关于非局部场论的两点注记

研究了非局部场论中尚未完全解决的两个基本问题：其一为局部化体力，力矩残余之间的相关性，由此得到了一个描述两者关系的定理；其二为线性非局部弹性理论的应力边界条件的提法；文中所得到的应力边界条件不仅解释了在裂纹混合边界值问题中线性非局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以给出裂纹尖端的分子内聚力模型。     

关于线性非局部弹性理论的应力边界条件

应力边界条件的提法是线性非局部弹性理论尚未解决的一个理论问题。文中针对这一问题进行了研究，所导出的应力边界条件包含了物体微观结构的长程相互作用，这个结果不仅解释了在裂纹混合边界值问题中非线性局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的分子内聚力模型。     

线性非局部弹性理论的修正

本文通过考虑局部化残余力的影响对线性非局部弹性理论进行了修正，由修正后的理论所导出的应力边界条件包含了物体微观结构的长程力的作用，这个结果不仅解释了在裂纹混合边界值问题中线性非局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的Ｂａｒｅｎｂｌａｔｔ分子内聚力模型。     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS (Ⅰ)──FUNDAMENTAL THEORY

In the linear nonlocal elasticity theory, the solution to the boundary-value problem of the crack with a constant stress boundary condition does not exist. This problem has been studied in this paper. The contents studied contain of examining objectivity of the energy balance, deducing the constitutive equations of nonlocal thermoelastic bodies, and determining nonlocal force and the linear nonlocal elasticity theory. Some new results are obtained. Among them, the stress boundary condition derived from the linear theory not only solves the problem mentioned at the beginning, but also contains the model of molecular cohesive stress on the sharp crack tip advanced by Barenblatt.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS (Ⅰ)──FUNDAMENTAL THEORY

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Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

The Stress–Strain State of Geometrically and Physically Nonlinear Plates with One-Sided Corrosive Wear

A technique is proposed to investigate one-sided corrosive wear. The problem is solved with regard for geometric and physical nonlinearity. Two, Dolinskii's and Gutman's corrosion models are considered. The quasistatic problem is solved by the method of variational iterations, which reduce ordinary differential equations to a system of nonlinear equations with approximation o(h ²) to be solved by Newton's method. At each step, to allow for physical nonlinearity, the method of variable elastic parameters is used. Also a technique is developed to consider various boundary conditions and ⁱ(e ⁱ) diagrams. Specific numerical results are presented.     

Application of complex path-independent integrals to the solution of the problem of a straight crack in a finite plane isotropic elastic medium

The classical problem of a straight crack in a finite, plane, isotropic, elastic medium of arbitrary shape is reconsidered by the well-known method of Muskhelishvili for such a crack (but in an infinite medium). Both the crack and the boundary of the medium are assumed loaded in an arbitrary way. It is shown that this problem can be completely solved if the numerical values of the first complex potential (z) of Muskhelishvili are known along a closed contour surrounding the crack, probably along the boundary of the medium. To this end, complex path-independent integrals associated with (z) and Chebyshev polynomials have been used. Numerical results for the stress intensity factors are displayed in an application. Generalizations of the method are also proposed and the second fundamental crack problem, the problem of a crack in an anisotropic medium and the problem of an interface crack between two isotropic media are considered in some detail.     

