The purpose of this research is to study systematically compatibility conditions and their implications for the microstructure of a phase-transforming material. The phase transformation in this thesis is restricted to crystalline solid-to-solid phase transformation. The conditions of compatibility refer to compatibility in the sense of the nonlinear elastic theory of martensite. Different versions of these conditions of compatibility are studied in this thesis, ranging from "weak compatibility'' (continuity along lines aligned with precipitates) to very strong conditions of compatibility as expressed by the "cofactor conditions''. In the case of a diffusionless, reversible martensitic phase transformation, the free energy of the undistorted body is described as the volume integral of the free energy density function, which depends on the temperature and deformation gradient of the continuous body. This free energy at continuum level describes the elastic and chemical energy stored in the lattice. Macroscopic deformations are related to lattice deformation by the Cauchy-Born rule.This rule yields a deformation gradient <bold>F</bold> relating a sublattice of the austenite phase to the primitive lattice of the martensite phase. We derive a heuristic algorithm to find <bold>F</bold> directly from X-ray diffraction measurement for both phases. For such a transformation both the lattice parameters and the symmetry of the crystal structure change. We assume that the free energy is invariant under rigid rotations and symmetry operations. The <italic>transformation stretch matrix</italic> <bold>U</bold> is calculated from the deformation gradient <bold>F</bold> by polar decomposition. The associated crystallographically equivalent variants <bold>U</bold><sub>1</sub>, ..., <bold>U<bold><sub>n</sub> are determined by symmetry arguments (We can choose <bold>U</bold><sub>1</sub> = <bold>U</bold>.). The matrices <bold>I</bold> (austenite) and <bold>U</bold><sub>1</sub>, ..., <bold>U<bold><sub>n</sub> (variants of martensite) determine the_para>energy wells of the free energy density. The formation of microstructure arises from the simultaneous requirements of energy minimization, i.e., being near the energy wells, and compatibility. The Widmanstatten type precipitation process produces a microstructure of elongated precipitates. For this microstructure we propose a weaker condition of compatibility than is used in the study of martensite. This weaker condition implies a rank-two connection between energy wells and predicts directions of elongation for the precipitates. This condition can be interpreted as a mathematical condition of semi-coherence. The transformation stretch matrix is calculated by the same algorithm mentioned above. The weak compatibility condition is equivalent to the statement that the smallest and largest eigenvalues of <bold>U</bold> satisfy &lambda<sub>1</sub> &le 1 &le &lambda<sub>3</sub>, which in turn implies that there is an undistorted direction <bold>e</bold>. We study this condition in the thermoelectric material PbTe/Sb<sub>2</sub>Te<sub>3</sub>,</DISS_para> <DISS_para>which consists of Sb<sub>2</sub>Te<sub>3</sub> precipitates in a PbTe matrix. This material shows typical Widmanst"atten microstructure. The satisfaction of the rank-two condition for this material implies that the undistorted directions of the precipitates lie on the lateral surface of a cone determined by the eigenvalues of <bold>U</bold>. By symmetry, there are four crystallographically equivalent cones, that all together restrict the spacial distribution of the Widmanstatten precipitates Sb<sub>2</sub>Te<sub>3</sub>. A 3D image reconstructed from a set of SE images of the precipitates by means of slice-and-view technique shows a good agreement with this theory. For the martensitic phase transformation, we discuss the cofactor conditions. These are currently the strongest achievable conditions of compatibility for the formation of microstructure of austenite and twinned martensite. The satisfaction of the cofactor conditions implies the existence of infinitely many compatible ways that twinned martensite laminates of any volume fraction can coexist with austenite at a low-energy interface. In this thesis we show that, in fact, many of these energy minimizing microstructures have <italic>zero elastic energy at all length scales</italic>. Experimentally, we have successfully achieved the first example Zn<sub>45</sub>Au<sub>30</sub>Cu<sub>25</sub> whose lattice parameters closely satisfy the cofactor conditions for both Type I and Type II twin systems. This material shows enhanced reversibility and extremely low hysteresis upon cyclic phase transformation. Strikingly, the martensitic microstructure has no reproducibility from cycle to cycle.This phenomenon contrasts sharply with the traditional martensite for a polycrystalline solid, which shows a detailed martensite memory effect for cyclic phase transformation. The zero elastic energy microstructures can be used as the building blocks of a set of compatible triple junctions between a pair of Type I twin/austenite and quad junctions consisting of a pair of Type I twin/Type II twins. From X-ray diffraction measurements, we calculate these building blocks for Zn<sub>45</sub>Au<sub>30</sub>Cu<sub>25</sub>, which are then used to construct a complex mosaic of microstructure. This microstructure is apparently observed under optical microscopy, but this awaits detailed confirmation by Electron Backscatter Diffraction (EBSD) and Transmission Electron Microscopy (TEM), currently underway by the author in collaboration with researchers at Carnegie Mellon University and the University of Antwerp.
University of Minnesota Ph.D. dissertation. July 2013. Major: Aerospace Engineering and Mechanics. Advisor: Richard D. James. 1 computer file (PDF); x, 118 pages, appendix A.
Influence of compatibility conditions on the microstructure at phase transformation.
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