The Moment Finite-Element Scheme in Problems of Nonlinear Continuum Mechanics

The conceptual and theoretical fundamentals of the original moment finite-element scheme (MFES) developed to solve problems of nonlinear continuum mechanics are presented. Typical examples of model problems are given to illustrate the advantages of the MFES over other traditional finite-element schemes. The basic relationships for studying the nonlinear deformation and distortion of mechanical continuum systems are formulated in an invariant tensor form. Also, some mathematical algorithms specially developed to solve systems of nonlinear equations of high order are described. The numerical solutions obtained are proved reliable and rapidly converging to the exact solutions for a sufficient number of test problems. Results of strength, postbuckling stability, vibration, fracture, and high-speed influence analyses of real mechanical systems are illustrated.     

Functional Concepts in Continuum Mechanics

This paper is an attempt to give a concise presentation of the main concepts of continuum mechanics and to show their articulation. Functional definitions have been favoured.The first section is devoted to a review of continuum mechanics.The second section deals with the mechanics of materials.Constitutive equations of the material are given first by equations of state and then by complementary equations written in order to fulfil the fundamental inequality concerning the production of entropy and the physical properties of the material (viscosity, plasticity, damage etc.).Section 3 gives the Lagrangian and the Hamiltonian formulations for a moving body. Section 4 is devoted to the motion of surfaces through which discontinuities appear, to show briefly two examples of application of the previous concepts. One can easily define the source of intrinsic inhomogeneity, of heat, of irreversible entropy on a surface of phase transition and also for a shockwave.SommarioIn questo lavoro si cerca di dare una presentazione concisa dei concetti fondamentali della meccanica dei continui e di mostrare le loro articolazioni,con particolare rilievo alle definizioni funzionali. La prima sezione è dedicata ad un esame generale introduttivo. La seconda alla meccanica dei materiali, le cui equazioni costitutive sono date come equazioni di stato e come equazioni complementari scritte al fine di soddisfare le diseguaglianze concernenti la produzionedi entropia e le proprietà fisiche dei materiali (qualiviscosità, plasticità, guasti...).La terza sezione dà le formulazioni Lagrangiana ed Hamiltoniana per il moto di un corpo.La quarta sezione è dedicata al moto di superfici di discontinuità per mostrare brevemente due esempi di applicazione dei concetti precedenti.Viene quindi definita la sorgente di disomogeneità intrinseca, di calore, di entropia irreversibile su di una superficie, di transizione di fase ed anche di un' onda d'urto.     

Constitutive Equations in Continuum Mechanics

One of the main research areas in continuum mechanics is constitutive equations. There is no generally accepted viewpoint on or method for this research. Several concepts have appeared, but further studies are still necessary. The paper elucidates some problems and viewpoints of such research     

Modeling Creep and Continuous Fracture Process Zones in Spatial Prismatic Bodies

The semianalytic finite-element method is used to develop a method for solving problems of creep and continuous fracture of complex spatial bodies. The method allows modeling the variation of the stress-strain state during creep, accompanying accumulation of dispersed microdamages, and development of macroscopic effects—continuous fracture process zones. The growth of a continuous fracture process zone is modeled. A criterion is formulated for determination of the applicability limits of continuum damage mechanics. The method is exemplified by the problem of deformation and continuous fracture of a gas turbine blade __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 70–86, September 2005.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 73–82, July 2005.     

Nonlinear dynamical characteristics of piles under horizontal vibration

The pile-soil system is regarded as a visco-elastic half-space embedded pile. Based on the method of continuum mechanics, a nonlinear mathematical model of pile-soil interaction was established-a coupling nonlinear boundary value problem. Under the case of horizontal vibration, the nonlinearly dynamical characteristics of pile applying the axis force were studied in horizontal direction in frequency domain. The effects of parameters, especially the axis force on the stiffness were studied in detail. The numerical results suggest that it is possible that the pile applying an axis force will lose its stability. So,the effect of the axis force on the pile is considered.     

Two Nonlinear Models of Brittle Fracture for Solids

This paper considers isotropic and orthotropic nonlinear constitutive relations for brittle materials in the case of plane stresses. Numerical solution algorithms based on the finite-element method are developed. The resulting material models are incorporated in the PIONER software. The correctness of crack path determination is examined by solving a test problem of crack propagation. The isotropic model gives mesh-dependent results, whereas the orthotropic model provides an adequate solution. It is shown that solutions obtained for the isotropic model are close to those obtained by eliminating failed elements.     

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｄｙｎａｍｉｃｓｙｓｔｅｍｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ : ｘ=ｆ(ｘ) ,( 1 )ｗｈｅｒｅｘａｎｄｆａｒｅｎ＿ｄｉｍｅｎｓｉｏｎａｌｒｅａｌｖｅｃｔｏｒｓ,ａｎｄｔｈｅｄｏｔｉｎｄｉｃａｔｅｓｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｉｍｅ ,ｔ.Ｓｕｐｐｏｓｅｔｈａｔｘ=ｕ(ｔ)ｉｓａｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆＥｑ .( 1 )ｗｉｔｈｐｅｒｉｏｄｉｃ,ｓａｙ ,Ｔ .ＫｒｙｌｏｖａｎｄＢ…     

计算力学非线性分析的现状与展望

本文对计算力学非线性分析近年来的发展做了简单回顾。文中说明了非线性数值分析日益受到重视并且还需要走很长的路,进而介绍了近廿年来非线性问题算法的两个最要方面:延续算法和单纯形算法。文末对数值方法未来十年的可能发展做了展望。     

Application of Discrete Element Method for Continuum Dynamic Problems

A new method based on the principle of minimum potential energy is presented, aiming to overcome some weakness of the present discrete element method (DEM). Our primary research is to put forward the DEM with a tight theory base and a fit technique for treating continuum dynamic problems. By using this method, we can not only extend the existing seven-disc model, but also establish a new nine-disc model in a general way. Moreover, the equivalences of two kinds of models have been verified. In addition, three numerical examples of stress wave propagation problems are given in order to validate accuracy and efficiency of the present DEM models and their algorithms. Finally, the dynamic stress concentration problem of a square plate with a circular hole is analyzed.This work was supported by Nation Natural Science Foundation of China (nos. 10232040 and 10572002).     

空间网格结构动力分析的非线性模态方法

空间网格结构因自由度数多且无简化的力学模型,非线性动力分析通常要耗费大量时间．传统的非线性模态方法用于求解多高层结构的局部非线性问题已获得良好的效果,但对系统非线性问题的应用尚缺少研究．对比分析多高层结构和空间网格结构动力性能差异,指出网格结构动力非线性分析存在的问题．以主振型理论和切线刚度分离法为基础,将非线性模态方法用于几何非线性效应显著的空间网格结构动力分析．通过对运动方程的非线性恢复力进行拆分,形成线性表达形式,然后解耦到主振型所在的广义坐标系,以达到缩减自由度数量的目的．并通过实例验证非线性模态方法的高效性与适用性．     

基于材料力学弯曲理论求解非线性梁的探讨

基于材料力学弯曲理论,利用广义变分原理研究了非线性本构关系静不定梁支承反力的求解,指出了有关文献求解非线性本构关系静不定梁支承反力存在的错误．研究结果表明：本构关系为的静不定梁弯曲时,会同时存在曲率为正、曲率为负的梁段．当n为奇数时,曲率为正梁段的余能表达形式与曲率为负的余能表达式相同．当n为偶数时,曲率为正梁段的余能表达式与曲率为负梁段的余能表达式则不相同,存在正负号的差异．     

一类非线性周期系统响应的精细积分法

对于一类非线性周期／变系数微分方程,提出基于精细积分法的数值解法,处理非线性周期／变系数微分方程系统的响应问题,其积分策略是：采用精细积分格式处理常系数部分;采用线性插值格式处理非线性周期／变系数部分,既继承精细积分方程高度准确的特点,又保证足够的精度与较小的计算量。通过数值算例,与以往与用的微分方程直接数值积分法（如预估－校正哈明法）求得的解加以比较表明,对于给定的精度要求,精细积分法更经济有效     

一类弱非线性振动问题的插值摄动解法

用插值摄动法［１］求解一类有阻尼的弱非线性振动问题。算例表明，当小参数很小时，本文结果和多尺度法的一级近似结果十分接近。当小参数不是很小时（即接近于强非线性振动时），本文结果，仍然相当准确，并优于多尺度法的一级近似结果。     

Method to evaluate the invariant J-integral in finite-element models of prismatic bodies

A new method to evaluate the J-integral in finite-element models is developed and implemented. It is shown that in contrast to well-known approaches, this method ensures the invariance of the J-integral in finite-element solution of linear and nonlinear problems Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 70–82, December 2008.     

损伤力学中的能量等效假设及其实验验证

大多数损伤模型在构造受损材料本构方程时所采用的应变或应力等效假设在一般情况下并不能满足热力学理论的基本规则.本文提出的能量等效假设,在弹性、塑性和蠕变损伤等普遍情形均证明了其有效性.同时,能量等效假设配合以应变能密度准则,可以很好地描述穿孔铜薄板试件的破断行为.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 40–51, May 2005.