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Particle shape measurement is an important analysis for a number of research and industrial disciplines. Most of the presently available 2-Dimensional shape analysis methods commonly suffer from at least two shortcomings: (1) particles, which are visually distinct, are not easily distinguished and (2) the parameter value is difficult to relate to physical attributes. This contribution seeks to address these issues by introducing the Curvature (κ) plot which can be used to identify particle corners and measure their sharpness. Using the κ plot, four shape parameters are proposed: number of corners, cumulative angularity, sharpest corner and straight fraction. This methodology is applied to synthetic geometrical shapes and 4000 natural sand particle data. The results demonstrate that the four new shape parameters can be collectively used to quantitatively describe particle shape which correlates closely with visual perceptions of particle boundary.

We present preliminary results on an all-Ireland network modelling approach to simulate the spreading the Severe Acute Respiratory Syndrome CoronaVirus 2 (SARS-CoV-2), commonly known as the coronavirus. In the model, nodes correspond to locations or communities that are connected by links indicating travel and commuting between different locations. While this proposed modelling framework can be applied on all levels of spatial granularity and different countries, we consider Ireland as a case study. The network comprises 3440 electoral divisions (EDs) of the Republic of Ireland and 890 superoutput areas (SOAs) for Northern Ireland, which corresponds to local administrative units below the NUTS 3 regions. The local dynamics within each node follows a phenomenological SIRX compartmental model including classes of Susceptibles, Infected, Recovered and Quarantined (X) inspired from Science 368, 742 (2020). For better comparison to empirical data, we extended that model by a class of Deaths. We consider various scenarios including the 5-phase roadmap for Ireland. In addition, as proof of concept, we investigate the effect of dynamic interventions that aim to keep the number of infected below a given threshold. This is achieved by dynamically adjusting containment measures on a national scale, which could also be implemented at a regional (county) or local (ED/SOA) level. We find that – in principle – dynamic interventions are capable to limit the impact of future waves of outbreaks, but on the downside, in the absence of a vaccine, such a strategy can last several years until herd immunity is reached.

Temperature history is one of the most important factors driving subsidence and the overall tectono-stratigraphic evolution of a sedimentary basin. The McKenzie model has been widely applied for subsidence modelling and stretching factor estimation for sedimentary basins formed in an extensional tectonic environment. Subsidence modelling requires values of physical parameters (e.g., crustal thickness, lithospheric thickness, stretching factor) that may not always be available. With a given subsidence history of a basin estimated using a stratigraphic backstripping method, these parameters can be estimated by quantitatively comparing the known subsidence curve with modelled subsidence curves. In this contribution, a method to compare known and modelled subsidence curves is presented, aiming to constrain valid combinations of the stretching factor, crustal thickness, and lithospheric thickness of a basin. Furthermore, a numerical model is presented that takes into account the effect of sedimentary cover on thermal history and subsidence modelling of a basin. The parameter fitting method presented here is first applied to synthetically generated subsidence curves. Next, a case study using a known subsidence curve from the Campos Basin, offshore Brazil, is considered. The range of stretching factors estimated for the Campos basin from this study is in accordance with previous work, with an additional estimate of corresponding lithospheric thickness. This study provides insight into the dependence of thermal history and subsidence modelling methods on assumptions regarding model input parameters. This methodology also allows for the estimation of valid combinations of physical lithospheric parameters, where the subsidence history is known.

