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.     

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.     

The evolution of microstructure during twinning: Constitutive equations,finite-element simulations and experimental verification

In this work, we develop a rate-dependent, finite-deformation and crystal-mechanics-based constitutive theory which describes the twinning in single-crystal metallic materials. Central to the derivation of the constitutive equations are the use of fundamental thermodynamic laws and the principle of micro-force balance [Fried, E., Gurtin, M., 1994. Dynamic solid–solid transitions with phase characterized by an order parameter. Physica D 72, 287–308]. A robust numerical algorithm based on the constitutive model has also been written and implemented in the ABAQUS/Explicit [Abaqus reference manuals, 2007. SIMULIA, Providence, R.I.] finite-element program.     

Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys

Biaxial proportional loading such as tension (compression)–internal pressure and bi-compression tests are performed on a Cu-Zn-Al and Cu-Al-Be shape memory polycrystals. These tests lead to the experimental determination of the initial surface of phase transformation (austenite→martensite) in the principal stress space (σ₁,σ₂). A first “micro–macro” modeling is performed as follows. Lattice measurements of the cubic austenite and the monoclinic martensite cells are used to determine the “nature” of the phase transformation, i.e. an exact interface between the parent phase and an untwinned martensite variant. The yield surface is obtained by a simple (Sachs constant stress) averaging procedure assuming random texture. A second modeling, performed in the context of the thermodynamics of irreversible processes, consists of a phenomenological approach at the scale of the polycrystal. These two models fit the experimental phase transformation surface well.     

Modeling the evolution of crystallographic dislocation density in crystal plasticity

Dislocations are the most important material defects in crystal plasticity, and although dislocation mechanics has long been understood as the underlying physical basis for continuum crystal plasticity formulations, explicit consideration of crystallographic dislocation mechanics has been largely absent in working constitutive models. Here, dislocation density state variables evolve from initial conditions according to equations based on fundamental concepts in dislocation mechanics such as the conservation of Burgers vector in multiplication and annihilation processes. The model is implemented to investigate the polyslip behavior of single-crystal aluminum. The results not only capture the mechanical stress/strain response, but also detail the development of underlying dislocation structure responsible for the plastic behavior.     

Thermally actuated shape-memory polymers: Experiments, theory, and numerical simulations

With the aim of developing a thermo-mechanically coupled large-deformation constitutive theory and a numerical-simulation capability for modeling the response of thermally actuated shape-memory polymers, we have (i) conducted large strain compression experiments on a representative shape-memory polymer to strains of approximately unity at strain rates of 10⁻³ and 10⁻¹ s⁻¹, and at temperatures ranging from room temperature to approximately 30 °C above the glass transition temperature of the polymer; (ii) formulated a thermo-mechanically coupled large-deformation constitutive theory; (iii) calibrated the material parameters appearing in the theory using the stress-strain data from the compression experiments; (iv) numerically implemented the theory by writing a user-material subroutine for a widely used finite element program; and (v) conducted representative experiments to validate the predictive capability of our theory and its numerical implementation in complex three-dimensional geometries. By comparing the numerically predicted response in these validation simulations against measurements from corresponding experiments, we show that our theory is capable of reasonably accurately reproducing the experimental results. As a demonstration of the robustness of the three-dimensional numerical capability, we also show results from a simulation of the shape-recovery response of a stent made from the polymer when it is inserted in an artery modeled as a compliant elastomeric tube.     

Cyclic thermomechanical behavior of a polycrystalline pseudoelastic shape memory alloy

In this work, 3D finite element modeling is employed to examine the thermomechanical behavior of a polycrystalline Ni-Ti shape memory alloy in the pseudoelastic regime. It is shown that the tension-compression asymmetry during uniaxial cyclic loading is due to a preferred orientation of the crystallographic texture. In pure shear loading, the thermomechanical behavior exhibits symmetry in both senses of shear, due to the fiber texture of the specimen bar stock. It is also shown that the apparent strain rate-dependence is due to thermomechanical coupling with latent heat generation/absorption during phase transformation.     

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.     

