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Multiscale Modeling Phase Transformations

Nanoscale Theory and Phase Field Modeling

A three-dimensional Landau theory for multivariant stress-induced PTs between high temperature phase (austenite) and variants of low temperature phase (martensitic variants) and between martensitic variants has been developed]. The model can incorporate all temperature-dependent thermomechanical properties of both phases for arbitrary crystal symmetries, including higher-order elastic constants, and it correctly describes the characteristic features of stress-strain curves for materials with strong martensitic PT including shape memory alloys and steels. These features include the constant transformation strain and constant (or weakly temperature-dependent) stress hysteresis, and nonzero tangent elastic moduli near the PT point. In contrast, previous Landau potentials did not have enough degrees of freedom to account for all material properties of both austenite and martensite, and cannot describe the above typical features of stress-strain curves for shape memory alloys and steels even in the case of one-dimensional loading. This makes the theory applicable to real materials. As an example, all parameters of our model are found for NiAl alloy using the results of molecular dynamics simulations available in the literature.

For one-dimensional spatial case, analytical solutions of time-independent Ginzburg-Landau equation under constant three-dimensional stress tensor were found. The structure and energetics of martensitic and austenitic critical nuclei, and diffuse martensite-austenite and martensite-martensite interfaces were derived. In particular, width of austenitic and martensitic nuclei and various interfaces, energy of nuclei and interfaces are determined in terms of two parameters, one of which is the driving force for austenite-martensite PT and another one characterizes the stability of austenite. Relation between energy and width of equilibrium austenite-martensite interface and stress hysteresis is found. A splitting of martensite M- - martensite M+ interface into two austenite-martensite interfaces is interpreted as a new mechanism of barrierless austenite nucleation. Experimental data supporting this mechanism are found in literature. Finally, our approach is applied for the development of phase field theory of dislocations, which is free of drawbacks similar to those we corrected for PTs.
The quantitative study of a number of important problems concerning temperature- and stress-induced martensitic PT in real materials can be carried out using our theory. Problems of interest include heterogeneous martensite nucleation at dislocations and cracks, formation of twinned microstructure, the structure of interfaces, solitary waves, and the interaction of moving interfaces with defects. The obtained results are applicable to a wide class of materials, including shape memory alloys, steels and ceramics.


The main drawback of our papers [4,5] and all other Landau models is their limitation to small transformation strains (smaller than 0.1). At the same time, the transformation shear is 0.2 for PTs in steels and shape memory alloys, the volumetric transformation strain for the graphite-diamond and BN rhombohedral-cubic PTs is 0.54, the transformation   shear for twinning in bcc and fcc lattices is 0.71, and for dislocations it is ~1. Moreover, finite rotations of the crystal  lattice, which can even occur at small transformation strain, can crucially affect the PT conditions. PT conditions in phase field theory are conditions for instability of the crystal lattice. Recent research on lattice instabilites at finite strain has focused on the ideal strength of crystals and solid-solid, amorphization, and melting PTs. A general approach to stability at finite strain has been developed on the basis of the Lagrange-Dirichlet criterion for conservative systems.
Most stability studies have addressed only intrinsic stability, where the applied stress ''follows'' the material during deformation. However, intrinsic stability may differ significantly from stability in laboratory experiments, where finite rotation of the specimen is involved.

In our letter [3], the Landau theory for stress-induced PT is developed for large strain and rotation for the first time. We derive for nonconservative systems a thermomechanical lattice instability criterion with respect to change in order parameters using the second law of thermodynamics. It accounts for finite elastic and transformation strains and lattice rotations. Invariance of the instability criterion and Gibbs potential under changes in stress and strain measures is proved. The Gibbs potential, criteria for PT, twinning, and dislocation nucleation, and equilibrium conditions are derived for a prescribed nonsymmetric Piola-Kirchoff (nominal) stress tensor, i.e., the force per unit area in the undeformed state, and small elastic strains but finite transformation strains and rotations. In contrast to other formulations, it is not necessarily the case that the instability occurs at a stationary value of some stress measure. PTs in NiAl, BN and graphite to diamond are analyzed and significant differences between the small- and finite-strain Gibbs potentials are found. All of these results are obtained for the first time.

