A viscoplastic constitutive model for nickel-base superalloy,part 2: modeling under anisothermal conditions

In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 °C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 °C, while it showed distinct rate/time-dependency above 800 °C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 °C and the imposed strain rate was fast, whereas in the case of the temperature above 800 °C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y.     

Kinematic hardening model suitable for ratchetting with steady-state

A new kinematic hardening model useful for simulating the steady-state in ratchetting is developed within the framework of the strain hardening and dynamic recovery format. The model is formulated to have two kinds of dynamic recovery terms, which operate at all times and only in a critical state, respectively. The model is examined on the basis of nonproportional experiments of Modified 9Cr–1Mo steel at 550°C and IN738LC at 850°C. The experiments include multiaxial, as well as uniaxial, ratchetting, multiaxial cyclic stress relaxation, and nonproportional cyclic straining along a butterfly-type strain path. It is shown that the model is successful in simulating the experiments, and that the model is featured by the capability of representing appropriately the steady-state in ratchetting under multiaxial and uniaxial cyclic loading.     

Study of plastic/viscoplastic models with various inelastic mechanisms

This paper describes a general framework for the development of plastic or viscoplastic constitutive equations. As the applications are focused on cyclic loadings, only small strains are considered, with an additive decomposition of the total strain into a thermo-elastic part, and several inelastic parts, the evolution of which is determined by several plastic or viscoplastic criteria. Quadratic or linear (crystallographic) criteria could be used, so that the approach is able to describe the contribution of several physical levels, or deformation mechanism, to the inelastic behavior. The present work is restricted to the case of quadratic criteria, and specially to the study of the various interactions which can be introduced between the mechanisms. The most important case is the coupling between kinematic hardening variables which allows to describe: (1) either normal rate sensitivity or inverse rate sensitivity; (2) plasticity-creep interaction; (3) ratcheting for high mean stress but either adaptation or plastic shakedown for lower mean stress.     

Uniaxial and non-proportionally multiaxial ratcheting of SS304 stainless steel at room temperature: experiments and simulations

The uniaxial and non-proportionally multiaxial ratcheting behaviors of SS304 stainless steel at room temperature were initially researched by experiment and then were theoretically described by a cyclic constitutive model in the framework of unified visco-plasticity. The effects of cyclic stress amplitude, mean stress, and their histories on the ratcheting were experimentally investigated under uniaxial and different multiaxial loading paths. The shapes of non-proportional loading paths were linear, circular, elliptical and rhombic, respectively. In the constitutive model, the rate-dependent behavior of the material was reflected by a viscous term; the cyclic flow and cyclic hardening behaviors of the material under asymmetrical stress-controlled cycling were reflected by the evolution rules of kinematic hardening back stress and isotropic deforming resistance, respectively. The effect of loading history on the ratcheting was also considered by introducing two fading memorization functions for maximum inelastic strain amplitude and isotropic deformation resistance, respectively, into the constitutive model. The effect of multiaxial loading path on the ratcheting was reflected by a non-proportional factor defined in this work. The predicting ability of the developed model was proved to be good by comparing the simulations with corresponding experiments.     

一个非比例循环粘塑性本构模型

本文提出地一个考虑材料非比例循环附加强效应，非比例循环加载历史产应和应变幅值历史效应的粘塑性体构模型。在该模型中，引入了对加载过程非常弹性应变幅值的记忆变量ｑ；定义了新的非比例度；引入了考虑材料非比例度的循环饱和各向同性变形阻力参量Ｑｓ；对各向同性变开引入了具有先前加载历史记忆的演化方程，将本文模型用于１Ｃｒ１８Ｎｉ９Ｔｉ不锈钢高温循环变形行为描述，其预言结果与实验结果吻合得很好，表明该模型能很好     

Uniaxial ratcheting of SS304 stainless steel at high temperatures: visco-plastic constitutive model

The uniaxial ratcheting of SS304 stainless steel at high temperatures (300, 600 and 700 °C) were analyzed experimentally, and described by a cyclic constitutive visco-plasticity model. The rate dependence of the material was accounted for by introducing a viscous term. The cyclic hardening and cyclic flow behavior of the material under asymmetrical stress-controlled cycling were described by the evolution rules of kinematic hardening back stress and isotropic deforming resistance. Under the isothermal condition, temperature effect was included by terms involving temperature in the evolution equations of isotropic deforming resistance. The effect of load history on ratcheting was also considered by introducing a fading memory function of the maximum inelastic strain amplitude and isotropic deformation resistance. After the material constants were determined from the experimental data, the uniaxial ratcheting of SS304 stainless steel was numerically simulated and compared with the corresponding experimental results at high temperatures. The predicted results agree well with the experimental ones.     

