On shakedown of three-dimensional elastoplastic strain-hardening structures

The present article considers the shakedown problem of structures made of either kinematic or mixed strain-hardening materials. Some basic and useful shakedown properties of elastoplastic strain-hardening structures are proved mathematically. It is impossible for a kinematic strain-hardening structure to be involved in incremental plastic collapse, and so its only possible failure mode is that of alternating plasticity. A time-independent self-equilibrium stress field has no influence on the shakedown of a kinematic strain-hardening structure although it contributes to the magnitude of plastic deformation. The sufficient shakedown conditions for either kinematic or mixed strain-hardening structures are deduced, from which the lower bound of shakedown load domain can be obtained via a mathematical programming problem. It should be pointed out that, to guarantee the safety of an elastoplastic strain-hardening structure, the damage analysis is also necessary to determine the maximum load factor the structure can bear. The shakedown analysis of strain-hardening structures can be simplified by the conclusions obtained in this article, as is illustrated by two simple examples.     

Lower bound for LCF lifetime and its application to safe design of elastic viscoplastic structures

The theory of shakedown is applied to obtain an upper estimation of LCF lifetime of structures. A model of elastic viscoplastic material similar to the Perzyna one with isotropic strain hardening and isotropic damage is adopted. Assumptions: viscoplastic strain rate is proportional to the access of the yield function over zero; the rate of damage evolution is equal to a function of hardening and damage parameters with the coefficient of fluidity, as a factor of proportionality; damage process is coupled with the viscoplastic deformation process; the hardening parameter is equal to accumulated viscoplastic deformation. The yield surfaces form a family of self-similar surfaces with the diameter as the parameter. The shakedown condition of the Melan type is formulated relatively to the initial yield surface. Features of the stress path lead to an equation with min–max problem of the mathematical programming in the left side, which determines a safe value of the virtual residual stress. The equation provides an opportunity to compute the maximal value of the strain hardening parameter possible under the prescribed loading program. This value allows to obtain an upper estimate to safe work time of the structure, which results in a sufficient condition of the structure integrity during the prescribed time period. An example of the developed theory application to resolve various problems arising from designing of structures is considered.     

The influence of strain-hardening and grain size on early fatigue damage based on a micromechanics theory

Early fatigue damage is assumed to correspond to the build-up of local plastic shear strain. The influence of strain-hardening and grain size on the early stage of fatigue damage of a polycrystal subjected to fluctuating stress is considered. The calculations are based on a micromechanics theory proposed recently by the authors. It is shown that the increase of the strain-hardening rate and/or the decrease of the grain size decreases the rate of early fatigue damage. In order to produce 100 per cent local plastic shear strain for a given number of loading cycles, the range of the alternating stress decreases with increasing amounts of mean stress. This result is shown to be rather insensitive to the rate of strain-hardening and is found to lie between the values predicted by Gerber's parabolic law and by the modified Goodman linear law for fatigue failure.     

数学规划加权残值法及其进展

微分方程问题双边不等式数学规划解法与现有各种数值方法不同．该方法在不知道精确解的情况下,可首先求得近似解的最小上界和最大下界．计算量小,精度高．算题过程中可随时控制计算精度,不浪费机时,也不盲停机,使计算终止于“经济精度”上．      

应变强化模型的安定准则研究

基于Melan经典的安定理论和von Mises屈服准则,建立了塑性应变强化条件下结构安定的数学模型,根据与时间无关的应力场的特性,对结构中与时间无关的应力场进行了合理的数学变换,将其与载荷变化系数联系起来,推导出与其对应的结构安定极限范围的表达式,给出塑性应变强化模型安定性存在的简化条件.该结论有利于简化应变强化条件下结构的安定分析.     

数学规划加权残值法分析薄板与薄壳的几何非线性问题

使用数学规划加权残值涯分析了薄板大挠度及薄壳的非线性稳定问题。在两者应变协调方程严格满足的情况下,论证并利用平衡方程的单调性,建立数学规划问题,首先得到无布载荷四边简支方板中心挠度的最小上界及最大下界,经典Ｌｅｖｙ解位基其间。     

含缺陷结构的塑性极限分析

结合极限分析中的数学规划理论和有限元技术,提出了三维含结构极限分析的数学规划方法,并采用罚函数法引入塑性不可压条件,对于考虑多组独立变化载荷作用的情况,提出了加载路径射线辐射求解方案,并基于这种射线射状的加载路径,推导了多组载荷联合作用下结构塑性极限上限分析的数学规划格式,编制了相应的有限元程序,文中的数值了该方法的正确性与有效性。     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.     

A mathematical programming algorithm for limit analysis

This paper deals with the limit analyses of perfect rigid-plastic continua. Based on the kinematic theorem of the limit analysis theory, a mathematical programming finite element formula for determining the upper bound load multiplier has been established, and an iteration algorithm proposed accordingly. In this algorithm the plastic and rigid zones are distinguished for every iteration step, and the goal function is modified gradually. The difficulties caused by the nonsmoothness of the goal function are overcome. Some examples solved by this algorithm are presented. The project supported by National Natural Science Foundation of China.     