We present preliminary results on an all-Ireland network modelling approach to simulate the spreading the Severe Acute Respiratory Syndrome CoronaVirus 2 (SARS-CoV-2), commonly known as the coronavirus. In the model, nodes correspond to locations or communities that are connected by links indicating travel and commuting between different locations. While this proposed modelling framework can be applied on all levels of spatial granularity and different countries, we consider Ireland as a case study. The network comprises 3440 electoral divisions (EDs) of the Republic of Ireland and 890 superoutput areas (SOAs) for Northern Ireland, which corresponds to local administrative units below the NUTS 3 regions. The local dynamics within each node follows a phenomenological SIRX compartmental model including classes of Susceptibles, Infected, Recovered and Quarantined (X) inspired from Science 368, 742 (2020). For better comparison to empirical data, we extended that model by a class of Deaths. We consider various scenarios including the 5-phase roadmap for Ireland. In addition, as proof of concept, we investigate the effect of dynamic interventions that aim to keep the number of infected below a given threshold. This is achieved by dynamically adjusting containment measures on a national scale, which could also be implemented at a regional (county) or local (ED/SOA) level. We find that – in principle – dynamic interventions are capable to limit the impact of future waves of outbreaks, but on the downside, in the absence of a vaccine, such a strategy can last several years until herd immunity is reached.

Bovine viral diarrhea (BVD) is a disease in cattle with complex transmission dynamics that causes substantial economic losses and affects animal welfare. The infection can be transient or persistent. The mostly asymptomatic persistently infected hosts are the main source for transmission of the virus. This characteristic makes it difficult to control the spreading of BVD. We develop a deterministic compartmental model for the spreading dynamics of BVD within a herd and derive the basic reproduction number. This epidemiological quantity indicates that identification and removal of persistently infected animals is a successful control strategy if the transmission rate of transiently infected animals is small. Removing persistently infected animals from the herd at birth results in recurrent outbreaks with decreasing peak prevalence. We propose a stochastic version of the compartmental model that includes stochasticity in the transmission parameters. This stochasticity leads to sustained oscillations in cases where the deterministic model predicts oscillations with decreasing amplitude. The results provide useful information for the design of control strategies. Dynamical systems are often described by deterministic mathematical models, where the state of the system is determined by the initial conditions. Many real-world processes, however, include an element of probability or randomness. Even so, deterministic models might still be able to reproduce the main trend as a mean-field approximation but fail to capture the spectrum of possible dynamical scenarios of individual realizations. In addition, stochastic input such as noise can trigger the emergence of hidden dynamical features with surprising effects such as stochastic resonance, coherence resonance, or other noise-induced changes of dynamical behavior. Here, we present the example of a cattle disease that is realized as an extended susceptible-infected-recovered model. To explore the impact of stochasticity on the temporal behavior of the dynamics, we consider a stochastic transmission coefficient and systematically investigate the interplay between parameter noise and the intrinsic time scales of the underlying deterministic system.

Shape analysis can provide vital information regarding the origin, transport and deposition history of grains. Particle shape measurement has been an active area of research for sedimentologists since the 20th century. With advancement in the field of computation and image analysis, shape analysis can be done in a faster and much more accurate way compared to manual measurements. The results obtained are reproducible as compared to visual qualitative analysis. However, there is a lack of image analysis software tools aimed at the field of sedimentology where the fine details of a particle boundaries are required. Image based Particle Shape Analysis Toolbox (IPSAT) developed in the Mathematica environment for the quantitative characterisation of sedimentary grains in 2-dimensions is presented here. This image analysis toolbox can be used to analyse consolidated as well as loose sediment samples. A total of 12 parameters are available for shape measurement comprising conventional shape parameters (roundness, angularity, circularity and irregularity), mathematically complex shape parameters (fractal dimension and Fourier descriptors) and common geometrical shape parameters (aspect ratio, convexity, solidity, mod ratio, rectangularity and compactness). Additionally, IPSAT offers to compute 6 particle size measurement parameters. Furthermore, 2-D particle size distribution can be transformed to a 3-D size distribution for thin section analysis. Example analyses have been carried out on a sandstone and a loose sediment sample. The toolbox presented here aims to establish a textural analysis methodology to be used by geologists and sedimentologists in particular. It will allow users to quantitatively characterise a large set of grains with a fast, cheap and robust methodology.

