Shakedown analysis of shell structures of kinematic hardening materials

It is of great practical importance to analyze the shakedown of shell structures under cyclic loading, especially of those made of strain hardening materials.In this paper, some further understanding of the shakedown theorem for kinematic hardening materials has been made, and it is applied to analyze the shakedown of shell structures. Though the residual stress of a real state is related to plastic strain, the time-independent residual stress field as we will show in the theorem may be unrelated to the time-independent kinematically admissible plastic strain field . For the engineering application, it will be much more convenient to point this out clearly and definitely, otherwise it will be very difficult. Also we have proposed a new method of proving this theorem. The above theorem is applied to the shakedown analysis of a cylindrical shell with hemispherical ends. According to the elastic solution, various possible residual stress and plastic strain fields, the shakedown analysis of the structure can be reduced to a mathematical programming problem. The results of calculation show that the shakedown load of strain hardening materials is about 30–40% higher than that of ideal plastic materials. So it is very important to consider the hardening of materials in the shakedown analysis, for it can greatly increase the structure design capacity, and meanwhile provide a scientific basis to improve the design of shell structures.     

Shakedown analysis of limited-ductility structures

Summary The paper deals with the shakedown analysis problem of discrete elastoplastic structures, subjected to variable and repeated quasi-static loads, account being given to the limited structural ductility by imposing a-priori limit values to some relevant parameters of plastic deformation. The key-concept of limited-ductility load amplifier is assessed and the problem is given a unified formulation as a nonlinear mathematical programming problem, where ductility constraints — of generalized character — are obtained by using a so-called perturbation method which leads to a-priori upper bounds of the relevant plastic deformation measures. Lower bounds to the load amplifier are so obtained and three different computationals stratagies are herein outlined: the linear parametric programming one is applied to three case-study structures.

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Sommario La memoria affronta lo studio del problema di adattamento elastoplastico di strutture discrete, sottoposte a carichi comunque variabili in modo quasi-statico, tenendo conto di un comportamento a limitata duttilità della struttura, il quale richiede che uno o più parametri della deformazione plastica risultino non superiori di assegnati valori. Del problema di analisi posto, utilizzando il concetto di moltiplicatore dei carichi per l'adattamento con limitata duttilità, viene data una formulazione unitaria come problema di programmazione matematica non lineare, i cui vincoli di duttilità sono ottenuti con tecniche perturbative di maggiorazione a-priori delle deformazioni plastiche. Il metodo conduce a limiti inferiori del moltiplicatore dei carichi dipendenti dal livello assegnato alla deformazione plastica. Tre differenti strategie computazionali vengono delineate per la soluzione del problema, una delle quali di tipo programmazione lineare parametrica viene applicata a tre esempi.

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Ricerca finanziata dal CNR e dal MPI.     

基于能量原理的连续体结构安定分析

基于安定分析的静力定理,建立了一种以结构吸收的塑性能为安定判断准则,应用一种特殊的加载方式求解连续体结构安定问题的数值方法.应用这种能量方法对两个经典安定分析算例进行了计算,与相关结果进行对比表明能量法的有效性和可靠性.该方法摆脱了传统分析方法中的数学规划运算,有较高的求解效率,可方便地应用于实际工程中.     

Shakedown analysis of shape memory alloy structures

Phase transformational shakedown of a structure refers to a status that plastic strains cease developing after a finite number of loading cycles, and subsequently the structure undergoes only elastic deformation and alternating phase transformations with limited magnitudes. Due to the intrinsic complexity in the constitutive relations of shape memory alloys (SMA), there is as yet a lack of effective methods for modeling the mechanical responses of SMA structures, especially when they develop both phase transformation and plastic deformation. This paper is devoted to present an algorithm for analyzing shakedown of SMA structures subjected to cyclic or varying loads within specified domains. Based on the phase transformation and plastic yield criteria of von Mises-type and their associated flow rules, a simplified three-dimensional phenomenological constitutive model is first formulated accounting for different regimes of elastic–plastic deformation and phase transformation. Different responses possible for SMA bodies exposed to varying loads are discussed. The classical Melan shakedown theorem is extended to determine a lower bound of loads for transformational shakedown of SMA bodies without necessity of a step-by-step analysis along the loading history. Finally, a simple example is given to illustrate the application of the present theory as well as some basic features of shakedown of SMA structures. It is interesting to find that phase transformation may either increase or decrease the load-bearing capacity of a structure, depending upon its constitutive relations, geometries and the loading mode.     

