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.     

Transformation-induced plasticity in ferrous alloys

We study the mechanical behavior of a class of multiphase carbon steels where metastable austenite at room temperature is found in grains dispersed in a ferrite-based matrix. During mechanical loading, the austenite undergoes a displacive phase change and transforms into martensite. This transformation is accommodated by plastic deformations in the surrounding matrix. Experimental results show that the presence of austenite typically enhances the ductility and strength of the steel. We use a recently developed model (Turteltaub and Suiker, 2005) to analyze in detail the contribution of the martensitic transformation to the overall stress-strain response of a specimen containing a single island of austenite embedded in a ferrite-based matrix. Results show that the performance of the material depends strongly on the lattice orientation of the austenite with respect to the loading direction. More importantly, we identify cases in which the presence of austenite can in fact be detrimental in terms of strength, which is relevant information in order to improve the behavior of this class of steels.     

Strain gradient plasticity analysis of transformation induced plasticity in multiphase steels

“To what extent do plastic strain gradients affect the strengthening resulting from the transformation of small metastable inclusions into hard inclusions within a plastically deforming matrix?” is the central question addressed here. Though general in the approach, the focus is on the behavior of TRIP-assisted multiphase steels. A two-dimensional embedded cell model of a simplified microstructure composed of a single metastable austenitic inclusion surrounded by a soft ferritic matrix is considered. The cell is inserted in a large homogenized medium. The transformation of a fraction of the austenite into a hard martensite plate is simulated, accounting for a transformation strain, and leading to complex elastic and plastic accommodation. The size of a transforming plate in real multiphase steels is typically between 0.1 and 2 μm, a range of size in which plastic strain gradient effects are expected to play a major role. The single parameter version of the Fleck–Hutchinson strain gradient plasticity theory is used to describe the plasticity in the austenite, ferrite and martensite phases. The higher order boundary conditions imposed on the plastic flow have a large impact on the predicted strengthening. Using realistic values of the intrinsic length parameter setting the scale at which the gradients effects have an influence leads to a noticeable increase of the strengthening on top of the increase due to the transformation of a volume fraction of the retained austenite. The geometrical parameters such as the volume fraction of retained austenite and of the transforming zone also bring significant strengthening. Strain gradient effects also significantly affect the stress state inside the martensite plate during and after transformation with a potential impact on the damage resistance of these steels.     

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.     

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.     

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.     

A Simplified Model for Calculating Hydrodynamic Lubrication Film Thickness in Elastoplastic Line Contacts

The present paper proposes a simplified model for calculating hydrodynamic lubrication film thickness in elastoplastic line contacts. According to the Saint-Venant’s principle, the pressure in the contact is taken as uniformly distributed, this gives the contact surface elastic deformations in the inlet zone far away from the contact center close to real ones while gives those close to the contact center greater than real ones. This treatment is validated for hydrodynamic lubricated elastic contacts for relatively light loads and high rolling speeds. It gives the film thickness at the contact center a little higher than that calculated based on the real elastic model. The treatment is extended to a hydrodynamic lubricated elastoplastic line contact. The contact surfaces in the inlet zone are assumed as elastic and their deformations are calculated based on the uniform pressure distribution in the elastoplastic contact area. An inlet zone analysis is taken for obtaining the calculating equation of the hydrodynamic film thickness at the contact center. The equation overestimates the central film thickness but gives a satisfactory film thickness prediction for the heavy load which gives significant plastic deformations in the elastoplastic contact. It is found that when the load is lighter than 0.6 w _pc , the contact can be taken as elastic when calculating the central film thickness, while when the load is heavier than 0.6 w _pc , the contact can be taken as fully plastic; Here w _pc is the critical load for the contact fully plastic deformation. The plastic deformation in an elastoplastic line contact is found to reduce the hydrodynamic lubrication film thickness in the contact. This reduction is greater for higher rolling speeds and heavier loads. However, it is significantly dropped with increasing surface hardness.     

A constitutive model for the formation of martensite in austenitic steels under large strain plasticity

A constitutive model for diffusionless phase transitions in elastoplastic materials undergoing large deformations is developed. The model takes basic thermodynamic relations as its starting point and the phase transition is treated through an internal variable (the phase fractions) approach. The usual yield potential is used together with a transformation potential to describe the evolution of the new phase. A numerical implementation of the model is presented, along with the derivation of a consistent algorithmic tangent modulus. Simulations based on the presented model are shown to agree well with experimental findings. The proposed model provides a robust tool suitable for large-scale simulations of phase transformations in austenitic steels undergoing extensive deformations, as is demonstrated in simulations of the necking of a bar under tensile loading and also in simulations of a cup deep-drawing process.     

Analysis of size effects associated to the transformation strain in TRIP steels with strain gradient plasticity

The size dependent strengthening resulting from the transformation strain in Transformation Induced Plasticity (TRIP) steels is investigated using a two-dimensional embedded cell model of a simplified microstructure composed of small cylindrical metastable austenitic inclusions within a ferritic matrix. Earlier studies have shown that within the framework of classical plasticity or of the single length parameter Fleck–Hutchinson strain gradient plasticity theory, the transformation strain has no significant impact on the overall strengthening. The strengthening is essentially coming from the composite effect with a marked inclusion size effect resulting from the appearance during deformation of new boundaries constraining the plastic flow. The three parameters version of the Fleck–Hutchinson strain gradient plasticity theory is used here in order to better capture the effect of the plastic strain gradients resulting from the transformation strain. The three parameters theory incorporates separately the rotational and extensional gradients in the formulation, which leads to a significant influence of the shear component of the transformation strain, not captured by the single-parameter theory. When the size of the austenitic inclusions decreases, the overall strengthening increases due to a combined size dependent effect of the transformation strain and of the evolving composite structure. A parametric study is proposed and discussed in the light of experimental evidences giving indications on the optimization of the microstructure of TRIP-assisted multi-phase steels.     

