Takeshi Takaishi

Two examples of extention of crack growth model with phase field approach are derived. A crack growth model for visco-elastic material is introduced from exctended free energy which add effect of Maxwell fluid. Another model with effect of H-embrittlement is introduced by adding some equations. Ability of phase field model which can describe crack growth phenomena of various material is shown.

A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic plate is proposed. This model is derived as a gradient flow of the reagularized energy, that is based on the Francfort-Marigo type energy. This model is a kind of the reaction diffusion system with two scalar variables, the anti-plane displacement and the phase field to represent the shape of the crack with a regularization parameter. This model has no numerical difficulties for computer simulation and it is expected to obtain more analytical information.

A complete classification of dynamics of a population of a inhibitory pulse-coupled oscillators is presented. The model is based on the work of Mirollo and Strogatz, but our model has an inhibitory coupling between oscillators which makes a sharp contrast with the dynamics of the above authors' model. The main result is that for a large class of initial conditions, the population approaches a periodic state in which all the oscillators keep finite size of phase difference (we call it "phase locking solution" here). For the remaining class of initial data except for nongeneric ones, it evolves to a periodic state with a cluster or a synchronous state depending on a size of cluster. The criterion for the classification is explicitly given and can be judged easily only by the initial condition.