A thermodynamics framework and numerical aspects for transformation-induced plasticity at large strains

In this work, we present a macroscopic material model for simulation transformation-induced plasticity, which is an important phenomenon in metal forming processes. The model is formulated within a thermodynamic framework at large strains. In order to account for both, phase transformation and plasticity, yield functions are related to these effects. Then, applying the concept of maximum dissipation evolution equations are obtained for the inelastic strains, the transformation strains, a hardening variable and the volume fraction of martensite. Furthermore the numerical implementation of the constitutive equations into a finite element program is described. In a numerical example we investigate the austenite-to-martensite phase transformation in a shaft subjected to thermo-mechanical loading in a hybrid-forming process.     

A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite

A 3-D constitutive model for polycrystalline shape memory alloys (SMAs), based on a modified phase transformation diagram, is presented. The model takes into account both direct conversion of austenite into detwinned martensite as well as the detwinning of self-accommodated martensite. This model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stress–temperature space. The model is based on thermodynamic potentials and utilizes three internal variables to predict the phase transformation and detwinning of martensite in polycrystalline SMAs. Complementing the theoretical developments, experimental data are presented showing that the phase transformation temperatures for the self-accommodated martensite to austenite and detwinned martensite to austenite transformations are different. Determination of some of the SMA material parameters from such experimental data is also discussed. The paper concludes with several numerical examples of boundary value problems with complex thermomechanical loading paths which demonstrate the capabilities of the model.     

Micromechanics of plastic deformation and phase transformation in a three-phase TRIP-assisted advanced high strength steel: Experiments and modeling

The micromechanics of plastic deformation and phase transformation in a three-phase advanced high strength steel are analyzed both experimentally and by microstructure-based simulations. The steel examined is a three-phase (ferrite, martensite and retained austenite) quenched and partitioned sheet steel with a tensile strength of ~980 MPa. The macroscopic flow behavior and the volume fraction of martensite resulting from the austenite–martensite transformation during deformation were measured. In addition, micropillar compression specimens were extracted from the individual ferrite grains and the martensite particles, and using a flat-punch nanoindenter, stress–strain curves were obtained. Finite element simulations idealize the microstructure as a composite that contains ferrite, martensite and retained austenite. All three phases are discretely modeled using appropriate crystal plasticity based constitutive relations. Material parameters for ferrite and martensite are determined by fitting numerical predictions to the micropillar data. The constitutive relation for retained austenite takes into account contributions to the strain rate from the austenite–martensite transformation, as well as slip in both the untransformed austenite and product martensite. Parameters for the retained austenite are then determined by fitting the predicted flow stress and transformed austenite volume fraction in a 3D microstructure to experimental measurements. Simulations are used to probe the role of the retained austenite in controlling the strain hardening behavior as well as internal stress and strain distributions in the microstructure.     

SUPER-ELASTIC CONSTITUTIVE MODEL CONSIDERING PLASTICITY AND ITS FINITE ELEMENT IMPLEMENTATION

Based on the experimental results of super-elastic NiTi alloy, a three-dimensional super-elastic constitutive model including both of stress-induced martensite transformation and plasticity is constructed in a framework of general inelasticity. In the proposed model, transformation hardening, reverse transformation of stress-induced martensite, elastic mismatch between the austenite and martensite phases, and temperature-dependence of transformation stress and elastic modulus of each phase are considered. The plastic yielding of martensite occurred under high stress is addressed by a bilinear isotropic hardening rule. Drucker-Prager-typed transformation surfaces are employed to describe the asymmetric behavior of NiTi alloy in tension and compression. The prediction capability of the proposed model is verified by comparing the simulated results with the correspondent experimental ones. Based on backward Euler＇s integration, a new expression of consistent tangent modulus is derived. The proposed model is then implemented into a finite element package ABAQUS by user-subroutine UMAT. Finally, the validity of such implementation was verified by some numerical samples.     

A tensorial description of the transformation kinetics of the martensitic phase transformation

Based on a local examination of the phase transition front, a macroscopic second order tensor describing the thermodynamic force for the phase transformation is proposed. Consequently, an associated thermodynamic flux is introduced. These tensorial variables are embedded into a material law which describes the behavior of steels during the austenite–martensite phase transformation. The material law is implemented into a finite element formulation. Homogeneous tests in pure tension/compression and torsion are performed to verify the behavior of the material law. Due to the independent modeling of the behavior of the phases, the influence of the yield stress of the austenite on the transformation kinetics can be verified. A classical example is presented to show the ability of the model to calculate large structural problems.     

Crystal plasticity finite element modeling of mechanically induced martensitic transformation (MIMT) in metastable austenite

A new crystal plasticity model incorporating the mechanically induced martensitic transformation in metastable austenitic steel has been formulated and implemented into the finite element analysis. The kinetics of martensite transformation is modeled by taking into consideration of a nucleation-controlled phenomenon, where each potential martensitic variant based on Kurdjumov–Sachs (KS) relationship has different nucleation probability as a function of the interaction energy between externally applied stress and lattice deformation. Therefore, the transformed volume fractions are determined following selective variants given by the crystallographic orientation of austenitic matrix and applied stress in the frame of the crystal plasticity finite element. The developed finite element program is capable of considering the effect of volume change by the Bain deformation and the lattice-invariant shear during the martensitic transformation by effectively modifying the evolution of plastic deformation gradient of the conventional rate-dependent crystal plasticity finite element. The validation of the proposed model has been carried out by comparing with the experimentally measured data under simple loading conditions. Good agreements with the measurements for the stress–strain responses, transformed martensitic volume fractions and the influence of strain rate on the deformation behavior will enable the model to be promising for the future applications to the real forming process of the TRIP aided steel.     

Experimental characterization and micromechanical modeling of superelastic response of a porous NiTi shape-memory alloy

Porous shape-memory alloys are usually brittle due to the presence of various nickel-titanium intermetallic compounds that are produced in the course of most commonly used synthesizing techniques. We consider here a porous NiTi shape-memory alloy (SMA), synthesized by spark-plasma sintering, that is ductile and displays full shape-memory effects over the entire appropriate range of strains. The porosity however is only 12% but the basic synthesizing technique has potential for producing shape-memory alloys with greater porosity that still are expected to display superelasticity and shape-memory effects. The current material has been characterized experimentally using quasi-static and dynamic tests at various initial temperatures, mostly within the superelastic strain range, but also into the plastic deformation regime of the stress-induced martensite phase. To obtain a relatively constant strain rate in the high strain-rate tests, a novel pulse-shaping technique is introduced. The results of the quasi-static experiments are compared with the predictions by a model that can be used to calculate the stress-strain response of porous NiTi shape-memory alloys during the austenite-to-martensite and reverse phase transformations in uniaxial quasi-static loading and unloading at constant temperatures. In the austenite-to-martensite transformation, the porous shape-memory alloy is modeled as a three-phase composite with the parent phase (austenite) as the matrix and the product phase (martensite) and the voids as the embedded inclusions, reversing the roles of austenite and martensite during the reverse transformation from fully martensite to fully austenite phase. The criterion of the stress-induced martensitic transformation and its reversal is based on equilibrium thermodynamics, balancing the thermodynamic driving force for the phase transformation, associated with the reduction of Gibbs’ free energy, with the resistive force corresponding to the required energy to create new interface surfaces and to overcome the energy barriers posed by various microstructural obstacles. The change in Gibbs’ free energy that produces the driving thermodynamic force for phase transformation is assumed to be due to the reduction of mechanical potential energy corresponding to the applied stress, and the reduction of the chemical energy corresponding to the imposed temperature. The energy required to overcome the resistance imposed by various nano- and subnano-scale defects and like barriers, is modeled empirically, involving three constitutive constants that are then fixed based on the experimental data. Reasonably good correlation is obtained between the experimental and model predictions.     

A CONSTITUTIVE MODEL FOR TRANSFORMATION, REORIENTATION AND PLASTIC DEFORMATION OF SHAPE MEMORY ALLOYS

A constitutive model is developed for the transformation, reorientation and plastic deformation of shape memory alloys (SMAs). It is based on the concept that an SMA is a mixture composed of austenite and martensite, the volume fraction of each phase is transformable with the change of applied thermal-mechanical loading, and the constitutive behavior of the SMA is the combination of the individual behavior of its two phases. The deformation of the martensite is separated into elastic, thermal, reorientation and plastic parts, and that of the austenite is separated into elastic, thermal and plastic parts. Making use of the Tanaka’s transformation rule modified by taking into account the effect of plastic deformation, the constitutive model of the SMA is obtained. The ferroelasticity, pseudoelasticity and shape memory effect of SMA Au-47.5 at.%Cd, and the pseudoelasticity and shape memory effect as well as plastic deformation and its effect of an NiTi SMA, are analyzed and compared with experimental results.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.     

具有非线性表面换热系数半圆柱薄壳在快速冷却过程中的热分析

用实验和数值计算相结合的方法，得到半圆柱壳体快速冷却过程中内外表面的非线性表面换热系数。在此基础上，用有限单元法对半圆柱壳体的热应力和变形进行了分析。在数值计算中，模拟钢的CCT图，计算了奥氏体、珠光体、贝氏体和马氏体的体积百分比，并将热物理性质和力学性能处理为相变体积百分比和温度的函数。所得结果表明，在半圆柱壳体快速冷却过程中的热应力和变形计算中，有必要考虑非线性表面换热系数、相变等非线性效应。     

Functionally graded structural members obtained via the low temperature strain induced phase transformation

The notion of functionally graded materials (FGM) covers all domains of discrete and smooth gradation of material microstructure designed in order to obtain macroscopic features suitable for a given application. A special class of multi-phase materials with graded microstructure can be obtained at cryogenic temperatures as a result of smooth transition from the parent phase to the secondary phase. The required continuously graded material features are obtained at low temperatures via the mechanism of controlled strain induced phase transformation from the purely austenitic to the martensitic lattice (γ → α′). Several families of ductile materials are known to behave in a metastable way when strained at very low temperatures. Among them the austenitic stainless steels are extensively used to construct components of the superconducting magnets, cryogenic transfer lines and other structural members loaded in cryogenic conditions. The constitutive model used to describe mathematically the plastic strain induced phase transformation at low temperatures involves strain hardening where two fundamental effects play an important role: interaction of dislocations with the martensite inclusions and increase in material tangent stiffness due to the mixture of harder martensite with softer austenite. The interaction of dislocations with the martensite inclusions is reflected by the hardening modulus that depends on the volume fraction of martensite. Here, a linear approximation, based on the micro-mechanics analysis, is used. On the other hand, evaluation of the material tangent stiffness of two-phase continuum is based on the classical homogenization scheme and takes into account the local tangent moduli of the components, as postulated by Hill [Hill, R., 1965. A self consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222]. In the present paper, the Mori–Tanaka homogenisation scheme is applied. Both effects contribute to strong nonlinear hardening that occurs as soon as the phase transformation process begins. The material model is suitable for a wide range of temperatures, however the best results are obtained at very low temperatures, where the linearized kinetic law of phase transformation is valid [Garion, C., Skoczeń B., 2002. Modeling of plastic strain induced martensitic transformation for cryogenic applications. J. Appl. Mech. 69 (6), 755–762]. As the application field the structural members in the form of rods (cylinders) of circular cross-section, used as parts of the carrying structures, are analyzed. The required graded microstructure of the material is obtained by imposing torsion at cryogenic temperatures. Both the intensity of the phase transformation and the depth of the transformed zone is obtained by suitable kinematic control (angle of twist). The closed form solutions for the stress state and torque as a function of the angle of twist are shown.     

Modelling of the plastic flow of trip-aided multiphase steel based on an incremental mean-field approach

An incremental mean-field model is developed for the prediction of transformation induced plasticity (TRIP) in multiphase steel. The partitioning of strain between softer and harder constituents is computed based on an elastic-plastic Mori–Tanaka approach that accounts for the progressive transformation of austenite into martensite. The latter transformation is predicted using an energy-balance criterion that is formulated at the level of individual austenite grains. The model has been tested against experimental data. Macroscopic stress-strain curves and rate of martensite formation have been measured on sheet samples subjected to various loading modes: uniaxial tension, simple shear, and (in-plane) uniaxial compression. These experiments were performed at 20 °C and the uniaxial tensile test was repeated at ?30 °C. The mean-field model produces fair predictions of the macroscopic hardening resulting from TRIP on the condition that a sufficient proportion of the load is carried by the very hard martensite inclusions. Such prediction implies that one accounts for the stress heterogeneity across the ferrite-based matrix. At the same time, the model reproduces the elastic lattice strains and the plastic elongation which are measured within the phases by neutron diffraction and by image correlation in a scanning electron microscope, respectively. The model can be used in finite element simulations of forming processes which is illustrated in a study of necking of a cylindrical bar under uniaxial tension.     

