Mathematical modelling of transformation plasticity in steels II: Coupling with strain hardening phenomena

The models for the plastic behaviour of steels during phase transformations proposed in Part I and in a previous paper ( et al. [1986b]) for the case of ideal-plastic phases are extended to include strain-hardening effects (isotropic or kinematic hardening). An expression for the transformation plastic strain rate is obtained by modifying the treatment of Part I in a suitable manner. The classical plastic strain rate is also studied in a similar way. Complementary evolution equations for the hardening parameters are finally given, taking into account the possible “recovery” of strain hardening during transformations (i.e., the fact that the newly formed phase can “forget,” partially or totally, the previous hardening).     

对图乘法的发展

将能量偏导数定理应用于力法中,探索了超静定结构力法计算的新思路．传统的力法是利用基本体系在基本未知力处的变形(连续)条件来建立力法方程的．为了克服传统力法中弯矩图乘时由于Mi和Mp图取自同一基本结构而产生的不便,利用卡氏第二定理,得到了力法的基本方程,证明了在力法中可以混合采用不同的基本结构来分别计算Mi和Mp,并分析了所得基本方程的物理意义．通过实例计算和比较,证明该方法完全正确,且大大简化了计算,使力法的实用性更进一步加强,有较大的使用价值．     

Transformation hardening and kinetics for stress assisted and temperature driven martensitic transformation in steels

In the present study a new phase fraction based transformation hardening and further a stress and temperature dependent transformation kinetics following the postulates in internal variable thermodynamics is presented. The generally formulated kinetics equation is applied to describe the progress of phase transformation in a Cr–Ni–Mo–Al–Ti maraging steel and represents a good tool to predict the continuous change of the slope of isovolume fractions.     

完全变换法在无网格伽辽金方法中的应用

由于移动最小二乘形函数一般不具有常规有限元或边界元形函数所具有的插值特征，本质边界条件的处理成为无网格伽辽金法实施中的一个难点。本文通过建立节点位移和广义位移之间的关系对移动最小二乘形函数进行修正，给出了修正的移动最小二乘形函数；以二维问题为例，对完全交换法在无网格伽辽金方法中的应用进行了研究，实现了本质边界条件在节点处的精确施加。数值计算结果表明该方法不仅简单合理，而且具有较高的精度、收敛性和稳定性。     

抗尺度变换的矢量地图匹配导航方法

相比于景象匹配导航,矢量地图匹配导航具有无损变换、空间效率和时间效率高等优势。针对矢量地图匹配导航过程中实时图与基准图尺度不一致的情况,提出了一种抗尺度变换的位姿解算算法,并将其应用到矢量地图匹配导航系统中。该算法将尺度因子作为未知参数,与位姿参数一起进行估计,从而避免了尺度差异造成的位姿解算误差较大的问题。仿真结果表明,抗尺度变换的位姿解算算法用于小尺度变换下的矢量地图匹配时,经度标准差和纬度标准差均小于0.06″,航向角标准差小于0.06°。     

Numerical integration method for kinematic hardening rules with partial activation of dynamic recovery term

In order to express steady state ratcheting, Abdel-Karim and Ohno formulated kinematic hardening rules on the assumption that dynamic recovery term of back stress is partially activated before it reaches its critical value. These rules have a feature that only the projection of plastic strain rate into the direction of back stress contributes to the dynamic recovery. They are in contrast with the other rules in which either/both accumulated plastic strain rate or/and plastic strain rate enters into the dynamic recovery term. In this paper, another feature, which has not been investigated by Abdel-Karim and Ohno, is introduced. Discussing this feature, characterized by decomposition of back stress into radial and tangential components, a numerical integration method based on radial-return mapping using radial components only is developed. This method consists of a trial predictor and final corrector depending on the current state of back stress and plastic strain increment. The applicability of the method is examined carefully and critically. It is shown that this method is effective and its accuracy is indistinguishable to the radial-return method for elastic-perfectly plastic materials.     

Comparison of isotropic hardening and kinematic hardening in thermoplasticity

A coupled thermo-mechanical problem is presented in this paper. The constitutive model is based on thermoplastic model for large strains where both kinematic and isotropic hardening are included. It is shown that a non-associated plasticity formulation enables thermodynamic consistent heat generation to be modeled, which can be fitted accurately to experimental data. In the numerical examples the effect of heat generation is investigated and both thermal softening and temperature-dependent thermal material parameters are considered. The constitutive model is formulated such that pure isotropic and pure kinematic hardening yield identical uniaxial mechanical response and mechanical dissipation. Thus, differences in response due to hardening during non-proportional loading can be studied. Thermally triggered necking is studied, as well as cyclic loading of Cook’s membrane. The numerical examples are solved using the finite element method, and the coupled problem that arises is solved using a staggered method where an isothermal split is adopted.     

