According to existing experiments on fine-grained polycrystalline mantle materials, in the seismic frequency band, mechanical loss Q-1 decreases with increasing angular frequency &omega in an absorption background; roughly Q-1 &sim &omega&alpha with different investigators reporting values of &alpha ranging from &sim -0.35 to &sim -0.2. There is inconclusive evidence that, under some conditions, a weak local maximum may be superposed on that absorption background. To understand this behaviour, we use a combination of analytical and numerical methods to analyze the Raj-Ashby bicrystal model of diffusionally-accommodated grain boundary sliding on a finite slope interface. In that model, two perfectly elastic layers of finite thickness are separated by a given fixed spatially periodic interface; dissipation is confined to that interfacial (grain boundary) region having an effective viscosity. It occurs by two processes: time-periodic shearing of the interfacial region; and time-periodic diffusion of matter along the interface. Two timescales govern these processes; namely, a characteristic time t&eta taken for the interfacial shear stress to relax and a characteristic time tD taken for matter to move by grain-boundary diffusion over distances of order the grain size.
Of particular interest is the case when the timescales are widely separated. Under that condition, we established two previously unrecognized features of the mechanical loss spectrum. First, the mechanical loss Q-1 in the seismic frequency band &omega tD >> 1 can be described by a strict power--law Q-1 &sim &omega&alpha if corners along the interface are geometrically identical. For the two orthogonal sliding modes found in a regular array of hexagonal grains, the values of &alpha is roughly -0.3. Second, our analysis reveals a mechanism allowing the magnitude of &alpha to decrease slowly as &omega is increased; when the corner angle varies from one corner to another along the interface , the rate of decrease in Q-1 gradually slows. Ultimately Q-1 is controlled by the corner having the most singular stress behaviour. Though these results are obtained from the idealized bicrystal model, we argue physically that similar behaviour will be found in numerical models of polycrystal. Overall, our analysis suggests that the range of &alpha -values found empirically may, in part, reflect the differing ranges of &omega tD covered in different experiments.
Because in experiments conducted on certain materials, a weak and broad peak superposed onto the power--law absorption background is observed in the loss spectrum whereas in others, the peak is completely absent, we evaluate three proposed factors that may weaken and broaden the peak. We show that the peak can be weaken moderately by (i) sharpening of corners along the interface, (ii) spatial variation in grain size and (iii) spatial variation in interfacial (grain boundary) viscosity. Reduction of the peak by these factors, however, does not suggest it to be completely hidden in the absorption background. By contrast, we show that the loss peak can be markedly broadened if the interfacial viscosity differs by an order of magnitude across adjacent interfaces. The shape of the loss peak is insensitive to the other two factors.
