Formulation Of Phase-field Model Research Articles

Linear and Nonlinear Formulation of Phase Field Model with Generalized Polynomial Degradation Functions for Brittle Fractures

AbstractThe classical phase field model has wide applications for brittle materials, but nonlinearity and inelasticity are found in its stress–strain curve. The degradation function in the classical phase field model makes it a linear formulation of phase field and computationally attractive, but stiffness reduction happens even at low strain. In this paper, generalized polynomial degradation functions are investigated to solve this problem. The first derivative of degradation function at zero phase is added as an extra constraint, which renders higher-order polynomial degradation function and nonlinear formulation of phase field. Compared with other degradation functions (like algebraic fraction function, exponential function, and trigonometric function), this polynomial degradation function enables phase in [0, 1] (should still avoid the first derivative of degradation function at zero phase to be 0), so there is no $$\Gamma $$ Γ convergence problem. The good and meaningful finding is that, under the same fracture strength, the proposed phase field model has a larger length scale, which means larger element size and better computational efficiency. This proposed phase field model is implemented in LS-DYNA user-defined element and user-defined material and solved by the Newton–Raphson method. A tensile test shows that the first derivative of degradation function at zero phase does impact stress–strain curve. Mode I, mode II, and mixed-mode examples show the feasibility of the proposed phase field model in simulating brittle fracture.

A Three-Dimensional Formulation of Phase Field Model Representing Polycrystalline Solidification

In this study, a novel formulation of phase field model for representing solidification of polycrystalline metals is proposed. A quaternion-based crystal orientation representation is introduced into the KWC-type phase field model. Using this approach, three-dimensional rotation of crystal orientation can be represented by a direct extension of the conventional KWC model. A numerical simulation by the presented model is conducted, and the characteristics and adequacy of the model are discussed.

Phase field modeling with large interface thickness and undercooling

The interface thickness is the most crucial parameter in the phase field model (PFM) for the accuracy and the computability. However, the Gibbs–Thomson equation can be satisfied only when the interface thickness is sufficiently small, especially with a large undercooling, but this greatly limits the applications of PFM to a realistic problem. The temperature correction in the thin-interface model partially resolves the problem, but the range is rather limited. In this report, we propose a new formulation of PFM by adding extra terms stabilizing the hyperbolic tangent profile of the phase-field, and this allows us to use a much larger interface thickness for simulation, even with a large undercooling. Several benchmark comparisons with analytical solutions are carried out and discussed.