A simplified technique for shakedown limit load determination of a large square plate with a small central hole under cyclic biaxial loading

A simplified technique for determining the shakedown limit load of a structure was previously developed and successfully applied to benchmark shakedown problems involving uniaxial states of stress ( [Abdalla et al., 2007a], [Abdalla et al., 2007b] and [Abdalla et al., 2007c]). In this paper, the simplified technique is further developed to handle cyclic biaxial loading resulting in multi-axial states of stress within the large square plate with a small central hole problem. Two material models are adopted namely: an elastic-linear strain hardening material model obeying Ziegler's linear kinematic hardening (KH) rule and an elastic-perfectly-plastic (EPP) material model. The simplified technique utilizes the finite element (FE) method and employs small displacement formulation to determine the shakedown limit load without performing lengthy time consuming full elastic-plastic cyclic loading FE simulations or conventional iterative elastic techniques. The simplified technique is utilized to generate the shakedown domain for the plate problem subjected to cyclic biaxial tension along its edges. The outcomes of the simplified technique showed very good correlation with the results of analytical solutions as well as full elastic-plastic cyclic loading FE simulations. Material hardening showed no effect on the shakedown domain of the plate in comparison to employing EPP-material.     

304不锈钢在单轴应变棘轮变形下的随动硬化实验研究

对304不锈钢在室温下进行了单轴应变控制下的应变棘轮变形与失效以及低周疲劳试验研究,对材料在循环过程中材料的硬化行为进行了系统的揭示。在对称应变循环下,研究了不同应变幅值下弹性区尺寸和背应力演化规律;在给定工程应变幅值和循环棘轮应变增量组合的应变棘轮变形下进行了弹性区尺寸和背应力演化研究。观察到了各向同性硬化和随动硬化演化对加载历史的依赖性。     

1Cr18Ni9Ti不锈钢低周疲劳随动硬化实验研究

为了研究单调加载对1Cr18Ni9Ti不锈钢材料后继循环塑性硬化和流动特性的影响,对该材料进行了带平均应变的低周疲劳实验研究.对不同应变幅值的对称应变及非对称的应变循环进行了屈服面半径和背应力演化分析,揭示了材料的随动硬化和各向同性硬化演化对应变幅值和平均应变的依赖性.实验结果为建立循环加载和单调加载耦合的复杂加载条件下的本构模型提供了实验基础.     

304不锈钢的非饱和循环硬化行为及其本构描述

针对循环硬化行为的应变幅值依赖性和相对大应变幅值下的非饱和特性．建立了新的粘塑性循环本构模型,在本构模型中．引入新的变量来表征材料的循环硬化特性,该变量的演变方程中引入一个临界状态来反映循环硬化对应变幅值的依赖性：同时,将该变量分解为2个具有不同演变规律的分量．以此来描述相对大应变幅值下的非饱和循环硬化特性．结果表明．新建模型能够很好地描述304不锈钢循环硬化行为的应变幅值依赖性和非饱和特性．     

On behaviour of fuel elements subject to combined cyclic thermomechanical loads

This paper presents detailed finite element formulations on the kinematic hardening rule of plasticity included in an existing thermoelastoplastic stress analysis code primarily designed to predict the thermomechanical behaviour of nuclear reactor fuel elements. The kinematic hardening rule is considered to be important for structures subject to repeated (or cyclic) loads. The start-up/operation/shut-down and various power excursions in a reactor all can be classified as cyclic loadings. In addition to the shifting of material yield surfaces as usually handled by the kinematic hardening rule, the thermal effect and temperature-dependent material properties have also been included in the present work for the first time.A case study related to an in-reactor experiment on a single fuel element indicated that significantly higher cumulative sheath residual strains after two load cycles was obtained by the present scheme than those calculated by the usual isotropic hardening rule. This observation may alert fuel modellers to select proper hardening rules in their analyses.     

Interaction buckling-progressive deformation

The interaction between buckling and ratchetting under cyclic additional loading is discussed for elastic-plastic structures. It is shown that progressive plastic deformation can lead to the buckling of a structure as a consequence of the resulting additional displacement. This interaction is particularly strong in perfect plasticity. In the case of kinematic hardening materials, an estimate of the primary loading is given in order to prevent the risk of ratchetting. It is shown that elastic and plastic shake-down theorems still hold when the primary load is smaller than the smallest critical load of tangent modulus.     

Modelization of ratcheting in biaxial experiments

A new constitutive equation has been developed on a purely phenomenological basis in order to interpret ratcheting experiments. The model is based on a generalized Armstrong-Frederick equation for the kinematic variable; the coefficients of this equation are functions of both instantaneous and accumulated plastic strain. The experiments described relate to austenitic stainless steel (17-12 SPH) tubular specimens subjected to cyclic torsional loading under constant tensile stress at 600°C. Comparisons between experimental and theoretical results show that the model reasonably well predicts not only ratcheting but also hysteresis loops and the cyclic hardening curve.     

贯穿件J形坡口焊接残余应力分析

核电站反应堆压力容器(RPV)顶盖控制棒驱动机构(CRDM)管座J形坡口焊缝在一回路高温高压水环境下存在应力腐蚀开裂(SCC)的风险,而焊接残余应力是SCC的主要驱动力。使用二维轴对称模型有限元方法对CRDM中心管座J形坡口进行焊接残余应力分析。为了探索一种简单、高效和保守的方法,研究了热源简化、焊缝形状简化、屈服强度、相变和强化行为对焊接残余应力的影响。结果表明:双椭球热源与均匀热源得到的残余应力结果基本一致;焊缝形状由鱼鳞状简化为方块模型对焊接残余应力结果影响不大,但是与合并焊道的结果相差较大;采用低屈服强度得到的残余应力结果并不保守;在ANSYS软件中,固液相变对残余应力结果影响不大;等向强化模型的结果比随动强化模型的结果保守;在工程上,建议采用均匀热源、方块焊道模型和等向强化模型进行焊接模拟。     

A two-surface cyclic plasticity model consistent with fundamental stress-strain equations of the power-law type

A plasticity constitutive model which can describe primary features of stress-strain behaviors of cyclically loaded metallic materials under isothermal conditions is developed in the framework of the two-surface plasticity theory. A special plastic hardening modulus function, which is consistent with the power-law type stress-strain expressions frequently used for both monotonic and cyclic stable stress-strain curves, is introduced to represent cyclic hardening as well as saturated behaviors more realistically. The model is validated through its application to the simulation of uniaxial cyclic and biaxial behaviors of cyclic hardening materials. This model is then implemented into a general purpose finite element computer program and applied to the analysis of a notched plate subjected to cyclic loading.     

An indentation system for determination of viscoplastic stress-strain behavior of small metal volumes before and after irradiation

The development of fusion materials for the first wall in future fusion reactors requires methods for the investigation of irradiation effects on the mechanical properties of materials which are only available in small volumes. Depth and force reading hardness measurement (or indentation) is one of the candidates that have the potential to extract valuable information on the stress-strain behavior of a material. A modified commercial indentation device installed in a hot cell of a fusion materials laboratory (FML) in combination with a neural network based analysis method allows identifying the material parameters of a unified viscoplasticity model with nonlinear isotropic and kinematic hardening from small metal specimens. By investigation of the same material before and after irradiation the method provides the possibility to separate irradiation effects on modulus, hardening and viscous behavior.     

A polycrystalline modeling of the mechanical behavior of neutron irradiated zirconium alloys

Zirconium alloys used as fuel cladding tubes in the nuclear industry undergo important changes after neutron irradiation in the microstructure as well as in the mechanical properties. However, the effects of the specific post-irradiation deformation mechanisms on the mechanical behavior are not clearly understood and modeled. Based on experimental results it is discussed that the kinematic strain hardening is increased by the plastic strain localization inside the dislocation channels as well as the only basal slip activation observed for specific mechanical tests. From this analysis, the first polycrystalline model is developed for irradiated zirconium alloys, taking into account the irradiation induced hardening, the intra-granular softening as well as the intra-granular kinematic strain hardening due to the plastic strain localization inside the channels. This physically based model reproduces the mechanical behavior in agreement with the slip systems observed. In addition, this model reproduces the Bauschinger effect observed during low cycle fatigue as well as the cyclic strain softening.     

