On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.     

张力索杆结构最小势能分析方法与结构特性研究

张力索杆结构为柔性多态体系,计算方法复杂。本文基于势能最小化迭代建立平衡态的最小势能方法,推导了体系势能、下降向量及步长列式,建立势能最小化共轭梯度法迭代格式,并用VC++编程实现了算法。通过正交四边形网格索网数值分析算例验证了程序正确性,并分析了新型六边形网格马鞍形双曲索网和索杆张拉整体结构的非线性荷载特性。研究表明最小势能迭代分析是一种有效的柔性张力结构非线性分析方法。     

索牵引并联机器人静态平衡位置的正解研究

研究了一种大跨度但运动缓慢的索牵引并联机器人的位置正解问题,此时可认为系统处于静平衡状态,但必须考虑柔索垂度的影响。索牵引并联机器人的静态平衡位置的正解就是确定机器人末端执行器的位置和姿态,使得系统能在给定的柔索长度下处于静平衡状态。这涉及到系统的非线性力学分析而不仅仅是一个几何问题。本文采用悬链线方程描述柔索的力学特性,结合索端位移与机器人末端执行器位移之间的关系推导出机器人的末端执行器位移与索长变化之间的增量关系,据此提出一种迭代策略完成正解计算。数值算例验证了本文方法的有效性。     

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。     

Stress as a Constraint Reaction in Rigid Bodies

The unique construction of stress as a constraint reaction in a rigid body loaded on its boundary in a state of equilibrium is described through the use of the elementary variational problem of minimum potential energy. We suppose that the body is naturally constrained to be strain free and this leads to a non-trivial constrained minimization problem. Our construction is based on an application of a Riesz-like representation theorem within the context of the classical Lagrange multiplier theorem of variational calculus.     

A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan’s problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane.     

Large deformation and postbuckling behavior of an initially deformed rod

Analytical and experimental procedures are used to investigate the large deformation of an initially deformed elastic rod subject to longitudinal and gravitational loads. The ends of the rod rest on simple supports and are connected by a linear spring. Predictions of the system's large deformation behavior, including snap-through from a buckled configuration to a lower-energy equilibrium state, are obtained through a direct application of the minimum potential energy principle. The minimization is performed using Powell's optimization routine. Analytically determined values of displacement are compared with the exact solution for the special case of an ideal column, and with experimental results for various rod/spring configurations. A simplified approach for predicting snap-through is described in the Appendix.     

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

可变体系的几何稳定平衡状态及受力分析

索和撑杆组成的张拉集成体系，膜结构及悬索体系，一般都要施加预应力使其成为能承受荷载的几何不变体系，整体受力表现为几何非线性。本文将其受力状态分解为两部分，一是几何稳定平衡状态，二是弹性稳定平衡状态。文中提出几何稳定平衡状态的分析计算方法，证明增量迭代法的收敛性。结果表明，所提出的方法收敛快，精度高。     

Using discrete optimization algorithms to find minimum energy configurations of slender cantilever beams with non-convex energy functions

This paper deals with a new solution technique for approximately solving certain variational problems in elasticity by using discrete optimization techniques that were originally used in information theory. This allows us to easily and approximately solve large deformation buckling problems for slender cantilever beams (including post-buckling behavior) as well as problems where the strain energy function is non-convex.The core idea is to quantize or discretize the variables describing the possible configurations of the body. This, when combined with the fact that the variational problem has an inherent Markov structure allows us to use computationally efficient search techniques based on dynamic programming (equivalent to finding the shortest path in a weighted directed graph) to find optimal solutions within the quantized state space. The results can be used in two ways: (1) directly as a fast approximate solution to the variational problem (2) As a means for finding very good (nearly minimum energy) initial configurations for application of conventional minimization techniques, which might otherwise fail because of a poor starting configurations which are far from the global minimum. We demonstrate both these uses in the paper.     

空间曲线索的平衡方程及单索几何判定

空间曲线索平衡方程可在理想柔索假定的基础上进行推导,也可以由空间曲线梁平衡方程退化得到。本文给出后者的连续化分析过程并进行了初步的讨论,目的在于将空间曲线索的各种力学模型统一起来。采用理想柔索假定后单索只能是平面曲线,自然坐标系和整体直角坐标系下平面曲线索的平衡方程是等价的。同时,针对以往单索连续化分析中几何判定的不足,基于空间曲线的弧长独立参数概念,指出单根理想柔索本质上为静不定和动不定体系。     

Extremum principles for certain hyperelastic materials

A hyperelastic material is here said to be of class H_m if the elastic potential is a homogeneous function of order m + 1 in the components of the Lagrangian displacement gradient. It is shown that a single solution to a boundary value problem generates an infinite family of solutions to a family of related boundary value problems. Assuming that a solution to a boundary value problem exists, it is shown that it is unique provided that the material is stable in the sense of Hill in a deleted neighbourhood of the stress-free state. A minimum theorem concerning the strain energy and the virtual work of the prescribed forces is established for the equilibrium configurations, and a maximum theorem concerning the virtual work of the prescribed surface displacements and the complementary stress energy is established for compatible stress fields. As an application, upper and lower bounds are found for the torsional stiffness of a cylindrical bar of square cross section under infinitesimal deformation.     

Statics of nonlinear elastic cable-stayed systems with slipping flexible cables

A design model is proposed to describe spatial cable-stayed systems with slipping cables. The basic variables of the displacement method are selected. A system of nonlinear equilibrium equations is derived from the condition of minimum potential energy, taking into account the large displacements of joints and the large deformations of cables. A method and formulas needed to iteratively solve the system of nonlinear equations are presented. Individual consideration is given to a cable net with prescribed tension. The computed results make it possible to fabricate and mark out cable rods and to assemble cable nets based on their geometry alone, without the need for force measurement __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 79–87, May 2006.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

On stretching, bending, twisting and flexure of cylindrical shells

The first objective of this work is to use the differential equations of equilibrium of cylindrical shells in conjunction with the principle of minimum complementary energy for the derivation of constitutive equations for coupled stretching, bending and twisting of straight thin-walled open- and closed-cross section beams.The second objective is to use appropriate solutions of the cylindrical-shell differential equations, again in conjunction with the principle of minimum complementary energy, for the sake of deriving values for flexibility and stiffness coefficients of end-loaded, thin-walled cantilever beams, with applications to the problem of shear center and twist center location, including an analysis of the effect of shear lag on the location of these centers.     

考虑损伤的内变量黏弹-黏塑性本构方程

基于Rice 不可逆内变量热力学框架,在约束构型空间中讨论材料的蠕变损伤问题. 通过给定具体的余能密度函数和内变量演化方程推导出考虑损伤的内变量黏弹-黏塑性本构方程. 通过模型相似材料单轴蠕变加卸载试验对一维情况下的本构方程进行参数辨识和模型验证,本构方程能很好地描述黏弹性变形和各蠕变阶段.不同的蠕变阶段具有不同的能量耗散特点. 受应力扰动后,不考虑损伤的材料系统能自发趋于热力学平衡态或稳定态. 在考虑损伤的整个蠕变过程中,材料系统先趋于平衡态再背离平衡态发展. 能量耗散率可作为材料系统热力学状态偏离平衡态的测度;能量耗散率的时间导数可用于表征系统的演化趋势;两者的域内积分值可作为结构长期稳定性的评价指标.      

Variational Characterization of a Quasi-rigid Body

The rigidity of a body usually is characterized by the kinematical assumption that the mutual distance between any two of its particles remains unaltered in any possible deformation. However, from this alone nothing can be said about the internal contact forces exerted between adjacent sub-bodies. Therefore, the determination and form of an internal state of stress for a rigid body is problematical. Here, we will show that by considering such a kinematical characterization as an internal constraint for an elastic body, the constrained body inherits the mechanical structure of the elastic parent theory, i.e., the internal constraint generates an associated set of Lagrange multiplier fields which can be interpreted as an internal constraint reaction pseudo-stress field with the same structure as the state of stress in the parent elastic body. Thus, although the final deformation is the same for both the rigid body and the rigidly constrained elastic body, the latter corresponds to a richer model and, to emphasize this distinction, we refer to it as a quasi-rigid body. While in equilibrium the pseudo-stress field of a quasi-rigid body will satisfy equations identical to the equilibrium equations for the stress field in the elastic parent theory, such equations are not, in general, sufficient to assure uniqueness. In order to overcome this indeterminacy, we consider the quasi-rigid body as the limit of a sequence of deformable bodies, where each member of the sequence is identified by a material parameter such that, as this parameter tends to infinity, the body to which it refers is rigidified. Our approach is variational, i.e., we consider a sequence of minimization problems for hyperelastic bodies whose elastic strain energy is multiplied by a penalty term, say 1/ε . As ε→?0, body distortions are more and more penalized so that the sequence of the displacement fields tends to a rigid displacement field, whereas the sequence of the associated stress fields tends to a definite non-zero limit. It will be shown that among all pseudo-stress fields that satisfy the equilibrium equations for the quasi-rigid body, the unique limit of the sequence as ε→0 minimizes a functional analogous to the complementary energy functional in classical linearized elasticity. This result permits its unique determination without having to consider the whole sequence of penalty problems.     

Stationary and extremum variational formulations for the elastostatics of cable networks

Summary The static analysis of elastic cable networks submitted to generic, conservative loads and prescribed dislocations is considered in this paper. The cables are assumed as stress-unilateral (only tensile stresses are admitted), and represented according to a Lagrangian standpoint, under the customary large displacements hypothesis. A variational formulation of the problem is given, as the stationary of a saddle functional with respect to displacements and (signconstrained) tractions. Uniqueness properties for the solution are derived from this statement, together with two complementary (constrained) minimum formulations, which correspond to the well-known extremum principles of the total potential energy and the complementary energy. The case of a network loaded only at the nodes is exposed as a specialization.

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Sommario Si considera il problema dell’equilibrio delle reti di funi elastiche sottoposte a carichi conservativi, distribuiti o concentrati, e a distorsioni impresse. Alla singola fune è attribuito comportamento unilaterale (reagenza a sola trazione). La trattazione è condotta in forma lagrangiana, sotto la naturale ipotesi di grandi spostamenti. Si perviene ad una formulazione variazionale di stazionarietà di un funzionale sella nelle variabili di spostamento e di sforzo, queste ultime vincolate in segno, e sulla base di questa formulazione vengono discusse le proprietà di unicità della soluzione. Si deducono inoltre due formulazioni complementari di estremo vincolato, corrispondenti agli usuali principi di minimo dell’energia complementare e dell’energia potenziale totale. Il problema delle reti caricate solo nei nodi è esposto come caso particolare.

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The content of this paper was partly presented at the sixth congress AIMETA, held in Genova (Italy), October 7–9, 1982.     

含一个无外力圆柱面带转动的三维杂交应力元

根据一种修正的余能原理,建立了一类具有一个无外力圆柱表面及结点含转动自由度的8 结点新型三维杂交应力元. 单元边界位移场选择二次位移插值函数,且与相邻元协调;单元内假定应力场满足以柱坐标表示的平衡方程及圆柱面上无外力边界条件. 数值算例表明,这种特殊杂交应力元在相当粗的网格下即能十分准确地分析圆弧形槽口附近及曲梁的三维（及二维）的孔边应力分布.