Measurement and modeling of back stress at intermediate to high homologous temperatures

Unified creep-plasticity models often require a number of internal state variables to accurately capture the path dependence of rate- and temperature-dependent alloys especially at intermediate to high homologous temperatures. However, it is impossible to fully measure the internal state variables directly. Consequently, assumptions on the relative importance of the various state variables must be made when determining the material parameters and their evolution must be inferred. Any enhancements designed to better capture the evolution of the state variables are dependent on these assumptions and are generally limited. The strain transient dip test and the rapid load/unload test are two indirect experimental methods used to obtain an approximate measure of the evolutionary nature of the back stress in 60Sn–40Pb, a common rate- and temperature-dependent solder alloy. The rapid load/unload test gives a better measure of back stress because there is less time for the internal state to evolve during data acquisition. Modifications to the back stress evolution equations in the McDowell unified creep-plasticity model are proposed using these new measurements as guidance. The modified model can better capture the transients under cyclic loading and following strain-rate jumps.     

316L不锈钢随动强化模型改进研究

本文对316L不锈钢进行了单轴与多轴非比例路径下的应力控制棘轮试验,考察了应力幅值、平均应力和加载历程对棘轮特性的影响。同时进行了应变控制循环试验以研究材料的应力松弛特性。试验结果表明轴向棘轮效应在对称剪切荷载下效果明显,同时棘轮应变随应力幅值和平均应力的增加而增加。研究了Chen-Jiao随动强化模型与Jiang-Sehitoglu随动强化模型采用的单轴与多轴参数对背应力分量增量方向的影响,将Chen-Jiao模型中的多轴系数替换为界面饱和率,并在此基础上引入新的参数对塑性模量系数进行修正,计算结果表明修正后的模型能提升应力控制下多轴棘轮的预测精度,并能很好的预测应力松弛现象,表明了新模型的正确性与有效性。     

显式方法精确模拟形状记忆合金循环荷载下的变形行为

本文提出全新的有限弹塑性J2流方程,用来显式、精确地模拟SMAs（形状记忆合金）材料在循环加载-卸载条件下从塑性逐渐转变为伪弹性的变形行为．首先,改进流动法,使得本构方程耦合屈服中心的移动和屈服面的增大,并改进背应力演化方程,使模型可以产生强烈的包辛格效应,从理论上具备模拟SMAs独特变形行为的能力;其次,构造全过程下的统一硬化函数显式表达式,代入本构方程后能得到符合要求的形函数;再次,利用选定的数据点构造统一光滑的上屈服函数,再利用上下屈服应力之间的一种线性关系,推导得到下屈服阶段的形函数;最后,只需要给定一个参数就可以得到单个循环结果,利用拉格朗日插值方法构建参数随循环次数变化的函数,就可以模拟任意循环荷载下的变形行为．通过模型结果和实验数据对比证明新方法的有效性．     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Representation of cyclic hardening and softening properties using continuous variables

Our aim is the modeling of cyclic hardening, cyclic softening, cyclic mean stress relaxation, and additional nonproportional cyclic hardening. We do so by means of hardening functionals for back stress and yield stress without employing additional memory surfaces. Rather, we suppose all quantities to evolve simultaneously during elastic-plastic loading in a continuous manner. The basic idea is to formulate evolution equations for the hardening variables, which are of the “hardening/dynamic recovery” format with respect to a transformed arc length. The corresponding transformation is influenced by continuously evolving parameters, measuring strain amplitude and nonproportionality during the recent process history. Although the resulting, model has a very simple structure, it is capable of describing the basic phenomena under quite general loading conditions.     

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.     

An orthotropic yield criterion in a 3-D general stress state

An anisotropic yield criterion with a general representation was suggested. The yield criterion was derived from the use of the invariants of the stress tensor, similar in constructing an isotropic yield criterion, but which contains a “three-yield-system hypothesis” specifying the state of anisotropy. When applied to rolled sheet metals, such as high strength steels and aluminum alloys, the criterion can be treated in an analytical form to facilitate analyses of engineering problems under a general triaxial stress state. For this specified form, anisotropic properties of the predicted yield surface were characterized by seven experimental results obtained from three standard uniaxial-tension tests and one equibiaxial-tension test. When the applied material becomes isotropic it is transformed back to the form of the von-Mises’s criterion. Since the convexity of the yield criterion was proven in its general type, the characterized criterion is valid as a plastic potential in the implementation of finite element programs. It was shown, in full agreement with experimental data, that the accuracy of predicted yield surface was similar to that of predicted by the polycrystal model. Considering the equibiaxial-tension data, in general, may be not available from material supplies, a formulated relation covered variables of the equibiaxial tension and uniaxial tension was proposed. The relation can be used to calculate the equibiaxial-tension yield stress from the experimental data in uniaxial tensions. Several calculated results showed very close to the experimental results.     