The application of a kinematic hardening model for assessing ratchetting in pressurized pipes

Concerns within the nuclear power generation industry regarding the possibility of incremental collapse or ratchetting incurred in pressure vessels and large pressurized piping runs during seismic disturbance has led to a programme of experimental work to simulate component and material behaviour under such conditions.As part of this programme, the plastic deformation of thin-walled cylinders has been experimentally examined for the loading conditions of ±1% cyclic axial strain with hoop stresses of approximately of the initial unixial yield stress.Two materials similar to those used in the pipework of pressurized-water reactor nuclear plant in the UK have been tested, namely type 304S11 stainless steel and En6 low carbon steel. Under the loading conditions, both materials incurred plastic hoop ratchet strains to varying degrees. These ratchet strains were compared with the limiting ratchet strains predicted by the Prager-Ziegler model of kinematic work hardening.It was concluded that this model could not be satisfactorily used for design purposed as it did not consistently either overestimate or underestimate the measured ratchet strains. Furthermore, the manner in which the model reaches a limit is not observed in the experimental results.     

The ratchetting behavior of pressurized plain pipework subjected to cyclic bending moment with the combined hardening model

A series of six tests have been conducted using carbon steel and stainless steel cylindrical specimens having mean diameter/thickness ratios in the range 8 ≤ D_m/t ≤ 28. Each cylinder is pressurized up to its calculated design pressure and is loaded with an alternative bending moment at frequencies typical of seismic events simultaneously. Ratchetting of the cylinder wall has been observed and recorded in the hoop direction. A finite element analysis with the nonlinear isotropic/kinematic (combined) hardening model has been used to evaluate ratchetting behavior of the cylinder under mentioned loading condition. Stress-strain data and material parameters have been obtained from several stabilized cycles of specimens that are subjected to symmetric strain cycles. The finite element results are compared with those obtained from experimental set-up. The results show that initial the rate of ratchetting is large and then it decreases with the increasing cycles. The FE model predicts the hoop strain ratchetting rate to be near that found experimentally in all cases that M/M_P0.2 ≤ 1. Also, M/M_y ratios for the onset of ratchetting in stainless steel specimens are less than carbon steel specimens with same D_m/t ratios.     

Elastic-plastic deformation of cylindrical pressure vessels under cyclic loading

A numerical procedure based on the finite element method and incremental solution approach is presented for analyzing cylindrical pressure vessels deformed in the state of generalized plane strain. The structures considered are subjected to internal pressure, thermal gradients and axial forces, which are cyclic in nature. The materials are assumed to obey the von Mises yield criterion with kinematic strain hardening which, during the process of plastic deformation, the yield surface is permitted to translate in the stress space and may change its size as a function of temperature. Based on Drucker's postulates for stable materials in the incremental theory of plasticity, a noniso-thermal flow rule is deduced in the cylindrical coordinates. By use of the virtual work principle, the finite element equations with displacement formulations are derived. The numerical solution is then carried out by a step-by-step approach for small loading increments and an iteration scheme is employed for each loading step. Furthermore, the analysis method is extended to the determination of limit load and shakedown load of a vessel. Several problem cases are presented.     

Some recent developments of the endochronic theory with applications

In this paper we presented a comprehensive review of recent developments of the endochronic theory of plasticity. The endochronic theory was first proposed by Valanis in 1971 with the aim of circumventing some of the difficulties associated with classical theories of plasticity, such as the concept of a yield surface and its motion in the stress space, criteria for unloading, hardening rules, etc. The theory is developed by using irreversible thermodynamics of internal variables. In the early version of the theory the intrinsic time was defined as a measure of length in strain space. This version is now called the simple endochronic theory. The derived constitutive equations have been applied with success to a number of problems of practical interest, e.g. cyclic response, cross hardening, cross ratcheting, etc. However, it has been shown by the first author that the simple theory leads to an unloading response which is not elastic at its onset. As a consequence infinitesimal hysteresis loops in the first quadrant of the stress-strain space are open, in disagreement with observed behavior in metals. Recently Valanis introduced a new intrinsic time scale which is a measure of length in the plastic strain space. As a result, a new model of endochronic theory has been developed which leads to closure of hysteresis loops in the small as well as the large. It is also shown that the new model predicts the existence of a yield surface. Hardening rules proposed in the classical theory of plasticity appear as special results.It is also shown that the constitutive equations derived from the new model are very powerful in their quantitative prediction of steady cyclic response under constant strain amplitude conditions. In the case of cyclic creep the theory predicts the dependence of cumulative axial creep on the plastic shear strain amplitude but that it overestimates the dependence of the latter on the number of cycles. At the present time we think that this difficulty cannot be overcome on the basis of an isotropic theory. This suggests a future area of investigation and application of the endochronic theory.     

