Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higher-dimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and free-surface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higher-order interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.Comment: 28 pages, 9 figures, LaTeX; Accepted for publication in Computer Methods in Applied Mechanics and Engineering. [Style and formatting modifications made, references added.

A continuum approach to the modeling of microstructural evolution in polycrystalline solids.

Interest in microstructural evolution of polycrystalline materials stems from the multiplicity of interrelated phenomena that contribute to this evolution, and from the impact that such phenomena have on the performance and reliability of these materials, especially in applications such as microelectronic devices. In this work, a continuum field formulation developed to study this type of phenomena is presented. The formulation accounts fully for the coupling between mechanical behavior, self diffusion, electric effects and interface migration. Each phenomenon being modeled is treated as a coupled initial and boundary value problem, consisting of these four component problems. Atomic-level mechanisms are taken into consideration while developing the thermodynamic basis of the formulation, from which the constitutive relations are derived. The computational framework used to solve the resulting coupled field equations is described in detail. This framework is built around a staggered solution scheme in which the finite element method is used to solve each governing differential equation individually. Additional computational techniques utilized in the implementation are also discussed. Examples of these include the level set method, a least-squares projection/smoothing technique and a modified form of the Galerkin/least-squares stabilization method. To study the problem of void nucleation in polycrystals, the stability of the atom-vacancy system is examined closely. A thermodynamic instability which could lead to spinodal decomposition of this system is identified. The amplications of this result in the context of void nucleation are explored and its relation to classical nucleation theory is realized. All quantities needed to calculate void formation rates are obtained from the coupled field calculations in a consistent mannen As expected, the results indicate that a high tensile stress can lead to void nucleation in the presence of impurities. The problem of grain-boundary migration is also treated. The level set method is used successfully to track the moving boundary without remeshing. Thermodynamic driving forces for boundary migration arising from boundary curvature, stress-driven diffusion and electromigration are accounted for. In this case also, the strongly coupled nature of the problem is captured, the calculations remain stable and physically-meaningful results are obtained.Ph.D.Applied SciencesMaterials scienceMechanical engineeringMechanicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/124515/2/3150048.pd

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton's principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee $C^0$-continuity of the solution at inter-element boundaries, and enforces the required $C^1$ regularity with the aid of additional variational and interior penalty terms in the weak form. The phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods