Success has been obtained in predicting the dynamic evolution of microstructures during phase transformation or cracking propagation by using the time-dependent phase field methodology ͑PFM͒. However, most efforts of PFM were made in the elastic regime. In this letter, stress distributions around defects such as a hole and a crack in an externally loaded two-dimensional representative volume element were investigated by a proposed phase field model that took both the elastic and plastic deformations into consideration. Good agreement was found for static cases compared to the use of finite element analysis. Therefore, the proposed phase field model provides an opportunity to study the dynamic evolution of microstructures under plastic deformation. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.2138358͔It is well known that microstructures of materials play a crucial role in determining the properties of materials. As a powerful computational approach to predicting mesoscale morphological and microstructure evolution in materials, the phase field method has attracted a considerable amount of research effort and has found wide application in various fields. 1,2 In a phase field model ͑PFM͒, the evolution of structural variables and chemical compositions can be described by time-dependent Ginzburg-Landau ͑TDGL͒ equations and by the Cahn-Hilliard diffusion equation, respectively. In general, the evolution of microstructures will result in the minimization of the free energy of the whole system, which may consist of the bulk chemical free energy, elastic strain energy, interfacial energy, electric and magnetic energy, and work done by applied external fields. Currently, most efforts of PFM were made in the elastic regime. However, both experimental and computational results have shown that the stresses around defects, crystal interfaces, or precipitates can significantly exceed the elastic limit. Therefore, for many applications, it is necessary to consider the contribution from plastic deformation during microstructure evolution. This is especially true when cracks are present in metals. In this situation, plastic energy could be the dominating factor controlling the initiation and propagation of a crack. In this letter, we propose to treat both elastic and plastic deformations around a void or crack as phase variables. The proposed PFM can also be applied to other microstructure analyses such as phase transformation and ordering, 3,4 defect dynamics, and pattern formation, 5 when plastic deformation is involved.An equivalent description can be made of the displacement and strain energy of an anisotropic discontinuous body with cracks under applied stress by an anisotropic continuous noncracked body of the same macroscopic size and shape, but with the heterogeneous misfit stress-free strain. The Khachaturyan-Shatalov ͑KS͒ theory 6 gives the exact elastic strain and strain energy of a system for a given set of stressfree strains, ij 0 ͑r͒. The strain energy of the system is a function of ij 0 ͑r͒ and an averag...
Free vibration problem of rectangular plates with multiple point supports is analyzed by a discrete method. The concentrated loads with Dirac's delta functions are used to simulate the point supports which limit the displacements of the plate but don't offer constraint on the slopes. The fundamental differential equations are established for the bending problem of the plate with point supports. The solution of these equations is obtained by transforming these differential equations into integral equations and using numerical integration. Green function which is the solution of deflection is used to obtain the characteristic equation of the free vibration. The effects of the number and positions of point supports, the boundary condition and the aspect ratio on the frequencies are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.
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