Reversible stress-induced martensitic phase transformations in a bi-atomic crystal

In an earlier work, Elliott et al. [2006a, Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. Journal of the Mechanics and Physics of Solids 54(1), 193-232], the authors used temperature-dependent atomic potentials and path-following bifurcation techniques to solve the nonlinear equilibrium equations and find the temperature-induced martensitic phase transformations in stress-free, perfect, equi-atomic binary B2 crystals. Using the same theoretical framework, the current work adds the influence of stress to study the model's stress-induced martensitic phase transformations.The imposition of a uniaxial Biot stress on the austenite (B2) crystal, lowers the symmetry of the problem, compared to the stress-free case, and leads to a large number of stable equilibrium paths. To determine which ones are possible reversible martensitic transformations, we use the (kinematic) concept of the maximal Ericksen-Pitteri neighborhood (max EPN) to select those equilibrium paths with lattice deformations that are closest, with respect to lattice-invariant shear, to the austenite phase and thus capable of a reversible transformation. It turns out that for our chosen parameters only one stable structure (distorted αIrV) is found within the max EPN of the austenite in an appropriate stress window. The energy density of the corresponding configurations shows features of a stress-induced phase transformation between the higher symmetry austenite and lower symmetry martensite paths and suggests the existence of hysteretic stress-strain loops under isothermal load-unload conditions. Although the perfect crystal model developed in this work over-predicts many key material properties, such as the transformation stress and the Clausious-Clapeyron slope, when compared to real experimental values (based on actual polycrystalline specimens with defects), it is—to the authors' knowledge—the first atomistic model that has been demonstrated to capture all essential trends and behavior observed in shape memory alloys.     

Stability and Elastic Properties of the Stress-Free B2 (CsCl-type) Crystal for the Morse Pair Potential Model

Solid-to-solid martensitic phase transformations are responsible for the remarkable behavior of shape memory alloys. There is currently a need for shape memory alloys with improved corrosion, fatigue, and other properties. The development of new accurate models of martensitic phase transformations based on the material’s atomic composition and crystal structure would lead to the ability to computationally discover new improved shape memory alloys. This paper explores the Effective Interaction Potential method for modeling the material behavior of shape memory alloys. In particular, an extensive parameter study of the Morse pair potential model of the stress-free B2 cubic crystal is performed. Results for the stability, potential energy, current unit cell volume, instantaneous bulk modulus, and the two instantaneous cubic shear moduli are presented and discussed. It is found that an Effective Interaction Potential model based on the Morse potential is appropriate for modeling transformations between the B2 cubic structure and the B19 orthorhombic structure, but is not likely to be capable of simulating the B2 cubic to B19′ monoclinic transformation found in the popular shape memory alloy NiTi. In fact, this conclusion may be extended to all types of pair interaction potential models.      

An effective interaction potential model for the shape memory alloy AuCd

The unusual properties of shape memory alloys (SMAs) result from a lattice level martensitic transformation (MT) corresponding to an instability of the SMAs crystal structure. Currently, there exists a shortage of material models that can capture the details of lattice level MTs occurring in SMAs and that can be used for efficient computational investigations of the interaction between MTs and larger-scale features found in typical materials. These larger-scale features could include precipitates, dislocation networks, voids, and even cracks. In this article, one such model is developed for the SMA AuCd. The model is based on effective interaction potentials (EIPs). These are atomic interaction potentials that are explicit functions of temperature. In particular, the Morse pair potential is used and its adjustable coefficients are taken to be temperature dependent. An extensive exploration of the Morse pair potential is performed to identify an appropriate functional form for the temperature dependence of the potential parameters. A fitting procedure is developed for the EIPs that matches, at suitable temperatures, the stress-free equilibrium lattice parameters, instantaneous bulk moduli, cohesive energies, thermal expansion coefficients, and heat capacities of FCC Au, HCP Cd, and the B2 cubic austenite phase of the Au-47.5at%Cd alloy. The resulting model is explored using branch-following and bifurcation techniques. A hysteretic temperature-induced MT between the B2 cubic and B19 orthorhombic crystal structures is predicted. This is the behavior that is observed in the real material. In addition to reproducing the important properties mentioned above, the model predicts, to reasonable accuracy, the transformation strain tensor and captures the latent heat and thermal hysteresis to within an order of magnitude.     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

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.     

