The work carried out by Richard Lisle, stemming from his stay in Oviedo and his integration into the fold analysis group at the local university, addressed a wide range of topics concerning fold geometry and kinematics. Regarding 2D fold geometry, Richard led the development of a method to relate the various classification schemes based on fold fitting functions, using a parameter defined as the normalized area bounded by the folded surface profile. Non-cylindrical folds were analysed based on the properties of developable surfaces and curvature analysis. In the field of kinematics, special attention was given to the neutral surface folding, which led to the distinction between two modes: “parallel tangential longitudinal strain” and “equiareal tangential longitudinal strain”. A method was also developed to estimate the total bulk shortening associated with the formation of flattened parallel folds. The 2D kinematics of various fold types was explored through numerical simulations, based on the displacement laws governing different folding mechanisms. This methodology was applied to the study of chevron folds, folded oblique surfaces, asymmetric folds, recumbent folds, and similar folds. The investigation led to the resolution of certain problems related to rock deformation and resulted in the description of a new deformation mode: “rotation shear”. The study of these topics also involved the development of computational tools to automate and facilitate the strain calculations. The formation mechanisms of several structures associated with folding were likewise examined, including crenulation cleavage, so-called “saw-tooth structures”, curved veins related to folds, and cleavage in pre-existing folds. Finally, we evaluate the impact of this body of work, its limitations, and future prospects for extending the research, particularly by using the results obtained as a foundation for dynamic modelling.

On the 50th anniversary of the publication of the book Folding and Fracturing of Rocks by John G. Ramsay we use an analysis of citations to assess the book's impact on research in the field of structural geology. Five topics dealt with in the book have had special success in terms of the number of citations: (1) superimposed folding; (2) fold classification based on layer thickness variation; (3) simple shear deformation; (4) the R-f/phi method; and (5) the flexural-slip mechanism of folding. The great impact of this book can be seen in several examples where the book has inspired new lines of research, such as: progressive deformation, strain analysis, shear zones and folding analysis. Ramsay's great merit in writing his book was to have wisely known to apply the quantitative methodology of continuum mechanics to the analysis of the geological structures. We conclude that Folding and Fracturing of Rocks is the most influential textbook in the field of structural geology research.

A field analysis of kink bands developed in slates from three areas (Grandas, Boal and Luarca areas) of the Westasturian-Leonese Zone (Iberian Variscan belt) is presented. The analysis of the main parameters that characterize the geometry of the studied kink bands shows that those of the Grandas and Luarca areas exhibit a different evolution than those of the Boal area. In this latter area, the interlimb angle of the kink bands has lower values than those developed in the former areas and it involves rotation of the foliation inside and outside the band. In the areas with higher bulk shortening associated with the development of kink bands, chevron folds formed by juxtaposition of kink bands. Slip between folia and their rotation was probably the dominant mechanism in the formation of the kink bands, as deduced from the different values of the angle between the kink plane and the foliation inside (φK) and outside (φ) the band, and the occurrence of fractures along the kink planes and small steps between folia cross-cutting these fractures planes. The fractures along the kink planes prevented subsequent hinge migration. Geometrical analysis of kink bands formed by slip between folia and their rotation provides an estimation of the changes in area and thickness, and the strain inside the kink band. For angles of folia rotation ψ < 50°, the ratio between the strain ellipse axes is < 3 inside the band; this ratio is almost independent of the orientation of the kink planes with respect to the foliation outside the band (angle φ).

The geometric evolution associated with the development of crenulation cleavage is analysed. The strong pre-existing anisotropy of the rocks in which this type of cleavage develops, together with the similar or sub-similar geometry of the microfolds, indicates that heterogeneous rotation shear is the main deformation type involved in their formation. In order to mitigate the high volume loss in the microfolds and the lack of a volume increase in the hinge zones that this mechanism entails, a volume correction is introduced, involving the occurrence of a constant stretching in the direction of the axial traces of the microfolds. In addition, a small component of flexural flow can act in the first stages of the microfolding. The analysis has been applied to a case of symmetric crenulation cleavage whose geometry has been fitted to two models. The most suitable of these involves heterogeneous rotation shear with a constant volume correction and implies a variable bulk shortening and volume loss in the different limbs of the microfolds. The values obtained range in most cases between 30 and 50% of bulk shortening and 10 and 30% of bulk volume loss.

