Mesoscale modeling of grain boundary migration under stress using coupled finite element and meshfree methods. Page: 2 of 8
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Probabilistic models are generally of Monte-Carlo type and have their basis in the classical spin
models of statistical physics; the most investigated is Potts model (Anderson et al., 1984). In the
Potts model approach grains are subdivided into small-area elements and growth dynamics are
simulated by exchange of area elements between grains. Growth takes place as a consequence of
the minimization of the internal energy of the system. The exchange step of area elements from one
grain to the neighboring grain is carried out using a Monte Carlo algorithm. The advantage of this
method is its simplicity and the ease of its implementation in two and three-dimensional systems.
However, in this method the origin of the stochastic aspect is not clear, nor is the relation between
the Monte Carlo time step and the physical time.
In the deterministic models, the motion of grain boundaries is followed by time integration of
their position assuming the normal velocity of the grain boundary to be proportional to the
boundary curvature. A purely deterministic approach was proposed first by Fullman (1952) and is
referred to as "vertex model". Later, this was improved by Soares et al. (1985) and Kawasaki et al.
(1989) assuming straight grain boundaries, and by Frost et al. (1988), Cocks and Gill (1996),
Weygand et al. (1998) by extending it to curved grain boundaries. Using the theoretical approach
of Needleman and Rice (1980) based on a variational principle for dissipative systems, Cocks and
Gill (1996) have proposed a new method to simulate curvature-driven grain growth. Their
modification describes the rate of power dissipation due to the competition between the reduction
in the grain boundary energy and the viscous drag during grain boundary migration. Moreover, the
grain boundaries are discretized using finite elements.
In general in a polycrystalline microstructure subject to an externally applied stress an
additional driving force to that given by the grain boundary curvature has to be considered. This is
due to the elastic anisotropy of the grains comprising the microstructure, which in general store
different amounts of elastic energies. Our focus in this study is to investigate the grain growth in
the presence of both curvature driven and stress induced grain boundary migration. This requires
the coupling of elastic deformation of grains with grain boundary migration and thus necessitates
the discretization of grain boundaries and grain domains. Using finite element method, the
migration of grain boundary leads to a severe mesh distortion in each grain, and the topological
changes of grain structures further demand a complete remeshing. In this work, a double-grid
method is proposed. The elastic deformation of grains is modeled by reproducing kernel
discretization with built-in strain discontinuities along the grain boundaries (Chen et al. 1996,
2002), whereas the migration kinematics of discretized grain boundaries is modeled using the
standard finite element formulation.
The numerical examples we provide in this study demonstrate that the evolution of grain
growth can be effectively simulated without any remeshing. Moreover, the study also shows the
proper time evolution of the grain structures in an idealized grain network with an imperfection
using the proposed methods.
GRAIN GROWTH KINEMATICS
In general, the grain boundaries migrate at a wide range of velocities, which depend on the
magnitude of both the driving force and the grain boundary mobility (which dependent on
temperature, and impurities concentration). Using a simplified model, Burke and Turnbull (1952)2
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Chen, J.-S.; Lu, H.; Moldovan, D. & Wolf, D. Mesoscale modeling of grain boundary migration under stress using coupled finite element and meshfree methods., article, May 24, 2002; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc742790/m1/2/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.