Variational principles for Cosserat body

Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity.     

Minimum principle of complementary energy of cable networks by using second-order cone programming

The minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity. It is rather amazing that the complementary energy always attains minimum value at the equilibrium state irrespective of the stability of cable networks, contrary to the fact that only the stationary principles have been presented for elastic trusses and continua even in the case of stable equilibrium state. In order to show the strong duality between the minimization problems of total potential energy and complementary energy, the convex formulations of these problems are investigated, which can be embedded into a primal–dual pair of second-order cone programming problems. The existence and uniqueness of solution are also investigated for the minimization problem of complementary energy.     

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.     

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of doubledeck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.     

On the unilateral contact problem of structures with a non quadratic strain energy density

The analysis of structures with “unilateral contact” boundary conditions is considered. The stress-strain relations are nonlinear and they are derived from a non quadratic strain energy density by “subdifferentiation”. It is proved that for the inequality constrained boundary value problem the “principles” of virtual and of complementary virtual work hold in an inequality form constituting a variational inequality. The theorems of minimum potential and complementary energy are proved to be valid to account for this type of boundary conditions. These theorems are used to formulate the analysis as a nonlinear programming problem. A numerical example of a structure having the “unilateral contact” boundary condition illustrates the theory.     

The solution of elastostatic problems and the principles of minimum potential energy,minimum complementary work,under fuzzy boundary conditions

A Solution of elastostatic problem is defined on the basis of set theory and extended to the cases with fuzzy boundary conditions. Extension is also given for the principles of minimum potential energy and minimum complementary work with fuzzy boundary conditions. A quasisolution of an elastostatic problem is defined as an approximate solution with boundary conditions most close to the original. And the existance of quasisolution of an elastostatic problem can be proved on the basis of certain assumptions and the theorem of minimum elementary potential energy.     

The principle of complementary energy in nonlinear plate theory

In this paper, the priciple of complementary energy is given for the von Kármán nonlinear plate theory. Thenecessary conditions are three linear and static equilibrium equations in the interior and static boundary conditions on that part of the boundary surface, where forces are prescribed. The stationary value of the complementary energy functional leads to the stress-displacement relations and the geometric boundary conditions.

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Zusammenfassung In dieser Arbeit wird das komplementäre Variationsprinzip für die nichtlineare Plattentheorie nach von Kármán untersucht. Als notwendige Bedingungen ergeben sich drei lineare statische Gleichgewichtsbedingungen sowie lineare statische Randbedingungen auf dem Teil des Randes, auf dem die Kräfte vorgegeben sind. Für den stationären Wert des Funktionals erhält man die Schnittgrößen-Verformungsbeziehungen sowie die geometrischen Randbedingungen.

     

On the use of the complementary energy in the solution of buckling problems

A systematic derivation of the expression for the complementary energy in elastic buckling problems is presented. Compatibility is identified with variation with respect to the stress components, and the resulting eigenvalue problem is shown to be equivalent to, and sometimes more convenient than, the corresponding formulation in terms of the potential energy. Similarly, approximate techniques may lead to better as well as simpler estimates, whose upper bound property can, however, be assured only through appropriate safeguards.The method is applied in some detail to buckling of columns of arbitrary boundary conditions and axial force distribution. Another example is the problem of lateral beam buckling, with the effect of warping restraint included. In both cases (and presumably in many others) the complementary energy formulation is of lower order than the conventional potential energy formulation, and it is clear that the same simplification should also apply to finite elements or other discrete formats. The method is restricted to the (technically significant) case of a linear prebuckling state.     

基于余能原理的有限变形问题有限元列式

利用基面力概念,推导了一种基于余能原理的有限变形问题显式有限元列式,可应用于结构的大位移、大转动问题。以基面力为状态变量来表达单元的余能,将有限变形情况下的单元余能分解为变形余能部分和转动余能部分,利用Lagrange乘子法推导出余能原理有限元的控制方程,编制了相应的非线性有限元程序。通过算例分析,说明该列式和程序的可靠性和精确性。     

Analytical solutions to general anti-plane shear problems in finite elasticity

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre–Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

浅曲梁大挠度分析的余能有限元方法

以高玉臣提出的弹性大变形余能原理为基础,利用Lagrange乘子,放松平衡方程和力边界条件对余能泛函的约束,推导出广义的余能原理.根据极分解定理,将变形分为刚性转动和纯变形两部分,则余能也包含相应的两部分,一部分与刚性转动有关,而另一部分与纯变形有关.使用线弹性本构关系,建立了可用于几何非线性计算的有限元模型.应用更新的Lagrange列式法,给出了增量形式的有限元公式.数值计算结果表明,该方法可用于浅曲粱的几何大变形计算.     