Ratchetting of viscoplastic material with cyclic softening,part 1: experiments on modified 9Cr–1Mo steel

Uniaxial and multiaxial ratchetting tests were conducted at temperatures between 200 and 600 °C on modified 9Cr–1Mo steel, which exhibits both viscoplastic and cyclic softening behavior. Anomalous behavior was observed in the stress-controlled uniaxial ratchetting tests; the material exhibited outstanding ratchetting in the tensile direction under zero mean stress. Under the uniaxial conditions, the ratchetting deformation significantly depended on the loading rate and hold time in addition to parameters such as the maximum stress and stress ratio. The uniaxial ratchetting was also accelerated to a great extent when cyclic deformation was given before the ratchetting tests. Under the multiaxial conditions, the ratchetting depended on the steady stress, cyclic strain range and strain rate. The ratchetting progressed faster as the steady stress or strain range became larger, or the strain rate became smaller, as expected. Monotonic compression tests were carried out to investigate the reason for the rachetting under no mean stress. Strain range change tests were also conducted to investigate the effect of strain range on the cyclic softening behavior of the material in detail.     

Experimental investigation of cyclic and time-dependent deformation of titanium alloy at elevated temperature

A novel cyclic deformation test program was undertaken to characterize macroscopic time dependent deformation of a titanium alloy for use in viscoplastic model development. All tests were conducted at a high homologous temperature, 650 °C, where there are large time dependent and loading rate dependent effects. Uninterrupted constant amplitude tests having zero mean stress or a tensile mean stress were conducted using three different control modes: strain amplitude and strain rate, stress amplitude and stress rate, and a hybrid stress amplitude and strain rate. Strain ratcheting occurred for all cyclic tests having a tensile mean stress and no plastic shakedown was observed. The shape of the strain ratcheting curve as a function of time is analogous to a creep curve having primary, steady state and tertiary regions, but the magnitude of the ratchet strains are higher than creep strains would be for a constant stress equal to the mean stress. Strain cycles interrupted with up to eight 2-h stress relaxation periods around the hysteresis loop, including hold times in each quadrant of the stress–strain diagram, were also conducted. Stress relaxation was path-dependent and in some cases the stress relaxed to zero. The cyclic behavior of these interrupted tests was similar even though each cycle was very complex. These results support constitutive model development by providing exploratory, characterization and validation data.     

对316L不锈钢的非比例循环粘塑性本构描述

对循环硬化的３１６Ｌ不锈钢提出了一个考虑非比例循环加载下流动和硬化特性的粘塑性本构模型。模型中，通过随动硬化的背应力演化以各向同性阻力演化非比例循环路径及其历史的依赖关系来表征材料的非比例循环附加硬化和非比例循环流动特性，将模型用于预测３１６Ｌ不锈钢的圆形，正菱形应变路径的复杂循环变形行为，其预言结果与实验结果吻合很好。     

On the evolution equation for the rest stress in rate-dependent plasticity

Three different structures of the evolution equation for the rest stress are derived. They correspond to different models of viscoplastic response, constructed by three-dimensional generalization of simple one-dimensional rheological models. Isotropic, kinematic and combined isotropic–kinematic hardening, with an evolution equation for the back stress, are incorporated in the constitutive framework. The results may be of interest in the analytical and numerical studies of the rate-dependent inelastic response.     

一个考虑循环应变幅值历史效应的粘塑性本构模型

本文提出了一个考虑材料应变幅值历史效应的粘塑性本构模型。在该模型中，引入了三个具有不同演化速率的背应力演化方程；建立了非弹性应变幅值历史记忆模型，对各向同性变形阻力，引入了具有先前加载历史记忆的演化方程。将本文模型用于１Ｃｒ１８Ｎｉ９Ｔｉ不锈钢循环变形行为描述中，其预言结果与实验结果吻合得很好，表明该模型能很好地描述材料的循环应变幅值历史下的循环变形行为。     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

Effects of matrix inelasticity on the overall hysteretic behavior of TiNi-SMA fiber composites

The effects of the inelastic deformation of the matrix on the overall hysteretic behavior of a unidirectional titanium–nickel shape-memory alloy (TiNi-SMA) fiber composite and on the local pseudoelastic response of the embedded SMA fibers are studied under the isothermal loading and unloading condition. The multiaxial phase transformation of the SMA fibers is predicted using the phenomenological constitutive equations which can describe the two-step deformation due to the rhombohedral and martensitic transformations, and the inelastic behavior of the matrix material using the standard nonlinear viscoplastic model. The average behavior of the SMA composite is evaluated with the micromechanical method of cells. It is observed that the inelastic deformation of the matrix due to prior tension results in a compressive stress in the matrix after unloading of the SMA composite and this residual stress impedes the complete recovery of the pseudoelastic strain of the SMA fibers. This explains that a closed hysteresis behavior of the SMA composite is no longer observed in contrast with the case that an elastic behavior of matrix is assumed. The predicted local stress–strain behavior indicates that the cyclic response of matrix is crucial to the design of the hysteretic performance of the SMA composite under the repeated loading conditions.     

Strain-rate effects in rheological models of inelastic response

Stress–strain response under constant and variable strain-rate is studied for selected models of inelastic behavior. The derived closed-form solutions for uniaxial loading enable simple evaluation of the strain-rate effects on the material response. The effect of an abrupt change of strain-rate is also examined. Non-Newtonian viscosity which decreases with an increasing strain-rate is incorporated in the analysis. Parabolic and hyperbolic hardening are used to describe the plastic response in monotonic loading. A three-dimensional generalization of an elastic–viscoplastic model is employed to study the stress relaxation in simple shear. A combined isotropic–kinematic hardening and the concept of overstress are used in the analysis. The unloading nonlinearity of the stress–strain curve is then discussed.     

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

Constitutive modeling of strain range dependent cyclic hardening

In this paper, a new approach for constitutive modeling of strain range dependent cyclic hardening is proposed by extending the kinematic hardening model based on the critical state of dynamic recovery. It is assumed that isotropic, as well as kinematic, hardening consists of several parts, and that each part of isotropic hardening evolves when the corresponding part of kinematic hardening is in the critical state of dynamic recovery. The extended model is capable of simulating the cyclic hardening behavior in which different characteristics of cyclic hardening appear depending on strain range. The model is verified by simulating the relatively large cyclic straining tests of 304 stainless steel at ambient temperature, in which cyclic hardening does not stabilize before rupture if strain range exceeds a certain value. The model is further verified by predicting the history dependence of cyclic hardening under incremental cyclic loading and the maximum plastic strain dependence of strain hardening in cyclic tension.     

Modelling paper as a two-dimensional elastic–plastic stochastic network

Stochastic two-dimensional elastic–plastic network models are used to represent the inelastic deformation behavior of well-bonded paper. Linear kinematic hardening is employed with an initial non-zero back stress to represent anisotropic fiber yield. Network models are used to simulate simple monotonic tension and simple cyclic tension of paper materials. The performance of the models is compared to experimental results and found to perform reasonably well. The results suggest that interfiber bonding must be explicitly accounted for to adequately describe the material. Some discrepancy between the model and experimental cyclic tension results is believed to be due to time-dependent strain recovery in the material which is not represented in the network models. Experimental results are also presented which show that simple tension failure in these materials occurs along a line of localized deformation in a majority of the samples. This line is generally observed to form immediately prior to failure and is oriented at a well-defined angle with respect to the loading direction.     

An experimental study of cyclic deformation of extruded AZ61A magnesium alloy

Thin-walled tubular specimens were employed to study the cyclic deformation of extruded AZ61A magnesium alloy. Experiments were conducted under fully reversed strain-controlled tension-compression, torsion, and combined axial-torsion in ambient air. Mechanical twinning was found to significantly influence the inelastic deformation of the material. Cyclic hardening was observed at all the strain amplitudes under investigation. For tension-compression at strain amplitudes higher than 0.5%, the stress-strain hysteresis loop was asymmetric with a positive mean stress. This was associated with mechanical twinning in the compression phase and detwinning in the subsequent tension phase. Under cyclic torsion, the stress-strain hysteresis loops were symmetric although mechanical twinning was observed at high shear strain amplitudes. When the material was subjected to combined axial-torsion loading, the alternative occurrence of twinning and detwinning processes under axial stress resulted in asymmetric shear stress-shear strain hysteresis loops. Nonproportional hardening was not observed due to limited number of slip systems and the formation of mechanical twins. Microstructures after the stabilization of cyclic deformation were observed and the dominant mechanisms governing cyclic deformation were discussed. Existing cyclic plasticity models were discussed in light of their capabilities for modeling the observed cyclic deformation of the magnesium alloy.