Shear zones are important phenomena in the Earth’s middle and lower crust and are of great interest to structural geologists. Models involving rigid boundaries moving parallel to themselves are extended here to include the case where (i) walls are deformable and (ii) mobile rigid walls approach each other. These models are combined with Couette and Poiseuille flow to define a broad range of kinematic possibilities. Deformable wall models lead to smooth transitions from deformed to undeformed materials as well as with the zone transitions to gentler and more spread out deformation. Mobile walls, on the other hand, lead to shear zones where shear sense can change along a shear zone boundary.

Most mountain belts on Earth show some degree of curvature in plan view, from a slight bend to horseshoe shapes. Such curvatures may occur on different scales, from individual thrust sheets to entire plate boundaries. Curvature may be acquired by vertical-axis rotation during or after orogenesis, or reflect primary lateral variations in shortening directions or physiographical features. Quantifying the amount of vertical-axis rotations of plan-view curvature is therefore helpful to our understanding of orogenesis, geodynamics, and paleogeography. The orocline test assesses to what extent vertical-axis rotations have played a role in the acquisition of an orogen’s curvature. The test quantifies through linear regression the relationships between changes in structural trends and the orientations of a geologic fabric. However, the current mathematical approaches to the orocline test show potential biases. In this paper we aim to overcome such biases by developing a novel orocline test that applies total least squares (TLS) regression combined with a novel approach to bootstrapping. This bootstrap TLS orocline test can be used with all types of directional data acquired from structural geology, paleomagnetism, or sedimentology. It quantifies, for the first time, secondary curvature with confidence bands. We also provide several graphical and analytical tests to evaluate the statistical significance of the result. An open source online application implementing this method is available for use on www.paleomagnetism.org. We illustrate the use of the methodology by reanalyzing published data sets from two well-known oroclines in the Cantrabrian (northwest Iberia) and Aegean (Greece) regions.

The probability density function associated with the March model is derived and used in a maximum likelihood method to estimate the best fit distribution and 3D strain parameters for a given set of linear or planar data. Typically it is assumed that in the initial state (pre-strain) linear or planar data are uniformly distributed on the sphere which means the number of strain parameters estimated needs to be reduced so that the numerical technique succeeds. Essentially this requires that the data are rotated into a suitable reference frame prior to analysis. The method has been applied to a suitable example from the Dalradian of SW Scotland and results obtained are consistent with those from an independent method of strain analysis. Despite March theory having been incorporated deep into the fabric of geological strain analysis, its full potential as a simple direct 3D strain analytical tool has not been achieved. The method developed here may help remedy this situation.

Introduction: Covid-19 was declared a pandemic in March 2020. Since then, governments have implemented unprecedented public health measures to contain the virus. This study will provide evidence to inform responses to the pandemic by: i) estimating population prevalence and trends of self-reported symptoms of Covid-19 and the proportions of symptomatic individuals and household contacts testing positive for Covid-19; ii) describing acceptance and compliance with physical-distancing measures, explore effects of public health measures on physical, mental and social wellbeing; iii) developing a mathematical network model to inform decisions on the optimal levels of physical distancing measures. Methods: Two cross-sectional nationally-representative telephone surveys will be conducted in Ireland using random digit-dialling, with response rates estimates based on proportion of non-operational and non-answering numbers. The first survey with four waves in May and June will address adherence to social distancing measures and whether the respondent or other household members are or have been unwell during the preceding two weeks with one or more symptoms of Covid-19. The second survey with three waves in June, July and September will address knowledge, attitudes, and compliance towards physical-distancing measures and physical, mental and social wellbeing. The mathematical network model will be developed for all-Ireland (on various levels of spatial granularity including the scale of counties and electoral divisions) based on outputs from both cross-sectional surveys and relevant publicly available data to inform decisions on optimal levels and duration of physical distancing measures. Discussion: This study will contribute to our understanding of the impact and sustainability of public health measures of the Covid-19 pandemic. Findings will have long-lasting benefits, informing decision-making on the best levels, and duration of physical-distancing measures, balancing a range of factors including capacity of the health service with the effects on individuals’ wellbeing and economic disruption. Findings will be shared with key policy-makers.