Shakedown and ratchetting under tension–torsion loadings: analysis and experiments

Structural design analyses are conducted with the aim of verifying the exclusion of ratchetting. To this end it is important to make a clear distinction between the shakedown range and the ratchetting range. The performed experiment comprised a hollow tension specimen which was subjected to alternating axial forces, superimposed with constant moments. First, a series of uniaxial tests has been carried out in order to calibrate a bounded kinematic hardening rule. The load parameters have been selected on the basis of previous shakedown analyses with the PERMAS code using a kinematic hardening material model. It is shown that this shakedown analysis gives reasonable agreement between the experimental and the numerical results. A linear and a nonlinear kinematic hardening model of two-surface plasticity are compared in material shakedown analysis.     

Development of inelastic constitutive model for austenitic stainless steel for design use

To prevent creep-fatigue failure or excessive deformation in high-temperature components of fast reactor plants, accurate estimation of inelastic deformation is essential. In performing inelastic analysis, employment of constitutive models, which can precisely reproduce inelastic deformation of the material is of critical importance. The authors have been engaged in the development of inelastic constitutive model for the use in structural design assessment of liquid metal-cooled fast reactor plants. Various improvements were made on the nonlinear hardening model proposed by Ohno and Wang, placing an emphasis on capability to simulate inelastic deformation behavior of austenitic stainless steels, under regular or irregular cyclic loading possibly with temperature variation and hold time. It was demonstrated that the model can simulate the inelastic deformation behavior under various loading conditions with a sufficient accuracy.     

Elastic-plastic creep response of structures under composite time history of loading

High temperature nuclear reactor components are subject to a complex history of thermal and mechanical loading cycles. To evaluate the adequacy of such components, detailed information on the accumulated inelastic strains and strain cycling is required. This paper presents the theory, describes efficient numerical techniques accounting for plasticity, creep and overall equilibrium, describes the overall structure of the resulting computer program, and demonstrates the capability of the analysis method on a real three-dimensional structure. Starting with the principle of virtual work, exact equilibrium equations are derived for a stepwise Lagrangian formulation. The resulting equilibrium equations are then specialized to the incremental Piola-Kirchhoff stress computation and to small incremental strain formulation. Classical plasticity theory is used to develop a novel method based on the concept of ‘plastic stress’ for consideration of inelastic behavior. It is shown that the material's stress-strain curves can be followed to any desired degree of accuracy both for isotropic and kinematic hardening. It is further shown that for kinematic hardening it is necessary to base the incremental change on the state corresponding to the mean of the initial and the final states to satisfy the yield condition at the final state. The equation of state and strain hardening is used to describe the creep behavior. A novel numerical technique to describe a complex load history is developed by using time as a parameter, history breakpoint determination by scanning of various load vectors, and by linear interpolation between any two breakpoints in the load history. Efficient criteria for load incrementation in the form of a fraction of the total ‘plastic stress’ for any sequence of two load history break points are developed and made an internal function of the program. This saves the user significant hardship when faced with guessing the load increment for an unknown state of the solution at any of the load history breakpoints. The ‘plastic stress’ load vector concept is utilized with interation and extrapolation to converge to the equilibrium states with simultaneous satisfaction of the stress-strain relations for each of the iterated states. The essential features of the computer program DYPLAS-FSH, based on the theory and techniques described above, and a postprocessor program POR-FSH, based on RDT F9-5T for ratcheting and fatigue evaluation, are identified and discussed. In summary, the new results of this work are the efficient handling of an arbitrary load history, introduction of the ‘plastic stress’ concept for inelastic computation, novel implementation of classical plasticity with recognition of incrementation conditions for the kinematic hardening, use of the load incrementation algorithm based on the ‘plastic stress’ concept, and development of a computer code capable of solving practical three-dimensional problems.