Yunzhi Wang

Three-dimensional field model and computer modeling of martensitic transformations

A three-dimensional (3D) continuum stochastic field kinetic model of martensitic transformations which explicitly takes into account the transformation-induced elastic strain is developed. The model is able to predict the major structural characteristics of martensite during the entire transformation including nucleation, growth and eventually formation of internally twinned plates which are in thermoelastic equilibrium with the parent phase. No a priori constraints are made on the possible configurations and sequences of structural patterns formed by orientation variants of the martensite. 3D computer simulations are performed for a generic cubic → tetragonal martensitic transformation in a prototype crystal which is elastically isotropic and elastically homogeneous. The simulations predict that (i) nucleation of martensite in a perfect crystal occurs collectively to accommodate the coherency strain, e.g. the critical nuclei are formed by two internally twinned orientation variants; (ii) the ultimate structure consists of plate-like martensite and retaining parent phase. The martensitic plates consist of twin-related platelets of two orientation variants and the habits of the plates meet the invariant plane requirement. These simulation results are in good agreement with experimental observations.

Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap

Morphological evolutions controlled by a transformation-induced elastic strain during a solid state precipitation are systematically investigated using a prototype binary alloy as a model system. A computer simulation technique based on a microscopic kinetic model including the elastic strain effect is developed. Without any a priori assumptions concerning shapes, concentration profiles and mutual positions of new phase particles, various types of coherent two-phase morphologies such as basket-weave structures, sandwich-like multi-domain structures, precipitate macrolattices and GP zones are predicted. A wide variety of interesting strain-induced kinetic phenomena are observed during development of the above microstructures, including selective and anisotropic growth, reverse coarsening, particle translational motion, particle shape transition and splitting. In spite of all simplifications of the model, most of the simulation results are confirmed by experimental observations in various alloy systems, indicating that this kinetic model can be efficiently used for understanding, interpreting and predicting structural evolutions in real alloys.

THREE-DIMENSIONAL PHASE FIELD MODEL AND SIMULATION OF MARTENSITIC TRANSFORMATION IN MULTILAYER SYSTEMS UNDER APPLIED STRESSES

Abstract The phase field microelasticity theory is used to formulate a three-dimensional phase field model of a multivariant martensitic transformation under external load. The model is based on the exact solution of the elasticity problem in the homogeneous modulus approximation. The transformation-induced coherency strain and applied stress are explicitly taken into account. Computer simulations are performed for a generic cubic→tetragonal martensitic transformation in a multilayer system consisting of alternating active and inert layers. The development of the martensitic transformation through nucleation, growth and coarsening of orientation variants is simulated at different levels of the applied stress. The simulated martensitic structure has a complex polytwinned morphology. The simulation predicted such effects as the formation of texture and the stress-induced transformation that are in a general agreement with the experimental observations. The simulation produced realistic stress–strain hysteresis loops, which, in principle, can be used for the formulation of the constitutive equations of the macroscopic mechanics for the active system.

Shape instability during precipitate growth in coherent solids

Abstract The effect of coherent elastic strain on shape instabilities during growth of a single precipitate in an elastically anisotropic cubic system is examined. A general phenomenological field approach to phase transformation kinetics is employed. Emphasis has been given to understanding the formation of concave interfaces of a coherent γ′ particle in the Ni-based superalloys. It is shown by a two-dimensional computer simulation that the infinite-range and highly anisotropic strain-induced interaction results in a shape transition from circle to faceted square with 10 habits and rounded corners at the early stage of growth. Then the divergence of solute atom supply at the corners enhances their growth which results in a concave morphology. Vanishing of the supersaturation (transition to the coarsening stage) causes a diffusional relaxation of the concave interfaces back into flat ones, indicating that the concave square is a nonequilibrium shape. A semi-quantitative criterion for the formation of concave shapes is derived. The concentration profile inside the misfitting particle is found to be nonuniform during its growth and coarsening.

