) is developed and the temperature dependency of the thermal conductivity is considered to derive the equations. In order to solve the time-dependent and nonlinear equations, Newmark's numerical time integration technique and an updated finite element algorithm is applied and to ensure achieving accurate continuity of the results, the Hermitian elements are used instead of Lagrangian's. The numerical responses for different factors such as input heat flux and nonlinear terms are expressed graphically and their impacts on the system's reaction are discussed in detail.The results of the study are presented for Green–Lindsay model and the findings are compared with Lord-Shulman model especially with regards to heat wave propagation. It is shown that the nature of the laser's thermal shock and its geometry are particularly determinative in the final stage of deformation. The research also concluded that employing FST leads to achieving more accuracy in terms of elastic deformations; however, the thermally nonlinear analysis does not change the results markedly. For this reason, the nonlinear theory of deformation is required in laser related reviews, while it is reasonable to ignore the temperature changes compared to the reference temperature in deriving governing equations.     

Generalization of the method of full approximation and its applications

This paper presents a generalized form of the method of full approximation.By usingthe concept of asymptotic linearization and making the coordinate transformationsincluding the nonlinear functionals of dependent variables,the original nonlinear problemsare linearized and their higher-order solutions are given in terms of the first-termasymptotic solutions and corresponding transformations.The analysis of a model equationand some problems of weakly nonlinear oscillations and waves with the generalized methodshows that it is effective and straightforward.     

GENERALIZED NOETHER’S THEOREM OF NONHOLONOMIC NONPOTENTIAL SYSTEM IN NONINERTIAL REFERENCE FRAMES

The new Lagrangian of the relative motion of mechanical system is constructed,thevariational principles of Jourdain’s form of nonlinear nonholonomic nonpotential system innoninertial reference frame are established,the generalized Noether’s theorem of thesystem above is presented and proved,and the conserved quantities of system are studied.     

APPLICATION OF THE METHOD OF THE RECIPROCAL THEOREM TO FINDING DISPLACEMENT SOLUTIONS OF CUBES

In this paper the method of reciprocal theorem is extended to find solutions of three-Dproblems of elasticity.First we give the basic solution of the cube with six surfaces fixed as the basic systemand then using the reciprocal theorem between the basic system acted on by unitconcentrated loads and the actnal system with Precribed surface displacements,we finddisplacement solution of the actual system.     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

THEOREMS ON THE LIMIT ANALYSIS OF FINITE DEFORMATION

A generalized variational principle (theorem 1) which is equivalent mathematically to the whole set of equations and conditions and must be satisfied by the limit analysis of finite deformation is proposed in this paper. It is also proved that the limit load deduced from theorem 1 will lie between the lower and upper bounds given by the bound theorems of finite deformation.     

EXTENDING THE KNOWLEDGE BASE OF CHEMICAL ENGINEERING

The obvious current reversion to micro-scale investigations in basic chemical engineering, combined with the need, of a quite different nature, in the rapid growth of high added-value and small-lot functional materials, have been pointing to an area not yet sufficiently covered by the unit operations, transport phenomena and chemical reaction engineering. Although it is difficult to define accurately this area, a cursory scan of the activities already in progress has revealed a few common attributes: multi-phased (structured), multi-scaled, multi-disciplined, nonlinear, needs for resolution to reductionism-solvable subsystems, and pervasive in the process industry. From these activities, the present paper drafts a tentative scheme for studying the related problems: first to dissect a problem into various scales -- spatial, temporal or otherwise as best suits the case in hand -- in order to identify pertinent parameters which are then organized into model formulations. Together with inter-scale model formulations, a zoom-in/zoom-out process is carried out between the scales, by trial-and-error and through reasoning, to arrive at a global formulation of a quantitative solution, in order to derive, eventually, the general from the particular.     

A universal bifurcation mechanism arising from progressive hydroelastic waves

A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schr?dinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.     

A CRITERION FOR THE STABILITY OF MOTION OF NONLINEAR SYSTEM

As to an autonomous nonlinear system, the stability of the equilibrium state in a sufficiently small neighborhood of the equilibrium state can be determined by eigenvalues of the linear part of the nonlinear system provided that the eigenvalues are not in a critical case. Many methods may be used to detect the stability for a linear system. A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems. Till now, none of these methods can be conveniently applied to all nonlinear systems. The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system. The familiar Krasovski's method is a special case of this method     

THE EXISTENCE AND UNIQUENESS OF THE SCREW TENSOR FOR THE FINITE DISPLACEMENT OF A RIGIDBODY

The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a consequence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.     

BASIC THEOREMS IN LINEAR MICROMORPHIC THERMOELECTROELASTICITY AND THEIR PRIMARY APPLICATION

As a natural extension of the micromorphic continuum theory, the linear theory of micromorphic thermoelectroelasticity is developed to characterize the nano-micro scale behavior of thermoelectroelastic materials with remarkable microstructures. After the basic governing equations are given and the reciprocal theorem is deduced, both the generalized variational principle and the generalized Hamilton principle for mixed boundary-initial value problems of micromorphic thermoelectroelastodynamics in convolution form are established. Finally, as a primary application, steady state responses of an unbounded homogeneous isotropic micromorphic thermoelectroelastic body to external concentrated loads with mechanical, electric, and thermal origins are analyzed.     

THE DYNAMIC ANALYSIS OF THE VASOMOTION IN MICROCIRCULATION

The vasomotion of the microcirculation of the microcirculation which has bee observed in experiments are studied with a simple model of the fluid dynamics. The influence of the vasomotion on the blood flow in microvessels is analysed and the strecmlines, velocity and pressurc distribution and the variation of thc flow rate are presented.