Estimation of three-phase relative permeabilities

Starting from the statistical structural model of Alemánet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).Notation a free parameter in Equation (19) - B(m, n) Beta function defined by Equation (17) - F ^(w), F^(nw) defined by Equations (31) and (27), respectively - G ⁽ⁱ⁾ defined by Equations (37) and (39) - H ⁽ⁱ⁾ defined by Equations (38) and (40) - k ⁽ⁱ⁾ three-phase relative permeability fo phasei - k ^(i)* defined by Equations (34) through (36) - k ^(i,j) relative permeability to phasei during a two-phase flow with phasej, possibly in the presence of an immobile phase - k ^(i,j)* defined by analogy with Equations (41) and (42) - k ^(i,j)** defined by Equations (49), (50), (53), and (54) - k _max ⁽ⁱ⁾ defined by Equation (11) - k ₁₉₇₀ ^(iw) defined by Equation (10) - k ₁₉₇₃ ^(iw)* defined by Equation (58) - k ₁₉₇₃ ^(iw) defined by Equation (13) - L length and diameter of cylindrical averaging surfaceS - L _t length of an individual capillary tube enclosed byS - L _t ^* defined by Equation (19) - L _t,min length of pore whose radius isR _max - N total number of pores contained within the averaging surfaceS - p ₁ ⁽ⁱ⁾ ,p ₂ ⁽ⁱ⁾ pressure of phasei at entrance and exit of averaging surfaceS, respectively - p defined by Equation (21) - p _c ^(i,j) capillary pressure function - p _c ^(i,j)* defined by Equations (23), (29), and (32) - p ⁽ⁱ⁾ intrinsic average of pressure within phasei defined by Alemánet al. (1988) - R pore radius - R ^* defined by Equation (18) - R _max maximum pore radius that occurs withinS - s ⁽ⁱ⁾ local saturation of phasei - s ^(i)* defined by Equation (7) - s _min ⁽ⁱ⁾ minimum or immobile saturation of phasei - S averaging surface introduced in local volume averaging - V ⁽ⁱ⁾ volume of phasei occupying the pore space enclosed byS Greek Letters , parameters in the Beta distribution defined by Equation (16) - ^(w), ^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (6) - ^(i,j) interfacial tension between phasesi andj - (x) Gamma function - defined by Equation (57) - , spherical coordinates in system centered upon the axis of the averaging surfaceS - _max maximum value of , 45 °, in view of assumption (9) - ^(i,j) contact angle between phasesi andj measured through the displacing phase - ^(w),^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (12) Other gradient operator Amoco Production Company, PO Box 591 Tulsa, OK 74102, U.S.A.     

Formulation of electro-chemico-osmotic processes in soils

Electrokinetic techniques have been used for various purposes including consolidation of soils, dewatering of sludges, and hazardous waste remediation among others. Estimating the feasibility of employing electro-osmosis in a particular operation depends on the ability to predict the outcome under a variety of conditions. Predictions of this type are frequently facilitated by the use of a mathematical model designed to represent the physical system under consideration in a rigorous fashion. First, a review of fundamental aspects of electro-chemico-osmotic flow in soils is presented. Following a brief outline of previous studies, identification and quantification of the significant processes, and the construction of mathematical representations are given. This is achieved using an approach based on the macroscopic conservation of mass equations and the principle of a continuum, in contrast to an approach based on the irreversible thermodynamics of coupled flows. Special emphasis is given to coupling effects on transport processes. A complete model and associated boundary conditions are then obtained for electrokinetic processes in a compressible porous medium. The proposed model takes into consideration the migration of a contaminant plume in a flow field generated by an applied electric potential.Symbols a _v soil compressibility - A an entity - C _w mass fraction of water component in the water phase - C _s mass fraction of chemical component in the water phase - C ^* capacitance of the porous medium per unit volume of porous volume - D mechanical dispersion coefficient - D _fw ^ps hydrodynamic diffusion tensor for the chemical component in the water phase - D _fw ^pw hydrodynamic dispersion coefficient for the water component in the water phase - D _f( )/Dt material derivative with respect to an observer moving at the water phase velocity V _f - D _s( )/Dt material derivative with respect to moving solids - e void ratio - f a function - F = 0 equation of a moving boundary - g gravitational acceleration - k permeability tensor of the porous medium - k _e coefficient of electro-osmotic permeability - k _ec coefficient of migration potential - k _hc chemico-osmotic coupling coefficient - m _i number of moles of the ith component - m _i0 number of moles of the ith component at a reference level - n porosity - p pore pressure - p _oi pore pressure at a reverence level - q specific discharge of water phase - q _e current density - q _fe ^p0 constant current density applied at a boundary - q ₀ constant flow rate - q _r specific discharge of the water phase relative to the moving solid matrix - R net mass transfer rate of the chemical component in the water phase - t time - u velocity of a moving surface - _i partial molar density of ith component - V _f velocity of the water phase - V _s velocity of the solid (rate of deformation) - x vertical coordinate - coefficient of matrix compressibility - _p compressibility of water phase in motion - total (overburden) stress tensor - effective stress tensor - _h streaming current conductivity - _e electrical conductivity - electrical potential - _f viscosity of the water phase - _hf density of the water phase     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