有效集法在数学规划加权残值法中的应用

本文研究有效集法在数学规划加权残值法求解微分方程问题中的应用。从理论上和实际应用方面 阐明，对于数学规划加权残值法和线性规划问题，有效集法比常用的单纯形法更简便、易程序化、计算速度更快，因此更适用于数学规划加权残值法求解微分方程问题。     

灰色相关性分析在结构静力损伤识别中的应用

基于灰色理论的相关性分析方法，首次提出了灰色曲率关联系数的概念并将其应用到结构的静力损伤识别中，提出了对局部损伤十分敏感的静态位移曲率置信因子SDCACi，通过该因子的大小对各节点所连接的单元是否会发生损伤进行精确的判断，然后运用最小二乘法对损伤区域的损伤程度进行识别．并将该方法应用于两端固支梁的损伤识别中，由识别结果可以证明：不论测量数据(用有限元仿真计算并考虑了测量误差)的多少，该方法对结构中的单损伤和多损伤都能进行准确的定位，因此该方法在大型结构及复杂结构的损伤识别中具有广阔的应用前景．     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

The evolution of voids in the plasticity strain hardening gradient materials

The main aim of this paper is to opens out the meso-mechanism of void growth and coalescence in the matrix materials with graded strain-hardening exponent distribution. For this end, detailed finite element computations of a representative cylindrical cell containing a spherical void have been carried out. According to the FE analyses, significant effects of the strain-hardening exponent gradient (SEG) in the matrix on the void growth and coalescence are revealed: (1) In the homogeneous materials, the void growth and coalescence are slightly dependent on the strain-hardening exponent, however, the SEG distribution in the matrix can increase remarkably the void growth rate and decrease seriously the void coalescence strain. (2) The critical void shapes in the homogeneous materials are mainly governed by the macroscopic stress triaxiality, but due to earlier plastic flow localization in the softer matrix layer, the SEG distribution in the matrix has very significant effects on the deformed void shapes, especially when the stress triaxiality is lower. (3) When the triaxial stress levels are lower, in the homogeneous materials, the shape change mode of the void evolution is dominate so the void growth rate is very low; however, the SEG distribution in the matrix can bring the volume change mode out, as a result of increasing the void growth rate. (4) Comparisons of the numerical results with the existing damage model indicate that the classic damage model cannot give satisfying prediction to the void growth in both the homogeneous strain-hardening matrix and the SEG materials. On the basis of large numbers of numerical computations, a new damage model, which can uniformly describe the void growing in the homogeneous and plasticity gradient materials, is suggested. A mass of element computations have validated that the new damage model can give satisfying agreement with the FE results of cell model.     

下限原理在岩土工程中的应用

采用极限分析下限原理求解了岩土工程中基础的极限载力和边坡的安全系数。求解过程中把有限元法和非线性规划相结合,把整个结构离散化,设定每个结点的应力,把原问题变成一个以边坡的稳定安全系数或基础的极限承载力为目标函数,以结点应力为优化变量,以对可静应力场的各种制约为约束条件的非线性规划问题。采用序列二次规划法求解该非线性规划问题,得到了人为构造的严格满足应力边界条件、平衡微分方程、不违反Mohr-Coulomb或Drucker-Prager屈服准则的应力场,解决了三维可静应力场的构造问题。算例分析表明,本文的方法是正确、可行的。     

Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.     

数学规划加权残值法在非线性微分方程中的应用

本文研究数学规划加权残值法在非线性微分方程求解中的应用,利用数学规划加权残值法和ＬＰ模理论,把非线性微分方程边值问题转化为一个可微分的无约束非线性优化问题,从而运用成熟稳定的寻优方法求得问题的解。文中数字计算例子表明本文方法可以快速有效地求解非线性微分方程。     

轴对称结构极限下限分析的自然单元法

作为一种基于自然邻近插值的新型无网格法,自然单元法克服了大多数无网格法难以施加本质边界条件的困难.将自然单元法与减缩基技术相结合,建立了一种轴对称结构极限下限分析的数值格式和求解算法.通过不断修正自平衡应力场,轴对称结构极限下限分析可转化为一系列的非线性数学规划子问题,并由复合形法求解.在每个非线性规划子问题中,自平衡应力场表示为一组带有待定系数的自平衡应力场基矢量的线性组合,并且这些自平衡应力场基矢量可由弹塑性增量分析的平衡迭代结果得到.算例结果表明,本文所提的轴对称结构极限下限分析方法行之有效.     

A NEW LINEARIZATION METHOD OF MOHR-COULOMB YIELD SURFACE FOR LOWER BOUND PROBLEMS

相比极限平衡法,基于下限原理的极限分析法具有更严谨的力学基础,且得到的安全系数偏于安全,更具有实用价值. 尽管很多学者对其进行了有益的研究,然而经典的线性化方法不能解决一般的强度各向异性问题. 在方位角离散化的基础上,建立各离散方位平面上的屈服条件,同时引入伪黏聚力以保证其具有下限性质.算例表明,该方法可以稳定地从极限解的下方收敛. 该方法不仅丰富了基于线性规划模型的下限有限元理论,而且为材料各向异性本构问题的计算打下了理论基础.     

结构安定分析的Galerkin边界元方法

基于Melan静力安定定理,利用Galerkin边界元方法建立了多组交变载荷作用下结构安定分析的下限计算格式.在给定载荷域的载荷角点所对应载荷作用下,采用Galerkin边界元法计算相应的虚拟弹性应力场,并且利用结构在Galerkin边界元弹塑性增量计算中同一增量步中不同迭代步之间的应力差作为自平衡应力场的基矢量,通过这些基矢量的线性组合构造了自平衡应力场,大大降低了所形成的数学规划问题的未知变量数.并通过复合形法对非线性规划问题直接进行求解,得到了结构在交变载荷作用下的下限安定乘子.计算结果表明,所采用的方法具有较高的精度和计算效率.     

Deflection Analysis of Elastic-Plastic Frames at Shakedown

ABSTRACT

A method is presented providing an upper bound solution to the maximum deflections which may occur in elastic-plastic frames under loadings that vary arbitrarily within prescribed limits.

The problem is reduced to linear programming; a numerical example is given. The results show that the theory of shakedown remains valid even for very low values of the safety factor against incremental collapse.