Understanding the finite strain state of a rock or its spatial variation is an important component of structural geology. In this chapter the problem of calculating finite strain using shape data and point data is considered in detail. The first step in strain analysis is data acquisition, and the first section is concerned with semiautomatic extraction of data for strain analysis from digital images. Two methods for strain analysis are considered: the first method (MRL, mean radial length) uses the shapes of strain markers, and the second method (Delaunay triangulation nearest neighbor method) uses the relative positions of strain marker centroids. In each case, relevant image files, data, and C++ code are provided. Background theory is discussed, and an example application is considered emphasizing that finite strain analysis is best used in conjunction with a range of structural observations.

Shape fabrics of elliptical objects in rocks are usually assumed to develop by passive behavior of inclusions with respect to the surrounding material leading to shape-based strain analysis methods belonging to the Rf/ϕ family. A probability density function is derived for the orientational characteristics of populations of rigid ellipses deforming in a pure shear 2D deformation with both no-slip and slip boundary conditions. Using maximum likelihood a numerical method is developed for estimating finite strain in natural populations deforming for both mechanisms. Application to a natural example indicates the importance of the slip mechanism in explaining clast shape fabrics in deformed sediments.

If the deformation history of individual layers in a deformed rock mass can be appropriately analysed and categorised then the surface of no elongation (snoe) and the surface of no infinitesimal elongation (snoie) may be identified. A computational approach to automatically identify the polar snoe and snoie is presented and tested on synthetic and natural examples. Additionally the equations that allow extraction of parameter values, which quantify the deformation, are derived here. Two fundamental assumptions are made: 1) the progressive deformation is pure shear and 2) the associated volume change is isotropic. The proposed method is tested using data generated from known deformation sequences and is found to work reasonably well. However, as is the case with most methods of analysis, suitable data with a wide range of orientations is required for reliable results.

A scale free representation of a general non-isochoric 2D deformation is presented which is amenable to mathematical analysis. By describing deformation in 2D in terms of polar coordinates the stretching and rotational histories of linear elements separate and are easily analysed both qualitatively and quantitatively. An analysis of finite strain combined with dynamical considerations allows the derivation of equations which may be used to estimate finite strain, area change and kinematic vorticity number. Numerical investigation of method developed here was carried out and it was found to perform well unless large area changes occur in combination with large components of simple shear. A re-analysis of natural data indicates the method is consistent.

Shear folding, which is also referred to as slip folding, involves shear along planes that are oriented approximately parallel to the axial plane of the fold structure. These planes, which are typically axial-planar cleavage planes, facilitate high-angle reverse slip leading to fold limb rotation and amplification. This study builds on recent advances in our understanding of the role of weak fault zones in facilitating slip on misoriented faults; i.e., faults at a high angle to the maximum principal tectonic stress (sigma(1)). Analysis of folded marine sedimentary rocks from the Variscan of southern Ireland provides unambiguous microstructural evidence for reverse shear on chemically weakened cleavage domains. Significant silica loss in these cleavage domains, and as a consequence marked mechanical weakening, is seen as the primary cause for the reverse slip associated with the shear folding of these sedimentary rocks.

An objective method is developed for identifying the shape of the void generated by analysis of the spatial arrangement of point distributions using the Fry method and in particular Delaunay triangulation nearest neighbour data. The method works by numerically minimising a weighted non-linear least-squares formulation which tends to selectively apply higher weights to points below the boundary and less weight to points above the boundary. Sampling errors are estimated using a bootstrap procedure. A simulation study indicates that the method works best for closely packed object arrangements and tends to underestimate the strain axial ratio at high imposed strains. Overall the method is found to be objective and consistent.