Theoretical properties of cubic crystals at arbitrary pressure—III. Stability

A complete account is presented of the application of the principles of bifurcation analysis for general materials to the particular case of cubic crystals subjected to hydrostatic loading. The treatment of crystal stability is classical in that (i) the loading environment is fully specified, to sufficient order and in both its active and passive modes, and (ii) the potential energy of the system as a whole is examined in all the nearby, possibly inhomogeneous, configurations allowed by the kinematic constraints. Computations are made of the pressures and the bulk and shear moduli of the entire Morse-model family of fcc, bcc, and sc monatornic crystals under pure hydrostatic compression and tension. The stable range of each lattice as well as the potential bifurcations at the range limits are presented and discussed in terms of the role of the particular lattice structure and the effective range of the interatomic potential function (as specified by the parameter log β). The fee lattices are stable in compression and in tension up to an all-round stretch λ = Λ_k, at which point the bulk modulus vanishes; Λ_k, is a monotonically decreasing function of log β. The bcc lattices are stable for λ < Λ_CR, where the bulk modulus or the shear modulus μ vanishes (depending upon the value of log β) at λ = Λ_CR. For very large values of log β a second range of bcc stability is located in a region of hydrostatic expansion. The sc crystals are stable only in a range of hydrostatic tension and only for relatively short-range interatomic interactions (large log β); the present work appears to be the first in which a theoretical range of stability of sc crystals has been revealed. The question of the possibility of assessing lattice stability under load with the aid of higher order moduli at zero load is given consideration quantitatively for the fee lattices and the bcc lattices that are stable at zero load. Finally, the present approach to crystal stability is distinguished from some simplistic notional criteria based upon local convexity of strain energy and, for the Morse-model cubic crystals, quantitative comparisons are made with the present classical treatment of stability in a hydrostatic environment.     

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic‐to‐Orthorhombic Phase Transformation

. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic‐to‐orthorhombic transformation of type ${\mathcal P}^{(432)} \to {\mathcal P}^{(222)'}We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic-to-orthorhombic transformation of type P⁽⁴³²⁾ ? P^(222)￠{\mathcal P}^{(432)} \to {\mathcal P}^{(222)'}. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank-one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers.     

Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity

This paper focuses on the issue plasticity within the framework of a micromechanical model for single-crystal shape-memory alloys. As a first step towards a complete micromechanical formulation of such models, we work with classical J₂-von Mises-type plasticity for simplicity. The modeling of martensitic phase transitions is based on the concept of energy relaxation (quasiconvexification) in connection with evolution equations derived from inelastic potentials. Crystallographic considerations lead to the derivation of Bain strains characterizing the transformation kinematics. The model is derived for arbitrary numbers of martensite variants and thus can be applied to any shape-memory material such as CuAlNi or NiTi. The phase transition model captures effects like tension/compression asymmetry and transformation induced anisotropy. Additionally, attention is focused on the interaction between phase transformations and plasticity in terms of the inheritance of plastic strain. The effect of such interaction is demonstrated by elementary numerical studies.     

Stability of crystalline solids—I: Continuum and atomic lattice considerations

Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material's crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum-stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under “hard-device” loading (i.e., all around displacement type boundary conditions) is hereby called “constrained material stability”. A more general “material stability” criterion, motivated by considering “soft” loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all “quasi-uniform” perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion.     

Fracture of shape memory CuAlNi single crystals

The fracture behavior of shape memory CuAlNi single crystals loaded in tension is studied. Specimens cut from a single crystal are notched and loaded in tension until final fracture. Eight different crystallographic orientations of the notch and tensile axes are considered. The stress field at the notch tip triggers a cubic to orthorhombic phase transition in the crystal, which results in a set of twinned martensite plates emanating from the notch tip. As loading increases, a crack forms and grows off the notch tip, with the martensite plates continuing to appear at the growing crack. Details of the crack growth depend strongly on both the type of singular microstructures that forms and how this microstructure interacts with the growing crack. In one group of orientations a distinct transformation zone forms along one flank of the crack and the motion of this zone is directly connected to the crack growth. In a second group of orientations, the microstructure formation is not as strongly tied to the crack. Interestingly, in all specimens studied, the final crack direction is approximately 80° from the direction of the martensite plates.     

Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.     

