Progressive Strain Localization Research Articles

Finite Element Analysis of Progressive Failure and Strain Localization of Carbon Fiber/Epoxy Composite Laminates by ABAQUS

Interaction mechanism between the intralaminar damage and interlaminar delamination of composite laminates is always a challenging issue. It is important to consider the progressive failure and strain softening behaviors simultaneously during the damage modeling and numerical simulation of composites using FEA. This paper performs three-dimensional finite element analysis of the progressive failure and strain localization of composites using FEA. An intralaminar progressive failure model based on the strain components is proposed and the nonlinear cohesive model is used to predict the delamination growth. In particular, the nonlocal integral theory which introduces a length scale into the governing equations is used to regularize the strain localization problems of composite structures. Special finite element codes are developed using ABAQUS to predict the intralaminar and interlaminar damage evolution of composites simultaneously. The carbon fiber/epoxy composite laminates with a central hole demonstrates the developed theoretical models and numerical algorithm by discussing the effects of the mesh sizes and layups patterns. It is shown the strain localization problem can be well solved in the progressive failure analysis of composites when the energy dissipation due to the damage of the fiber, matrix and interface occurs at a relatively wide area.

Localized vs distributed deformation associated with the linkage history of an active normal fault, Whakatane Graben, New Zealand

The deformation associated with an active normal fault is investigated at a high temporal resolution (c. 104 yr). The Rangitaiki Fault (Whakatane Graben, New Zealand) and its adjacent faults accommodated an overall extension of ∼0.83% oriented at ∼N324°E over the past 17 kyr. This is consistent along strike, but the pattern of faulting that accommodates this strain defines two different spatial domains. To the SW, one domain is characterized by a few large faults, with >80% of strain localized onto geometrically and kinematically linked segments of the main fault. This produces marked heterogeneity in the spatial distribution of strain across the graben. In contrast, to the NE, a domain of distributed faulting is characterized by numerous small faults contributing to the overall deformation, with only ∼35% of strain localized onto the Rangitaiki Fault. The transition from distributed to localized deformation is attributed to an increase in linkage maturity of the Rangitaiki Fault. Progressive strain localization has been ongoing within the network over the last 17 kyr, with localization of fault activity increasing by ∼12%, indicating this process occurs over kyr time periods that only reflect a few earthquake events.

Microscopic Strain and Crystal Rotation Measurement within Metallurgical Grains

This work describes an experimental procedure to measure the progressive strain localization and crystal lattice rotation within metallurgical grains. A digital image correlation software was implemented and associated with mechanical tests carried out inside a scanning electron microscope on specimens exhibiting nanometric grainy patterns. Cross-correlation analyzes between electron backscattering diffraction maps were also developed to quantify the corresponding local crystal rotation relative to the original structure. The microscale strain and rotation fields on the surface of a tensile-loaded specimen made of austenitic stainless steel 316L are presented as an illustration. Their direct spatial correlation between strain heterogeneities and the progressive activation of slip systems is put into evidence and discussed.

Failure-Oriented Multi-scale Variational Formulation: Micro-structures with nucleation and evolution of softening bands

This contribution presents the theoretical foundations of a Failure-Oriented Multi-scale variational Formulation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phenomena. The multi-scale model considers two coupled mechanical problems at different physical length scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linked to a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homogenization of the micro-mechanical problem.At the macroscopic level, the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled.The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical information transference from the macro-to-micro scales during the complete loading history, even in those points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-models consistently coupled:(i)a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response.(ii)A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated, for modeling the macro-crack evolution.When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a change in the rule that defines how the increments of generalized macro-strains are inserted into the micro-scale and (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures are obtained. Then, as a consequence of the Hill–Mandel Variational Principle and the proposed kinematical assumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuum part of the body, as well as, for the traction acting on the macro-discontinuity surface.The assumed macro-to-micro mechanism of kinematical coupling defines a specific admissible RVE-displacement space, which is obtained by incorporating additional boundary conditions, Non-Standard Boundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Boundary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we call through the paper “objectivity” of the homogenized constitutive response.Several numerical examples are presented showing the objectivity of the formulation, as well as, the capabilities of the new multi-scale approach to model material failure problems.

Synkinematic magmatism, heterogeneous deformation, and progressive strain localization in a strike‐slip shear zone: The case of the right‐lateral Karakorum fault

New structural observations coupled with 15 U/Pb and 24 Ar/Ar new ages from the Karakorum shear zone (KSZ) constrain the timing and slip rate of the right‐lateral Karakorum fault zone (KFZ), one of the great continental Asian strike‐slip faults. In the Tangtse‐Darbuk area, the Tangtse (SW) and Muglib (NE) mylonitic strands of the KSZ frame the less deformed Pangong Range. Inherited U/Pb ages show that granitic protoliths are mostly from the Karakorum and Ladakh batoliths, with a major Miocene melting event lasting from ≥21.5 to 13.5 Ma. Some of the Miocene granitic bodies show structural evidence for intrusion synkinematic to the KSZ. The oldest of these granitoids is 18.8 ± 0.4 Ma old, implying that deformation started prior to ∼19 Ma. Microstructural data show that right‐lateral deformation pursued during cooling. Ar/Ar data show that ductile deformation stopped earlier in the Tangtse (∼11 Ma) than in the Muglib strand (∼7 Ma). Deformation ended at ∼11 Ma in the Tangtse strand while it is still active in the Muglib strand, suggesting a progressive localization of deformation. When merged with published observations along the KFZ, these data suggest that the KFZ nucleated in the North Ayilari range at least ∼22 Ma ago. The long‐term fault rate is 0.84 to 1.3 cm/yr, considering a total offset of 200 to 240 km. The KSZ collected magma produced within the shear zone and/or deeper in crust for which the producing mechanism stays unclear but was not the lower crustal channel flow.

Evolution of a polymineralic mantle shear zone and the role of second phases in the localization of deformation

[1] The influence of second phases (e.g., pyroxenes) on olivine grain size was studied by quantitative microfabric analyses of samples of the Hilti massif mantle shear zone (Semail ophiolite, Oman). The microstructures range from porphyroclastic tectonites to ultramylonites, from outside to the center of the shear zone. Starting at conditions of ridge-related flow, they formed under continuous cooling leading to progressive strain localization. The dependence of the average olivine grain size on the second-phase content can be split into a second-phase controlled and a dynamic recrystallization–controlled field. In the former, the olivine grain size is related to the ratio between the second-phase grain size and volume fraction (Zener parameter). In the latter, dynamic recrystallization manifested by a balance between grain growth and grain size reduction processes yields a stable olivine grain size. In both fields the average olivine and second-phase grain size decreases with decreasing temperature. Combining the microstructural information with deformation mechanism maps suggests that the porphyroclastic tectonites (∼1100°C) and mylonites (∼800°C) formed under the predominance of dislocation creep. Since olivine-rich layers are intercalated with layer parallel, polymineralic bands in the mylonites, nearly equiviscous conditions can be assumed. In the ultramylonites, diffusion creep represents the major deformation mechanism in the polymineralic layers. It is this switch in deformation mechanism from dislocation creep to diffusion creep that forces strain to localize in the fine-grained polymineralic domains at low temperatures (<∼700°C), underlining the role of the second phases on strain localization in cooling mantle rocks.

Progressive strain localization in a major strike‐slip fault exhumed from midseismogenic depths: Structural observations from the Salzach‐Ennstal‐Mariazell‐Puchberg fault system, Austria

Analysis of a strike‐slip fault exhumed from midseismogenic depths reveals that the fault experienced progressive strain localization toward a high‐strain fault core. We focus on the Ennstal segment of the 400‐km‐long Salzach‐Ennstal‐Mariazell‐Puchberg (SEMP) strike‐slip fault system in the Eastern Alps, which accommodated ∼60 km of left lateral displacement during Oligo‐Miocene time. Macroscopic and microscopic observations reveal a zoned fault featuring a high‐strain core at least 10 m wide within a fault zone that is at least 150 m wide. Grain‐size distribution analysis shows how the Ennstal segment of the SEMP evolved. Our data reveal a 10‐m‐wide high‐strain fault core (characterized by a power law relationship of grain sizes, D2 ≈ 2.0) bordered by a 54‐m‐wide “transition zone” where the largest and smallest grains are characterized by two power law relationships (D2 ≈ 2.0 and 1.6, respectively). This zone is in turn bordered by a region with grain sizes that show a single low‐strain power law relationship of D2 ≈ 1.6. We interpret these relationships to be the result of concentrated shear overprinting an initial low‐strain, power law grain‐size distribution before strain localized to the core. This is consistent with the theory that faults mature by smoothing geometrical complexities, forming a highly localized, high‐strain fault core. The data do not support the idea that damage forms primarily in response to dynamic stresses during seismic rupture, although they do suggest that this mechanism may operate within tens of meters of the fault once it has developed its zoned structure.

