Molecular dynamics simulation study of microstructure evolution during cyclic martensitic transformations

Shape memory alloys (SMA) exhibit a number of features which are not easily explained by equilibrium thermodynamics, including hysteresis in the phase transformation and “reverse” shape memory in the high symmetry phase. Processing can change these features: repeated cycling can “train” the reverse shape memory effect, while changing the amount of hysteresis and other functional properties. These effects are likely to be due to formations of localised defects and these can be studied by atomistic methods. Here we present a molecular dynamics simulation study of such behaviour employing a two-dimensional, binary Lennard-Jones model. Our atomistic model exhibits a symmetry breaking, displacive phase transition from a high temperature, entropically stabilised, austenite-like phase to a low temperature martensite-like phase. The simulations show transformations in this model material proceed by non-diffusive nucleation and growth processes and produce distinct microstructures. We observe the generation of persistent lattice defects during forward-and-reverse transformations which serve as nucleation centres in subsequent transformation processes. These defects interfere the temporal and spatial progression of transformations and thereby affect subsequent product morphologies. During cyclic transformations we observe accumulations of lattice defects so as to establish new microstructural elements which represent a memory of the previous morphologies. These new elements are self-organised and they provide a basis of the reversible shape memory effect in the model material.     

Bubbles in liquids with phase transition

In the forthcoming second part of this paper a system of balance laws for a multi-phase mixture with many dispersed bubbles in liquid is derived where phase transition is taken into account. The exchange terms for mass, momentum and energy explicitly depend on evolution laws for total mass, radius and temperature of single bubbles. Therefore in the current paper we consider a single bubble of vapor and inert gas surrounded by the corresponding liquid phase. The creation of bubbles, e.g. by nucleation is not taken into account. We study the behavior of this bubble due to condensation and evaporation at the interface. The aim is to find evolution laws for total mass, radius and temperature of the bubble, which should be as simple as possible but consider all relevant physical effects. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves and phase transition on the evolution of the bubble. We find ordinary differential equations that describe the bubble dynamics. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble. It is shown that a rebounding bubble is only possible, if it contains in addition an inert gas. In Part 2 of the current paper the equations derived are sought in order to close the system of equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change.     

Fluidization of a granular medium in a viscous fluid under vertical vibration

The dynamics of a granular medium in a cavity filled with incompressible viscous fluid under harmonic vertical vibration are studied experimentally. The sand is fluidized in a relatively thin sublayer of the granular layer near the interface between the media. The fluidization is of the threshold type and is accompanied by intense parametric oscillations of the interface. For viscous fluids, the transition of the sand from a quasi-solid to a fluidized state and the reverse transition associated with a decrease in the oscillation rate occur with hysteresis. The nondimensional governing parameters determining the sand dynamics are established. The analysis is focused on the case of low nondimensional frequencies. Perm’, Paris. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–122, May–June, 2000.     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic ‘butterfly’ curves; and others.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Slip coefficients and macroparameter jumps of a diatomic gas with rotational degrees of freedom on a weakly curved spherical surface

Using the half-space moment method, the problem of the slip of a diatomic gas along a rigid spherical surface is solved within the framework of a model kinetic equation previously proposed which takes into account the rotational degrees of freedom of the gas. Second-order slip coefficients (correctionsC _m ^′ , β _R ^′ , and β _R to the isothermal and thermal slip which are linear with respect to the Knudsen number Kn) are obtained. The gas macroparameter jump coefficientsC _v andC _q, which are of the second order in the Knudsen number and characterize the discontinuity of the normal mass and heat fluxes on the gas-rigid phase interface, are calculated. These coefficients are given as functions of the tangential momentum accommodation coefficient, the translational and rotational energy accommodation coefficients, and the Prandtl number. The coefficients are calculated for certain diatomic gases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–173, January–February, 2000.     

Simulation of bacterial flagellar phase transition by non-convex and non-local continuum modeling

Bacterial flagellar filament can undergo a stress-induced polymorphic phase transition in both vitro and vivo environments. The filament has 12 different helical forms (phases) characterized by different pitch lengths and helix radii. When subjected to the frictional force of flowing fluid, the filament changes between a left-handed normal phase and a right-handed semi-coiled phase via phase nucleation and growth. This paper develops non-local finite element method (FEM) to simulate the phase transition under a displacement-controlled loading condition (controlled helix-twist). The FEM formulation is based on the Ginzburg-Landau theory using a one-dimensional non-convex and non-local continuum model. To describe the processes of the phase nucleation and growth, viscosity-type kinetics is also used. The non-local FEM simulation captures the main features of the phase transition: two-phase coexistence with an interface of finite thickness, phase nucleation and phase growth with interface propagation. The non-local FEM model provides a tool to study the effects of the interfacial energy/thickness and loading conditions on the phase transition.     