A micromechanical model of hardening, rate sensitivity and thermal softening in BCC single crystals

The present paper is concerned with the development of a micromechanical model of the hardening, rate-sensitivity and thermal softening of bcc crystals. In formulating the model, we specifically consider the following unit processes: double-kink formation and thermally activated motion of kinks; the close-range interactions between primary and forest dislocations, leading to the formation of jogs; the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; dislocation multiplication due to breeding by double cross-slip; and dislocation pair annihilation. The model is found to capture salient features of the behavior of Ta crystals such as: the dependence of the initial yield point on temperature and strain rate; the presence of a marked stage I of easy glide, specially at low temperatures and high strain rates; the sharp onset of stage II hardening and its tendency to shift towards lower strains, and eventually disappear, as the temperature increases or the strain rate decreases; the parabolic stage II hardening at low strain rates or high temperatures; the stage II softening at high strain rates or low temperatures; the trend towards saturation at high strains; the temperature and strain-rate dependence of the saturation stress; and the orientation dependence of the hardening rate.     

Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems.     

Experimental investigation and modeling of non-monotonic creep behavior in polymers

The deformation behavior of two unfilled engineering thermoplastics, ultra high molecular weight polyethylene (UHMWPE) and polycarbonate (PC), has been investigated in creep test conditions. It has been found that a loading history (prior to the creep test) comprising of loading to a maximum stress or strain value followed by partial unloading to arrive at the target stress value can greatly modify the strain-time behavior. Under such a test protocol, while the expected increase in strain during creep (constant tensile load) is observed, at relatively low creep stresses specimens have also demonstrated a monotonic decrease in strain. In an intermediate stress range, specimens have demonstrated time dependent behavior comprising of a transition from decreasing to increasing strain during creep in tension. This paper presents experimental results to delineate these findings and explore the effect of prior strain rate on the qualitative and quantitative changes in the output (strain-time) behavior. Furthermore, modification of the viscoplasticity theory based on overstress (VBO) model into a double element configuration is introduced. These changes confer upon the model the ability to yield non-monotonic behavior in creep, and supporting simulation results have been included. These changes, therefore, allow the model to simulate strain rate sensitivity, creep, relaxation, and recovery behavior, but more importantly address the issue of non-monotonic changes in creep and relaxation when a loading history involves some degree of unloading.     

Multiscale modeling of the plasticity in an aluminum single crystal

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.     

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic-plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress-strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress-strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.     

Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading

This paper is concerned with the experimental behaviour of a 16MND5 steel (french vessel steel) under complex loading. A particular attention is paid to plasticity induced by phase transformation. We present an experimental set-up to apply thermo-mechanical loads under tension-torsion. This apparatus enables us to reach temperature of 1200 °C at a maximum heating rate of 60 °C s⁻¹ and a high cooling rate of −30 °C s⁻¹. A series of tests is performed in order to show the rule of loading on transformation plasticity.     

Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation

Presented is a constitutive framework for modeling the dynamic response of polycrystalline microstructures, posed in a thermodynamically consistent manner and accounting for finite deformation, strain rate dependence of flow stress, thermal softening, thermal expansion, heat conduction, and thermoelastic coupling. Assumptions of linear and square-root dependencies, respectively, of the stored energy and flow stresses upon the total dislocation density enable calculation of the time-dependent fraction of plastic work converted to heat energy. Fracture at grain boundary interfaces is represented explicitly by cohesive zone models. Dynamic finite element simulations demonstrate the influences of interfacial separation, random crystallographic orientation, and grain morphology on the high-rate tensile response of a realistic two-phase material system consisting of comparatively brittle pure tungsten (W) grains embedded in a more ductile matrix of tungsten-nickel iron (W-Ni-Fe) alloy. Aspects associated with constitutive modeling of damage and failure in the homogenized material system are discussed in light of the computational results.     

Modelling of laminated microstructures in stress-induced martensitic transformations

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro-macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β₁ (austenite) to orthorhombic γ₁′ (martensite) phase transformation in a single crystal of Cu-Al-Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.     

A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed.