Configurational Mechanics of Materials pp 221-272 | Cite as

# Morphological Equilibrium and Kinetics of Two-Phase Materials

## Abstract

Many engineering and natural crystalline materials are microscopically not homogeneous but consist of two ore more phases with different material properties. The morphology, i.e. the geometric shape of the phases, can be regarded as a result of stresses and strains induced by their different crystalline structure and by the loading history. On account of diffusive mass transport, the morphology may change with time and eventually tend to a final equilibrium state.. It is self-evident that in such multiphase materials the overall properties are found to depend not only on the properties of each phase, but also on the morphology of the microstructure and its evolution. As an example, Fig. la shows the lens-shaped microstructure of Mg-stabilized Zirconia (ZrO_{2}) formed by a tetragonal in a cubic phase. Similarly, a lamellar structure is found in a geomaterial consisting of a pigeonite in an augite phase, see Fig. lb. In high temperature applications, such as turbine blades, Ni-base alloys are becoming more widely used. Microscopically they consist of Ni_{3}X precipitates, the so-called *γ*′-phase (where *X* stands for Al, Ti, Si etc.), embedded in a Ni matrix, the *γ*-phase. In Fig. 2 the microstructure of such materials is displayed for different heat treatment and loading histories. It can be seen that, depending on the specific parameters, the precipitates may be sphere or cube shaped, they may be oblate or prolate. Furthermore they align along the crystallographic axis and show the so-called *rafting* behaviour under elevated temperature and external loads. The binary system Ni — Ni_{3}Al has been under careful investigation for several years. Materials science has described the material properties very thoroughly. It was observed that the microstructure of those materials is of great importance for their overall properties, see e.g. Hornbogen and Roth (1967), Ardell and Nicholson (1966), Ardell and Meshkinpour (1994). It is characteristic not only for Ni-base alloys that the crystallographic planes of the two phases are continuous. Such phase boundaries are called *coherent.* Beside these empirical aspects, progress has been made in understanding the mechanics and thermodynamics of two-phase materials, e.g. Leo and Sekerka (1989), Gurtin and Voorhees (1993), Gurtin (1995), Schmidt and Gross (1997). Herein the *configurational forces* play a central role. These more theoretical works lay the basis for understanding how the microstructures form and how different loading situations influence the morphology on the micro level. However, pure analytical investigations can only treat very simple phenomena or predict qualitative results, see Johnson and Cahn (1984), Kaganova and Roitburd (1988). Numerical investigation, on the other hand, are capable of overcoming these limitations.

## Keywords

External Load Boundary Element Method Interfacial Energy Boundary Integral Equation Elastic Strain Energy## Preview

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## References

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