Analysis of near-electrode plasma perturbation region

The problem of parameter distribution in the plasma perturbation region near an electrode surface is considered on the basis of the diffusion equations.Depending on the parameter values, in the solution there may exist two regions: laminar diffusion and a space charge layer.Approximate solutions are obtained for these regions in the form of the principal term of the expansion in terms of the parameter ==(h/l _i)², where h is the Debye length andl _i is the dimension of the perturbation region. Under certain conditions these solutions may be obtained in closed form. The conditions for matching the solutions at the boundary of these regions are discussed.If the parameters defining the problem are such that h_m orl _im, where _m is the mean free path, then the plasma is separated from the electrode by a dielectric layer of free-molecular particle motion. The presence of this layer must be taken into account in formulating the boundary conditions for the continuum equations.The results are used to determine the possible parameter variation in the perturbation region for specific gas mixtures and different values of the defining parameters.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)--RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NON-LOCAL THERMOELASTIC BODIES

In this paper, nonlinear constitutive equations are deduced strictly according to the constitutive axioms of rational continuum mechanics. The existing judgments are modified and improved. The results show that the constitutive responses of nonlocal thermoelastic body are related to the curvature and higher order gradient of its material space, and there exists an antisymmetric stress whose average value in the domain occupied by thermoelastic body is equal to zero. The expressions of the antisymmetric stress and the nonlocal residuals are given. The conclusion that the directions of thermal conduction and temperature gradient are consistent is reached. In addition, the objectivity about the nonlocal residuals and the energy conservation law of nonlocal field is discussed briefly, and a formula for calculating the nonlocal residuals of energy changing with rigid motion of the spatial frame of reference is derived. Foundation item: the Natural Science Foundation of Province Jiangshu (BK97063)     

Recurrence formula for B-splines with respect to a class of differential operators

In this paper, we give recurrence formula for normalized B-Splines with respect to standard system { ^i–1(x)} _m ⁱ⁼¹ which is the basis set of solutions to a class of differential operators in view of generalized divided difference—Green function—B-Splines.     

The Evolution of Stress Intensity Factors and the Propagation of Cracks in Elastic Media

When a crack Γ _s propagates in an elastic medium the stress intensity factors evolve with the tip x(s) of Γ _s . In this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions. Denoting by ψ=ψ(x,s) the stress potential (ψ is biharmonic and has zero traction along the crack Γ _s ) and by κ(s) the curvature of the crack at the tip x(s), we prove that the stress intensity factors A ₁(s), A ₂(s), as functions of s, satisfy:

Self-similar solution for deep-penetrating hydraulic fracture propagation

The propagation of a vertical hydraulic fracture of a constant height driven by a viscous fluid injected into a crack under constant pressure, is considered. The fracture is assumed to be rectangular, symmetric with respect to the well, and highly elongated in the horizontal direction (the Perkins and Kern model). The fracturing fluid viscosity is assumed to be different from the stratum saturating fluid viscosity, and the stratum fluid displacement by a fracturing fluid in a porous medium is assumed to be piston-like. The compressibility of the fracturing fluid is neglected. The stratum fluid motion is governed by the equation of transient seepage flow through a porous medium.A self-similar solution to the problem is constructed under the assumption of the quasi-steady character of the fracturing fluid flow in a crack and in a stratum and of a locally one-dimensional character of fluid-loss through the crack surfaces. Crack propagation under a constant injection pressure is characterized by a variation of the crack sizel in timet according to the lawl(t)=l _o (1+At)^1/4, where the constantA is the eigenvalue of the problem. In this case, the crack volume isVl, the seepage volume of fracturing fluidV _f l ³, and the flow rate of a fluid injected into a crack isQ ₀l ^–1.     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Effect of hydrodynamic forces on the pressure-difference equation

Because of the influence of hydrodynamic forces, the difference in macroscopic pressure which exists, at static equilibrium, between two immiscible phases located in a porous medium may be different from that which pertains during flow. In this paper, the concept of relative pressure difference, together with a new pressure-difference equation, is used to investigate the impact that the hydrodynamic forces have on the difference in macroscopic pressure which pertains when two immiscible fluids flow simultaneously through a homogeneous, water-wet porous medium. This investigation reveals that, in general, the equation defining the difference in pressure between two flowing phases must include a term which takes proper account of the hydrodynamic effects. Moreover, it is pointed out that, while neglect of the hydrodynamic effects introduces only a small amount of error when the two fluids are flowing cocurrently, such neglect is not permissible during steady-state, countercurrent flow. This is because failure to include the impact of the hydrodynamic effects in the latter case makes it impossible to explain the pressure behaviour observed in steady-state, countercurrent flow. Finally, the results of this investigation are used as a basis for arguing that, during steady-state, countercurrent flow, saturation is uniform, as is the case of steady-state, cocurrent flow.Roman Letters a parameter in Equation (18) - k absolute permeability, m² - k _i effective permeability to phasei;i=1, 2, m² - k _ij generalized effective permeability for phasei;i, j=1, 2, m² - p _d p ₂–p ₁=difference in macroscopic pressure between two flowing phases, N/m² - p _i pressure for phasei;i=1, 2, N/m² - p _h hydrodynamic contribution to difference in macroscopic pressure which exists during flow, N/m² - P _c macroscopic static capillary pressure, N/m² - R ₁₂ function defined by Equation (18) - S _i saturation of phasei;i=1, 2 - S _n normalized saturation of phase 1 - t time, s - u _i flux of phasei;i=1, 2m³/m²/s - x distance in direction of flow, m Greek Letters _R relative pressure difference - _i k _i / _i =mobility of phasei;i=1, 2m²/Pa·s - _ij k _ij / _j =generalized mobility of phasei;i, j=1, 2m²/Pa·s - _i viscosity of phasei;i=1, 2, Pa·s - porosity     

Lattice gas simulation of Darcy flow with memory-free collisions

A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols c ⁽ⁿ⁾ constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c ⁽ⁿ⁾ 1 - D diffusion constant - D vertical extent of the system, measured in cells - E _i vector connecting a site to its neighbour in directioni - i direction of a nearest neighbour site,i = 1,..., 4 - j direction of a nearest neighbour site,j = 1,..., 4 - j mass transport (fluid flow),j = v - j ^x x-component of the flowj - k(x) spatial dependence of the permeability, user defined under the constraint 0 k 1 - k () the part of the permeability which depends on the degree of saturation (seek) - k ⁽ⁿ⁾ (x) effective permeability at a sitex that holdsn particles - L horizontal extent of the system, measured in cells - l _mac macroscopic length scale, e.g. one meter - l _mic microscopic length scale (one lattice constant) - m integer number of time steps - n (x) number of particles at the lattice sitex - N _A total number of particles on all A-sites - P probability for rejection of a randomly selected direction or set of directions - p arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours) - p _i ⁽ⁿ⁾ probability that one out ofn particles at a site is sent in directioni - p _ij ⁽²⁾ probability that the two particles at a site are sent in directionsi andj - t time - t _mac macroscopic time scale, e.g. one day - t _mic microscopic time scale (one time step) - v fluid velocity - x space vector, mostly two-dimensional:x = (x, y) - x horizontal component ofx - y vertical component ofx - quotient of microscopic and macroscopic time scales,t _mic /t _mac - quotient of microscopic and macroscopic length scales,l _mic /l _mac - _i p + _i is the probability that a particle is received from the neighbour atx +E _i - K(X, ) effective permeability,k =k(x)k () - correlation length - degree of saturation, used synonymously with density (homogeneous porosity) - ₀ value of a homogeneous particle density - ø(x) external potential (user defined), ø = ^gr + ^mat - ø(x) arithmetic mean of the external potential at the four sites surroundingx - ø _i external potential at the sitex +E _i - total potential, = ø + ^den - ^gr(x) gravitational potential - ^mat(x) matrix potential - ^den() density-dependent potential - _n potential depending on the occupation number - ⁽ⁿ⁾ (x) probability that sitex is occupied byn particles - ₀ ^{(n) (n)} in a system with homogeneous particle density - mac macroscopic - mic microscopic     

A DEVELOPMENT OF BEM TO TWO DIMENSIONAL ELASTICITY

This paper presents a new boundary integral equatiòn for two-dimensional elasticity with the stress component σ_(ij)t_it_j as one of the boundary values,where t_i is the direction cosine of the tangent on the boundary.This form of BEM hasan advantage in that the stress component σ_(ij)t_it_j on the boundary can be calculateddirectly from the numerical solution.The present formulation for plane problems usestwo kernels,the one is logarithmically singular and the other is 1/r singular.Theeffectiveness of the approach is also discussed through test examples.