We define a simple two-dimensional deformation called "rotation shear". It has one line of no finite longitudinal strain with invariant direction and another one that rotates with the deformation. An analysis of this deformation is carried out. Rotation shear superficially resembles simple shear but the analysis reveals that the two deformations have very different properties. In general, lines deformed by simple shear show a more complex deformation history and undergo greater longitudinal strain, i.e. are more extended, than lines deformed by rotation shear. Rotation shear is used to explain the development of geological structures such as kink bands, ideal similar folds, crenulation and crenulation cleavage and shear zones.

The paleothermometric evolution of the Ponga unit at the core of the Cantabrian Zone, the arcuate foreland fold and thrust belt of the Variscan orogen in NW Spain, is analyzed using the conodont Color Alteration Index and the Kübler Index of illite. The results indicate that the distribution of the thermal maturity patterns is independent of the stratigraphy and the structure of the unit. These include the occurrence of dominant diagenetic values and very low-to-low-grade metamorphism in the southern area. The main thermal event is interpreted to be related to hydrothermal fluids ascending through late-Variscan extensional faults. The anomalous high heat flow in the southern sector of the Ponga unit is part of a previously identified large band in the Cantabrian Zone, where the common occurrence of low-grade metamorphic rocks, hydrothermal dolostones, and ore mineralizations is related to an extensional orogenic episode that started near the Carboniferous–Permian boundary, soon after the closure of the Asturian arc.

A new mechanism, named “heterogeneous rotation shear”, is defined to explain the development of similar folds. It is based on the heterogeneous distribution of a type of deformation in which a line rotates without length change and another maintains constant its direction and length (pure rotation shear) or undergo a constant stretching through the rock (rotation shear with area correction). With these constraints, a kinematical numerical model of similar folds is developed. The mechanism produces a divergent pattern of the major axis directions of the strain ellipses throughout the fold, which does not agree with the usual axial plane cleavage of natural similar folds. The operation of layer parallel shortening is necessary to solve this problem; this shortening agrees with the incompetent character of the rocks where similar folds develop. Eventually fold flattening can modify the fold shape whilst maintaining the similar geometry. The model is applied to explain the development of a natural similar fold, indicating that this could be formed by a combination of layer parallel shortening and heterogeneous rotation shear. Finally, three types of similar or sub-similar folds have been distinguished: a) sub-similar folds formed by intense flattening of parallel folds; b) sub-similar folds formed by simple shear acting on previous perturbations, and c) similar folds produce by layer-parallel shortening and heterogeneous rotation shear.

An outcrop on the Cantabrian coast (Burela section) shows a long train of tight meter-scale folds developed in Cambrian siliciclastic rocks. These folds have been shortened in the axial trace direction on the fold profile, developing a cleavage in the incompetent layers which obliterates the primary cleavage and crosscuts the folds. Several mechanisms have been analyzed to explain the development and attitude of this cleavage, some of them being the same as those that have previously been proposed to form folds but operating in a reverse sense. They are: anti-flexural flow, anti-reverse tangential longitudinal strain and homogeneous strain. The sole operation of these mechanisms cannot explain this cleavage and a new one has been defined with this aim. This mechanism consists of deformation of the incompetent layers by translation of the competent ones (translation mechanism), and it involves an area decrease within the incompetent layers in the fold profile plane and, if there is no important volume decrease, a stretching in the hinge direction that must affect both competent and incompetent layers. The geometrical properties of this mechanism have been analyzed in detail and it is concluded that, combined with a small amount of homogeneous flattening, this mechanism can explain the distribution of the cleavage through the folds.