Boundary Mobility and Energy Anisotropy Effects on Microstructural Evolution During Grain Growth

We have performed mesoscopic simulations of microstructural evolution during curvature driven grain growth in two-dimensions using anisotropic grain boundary properties obtained from atomistic simulations. Molecular dynamics simulations were employed to determine the energies and mobilities of grain boundaries as a function of boundary misorientation. The mesoscopic simulations were performed both with the Monte Carlo Potts model and the phase field model. The Monte Carlo Potts model and phase field model simulation predictions are in excellent agreement. While the atomistic simulations demonstrate strong anisotropies in both the boundary energy and mobility, both types of microstructural evolution simulations demonstrate that anisotropy in boundary mobility plays little role in the stochastic evolution of the microstructure (other than perhaps setting the overall rate of the evolution. On the other hand, anisotropy in the grain boundary energy strongly modifies both the topology of the polycrystalline microstructure the kinetic law that describes the temporal evolution of the mean grain size. The underlying reasons behind the strongly differing effects of the two types of anisotropy considered here can be understood based largely on geometric and topological arguments.

The continuum field approach to modeling microstructural evolution

The majority of advanced engineering materials contain multiphase and/or multidomain structures. Their physical and mechanical properties depend strongly on the number of phases present and their mutual arrangement; the volume fraction of each phase; and the shape, size, and size distribution of domains (or grains). This article describes a continuum diffuse-interface field approach to modeling microstructural evolution and its application to a number of different processes, including precipitation reactions through nucleation and growth, structural transformations involving symmetry changes, and curvature-driven grain growth.

Phase field model of dislocation networks

Abstract In phase field model of dislocations, the short-range, non-linear core–core interactions are characterized through the crystalline energy and the gradient energy. In this article we extend and generalize the approximations of these energies employed in previous phase field models to account for dislocation reactions leading to network formation. In order to characterize dislocation activities involving all slip planes, we suggest the crystalline energy to be a function of a general plastic strain tensor produced by arbitrary linear combinations of simple shears associated with each slip system. For the four {111} slip planes in an fcc crystal, a particular form of such a crystalline energy is formulated by simple linear superposition of the interplanar potential of each individual slip plane. A more detail and general form of the gradient energy is derived from the consideration of the total Burgers vector dependence of dislocation line energy. Examples of applications are presented for interactions between two dislocation loops expanding on either a single slip plane or two intersecting slip planes, as well as for more complex reactions taking place in dislocation networks. It is shown that the generalized expressions are able to handle self-consistently reactions among dislocations of all slip systems in accord with Frank’s rule. These extensions are necessary steps toward advanced applications of the phase field method to dislocation substructure formation and coarsening.

Kinetics of tweed and twin formation during an ordering transition in a substitutional solid solution

Abstract A kinetic model based on the theory of strain-induced interactions is developed to describe the effect of elastic strain on structural transformation kinetics. An ordering transition resulting in a reduction in the crystal lattice point-group symmetry in a substitutional solid solution is considered. Our computer simulation of a two-dimensional model binary alloy revealed the detailed kinetics of formation of tweed morphologies and the tweed → twin coarsening process driven by the transformation-induced elastic strains.

Phase-field modeling of bimodal particle size distributions during continuous cooling

Abstract Microstructures in Nickel-base alloys typically contain a two-phase mixture of γ/γ′. The microstructure having a bimodal size distribution of γ′ is of particular interest because it has important property consequences [1] . In this paper, the phase-field method with an explicit nucleation algorithm is employed to investigate the microstructural development during a continuous cooling with various cooling rates. It is demonstrated that bimodal particle size distributions can be achieved at an intermediate cooling rate due to a coupling between diffusion and undercooling, in which the system experiences two peaks of well-isolated nucleation events. It is suggested that this is caused by soft impingement, followed by a renewal of driving force for nucleation, followed by a subsequent soft impingement. Under very high cooling rates, the microstructure becomes unimodal, because undercooling always outruns diffusion and the microstructure never reaches soft impingement.

Sparse data structure and algorithm for the phase field method

The concepts of sparse data structures and related algorithms for phase field simulations are discussed. Simulations of polycrystalline grain growth with a conventional phase field method and with sparse data structures are compared. It is shown that memory usage and simulation time scale with the number of nodes but are independent of the number of order parameters when a sparse data structure is used.