Elastoplastic bending of a sandwich bar on an elastic foundation

The elastoplastic bending of a sandwich bar with a stiff compressible core on an elastic foundation is studied. The kinematics of the bar, which is asymmetric across the thickness, is described adopting Bernoulli’s hypotheses for the face layers. The displacements of the core are assumed to vary linearly across the thickness. The foundation is described by the Winkler model. A system of equilibrium equations for displacements is derived and solved. Numerical results for a metal-polymer sandwich bar are presented __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 110–120, April 2007.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Analysis of coupled-mode flutter of pipes conveying fluid on the elastic foundation

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马氏体相变微结构演化过程的模拟与分析

实验中观察到形状记忆合金在应力诱发马氏体相变过程中,出现多界面的微结构,马氏体相会逐渐长大变粗,同时会出现由马氏体形核造成的应力突然降低.用多阱的弹性能函数来刻画此相变与微结构演化过程,发现相变时会出现多界面的微结构且伴随着马氏体相的形核至奥氏体相的消失过程,出现了界面数先增后减的变化,同时应力会出现跳跃而不连续.相对应的动力学模型的有限差分的计算结果同样显示形核时出现了多界面的微结构并伴随着应力的大幅振荡,随着载荷的增加界面位置随之移动,使得马氏体相区域逐渐长大.理论分析与数值模拟的结果较好地刻画了实验中观察到的马氏体相变过程中的形核,产生多界面,再到马氏体逐渐长大这一微结构的演化过程.     

A discrete dislocation analysis of bending

Bending of a strip in plane strain is analyzed using discrete dislocation plasticity where the dislocations are modeled as line defects in a linear elastic medium. At each stage of loading, superposition is used to represent the solution in terms of the infinite medium solution for the discrete dislocations and a complementary solution that enforces the boundary conditions, which is non-singular and obtained from a linear elastic, finite element solution. The lattice resistance to dislocation motion, dislocation nucleation and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. Solutions for cases with multiple slip systems and with a single slip system are presented. The bending moment versus rotation relation and the evolution of the dislocation structure are outcomes of the boundary value problem solution. The effects of slip geometry, obstacles to dislocation motion and specimen size on the moment versus rotation response are considered. Also, the evolution of the dislocation structure is studied with emphasis on the role of geometrically necessary dislocations. The dislocation structure that develops leads to well-defined slip bands, with the slip band spacing scaling with the specimen height.     

Landau-Ginzburg model for a deformation-driven experiment on shape memory alloys

A Landau-Ginzburg model describing first order martensitic phase transitions in shape memory alloys is considered. The model developed by Falk is transformed in order to simulate deformation-driven experiments done by I. Müller and his co-workers. In these experiments, they do not only observe load-deformation hysteresis loops but also small loops inside these hysteresis loops. Numerical simulations for a CuZnAl single crystal show good agreement with the experiment. We find, for example, nucleation processes, moving phase boundaries, rate-independent hysteresis loops and interior loops. Received January 25, 1996     

Material forces and phase transitions in elasticity

A computational scheme for the determination of the interface in a strain-induced phase-transition problem for an elastic bar is proposed. The algorithm is based on the material force notion and more specifically on the simultaneous solution of equilibrium equations for the physical and material forces. The weak form of both equations is derived with the aid of a variational principle that accounts for the variations of the dependent and the independent variables. The whole scheme concludes in a nonlinear algebraic system which is numerically solved by the Newton method. The numerical results thus derived seem to be quite encouraging for further application of the concept of material forces in computations related to phase transition problems. The austenite–martensite transformation could be a possible application of the proposed model.     

Bifurcations in the Regularized Ericksen Bar Model

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller’s conjecture (Müller, Calc. Var. 1:169–204, 1993) concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Müller’s conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that characterizes (k,3k) bifurcations.      

Dislocation nucleation in Σ3 asymmetric tilt grain boundaries

Atomistic simulations were used to investigate dislocation nucleation from Σ3 asymmetric (inclined) tilt grain boundaries under uniaxial tension applied perpendicular to the boundary. Molecular dynamics was employed based on embedded atom method potentials for Cu and Al at 10 K and 300 K. Results include the grain boundary structure and energy, along with mechanical properties and mechanisms associated with dislocation nucleation from these Σ3 boundaries. The stress and work required for dislocation nucleation were calculated along with elastic stiffness of the bicrystal configurations, exploring the change in response as a function of inclination angle. Analyses of dislocation nucleation mechanisms for asymmetric Σ3 boundaries in Cu show that dislocation nucleation is preceded by dislocation dissociation from the boundary. Then, dislocations preferentially nucleate in only one crystal on the maximum Schmid factor slip plane(s) for that crystal. However, this crystal is not simply predicted based on either the Schmid or non-Schmid factors. The synthesis of these results provides a better understanding of the dislocation nucleation process in these faceted, dissociated grain boundaries.