The tectonothermal evolution of an area located in the core of the Ibero-Armorican Arc (Variscan belt) has been determined by using the conodont colour alteration index (CAI), Kübler index of illite (KI), the Árkai index of chlorite (AI) and the analysis of clay minerals and rock cleavage. The area is part of the Cantabrian Zone (CZ), which represents the foreland fold and thrust belt of the orogen. It has been thrust by several large units of the CZ, what resulted in the generation of a large number of synorogenic Carboniferous sediments. CAI, KI and AI values show an irregular distribution of metamorphic grade, independent of stratigraphic position. Two tectonothermal events have been distinguished in the area. The first one, poorly defined, is mainly located in the northern part. It gave rise to very-low-grade metamorphism in some areas and it was associated with a deformation event that resulted in the emplacement of the last large thrust unit and development of upright folds and associated cleavage (S1). The second tectonothermal event gave rise to low-grade metamorphism and cleavage (S2) crosscutting earlier upright folds in the central, western and southern parts of the study area. The event continued with the intrusion of small igneous rock bodies, which gave rise to contact metamorphism and hydrothermal alteration. This event was linked to an extensional episode due to a gravitational instability at the end of the Variscan deformation. This tectonothermal evolution occurred during the Gzhelian–Sakmarian. Subsequently, several hydrothermal episodes took place and local crenulation cleavage developed during the Alpine deformation.

The tectonothermal evolution of an area located in the core of the Ibero-Armorican arc (Variscan belt) 7 has been determined by using the conodont color alteration index (CAI), Kübler index of illite (KI), the Árkai 8 index of chlorite (AI), and the analysis of clay minerals and rock cleavage. The area is part of the Cantabrian 9 Zone (CZ), which represents the foreland fold and thrust belt of the orogen. It has been thrust by several large 10 units of the CZ, what resulted in the generation of a large amount of synorogenic Carboniferous sediments. CAI, 11 KI and AI values show an irregular distribution of metamorphic grade, independent of stratigraphic position. 12 Two tectonothermal events have been distinguished in the area. The first one, poorly defined, is mainly located 13 in the northern part. It gave rise to very low-grade metamorphism in some areas and it was associated with a 14 deformation event that resulted in the emplacement of the last large thrust unit and development of upright folds 15 and associated cleavage (S1).The second tectonothermal event gave rise to low-grade metamorphism and 16 cleavage (S2) crosscutting earlier upright folds in the central, western and southern parts of the study area. The 17 event continued with the intrusion of small igneous rock bodies, which gave rise to contact metamorphism and 18 hydrothermal alteration. The second event was linked to an extensional episode due to a gravitational instability 19 at the end of the Variscan deformation. This tectonothermal evolution occurred during the Gzhelian-Sakmarian. 20 Subsequently, several hydrothermal episodes took place, in association with local development of crenulation 21 cleavage during the Alpine deformation. 22

StrainModeler is a program constructed in the Mathematica™ environment that performs 3D progressive strain calculations for lines and planes undergoing any sequence of homogeneous deformations. The main inputs to the system define the initial line or plane to be deformed and the deformation sequence to be applied, including combinations of simple shear, pure shear and volume change. For the deformation of lines, the output of the program is the change of attitude of the initial line, which can be represented by graphics or plotted in an equal-area projection. For the deformation of planes, the program has several outputs: (i) change of attitude of the initial plane; (ii) magnitudes and ratio of the semi-axes of the strain ellipse on the deformed plane; (iii) orientation of the major and minor axes of the strain ellipse on the deformed plane; (iv) orientations of the axial planes of the folds formed on the deformed plane, and (v) area change on the deformed plane. The variation of any of these parameters can be shown against a linear parameter only linked to the number of steps involved in the deformation, as a kind of “time” line, or it can be shown against the variation of a parameter of the strain ellipsoid (e. g.: major axis/minor axis ratio). A sequence of directions can be also visualized as a curve in an equal-area plot. Three applications of the program are presented. In the first, the deformation by simple shear of a plane with any orientation is analyzed. In the second, we explore the formation of recumbent folds in layers with different initial orientations for simple shear and pure shear deformations. In the third, we use StrainModeler to analyze the deformation of a set of folds located in a ductile shear zone in the Variscan Belt of NW Spain.

