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Greens

Greens function methods
Definition:Computational method for simulating microstructure evolution and solving linear problems.
Explanation:Greens functions methods are employed in the SFB with different aspects: firstly for solving linear elastic problems in multiphase materials, and secondly for predicting diffusion limited microstructure evolution during solidification and melting problems.

For the first application, the misfit stresses at coherent interfaces are expressed in terms of localized force terms at the interfaces. They act as point forces, for which the solution of the elastic problem in an infinite sample is known via Greens function. Therefore, the total deformation is a superposition of the forces from all interfaces. Similar considerations are possible for specific finite geometries, which typically lead to image forces at free surfaces. For arbitrary geometries, the description results in boundary integral methods.

For the second aspect of predicting microstructure evolution, the Greens function methods are a complementary approach to phase field modeling. For an example of solidification, the partitioning at the moving interfaces under local equilibrium effectively leads to source terms for the diffusion problem in the bulk, in analogy to the mechanical problems discussed above. The additional challenge here is that the location of the interfaces is not known in advance and it needs to be solved as the part of the problem. The entire problem would then be reformulated as integro-differential equations, which have to be solved self-consistently within the sharp interface concept. This is a particularly useful approach for the steady state regime, where it can often lead to more efficient computation and precise descriptions than phase field modeling and allows to extract analytical scaling relations. In the combination with the diffuse interface description of pphase field modeling which can decide about the stability of the steady state solutions, it becomes a powerful way to simulate microstructure evolution.
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Diagram:

Cellular structures during isothermal solidification in peritectic systems. Depending on the lamella spacing of the primary solidification, either dendritic or cellular microstructures are kinetically preferred.
SFB-Link:In the project A9, Greens function methods are used for modeling peritectic solidification, grain boundary melting and hydride formation near surfaces.
References:G. Boussinot, C. Hüter, R. Spatschek, E. A. Brener, Acta Mat. 75 (2014), 212. C.
Hüter, et al., Phys. Rev. B, 89 (2014), 224104.