Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

Size effects on the martensitic phase transformation of NiTi nanograins

The analysis of nanocrystalline NiTi by transmission electron microscopy (TEM) shows that the martensitic transformation proceeds by the formation of atomic-scale twins. Grains of a size less than about 50 nm do not transform to martensite even upon large undercooling. A systematic investigation of these phenomena was carried out elucidating the influence of the grain size on the energy barrier of the transformation. Based on the experiment, nanograins were modeled as spherical inclusions containing (0 0 1) compound twinned martensite. Decomposition of the transformation strains of the inclusions into a shear eigenstrain and a normal eigenstrain facilitates the analytical calculation of shear and normal strain energies in dependence of grain size, twin layer width and elastic properties. Stresses were computed analytically for special cases, otherwise numerically. The shear stresses that alternate from twin layer to twin layer are concentrated at the grain boundaries causing a contribution to the strain energy scaling with the surface area of the inclusion, whereas the strain energy induced by the normal components of the transformation strain and the temperature dependent chemical free energy scale with the volume of the inclusion. In the nanograins these different energy contributions were calculated which allow to predict a critical grain size below which the martensitic transformation becomes unlikely. Finally, the experimental result of the atomic-scale twinning can be explained by analytical calculations that account for the transformation-opposing contributions of the shear strain and the twin boundary energy of the twin-banded morphology of martensitic nanograins.     

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.     

An upper bound to the free energy of n-variant polycrystalline shape memory alloys

In the last two decades, the problem of computing the elastic energy of phase transforming materials has been studied by a variety of research groups. Due to the non-quasiconvexity of the underlying multi-well landscape, different relaxation methods have been used in order to estimate the quasiconvex envelope of the energy density, for which no explicit expression is known at present.This paper combines a recently developed lamination bound for monocrystalline shape memory alloys which relies on martensitic twinned microstructures with the work of Smyshlyaev and Willis [1998a. A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. London A 454, 1573–1613]. As a result, a lamination upper bound for n-variant polycrystalline martensitic materials is obtained.The lamination bound is then compared with Reuß- and Taylor-type estimates. While, for given volume fractions, good agreement of lamination upper and convexification lower bounds is obtained, a comparison using energy-minimizing volume fractions computed from the various bounds yields larger differences. Finally, we also investigate the influence of the polycrystal's texture. For a strong ellipsoidal texture, we observe even better agreement of upper and lower bounds than for the case of isotropic statistics.     

Coupling between martensitic phase transformations and plasticity: A microstructure-based finite element model

A microstructural finite element (MFE) model is developed to capture the interaction between martensitic transformations and plasticity in NiTi shape memory alloys (SMAs). The interaction is modeled through the grain-to-grain redistribution of stress caused by both plasticity and phase transformation, so that each mechanism affects the driving force of the other. A unique feature is that both processes are modeled at a crystallographic level and are allowed to operate simultaneously. The model is calibrated to pseudoelastic data for select single crystals of Ti–50.9at.%Ni. For polycrystals, plasticity is predicted to enhance the overall martensite volume fraction at a given applied stress. Upon unloading, residual stress can induce remnant (retained) martensite. For thermal cycling under load bias, plasticity is observed to limit the net transformation strain/cycle and increase the hysteretic width. Deformation processing, via plastic pre-straining at elevated temperature, is shown to dramatically alter subsequent pseudoelastic response, as well as induce two-way shape memory behavior during no-load thermal cycling. Overall, the model is suitable at smaller imposed strains, where martensite detwinning is not expected to dominate.     

The influence of surface energy on stress-free microstructures in shape memory alloys

Certain alloys such as In-Tl, Ni-Ti, Ag-Cd or Cu-Al-Ni display remarkable mechanical properties such as the shape memory effect or pseudo-elasticity. This behaviour is related to a solid-solid phase transformation which leads to a complicated microscopic arrangement of different phases. In recent studies such microstructures have been analyzed by the minimization of elastic energy. We discuss the influence of additional surface energy terms on the existence of stress-free microstructures both in the nonlinear and a geometrically linear setting.

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Sommario Certe leghe come quelle di In-Tl, Ni-Ti, Ag-Cd o Cu-Al-Ni mostrano proprietà meccaniche notevoli quali la memoria di forma o la pseudoelasticità. Questo comportamento è determinato da una trasformazione di fase solido-solido che conduce a complicati arrangiamenti a livelo microscopico. In studi recenti tali microstrutture sono state analizzate attraverso la minimizzazione dell'energia elastica. Noi discutiamo l'influenza di termini addizionali di energia superficiale sull'esistenza di microstrutture in uno stato naturale sia in un contesto lineare che non lineare.