This review has two main parts. The first of them presents existing ideas and data related to recumbent folds, reviewing aspects such as the physical conditions of the development of these folds, the strain inside the folded layers, the kinematic mechanisms of their formation, the role of gravitational forces, the tectonic context of their development and the structures associated with them. In the second part, the above ideas are discussed and possible mechanisms for the development of these folds are presented. It is proposed that initial perturbations of the layers are essential to give rise to the asymmetry of recumbent folds. These perturbations may be non-planarities of the layering or may be linked to the existence of a core or basement of competent rock that hinders the normal propagation of the deformation. This could explain why many large recumbent folds have a root zone.Deformation with an important component of simple shear is a general condition for the formation of recumbent folds. In areas with very low grade metamorphism, competent layers often play an active role during the deformation and undergo buckling with the development of an overturned fold limb, which can be stretched and thinned to finally produce a pair of recumbent folds separated by a thrust. In areas with low or medium metamorphism, buckling under a simple shear regime is probably the most important mechanism for producing large folds with gentle or moderately dipping axial surfaces; subsequent kinematic amplification by coaxial strain components with vertical maximum shortening is important for the formation of recumbent folds. These components involve a sub-horizontal stretching that can cause a problem of strain compatibility and give rise to a basal thrust In areas deformed under high P and T conditions, recumbent folds can develop by flow perturbations and kinematic amplification of folds; this is probably a common mechanism in ductile shear zones. (C) 2014 Elsevier B.V. All rights reserved.

Abstract The tectonothermal evolution of a unit in the foreland fold-and-thrust belt of the Iberian massif is established using the conodont colour alteration index (CAI). The unit consists of two parts with different tectonothermal histories – the Esla nappe region and the Valsurbio region – separated by a synorogenic Carboniferous basin (Guardo–Valderrueda basin). The Esla nappe region evolved in diacaizonal conditions (corresponding to the diagenetic conditions of the pelites) whose palaeotemperatures were controlled by rock burial. Maximum values were reached before the emplacement of the thrust nappes, so tectonic superimposition is not registered by the CAI. Overburial due to the emplacement of the thrust units was prevented by simultaneous intense erosion. The geothermal gradient obtained for burial was c. 35 °C km−1 and the temperature reached by the older Cambrian rocks was c. 210–230 °C. The Valsurbio region was affected by an extensional tectonothermal post-orogenic event that gave rise to metamorphism with ancaizonal or epicaizonal conditions (corresponding to anchizone or epizone of the pelites). The most common maximum palaeotemperatures reached in this event fall within the range 305–415 °C, although higher palaeotemperatures could be reached locally as a consequence of contact metamorphism. This event gave rise to subhorizontal cleavage that cuts the main Variscan folds. Coal rank data indicate an increase in maximum palaeotemperatures eastwards from the Esla nappe region to the Valsurbio region through the Guardo–Valderrueda basin.

Two generation of folds (F-1 and F-2) and associated structures, developed in the Eocene turbidites of the south-central Pyrenees, are analyzed in this paper. F-1 folds are close, have sub-horizontal axes and southwards vergence. They have an associated cleavage S-1. Competent layers were folded by layer-parallel shortening, tangential longitudinal strain, some possible flexural flow and an obliquely superimposed homogeneous strain due mainly to simple shear. Flexural slip is also an important mechanism in the whole multilayer. F-2 folcs are gentle and scarce; they fold the S-1 cleavage.Among the structures associated with F-1 folds, there are sets of veins with curved form in the competent layers. The displacement of each vein gave rise usually to a step in the layer boundary, so that a set of veins produces a structure that is named "saw-tooth structure". The veins initiated as small faults that made flexural slip difficult and gave rise to a concentration of stress on the steps, leading to an opening of the fractures and a propagation of them along a curved path, as suggested by a simple mechanical model. This propagation agrees with finite element models developed by other authors. (C) 2011 Elsevier Ltd. All rights reserved.

