均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Vibration analysis of a rod with complex boundary conditions

This paper deals with the forced longitudinal vibration of a rod carrying a concentrated mass and supported by a spring at one end. The vibration of the rod is excited by the motion of the support point at the other end. Since the boundary conditions of the problem are complex and it is necessary to consider the damping, we determine only the steady state periodic solution. First the linear system is analysed; then the material nonlinearity is considered and the approximate analytic solution of nonlinear partial differential equation with nonlinear boundary conditions is obtained by the perturbation method.     

Dynamics of unidirectional glass-fiber-reinforced plastic

The motion problem of unidirectional glass-fiber-reinforced plastic is formulated under the assumption that the fibers are under stress-strain only, while the binder is under shear stress only. The binder and fiber inertia is calculated along a direction parallel to the fibers. The system of equations in partial derivatives obtained is reduced by Laplace transformation with respect to time to a system of ordinary differential equations in which only the fiber displacements occur. As illustration, the effect of a normal stress wave on a half space is solved. The solution is obtained in the form of an infinite series provided with an explicit law by which the terms are obtained. Curves are presented for the distribution of the normal and shearing stresses at different moments of time. The binder inertia reduces to the appearance of tangential stresses at the fiber-binder boundary, which can explain the tendency towards stratification in constructions made of glass fiber-reinforced plastic.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 139–145, July–August, 1974.     

A Refined Shear Deformation Theory of Bending of Isotropic Plates*

ABSTRACT

A nonlinear, in-plane displacement assumption is proposed, based on an undetermined variation df/dz of transverse shear strains through the plate thickness. A second-order ordinary differential equation for f(z) and two surface conditions, as well as a set of eighth-order partial differential equations and four associated boundary conditions, are derived from the principle of minimum potential energy. Coupling exists between the partial and ordinary differential equations. In the homogeneous solutions for the former, in addition to an interior solution contribution, there exist two edge-zone solution contributions, one of which induces self-equilibrated (in the thickness direction) boundary stresses. Three examples are calculated using the present theory. The last gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. Numerical results for the examples are compared with those given by three-dimensional elasticity theory and several two-dimensional theories. It is found that the present theory can accurately predict nonlinear variations of in-plane stresses through the thickness of a plate.     

Mathematical solution for nonlinear cylindrical bending of sigmoid functionally graded plates

Problems of nonlinear cylindrical bending of sigmoid functionally graded plates in which material properties vary through the thickness are considered. The variation of the material properties follows two power-law distributions in terms of the volume fractions of constituents. The nonlinear strain-displacement relations in the von Kármán sense are used to study the effect of geometric nonlinearity. The governing equations are reduced to a linear differential equation with nonlinear boundary conditions, yielding a simple solution procedure. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

The exact solutions of von mises yielding criterion for ideally plastic body under uniform pressure in case of plane strain

This problem is solved by dividing the quadratic yielding criterion into two linear partial differential equations. With the help of Cauchy’s integral, these two linear equations can be easily solved. An example is given to show the calculation of the stress components in the plastic domain and the determination of equation of the boundary line between the plastic and elastic domains.     

Cylindrical void in a rigid-ideally plastic single crystal III: Hexagonal close-packed crystal

The analytical solution is derived for the plane strain stress field around a cylindrical void in a hexagonal close-packed single crystal with three in-plane slip systems oriented at the angle π/3 with respect to one another. The critical resolved shear stress on each slip system is assumed to be equal. The crystal is loaded by both internal pressure and a far-field equibiaxial compressive stress. The deformation field takes the form of angular sectors, called slip sectors, within which only one slip system is active; the boundaries between different sectors are radial lines. The stress fields are derived by enforcing equilibrium and a rigid, ideally plastic constitutive relationship, in the spirit of anisotropic slip line theory. The results show that each slip sector is divided into smaller regions denoted as stress sectors and the stress state valid within each stress sector is derived. It is shown that stresses are unique and are continuous within stress sectors and across stress sector boundaries, but the gradient of stresses is not continuous across the boundaries between stress sectors. The solution shows self-similarity in that the stresses over the entire domain can be determined from the stresses within a small region adjacent to the void by invoking certain scaling and symmetry properties. In addition, the stress state exhibits periodicity along logarithmic spirals which emanate from the void. The results predict that the mean value of in-plane pressure required to activate plastic deformation around a void in a single crystal can be higher than that necessary for a void in an isotropic material and is sensitive to the orientation of the slip systems relative to the void.     

功能梯度板的非线性动力分析

非线性材料功能梯度板件的动力分析是属于在数学方程上同时具有变系数、非线性、非定常特征的固体力学问题.文中首先将问题的变系数非线性偏微分方程组转化为各向异性常系数非线性常微分方程,然后用小参数法求得解析解,适用于各种形状、边界及功能梯度分布的板件非线性弹性振动分析.     

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

Elastoplastic state of flexible cylindrical shells with two circular holes

The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 107–112, October 2004.     

Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity

For high Reynolds numbers asymptotic expansions are constructed of the solution of the axially symmetric wave problem on the surface of a viscous incompressible fluid of infinite depth under the assumption that the tangential stresses on the free surface are of the order 0(1/Re). The principal terms of the asymptotic expansion are solutions of linear partial differential equations. The obtained result is then adapted to the case in which the fluid fills a bounded region whose boundary is a free surface. Some examples are given.     