The Cantabrian Arc of the Variscan Belt has been recently defined as a true orocline, constraining kinematics and deformation timing. This curved sector of the orogenic belt is characterized by two different fold sets: (1) one runs parallel to the outcrops of the main thrusts and describes a horseshoe shape concave towards the east, and (2) another is radial to the arc. A detailed geometric study of the fold interference patterns in the Cantabrian Arc revealed the conical nature of the folds belonging to the radial set. These conical folds developed with different geometrical characteristics (semi-apical angles and axis attitudes) depending on the initial orientation and geometry of the folded surfaces. They are interpreted to result from a vertical axis rotation during oroclinal buckling of the Variscan Belt in NW Iberia. This study of conical folds in the Cantabrian Arc highlights that conical folds in curved orogenic arcs are a powerful tool for establishing the sequence of tectonic events because interference patterns due to vertical axis secondary differential rotations provide unique geometrical characteristics observed in the Cantabrian Arc that can be extrapolated to other oroclines.

The Tso Morari crystalline (TMC) gneiss dome in the Indian Himalaya extruded from a depth of ~120 km through an inclined subduction channel of sub-elliptical cross-section at the leading edge of the Indian plate. The velocity profile of this gneiss dome is derived after (1) presuming its incompressible Newtonian rheology, (2) finding the “best fit” of the outcrop of the gneiss dome to an ellipse, (3) taking into account different lithologies to have existed at the top of the extruding gneiss body, (4) considering the extrusion to have been driven by the buoyant push of the denser mantle beneath the lighter gneiss, and (5) assigning a range of plausible densities for different litho-units. Fitting the known rates of extrusion—from a few centimetres up to about one-hundredth of a millimetre per year—from ~53 Ma onwards of this gneiss dome to its velocity profile constrains its maximum possible viscosity to ~7.5 × 1022 Pa s. This magnitude is 102–104 times higher than previous estimates for gneisses and granites. Alternative explanations of our data are the following: (1) There was a fall in extrusion rates of the TMC gneiss from 53 to

Samples in this paper have been assigned formations based on the Geological Survey of Ireland shapefile released prior to the commencement of the study. However, the authors were not aware that, since obtaining the samples, an updated shapefile had been released. This update affects three of the four apatite samples assigned to the Lower Devonian Ballymore Formation. The location of samples Mb-1, Mb-4 and Mb-5 now places them well within the undifferentiated, Upper Devonian Slieve Mish Group. As outlined in our paper, the apatite ages were originally produced concurrently with apatite fission track analysis and were later used in our study to provide additional provenance information in support of the detrital zircon geochronological data. In the second paragraph of the discussion section we say the following: "Williams et al. (1999) obtained an age of 411 Ma for the Cooscrawn Tuff Bed in the Ballymore Formation, which is older than 22 of the 70 detrital apatites analysed in this formation". The reassignment of the three samples to the Upper Devonian Slieve Mish Group nullifies the above statement. However, our interpretation that the depositional age of the Ballymore Formation is younger than the 411 Ma age given by Williams et al. (1999) is predominantly based upon the evidence given by the six youngest detrital zircons from the formation which underlies the Ballymore Formation (i.e. the Slea Head Formation). These zircons give a concordia age of 405 ± 4 Ma. This suggests that the Ballymore Formation was more than likely deposited after 409 Ma. We do not believe that the reassignment of the three samples has any major impact on our provenance interpretations. The ∼420 Ma age of the majority of the apatites in samples Mb-1, Mb-4 and Mb-5 actually fits with the range of Palaeozoic detrital zircons in sample AK-17 which was taken from the Slieve Mish Group, thereby supporting minor input of rocks affected by end-Scandian metamorphism.

