Evolution of elliptic-cylindrical and circular-cylindrical voids inside power-law viscous solids

Abstract

The evolution of voids inside power-law viscous solids is investigated. A representative volume element (RVE) model of an infinite matrix containing an isolated void is applied. In the RVE model, the void is assumed to be elliptic-cylindrical or circular-cylindrical, and the matrix is considered as the isotropic and incompressible power-law viscous material. To obtain the velocity field of RVE, a Ritz procedure is developed using the method proposed by Lee and Mear (1992). Moreover, the results obtained from the Ritz procedure are verified by the finite element simulations. Based on the data obtained from RVE models, the effects of material Norton exponent, remote stress field and void aspect ratio on the changing rate of void aspect ratio are discussed. Especially, when the void’s principal axes are parallel to the principal axes of the remote stress, the mathematical models are proposed to relate the changing rate of void aspect ratio to the void aspect ratio and material Norton exponent. The results show that the material Norton exponent, remote stress field and void aspect ratio have a great influence on the changing rate of void aspect ratio. For the remote shear stress and uniaxial compression stress fields, the changing rate of void aspect ratio increases with the increase of void aspect ratio and material Norton exponent. Furthermore, the relationships between the changing rate of void aspect ratio and the void aspect ratio can be represented as the parabolic function and linear function for the remote shear stress field and uniaxial compression stress field, respectively. While the relationships between the changing rate of void aspect ratio and material Norton exponent can be expressed as the first order exponential function for these two remote stress fields. Besides, the changing rate of void aspect ratio can also be expressed as a unified function of void aspect ratio and material Norton exponent. For the remote biaxial compression stress field, the relationships between the changing rate of void aspect ratio and the void aspect ratio can be represented as the parabolic function, in which the coefficients can be expressed as functions of material Norton exponent and remote stress field. The findings of this study can be mainly used to evaluate the aspect ratio of voids inside large ingots during hot working, as well as to model the final densification stage of powder metal compacts.