     

Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution

In this work recent results regarding bounds of the relaxed free energy functions of a broad class of shape memory materials of arbitrary symmetry are leveraged to develop a simple and efficient numerical method to analyze several aspects of the thermomechanical response of such materials. This approach is shown to be useful for construction of the austenitic phase diagram used in many early phenomenological models. It is also demonstrated that the resulting implementation is suitable for finite element analysis, has desirable numerical properties, and makes realistic quantitative predictions regarding the evolution of the lattice correspondence variants, transformation stress and strain levels, and orientation dependence.     

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.     

Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations

Analytical solutions for diffuse interface propagation are found for two recently developed Landau potentials that account for the phenomenology of stress-induced martensitic phase transformations. The solutions include the interface profile and velocity as a function of temperature and stress tensor. An instability in the interface propagation near lattice instability conditions is studied numerically. The effect of material inertia is approximately included. Two methods for introducing an athermal interface friction in phase field models are discussed. In the first method an analytic expression defines the location of the diffuse interface, and the rate of change of the order parameters is required to vanish if the driving force is below a threshold. As an alternative and more physical approach, we demonstrate that the introduction of spatially oscillatory stress fields due to crystal defects and the Peierls barrier, or to a jump in chemical energy, reproduces the effect of an athermal threshold. Finite element simulations of microstructure evolution with and without an athermal threshold are performed. In the presence of spatially oscillatory fields the evolution self-arrests in realistic stationary microstructures, thus the system does not converge to an unphysical single-phase final state, and rate-independent temperature- and stress-induced phase transformation hysteresis are exhibited.     

Phase transitions induced by extension in a slender SMA cylinder: Analytical solutions for the hysteresis loop based on a quasi-3D continuum model

In this paper, we propose a quasi-3D continuum model to study the rate-independent hysteresis phenomenon in phase transitions of a slender shape memory alloy (SMA) cylinder subject to the uniaxial tension. Based on the three-dimensional field equations and the traction-free boundary conditions, by using a coupled series-asymptotic expansion method, we manage to express the total elastic potential energy of the cylinder in terms of the leading order term of the axial strain. We further consider the rate-independent dissipation effect in a purely one-dimensional setting. The mechanical dissipation functions are also expressed in terms of the axial strain. The equilibrium configuration of the cylinder is then determined by using the principle of maximizing the total energy dissipation. An illustrative example with some special chosen material constants is further considered. Free end boundary conditions are proposed at the two ends of the cylinder. By conducting a phase plane analysis and through some calculations, we obtain the analytical solutions of the equilibrium equation. We find that the engineering stress–strain curves corresponding to the obtained solutions can capture some important features of the experimental results. It appears that the analytical results obtained in this paper reveal the multiple solutions nature of the problem and shed certain light on the instability phenomena during the phase transition process.     

Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation

The universal (i.e. independent of the constitutive equations) thermodynamic driving force for coherent interface reorientation during first-order phase transformations in solids is derived for small and finite strains. The derivation is performed for a representative volume with plane interfaces, homogeneous stresses and strains in phases and macroscopically homogeneous boundary conditions. Dissipation function for coupled interface (or multiple parallel interfaces) reorientation and propagation is derived for combined athermal and drag interface friction. The relation between the rates of single and multiple interface reorientation and propagation and the corresponding driving forces are derived using extremum principles of irreversible thermodynamics. They are used to derive complete system of equations for evolution of martensitic microstructure (consisting of austenite and a fine mixture of two martensitic variants) in a representative volume under complex thermomechanical loading. Viscous dissipation at the interface level introduces size dependence in the kinetic equation for the rate of volume fraction. General relationships for a representative volume with moving interfaces under piece-wise homogeneous boundary conditions are derived. It was found that the driving force for interface reorientation appears when macroscopically homogeneous stress or strain are prescribed, which corresponds to experiments. Boundary conditions are satisfied in an averaged way. In Part 2 of the paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples. Int. J. Plasticity (2008)], the developed theory is applied to the numerical modeling of the evolution of martensitic microstructure under three-dimensional thermomechanical loading during cubic-tetragonal and tetragonal-orthorhombic phase transformations.     

Internal loops in pseudoelastic behaviour of Ti-Ni shape memory alloys: Experiment and modelling

The stress-strain isothermal hysteresis loops due to the incomplete martensitic transformation are analysed for Ti-Ni shape memory alloys. Experiments show the existence of two distinct yield lines for phase transition; one for the forward transformation austenitemartensite (AM), the other for the reverse transformation MA. The tensile behaviour of single crystals with only one yield line (AM) [1] can be considered as an ideal case. An extension of a thermodynamic model for pseudoelasticity [2] allows these two yield lines to be taken into account.

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Sommario Per leghe Ti-Ni con memoria di forma vengono analizzati i cicli di isteresi isotermici tensione-deformazione prodotti da una incompleta trasformazione martensitica. Gli esperimenti mostrano l'esistenza di due distinte linee di snervamento per la transizione di fase, una verso la trasformazione austenitemartensite (AM), l'altra per la trasformazione inversa MA. Il comportamento a trazione di un singolo cristallo con una sola linea di snervamento (AM) [1], può essere considerato un caso ideale. L'estensione ad un modello termodinamico pseudo-elastico [2] consente di analizzare queste due linee di snervamento.