The computer strain analysis methods SAPE, MRL and DTNNM have permitted the characterization of finite strain in two different regions with contrasting geodynamic scenarios; (1) the Talas Ala Tau (Tien Shan, Kyrgyzs Republic) and (2) the Somiedo Nappe and Narcea Antiform (Cantabrian to West Asturian-Leonese Zone boundary, Variscan Belt, NW of Iberia). The performed analyses have revealed low-strain values and the regional strain trend in both studied areas. This study also investigates the relationship between lithology (grain size and percentage of matrix) and strain estimates the two methodologies used. The results show that these methods are comparable and the absence of significant finite strain lithological control in rocks deformed under low metamorphic and low-strain conditions.

The evolution of resonant oscillations in closed containers shaped like a cone or a bulb, as in the defining experiments of Lawrenson et al. [C.C. Lawrenson, B. Lipkens, T.S. Lucas, D.K. Perkins, T.W. Van Doren, Measurements of macrosonic standing waves in oscillating closed cavities, J. Acoust. Soc. Am. 104 (2) (1998) 623–636], is investigated theoretically. These shapes ensure the resonant frequencies are incommensurate, and hence that the resonant motions are shockless. The containers are oscillated parallel to the axis of symmetry at or near the fundamental frequency. The relationship between shocks and inhomogeneities in elastic panels is a related problem. A perturbation scheme that combines the method of multiple scales with a Duffing-type expansion yields a set of coupled nonlinear ordinary differential equations for the slow variation of the amplitude and phase of the fast resonant oscillations. The inclusion of damping ensures the oscillations approach a steady state, and the parameters in the steady state amplitude–frequency relation appear in the evolution equations.

Existing methodologies that use ellipsoidal objects for the analysis of geological strain typically assume that these objects acted passively during deformation. This assumption, when not valid, can lead to significant underestimates of strain in rocks deformed under low-grade conditions in orogenic forelands. This is especially true when clastic sedimentary rocks are utilized to measure strain; the competency contrast between clasts and matrix possibly leading to marked ‘non-passive’ behaviour. The systematic nature of this finite strain underestimation may allow for a correction of strain estimates when this type of behaviour is evident.

The current view regarding the timing of regionally developed penetrative tectonic fabrics in sedimentary rocks is that their development postdates lithification of those rocks. In this case, fabric development is achieved by a number of deformation mechanisms, including grain rigid body rotation, crystal-plastic deformation, and pressure solution. The latter is believed to be the primary mechanism responsible for the domainal structure of cleavage in low-grade metamorphic rocks. In this study we combine field observations with strain studies to characterize considerable (>50%) Acadian crustal shortening in a Devonian clastic sedimentary sequence from southwest Ireland. Despite these high levels of shortening there is a marked absence of the domainal cleavage structure and intraclast deformation that are expected with this level of deformation. Fabrics in these rocks are predominantly a product of rigid body rotation and repacking of extraformational clasts during deformation of a clastic sedimentary sequence before lithification was complete.

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.

Geological materials are largely heterogeneous and are typically com- prised of approximately ellipsoidal objects immersed in a matrix with different phys- ical properties. Methodologies for the identification of ancient regional tectonic pat- terns may be developed based on an understanding of the behaviour of heterogeneous materials. In this contribution, the differential equation governing the rotation of a de- formable ellipse immersed in a viscous fluid is considered and is found to contain a singularity when the ellipse becomes circular in shape. This problem is avoided by reformulating the equations using the standard algebraic representation of an ellipse. Thus, the equations can be numerically solved without difficulty.

Shape fabrics of elliptical objects in rocks are usually assumed to develop by passive behaviour of inclusions together with the surrounding material leading to shape-based strain analysis methods belonging to the Rf/ϕ family. By deriving the probability density function for populations of rigid ellipses deforming in a general 2D deformation, a method is developed which can be used to estimate both finite strain and the kinematic vorticity number. Statistical parameters are theoretically derived and their behaviours under various kinematic conditions are investigated. The maximum likelihood method from statistics is used to produce a numerical method for estimating deformation parameters from natural populations. A simulation study demonstrates that finite strain can be estimated well for both low and high applied finite strains, whereas the kinematic vorticity number is well estimated only in the case of high finite strains (Rs > 40), and that large sample numbers (≈1000) are required.