Large Plastic Deformations Accompanying the Growth of an Elliptical Hole in a Thin Plate

Within the context of plane stress assumptions and approximations, an analytical solution is derived for the finite deformation of a traction-free elliptical hole in an infinite plate with tensile tractions at infinity. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under a deformation theory of plasticity. The governing partial differential equations are parabolic in nature and consequently have a single family of mathematical characteristics or slip lines associated with them. Each particle of mass follows a rectilinear path in the plane defined by its slip line which emanates orthogonally from the elliptical hole. By assuming a constant speed for each particle in the plane, a state of plane equilibrium is realized. The originally elliptical hole expands in the shape of an oval which is a parallel curve to the original ellipse. The slip lines remain orthogonal to the evolving oval hole as a necessary condition for a traction-free interior boundary. This solution also satisfies the material stability criterion that the rate of plastic work be positive throughout the entire body for all time. As this solution has some features associated with large deformation crack problems at elevated temperatures, possible applications include secondary or steady-state creep.     

Large deflections of heterogeneous anisotropic rectangular plates

An approximate solution is presented for large deflections of clamped, uniformly loaded, unsymmetrically laminated, anisotropic, rectangular plates. Expressing the load and displacements in the form of series, the von Karman-type nonlinear differential equations and immovable boundary conditions are reduced to a series of linear partial differential equations and boundary conditions. The solution obtained by successive approximations can reduce to some existing solutions for large deflections of homogeneous plates. Numerical results based on the first three terms of the truncated series are graphically presented for unsymmetrical cross-ply and angle-ply plates having various values of fiber-reinforced material, number of layers, and aspect ratio. The results in small deflections of coupled laminates are compared with available data.     

Karman型正交异性矩形薄板非线性弯曲的统一求解方法

本文对Karman型四边支承正交异性薄板在5种不同边界条件下的几何非线性弯曲进行了统一分析。所设的位移函数均为梁振动函数。它们精确地满足边界条件，利用Galerkin方法和位移函数的正交属性，转换控制方程为非线性代数方程。用“稳定化双共轭梯度法”求解稀疏矩阵线性方程组以及“可调节参数的修正迭代法”求解非线性代数方程组，最后给出了相应的数值结果。     

Reduction Principle and Asymptotic Phase for Center Manifolds of Parabolic Systems with Nonlinear Boundary Conditions

We prove the reduction principle and study other attractivity properties of the center and center-unstable manifolds in the vicinity of a steady-state solution for quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains.     

Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.     

SOLUTION AND ANALYSIS OF CIRCULAR TUNNEL FOR THE STRAIN-SOFTENING ROCK MASSES CONSIDERING THE AXIAL IN SITU STRESS AND SEEPAGE FORCE

为了研究轴向应力和渗透力共同作用下软化围岩的应力与位移的变化及分布规律. 基于摩尔-库伦屈服准则及应力-应变软化模型并考虑轴向应力和渗透力的共同作用,将整个塑性区分为有限个同心圆环,以弹塑性交界面处的应力、应变为初始值,并采用微小径向应力增量逐步求出各个圆环上的应力应变及塑性区半径,据此重构了考虑渗透力和轴向力共同作用下软化围岩应力应变特性的逐步求解方法. 利用该方法,推导出软化围岩应力应变的解. 计算结果表明：在考虑轴向应力作用下,塑性区半径和隧道围岩位移都随着渗透力的增加而有所减小;当轴向应力为最小主应力时,渗透力的影响更为显著. 这说明渗透力的存在对于隧道围岩的应力应变分布以及塑性半径和围岩的位移有不可忽略的影响.     

考虑轴向力和渗透力时软化围岩隧道解析

为了研究轴向应力和渗透力共同作用下软化围岩的应力与位移的变化及分布规律. 基于摩尔-库伦屈服准则及应力-应变软化模型并考虑轴向应力和渗透力的共同作用,将整个塑性区分为有限个同心圆环,以弹塑性交界面处的应力、应变为初始值,并采用微小径向应力增量逐步求出各个圆环上的应力应变及塑性区半径,据此重构了考虑渗透力和轴向力共同作用下软化围岩应力应变特性的逐步求解方法. 利用该方法,推导出软化围岩应力应变的解. 计算结果表明:在考虑轴向应力作用下,塑性区半径和隧道围岩位移都随着渗透力的增加而有所减小;当轴向应力为最小主应力时,渗透力的影响更为显著. 这说明渗透力的存在对于隧道围岩的应力应变分布以及塑性半径和围岩的位移有不可忽略的影响.      

Similarity Solution of Unsteady Boundary Layer Equations with a Magnetic Field

The unsteady laminar incompressible boundary layer flow of an electricallyconducting fluid in the stagnation region of two-dimensional and axisymmetricbodies with an applied magnetic field has been studied. The boundary layerequations which are parabolic partial differential equations with threeindependent variables have been reduced to a system of ordinary differential equations by using suitable transformations and then solved numerically using a shooting method. Here, we have obtained new solutions which are solutions of both the boundary layer and Navier-Stokes equations.     

Separation of principal stresses by SOR technique over arbitrary boundaries

A computerized method is presented that generates a grid mesh within the digitized boundary of a photoelastic specimen as it appears in the single viewing through an overhead polariscope. The second-order partial differential equation for the first linear invariant of stress which satisfies the Laplace equation is solved from the boundary values for the digitized domain by the finite-difference method. Connectivity and the weighting functions that are required for the iterative solution of the systems of linear equations are generated from the digitized information along the boundary. Isochromatic values at each nodal point within the boundary are estimated from the digitized fringe patterns by a scanning technique, and the individual values of principal stresses are determined. To enhance convergence, the method of successive over relaxation is applied with an optimum accelerating factor determined in the course of the solution process. The accuracy and the speed of the solution are tested with three different examples. Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.