A finite-deformation-based phenomenological theory for shape-memory alloys

In this work we develop a finite-deformation-based, thermo-mechanically-coupled and non-local phenomenological theory for polycrystalline shape-memory alloys (SMAs) capable of undergoing austenite ↔$\leftrightarrow$ martensite phase transformations. The constitutive model is developed in the isotropic plasticity setting using standard balance laws, thermodynamic laws and the theory of micro-force balance (Fried and Gurtin, 1994). The constitutive model is then implemented in the ABAQUS/Explicit (2009) finite-element program by writing a user-material subroutine. Material parameters in the constitutive model were fitted to a set of superelastic experiments conducted by Thamburaja and Anand (2001) on a polycrystalline rod Ti–Ni. With the material parameters calibrated, we show that the experimental stress-biased strain–temperature-cycling and shape-memory effect responses are qualitatively well-reproduced by the constitutive model and the numerical simulations. We also show the capability of our constitutive mode in studying the response of SMAs undergoing coupled thermo-mechanical loading and also multi-axial loading conditions by studying the deformation behavior of a stent unit cell.     

An isotropic-plasticity-based constitutive model for martensitic reorientation and shape-memory effect in shape-memory alloys

In this work, we develop an isotropic-plasticity-based constitutive model for initially martensitic shape-memory alloys (SMA) which exhibit martensitic reorientation and the shape-memory effect. The constitutive model is then implemented in the [Abaqus reference manuals. 2006. Providence, R.I.] finite-element program by writing a user-material subroutine. The results from the constitutive model and numerical procedure are then compared to representative physical experiments conducted on polycrystalline rod and sheet Ti–Ni. The constitutive model and the numerical simulations are able to reproduce the stress–strain responses from these physical experiments to good accuracy. Finally, two different boundary value problems utilizing the one-way shape-memory effect are studied: (a) the deformation of an arterial stent, and (b) a micro-clamper. We show that our constitutive model can be used to model the response of the aforementioned boundary value examples.     

Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys

A crystal-inelasticity-based constitutive model for martensitic reorientation and detwinning in shape-memory alloys (SMAs) has been developed from basic thermodynamics principles. The model has been implemented in a finite-element program by writing a user-material subroutine. We perform two sets of finite-element simulations to model the behavior of polycrystalline SMAs: (1) The full finite-element model where each finite element represents a collection of martensitic microstructures which originated from within an austenite single crystal, chosen from a set of crystal orientations that approximates the initial austentic crystallographic texture. The macroscopic stress-strain responses are calculated as volume averages over the entire aggregate: (2) The Taylor model (J. Inst. Metals 62 (1938) 32) where an integration point in a finite element represents a material point which consist of sets of martensitic microstructures which originated from within respective austenite single-crystals. Here the macroscopic stress-strain responses are calculated through a homogenization scheme.Experiments in tension and compression were conducted on textured polycrystalline Ti-Ni rod initially in the martensitic phase by Xie et al (Acta Mater. 46 (1998) 1989). The material parameters for the constitutive model were calibrated by fitting the tensile stress-strain response from a full finite-element calculation of a polycrystalline aggregate to the simple tension experiment. With the material parameters calibrated the predicted stress-strain curve for simple compression is in very good accord with the corresponding experiment. By comparing the simulated stress-strain response in simple tension and simple compression it is shown that the constitutive model is able to predict the observed tension-compression asymmetry exhibited by polycrystalline Ti-Ni to good accuracy. Furthermore, our calculations also show that the macroscopic stress-strain response depends strongly on the initial martensitic microstructure and crystallographic texture of the material.We also show that the Taylor model predicts the macroscopic stress-strain curves in simple tension and simple compression reasonably well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory materials.     

Superelastic behavior in tension–torsion of an initially-textured Ti–Ni shape-memory alloy

A recently-developed crystal-mechanics-based constitutive model for polycrystalline shape-memory alloys [J. Mech. Phys. Solids 49 (2001) 909] is shown to quantitatively predict the superelastic response of an initially-textured Ti–Ni alloy in (i) a proportional-loading, combined tension–torsion experiment, as well as (ii) a path-change, tension–torsion experiment.     