Flanking structures develop locally around material inhomogeneities. The evolution of these structures is investigated here by modelling peturbations around a deformable ellipse using a 2D analytical solution. Non-unique instantaneous and finite geometries are predicted, and it is shown that in terms of slip and curvature, complex histories are possible under simple shear, whereas under pure shear instantaneous geometries map directly into finite state geometries. Single flanking structures are of limited use in kinematic analysis but can help constrain the kinematics when interpreted in conjunction with other structural features. Flanking structures exhibiting a range of CE (cross-cutting element) orientations have more potential as kinematic indicators. Flanking structures serve as an excellent example of the role that material homogeneity can play in locally producing complex structures in a relatively simple bulk flow field.

A mathematical model for rigid inclusions with a slipping interface immersed in a general 2D homogeneous deformation is developed. Under bulk pure shear inclusions are expected to rapidly approach the stretching axis when compared to the behaviour of inclusions with no slip at the interface. The derived model predicts synthetic and antithetic motion into a stable orientation under simple shear, and thereafter the inclusion makes an antithetic angle with the shear direction. Under simple shear rotation rates can be higher or lower than those of no-slip inclusions, depending on orientation. A direct relationship between object inclination to the shear direction and the vorticity of the bulk flow is predicted. The model compares well with published analogue and numerical experiments.

A solution for the deformation of viscous layers during a two-dimensional general deformation is derived. Layer deformation is found to consist of components of pure shear, simple shear and pure rotation when considered with respect to the layer orientation. It is shown that for external bulk pure shear the layer deformation is typically vastly different to that outside the layer, ranging from pure to simple to super shear depending on layer orientation and viscosity contrast. Treagus' method for determining viscosity ratio from cleavage refraction is re-evaluated and is found to applicable under bulk pure shear. However, the presence of earlier LPS (layer-parallel shortening) can lead to erroneous viscosity ratio estimates. A new approach is developed, relying on the variation of cleavage orientation with layer orientation that is found to be robust with respect to LPS. The method is applied to examples from the Irish Variscides and the new approach has implications for determining whether or not LPS occurred in this foreland setting.

This paper presents a statistical analysis of the algebraic strain estimation algorithm of Shimamoto and Ikeda [Shimamoto, T., Ikeda, Y., 1976. A simple algebraic method for strain estimation from deformed eillipsoidal objects: 1. Basic theory. Tectonophysics 36, 315–337]. It is argued that the error in their strain estimation procedure can be quantified using an expected discrepancy measure. An analysis of this measure demonstrates that the error is inversely proportional to the number of clasts used. The paper also examines the role of measurement error, in particular that incurred under (i) a moment based and (ii) manual data acquisition methods. Detailed analysis of these two acquisition methods shows that in both cases, the effect of measurement error on the expected discrepancy is small relative to the effect of the sample size (number of objects). Given their relative speed advantage, this result favours the use of automated measurement methods even if they incur more measurement error on individual objects. Validation of these results is carried out by means of a simulation study, as well as by reference to studies appearing in previous literature. The results are also applied to obtain an upper bound on the error of strain estimation for various studies published in the literature on strain analysis.

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.

