The mechanisms of ductile flow in crystalline solids have been deduced from studies of metals, which have the advantage that they flow easily at low temperatures and pressures. In general, where the temperature of a material is less than about half its melting temperature (Tm in Kelvin), materials react to low stresses by flowing slowly, or creeping, in the solid state. At high temperatures and pressures, the strength and flow of silicate minerals that characterize the crust (Tullis, 2002) and mantle (Li et al., 2004) have been studied using experimental apparatus.
There are several types of ductile flow that may occur in the crust and mantle (Ashby & Verrall, 1977). All are dependent upon the ambient temperature and, less markedly, pressure. Increased temperature acts to lower the apparent viscosity and increase the strain rate, while increased pressure produces a more sluggish flow. In general, for ductile flow, the differential stress (Ao) and the strain rate (8e/8x) are related through a flow law of the form:
Ao = [(8e/8x)/A]1/n exp[E/nRT], where E is the activation energy of the assumed creep process, T is temperature, R is the universal gas constant, n is an integer, and A is an experimentally determined constant.
Plastic flow occurs when the yield strength of the material is exceeded. Movement takes place by the gliding motions of large numbers of defects in the crystal lattices of minerals. Slip within a crystal lattice occurs as the individual bonds of neighboring atoms break and reform across glide planes (Fig. 2.23). This process results in linear defects, called dislocations, that separate slipped from unslipped parts of the crystal. The yield strength of materials deforming in this way is controlled by the magnitude of the stresses required to overcome the resistance of the crystal framework to the movement of the dislocations. The strain produced tends to be limited by the density of dislocations. The higher the density, the more difficult it is for dislocations to move in a process known as strain-or work-hardening.
Power-law creep (also known as dislocation creep) takes place at temperatures in excess of 0.55 Tm. In this form of creep the strain rate is proportional to the nth power of the stress, where n > 3. Power-law creep is similar to plastic flow, where deformation takes place by dislocation glide. However, in addition, the diffusion of atoms and of sites unoccupied by atoms called vacancies is permitted by the higher temperatures (Fig. 2.24). This diffusive process, termed dislocation climb, allows barriers to dislocation movement to be removed as they form. As a result work-hardening does not occur and steady state creep is facilitated. This balance results in dynamic recrystallization whereby new crystal grains form from old grains. Because of the higher temperature the yield strength is lower than for plastic flow, and strain results from lower stresses. Power-law creep is believed to be an important form of deformation in the upper mantle where it governs convective flow (Weertman, 1978). Newman & White (1997) suggest that the rheology of continental lithosphere is controlled by power-law creep with a stress exponent of three.
Diffusion creep dominates as temperatures exceed 0.85 Tm, and results from the migration of individual atoms and vacancies in a stress gradient (Fig. 2.25). Where the migration occurs through a crystal lattice it is known as Nabarro-Herring creep. Where it occurs along crystal boundaries it is known as Coble creep. In both forms of creep the strain rate (8e/8x) is proportional to the differential stress (Ao) with the constant of proportionality being the dynamic viscosity (n). This relationship is given by:
The viscosity increases as the square of the grain radius so that a reduction in grain size is expected to result in rheological weakening. Diffusion creep is believed to occur in the asthenosphere (Section 2.12) and in the lower mantle (Section 2.10.6).
Superplastic creep has been observed in metals and may also occur in some rocks. This type of creep results from the coherent sliding of crystals along grain boundaries where the movement occurs without opening up gaps between grains. The sliding may be accommodated by both diffusion and dislocation mechanisms. Superplastic creep is characterized by a power-law rheology with a stress exponent of one or two and is associated with high strain rates. Some studies (e.g. Karato, 1998) have inferred that superplastic creep contributes
B plane B
B plane B
Figure 2.23 Plastic flow by the migration of a linear edge dislocation through a crystal (from Structural Geology by Robert J. Twiss and Eldridge M. Moores. © 1992 by W.H. Freeman and Company. Used with permission).
to deformation in the lower mantle, although this interpretation is controversial.
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