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.     

Theoretical and computational aspects in the shakedown analysis of finite elastoplasticity

A fully nonlinear shakedown analysis is considered for structures undergoing large elastic-plastic strains. The underlying kinematics of finite elastoplasticity are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. It is Shown that the notion of a fictitious, self-equilibrated residual stress field of Melan's linear shakesdown theorem has to be replaced by the notion of real, self-equilibrated residual state. Path-dependent and path-independent shakedown theorems are presented that can be realized in an incremental step-by-step procedure using Finite Element codes. The numerical implementation is considered for highly nonlinear truss structures undergoing large cyclic deformations with ideal-plastic, isotropic and kinematic hardening material behavior. Path-dependency of the residual states in the case of non-adaptation and path-independency in the case of shakedown are shown, and the shakedown domain is determined taking into account also the stability boundaries of the structure.     

Research note on a reexamination of neutral loading experiments

In the examination of the published results from neutral loading experiments, the question as to whether plastic deformation occurs is found to depend on both the material and initial loading strain. Provided that initial loading is elastic, then a subsequent stress path that follows the boundary of the initial yield surface for a hardening material is truly neutral with a wholly elastic response. However, when initial loading is elastic-plastic, then further plastic deformation is produced from a subsequent stress path that follows an isotropic expansion of the initial yield surface. These results enable the appropriateness of the kinematic hardening rule and more recent developments in plasticity theory to be appraised. Neutral loading of a non-hardening material produces plastic flow. Whether the absence of hardening is inherent or induced by plastic prestrain, it is shown that the Prandtl-Reuss theory then represents the observed behavior. In general, the purely elastic and nonhardening solutions provide respectively lower and upper bounds on the deformation.     

Shakedown kinematic theorem for elastic–perfectly plastic bodies

The classical shakedown kinematic theorem due to Koiter for elastic–perfectly plastic bodies is re-examined and divided into separated shakedown and nonshakedown theorems. While the shakedown theorem is based on the set of Koiter's plastic strain rate cycles, the non-shakedown one involves a broader set of admissible plastic strain rate cycles, the end-cycle accumulated strains of which are deviatoric parts of compatible strain fields. For certain broad classes of practical problems the two statements are unified to yield the unique theorem in Koiter's sense.     

A kinematic elastic nonshakedown theorem for implicit standard materials

Summary  Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxcé's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated. Received 15 February 2001; accepted for publication 18 October 2001     

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 reduced kinematic formulation,separated collapse modes,and numerical implementation

Our shakedown reduced kinematic formulation is developed to solve some typical plane stress problems, using finite element method. Whenever the comparisons are available, our results agree with the available ones in the literature. The advantage of our approach is its simplicity, computational effectiveness, and the separation of collapse modes for possible different treatments. Second-order cone programming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode. The approach is ready to be used to solve general shakedown problems, including those for elastic–plastic kinematic hardening materials and under dynamic loading.     

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle minimum principle, integrated form of the aforementioned minimum net resistant power theorem, is provided, which characterizes the structure’s steady state response (steady cycle) and proves to be an extension to the present context of known principles of perfect plasticity. The optimality equations of this minimum principle are studied and two particular cases are considered: (i) loads not exceeding the shakedown limit (so recovering known results of shakedown theory) and (ii) specimen under uniform cyclic stress (or strain). Criteria to assess the structure’s ratchet limit loads are given. These, together with some insensitivity features of the structure’s alternating plasticity state, provide the basis to the ratchet limit load analysis problem, for which solution procedures are discussed.     

Constitutive modelling of cyclic plasticity of metal particulate-reinforced brittle matrix composites under compression-compression loading

The Mori-Tanaka approach is used to modelling metal particulate-reinforced brittle matrix composites under cyclic compressive loading. The J₂-flow theory is considered as the relevant physical law of plastic flow in inclusions. Ratchetting of the composite is prevented by the strong constraint exerted by the matrix on the inclusions, even under the assumption of evanescent kinematic hardening. However, the weakening constraint power of the matrix caused by microfracture damage around inclusions is closely coupled with the plasticity of inclusion and leads to ratchetting even when the plastic deformation of inclusions is described by an isotropic hardening rule. A detailed parametric study has revealed that ratchetting is followed by either plastic or elastic shakedown, depending on the load amplitude, composite parameters and the mean length of microcracks.     

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.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

On loading,yield and quasi-yield hypersurfaces in plasticity theory

In the context of the author's previously published “simple” theory of plasticity[1] in which no loading or yield surfaces are assumed to exist, it is shown that (a) loading surfaces must exist for a plastic material as a result of Caratheodory's theorem on Pfaffian forms, and that (b) a yield hypersurface in state space may be defined as the boundary of the region in which no loading surfaces exist (the elastic region) if this region has a positive volume, otherwise this region degenerates into the quasi-yield hypersurface. The significance of loading and yield (or quasi-yield) hypersurfaces is further explored for one-component loadings, with particular attention to the Bauschinger effect and kinematic hardening.     

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.     

Multiple loading–unloading of an elastic–plastic spherical contact

A model for multiple repeated loading and unloading of an elastic–plastic sphere and a rigid flat is presented to cover a wide range of loading conditions far beyond the elastic limit. The sphere material is modeled as elastic linear isotropic hardening and follows the von Mises yield criterion. It is shown that although most of the plastic deformation occurs during the first loading, secondary plastic flow may evolve during the first unloading. The occurrence of this secondary plastic flow depends on the level of first loading and is strongly affected by the Poisson’s ratio and material hardening. The region of secondary plastic flow may propagate during the very first loading–unloading cycles, reaching a steady state after which the following loading–unloading cycles become fully elastic.     

Shape optimization for elasto-plastic deformation under shakedown conditions

An integrated approach for all necessary variations within direct analysis, variational design sensitivity analysis and shakedown analysis based on Melan’s static shakedown theorem for linear unlimited kinematic hardening material behavior is formulated. Using an adequate formulation of the optimization problem of shakedown analysis the necessary variations of residuals, objectives and constraints can be derived easily. Subsequent discretizations w.r.t. displacements and geometry using e.g. isoparametric finite elements yield the well known ‘tangent stiffness matrix’ and ‘tangent sensitivity matrix’, as well as the corresponding matrices for the variation of the Lagrangian-functional which are discussed in detail. Remarks on the computer implementation and numerical examples show the efficiency of the proposed formulation. Important effects of shakedown conditions in shape optimization with elasto-plastic deformations are highlighted in a comparison with elastic and elasto-plastic material behavior and the necessity of applying shakedown conditions when optimizing structures with elasto-plastic deformations is concluded.     

QUASI－FLOW CORNER THEORY ON LARGE PLASTIC DEFORMATION OF DUCTILE METALS AND ITS APPLICATIONS

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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.     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

Structural shakedown for elastic–plastic materials with hardening saturation surface

An elastic–plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is assumed as a basis to formulate a statical Melan-type shakedown theorem. Grounding on the optimality conditions relative to the shakedown load multiplier problem for a structure subjected to cyclic loads, the impending inadaptation collapse mechanism at the shakedown limit state is analyzed and discussed. It is shown that the adopted model is able to catch ratchetting collapse mode at a structural level. Numerical results for a simple structure are finally reported.