Optimal material orientation of linear and non-linear elastic 3D anisotropic materials

Optimal material orientation problems of linear and non-linear elastic three-dimensional anisotropic materials are studied. Most commonly, the energy based formulation is applied for solving orientational design problems of anisotropic materials, considering elastic energy density as a measure of the stress strain state. The same approach is used in the current study, but the strength criteria based approaches are also discussed. A simple relation between the stationary conditions in terms of Euler angles and the optimality conditions in terms of strains is pointed out. The complexity analysis of the different existing optimality conditions has been performed. The solution of the posed optimization problem is decomposed into the strain level solution, search for global extremes and evaluation of Euler angles (parameters). The results obtained are extended to some nonlinear elastic material models.     

索杆体系的机构运动及其与弹性变形的混合问题

基于有限单元平衡方程推导了索杆体系机构运动的通解、特解及其路径跟踪的迭代求解列式，以及机构运动与弹性变形混合问题的迭代求解列式。从机构运动和弹性变形问题的比较角度出发，解释了机构运动中的若干基本定义，对索杆体系的机构运动和弹性变形混合问题作了分类。数值算例表明：采用本文方法可以非常有效地跟踪求解索杆体系机构运动、弹性变形及其混合问题的全过程路径，得到机构运动的静定状态和混合问题的平衡状态。     

Applying the calculus of variations to circular motion

We demonstrate that having found a condition for the stationary points in multivariable calculus, that condition may be substituted back into the original equation and still yield the correct stationary points. With that, we emphasise the conditions that must be met in solving multivariable stationary point problems. We further use the analogy of the stationary points problem with finding stationary paths in calculus of variations to apply the latter to circular paths in an axisymmetric potential. Surprisingly, we find that this classical problem does not meet the required conditions. We subsequently derive new conditions that must be met and suggest a possible application.     

Generalized variational principles of symmetrical elasticity problem of large deformations

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Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

The present paper concerns the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one-dimensional heat-conductive hydrodynamic model for semiconductors. It is important to analyze thermal influence on the motion of electrons in semiconductor device to improve the reliability of devices by handling a hot carrier problem. We show the unique existence of the stationary solution satisfying a subsonic condition by using the Leray–Schauder and the Schauder fixed-point theorems. Then the asymptotic stability of the stationary solution is proved by deriving the a priori estimate uniformly in time. Here an energy form plays an essential role. We also prove that the solution converges to the stationary solution exponentially fast as time tends to infinity.     

Bayesian System Identification using auxiliary stochastic dynamical systems

Bayesian approaches to statistical inference and system identification became practical with the development of effective sampling methods like Markov Chain Monte Carlo (MCMC). However, because the size and complexity of inference problems has dramatically increased, improved MCMC methods are required. Dynamical systems based samplers are an effective extension of traditional MCMC methods. These samplers treat the posterior probability distribution as the potential energy function of a dynamical system, enabling them to better exploit the structure of the inference problem. We present an algorithm, Second-Order Langevin MCMC (SOL-MC), a stochastic dynamical system based MCMC algorithm, which uses the damped second-order Langevin stochastic differential equation (SDE) to sample a posterior distribution. We design the SDE such that the desired posterior probability distribution is its stationary distribution. Since this method is based upon an underlying dynamical system, we can utilize existing work to develop, implement, and optimize the sampler's performance. As such, we can choose parameters which speed up the convergence to the stationary distribution and reduce temporal state and energy correlations in the samples. We then apply this sampler to a system identification problem for a non-linear hysteretic structure model to investigate this method under globally identifiable and unidentifiable conditions.     

以优化方法求解布料无摩擦接触变形问题

利用作者所提出的布料质点模型,用罚函数法解除布料和其它物体之间的接触约束,把接触约束条件下的极值问题转化为无约束极值问题,用F-R共轭梯度法求解,并用三次插值方法进行一维搜索．计算了有缝摺和裁剪开口的布料与球形物体的接触变形问题,得到的结果符合人们的日常实际观察．