Kieran F. Mulchrone

Fitting an ellipse to an arbitrary shape: implications for strain analysis

An ellipse can be fit to an arbitrary shape using a linear least squares approach applied to boundary data. Alternatively, this problem can also be solved by calculating the second moments of the entire region, a technique popular in image analysis applications. If the irregular shape can be approximated by a polygon then Greens theorem allows efficient calculation of the second moments. If the shape is pixelated then the second moments can be calculated by a simple summation process. By considering the behaviour of these fitting methods with increasing deformation it is shown that as an arbitrary shape passively deforms, the best-fit ellipse also behaves as if it were deforming passively. This implies that all techniques of strain analysis that were previously restricted to populations of elliptical objects may now be applied to populations of arbitrary shapes, provided the best-fit ellipse is calculated by one of the methods described here. Furthermore it implies that selective sampling based on shape or methods of weighting based upon shape are invalid and tend to bias the raw data.

Finite strain estimation using the mean radial length of elliptical objects with bootstrap confidence intervals

A new method for calculating finite sectional strain from distributions of elliptical objects is presented. The only assumptions required are that before deformation (1) long axis orientations are uniformly distributed and (2) the distribution of axial ratios is independent of orientation. Importantly, an estimate of the orientation of the long axis of the strain ellipse is not required before the method can be applied. The method is based on the conceptually simple fact that the mean radial length of a set of uniformly oriented ellipses in the unstrained state equates to that of a circle, so that after strain, the mean radial length evaluates to the strain ellipse. Errors associated with the method are calculated from the bootstrap, and a simulation study verifies both the applicability of the new method for finite strain estimation and the accuracy of errors calculated with the bootstrap. The method is applied to a large set of sandstone quartz clast data from the Irish Variscides, whilst cross-checking of results with those from established methods also validates the approach taken.

Application of Delaunay triangulation to the nearest neighbour method of strain analysis

Abstract The nearest neighbour method of strain analysis is re-evaluated and a method for objectively determining nearest neighbours, namely the Delaunay triangulation, is applied. A simulation study and application to a real set of data demonstrates that this approach makes the NNM of strain analysis a practical (and computationally more efficient) alternative to the Fry and associated methods. Once nearest neighbours are selected centre–centre distances can be processed by normalisation and enhancement and the best fit ellipse is determined using a steepest gradient non-linear least squares algorithm applied to the polar equation of a centred ellipse. A simulation study indicates that the technique is a valid one and estimates the strain ellipse well at the 95% confidence interval. Application to a set of natural oolite data shows that there is a systematic variation of error with selection factor and it is suggested that the best estimate of the strain ellipse is obtained by choosing the selection factor which minimises the error.

Viscous dissipation pattern in incompressible Newtonian simple shear zones: an analytical model

An analytical model of shear heating in an inclined simple shear zone with Newtonian rheology under a reverse shear sense and an upward resultant pressure gradient is presented. Neglecting a number of secondary factors, the shear heat is expressed as a function of the total slip rates at the boundaries, pressure gradient, dip and thickness of the shear zone, and density, viscosity, and thermal conductivity of the rock. A quartic temperature profile develops with a point of maximum temperature near the bottom part of the shear zone in general. The profile is parabolic if pressure gradient vanishes leading to a Couette flow. The profile attains a bell shape if there is no slip at the boundaries, i.e., a true Pouseille flow. The present model of shear heating is more applicable in subduction channels and some extruding salt diapirs where the rheology is Newtonian viscous and pressure gradient drives extrusion.

Shear Senses and Viscous Dissipation of Layered Ductile Simple Shear Zones

Abstract Velocity profiles and shear heat profiles for inclined, layered Newtonian simple shear zones are considered. Reverse fault-like simple shear of the boundaries and upward net pressure gradient act together in such shear zones. As the velocity of the boundary increases, the point of highest velocity shifts from the lower layer of less viscosity into the upper layer. The shear heat profile shows a temperature peak inside the lower layer. For a more viscous upper layer, the point of highest velocity is located inside the upper layer and shifts towards the upper boundary of the shear zone. The shear heat profile shows a maximum temperature within the upper layer. Depending on the flow parameters of the two layers, the slip rate of the boundary, and the dip and thickness of the shear zone, a shear sense in reverse to the relative movement of the shear zone boundaries may develop. These models can decipher thermo-kinematics of layered shear zones in plate-scale hot orogens.

The effect of sample size on geological strain estimation from passively deformed clastic sedimentary rocks

Abstract The role of sample size in the estimation of geological strain, both finite strain ( R s ) and that of the orientation of the finite strain ellipse ( φ s ), is investigated for clastic sedimentary rocks. This study looks at four strain methods, the Robin method, the linearization method, the Mulchrone and Meere method and the mean radial length method that are initially tested using simulated strained data sets and subsequently by applying the methods to real data. It is found that the optimum strain analysis sample size for a clastic sedimentary rock is primarily dependant on the intensity of strain suffered by that rock because of the error behavior associated with R s estimates. An iterative process is therefore recommended starting with a minimum sample size of 150, which can be maintained or reduced based on the initial R s estimates.

Timing of deformation within Old Red Sandstone lithologies from the Dingle Peninsula, SW Ireland

Finite strain analysis of deformed clastic sedimentary rocks below and above the Acadian unconformity on the Dingle Peninsula, SW Ireland, reveals that this boundary represents a significant bulk strain discontinuity. The Late Emsian Acadian event is primarily responsible for penetrative cleavage development and high strain in rocks below the unconformity and not, as previously held, the later Variscan event, which overprinted the peninsula with a weak and localized disjunctive cleavage. The presence of apparent anticlockwise transecting cleavage and implied dextral closure of the Dingle Basin during the Acadian event is compatible with northward convergence of Armorica with respect to Eastern Avalonia.

Kinematics and shear heat pattern of ductile simple shear zones with ‘slip boundary condition’

Abstract Extrusion by Poiseuille flow and simple shear of hot lower crust has been deciphered from large hot orogens, and partial-slip boundary condition has been encountered in analogue models. Shear heat and velocity profiles are deduced from a simplified form of Navier–Stokes equation for simple shear together with extrusive Poiseuille flow and slip boundary condition for Newtonian viscous rheology. A higher velocity at the upper boundary of the shear zone promotes higher slip velocity at the lower boundary. The other parameters that affect the slip are viscosity and thickness of the shear zone and the resultant pressure gradient that drives extrusion. In the partial-slip case, depending on flow parameters (resultant pressure gradient, density and viscosity) and thickness of the shear zone, the velocity profiles can curve and indicate opposite shear senses. The corresponding shear heat profiles can indicate temperature maximum inside shear zones near either boundaries of the shear zone, or equidistant from them.

A statistic for estimating strain with confidence intervals from deformed line distributions with an application to schists and gneisses of the Western Gneiss Region, west central Norway

Abstract It is shown how to calculate the axial ratio of the strain ellipse ( R s ) from a deformed set of lines, assuming that they were initially uniformly distributed and deformed passively, using an exact but simple relationship between R s and the mean resultant length ( R ) of the distribution. The method applies to distributions of lines modified under a general 2D deformation (−∞ W k W k is the kinematical vorticity number. By using the circular variance for circular data, expressions for confidence intervals and confidence levels are calculated for strain-modified distributions of lines. These expressions indicate that confidence interval widths decrease with increasing R s and more importantly that the method is incapable of accurately estimating low strains. The method was applied to 18 gneiss and schist samples from Joa, west central Norway and found to produce realistic strain estimates, which are internally consistent and concur with field relationships.

Ductile shear zones : from micro- to macro-scales

Contributors vii Acknowledgments ix Introduction x Part I: Theoretical Advances and New Methods 1 From finite to incremental strain: Insights into heterogeneous shear zone evolution 3 Stefano Vitale and Stefano Mazzoli 2 How far does a ductile shear zone permit transpression? 14 Sujoy Dasgupta, Nibir Mandal, and Santanu Bose 3 2D model for development of steady ]state and oblique foliations in simple shear and more general deformations 30 Kieran F. Mulchrone, Patrick A. Meere, and Dave J. McCarthy 4 Ductile deformation of single inclusions in simple shear with a finite ]strain hyperelastoviscoplastic rheology 46 Christoph Eckart Schrank, Ali Karrech, David Alexandre Boutelier, and Klaus Regenauer ]Lieb 5 Biviscous horizontal simple shear zones of concentric arcs (Taylor Couette flow) with incompressible Newtonian rheology 59 Soumyajit Mukherjee and Rakesh Biswas Part II: Examples from Regional Aspects 6 Quartz ]strain ]rate ]metry (QSR), an efficient tool to quantify strain localization in the continental crust 65 Emmanuelle Boutonnet and Phillipe ]Herve Leloup 7 Thermal structure of shear zones from Ti ]in ]quartz thermometry of mylonites: Methods and example from the basal shear zone, northern Scandinavian Caledonides 93 Andrea M. Wolfowicz, Matthew J. Kohn, and Clyde J. Northrup 8 Brittle ]ductile shear zones along inversion ]related frontal and oblique thrust ramps: Insights from the Central Northern Apennines curved thrust system (Italy) 111 Paolo Pace, Fernando Calamita, and Enrico Tavarnelli 9 Microstructural variations in quartzofeldspathic mylonites and the problem of vorticity analysis using rotating porphyroclasts in the Phulad Shear Zone, Rajasthan, India 128 Sudipta Sengupta and Sadhana M. Chatterjee 10 Mineralogical, textural, and chemical reconstitution of granitic rock in ductile shear zones: A study from a part of the South Purulia Shear Zone, West Bengal, India 141 Nandini Chattopadhyay, Sayan Ray, Sanjoy Sanyal, and Pulak Sengupta 11 Reworking of a basement cover interface during Terrane Boundary shearing: An example from the Khariar basin, Bastar craton, India 164 Subhadip Bhadra and Saibal Gupta 12 Intrafolial folds: Review and examples from the western Indian Higher Himalaya 182 Soumyajit Mukherjee, Jahnavi Narayan Punekar, Tanushree Mahadani, and Rupsa Mukherjee 13 Structure and Variscan evolution of Malpica Lamego ductile shear zone (NW of Iberian Peninsula) 206 Jorge Pamplona, Benedito C. Rodrigues, Sergio Llana ]Funez, Pedro Pimenta Simoes, Narciso Ferreira, Carlos Coke, Eurico Pereira, Paulo Castro, and Jose Rodrigues 14 Microstructural development in ductile deformed metapelitic metapsamitic rocks: A case study from the greenschist to granulite facies megashear zone of the Pringles Metamorphic Complex, Argentina 224 Sergio Delpino, Marina Rueda, Ivana Urraza, and Bernhard Grasemann 15 Strike slip ductile shear zones in Thailand 250 Pitsanupong Kanjanapayont 16 Geotectonic evolution of the Nihonkoku Mylonite Zone of north central Japan based on geology, geochemistry, and radiometric ages of the Nihonkoku Mylonites: Implications for Cretaceous to Paleogene tectonics of the Japanese Islands 270 Yutaka Takahashi 17 Flanking structures as shear sense indicators in the Higher Himalayan gneisses near Tato, West Siang District, Arunachal Pradesh, India 293 Tapos Kumar Goswami and Sukumar Baruah Index 302