Time integration of a model for martensite detwinning and reorientation under nonproportional loading using Lagrange multipliers

The paper presents a consistent integration scheme for a model of shape memory alloys that takes into account the processes of detwinning and reorientation of martensite. Lagrange multipliers are used to account for the saturation condition on the martensitic phase, which expresses the requirement on the inelastic deformation of martensite not to exceed a material-specific value. A detailed procedure for implicit time integration of the constitutive equations is given, including a comprehensive derivation of a closed-form expression for the consistent tangent moduli. The equations are implemented into a commercial finite element software, which is used to simulate the response of martensitic SMA structures to complex loading. The results are shown to agree with experimental data taken from the literature.     

Micromechanical model of the high temperature cyclic behavior of 9–12%Cr martensitic steels

9–12%Cr quenched and tempered martensitic steels are known to soften under cyclic loadings at high temperature. The present article proposes a model based on physical mechanisms described at the scale of slip systems. This model describes explicitly the microstructural recovery (corresponding to a decrease of the dislocation density and subgrain coarsening) observed experimentally. The scale transition is carried out in the framework of self-consistent homogenization schemes. The model assumptions and its physical basis are explicitly discussed. The parameters are identified on a very limited amount of experimental data. The model turns out to give very good predictions and extrapolations for the cyclic softening effect observed in uniaxial tension–compression loadings for strain ranges larger than 0.3%. Stress–relaxation and creep behavior can also be simulated for high stresses. In addition the cyclic softening effect is reproduced for multiaxial tension–torsion loadings.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Thermal effects in the superelasticity of crystalline shape-memory materials

This paper develops a three-dimensional theory for the superelastic response of single-crystal shape-memory materials. Since energetic considerations play a major role in the phase transformations associated with the superelastic response, we have developed the theory within a framework that accounts for the laws of thermodynamics. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program, and using this program have simulated the superelastic response of a single crystal Ti-Ni shape-memory alloy under both isothermal and thermo-mechanically coupled situations. Both manifestations of superelasticity—stress-strain response at fixed temperature and strain-temperature response at fixed stress—are explored. The single-crystal constitutive-model is also used to discuss the superelastic response of a polycrystalline aggregate with a random initial crystallographic texture. The overall features of the results from the numerical simulations are found to be qualitatively similar to existing experimental results on Ti-Ni.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.     

Modelling the behaviour of polycrystalline austenitic steel with twinning-induced plasticity effect

A micromechanical model using the scale transition method in elastoviscoplasticity has been developed to describe the behaviour of those austenitic steels that display a TWIP effect. A physically based constitutive equation at the grain scale is proposed considering two inelastic strain modes: crystallographic slip and twinning. The typical organizations of microtwins observed in electron microscopy are considered, and the twin–slip as well as the twin–twin interactions are accounted for. The parameters for slip are first fitted on the uniaxial tensile response obtained at intermediate temperatures (when twinning is inhibited). Then, the parameters associated with twinning are identified using the stress–strain curve at room temperature. The simulated results in both macro and micro scales are in good agreement with experimentally obtained results.     

Gradient plasticity with the tangential-subloading surface model and the prediction of shear-band thickness of granular materials

An extended gradient elastoplastic constitutive equation is formulated, which is capable of describing the plastic strain rate due to the rate of stress inside the yield surface and the inelastic strain rate due to the stress rate component tangential to the subloading surface by incorporating the tangential-subloading surface model. Based on the extended constitutive equation, the post-localization analysis of granular materials is performed to predict the shear-band thickness. It is revealed that the shear-band thickness is almost determined by the gradient coefficient characterizing the inhomogeneity of deformation, although the stress–strain curve is strongly dependent on material properties.     

Cyclic multiaxial and shear finite deformation responses of OFHC Cu. Part II: An extension to the KHL model and simulations

In this part, the Khan–Huang–Liang (KHL) constitutive model was extended to account for kinematic hardening characteristic behavior of materials. The extended model is then generalized and used to simulate experimental response of oxygen free high conductivity (OFHC) copper under cyclic shear straining and biaxial tension–torsion (multiaxial ratchetting) experiments presented in Part I (Khan et al., 2007). In addition, a new modification for the non-linear kinematic hardening rule of Karim–Ohno (Abdel-Karim and Ohno, 2000) is proposed to simulate multiaxial ratchetting behaviors. Although, the kinematic hardening contributes the most to the response, it is shown that, the loading rate effect, and a coupled isotropic and kinematic hardening effect should also be considered while simulating the multiaxial ratchetting behavior of OFHC copper. Furthermore, the newly modified kinematic hardening rules is able to fairly well simulate the multiaxial ratchetting experiments under different loading conditions, irrespective of the value of applied axial tensile stress, shear strain amplitude, pre-cyclic hardening and/or loading sequence.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

