kT b Db exp( Q b /RT) D = Ω − δ (3) where Db is the grain boundary diffusion coefficient at a given temperature, T, Ω is the atomic volume, δb is the thickness of the diffusion layer, Qb is the activation energy, and R and k are the Gas and Boltzmann’s constant. The functional [1] Starts from Needleman and Rice [2], with appropriate modifications to account for material elasticity, etc. It can be easily shown that F as given in equation (1) is not only stationary but also a global minimum for the true field. Exercising the variational principle on F, it can be shown that the full set of field equations for the considered problem will result. Using the above formulation to develop the finite element equations (as elaborated in [3]), a special user element (UEL) was developed and used with the ABAQUS finite element software [4]. We have examined the transient effect for several metallic materials at a temperature, T = 0.6 Tm in this study. The properties of the materials are given in Table 1. We consider an average void spacing 10 µm (b = 5 µm). Using this value for b and a/ b = 0.1, we obtain a characteristic time, τ, also given in Table 1, for the materials considered in this study. The properties listed in Table 1 are obtained from Frost and Ashby [5]. In all the cases examined, the ratio of far-field stress to Young’s modulus, σ∞/E, was 10-3. The characteristic time is defined as: (4) ED a ) 3 ( b − τ = TABLE 1. MATERIAL PROPERTIES OF THE METALLIC MATERIALS USED. Material Property Aluminum Copper γ−Iron Young’s Modulus, E (MPa)* 75.30 124.0 205 Poisson’s Ratio, ν** 0.34 0.32 0.3 Atomic Volume, Ω (m3) * 1.66 x 10-29 1.18 x 10-29 1.21 x 10-29 Melting Point, Tm (K) * 933 1356 1810 Grain Boundary (GB) Diffusion Pre-exponential, δbDb (m3/s) * 5 x 10-14 0.5 x 10-14 7.5 x 10-14 Activation Energy for GB Diffusion Qb (kJ/mole) * 84 104 159 Characteristic Time, τ, secs 772 857 327 Grain Boundary Energy, γb (J/m2) ** 0.63 0.65 0.78 Surface Energy, γs (J/m2) ** - 1.73 1.95 In Figure 2, the ratio of the void growth rate predicted by FEM and by the Hull-Rimmer model (the HullRimmer classical solution neglect elastic accommodation, i.e., assumes rigid response [1]) is plotted against normalized time. The inset shows the transition from transient to steady state conditions. It can be seen that the transient time is larger for larger a/b values. The variation of the ratio of normal stress and applied stress along the grain boundary ahead of the cavity tip is shown in Figure 3 for a/b of 0.1. In the figure, X denotes the distance from the tip of the cavity along the grain boundary. While in the pure elastic case the peak stresses occur at the cavity tip, matter diffusing into the grain boundary from the cavity surfaces relaxes the stresses at the tip, even during the beginning stages of the transient. Consequently, the peak stress occurs away from the cavity during the transient stage. With increasing time, the peak stress decreases in magnitude and moves away from the cavity tip. As can be seen from the figure, a parabolic profile is
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