Shakedown analysis of structures made of materials with temperature-dependent yield stress

In this work, the shakedown of structures made of materials with temperature-dependent yield stress is considered. Under some restrictions on the thermal loading condition the yield stress is linearized and shakedown theorems are established. Based on these linearized shakedown theorems, the shakedown limit is formulated as a problem of convex optimization. An algorithm is built to compute shakedown limits. Numerical tests show good agreement with analytic solutions and experimental data.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

薄板结构的安定优化设计

本文通过对薄板的Ｂ样条函数插值及结构中残余应力场的温度参数模拟，研究了薄板安定分析的样条有限元列式，在分析结构安定分析的数学规划列式的基础上，解决了薄板结构在给定可变载荷作用下的安定最小厚度设计问题。由于温度参数的引用，使得研究的问题的自由度大幅度减少。算例表明本文所提出的方法及所编程序的可行和有效。     

Shakedown theorems for some classes of nonassociative hardening elastic-plastic material models

Criteria for quasistatic (elastic) shakedown are established by a static approach and discussed in the light of illustrative examples. Reference is made to elastoplastic solids either with nonassociative plastic flow rules or nonlinear hardening (governed by internal variables), or both. Nonassociativity, that is, lack of normality, is known to occur in the superposed spaces of stresses and strains for the constitutive models of materials with internal friction (like most geomaterials); it is shown also to manifest in the augmented space (of both measurable and internal variables) for realistically modelling a hardening behaviour of metals subjected to cyclic stressing processes. Lower and upper bounds on the shakedown limit derived for these materials from sufficient and necessary shakedown criteria, respectively, are distinct due to the nonassociativity. They suggest recourse to the notion of the reduced yield domain different from the conventional yield domain defined in the augmented space of static variables.     

SHAKEDOWN ANALYSIS OF ELASTO－PLASTIC STRUCTURES SUBJECTED TO EXTERNAL LOADING AND TEMPERATURE VARIATION

ＳＨＡＫＥＤＯＷＮＡＮＡＬＹＳＩＳＯＦＥＬＡＳＴＯ－ＰＬＡＳＴＩＣＳＴＲＵＣＴＵＲＥＳＳＵＢＪＥＣＴＥＤＴＯＥＸＴＥＲＮＡＬＬＯＡＤＩＮＧＡＮＤＴＥＭＰＥＲＡＴＵＲＥＶＡＲＩＡＴＩＯＮＤｕＳｅｎｔｉａｎ（杜森田）ＬｉｕＨａｎｂｉｎｇ（刘寒冰）Ｃｈｅｎ...     

Shakedown limits for reinforced beam structures under fluctuating loads

Shakedown of planar beam structures, with possible different reinforcements in the upper and lower layers of the beam members, subjected to quasi-statically as well as dynamically fluctuating loads is studied. With an assumption of elastic perfectly-plastic behaviour of the beams in bending, a reduced kinematic formulation for the safety factor determining shakedown limits is constructed, which is proved to be equivalent to (but simpler than) the original one, and ready for use in solving practical problems.     

A general kinematical analysis of double slip

General kinematic solutions for double slip in fcc and bcc crystals are presented which are free from constitutive assumptions: that is, the analysis does not presuppose equal amounts of slipping on equallystressed slip systems, in contrast to the standard solutions (wherein Taylor hardening is implicitly assumed). The axis rotations and limiting positions on a stereographic projection are illustrated for several different slip-system combinations, initial axis positions (none on a symmetry line), and proportional slip ratios in both fcc and bcc crystals. It is suggested that the solutions have particular application to the experimental study of double slip in the tensile test of prestrained crystals.     