Numerical investigation of the interaction between the martensitic transformation front and the plastic strain in austenite

Phase-field simulations of the martensitic transformation (MT) in an austenitic matrix which has already undergone the plastic deformation are carried out. For this purpose the elasto-plastic phase-field approach of incoherent MT developed in a previous work [Kundin et al., 2011. A phase-field model for incoherent martensitic transformations including plastic accommodation processes in the austenite. J. Mech. Phys. Solids 59, 2082–2012] is used. The evolution equation for the dislocation density field is extended by taking into account the thermal and athermal annihilation of the dislocations in the austenitic matrix and the athermal annihilation at the transformation front. It is shown that the plastic deformation in the austenite caused by the MT interacts with the dislocation field and the MT front that leads to an inhomogeneous increasing of the total dislocation density. During the phase transformation one part of the dislocations in the austenite is inherited by the martensitic phase and this inheritance depends on the kinetics and the crystallography of MT. Another part of dislocations annihilates at the transformation front and decreases the dislocation density in the growing martensite. Based on the simulation results the specific type of phenomenological dependency between the inherited dislocations, the martensite phase fraction and the plastic deformation is proposed.     

Grain size effects in multiphase steels assisted by transformation-induced plasticity

The influence of the austenitic grain size on the overall stress–strain behavior in a multiphase carbon steel is analyzed through three-dimensional finite element simulations. A recently developed multiscale martensitic transformation model is combined with a plasticity model to simulate the transformation-induced plasticity effects of a grain of retained austenite embedded in a ferrite-based matrix. Grain size effects are included via a surface energy term in the Helmholtz energy. Tensile simulations for representative orientations of the grain of retained austenite show that the initial stability of the austenite increases as the grain size decreases. Consequently, the effective strength is initially higher for smaller grains. The influence of the grain size on the evolution of the transformation process strongly depends on the grain orientation. For “hard” orientations, the transformation rate is higher for larger grains. In addition, the phase transformation is partially suppressed as the grain size decreases. In contrast, for “soft” orientations, the transformation rate is lower for larger grains. The phase transformation is more homogeneous for smaller grains and, consequently, the effective transformation strain is larger. Nevertheless, in multiphase carbon steels with a relatively low percentage of retained austenite, the influence of the austenitic grain size on the overall constitutive response is smaller than the influence of the austenitic grain orientation.     

A thermodynamic finite-strain model for pseudoelastic shape memory alloys

A thermodynamic finite-strain model describing the pseudoelastic response of shape memory alloys is proposed. The model is based on a self-consistent Eulerian theory of finite deformations using the logarithmic rate. Purely elastic material response is derived from a hyperelastic potential. The mass fraction of martensite is introduced as internal state variable to indicate the thermomechanical state of the phase transforming material. The evolution of martensite is governed by a kinetic law which is derived from the Helmholtz free energy of the two-phase solid and takes the heat generated during phase transition into account. The material model is implemented into a finite element code in an updated Lagrangian scheme and calibrated to experimental data. Simulations under different loading conditions illustrate the characteristics of the model.     

考虑塑性和刚度不匹配的焊接残余应力估算

现有残余应力计算方法未能考虑材料塑性变形和焊接接头刚度不匹配的影响,使得焊接残余应力计算结果和实际残余应力存在较大偏差.在2219-T87铝合金钨极氩弧焊焊接头残余应力测试基础上,提出一种基于非线性有限元和材料弹性模量分区的残余应力—释放应变曲线的残余应力计算方法,研究了材料塑性变形和接头刚度不匹配对焊接残余应力计算的影响.结果表明,焊接接头中非均质材料塑性不匹配可以引起对于残余应力计算的较大误差;材料塑性变形对残余应力的影响大于接头刚度不匹配对残余应力的影响.所提出方法修正了传统方法在焊接接头的残余应力计算中由于未考虑接头非均质材料塑性不匹配而引起的误差.     

Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys

Except for the recoverable strain induced by phase transformation, NiTi alloys are very ductile even in the martensite phase. The purpose of the present paper is to study the influence of permanent deformation, which results from plastic deformation of martensite, on the mechanical behaviour of pseudoelastic NiTi alloys. Based on phenomenological theory of martensitic transformation and crystal plasticity, a new three dimensional micromechanical model is proposed by coupling both the slip and twinning deformation mechanisms. The present model is implemented as User MATerial subroutine (UMAT) into ABAQUS/Standard to study the influences of plastic deformation on the stress and strain fields, and on the evolution of martensite transformation. Results show that with the increasing of plastic deformation the residual strain increases and the phase transformation stress–strain curves from the martensite to austenite become steeper and less obvious. Both characteristics, stabilisation of martensite and impedance of the reverse transformation, due to plastic deformation are captured.     

A thermodynamical constitutive model for shape memory materials. Part II. The SMA composite material

The phenomenological SMA equations developed in Part I are used in this second paper to derive the free energy and dissipation of a SMA composite material. The derivation consists of solving a boundary value problem formulated over a mesoscale representative volume element, followed by an averaging procedure to obtain the macroscopic composite constitutive equations. Explicit equations are derived for the transformation tensors that relate the composite transformation strain rate to the phase transformation rate in the fiber and matrix. Some key findings for the two-way SME in a SMA fiber/elastomer matrix composite are that processing-induced residual stresses alter the composite austenite start and martensite start temperatures, as well as the amount of composite strain recovered during a complete cycle of temperature and fiber martensite volume fraction. Relative to the two-way SME response of stiff-matrix composites, it was found that compliant-matrix composites: (1) complete the phase transformation over a narrower temperature range; (2) exhibit greater transformation strain during the reverse transformation; and (3) undergo an incomplete strain cycle during a complete cycle of temperature and fiber martensite volume fraction. Due to the interaction of the fiber and matrix during transformation, macroscopic proportional stressing of the composite results in non-proportional fiber stressing, which in turn causes a small amount of martensitic reorientation to occur simultaneously with the transformation.     

A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures

Here, the effects of localization and propagation of martensitic phase transformation on the response of SMA thin structures subjected to thermo-mechanical loadings are investigated using nonlocal constitutive model in conjunction with finite element method. The governing equations are derived based on variational principle considering thermo-mechanical equilibrium and the spatial distribution of the nonlocal volume fraction of martensite during transformation. The nonlocal volume fraction of martensite is defined as a weighted average of the local volume fraction of martensite over a domain characterized by an internal length parameter. The local version of the thermo-mechanical behavior model derived from micromechanics considers the local volume fraction of martensite and the mean transformation strain. A 4-noded quadrilateral plane stress element with three degrees of freedom per node accounting for in-plane displacements and the nonlocal volume fraction of martensite is developed. Numerical simulations are conducted to bring out the influence of material and geometrical heterogeneities (perturbations/defects) on the localization and propagation of phase transformation in SMA thin structures. Also, a sensitivity analysis of the material response due to the localization and the other related model parameters is carried out. The detailed investigation done here clearly shows that the localization of phase transformation has significant effect on the response of shape memory alloys.     

The influence of cooling rates on the distribution of the structure and phases in hardened cylindrical elements

Summary The paper presents an analysis of the temperature field and phase transformation kinetics for cylindrical steel elements with arbitrary shaped cross sections. The influence of different cooling rates as well as different time intervals to achieve the half of the full austenite into pearlite transformation τ_0.5 and weight fractions of pearlite and martensite are taken into account. The analysis is based on Lomakin's theory [2, 3, 4] and its modification [5] as well as the laws of phase transformation kinetics of [9]. The elaborated program of numerical calculations refers to steel grades characterized by C-shaped T-T-T curves and with carbon content close to that of an eutectoidal steel. The analysis is made for a two-dimensional region. As an example, a spline shaft with arbitrary shaped cross section is investigated. It is assumed that the heat exchange occurs on the boundary of the cross section alone. The solution of the problem is based on the variational difference method, being a combination of the finite element method and the finite difference method. Accepted for publication 6 August 1996