The weak compatibility equations of nonlinear elasticity and the insufficiency of the Hadamard jump condition for non-simply connected bodies

We derive the compatibility equations of L ² displacement gradients on non-simply connected bodies. These compatibility equations are useful for non-smooth strains such as those associated with deformations of multi-phase materials. As an application of these compatibility equations, we study some configurations of different phases around a hole and show that, in general, the classical Hadamard jump condition is not a sufficient compatibility condition.     

A variational formulation for plate buckling problems by the hybrid finite element method

A functional is derived for development of stress hybrid finite elements for plate buckling problems. The equilibrium equations inside the element are identically satisfied in terms of Southwell stress functionsand the transverse displacement. Along the boundary of the element further displacement and normal slope functions are employed. These functions are so chosen as to satisfy the interelement compatibility requirements when the elements are connected. The boundary and internal displacements are selected entirely independently and comments are made on the choice of interpolation functions for the internal displacement.The stationarity of the functional is shown to lead to satisfaction of the equilibrium conditions along interelement boundaries, and the compatibility conditions inside the elements. The paper includes the details of a simple rectangular element and the results of a number of plate buckling problems analysed by the developed element.     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

基于无网格法的弹性地基加肋斜板频率数值解

为了给实际地基工程中的经济效益提供技术指标参考，应用无网格法计算加肋斜板在地基上的自由振动行为，应用移动最小二乘近似MLS和一阶剪切变形原理描述加肋斜板的位移场，分别建立斜板与肋条的势能泛函，使用Winkler弹性地基模型处理加肋板与地基之间的接触势能。通过斜板与肋条的位移协调关系寻找斜板和肋条节点参数转换公式，确立加肋斜板的自由振动控制方程。本文运用了无网格法的优势，即使肋条位置改变也不需重置网格。与有限元解的对比验证了本文方法的有效性和精确性。     

Variational Convergences of Dual Energy Functionals for Elastic Materials with a ε-Thin Strong Inclusion

We give a new derivation, based on the complementary energy formulation, of a simplified model for a multi-structure made up of two anisotropic hyper-elastic bodies connected by a thin strong material layer. The model is obtained by identifying the Mosco-limit of the stored complementary energy functional when the thickness is of order ?? and the stiffness of order 1/?? where ?? is a positive real adimensional parameter. In order to prove the existence of the displacement associated with the stress we use a suitable weak version of the Saint-Venant compatibility condition also known as Donati??s theorem.     

On the load transfer from a rigid cylindrical inclusion into an elastic half space

The present paper examines the problems related to the axial, lateral, and rotational loading of a rigid cylindrical inclusion which is embedded in bonded contact at the boundary of an isotropic elastic half space. The rigid inclusion is modeled as a field of distributed forces which represent the normal and shear tractions that act on the inclusion-elastic-medium interface. The intensities of these distributed tractions are determined by enforcing displacement compatibility conditions at discrete locations of the interface. These compatibility conditions are derived from rigid-body displacement modes appropriate for each loading. The results derived from this numerical scheme are compared with equivalent results derived via analytical techniques which focus on the solution of the governing integral-equation schemes and other approximate-solution schemes. The numerical results presented in the paper illustrate the manner in which the generalized stiffnesses of the embedded inclusion are influenced by its geometry and Poisson's ratio of the half-space region.     

Sizing optimization of truss structures by method of centers and force formulation

This paper describes a new approach to optimum weight design of truss structures. The force method is incorporated in an optimization algorithm based on the method of center points. Design variables are the member cross-sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure. The optimization method utilizes the largest hyperspheres inscribed within the feasible space. The method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well. By including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time. The viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses, nodal displacement and minimum gage. Numerical results are compared with those reported in the literature.     

Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain

The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the RayleighRitz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth,the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental(AVMI)factor is used. The results show good agreement with those from the Rayleigh-Ritz method.     

Construction of rectangular displacement-based element for thick/thin plates

In this paper, a method of constructing displacement-based element for thick/thin plates is developed by using the technique of generalized compatibility, and a rectangular displacement based element with 12 degrees of freedom for thick/thin plates is presented. This method enjoys a good accuracy with simple formulation and is free of shear locking as the thickness of the plate approaches zero. The project supported by National Natural Science Foundation of China through Grant No. 59208075     

胶补加筋板应力强度因子及胶层剪应力计算

在铆钉力法的基础上，提出计算胶接修补加筋板结构应力强度因子及胶层剪应力的解析法，对分析胶接修补的加筋板强度具有现实意义．利用影响系数，可以得到筋条、补片及蒙皮位移的表达式，根据位移协调条件，建立联立方程，从而得到胶层剪应力及铆钉力．通过解析式，计算裂纹尖端的应力强度因子，分析补片对结构修补的效果，综合考虑了筋条及补片对应力强度因子的影响．     

Existence Results and Blow-Up Criterion of Compressible Radiation Hydrodynamic Equations

In this paper, we consider the 3D compressible radiation hydrodynamic equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions with vacuum is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. Moreover, we show that if the initial vacuum domain is not so irregular, then the compatibility condition is necessary and sufficient to guarantee the existence of the unique strong solution. Finally, a Beale–Kato–Majda type blow-up criterion is shown in terms of \((\nabla I,\rho ,\theta )\).     

一种薄板弯曲问题的四边形位移单元

本文基于修正势能泛函,引入广义协调的概念,导出了一种具有12个自由度的薄板弯曲四边形位移单元LGC—Q12,这种单元协调性好,列式简单,计算精度高。并且能近似地通过分片检验。     

Fatigue-crack-tip plastic strains by the stereoimaging technique

The stereoimaging technique is an accurate, high-resolution means of measuring the in-plane displacements resulting from the deformation of a specimen so that the corresponding components of the strain tensor can be computed independently of the stresses. The example used in this paper is a fatigue-cracked specimen of a microscopically homogeneous experimental powder-metallurgy aluminum alloy, analyzed to determine the displacement and strain fields accompanying the opening of the fatigue crack. The displacement measurements are processed by a computer program which compensates for measurement fluctuations in the displacement data by smoothing, and derives the strain magnitudes. The principal strains and the maximum shear strain are determined using Mohr's circle, and the latter strain is then used to estimate the plastic-zone size. The crack-opening mode may be inferred from the displacement map, and the state of stress (plane stress or plane strain) inferred by applying the in-plane compatibility equation.     

An orthogonal self-stress matrix for efficient analysis of cyclically symmetric space truss structures via force method

The analysis of structures is normally carried out through displacement method while the force method is considered as an alternative approach for this purpose and used on occasion. The generation of compatibility conditions (the transpose of self-stress matrix) is one of the major and complicated parts of any structural analysis using force method. In this paper, an efficient method is proposed for producing orthogonal self-stress matrix related to space truss structures with cyclic symmetry. This is actually performed by eigen-decomposition of a special matrix having the same null basis as in equilibrium matrix. Then, the advantages of the obtained compatibility conditions are demonstrated with respect to different formulations such as standard force method, eigen force method and integrated force method. Finally, the efficiency of the presented method is comprehensively compared with three well-known numerical methods and tested on a set of practical examples. The results indicate clearly the significant superiority of the proposed approach in terms of both computational time and the accuracy of the results.     

阶梯式Timoshenko梁自由振动的DCE解

本文基于微分容积法和区域叠加技术提出了微分容积单元法（Differential Cubature Element method，以下简称DCE方法），并用之求解阶梯式变截面Timoshenko梁的自由振动问题。根据梁的变截面情况将其划分为几个单元，在每个单元内应用微分容积法将梁的控制微分方程和边界约束方程离散成为一组关于该单元内配点位移的线性代数方程组，将这些方程组写在一起并在各单元之间应用连续性条件和平衡条件得到一组关于整个域内各点位移的齐次线性代数方程组，这是一广义特征值问题，由子空间迭代法求解该特征问题便可求得系统的自振动频率。数值算例表明，本方法能稳定收敛、并有较高的数值精度和计算效率。     

EIGEN THEORY OF VISCOELASTIC MECHANICS FOR ANISOTROPIC SOLIDS

Anisotropic viscoelastic mechanics is studied under anisotropic subspace. It is proved that there also exist the eigen properties for viscoelastic medium. The modal Maxwell's equation, modal dynamical equation (or modal equilibrium equation) and modal compatibility equation are obtained. Based on them, a new theory of anisotropic viscoelastic mechanics is presented. The advantages of the theory are as follows: 1) the equations are all scalar, and independent of each other. The number of equations is equal to that of anisotropic subspaces, 2) no matter how complicated the anisotropy of solids may be, the form of the definite equation and the boundary condition are in common and explicit, 3) there is no distinction between the force method and the displacement method for statics, that is, the equilibrium equation and the compatibility equation are indistinguishable under the mechanical space, 4) each model equation has a definite physical meaning, for example, the modal equations of order one and order two express the volume change and shear deformation respectively for isotropic solids, 5) there also exist the potential functions which are similar to the stress functions of elastic mechanics for viscoelastic mechanics, but they are not man-made, 6) the final solution of stress or strain is given in the form of modal superimposition, which is suitable to the proximate calculation in engineering.     

波数-频率域内地基土表面位移Green函数的理论分析

建立了柱面坐标系下分层弹性半空间地基土模型。利用钟阳刚度矩阵法和Haskell-Thomson传递矩阵法推导出所有分层土体之间的振动传递关系;根据Helmholtz定理将土体的位移向量分解成势函数的形式,推导出弹性半空间表面应力与位移之间的关系;再将分层土体和半空间地基土通过位移与应力之间的关系进行耦合,得到分层弹性半空间地基土模型表面位移与应力之间的关系。结合单位脉冲荷载作用下地基土表面的边界条件,推导出波数-频率域内地基土表面位移Green函数的解析解,用Matlab程序语言对理论进行实现并通过算例对地基土表面位移Green函数的特征进行了分析和总结。     

A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional (2-D) periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended aver-aging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell. In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellu-lar solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.     

Equivalence of u–p and ζ–ψ formulations of the time-dependent Navier–Stokes equations

This paper deals with the non-stationary incompressible Navier--Stokes equations for two-dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H¹ for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.     

压电材料平面裂纹尖端场的杂交应力有限元分析

基于复势理论和杂交变分原理建立了一种适用于力电耦合分析的杂交应力有限元模 型. 给出了建立刚度矩阵的主要公式和推导过程，单元内的位移场和应力场采用满足平 衡方程的复变函数级数解，假设的复变函数级数解事先精确满足裂纹的无应力和电位移法向 分量为零的条件，单元外边界的位移场假设按抛物线变化, 单元的刚度矩阵采用Gauss积分的方法得出. 通过对力电耦合裂尖场的数值计算验证了程序 的正确性和单元的有效性，同时也用所得结果校验了理论解.