An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations

Modeling the energetic behavior of martensitic (phase transforming) materials usually leads to non quasiconvex energy formulations. For this reason, researchers often employ quasiconvex relaxation methods to improve the character of the formulation. Unfortunately, explicit expressions for the relaxed free energy density for multi-variant martensitic materials are typically not available. Thus, some researchers have employed a Reuβ-like convex lower bound, which neglects compatibility constraints, as an estimate on the free energy of mixing. To be confident with such a technique, one needs a measure of the quality of the lower bound. In this paper, we seek such a measure by comparing the Reuβ-like lower bound to an upper bound. The upper bound is constructed upon assumptions on the type of microstructures that form in such alloys. In particular, we consider lamination type microstructures which form by temperature- or stress-induced transformation in monoclinic and orthorhombic Copper-based alloys with cubic austenitic symmetry. Our results display a striking congruence of upper and lower bounds in the most relevant cases.     

High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: Geometrically necessary dislocation densities

Experimental studies on indentation into face-centered cubic (FCC) single crystals such as copper and aluminum were performed to reveal the spatially resolved variation in crystal lattice rotation induced due to wedge indentation. The crystal lattice curvature tensors of the indented crystals were calculated from the in-plane lattice rotation results as measured by electron backscatter diffraction (EBSD). Nye's dislocation density tensors for plane strain deformation of both crystals were determined from the lattice curvature tensors. The least L²-norm solutions to the geometrically necessary dislocation densities for the case in which three effective in-plane slip systems were activated in the single crystals associated with the indentation were determined. Results show the formation of lattice rotation discontinuities along with a very high density of geometrically necessary dislocations.     

Theoretical properties of cubic crystals at arbitrary pressure—I. Density and bulk modulus

The theoretical elastic behaviour of simple monatomic cubic crystals at arbitrary pressure is to be presented in a series of papers. In the present paper, general expressions are derived for calculating the pressure P and the bulk modulus κ as a function of the all-round stretch λ for crystals in which the interatomic interaction energies are modelled by pairwise functions φ. With the aid of a particular family of functions φ, calculations are carried out for the three cubic structures, and a detailed study is made of the influence of crystal structure and explicit nature of φ upon the theoretical elastic behaviour under pressure loading. Calculations are also made of higher order derivatives of P(λ) and gk(λ) in the reference state (i.e. at λ = 1) and the “higher order moduli” thus calculated are used to formulate series expansion approximations to the functions P(λ) and κ(λ). Values of P(λ) and κ(λ) in the series approximations, based upon successively higher order moduli (evaluated in the reference state), are compared with the corresponding “exact” values evaluated in the current state. The theoretical results are useful as empirical relationships modelling the elastic behaviour of crystals at arbitrary pressure.     

Self-accommodation in martensite

The shape-memory effect is a phenomenon wherein an apparently plastically deformed specimen recovers all strain when heated above a critical temperature. This is observed in some crystalline solids that undergo martensitic phase transformation. The martensitic transformation is a temperature-induced, diffusionless solid-to-solid phase transformation involving a change in crystalline symmetry. Shape-memory materials are able to transform from the high-temperature austenite to the low-temperature martensite phase without any apparent change in shape. This is known as self-accommodation. Necessary and sufficient conditions that the lattice parameters of a material must satisfy for the material to form a self-accommodating microstructure are derived. The main result states that if the austenite is cubic, the material is self-accommodating if and only if the transformation is volume preserving. On the other hand, if the symmetry of the austenite is not cubic, it is not possible to construct any microstructure that is self-accommodating unless the transformation strain or the Bain strain satisfies additional, rather strict, conditions. These results show good agreement with the available experimental data. The analysis here is significantly different from previous studies because it makes no a priori assumption on the microstructure.     

A Variational Model for Reconstructive Phase Transformations in Crystals,and their Relation to Dislocations and Plasticity

1We study the reconstructive martensitic transformations in crystalline solids (i.e., martensitic transformations in which the parent and product lattices have arithmetic symmetry groups admitting no finite supergroup), the best known example of which is the bcc–fcc transformation in iron. We first describe the maximal Ericksen-Pitteri neighborhoods in the space of lattice metrics, thereby obtaining a quantitative characterization of the weak transformations, which occur within these domains. Then, focusing for simplicity on a two-dimensional setting, we construct a class of strain-energy functions admitting large strains in their domain, and which are invariant under the full symmetry group of the lattice. In particular, we exhibit an explicit energy suitable for the square-to-hexagonal reconstructive transformation in planar lattices. We present a numerical scheme based on atomic-scale finite elements and, by means of our constitutive function, we use it to analyze the effects of transformation cycling on a planar crystal. This example illustrates the main phenomena related to the reconstructive martensitic phase changes in crystals: in particular, the formation of dislocations, vacancies and interstitials in the lattice.