On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Reisner板的哈密尔顿体系及其辛正交解析法

基于Ｒｅｉｓｎｅｒ板理论，通过对混合能变分原理的修正，建立了更一般的哈密尔顿型广义变分原理，并给出了Ｒｅｉｓｎｅｒ板问题的哈密尔顿正则方程及其共轭辛正交解析法     

各向异性弹性力学问题Hamilton正则方程的一般形式

本文从修正后的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，导出了由２１个弹性常数组成的各向异性材料的混合方程，井证明它们即是Ｈａｍｉｌｔｏｎ正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Ｈａｎｉｌｔｏｎ正则形式。     

Research on a systematic methodology for theory of elasticity

IntroductionTraditionalsolutionmethodsforelastictheorygenerallyconformedtotwodifferentthoughts[1～ 12 ] .Onewasbaseduponpointsthatstartfromdifferentialequations.Theotherwasbaseduponwholethatstartsfromenergyprinciples .Inthispaper,theequivalenceisprovedbetweendifferentialformandintegralformofasystematicmethodologyfortheoryofelasticity .Auniformframeworkofthesystematicmethodologyisestablished .Innewsystemitisexpressedintheformsofdifferentialform ,integralformormixedform .Aseriesofdualrelations…     

相空间中非保守系统的Herglotz广义变分原理及其Noether定理

与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.     

弹性力学混合状态方程的弱形式及其边值问题

导出了弹性力学混合状态方程和边界条件弱形式的统一方程，此法使函数的选择无需事先完全满足边界条件，对于各种不同的边值问题可以用统一形式处理，这使得求解弹性力学问题的形式得以扩大和统一     

Numerical modelling of the stability of loaded shells of revolution containing fluid flows

A mixed finite-element algorithm is proposed to study the dynamic behavior of loaded shells of revolution containing a stationary or moving compressible fluid. The behavior of the fluid is described by potential theory, whose equations are reduced to integral form using the Galerkin method. The dynamics of the shell is analyzed with the use of the variational principle of possible displacements, which includes the linearized Bernoulli equation for calculating the hydrodynamic pressure exerted on the shell by the fluid. The solution of the problem reduces to the calculation and analysis of the eigenvalues of the coupled system of equations. As an example, the effect of hydrostatic pressure on the dynamic behavior of shells of revolution containing a moving fluid is studied under various boundary conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 185–195, March–April, 2008     

Continuum theory of dislocations revisited

The paper continues the discussion of continuum theory of dislocations suggested by Berdichevsky and Sedov (PMM 31(6): 981–1000, 1967). The major new points are: the choice of energy, the variational form of the governing dynamical equations, the variational principle for the final plastic state.     

A mixed system of equations of elasticity

A mixed system of six equations of elasticity is represented as a Hamiltonian (canonical) operator system in one of the spatial coordinates. It is shown that this system is the Euler equations for the Hellinger–Reissner principle with an appropriately modified integrand. One more functional with an operator integrand from which the canonical operator system can be derived is set up     

二维Nonlocal线弹性理论的变分原理与对偶方程

基于Eringen提出的Nonlocal线弹性理论的微分形式本构关系,导出了相应的能量密度表达式,进而得到二维Nonlocal线弹性理论的变分原理.利用变分原理导出了对偶平衡方程和相应的边界条件.进而给出了非局部动力问题的Lagrange函数,并引入对偶变量和Hamilton函数,得到了对偶体系下的变分方程.在Hamilton体系下,通过变分得到了二维Nonlocal线弹性理论的对偶平衡方程和相应的边界条件.     

COMPUTATION OF STRESS INTENSITY FACTORS BY THE SUB-REGION MIXED FINITE ELEMENT METHOD OF LINES

Based on the sub-region generalized variational principle,a sub-region mixed ver- sion of the newly-developed semi-analytical‘finite element method of lines’(FEMOL)is pro- posed in this paper for accurate and efficient computation of stress intensity factors(SIFs)of two-dimensional notches/cracks.The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used,with the sought SIFs being among the unknown coefficients.The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements.A mixed system of ordinary differential equations(ODEs) and al- gebraic equations is derived via the sub-region generalized variational principle.A singularity removal technique that eliminates the stress parameters from the mixed equation system even- tually yields a standard FEMOL ODE system,the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver.A number of numerical examples,including bi-material notches/cracks in anti-plane and plane elasticity,are given to show the generally excellent performance of the proposed method.     

Potentials for the modified Cam-Clay model

Energy and dissipation pseudo-potentials are employed to derive constitutive relationships, in the context of thermodynamic concepts, for the widely used Modified Cam-Clay (MCC) model for soil mechanics. A variational formulation of the MCC evolution equations is proposed in this paper. Since plastic collapse of MCC soils cannot be embedded in the classical limit analysis theory, finding the critical amplification of the load that produces plastic collapse is formulated in the form of a system of equations and inequalities. Then, a mixed minimization principle is proposed for the plastic collapse analysis of MCC soils. This principle is obtained by the application of the variational formulation for the flow law introduced in the first part of the article.     

广义对流扩散方程的隐式特征线混合元方法

提出了一个广义对流扩散方程的混合有限元方法,方程的基本变量及其空间梯度和流量在单 元内均作为独立变量分别插值. 基于胡海昌-Washizu三变量广义变分原理结合特征线法给 出了控制方程的单元弱形式. 混合元方法采用基于一点积分方案并结合可以滤掉虚假的 数值震荡的隐式特征线法. 数值结果证明了所提出的方法可以提供和四点积分同样的数 值计算结果,并能够提高计算效率.     

弹性力学Hamilton方法广义解的适定性

首先引入了Hamilton体系中平面应力弹性力学问题正则方程的Galerkin变分方程,证明了Galerkin变分方程和目前文献中所用的Ritz变分方程的等价性,以及相应广义解的适定性,从而为目前的数值方法提供了理论基础。从证明过程中可以看到广义解实际上是Ritz变分泛函的一个鞍点。     

A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials

In relation to electroelastic media with thermopiezoelectric coupling, the system of one-dimensional equations is consistently derived so as to accommodate the high-frequency vibrations of a rod with temperature-dependent material. In the first part of the paper, a unified variational principle of differential type is presented which describes the fundamental equations of thermopiezoelectricity with second sound, including the physical and geometrical non-linearities. In the second part, the hierarchic system of rod equations is systematically deduced from the three-dimensional fundamental equations by use of Mindlin's method of reduction. The hierarchic system of equations which is derived in both differential and variational forms is capable of predicting the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of the rod of uniform cross-section. All the higher-order effects are taken into account as deemed pertinent in any particular case. In the third part, attention is confined to certain cases involving special motions, materials and geometry. Besides, the uniqueness is investigated in solutions of the linearized system of rod equations and the sufficient conditions are enumerated for the uniqueness of solutions.     

On the fundamental equations of electromagnetoelastic media in variational form with an application to shell/laminae equations

The fundamental equations of elasticity with extensions to electromagnetic effects are expressed in differential form for a regular region of materials, and the uniqueness of solutions is examined. Alternatively, the fundamental equations are stated as the Euler–Lagrange equations of a unified variational principle, which operates on all the field variables. The variational principle is deduced from a general principle of physics by modifying it through an involutory transformation. Then, a system of two-dimensional shear deformation equations is derived in differential and fully variational forms for the high frequency waves and vibrations of a functionally graded shell. Also, a theorem is given, which states the conditions sufficient for the uniqueness in solutions of the shell equations. On the basis of a discrete layer modeling, the governing equations are obtained for the motions of a curved laminae made of any numbers of functionally graded distinct layers, whenever the displacements and the electric and magnetic potentials of a layer are taken to vary linearly across its thickness. The resulting equations in differential and fully variational, invariant forms account for various types of waves and coupled vibrations of one and two dimensional structural elements as well. The invariant form makes it possible to express the equations in a particular coordinate system most suitable to the geometry of shell (plate) or laminae. The results are shown to be compatible with and to recover some of earlier equations of plane and curved elements for special material, geometry and/or effects.     

Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

A mixed variational principle is derived by Hamilton’s method from the principle of minimum potential energy for thin anisotropic shells of revolution and is then used to derive a normal system of equations with complex coefficients. Discrete orthogonalization is used to solve this homogeneous system and the nonlinear system of equations that describes the precritical state of shells. A shell generated by revolving a circular arc around the axis parallel to its chord is analyzed for stability. The solution is compared with the approximate solution obtained assuming that the precritical state is membrane. It is established that the approximate problem formulation gives incorrect results for shells of negative Gaussian curvature     

An enhanced single-layer variational formulation for the effect of transverse shear on laminated orthotropic plates

A novel mixed formulation is derived by means of Reissner's variational approach-based on Castigliano's principle of least work in conjunction with a Lagrange multiplier method for the calculus of variations. The governing equations present an alternative theory for modeling the important three-dimensional structural aspects of plates in a two-dimensional form. By integrating the classical Cauchy's equilibrium equations with respect to the thickness co-ordinate, and enforcing continuity of shear and normal stresses at each ply interface, condenses the effect of the thickness. A reduced system of partial differential equations of sixth-order in one variable, is also proposed, which contains differential correction factors that formally modify the classical constitutive equations for composite laminates. The theory degenerates to classical composite plate analysis for thin configurations. Significant deviations from classical plate theory are observed when the thickness becomes comparable with the in-plane dimensions. A variety of case studies are presented and solutions are compared with other models available in the literature and with finite element analysis.     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

Plastic collapse in non-associated hardening materials with application to Cam-clay

This paper presents a variational formulation for the analysis of plastic collapse conditions for a class of hardening materials that accounts for some non-associated flow laws such as the modified Cam-clay model of soils. In this framework, classical statical and kinematical principles of limit analysis do not hold. The variational principle is formulated for the general class of materials whose flow equations are derived from a kind of generalized potentials named bipotentials by de Saxcé.The plastic collapse phenomenon for hardening materials is considered first and formulated as a system of equations. In particular, the case of the usual modified Cam-clay model is analyzed. The paper follows with the proposal of a minimization principle whose solution is then related to the solution of the plastic collapse problem. We demonstrate the use of this minimum principle in a simple example of triaxial compression of a modified Cam-clay material. Finally, we discuss the particular form of the proposed variational formulation for the case of associated plasticity.