Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

面内功能梯度三角形板等几何面内振动分析

基于平面应变理论,利用等几何有限元方法分析了弹性边界条件下面内功能梯度三角形板的面内振动特性.板的材料属性沿厚度方向呈均匀分布,而在面内方向呈任意指数梯度变化.采用非均匀有理B样条(NURBS)基函数对三角形结构进行等几何建模和位移描述,实现了三角形板几何设计和振动分析的无缝衔接.在三角形板边界上引入虚拟弹簧约束并通过调节虚拟弹簧刚度,实现任意边界条件的施加.通过不同的单元细化方案和对比算例,验证了等几何方法的灵活性、准确性和快速收敛性.系统研究了边界条件、材料属性和几何参数对三角形板振动特性的影响.同时给出了弹性边界条件下面内功能梯度三角形板的振动特性解,具有重要参考价值.     

Good ideals in Gorenstein local rings

Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.

     

Almost Connected Orders

One of the standard axioms for semiorders states that no three-point chain is incomparable to a fourth point. We refer to asymmetric relations satisfying this axiom as almost connected orders or ac-orders. It turns out that any relation lying between two weak orders, one of which covers the other for inclusion, is an ac-order (albeit of a special kind). Every ac-order is bracketed in a natural way by two weak orders, one the maximum in the set of weak orders included in the ac-order, and the other minimal, but not necessarily the minimum, in the set of weak orders that include the ac-order. The family of ac-orders on a finite set with at least five elements is not well graded (in the sense of Doignon and Falmagne, 1997). However, such a family is both upgradable and downgradable, as every nonempty ac-order contains a pair whose deletion defines an ac-order on the same set, and for every ac-order which is not a chain, there is a pair whose addition gives an ac-order.     

分次直投射模及应用

本文引进分次直投射模的概念,得到分次直投射模的一个判定定理,并利用分次直投射模刻划了分次左遗传环,分次左半遗传环,分次左半单环和分次左PP-环,     

Morita contexts and Galois theory for weak Hopf comodulelike algebras

We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.     

Nonexistence of Smooth Axially Symmetric Harmonic Maps from B^3 into S^2

Inspired by the construction of blow-up solutions of the heat flow of harmonic maps from D^2 into S^2 via maximum principle （Chang et al., J. Diff. Geom., 36, 1992, pp. 507-515.） we provide examples of nonexistence of smooth axially symmetric harmonic maps from B^3 into S^2 with smooth boundary maps of degree zero.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Elastic Analysis of Anisotropic Functionally Graded Rotating Disks With Non-Uniform Thicknesses北大核心CSCD

研究了任意梯度变化的变厚度各向异性转动圆盘的弹性问题.假设圆盘绕刚性轴匀速转动,其材料性能和厚度沿径向任意梯度变化.考虑圆盘在中心转轴处受位移约束,外侧自由,根据各向异性转动圆盘的平衡微分方程,得到关于径向应力的Fredholm积分方程,继而通过对Fredholm积分方程进行数值求解,得到结构的位移场和应力场.对具体梯度变化情况仅需代入相应梯度变化进行求解即可.数值算例部分,通过假设厚度、弹性模量等参数为特殊的幂函数形式,将由Fredholm积分方程求出的数值解与对应的精确解进行对比,以及针对常见的Voigt模型,将由该方法算得的数值解和ANSYS有限元计算结果进行对比,验证了该方法的准确性和精度.其次,针对Voigt模型,重点分析了厚度变化、材料性能梯度参数、各向异性度等对应力场和位移场的影响.提出了针对材料性能和厚度沿径向呈任意梯度变化的圆盘结构弹性分析方法,将为优化功能梯度圆盘的结构和材料参数、有效调整构件应力分布、提高结构安全性,提供强有力的工具;算例分析结果对功能梯度圆盘在复杂条件下的结构安全设计有重要的理论指导意义.     

A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X. Pan and Q. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.