A crystal plasticity model for strain-path changes in metals

A model is proposed that deals with the transient mechanical anisotropy during strain-path changes in metals. The basic mechanism is assumed to be latent hardening or softening of the slip systems, dependent on if they are active or passive during deformation, reflecting microstructural mechanisms that depend on the deformation mode rather than on the crystallography. The new model captures the experimentally observed behaviour of cross hardening in agreement with experiments for an AA3103 aluminium alloy. Generic results for strain reversals qualitatively agree with two types of behaviour reported in the literature – with or without a plateau on the stress–strain curve. The influence of the model parameters is studied through detailed calculations of the response of three selected parameter combinations, including the evolution of yield surface sections subsequent to 10% pre-strain. The mathematical complexity is kept to a minimum by avoiding explicit predictions related directly to underpinning microstructural changes. The starting point of the model is a combination of conventional texture and work hardening approaches, where an adapted full-constraints Taylor theory and a simple single-crystal work-hardening model for monotonic strain are used. However, the framework of the model is not restricted to these particular models.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

Tension–compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior

In-plane tension and compression experiments on copper alloy sheets (phosphor bronze) and 6000 series aluminum alloy sheets (AA6016-T4) were conducted using a specially designed testing apparatus. The apparatus is equipped with comb-type dies so that stress–strain curves of a sheet specimen subjected to tension followed by compression, and vice versa, can be measured without buckling of the specimen, as well as those for monotonic tension and compression. A difference was observed in the flow stresses between tension and compression for the as-received copper alloy, but not for the aluminum alloy. Moreover, stress reversal tests, such as tension followed by compression and compression followed by tension, were carried out in order to measure the Bauschinger effect. In the second part of the experiment, bending moment–curvature diagrams were measured for the as-received and pre-stretched specimens. The bending moment–curvature diagrams were compared with those calculated using the stress–strain curves obtained from the tension–compression tests, and were in good agreement with those calculated with the tension–compression asymmetry and the Bauschinger effect correctly reproduced.     

Effects of spatial grain orientation distribution and initial surface topography on sheet metal necking

The finite element method is used to numerically simulate localized necking in AA6111-T4 under stretching. The measured EBSD data (grain orientations and their spatial distributions) are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. We assume that localized necking is associated with surface instability, the onset of unstable growth in surface roughening. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and the artificial initial imperfection necessitated by the macroscopic M–K approach [Marciniak and Kuczynski (1967). Int. J. Mech. Sci. 9, 609–620] is not relevant in the present analysis. The effects of spatial orientation distribution, material strain rate sensitivity, texture evolution, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution. It is also demonstrated that the initial surface topography has only a small influence on necking.     

Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples

Based on the theory developed in Part 1 of this paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation. Int. J. Plasticity. doi:10.1016/j.ijplas.2008.02.004], various non-trivial examples of microstructure evolution under complex multiaxial loading are presented. For the case without interface rotation, the effect of the athermal thresholds for austenite (A)–martensite (M) and martensitic variant M_I–variant M_II interfaces and loading paths on stress–strain curves and phase transformations was studied. For coupled interface propagation and rotation, two types of numerical simulations were carried out. The tetragonal–orthorhombic transformation has been studied under general three-dimensional interface orientation and zero athermal threshold. The cubic–tetragonal transformation was treated with allowing for an athermal threshold and interface reorientation within a plane. The effect of the athermal threshold, the number of martensitic variants and an interface orientation in the embryo was studied in detail. It was found that an instability in the interface normal leads to a jump-like interface reorientation that has the following features of the energetics of a first-order transformation: there are multiple energy minima versus interface orientation that are separated by an energy barrier; positions of minima do not change during loading but their depth varies; when the barrier disappears (i.e. one of the minima transforms to the local saddle or maximum points), the system rapidly evolves toward another stable orientation. Depending on the loading and material parameters, we observed a large continuous change in interface orientation, a jump in interface reorientation, a jump in volume fractions and stresses, an expected stress relaxation during the phase transition and unexpected stress growth during the transition because of large change in elastic moduli.