Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

Abstract

Ductile shear zones are common features in Earth’s lithosphere. They vary from submeter to plate scales. As rheologically distinct elements in the lithosphere, they undoubtedly affect the rheology of the lithosphere. Shear zones also serve as conduits for the movement of fluids and melt and are often sites of economically significant minerals. In this chapter, we use Eshelby’s equivalent-inclusion approach to investigate the stress field and progressive deformation in a viscous shear zone embedded in the lithosphere. The situation that the lithosphere is also viscous but with a higher viscous stiffness is trivial, as the problem of viscous inhomogeneity in a viscous medium is solved in Chap. 11 . We consider the situation of a viscous shear zone enclosed in an elastic lithosphere instead. Past studies of ductile shear zones often treat them in isolation. In kinematic models (Chap. 6 ), the only connection between a shear zone and its country rocks is the boundary displacement or velocity condition, chosen to maintain compatibility between the country rock and the deforming zone. In mechanical models, shear zones are often assumed to be infinitely-extending zones of rheologically weaker rocks between parallel walls subjected to a boundary displacement or traction condition (Lockett and Kusznir 1982; Turcotte and Schubert 1982, p. 375–378; Robin and Cruden 1994). However, shear zones in Earth’s ductile lithosphere are finite in size. Small ductile shear zones, in particular, are commonly entirely enclosed in the lithosphere (Fig. 15.1). It is inappropriate to use a constant boundary condition for such a zone because the mechanical interaction between the zone and the country rock is dynamic during deformation. In this chapter, the ductile shear zone is regarded as a rheologically-distinct element (RDE, a heterogeneity) embedded in the lithospheric country rock. Microstructures of mylonites from natural ductile shear zones suggest that the dominant mechanism of deformation is dislocation creep which is associated with power-law viscous rheology (Tullis 2002; Kohlstedt et al. 1995). We regard ductile shear zones as Newtonian viscous in this chapter. The analysis here may be extended to power-law viscous rheology using the extended Eshelby theory to be elucidated in Chap. 17 . The simplest approach is to regard a ductile shear zone as a viscous inclusion in a viscous but stronger matrix because the partitioning Eq. ( 10.20 ) or interaction Eq. ( 11.24 ) can be applied directly. Here, we consider a more challenging scenario where the country rock is elastic. Most shear zones in the crust have country rocks that are either undeformed or much less deformed. The elasticity of the country rock must have been mechanically significant during shear zone evolution.