General analytical shakedown solution for structures with kinematic hardening materials

The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.     

Kinematic shakedown analysis under a general yield condition with non-associated plastic flow

A nonlinear, purely kinematic approach with the finite element implementation is developed to perform shakedown analysis for materials obeying a general yield condition with non-associated plastic flow. The adopted material model can be used for both isotropic materials (e.g. von Mises's, Mohr-Coulomb and Drucker-Prager criteria) and anisotropic materials (e.g. Hill's and Tsai-Wu criteria) with both associated and non-associated plastic flow. Nonlinear yield criterion is directly introduced into the kinematic shakedown theorem without linearization and instead a nonlinear, purely kinematic formulation is obtained. By means of mathematical programming techniques, the finite element model of shakedown analysis is formulated as a nonlinear programming problem subject to only a small number of equality constraints. The objective function corresponds to plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a structure can then be obtained by solving the minimum optimization problem. A direct, iterative algorithm is proposed to solve the resulting nonlinear programming problem, where a penalty factor based on the calculation of the plastic dissipation power is used to overcome the numerical difficulty caused by the non-differentiability of the objective function in elastic areas. The calculation is entirely based on a purely kinematical velocity field without calculation of stresses. Meanwhile, only a small number of equality constraints are introduced into the nonlinear programming problem. So the computational effort is very modest. Numerical applications prove that the developed algorithm has a very good numerical stability and computational efficiency. The proposed approach can capture different plastic behaviours of materials and therefore has a very wide applicability.     

Safety and collapse of elastic–plastic beams against dynamic loads

The dynamic load-bearing capacity of elastic–plastic beam structures is analysed by the apparatus of shakedown theory. The reduced kinematic formulation for bending beams, which is equivalently deduced from Koiter’s kinematic theorem, combined with the plastic collapse’s method of hinge mechanisms appears effective in solving practical problems. The safety limits on the quasiperiodic dynamic loads as well as respective collapse mechanisms for a number of practical beams are determined.     

Evaluation of shakedown loads for plates

A kinematic method is developed to evaluate the shakedown limits for plates subjected to variable (quasistatic as well as dynamic) loads. The shakedown kinematic theorem is transformed into a reduced form, to which the upper-bound method involving construction of kinematically admissible mechanisms with yield curves used in plastic limit analysis can be applied. As an example, inadaptation of a circular plate clamped at the edge under variable uniform loading is considered.     

大型复杂结构安定性数值分析的新方法及其应用

大型复杂结构承载后普遍存在局部高应力区,特别是某些巨型结构,由于受几何尺寸和制造能力的限制,其局部高应力往往会接近甚至超过材料的屈服极限,因而成为弹性范围内的设计难题。应用安定性分析设计准则,则可通过利用材料的塑性承载能力很好地解决这类问题。为此,在大型复杂结构的强度分析中引入安定性理论,应用Melan安定下限定理,通过将峰值载荷下的弹塑性应力场与弹性参考应力场相减,构造定理要求的自平衡应力场,并推导了相应的安定性判定条件。在此基础上,结合弹塑性增量有限元技术,通过不同载荷水平下的加载-卸载计算,计算相应载荷水平下的残余应力场。根据加载-卸载得到的残余应力场与构造的自平衡场的关系,按照安定性判定条件确定安定极限载荷,从而建立了一种适合于大型复杂结构安定性数值分析的新方法,并应用该方法对某125MN锻造液压机的主液压缸缸体结构进行了安定性分析。     

Adaptation of spherical and cylindrical vessels to variable internal pressure and temperature

The kinematic theorem is applied to solve some problems of shakedown of spherical and cylindrical vessels subjected to variable internal pressure and temperature, taking into account the temperature dependence of the yield stress.     

Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals

The present paper deals with the numerical simulation of the elastic–plastic deformation and localization behavior of solids which are plastically dilatant and sensitive to hydrostatic stresses. The model is based on a generalized macroscopic theory taking into account macroscopic as well as microscopic experimental data obtained from tests with iron-based metals. It shows that hydrostatic components may have a significant effect on the onset of localization and the associated deformation modes, and that they generally lead to a notable decrease in ductility. The continuum formulation relies on a generalized I₁–J₂–J₃ yield criterion to describe the effect of the hydrostatic stress on the plastic flow properties of metals. In contrast to classical theories of metal plasticity, the evolution of the plastic part of the strain rate tensor is determined by a non-associated flow rule based on a plastic potential function which is expressed in terms of stress invariants and kinematic parameters. Numerical simulations of the elastic–plastic deformation behavior of hydrostatic-stress-sensitive metals show the physical effects of the model parameters and also demonstrate the efficiency of the formulation. Their results are in excellent agreement with available experimental data. A variety of large-strain elastic–plastic problems involving pronounced localizations is presented, and the influence of various model parameters on the deformation and localization behavior of hydrostatic-stress-sensitive metals is discussed.     