     

Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces

In this paper, interfacial excess energy and interfacial excess stress for coherent interfaces in an elastic solid are reformulated within the framework of continuum mechanics. It is shown that the well-known Shuttleworth relationship between the interfacial excess energy and interfacial excess stress is valid only when the interface is free of transverse stresses. To account for the transverse stress, a new relationship is derived between the interfacial excess energy and interfacial excess stress. Dually, the concept of transverse interfacial excess strain is also introduced, and the complementary Shuttleworth equation is derived that relates the interfacial excess energy to the newly introduced transverse interfacial excess strain. This new formulation of interfacial excess stress and excess strain naturally leads to the definition of an in-plane interfacial stiffness tensor, a transverse interfacial compliance tensor and a coupling tensor that accounts for the Poisson's effect of the interface. These tensors fully describe the elastic behavior of a coherent interface upon deformation.     

Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

We consider dynamics of chains of rigid masses connected by links described by irreversible, piecewise linear constitutive relation: the force-elongation diagram consists of two stable branches with a jump discontinuity at the transition point. The transition from one stable state to the other propagates along the chain and excites a complex system of waves. In the first part of the paper (Cherkaev et al., 2004, Transition waves in bistable structures. I. Delocalization of damage), the branches could be separated by a gap where the tensile force is zero, the transition wave was treated as a wave of partial damage. Here we assume that there is no zero-force gap between the branches. This allows us to obtain steady-state analytical solutions for a general piecewise linear trimeric diagram with parallel and nonparallel branches and an arbitrary jump at the transition. We derive necessary conditions for the existence of the transition waves and compute the speed of the wave. We also determine the energy of dissipation which can be significantly increased in a structure characterized by a nonlinear discontinuous constitutive relation. The considered chain model reveals some phenomena typical for waves of failure or crushing in constructions and materials under collision, waves in a structure specially designed as a dynamic energy absorber and waves of phase transitions in artificial and natural passive and active systems.     

Effect of microstructure on the hardening and softening behaviors of polycrystalline shape memory alloys Part II: Numerical simulation under axisymmetrical loading

Based on the microstructure-based constitutive model established in Part I, a detailed numerical investigation on the role of each microstructure parameter in the kinematical and kinetic evolution of polycrystalline SMA under axisymmetrical tension loading is performed. Some macroscopic constitutive features of stress-induced martensite transformation are discussed. The subject supported by the Research Grant Committee (RGC) of Hong Kong SAR, the National Natural Science Foundation of China and the Provincial Natural Science Foundation of Jiangxi Province of China     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.     

Modeling Three-Dimensional Groundwater Flows by the Body-Fitted Coordinate (BFC) Method: II. Free and Moving Boundary Problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.     

Hysteresis loops predicted by isoenergy density theory for polycrystals. Part II: cyclic heating and cooling effects predicted from non-equilibrium theory for 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium cylindrical bars

Mathematical tools and physical models should fit like hand and glove. Traditionally, such an understanding has worked surprisingly well in mechanics and physics. Differential calculus enabled the determination of motion of celestial bodies. Heisenberg's use of Hilbert space applied to calculations in quantum mechanics. These foot steps, however, failed to continue in the field of material science when attempting to address the evolution of material damage. On one hand, the mathematical formulation of the Riemann–Hilbert problem achieved great success for solving macrocrack problems but whether the same tool could be used for lower scale defects seems to be of less concern. This is a surprise because it is not obvious by any means whether microcracks can be distinguished from macrocrack by size difference alone, particularly when they are both treated in a single formulation.There are several issues of the fatigue problem that must be addressed. To begin with, local failure initiating near a surface should be distinguished from global failure that correspond to separation of a specimen into two or more pieces. The interim stages of initiation and termination would depend on the ductility and brittleness of the material in addition to the history of cyclic loading. Three different materials 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium will therefore be analyzed to show how their uniaxial properties would compare with the dissipation energy density functions and oscillation in temperature. Hysteresis loops for the local material points are determined. Their averages for the ASTM standard hour-glass specimen are then found. They give the global hysteresis loops corresponding to measurements from the uniaxial tests. The local dissipative energy density loops represent the irreversible nature of the materials are also taken into consideration. They are calculated for the soft aluminum and the hard titanium with steel being in between.Obtained also are the cyclic cooling and heating characteristics of the material under fatigue; they fluctuate about the ambient temperature. The -function designating the order and disorder of the fatigue process is also found to be oscillatory in character. Unlike entropy, it is not non-increasing or non-decreasing. It can change sign in a given process. This implies that the order of a system can be enhanced and then impeded or vice versa. Such a behavior is shown to prevail during the initial stage of the fatigue.