A modified basis reduction method for limited kinematic hardening shakedown analysis under complex loads

A novel extension of the basis reduction method for kinematic hardening shakedown problem is presented. Firstly, the basis reduction method is implemented based on the modified Newton–Raphson (N-R) method. Then a new technique for the construction of back stress field is introduced, where the simultaneous influence of multiple load corners in shakedown is taken into consideration. Finally, two typical numerical examples are investigated. The results compared with previous works in literatures demonstrated that the proposed method is accurate and the performance in reducing of computation time is significant.     

Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures

Shakedown analysis is an extension of plastic limit analysis to the case of variable repeated loads and plays a significant role in safety assessment and structural design. This paper presents a solution procedure based on the meshless local Petrov–Galerkin (MLPG) method for lower-bound shakedown analysis of bounded kinematic hardening structures. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the targeted domain. Moreover, the natural neighbour interpolation (NNI) is employed to construct trial functions for simplifying the imposition of essential boundary conditions. The kinematic hardening behaviour is simulated by an overlay model and the numerical difficulties caused by the time parameter are overcome by introducing the conception of load corner. The reduced-basis technique is applied to solve the mathematical programming iteratively through a sequence of reduced residual stress subspaces with very low dimensions and the resulting non-linear programming sub-problems are solved via the Complex method. Numerical examples demonstrate that the proposed solution procedure is feasible and effective to determine the shakedown loads of bounded kinematic hardening structures as well as unbounded kinematic hardening structures.     

Plastic analysis of thin plates with anisotropic hardening

In this paper we discuss the adoption of the anisotropic hardening model due to the existence of Bauschinger effect when thin plate is applied by repeated loading. The loading condition of thin plates for linear kinematic hardening has been deduced in terms of generalized forces and generalized plastic deformation. And it can be extended to nonlinear kinematic hardening and mixed hardening. Finally as an example the numerical results are obtained for a circular plate.     

复杂结构静力分析和屈曲分析程序系统DDJTJQ

ＤＤＪＴＪＱ是一个国内开发的复杂结构静力分析和屈曲分析有限元程序系统，自１９８６年通过技术鉴定以来又有了许多发展．本文着重介绍该程序系统的屈曲和后屈曲分析能力以及在不同工程领域结构分析中的应用．给出了几个大型复杂工程结构的计算结果．     

Shakedown of unidirectional composites

The shakedown problem for a composite lamina made of an elastic-plastic matrix and elastic cylindrical fibers is studied. The plastic deformation modes of the lamina are reviewed, and it is concluded that significant shakedown effects can be caused only by the I₁ = 1/2(T₁₁ + T₂₂) and I₂ = T₃₃ components of the remotely applied stress field which are symmetric about the axis x₃ of the fiber; T₁₁ and T₂₂ are the normal composite stresses in the transverse plane. It is shown that the I₁I₂ stress system is needed also to represent thermal loads caused by a uniform change of temperature in the composite.Two methods for evaluation of shakedown limits in the I₁I₂-plane are described. First, the classical approach involving the determination of parametric families of self-stress fields and the solution of mathematical programming problems is used. Results are presented for selected B-Al, Be-Al, B-Ti and B-Mg composites.In the second method, the shakedown problem is related to the recently developed kinematic hardening rules for fibrous composites. It is shown that the composite will shake down for any loading program within a prescribed domain in the I₁I₂-plane, providing that the domain can be contained within a translated initial yield surface. This approach leads to a closed-form evaluation of shakedown limits for any arbitrary combination of mechanical and thermal variable cyclic loads in fibrous composites with temperaturedependent matrix yield strengths.The relationship between shakedown and fatigue in metal matrix composites is discussed.