Discrimination of annealed and dynamic fabrics: Consequences for strain localization and deformation episodes of large-scale shear zones

Large-scale shear zones are often characterized by long lasting deformation with complex strain localization behavior. Based on a drill core across polymineralic carbonate mylonites of the Doldenhorn nappe (Helvetic Alps, Switzerland), the present study uses spatially resolved microstructural variations to discriminate unambiguously deformation microstructures from annealed ones, to infer the deformation mechanisms, and to reconstruct the strain localization history. These goals require quantitative microstructural and textural analyses of sample suites in sections across the shear zone. Particularly the occurrence of two distinct Zener trends, i.e., the relationship between grain size and volume fraction of matrix and second phase grains, allows for the identification of annealed from dynamic microstructures. The data presented here indicate that continued shearing after peak metamorphic conditions and during exhumation-induced cooling resulted in progressive strain localization at the base of the shear zone, while the inactive parts of the formerly high temperature shear zone were progressively overprinted by annealing.

Fluid flow and rock alteration along the Glarus thrust

Chemical alteration of rocks along the Glarus overthrust reflects different stages of fluid rock interaction associated with thrusting. At the base of the Verrucano in the hanging wall of the thrust, sodium was largely removed during an early stage of fluid-rock interaction, which is ascribed to thrust-parallel fluid flow in a damage zone immediately above the thrust. This alteration leads to the formation of white mica at the expense of albite-rich plagioclase and potassium feldspar. This probably enhanced mechanical weakening of the Verrucano base allowing for progressive strain localization. At a later stage of thrusting, fluid-mediated chemical exchange between the footwall and the hanging wall lithologies produced a second generation of alteration phenomena. Reduction of ferric iron oxides at the base of the Verrucano indicates fluid supply from the underlying flysch units in the northern section of the thrust. Fluid supply from the footwall may have kept pore fluid pressure close to lithostatic and enhanced cataclastic deformation. The chemical characteristics of the Lochseiten calc-tectonite suggest its derivation from Mesozoic limestone. In the southern sections of the thrust, the major element and stable isotope compositions show continuous trends from the Cretaceous limestone in the footwall of the thrust up to the contact with the Verrucano, indicating that the calc-tectonite developed due to progressive deformation from the footwall units. In the northern sections of the thrust, the Lochseiten calc-tectonite has a distinct chemical and stable isotope signature, which suggests that it is largely derived from Infrahelvetic slices, i.e. decapitated fragments of the footwall limestone from the southern sections of the thrust, which were tectonically emplaced along the thrust further north. Only at the Lochseiten type locality the original chemical and stable isotope signatures of the calc-tectonite were completely obliterated during intense reworking by dissolution and re-precipitation.

Microstructural evolution within a partitioned midcrustal transpression zone, northeast Greenland Caledonides

The thrust‐related Imbricate Zone and the sinistral strike‐slip Storstrømmen Shear Zone (SSZ) are two major orogen‐parallel ductile shear zones that mark the western limit of intense Caledonian deformation in northeast Greenland and transect the orogenic hinterland, respectively. Detailed microstructural analysis of associated mylonitic fault rocks from Dronning Louise Land and Hertugen af Orléans Land has been carried out to provide insights into the evolution of these structures and to constrain the relative timing of movement. Amphibolite‐facies mylonites were developed across the entire ca. 8‐km‐wide SSZ and are currently exposed across an area ca. 3–4 km wide within the center of the shear zone. In contrast, retrograde greenschist‐facies mylonites are concentrated within a ca. 2‐ to 4‐km‐wide belt along the western margin of the shear zone. This metamorphic pattern is interpreted to reflect progressive localization of strain associated with exhumation and syntectonic cooling. The ca. 15‐km‐wide Imbricate Zone consists of numerous east‐dipping ductile thrust sheets. At low structural levels, mylonitic fabrics indicate that thrust‐related deformation occurred at middle‐greenschist facies. Toward higher levels, however, microstructures suggest that the metamorphic grade progressively increases. We propose that ductile thrusting across the Imbricate Zone was synchronous with uplift and exhumation of the orogenic hinterland and SSZ, accounting for the observed metamorphic variations. This implies that both structures were active broadly contemporaneously during regional partitioned transpression, an inference supported by scattered 40Ar/39Ar and recent U‐Pb zircon ages. The microstructural observations therefore support models that involve sinistrally oblique Silurian‐Devonian collision between Laurentia and Baltica.

