We describe a theoretical and numerical analysis of an existing model of anelasticity owing to grain boundary sliding. Two linearly elastic layers having finite thickness and identical material constants are separated by a given fixed spatially periodic interface across which the normal componentu * n of velocity is continuous, whereas the tangential componentu * s has a discontinuity determined by the shear stress s * ns and the boundary sliding viscosity h * . We derive asymptotic forms giving the complex rigidity for the extremes of low-frequency forcing and of high-frequency forcing. Using those forms, we create master variables allowing results for different interface shapes, and arbitrary forcing frequency, to be collapsed (very nearly) into a single curve. We then analyse numerically, with finite interface slope, three proposed factors that may weaken and broaden the theoretical prediction of a single Debye peak in the loss spectrum. They are, namely, stress concentrations at interface corners, spatial variation in grain size and spatial variation in boundary sliding viscosity h * . Our results show that all these factors can, indeed, contribute to a moderate weakening of the loss peak. By contrast, the loss peak markedly broadens only when the boundary sliding viscosity h * differs by an order of magnitude across adjacent interface. The shape of the loss spectrum (self-similar to a single Debye peak) is insensitive to the other two factors.

Using analytical and numerical methods, we analyse the Raj-Ashby bicrystal model of diffusionally accommodated grain-boundary sliding for finite interface slopes. Two perfectly elastic layers of finite thickness are separated by a given fixed spatially periodic interface. Dissipation occurs by time-periodic shearing of the viscous interfacial region, and by time-periodic grain-boundary diffusion. Although two time scales govern these processes, of particular interest is the characteristic time t D for grain-boundary diffusion to occur over distances of order of the grain size. For seismic frequencies ut D 1, we find that the spectrum of mechanical loss Q -1 is controlled by the local stress field near corners. For a simple piecewise linear interface having identical corners, this localization leads to a simple asymptotic form for the loss spectrum: for ut D 1, Q −1 ∼ const.u −a . The positive exponent a is determined by the structure of the stress field near the corners, but depends both on the angle subtended by the corner and on the orientation of the interface; the value of a for a sawtooth interface having 120• angles differs from that for a truncated sawtooth interface whose corners subtend the same 120• angle. When corners on an interface are not all identical, the behaviour is even more complex. Our analysis suggests that the loss spectrum of a finely grained solid results from volume averaging of the dissipation occurring in the neighbourhood of a randomly oriented three-dimensional network of grain boundaries and edges.