激光光通量位移计在Hopkinson杆中的应用

简要介绍了激光位移测试技术的基本原理及实现方案,准静态和动态标定结果表明该系统可以用 于动态测试。给出了激光光通量位移计在Hopkinson杆实验中的3种应用:(1)用激光光通量位移计可以监 测试样的横向变形,从而得出试样的动态泊松比;(2)在动态断裂实验中,激光光通量位移计可以监测裂尖张 开位移,结合应变片的信息可以得出试样的动态断裂能;(3)将激光光通量位移计和石英晶体应力计相结合可 以直接在试样两端得到应力应变信息,用来测量超软材料在中应变率下的力学响应。     

Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening

The virtual fields method (VFM) has been specifically developed for solving inverse problems from dense full-field data. This paper explores recent improvements regarding the identification of elasto-plastic models. The procedure has been extended to cyclic loads and combined kinematic/isotropic hardening. A specific attention has also been given to the effect of noise in the data. Indeed, noise in experimental data may significantly affect the robustness of the VFM for solving such inverse problems. The concept of optimized virtual fields that minimize the noise effects, previously developed for linear elasticity, is extended to plasticity in this study. Numerical examples with models combining isotropic and kinematic hardening have been considered for the validation. Different load paths (tension, compression, notched specimen) have shown that this new procedure is robust when applied to elasto-plastic material identification. Finally, the procedure is validated on experimental data.     

Applications of maximum entropy method in capillary viscometry

Maximum entropy method (MEM) is presented as a technique for processing data obtained from capillary viscometers. The performance of MEM is assessed by comparing the viscosity versus shear rate curves generated by MEM against that obtained by the standard method based on the Weissenberg-Rabinowitsch-Mooney equation. In all the cases in vestigated, MEM proved to be a reliable technique in coping with the experimental noise in the capillary data.     

A new exact integration method for the Drucker–Prager elastoplastic model with linear isotropic hardening

This paper presents the exact stress solution of the non-associative Drucker–Prager elastoplastic model governed by linear isotropic hardening rule. The stress integration is performed under constant strain-rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory. Based on the time-continuous stress solution, a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex. Explicit expression for the algorithmically consistent tangent tensor is also derived. In addition, a fully analytical strain solution is also derived for the stress-driven case using constant stress-rate assumption. In order to get a deeper understanding of the features of these solutions, two numerical test examples are also presented.     

Characterization of the post-necking strain hardening behavior using the virtual fields method

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.     

等温淬火球墨铸铁渣浆泵叶片磨损试验的正交分析

采用正交试验分析了在实验室条件下影响奥贝球铁叶片磨损的因素,探讨了决定叶片材料特性的成分和热处理工艺以及叶、磨料种类等水力学条件对叶片磨损的综合影响,研究结果表明,渣浆泵叶片的磨损失效受到材料特性和水力条件的综合影响;叶片材料成分和热处理工艺交互作用是影响叶磨损的主要因素,其次是磨料种类及叶型症 交试验是研究 渣浆泵磨损规律的有效方法 。     

Modeling of intra-crystalline hardening of materials with particles

A two-level homogenization approach is developed for the micromechanical modeling of the elastoplastic behavior of polycrystals containing intracrystalline non-shearable particles. First, a micro-meso transition is employed to establish a constitutive relation for a single crystal containing particles. The behavior of an equivalent single crystal with particles is derived from the classical formulation of plasticity of the single crystal based on the Schmid's law and crystallographic gliding. Then, the transition to the macroscopic scale is performed with a self-consistent scheme to determine the elastoplastic behavior of the macro homogeneous material. The obtained global behavior is characterized by a mixed anisotropic and kinematic hardening related to an evolution of inter- and intra-granular material microstructure. Results have been analyzed in light of second and third order internal stresses developed during the plastic flow. Especially, yield surfaces have been determined for various preloadings and particle volume fractions.     

Plastic analysis of thin plates with anisotropic hardening

In this paper we discuss the adoption of the anisotropic hardening model due to the existence of Bauschinger effect when thin plate is applied by repeated loading. The loading condition of thin plates for linear kinematic hardening has been deduced in terms of generalized forces and generalized plastic deformation. And it can be extended to nonlinear kinematic hardening and mixed hardening. Finally as an example the numerical results are obtained for a circular plate.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

Nonlinear behavior of circular plates with work hardening

Tests were performed on two simply supported plates of aluminum alloy 2024-0, under a central concentrated load, with peak deflections up to 2.6 times the thickness. The load was provided by a small-diameter hard-steel rod. The plates had diameter-to-thickness ratios (D/h) of 20 and 41. Measurements were made of load, deflections and strains; membrane and bending strains were calculated from the test data. The test data are presented in comparison with theoretical predictions generated by the Grumman-developed finiteelement-computer code PLANS, which includes material and geometric nonlinearities. The theoretical prediction was excellent for deflections, and generally good for strains, when the central force was represented by a line load around the loading rod's contact circle. Using a uniform pressure as the central force caused the theory to slightly overpredict the peak deflections and greatly overpredict the peak strains at the plate center. The plates exhibited initial loss of stiffness under the plastic-bending behavior, followed by a restiffening when the large deflections caused a rapidly increasing membrane action that provided much additional resistance to the applied load.