Susanta Kumar Samanta

Boudinage in multilayered rocks under layer-normal compression: a theoretical analysis

Abstract This paper presents a dynamic analysis of boudinage in multilayers of alternate brittle and ductile layers under layer-normal compression. Based on the mode of fracturing of individual brittle layers, boudinage is classified into three types: tensile fracture boudinage (Type 1), shear fracture boudinage (Type 2a) and extensional shear fracture boudinage (Type 2b). The layer-thickness ratio, T r (= t b / t d ), and the strength ratio, F (= T /2 ηe ), between the brittle and the ductile units are the principal physical factors determining the type of boudinage. Type 1 boudinage develops rectangular boudins and occurs when T r is low ( F is high (>0.8). In contrast, Type 2a boudinage takes place when T r is high (>8.5) or F is low ( A r ) of all the types of boudins is inversely proportional to layer-thickness ratio ( T r ). However, Type 1 and Type 2 boudins, have contrasting aspect ratios, which are generally greater and less than 1, respectively.

Numerical modeling of heterogeneous flow fields around rigid objects with special reference to particle paths, strain shadows and foliation drag

Abstract With the help of two-dimensional numerical models this paper investigates three aspects of heterogeneous deformation around rigid objects: (1) the nature of particle paths ; (2) the development of strain shadow zones; and (3) the drag patterns of passive markers. In simple shear, spherical objects develop typically a concentric vortex motion, showing particle paths with an eye (double-bulge)-shaped separatrix. The separatrix has no finite dimension along the central line, parallel to the shear direction. Under a combination of pure shear and simple shear, the particle paths assume a pattern with a bow-tie shaped separatrix. With increase in the ratio of pure shear to simple shear ( S r ), the separatrix around the object shrinks in size. The axial ratio of the object ( R ) is another important factor that controls the geometry of particle paths. When R R , the loci form a doublet elliptical shell structure. Objects with R >3 do not show closed particle paths, but give rise to elliptical or circular spiral particle paths. The development of strain shadow zones against equant rigid bodies depends strongly on the strain ratio S r . When S r =0 (simple shear), they develop opposite to the extensional faces of the object, forming a typical σ -type tail. The structure has a tendency to die out with an increase in the pure shear component of the bulk deformation ( S r ). The initial angle of the long axis of the object with the shear direction ( φ ) and the axial ratio of the object ( R ) determine the development of strain shadow zones near inequant rigid objects. Objects with large R and φ between 60 and 120° form pronounced zones of low finite strain, giving rise to strain shadow structures. A geometrical classification of diverse drag patterns of passive markers around rigid objects is presented along with their conditions of formation.

Flow patterns around rigid inclusions in a multiple inclusion system undergoing bulk simple shear deformation

Abstract During deformation of an inclusion-matrix system, the velocity fields around individual inclusions mutually interfere with one another. Such interacting inclusions rotate at slower rates than non-interacting, single inclusions. This paper presents a theoretical model that describes the flow pattern of matrix (viscous) material around interacting rigid inclusions of spherical shape in bulk simple shear deformation. Numerical simulations based on the velocity functions reveal that the volume concentration of inclusions is a crucial parameter controlling the flow pattern around rotating inclusions under interacting conditions. At low volume concentrations (ρv 0.1), transforms into a pattern with a bow-tie shaped separatrix. At a large volume concentration (ρv=0.4) the separatrix assumes the geometry of a super-ellipse. We also present numerical models that illustrate the influence of volume concentration on the (1) nature of strain distribution, (2) distortion patterns of passive foliations ,and (3) mantle structures around inclusions in an interacting state. Based on this theory, it is shown that the rotational retardation of the inclusions slightly enhances the bulk viscosity of the inclusion-matrix system.

Progressive development of mantle structures around elongate porphyroclasts: insights from numerical models

Abstract This paper presents a generalized theoretical approach towards two-dimensional numerical modeling of the mantle geometry of inequant porphyroclasts of varying shapes within a Newtonian matrix during progressive, general type of bulk deformation. The analysis takes into account the effects of synkinematic size reduction of the porphyroclast with concomitant mantle development in response to dynamic recrystallization. Numerical simulations reveal that the principal factors governing the geometry of mantle structures are: (1) the initial aspect ratio of the porphyroclast (a/b), (2) the rate of clast-size reduction, and (3) the ratio of the rates of pure shear and simple shear (Sr) or the kinematic vorticity (Wk) in the general type of non-coaxial deformation. In general, porphyroclasts develop δ-, φ- and finally, σ-type mantle structures, as the rate of clast-size reduction is progressively increased. The tails of equant porphyroclasts tend to be characterized by wings with increase in bulk shear during progressive deformation. In contrast, inequant objects (a/b>1) develop composite tails with multiple wings, even at low finite shear strains. However, with increase in aspect ratio δ geometry tends to dominate the overall mantle structure. Porphyroclasts with a large aspect ratio (a/b=3) form tails with overturned δ-wings, as described in Passchier, C.W., Simpson, C., 1986. “Porphyroclast system as kinematic indicators”, Journal of Structural Geology 8, 831–844. In general the type of non-coaxial deformation, with decrease in kinematic vorticity (or increase in Sr), porphyroclasts irrespective of their initial shapes, tend to form atypical δ-like tails that do not cross the central reference plane.

Deformation of ductile inclusions in a multiple inclusion system in pure shear

This paper analyzes the deformational behavior of mutually interacting spherical inclusions in a multiple inclusion system, considering two physical factors: viscosity ratio between inclusion and matrix (m) and the ratio of inclusion diameter to mean inter-inclusion distance (a/b). For a given value of m, the strain partitioning between a stiff inclusion and the bulk system (i.e. ratio of their natural extension rates) increases non-linearly with increasing a/b ratios and the gradient of increase becomes steeper when the inter-inclusion distance is less than about twice their diameter (i.e. a/b>about 0.5). The strain distribution within a deformed inclusion is homogeneous when the a/b ratio is less than about 0.6. For larger values of a/b, the internal deformation becomes heterogeneous, with the strain increasing or decreasing towards the core in the case of stiff (m>1) and soft (m<1) inclusions, respectively. The deformed shape of inclusions in section also shows departure from an ideal ellipse with an increase in the a/b ratio. Stiff inclusions develop shapes similar to that of a super-ellipse in contrast to soft inclusions that resemble a sub-ellipse. The heterogeneity of internal deformation is also reflected in the distortion of passive foliations initially at right angles to the bulk extension direction, which become curved with convexity outward and inward, respectively, within stiff and soft inclusions.

Controls on the failure mode of brittle inclusions hosted in a ductile matrix

Plane strain deformation experiments were performed on elliptical inclusions of cohesive sand embedded within a slab of pitch, with the aim of investigating the mode of fracturing of brittle inclusions within a ductile matrix. Under pure and simple shear, the inclusions failed in three different modes: tensile fracturing (Mode 1), shear fracturing (Mode 2a) and extensional shear fracturing (Mode 2b). Jeffrey’s (1922) theory of the flow of a viscous medium around an ellipsoidal body was applied to the experimental results to determine the principal tensile and compressive stresses within an inclusion, and analyze the failure modes using Griffith’s Criterion. The analysis reveals that the aspect ratio (R) and the orientation (u ) of the inclusion control the principal tensile and compressive stresses within it, and in turn govern the mode of brittle deformation. At a particular inclusion orientation, the tensile stress increases, whereas the compressive stress decreases monotonically with increasing aspect ratio of the inclusion. The principal stresses also vary with inclusion orientation for a given aspect ratio, but not monotonically. The analysis delimits the fields of each mode of brittle deformation of inclusions in R- u space under pure shear and simple shear. q 2001 Elsevier Science Ltd. All rights reserved.

Modes of detachment at the inclusion-matrix interface

Abstract With the help of a 2D theoretical model, this paper analyses the modes of matrix detachment around a circular–elliptical rigid inclusion under pure and simple shear bulk deformations. Three modes of matrix detachment are possible: Mode 1 —the matrix is displaced normal to the inclusion boundary, forming fissures at the interface; Mode 2 —the matrix slips along the inclusion boundary without any separation; Mode 3 —the detachment occurs by a combination of Modes 1 and 2. In order to determine the onset of detachment at any point on the coherent inclusion–matrix interface, the tensile and shear stresses were derived at that point, and compared with imposed critical values. Numerical simulations based on the theoretical model reveal that the three modes occur systematically along the inclusion–matrix interface, the geometrical dispositions of which depend on the aspect ratio ( R ) and orientation ( φ ) of the inclusion. In the case of circular inclusions, Mode 1 and Mode 2 domains are separated by a Mode 3 domain and the disposition shows an internal symmetry. On the other hand, it is asymmetrical when the inclusions are elliptical and oriented oblique to the bulk extension direction in pure shear and to the shear direction in simple shear. In simple shear, Mode 2 detachment with synthetic slip occurs dominantly when φ is less than 45° and greater than 135°. The results obtained from numerical models are complemented with observations in physical experiments. The paper also determines theoretically the critical stresses for detachment to occur as a function of R for different φ values, revealing that for a given mechanical strength of the inclusion–matrix interface, a particular mode of detachment can take place only when the aspect ratio crosses a threshold value.

An analysis of anisotropy of rocks containing shape fabrics of rigid inclusions

Abstract This paper presents a theoretical basis for estimation of mechanical anisotropy in homogeneous rocks containing shape fabrics of rigid inclusions. The analysis is based on two types of viscous models: one containing linear fabrics of prolate ( a > b = c ) inclusions (cf. L -tectonite) and the other containing planar fabrics of oblate ( a b = c ) inclusions (cf. S -tectonite). Models show contrasting bulk viscosities in stretching ( normal viscosity ) and shearing ( shear viscosity ) parallel to the fabric. The axial ratio R (= a / b ) and the volume concentration ( ρ v ) of rigid inclusions appear to be the principal parameters in determining the viscosity contrast. In anisotropic models with linear fabrics, normal viscosity ( η p ) increases monotonically with increase in R , whereas shear viscosity ( η s ) increases to a maximum, and then drops down to a near-stationary value. In anisotropic models with planar fabrics, the normal viscosity increases little with increasing flatness of inclusions, but the variation assumes a steep gradient when the latter is large. Shear viscosity, on the other hand, is relatively less sensitive to the shape of inclusions. The ratio of normal and shear viscosities, conventionally described as anisotropy factor δ , in both the models is always greater than 1, indicating that normal viscosity will be essentially greater than shear viscosity, irrespective of the axial ratio of inclusions forming the fabric. Models with a linear fabric show contrasting normal viscosities in pure shear flow along and across the linear fabric. The anisotropy is expressed by the ratio of longitudinal and transverse normal viscosities ( anisotropic factor σ ). It is revealed that the transverse viscosity is essentially less than the longitudinal viscosity, as observed in test models.

Development of structures under the influence of heterogeneous flow field around rigid inclusions: insights from theoretical and numerical models

Rocks that are mechanically heterogeneous due to the presence of stiff or rigid inclusions floating in a ductile matrix, commonly show a variety of micro- to macro-scale structures developing under the influence of heterogeneous flow field in the neighbourhood of the inclusions. It is of fundamental importance to apprehend the nature of strain heterogeneity around inclusions to understand progressive development of structures associated with rigid inclusions such as strain shadow, foliation drag, porphyroclast mantle, porphyroblast inclusion trails, intragranular fractures, etc. The development of these diverse types of structures can be analyzed with the help of a suitable hydrodynamic theory. In this paper, we review different continuum models that have been proposed to characterize the heterogeneous flow field around rigid inclusions, focusing on recent developments. Recent studies reveal that Jefferys [Proc. R. Soc. Lond. A 120 (1922) 161.] theory dealing with the motion of ellipsoidal rigid bodies in an infinitely extended viscous medium is more general in nature, and applicable for modeling the heterogeneous flow around both equant and inequant shapes of inclusions and ideal or non-ideal shear deformation of the matrix. The application of this theory, therefore, has advantages over other models, based on Lambs [Lamb, H., 1932. Hydrodynamics. Cambridge University Press, Cambridge.] theory dealing with spherical inclusions. The review finally illustrates numerical simulations based on hydrodynamic theories, highlighting the controls of physical and kinematic factors on the progressive development of the structures mentioned above.

Flattening in shear zones under constant volume: a theoretical evaluation

Abstract This paper presents a theoretical model based on strain energy and work rate calculations that evaluates the possible degrees of flattening in finite, ductile shear zones with rigid and deformable walls under constant volume conditions. The principal parameters governing the ratio of bulk flattening and shear rates ( S r = ϵ b / γ b ) in shear zones with rigid walls are found to be: (1) the length to width ratio ( D f ; measured in the normal section parallel to the extrusion direction), and (2) the inclination of shear zone normal ( α ) with the bulk compression direction. Narrow and long shear zones ( D f >10) are characterized by low S r ratios, implying little flattening in the shear zone even when α is low (in the order of a few degrees). Accordingly, the kinematical vorticity number W k is close to one when D f is large (>10) or α is high (>20°), and is much less than one if D f or α are low. The stretching rate of shear zone walls relative to the shear zone ( R f ) is an additional parameter that controls the degree of flattening in shear zones with deformable walls. For given D f and α values the flattening rate increases with increasing relative stretching rate R f , and is significant at large values of R f . Likewise the kinematical vorticity number W k shows an inverse relation with the relative stretching rate of shear zone walls.