Theories and applications of earthquake-induced gravity variation:Advances and perspectives

Earthquake-induced gravity variation refers to changes in the earth’s gravity field associated with seismic activities.In recent years,development in the theories has greatly promoted seismic deformation research,laying a solid theoretical foundation for the interpretation and application of seismological gravity monitoring.Traditional terrestrial gravity measurements continue to play a significant role in studies of interseismic,co-seismic,and post-seismic gravity field variations.For instance,superconducting gravimeter networks can detect co-seismic gravity change at the sub-micro Gal level.At the same time,the successful launch of satellite gravity missions(e.g.,the Gravity Recovery and Climate Experiment or GRACE)has also facilitated applied studies of the gravity variation associated with large earthquakes,and several remarkable breakthroughs have been achieved.The progress in gravity observation technologies(e.g.,GRACE and superconducting gravimetry)and advances in the theories have jointly promoted seismic deformation studies and raised many new research topics.For example,superconducting gravimetry has played an important role in analyses of episodic tremor,slow-slip events,and interseismic strain patterns;the monitoring of transient gravity signals and related theories have provided a new perspective on earthquake early warning systems;the mass transport detected by the GRACE satellites several months before an earthquake has brought new insights into earthquake prediction methods;the use of artificial intelligence to automatically identify tiny gravity change signals is a new approach to accurate and rapid determination of earthquake magnitude and location.Overall,many significant breakthroughs have been made in recent years,in terms of the theory,application,and observation measures.This article summarizes the progress,with the aim of providing a reference for seismologists and geodetic researchers studying the phenomenon of gravity variation,advances in related theories and applications,and future research directions in this discipline.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes （MWCNTs） are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton＇s principle and Rayleigh＇s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes （SWCNTs） and double-walled carbon nanotubes （DWCNTs） with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

THE VARIATIONAL INEQUALITY FORMULATION FOR THE DEFORMATION THEORY IN PLASTICITY AND ITS NON-ITERATIVE SOLUTION

In this paper, the deformation theory in plasticity is formulated in the variational inequality, which can relax the constraint conditions of the constitutive equations. The new form makes the calculation more convenient than general energy forms and have reliable mathematical basis. Thus the plasticity theory may be solved by means of the quadratic programming instead of the iterative methods. And the solutions can be made in one step without any diversion of the load.

CONTACT PROBLEMS AND DUAL VARIATIONAL INEQUALITY OF 2－D ELASTOPLASTIC BEAM THEORY

In Order to study the frictional contact problems of the elastoplastic beam theory,an extended two-dimensional beam model is established, and a second order nonlinear equilibrium problem with both internal and external complementarity conditions is proposed. The external complementarity condition provides the free boundary condition. while the internal complemententarity condition gives the interface of the elastic and plastic regions. We prove that this bicomplementarity problem is equivalent to a nonlinear variational inequality The dual variational inequality is also developed.It is shown that the dual variational inequality is much easier than the primalvariational problem. Application to limit analysis is illustrated.

Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation

A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.

GURTIN-TYPE VARIATIONAL PRINCIPLES FOR DYNAMICS OF A NON-LOCAL THERMAL EQUILIBRIUM SATURATED POROUS MEDIUM

Based on the porous media theory and by taking into account the efects of the pore fuid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fuid phases, a mathematical model for the dynamic-thermo-hydro-mechanical coupling of a non-local thermal equilibrium fuid-saturated porous medium, in which the two constituents are assumed to be incompressible and immiscible, is established under the assumption of small de- formation of the solid phase, small velocity of the fuid phase and small temperature changes of the two constituents. The mathematical model of a local thermal equilibrium fuid-saturated porous medium can be obtained directly from the above one. Several Gurtin-type variational principles, especially Hu-Washizu type variational principles, for the initial boundary value problems of dy- namic and quasi-static responses are presented. It should be pointed out that these variational principles can be degenerated easily into the case of isothermal incompressible fuid-saturated elastic porous media, which have been discussed previously.