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.     

A phase-field model for incoherent martensitic transformations including plastic accommodation processes in the austenite

If alloys undergo an incoherent martensitic transformation, then plastic accommodation and relaxation accompany the transformation. To capture these mechanisms we develop an improved 3D microelastic–plastic phase-field model. It is based on the classical concepts of phase-field modeling of microelastic problems (Chen, L.Q., Wang Y., Khachaturyan, A.G., 1992. Philos. Mag. Lett. 65, 15–23). In addition to these it takes into account the incoherent formation of accommodation dislocations in the austenitic matrix, as well as their inheritance into the martensitic plates based on the crystallography of the martensitic transformation. We apply this new phase-field approach to the butterfly-type martensitic transformation in a Fe–30 wt%Ni alloy in direct comparison to recent experimental data (Sato, H., Zaefferer, S., 2009. Acta Mater. 57, 1931–1937). It is shown that the therein proposed mechanisms of plastic accommodation during the transformation can indeed explain the experimentally observed morphology of the martensitic plates as well as the orientation between martensitic plates and the austenitic matrix. The developed phase-field model constitutes a general simulations approach for different kinds of phase transformation phenomena that inherently include dislocation based accommodation processes. The approach does not only predict the final equilibrium topology, misfit, size, crystallography, and aspect ratio of martensite–austenite ensembles resulting from a transformation, but it also resolves the associated dislocation dynamics and the distribution, and the size of the crystals itself.     

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.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

Numerical implementation of a thermo-elastic–plastic constitutive equation in consideration of transformation plasticity in welding

Finite element analysis of welding processes, which entail phase evolution, heat transfer and deformations, is considered in this paper. Attention focuses on numerical implementation of the thermo-elastic–plastic constitutive equation proposed by Leblond et al. [J. Mech. Phys. Solids 34(4) (1986a) 395; J. Mech. Phys. Solids 34(4) (1986b) 411] in consideration of the transformation plasticity. Based upon the multiplicative decomposition of deformation gradient, hyperelastoplastic formulation is borrowed for efficient numerical implementation, and the algorithmic consistent moduli for elastic–plastic deformations including transformation plasticity are obtained in the closed form. The convergence behavior of the present implementation is demonstrated via a couple of numerical examples.     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

Computational simulation of the dependence of the austenitic grain size on the deformation behavior of TRIP steels

Due to the strain-induced martensitic transformation which occurs during plastic deformation, a transformation-induced plasticity (TRIP) phenomenon is generated. With the TRIP phenomenon, the TRIP steel possesses favorable mechanical properties such as high strength, ductility and toughness, and is frequently employed as a structural material. In the past, several researchers clarified experimentally that the strain-induced martensitic transformation and the deformation behavior of TRIP steel depend upon the austenitic grain size. In order to obtain the expected mechanical properties of TRIP steel through control of the austenitic grain size, prediction and control of the material characteristics in the deformation processes is essential. Here, the new strain-induced martensitic transformation kinetics model and constitutive equation of TRIP steels are proposed by considering the dependence of the austenitic grain size. Then, the deformation behavior of a type 304 austenitic stainless steel cylinder is simulated under different environmental temperatures with the various austenitic grain sizes by the finite-element method along with newly-proposed constitutive equations. Finally, the validity of proposed constitutive equations and the possibility of the improvement of the mechanical properties through control of the austenitic grain size are discussed.     

Couplings between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels

Several couplings between plasticity and martensitic phase transformation are at the origin of remarkable properties of ductility and toughness in the case of TRIP steels. A micromechanical model is developed to predict the conditions of nucleation and growth of a martensitic microdomain inside an inhomogeneous plastic strain field. More explicit relations are developed in the case of a simple shear test where a heterogeneous plastic strain field leads to a significant decrease of the critical stress for martensitic transformation. The obtained results are combined with a kinetics and kinematics studies to derive the constitutive equation of an austenitic single crystal from which the overall behavior of a polycrystalline steel is deduced using the self-consistent scale transition method. Comparison with experimental data shows a good agreement.     

Elastoplastic Analysis with Limited Plastic Deformations and Displacements*

ABSTRACT

This paper presents solution methods for elastoplastic and shakedown analysis of linearly elastic, perfectly plastic bodies for which the conventional classical formulations of these problems are completed by constraints on overall plastic deformation and elastoplastic displacement. The methods are described in terms of nonlinear mathematical programming and provide solutions when the plastic reserves of the body are not fully exhausted, and the plastic performance and the plastic deformations are controlled. Application of the method is illustrated by an example.     

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.     

Study of transformation plasticity in tetragonal zirconia polycrystals by moiré interferometry

Transformation plasticity in ceria-stabilized tetragonal zirconia polycrystals due to the stress-induced tetragonal-to-monoclinic martensitic transformation under tension and bending is studied by moiré interferometry. The whole fringe patterns includingu fields andv fields are acquired. According to these patterns, the distributions of transformation plasticity in transformation zones are obtained, and the phenomenon of plastic flow localization for transformation is revealed. The above work provides a significant experimental foundation for establishing transformation constitutive relations The project supported by the National Natural Science Foundation of China