**4. Conclusion and discussion**

In Sections 1 through 3 a new model on dynamic response of quasicrystals based on argument of Lubensky et al is formulated. This model is regarded as an elastohydrodynamics model for the material, or as a collaborating model of wave propagation and diffusion. This model is more complex than pure wave propagation model for conventional crystals, the analytic solution is very difficult to obtain, except a few simple examples introduced in Fan's monograph [1]. Numerical procedure based on finite difference algorithm is developed. Computed results confirm the validity of wave propagation behaviour of phonon field, and behaviour of diffusion of phason field. The interaction between phonons and phasons are also recorded.

The finite difference formalism is applied to analyze dynamic initiation of crack growth and crack fast propagation for two-dimensional decagonal Al-Ni-Co and three-dimensional icosahedral Al-Pd-Mn quasicrystals, the displacement and stress fields around the tip of stationary and propagating cracks are revealed, the stress present singularity with order 1/2 *r* , in which *r* denotes the distance measured from the crack tip. For the fast crack propagation, which is a nonlinear problem—moving boundary problem, one must provide additional condition for determining solution. For this purpose we give a criterion for checking crack propagation/crack arrest based on the critical stress criterion. Application of this additional condition for determining solution has helped us to achieve the numerical simulation of the moving boundary value problem and revealed crack length-time evolution. However, more important and difficult problems are left open for further study.

Up to now the arguments on the physical meaning of phason variables based on hydrodynamics within different research groups have not been ended yet, see e.g. Coddens [11], which may be solved by further experimental and theoretical investigations.
