Time-dependent formation of mantled inclusion structures for different rheologies under a simple shear
Introduction
Many geological structures may be considered as an inclusion set in a matrix with more or less uniform properties. The sizes of these structures are scaled from millimeters up to kilometers. On the smallest scales, they appear as individual grains of crystals. On the other end of the spectrum, they can be found as blocks of coexisting, but rheologically different rocks (for example, a magmatic body embedded in sediments). For many geological applications (e.g. [1]), it is very important to understand how the inclusions (heterogeneities) influence the flow and how the flow forms the observed structures. The final structures imprinted in the rock are a few precious evidences of the past deformation. It has long been recognized that the geometry involving the mantle surrounding the inclusion and the inclusion itself may serve as a gauge of deformation 2, 3, thus helping us to restore the past deformation regime or some of its parameters from the structures.
In order to interpret these deformation footprints, one must understand the relationships between the rheology, the deformation regime and the mantle/inclusion geometry [4]. Stresses, applied to mantled inclusion, result in structures, which obviously depend on the particular rheology and deformation regime. It is reasonable to expect that there is way back i.e. there is possibility to retrieve deformation regime from the geometry of the mantled inclusion. For a long time this problem has attracted a great deal of attention. The complex physics of the deformation phenomena requires very sophisticated mathematical methods, even for simplified models. N.C. Gay found an analytical solution in linear media with small deformation 5, 6 in system inclusion/matrix. T. Masuda and S. Ando [7] considered finite-amplitude flow around rigid inclusion in Newtonian matrix and later [10] in a power-law medium.
Another approach to this problem is to carry out analog experiments. Due to the complex nature of the phenomena, analogue experiment has been a favorite tool for decades. Based on experiment results, Passchier and his co-authors discussed a stair stepping of the wings as evidence of nonlinearity of the rheology of the matrix [3]. They have proposed several possible structural evolution scenarios. It was supposed that the ratio of effective viscosities of mantle and matrix was responsible for the wing structure of the mantle [8]. The conclusion was based on a number of high-quality analog experiments. Although experiments remain one of the common tools of investigation of complex media [8], the dramatic growth in computer power during the last decade has brought numerical approaches to a new level, which allows them to compete directly with analog modeling. The finite-difference technique was used to estimate the pressure field around a deformable inclusion and its structural behavior [9]. Finite-element technique in conjunction with the variational method has been applied for modeling the interaction of a rigid inclusion with a non-Newtonian matrix [10].
Rheology controls the path along which rocks are deformed and the formation of the structures 11, 3, 12. This paper is focused on the influence of the rheological parameters on the structural evolution of a mantled inclusion in a simple shear flow. The inclusion is assumed to be deformable. The rheological law, which can be Newtonian or non-Newtonian, is the same for both inclusion and matrix, but material constants are different. Such a simplification allows one to express the rheological contrast with the only parameter. This enables one to compare different systems more clearly and logically.
The comparison is based on results from numerical modeling. This model utilizes the 2D finite-difference approximation [13] coupled with an iterative method [14]. This problem has been formulated in primitive variables (velocity and dynamical pressure), rather than the stream function approach.
Section snippets
Model
Rocks under the typical crustal stress/temperature conditions deform with a broad spectrum of styles. According to laboratory experiments, the relation between stress and strain-rate may be described by linear, power or exponential laws [15]. It is generally believed that under moderate temperatures and lithostatic pressure, the rock rheology is well represented by both linear and a power law relationship. The value of the exponent ranges from 1 to 5 [16]. Thus the basic rheology of the model
Results and discussion
All of the calculations have been performed in a rectangular domain with a high-resolution grid consisting of 600×300 nodal points. Such a resolution in geological situations would translate to a scale of 1/100 of inclusion diameter. The diameter of the inclusion is taken to be about 10% of the size of the domain that strongly reduces the wall influence [22]. Both the matrix and the inclusion have the same rheology exponent, but their material constants are different. The formal viscosity
Concluding remarks
The dynamics of multimaterials media in natural and engineering systems are invariably very complex and the inclusion/matrix system is certainly no exception. The results of our numerical simulation and comparison of structural evolution shows that, in spite of the complexity, it is still possible to employ structures as a gauge of the deformation regime 8, 11, 12. We note that even though the inclusion is deformable, the model does not consider such complex smaller-scale process as grain size
Acknowledgements
We thank Vladimir V. Khlestov, Bobby Bolshoi and Yuri Yu. Podladchikov for their interest in this problem. The paper was greatly improved due to the useful comments of J. Wheeler and anonymous reviewer. This research was supported in part by the Geosciences Program of the Dept. of Energy, grant 96-05-66051 of Russian Foundation of Basic Research and a visiting scholarship from MSI to Arkady Ten. [RV]
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Behaviour of an isolated rimmed elliptical inclusion in 2D slow incompressible viscous flow
2013, Journal of Structural GeologyCitation Excerpt :For detachment of the interface allowing shear slip but no normal movement (i.e., Mode 2 detachment of Samanta and Bhattacharyya, 2003), the concentric elliptical shape will be maintained as long as the inclusion itself remains elliptical. However, in general, a rim of finite thickness will be deformed into an increasingly non-elliptical shape with increasing bulk strain (Figs. 15–17) (e.g., Passchier and Simpson, 1986; Passchier and Sokoutis, 1993; Passchier et al., 1993; Bjørnerud and Zhang, 1995; ten Brink and Passchier, 1995; Masuda and Mizuno, 1996; Ten and Yuen, 1999; Mandal et al., 2000; Passchier and Trouw, 2005). The question is how significantly this alters the rotational and (for non-rigid inclusions) stretching behaviour of the inclusion.
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