Plastic failure risk of a metal matrix composite structure under variable thermal loads

In the heat sink components reinforced with fibrous metal matrix composites under cyclic heat flux loads; their structural reliability is frequently limited by the low cycle fatigue of the composites. The stress evolution in the composites becomes complex. In order to investigate the plastic failure risk of the composites, the stress evolution has to be tracked on different length scales. In addition, a proper criterion of plastic failure is needed for the relevant composite geometry and loads. In this work, a computational methodology is presented to estimate the plastic failure risk of a composite heat sink structure. This method was based on a triple scale non-linear finite element analysis and a shakedown theorem. Average lamina stresses were assessed using a micro-mechanics based constitutive law and compared with a shakedown boundary predicted by a direct shakedown analysis. By this comparison both the critical locations and the relative failure risk at each lamina could be identified. A remarkable merit of the current approach was that the computational costs could be enormously reduced by the two employed numerical techniques.     

Upper bounds on plastic strains for elastic-perfectly plastic solids subjected to variable loads

The paper considers shakedown analysis problems for elastic-perfectly plastic solids subjected to quasi-static loads which vary arbitrarily within a given domain. It gives a general inequality which is able to generate Melan's theorem for shakedown, as well as bounds on plastic strains at any point of the solid. These bounds can be made the most stringent by solving a “perturbed” shakedown problem in “finite” or “holonomic” terms. The results presented in this paper are a generalization of those given in a previous paper by the present author[10].     

Shakedown of elastic-plastic solids with frictionless unilateral contact boundary conditions

Elastic perfectly plastic solids (or structures) in frictionless unilateral contact with a rigid obstacle and subjected to quasi-statically variable loads within a given domain are considered. In the hypothesis that the structure undergoes small displacements and complies with a d-stability requisite herein introduced, a Melantype shakedown theorem is presented. This theorem is conceptually similar to the classical one; namely, it requires that the unilateral-contact elastic stress response to the loads and to some initial plastic strains be plastically admissible everywhere in the body and for all load conditions. A method for evaluating the shakedown load boundary is also discussed. It is shown that, by virtue of the unilateral contact constraint, all the ratchetting collapse modes pertaining to the associated no-contact shakedown problem and characterized by ratchet normal displacements against the rigid obstacle are ruled out. A simple numerical example is presented.     

Theorems of restricted dynamic shakedown

Dynamic shakedown for a rate-independent material with internal variables is addressed in the hypothesis that the load values are restricted to those of a specified load history of finite or even infinite duration, thus ruling out the possibility—typical of classical shakedown theory—of indefinite load repetitions. Instead of the usual approach to dynamic shakedown, based on the bounded plastic work criterion, another approach is adopted here, based on the adaptation time criterion. Static, kinematic and mixed-form theorems are presented, which characterize the minimum adaptation time (MAT), a feature of the structure-load system, but which are also able to assess whether plastic work is finite or not in the case of infinite duration load histories, where they then prove to be equivalent to known shakedown theorems.     

An Engineering Model for Three-Dimensional Elastic-Plastic Rolling Contact Analyses

Fatigue life of heavily loaded rolling bearings is strongly dependent on elastic-plastic material properties. For bearing steels these elastic-plastic properties can be accurately obtained by performing monotonic or half-compressive tests. A three-dimensional strain deformation analysis based on the incremental theory of plasticity and the use of Prandtl-Reuss relations in conjunction with the von Mises yield criterion was developed in order to evaluate the permanent deformation in dry contacts loaded above the elastic limit in case of normal loading. The Ramberg-Osgood stress-strain relation for two martensitically hardened variants of SAE 52100 bearing steel considered the nonlinear kinematic and/or isotropic material behavior. Parameters describing the influence of retained austenite are modeled by using a nonlinear isotropic law. Pressure distribution and contact surface displacements during incremental loading are evaluated by using a conjugate gradient method and the internal stress field is derived by using the superposition principle. Further, a fast analysis of smooth surfaces in elastic-plastic static and rolling contact is developed based on analytical relations for the internal stress field. Cyclic evaluation of plastic strains and residual stresses is carried out until shakedown. In order to verify the theoretical model, rolling contact tests under high normal load were performed. Residual stresses and residual profiles measurements show excellent agreement between numerical and measured cyclic values.     

Lower-bound shakedown analysis of axisymmetric structures subjected to variable mechanical and thermal loads

This paper is concerned with axisymmetric structures subjected to various combinations of steady and (irregularly) pulsating mechanical and thermal loads. The numerical method is based on the static shakedown theorem, which deals with elastic-perfectly-plastic bodies. Linear temperature dependence of the yield condition in a 4D stress space is considered. The pseudo-residual stress field is simulated by the Temperature Parameter Method and the yield surface is linearized, so that the shakedown analysis is transformed into a linear programming problem and hence the computational difficulties otherwise encountered are overcome. The shakedown analysis of cylinders and spherical shells with and without defects is presented in this paper. Results presented show that the method is beneficial for the shakedown analysis of complicated structures subjected to various combinations of loads.     

The influence of repeated loading,residual stresses and shakedown on the behaviour of tribological contacts

Most tribological pairs carry their service load not just once but for a very large number of repeated cycles. During the early stages of this life, protective residual stresses may be developed in the near surface layers which enable loads which are of sufficient magnitude to cause initial plastic deformation to be accommodated purely elastically in the longer term. This is an example of the phenomenon of ‘shakedown’ and when its effects are incorporated into the design and operation schedule of machine components this process can lead to significant increases in specific loading duties or improvements in material utilization. Although the underlying principles can be demonstrated by reference to relatively simple stress systems, when a moving Hertzian pressure distribution in considered, which is the form of loading applicable to many contact problems, the situation is more complex. In the absence of exact solutions, bounding theorems, adopted from the theory of plasticity, can be used to generate appropriate load or shakedown limits so that shakedown maps can be drawn which delineate the boundaries between potentially safe and unsafe operating conditions. When the operating point of the contact lies outside the shakedown limit there will be an increment of plastic strain with each application of the load—these can accumulate leading eventually to either component failure or the loss of material by wear.     

基于安定理论的微小裂纹疲劳门槛值的分析

通过弹塑性有限元法 ,利用 Melan的安定理论 ,分析了单缺口试件 ,在含有与试件材料晶粒尺寸相当的微小裂纹下的疲劳门槛值。通过对航空齿轮材料试件进行算例分析 ,其结果与相应实验结论比较吻合 ,并且表明 :利用静力安定法所研究微小裂纹疲劳门槛值 ,预测航空齿轮的寿命是较安全的。     

An extension of the static shakedown theorem to inelastic cracked structures

Here we have investigated under which conditions elastic-perfectly plastic, cracked bodies subjected to variable loads shake down. For this purpose, an extension of the static shakedown theorem (Melan’s theorem) is presented by using a crack analysis developed by Nguyen Quoc Son in the framework of the concept of generalized standard materials.     

A min-max procedure for the shakedown analysis of skeletal structures

A solution procedure for elastic-plastic structures subjected to variable repeated loads is elaborated using a min-max formulation of the shakedown problem. This optimization procedure performed in a multi-dimensional space of parameters is transformed and further reduced to a solution of a set of algebraic equations and a one-dimensional minimization problem. To this purpose use is made of relationship between statically admissible residual stresses and plastic strains. The latter, treated in the analysis as free parameters, provide at the end of the optimization process some additional information on possible residual displacements prior to shakedown. Illustrative examples of space frames are presented to show the accuracy of the proposed procedure.     

Shakedown Criterion Employing Actual Residual Stress Field and Its Application in Numerical Shakedown Analysis

Construction of the static admissible residual stress field and searching the optimal field are key tasks in the shakedown analysis methods applying the static theorem. These methods always meet dimension obstacles when dealing with complex problems. In this paper, a novel shakedown criterion is proposed employing actual residual stress field based on the static shakedown theorem. The actual residual stress field used here is produced under a specified load path, which is a sequence of proportional loading and unloading from zero to all the vertices of the given load domain. This ensures that the shakedown behavior in the whole load domain can be determined based on the theorem proposed by K?nig. The shakedown criterion is then implemented in numerical shakedown analysis. The actual residual stress fields are calculated by incremental finite element elastic-plastic analysis technique for finite deformation under the specified load path with different load levels. The shakedown behavior and the shakedown limit load are determined according to the proposed criterion. The validation of the criterion is performed by a benchmark shakedown example, which is a square plate with a central hole under biaxial loading. The results are consistent with existing results in the literatures and are validated by full cyclic elastic-plastic finite element analysis. The numerical shakedown analysis applying the proposed criterion avoids processing dimension obstacles and performing full cyclic elastic-plastic analysis under arbitrary load paths which should be accounted for appearing. The effect of material model and geometric changes on shakedown behavior can be considered conveniently.     

考虑材料强化的结构安定性分析方法

应用统一强化模型和塑性应变模态的概念，建立了考虑材料强化时求解安定极限载荷下限的线性规划数值逼近法。另外，基于安定分析的机动学原理，建立了一种估计安定极限载荷上限的方法，可以和求解下限过程结合起来，从而估计误差的范围。最后给出了数值算例，证明了本文所提供方法的可靠性和适用性．     

A note on shakedown

The residual stresses developed beneath a rigid ball pressed into an ideally elastoplastic half-space are found on the centre-line of contact. Using the theoretical results of Johnson a shakedown limit based on the criterion that residual stresses are maintained at less than the yield criterion is found, and experimental results confirm that this limit obtains.