Constrained automated seeded region growing (CASRG) is an algorithm for automated grain boundary detection. It uses as input a single digitised microphotograph, such as ones obtained from a polarising microscope with an attached digital camera. In addition to this, it requires the user to click on the clasts within the microphotograph that the user wishes to obtain boundaries for. The algorithm requires no subsequent human input. The algorithm is based on the seeded region growing (SRG) algorithm of Adams and Bischof [Adams, R., Bischof, L., 1994. Seeded region growing. IEEE Transactions on Pattern Analysis Machine Intelligence 16, 641–647]. We have modified this algorithm to be guided by constraints and to adapt to the heterogeneity of colour information in the image. Imposition of these pre-determined additional conditions enables automated grain boundary detection without human intervention. The accuracy of CASRG has been validated through two benchmarking comparisons; one lithology with low tectonic strain and a second with high strain are used. The CASRG measurements are compared with those from hand drawn boundaries, which are used as a gold standard. Comparison is made using (a) a non-overlap statistic, (b) shape features, (c) strain estimates. In each case, the CASRG method compares very favourably with the gold standard.

A solution for the deformation of a non-rigid viscous elliptical inclusion in a matrix of differing viscosity is developed for the case of a general 2D deformation. A Newtonian rheology is assumed and velocities and stresses are equated at the boundary. An important parameter is the viscosity ratio given by the ratio of the external to the internal viscosities. The dynamics of the behaviour of such inclusions is examined for the cases of pure and simple shear and variable viscosity ratio. In general less viscous inclusions tend to accumulate finite strain more rapidly than more rigid inclusions. Large discordancies between the internal finite strain ellipse orientation and the bulk external finite strain ellipse are to be expected. It is also found that the kinematics of deformation inside an inclusion can often be one of super shear (i.e. kinematic vorticity number, Wk, greater than one) even though the external bulk kinematics is one of pure or simple shear (Wk=0 or 1). Objects tend to continuously rotate (the viscosity ratio must be less than 0.5) or asymptotically rotate (i.e. tend to ultimately align parallel to a fixed direction). This solution has many applications, some of which are briefly considered.

SAPE (Semi-Automatic Parameter Extraction) is a program for extracting information relevant to strain analysis from input digital images. Input images are manually produced by tracing the outlines of objects of interest. This is a less onerous, more objective and much faster task than manual measurement or digitisation of object data (i.e. aspect ratio R i and orientation ϕ i ). SAPE rapidly extracts the required data by using a simple region-growing algorithm to identify regions of interest. Subsequently, the second moments are calculated for each region enabling the common strain analysis parameters to be readily computed. The performance of SAPE was tested on three samples (microphotographs of deformed sandstone and oolite as well as a meter scale photograph of a deformed conglomerate) using a non-overlap statistic. This statistic is a normalised measure of overlap between two regions and can be used to compare different fits applied to the same region. In each case SAPE out-performed the manual method. SAPE offers significant advantages in terms of speed, objectivity and robustness and should facilitate the collection of large datasets for strain analysis.

An analytical expression for the error associated with estimating the axial ratio ( R s ) and orientation ( ϕ s ) of the finite strain ellipse with the mean radial length method of strain analysis is presented. Analytical errors are computationally efficient and compare excellently with errors calculated by the computationally intensive bootstrap approach.

High-pressure granulites of the Bacariza Formation (Cabo Ortegal Complex, NW Spain) exhibit spectacular examples of ductile shear zones developed at different scales in rocks containing pre-existing foliations. A detailed structural analysis was carried out on these shear zones in order to unravel and compare the role of various parameters controlling the deformation process (i.e. heterogeneous simple shear, components of homogeneous deformation, heterogeneous volume change and degree of non-coaxiality). Although heterogeneous simple shear largely dominated, negligible deviations from the ideal simple shear model were detected involving shortening along the structural directions perpendicular to the stretching axis (within the foliation plane) of the finite strain ellipsoid. The relationship between displacement parallel to a half-shear zone and the normal distance from its boundary provided the basis for the estimation of the stress exponent in the power-law constitutive flow equation associated with each shear zone, which is interpreted as a rheological indicator. These geometric and rheological results, and the thermobaric conditions of high-pressure shear zone deformation, indicate that these shear zones accommodated dominant plastic rock flow coeval with high-pressure and high-temperature deformations under moderate stress levels concomitant with elevated strain rates.