Takeshi Takaishi

<title>Abstract</title>Three new industrial applications of irreversible phase field models for crack growth are presented in this study. The phase field model for irreversible crack growth in an elastic material is derived as a unidirectional gradient flow of the Francfort–Marigo energy with the Ambrosio–Tortorelli regularization, which is known to be consistent with classic Griffith fracture theory. The obtained compact parabolic-elliptic system of PDEs including two regularization parameters for space and time enables us to simulate various kinds of crack behaviors with standard finite element software, without any geometric restriction on the crack path. We extend the irreversible phase field model to thermal cracking in solder and to cracking in a viscoelastic material, keeping the compact forms of the PDEs and the energy consistency. On the other hand, for hydrogen-assisted cracking in metal, we propose a compact phase field model focusing on a kinematic jamming effect of the hydrogen by a weak coupling approach. Several numerical experiments for these three models show not only their reasonableness and usefulness but also flexible extendability of the phase field approach in fracture mechanics.

As an application of the phase field model for crack propagation in elastic body, chemical-diffuse crack growth model with the effect of the hydrogen embrittlement is considered. Numerical results show the difference of crack path between data with the effects and data without the effect. Temporal evolution of the normalized difference of phase field depict the time when start the difference of crack path.

A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. We introduce a phase field to represent the shape of the crack with a regulaxization parameter epsilon &gt; 0 and we approximate the Francfort-Marigo type energy using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient flow of this regularized energy. We show several numerical examples of the crack growth computed with an adaptive mesh finite element method.

The phase field model which is suitable for the numerical simulation of mode III (anti-plane shear) crack growth is proposed. Using these model equations, some numerical results for crack growth are presented. Usefulness of this model is shown in the results, in the case that 2 cracks grow as various types, including the sub-crack. This model has many good features for numerical simulation, such as fixed computational domain, and it would be adapted to the well-known numerical methods. It is expected to be very useful to study the crack growth.

A general formula for the dispersion characteristics of electrostatic waves in dusty plasmas in the presence of charge fluctuation of dust particles is derived based on the linear response theory. The formula is applicable to arbitrary plasma models and waves. Charging equation due to plasma current on the dust surface is employed to describe the charge fluctuation effect. In particular, temperature fluctuation associated with the space charge fluctuation is taken into account in the charging equation and is found to make an important contribution to the damping or growth of the wave. Stability of low frequency waves where the electron response can be treated as static is discussed in detail. It is shown that the dust charge fluctuation contributes to damping when the dust is negatively charged, but to growth when the dust is positively charged. Results are applied to ion waves with positive dust charge.

Linear dispersion characteristics of ion acoustic waves in a dusty plasma is studied in the fluid model. It is shown that charge variation of dust particles has a destabilizing effect of the wave in the short wavelength region where the electron Debye shielding of the ion space charge becomes incomplete. Secondary electron emission effect is also investigated and the charge variation is shown to have a destabilizing effect at all wavelengths except the longest one, if the dust particles are positively charged.

The shadow system u(t) = epsilon(2)u(xx) + f(u) - xi, xi(t) = integral(t) g(u,xi) dx, is a scalar reaction diffusion equation coupled with an ODE. The extra freedom coming from the ODE drastically influences the solution structure and dynamics as compared to that of a single scalar reaction diffusion system. In fact, it causes secondary bifurcations and coexistence of multiple stable equilibria. Our long term goal is a complete description of the global dynamics on its global attractor A as a function of epsilon, f, and g. Since this is still far beyond our capabilities, we focus on describing the dynamics of solutions to the shadow system which are monotone in x, and classify the global connecting orbit structures in the monotone solution space up to the semi-conjugacy. The maximum principle and hence the lap number arguments, which have played a central role in the analysis of one dimensional scalar reaction diffusion equations, cannot be directly applied to the shadow system, although there is a Lyapunov function in an appropriate parameter regime. In order to overcome this difficulty, we resort to the Conley index theory. This method is topological in nature, and allows us to reduce the connection problem to a series of algebraic computations. The semi-conjugacy property can be obtained once the connectin problem is solved. The shadow system turns out to exhibit minimal dynamics which displays the mechanism of basic pattern formation, namely it explains the dynamic relation among the trivial rest states (constant solutions) and the event patterns (large amplitude inhomogeneous solutions). (C) 1997 Academic Press.

A simplified coupled reaction-diffusion system is derived from a diffusive membrane coupling of two reaction-diffusion systems of activator-inhibitor type. It is shown that the dynamics of the original decoupled systems persists for weak coupling, while new coupled stationary patterns of alternated type emerge at a critical strength of coupling and become stable for strong coupling independently of the dynamics of the decoupled systems. The approach which we use here is singular perturbation techniques and complementarily numerical methods. © 1995 JJIAM Publishing Committee.