ANALYSIS Consider a damaged material exposed to high temperature creep conditions in which voids were periodically arranged in parallel sheets. The initial shape of the voids was assumed to be spherical with a radius of ‘a’. Interaction effects between voids lying within a sheet, with initial mean center-to-center spacing of ‘b’, were accounted for by approximating the voided medium as consisting of cylindrical unit cells, each with a void located at is center, as shown in Figure 1. Assuming axisymmteric conditions, only one quarter of the cylindrical unit cell needs to be modeled. The axisymmetric unit cell is shown as hatched region in Figure 1. The far-field stress state was assumed to be uniaxial. The grain material was assumed to be elastically linear and isotropic. Diffusion of matter takes place along the grain boundaries. ψ σ ∞ b a Figure 1. Dimensions and discretization for model. We consider an incremental form of a functional, F, given by, ∫ ∆ •∆ ∫Γ σ • Γ ∫ •δ ∫ σ δ∆ dA + o m j d j j 2D 1 T T v + A F=V : ddV-S (1) for all kinematically associated fields, namely, the rate of deformation tensor, d, and the velocity, v, and the volumetric flux, j, crossing unit length in the grain boundary. In Equation (1), σ is the applied stress, T is the applied traction along the boundary S, A denotes the grain boundary area, and Γ denotes the collection of arcs where the grain boundaries meet the void surfaces. The normal stress, σo, on the grain boundary at the void tip, also known as the sintering stress, is given by, o= s( 1+ 2) σ γ κ κ (2) where γs is the surface energy, and κ1 and κ2 are the principal curvatures of the surface of the void. The diffusion parameter, D, used in Equation (1), is related to the grain boundary diffusion coefficient of the material by,
RkJQdWJsaXNoZXIy MjM0NDE=