ANALYTICAL VARIATIONAL METHOD OF SOLUTION FOR STRESS CONCENTRATION FACTORS OF FINITE PLATES WITH A PIN HOLE

First of all, Laurent series expansions of stress and displacement fields satisfying all the governing equations of plane problems in theory of elasticity are derived. Next, the variational method is applied to satisfy the boundary conditions of a plate with a pin hole. Then, the coefficients in Laurent series and the stress concentration factor can be determined. Finally, the convergency tests and systematical results are given for engineering applications.

Modeling and solving for transverse vibration of gear with variational thickness

A analyzed model of gear with wheel hub, web and rim was derived from the Mindlin moderate plate theory. The gear was divided into three annular segments along the locations of the step variations. Traverse displacement, rotation angle, shear force and fiexural moment were equal to ensure the continuity along the interface of the wheel hub, web and rim segments. The governing differential equations for harmonic vibration of annular segments were derived to solve the gear vibration problem. The influence of hole to diameter ratios, segment thickness ratios, segment location ratios, Poisson ratio on the vibration behavior of stepped circular Mindlin disk were calculated, tabletted and plotted. Comparisons were made with the frequencies arising from the presented method, finite elements method, and structure modal experiment. The result correlation among these three ways is very good. The largest error for all frequencies is 5.46%, and less than 5% for most frequencies.

An FEM approximation for a fourth-order variational inequality of second kind

A fourth-order variational inequality of the second kind arising in a plate frictional bending problem is considered. By using regularization method, the original problem can be formulated as a differentiable variational equation, and the corresponding discrete FEM variational equation is presented afterwards. Abstract error estimates and error estimates of the approximation are derived in terms of energy norm and L^2-norm.

A Generalized Variational Principle for 2-D Piezoelectricity with Surface Electrodes

It is difficult to establish a classical variational model for piezoelectricity. Following the semi inverse method of establishing generalized variational principles, an energy like trial functional with a certain unknown function is constructed. The unknown function is easily identified step by step. A family of variational principles for the static behavior of the elastic and electric variables in the vicinity of a surface electrode attached to a piezoelectric ceramic is established directly from its field equations and boundary conditions.

Multiplicity Results for Second Order Impulsive Differential Equations via Variational Methods

In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png" width="40" height="17" alt="" /> and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.

VARIATIONAL INEQUALITY IN ELASTOPLASTIC PROBLEMS

In this paper,the incremental theory of an elastoplastic problem is trans-formed to a variational inequality.The existence and uniqueness of the variationalinequality are proved for the materials in which the Drucker postulate is satisfied.And theconvergence of the finite element solutions is also discussed.

Variational principle for a special Cosserat rod

In this paper, we describe the nonlinear models of a rod in three-dimensional space based on the Cosserat theory. Using the pseudo-rigid body method and variational principle, we obtain the motion equations of a Cosserat rod including shear deformations.

Transmissivity of electromagnetic wave propagating in magnetized plasma sheath using variational method

A variational method is introduced to analyze the transmissivity of an electromagnetic wave propagating in the magnetized plasma sheath. The plasma density is modeled by two parabolic inhomogeneous regions separated by one homogeneous region. The Lagrangian density of the system is constructed based on the fluid energy density and the electromagnetic energy density.The total variation of the Lagrangian density is derived. The fluid and electromagnetic fields are numerically solved by expansion in piecewise polynomial function space. We investigate the effect of an external magnetic field on the transmissivity of the electromagnetic wave. It is found that the transmissivity is increased when an external magnetic field is applied. The dependence of transmissivity on the collision frequency between the electrons and the neutral particles has also been studied. We also show that the external magnetic field causes a shift in the critical frequency of the plasma sheath.

VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM

Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets. This description is based on the continuum model of the graphene sheets which can also take the small scale effects into account by employing a nonlocal theory. In the present article a variational principle is derived for the nonlocal elastic theory of rectangular graphene sheets embedded in an elastic medium and undergo- ing transverse vibrations. Moreover the graphene sheets are subject to biaxial compression. Rayleigh quotients are obtained for the frequencies of freely vibrating graphene sheets and for the buckling load. The influence of small scale effects on the frequencies and the buckling load can be observed qualiatively from the expressions of the Rayleigh quotients. Elastic medium is modeled as a combination of Winkler and Pasternak foundations acting on the top and bottom layers of the mutilayered nano-structure. Natural boundary con- ditions of the problem are derived using the variational principle formulated in the study. It is observed that free boundaries lead to coupled boundary conditions due to nonlocal theory used in the continuum formulation while the local （classical） elasticity theory leads to uncoupled boundary conditions. The mathematical methods used in the study involve calculus of variations and the semi-inverse method for deriving the variational integrals.

BASIC EQUATIONS, THEORY AND PRINCIPLE OF COMPUTATIONAL STOCK MARKET (Ⅲ)—BASIC THEORIES

By basic equations, two basic theories are presented: 1.Theory of stock's value v *(t)=v *(0) exp (ar * 2t); 2. Theory of conservation of stock's energy. Let stock's energy be defined as a quadratic function of stock's price v and its derivative , =Av 2+ Bv+C 2+Dv, under the constraint of basic equation, the problem was reduced to a problem of constrained optimization along optimal path. Using Lagrange multiplier and Euler equation of variation method, it can be proved that keeps conservation for any v,. The application of these equations and theories on judgement and analysis of tendency of stock market are given, and the judgement is checked to be correct by the recorded tendency of Shenzhen and Shanghai stock markets.

SOME GENERAL THEOREMS AND GENERALIZED AND PIECEWISE GENERALIZED VARIATIONAL PRINCIPLES FOR LINEAR ELASTODYNAMICS

From the concept of four-dimensional space and under the four kinds of time limit conditions, some general theorems for elastodynamics are developed, such as the principle of possible work action, the virtual displacement principle, the virtual stress-momentum principle, the reciprocal theorems and the related theorems of time terminal conditions derived from it. The variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are obtained. Hamilton's principle, Toupin's work and the formulations of Ref. [5], [17]-[24] may be regarded as some special cases of the general principles given in the paper. By considering three cases: piecewise space-time domain, piecewise space domain, piecewise time domain, the piecewise variational principles including the potential, the complementary and the mixed energy action fashions are given. Finally, the general formulation of piecewise variational principles is derived. If the time dimension is not considered, the formulations obtained in the paper will become the corresponding ones for elastostatics.

SUBSPACE VARIATIONAL PRINCIPLE OF RODS AND SHELLS

This paper builds the general forms of subspace variational principles of rods and shells which are taken as the controlled equations of the constitutive theories developed front the three-dimensional (non-polar) continuum mechanics. And the constitutive equations of rods and shells using the principles are satisfactory.

A UNIVERSAL VARIATIONAL FORMULATION FOR TWO DIMENSIONAL FLUID MECHANICS

A universal variational formulation for two dimensional fluid mechanics is obtained. which is subject to the so-called parameter-constrained equations (the relationship between parameters in two governing equations). By eliminating the constraints the generalized variational principle (GVPs) can be readily derived from the formulation The formulation can be applied to any conditions in case the governing equations can be converted into conservative forms. Some illustrative examples are given to testify the effectiveness and simplicity of the method.

Stream Function Wave Derived by Unified Variational Principle of Water Gravity Wave

Based on the linear wave, solitary wave and fifth order stokes wave derived by use of the Unified Variational Principle of Water Gravity Wave (UVPWGW), this paper derives stream function wave theory by using UVPWGW. This paper will handle the Kinematic Free Surface Boundary Condition (KFSBC) and Dynamic Free Surface Boundary Condition (DFSBC) directly and give the optimum solution, instead of the conditions Sigma(Q(av) - Q(i))(2) = min, and the related equations of stational condition. When the wave height H, period T and water depth D are given, the original stream function wave will be determined, and can not be adjusted if it does not agree with the real case; in the present method, the adjustment can be done by adding several constraint conditions, for example, the wave profile can be adjusted according to the condition of accurate peak position. The examples given in this paper show that for the original stream function wave, the DFSBC can be fairly well satisfied, but the KFSBC can not; however, the stream function wave derived by UVPWGW is better than the original one in the sense of minimum error squares in the aspect of the level at which KFSBC and DFSBC are satisfied.