Thomas Blesgen, Anja Schlömerkemper
We present an extension of the Allen-Cahn/Cahn-Hilliard system which
incorporates a geometrically linear ansatz for the elastic energy of the
precipitates. The model contains both the elastic Allen-Cahn system and the
elastic Cahn-Hilliard system as special cases and accounts for the
microstructures on the microscopic scale. We prove the existence of weak
solutions to the new model for a general class of energy functionals. We then
give several examples of functionals that belong to this class. This includes
the energy of geometrically linear elastic materials for D<3. Moreover we show
this for D=3 in the setting of scalar-valued deformations, which corresponds to
the case of anti-plane shear. All this is based on explicit formulas for
relaxed energy functionals newly derived in this article for D=1 and D=3. In
these cases we can also prove uniqueness of the weak solutions.
View original:
http://arxiv.org/abs/1202.5197
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