304不锈钢室温单轴循环棘轮行为的粘塑性本构描述

在统一粘塑性循环本构模型的框架下对循环硬化的304不锈钢的单轴棘轮行为进行了本构描述。模型中通过随动硬化背应力演化和各向同性变形阻力演化对304不锈钢在非对称应力循环下的循环附加硬化和循环流动特性进行了分析，同时考虑了加载历史对循环棘轮行为的影响。将模型应用于304不锈钢室温单轴循环棘轮行为及其对加载历史依赖性的描述中，预言结果与实验结果吻合较好。     

现代统一塑性理论

经典的本构理论不能精确描述材料表现出的率相关性和非线性本构关系,导致现代统一塑性理论的出现.各派现代统一塑性理论形式不相同,但最根本的观点接近:①材料变形由弹性变形与非弹性变形两部分组成,不再将非弹性变形区分为塑性变形与蠕变;②存在着两种物理参量:背应力X与粘滞系数K_0 σ—X构成非弹性形变的动力,K是非弹性应变的阻力,因而ε~(in)=f((σ-X)/K);③原始材料X= 0,K=K_0,K_0为较大的标量,所以加载初始表现出线弹性,续继加载X与K按演化规律变化导致非线性形变规律的出现.各派统一塑性理论的主要差异在于X与K的演化公式不同.各有其长亦有其短.在大量试验基础上取各派之长,舍各派之短,融合为一科学的理论是今后发展的方向.      

A non-equilibrium thermodynamic geometric structure for thermoviscoplasticity with maximum dissipation

A maximum dissipation principle induces a class of viscoplastic evolution equations within a non-linear, non-equilibrium thermodynamic structure which extends Gibbs thermostatics. The evolution of a relaxation process is determined through non-linear affinities, which generalize the linear Onsager construct, along with the maximum dissipation principle and the long term states. The relaxation modulus is the norm of the plastic strain rate. Forced processes, in which the control variables change with time, are assumed to be relaxation processes at each instant. This class of thermoviscoplastic models includes the three-dimensional Freed–Chaboche–Walker model in which the internal state variables are the back stress, the drag strength and the elastic yield strength for creep initiation.     

On finite plastic deformation of anisotropic metallic materials

The classical flow theory of plasticity has been extended to the large strain range for anisotropic metallic materials. The following concepts have been incorporated into the constitutive framework: (1) the convected coordinates and the contravariant true stress, (2) an observer independent yield function, (3) the convected rate for general kinematics of deformation, and (4) the rotation of material texture expressed by a constitutive spin. The theory has been applied to the problem of free-end torsion of a thin-walled tube. The predicted results of shear stress-strain curve, axial strain versus shear strain curve, back stress versus shear strain curve, and initial and subsequent yield surfaces compare favorably with experimental data obtained by the author and his co-workers. It has been shown that the yield function defined by the contravariant true stress can account for the distortion of the yield loci.     

Viscoelastic plastic rheological model for particle filled polymer melts

A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.     

A parameter for measuring the magnitude of a change of strain path: Validation and comparison with experiments on low carbon steel

Various tension-tension and tension-shear strain sequential experiments have been performed on low carbon steel sheet along different material axes. Owing to the rapid plastic instability that occurs during the reloading in uniaxial tension of prestrained samples, the results are focussed on the evolution of the macroscopic reloading yield stress (back extrapolated stress). For a given prestrain amount, the reloading stress is a function of the magnitude of the strain path change. A parameter is proposed, which allows the comparison of different sequential loading tests: the scalar product of the unit tensors corresponding to the prestrain and to the subsequent strain modes, respectively. For low carbon steel, a single curve is obtained when the reloading stress, normalized by the stress along the monotonic strain path is plotted against this parameter whatever the combination of loading sequences and the material direction of the prestrain.     