  1. Levitas V.I., Lee D-W.and  Preston D. Phase field theory of surface- and size-induced microstructures. Europhysics Letters (in press).
  2. Levitas V.I. , Preston D and Lee D-W. Ginzburg-Landau theory of microstructures: stability, transient
    dynamics, and functionally graded nanophases. Europhysics Letters, 2006, Vol. 75, No. 1, 84-90.  pdf
  3. Levitas V.I., Preston D.L. Thermomechanical lattice instability and phase field theory of martensitic
    phase transformations, twinning and dislocations at large strains. Physics Letters A,  2005, Vol. 343, 32-39. pdf
  4. Levitas V.I. and Preston D. Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. Part I and II. Phys. Rev. B, 2002, Vol. 66, 134206(1-9) and 134207(1-15).pdf-1 pdf-II
  5. Levitas V.I., Preston D and Lee D-W. Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. Part III. Alternative potentials, critical nuclei, kink solutions, and dislocation theory. Phys. Rev. B, 2003, Vol. 68, 134201 (1-24). pdf.file


Microscale and Macroscale Theory and Modeling

A new approach for modeling multivariant martensitic PTs and martensitic microstructure in
elastic materials is proposed. It is based on a thermomechanical model for PT that includes strain softening and the corresponding strain localization during PT. Mesh sensitivity in numerical simulations is avoided by using rate dependent constitutive equations in the model. Due to strain softening, a microstructure comprised of pure martensitic and austenitic domains separated by narrow transition zones is obtained as the solution of the corresponding boundary value problem. In contrast to Landau-Ginzburg models, which are limited in practice to nanoscale specimens, this new phase field model is valid for scales greater than 100 nm and without upper bound. A finite element algorithm for the solution of elastic problems with multivariant martensitic PT is developed and implemented into the software ABAQUS. Simulated microstructures in elastic single crystals and polycrystals under uniaxial loading are in qualitative agreement with those observed experimentally.

  1. Idesman A. V., Levitas V. I. , Preston D. L., and Cho J.-Y. Finite Element Simulations of Martensitic Phase Transitionsand Microstructure Based on Strain Softening Model. J. Mechanics and Physics of Solids, 2005,Vol. 53, No. 3, pp. 495-523. pdf 
  2. Levitas V. I., Idesman A. V. and Preston D. Microscale simulation of evolution of martensitic microstructure. Phys. Review Letters, 2004, 2004, 93, 105701 (Selected and reproduced in Virtual J. Nanoscale Science Technology, 2004, Sept. 5). pdf
  3. Mielke A., Theil F., Levitas V.I. A Variational Formulation of Rate-Independent Phase Transformations Using an Extremum Principle. Archive for Rational Mechanics and Analysis, 2002, Vol 162, 137-177 . pdf.file                 Essential Science Indicator: Emerging Research Fronts Paper in Mathematics in August 2006
  4. Leshchuk A. A., Novikov N. V., Levitas V. I. Thermomechanical Model of Phase Transformation Graphite to Diamond. J. of Superhard Materials, 2002, No. 1, pp. 49-57.pdf.file

 

Interaction between Phase Transformation and Plasticity

For the description of PT in an elastic solid, the principle of a minimum of Gibbs free energy is usually used. For inelastic materials a corresponding principle has been lacking. We develop a conceptually new approach and for the description of PT in an inelastic solid and verified it by explanation and interpretation of a number of experimental phenomena.

  1. Levitas V. I. Continuum Mechanical Fundamentals of Mechanochemistry. In: High Pressure Surface Science and Engineering. Section 3. Institute of Physics, Bristol, Eds. Y. Gogotsi and V. Domnich, 2004, pp. 159-292.pdf
  2. Levitas V.I., Idesman A.V., Olson G.B. and Stein E. Numerical Modeling of Martensite Growth in Elastoplastic Material. Philosophical Magazine, A, 2002, Vol. 82, No. 3, 429-462.pdf.file
  3. Levitas V.I. Critical Thought Experiment to Choose the Driving Force for Interface Propagation in Inelastic Materials. Int. J. Plasticity, 2002, Vol. 18, pp. 1499-1525. pdf. file