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.     

Model for 28CrMoV5-8 steel undergoing thermomechanical cyclic loadings

An anisothermal elastoviscoplastic three-dimensional model is proposed in order to predict the response of brake discs used in the French T.G.V. (high speed train). The brake disc is subjected simultaneously to mechanical and thermal cyclic loadings by the application of brake pads to the friction surface. This anisothermal law is based on the internal thermodynamic variables and takes into account the non-linear kinematic hardening, isotropic hardening (to describe cyclic softening) and plastic strain memory effect. Cyclic viscoplastic behavior under in-phase changes of temperature and strain is analysed by using this elaborate anisothermal model with its material constants determined from isothermal experiments. Good agreement is obtained between the predictions and experiments.     

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.     

非等温条件下非比例循环粘塑性本构描述

为了描述在非等温非比例循环加载下的循环变形行为，本文提出了一个考虑材料非比例循环附加硬化效应、非比例循环加载历史效应和温度历史效应的粘塑性本构模型．在该模型中，引入了具有三种不同演化速率的背应力演化方程；定义了新的非比例度；为了反映非比例循环历史和温度历史的影响，引入了表现各向同性变形阻力Ｑａｓｍ，并对各向同性的表现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．     

The viscoplastic behavior of hastelloy-X single crystal

A viscoplastic constitutive model for Hastelloy-X single crystal material is developed based on crystallographic slip theory. The constitutive model was constructed for use in a viscoplastic self-consistent model for isotropic Hastelloy-X polycrystalline material, which has been described in a recent publication. It is found that, by using the slip geometry known from the metallurgical literature, the anisotropic response can be accurately predicted. The model was verified by using tension and torsion data taken at 982°C (1800°F). The constitutive model used on each slip system is a simple unified visoplastic power law model in which weak latent interaction effects are taken into account. The drag stress evolution equations for the octahedral system are written in a hardening/recovery format in which both hardening and recovery depend on separate latent interaction effects between the octahedral crystallographic slip systems. The strain rate behavior of the single crystal material is well correlated by the constitutive model in uniaxial and torsion tests, but it is necessary to include latent information effects between the octahedral slip systems in order to obtain the best possible representation of biaxial cyclic strain rate behavior. Finally, it was observed that the single crystal exhibited dynamic strain aging at 871°C (1600°F). Similar dynamic strain aging occurs at 649°C (1200°F) in the polycrystalline version of the alloy.     

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.     

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.     

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     

Time-dependent ratchetting experiments of SS304 stainless steel

The time-dependent strain cyclic characteristics and ratchetting behaviours of SS304 stainless steel were investigated by uniaxial/multiaxial cyclic loading tests at room and elevated temperatures (350 and 700 °C). The effects of loading rate, peak/valley strain or stress holds, ambient temperature and non-proportional loading path on the cyclic softening/hardening and ratchetting behaviours of the material were discussed. It is shown that: the cyclic deformation of the material presents remarkable time-dependence at room temperature and 700 °C; the cyclic hardening feature and ratchetting strain depend significantly on straining or stressing rate, hold-time, ambient temperature and the non-proportionality of loading path; the time-dependent ratchetting is resulted from the slight opening of hysteresis loop and visco-plasticity together, and the viscosity is a dominating factor at 700 °C; at 350 °C, abnormal rate-dependence and quick shakedown of ratchetting are observed due to the dynamic strain aging of the material at this temperature. Some significant conclusions are obtained, which are useful to construct a constitutive model to describe the time-dependent ratchetting behaviour of the material. It is also stated that the unified visco-plastic constitutive model discussed here cannot provide reasonable simulation to the time-dependent ratchetting at 700 °C, especially to that with certain peak/valley stress hold, since the effect of the high viscosity on time-dependent ratchetting cannot be properly described by using a unified visco-plastic flow rule.     

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.     

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。     

Micromechanical analysis of the finite elastic–viscoplastic response of multiphase composites

Nonlinear thermoelastic–viscoplastic constitutive equations for large deformations with isotropic and directional hardening, are incorporated into a micromechanical finite strain analysis. As a result of this analysis, which is based on the homogenization technique for periodic microstructures, a global thermoinelastic constitutive law is established that governs the overall response of multiphase materials under finite deformations. This constitutive law is expressed in terms of the instantaneous effective mechanical and thermal stress tangent tensors together with the instantaneous global inelastic stress tensor that represents the viscoplastic effects. Results for a thermoinelastic matrix reinforced by a hyperelastic compressible material are given that illustrate the response of fibrous and particulate composites to various types of loading.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates

A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.     

Thermo-viscoelastic and viscoplastic behavior of high-density polyethylene

Observations are reported on high-density polyethylene in uniaxial tensile tests with constant strain rates and relaxation tests at various temperatures ranging from 25 to 90 °C. A constitutive model is derived for the nonlinear viscoelastic and viscoplastic behavior of semi-crystalline polymers at three-dimensional deformations. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is demonstrated that (i) the model correctly approximates the observations and (ii) material parameters are independent of strain rate and change consistently with temperature.     

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.     

Theory of creep deformation with kinematic hardening for materials with different properties in tension and compression

A constitutive model for creep deformation that describes the loading-history-dependent behavior of initially isotropic materials with different properties in tension and compression under stress vector rotations limited by 50–60° is presented within a thermodynamic framework. In the proposed constitutive model a kinematic hardening rule is adopted. This model also introduces an effective equivalent stress in the creep potential that is based on the first and second invariants of the effective stress tensor, and on the joint invariant of the effective stress tensor and eigenvector associated with the maximum principal Cauchy stress. The formulation of the kinematic hardening rule is presented and discussed. All the material parameters in the model have been obtained from a series of proposed basic experiments with constant stresses. These model parameters are then used to predict the creep deformation of the aluminum alloy under multiaxial loading with constant stresses, and under non-proportional uniaxial and non-proportional multiaxial loadings for both isothermal and nonisothermal processes.     

Cyclic thermo-viscoplasticity of high density polyethylene

Observations are reported in uniaxial cyclic tensile tests (loading–unloading with various maximum strains) on high density polyethylene at temperatures ranging from room temperature up to 90 °C. It is demonstrated that the maximum stress per cycle and an apparent residual strain (measured at the instant when the tensile force vanishes under retraction) strongly decrease with temperature. The latter seems unexpected as the interval of temperatures covers the α-relaxation temperature, which is conventionally associated with activation of additional mechanisms for inelastic flow. A model is developed that captures the decrease in residual strain with temperature. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. The effects of temperature and maximum strain per cycle on residual strains are studied numerically.