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.     

微构件自由弯曲回弹的数值模拟

回弹是影响弯曲成形精度的一个重要因素,由于尺度效应的存在,微构件的弯曲回弹问题更复杂.利用动态显式与静态隐式算法相结合的方法数值模拟了C1200黄铜V形微构件的自由弯曲,与其超薄板三点弯曲的实验结果吻合度很好,验证了该方法模拟微构件弯曲回弹的可行性.分析了板厚t、晶粒大小d、t/d(板厚与晶粒大小比值)和残余应力对回弹的影响,结果显示t/d可以作为表征回弹的重要参数,回弹量随着t/d减小而增大;随着t和d的增大,工件的残余应力也增大,会影响工件的质量.可以采用适当热处理工艺增大材料的晶粒尺寸来减少回弹,当然也要兼顾残余应力对工件质量的影响.     

超弹性镍钛形状记忆合金单轴相变棘轮行为的宏观唯象本构模型

超弹性镍钛形状记忆合金因其良好的力学性能以及独特的超弹性和形状记忆效应已广泛应用于土木工程、航空航天和生物医疗等多个领域,在实际服役环境中超弹性镍钛合金元件不可避免地会承受不同应力水平的循环载荷作用,亟待建立描述相变棘轮行为(即峰值应变和谷值应变随着正相变和逆相变循环的进行不断累积)的循环本构模型.为此,基于已有的超弹性镍钛形状记忆合金在不同峰值应力下的单轴相变棘轮行为实验研究结果,在广义黏塑性框架下,对Graesser等提出的通过背应力非线性演化方程反映超弹性镍钛形状记忆合金超弹性行为的一维宏观唯像本构模型进行了拓展,考虑了正相变和逆相变过程中特征变量的差异及其随循环的演化,以非弹性应变的累积量为内变量引入了正相变开始应力、逆相变开始应力、相变应变和残余应变的演化方程,同时通过峰值应力与正相变完成应力的比值来确定演化方程中的相关系数,建立了描述超弹性镍钛合金单轴相变棘轮行为的本构模型.将模拟结果与对应的实验结果进行对比发现,建立的宏观唯像本构模型能够合理地描述超弹性镍钛形状记忆合金的单轴相变棘轮行为及其峰值应力依赖性,模型的预测结果和实验结果吻合得很好.     

A mathematical theory of materials with elastic range and the definition of back stress tensor

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｇｅｎｅｒａｌ,ｔｈｅｋｉｎｅｍａｔｉｃａｌｈａｒｄｅｎｉｎｇｂｅｈａｖｉｏｒｏｆｍａｔｅｒｉａｌｓｉｓｄｅｓｃｒｉｂｅｄｂｙａｖａｒｉａｂｌｅｃａｌｅｄｂａｃｋｓｔｒｅｓｏｒｓｈｉｆｔｔｅｎｓｏｒ．Ｉｔｓｖａｌｕｅｒｅｐ...     

A multi-scale constitutive model for the sintering of an air-plasma-sprayed thermal barrier coating, and its response under hot isostatic pressing

A micromechanical model is developed for the sintering of an air-plasma-sprayed, thermal barrier coating, and is used to make predictions of microstructure evolution under free sintering and under hot isostatic pressing. It is assumed that the splats of the coating are separated by penny-shaped cracks; the faces of these cracks progressively sinter together at contacting asperities, initially by the mechanism of plastic yield and subsequently by interfacial diffusion. Diffusion is driven by the reduction in interfacial energy at the developing contacts of the cracks and also by the local contact stress at asperities. The contact stress arises from the remote applied stress and from mechanical wedging of the rough crack surfaces. Sintering of the cracks leads to an elevation in both the macroscopic Young's modulus and thermal conductivity of the coating, and thereby leads to a degradation in thermal performance and durability. An assessment is made of the relative roles of surface energy, applied stress and crack face roughness upon the sintering response and upon the evolution of the pertinent mechanical and physical properties. The evolution in microstructure is predicted for free sintering and for hot isostatic pressing in order to provide guidance for experimental validation of the micromechanical model.