Folding during the Variscan deformation in NW Iberia has been analysed from the foreland to the hinterland of the orogen using several geometrical techniques complemented by the numerical simulation of kinematical folding mechanisms with the aid of a computer program (‘FoldModeler’). In the foreland, folds are related to thrusts; in the inner zones, folds can be attributed to three deformation phases. D 1 and D 2 involved deformation with horizontal foreland‐directed displacements. D 1 gave rise to closed or tight folds ( F 1 ) and cleavage ( S 1 ); development of large recumbent F 1 folds in the hinterland required a simple shear regime and a final coaxial strain component with sub‐vertical maximum shortening. Strain incompatibilities at deeper levels, together with high temperatures, favoured the concentration of ductile deformation in shear zones and development of mylonites during D 2 . Local flow instabilities generated F 2 folds that were passively amplified by a combination of simple shear, coaxial strain and area change. D 3 involved a change to a regime with dominant coaxial deformation and a sub‐horizontal maximum shortening; it gave rise to upright or steeply inclined folds ( F 3 ). Development of D 3 structures was heterogeneous and depended on the previous dip of the bedding and S 1 , the presence or emplacement of granitoids, the stacking of thrust sheets or the previous development of large faults bringing into contact rocks with different competence. D 3 structures are mainly concentrated in metapelitic areas and appear distributed in bands on several scales. On a small scale, tectonic banding appears associated with small folds as a result of pressure‐solution processes. In areas with sub‐vertical S 1 , later sub‐horizontal kink bands were formed by a vertical compression. Copyright © 2010 John Wiley & Sons, Ltd.

Hinge lines are loci of high curvature points on folded surfaces. They are significant geometrical features of geological folds, and the arrangement of hinge lines constructed for the surface serves to characterize important aspects of the fold pattern. Since the current definition of hinge line is only appropriate for cylindrical folds, we propose a new definition for use with folds of general shape. Like the concept of ridge lines used in differential geometry, the new definition uses the lines of curvature (principal curvature trajectories) as a reference frame for comparing curvatures across the surface. A hinge line passes through points of extreme principal curvature magnitude observed along the corresponding principal curvature trajectory. Two types of hinge lines are defined and methods for constructing hinge lines are suggested.

Mathematical 2D modelling of asymmetric folds is carried out by applying a combination of different kinematic folding mechanisms: tangential longitudinal strain, flexural flow and homogeneous deformation. The main source of fold asymmetry is discovered to be due to the superimposition of a general homogeneous deformation on buckle folds that typically produces a migration of the hinge point. Forward modelling is performed mathematically using the software 'FoldModeler', by the superimposition of simple shear or a combination of simple shear and irrotational strain on initial buckle folds. The resulting folds are Ramsay class 1C folds, comparable to those formed by symmetric flattening, but with different length of limbs and layer thickness asymmetry. Inverse modelling is made by fitting the natural fold to a computer-simulated fold. A problem of this modelling is the search for the most appropriate homogeneous deformation to be superimposed on the initial fold. A comparative analysis of the irrotational and rotational deformations is made in order to find the deformation which best simulates the shapes and attitudes of natural folds.Modelling of recumbent folds suggests that optimal conditions for their development are: a) buckling in a simple shear regime with a sub-horizontal shear direction and layering gently dipping towards this direction: b) kinematic amplification due to superimposition of a combination of simple shear and irrotational strain with a sub-vertical maximum shortening direction for the latter component. The modelling shows that the amount of homogeneous strain necessary for the development of recumbent folds is much less when an irrotational strain component is superimposed at this stage that when the superimposed strain is only simple shear. In nature, the amount of the irrotational strain component probably increases during the development of the fold as a consequence of the increasing influence of the gravity due to the tectonic superimposition of rocks. (C) 2010 Elsevier Ltd. All rights reserved.