THE NORTH ANATOLIAN FAULT: A NEW LOOK

Dedicated to the memory of three pioneers, İhsan Ketin, Sırrı Erinç and Melih Tokay, and a recent student, Aykut Barka, who burnt himself out in pursuit of the mysteries of the North Anatolian Fault.▪ Abstract The North Anatolian Fault (NAF) is a 1200-km-long dextral strike-slip fault zone that formed by progressive strain localization in a generally westerly widening right-lateral keirogen in northern Turkey mostly along an interface juxtaposing subduction-accretion material to its south and older and stiffer continental basements to its north. The NAF formed approximately 13 to 11 Ma ago in the east and propagated westward. It reached the Sea of Marmara no earlier than 200 ka ago, although shear-related deformation in a broad zone there had already commenced in the late Miocene. The fault zone has a very distinct morphological expression and is seismically active. Since the seventeenth century, it has shown cyclical seismic behavior, with century-long cycles beginning in the east and progressing westward. For earlier times, the record is less clear but does indicate a lively seismicity. The twentieth century record has been successfully interpreted in terms of a Coulomb failure model, whereby every earthquake concentrates the shear stress at the western tips of the broken segments leading to westward migration of large earthquakes. The August 17 and November 12, 1999, events have loaded the Marmara segment of the fault, mapped since the 1999 earthquakes, and a major, M ≤ 7.6 event is expected in the next half century with an approximately 50% probability on this segment. Currently, the strain in the Sea of Marmara region is highly asymmetric, with greater strain to the south of the Northern Strand. This is conditioned by the geology, and it is believed that this is generally the case for the entire North Anatolian Fault Zone. What is now needed is a more detailed geological mapping base with detailed paleontology and magnetic stratigraphy in the shear-related basins and more paleomagnetic observations to establish shear-related rotations.

Analysis of progressive failure of pillar and instability criterion based on gradient-dependent plasticity

A mechanical model for strain softening pillar is proposed considering the characteristics of progressive shear failure and strain localization. The pillar undergoes elastic, strain softening and slabbing stages. In the elastic stage, vertical compressive stress and deformation at upper end of pillar are uniform, while in the strain softening stage there appears nonuniform due to occurrence of shear bands, leading to the decrease of load-carrying capacity. In addition, the size of failure zone increases in the strain softening stage and reaches its maximum value when slabbing begins. In the latter two stages, the size of elastic core always decreases. In the slabbing stage, the size of failure zone remains a constant and the pillar becomes thinner. Total deformation of the pillar is derived by linearly elastic Hooke’s law and gradient-dependent plasticity where thickness of localization band is determined according to the characteristic length. Post-peak stiffness is proposed according to analytical solution of averaged compressive stress-average deformation curve. Instability criterion of the pillar and roof strata system is proposed analytically using instability condition given by Salamon. It is found that the constitutive parameters of material of pillar, the geometrical size of pillar and the number of shear bands influence the stability of the system; stress gradient controls the starting time of slabbing, however it has no influence on the post-peak stiffness of the pillar.

Strain localization in the Sea of Marmara: Propagation of the North Anatolian Fault in a now inactive pull‐apart

The Sea of Marmara is presently the site of pure dextral strike‐slip faulting that connects the Izmit Fault in the east to the Ganos Fault in the west along a linear strike‐slip fault that is part of the northwestern branch of the North Anatolian Fault. This active Main Marmara Fault (MMF) cuts across a succession of depocenters and highs. Analysis of a large data set that includes multichannel seismic lines, sparker and deep towed high definition seismics, shows that most of the N120°E trending normal faults framing these depocenters are now inactive as they are sealed by an average thickness of 300 m of sediments. The highs correspond to a series of NE‐SW trending anticlines and reverse faults blanketed by a thin cover of undeformed sediments. We interpret these structures as inverted former transfer faults of the now inactive Sea of Marmara pull‐apart system. Depocenters and highs are the relicts of a former pull‐apart basins and transfer faults system that accommodated the North Anatolian Fault motion within the Sea of Marmara prior to the formation of the MMF. We propose that a relatively short (a few hundreds of thousands of years) period of basin inversion accompanied the localization of strain along the present Main Marmara Fault, until it was fully formed, 200 ± 100 Kyr ago. The preexisting pull‐apart basin system accounts for about 30 km of motion along this fault system and thus probably appeared sometime in late Pliocene or early Pleistocene as a reactivated portion of the old Thrace basin. During the Plio‐Quaternary the Sea of Marmara has been the site of progressive localization of strain.

Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah

Riedel shear structures are common fault patterns identified within shear zones and related to the embryonic stages of fault formation. This study focuses on the geometry of outcrop-scale natural shear zones consisting of different generations of Riedel structures, exposed in the Jurassic Navajo sandstone, Capitol Reef National Park, Utah. Geometric analysis of different structures shows that the spacing of synthetic R-deformation bands increases with the spacing of antithetic R′-deformation bands. Systematic correlation is found between the R-band spacing and the angles formed between R- and R′-bands. Examination of young Riedel structures shows their tendency to localize along narrow, elongated domains sub-parallel to the shear direction and create denser Riedel networks. We suggest that the evolution of Riedel structures is dominated by two mechanisms: (1) discrete faulting in the form of conjugate deformation bands, generally complying with the Mohr–Coulomb criteria, and (2) granular flow, which rotates mainly the R′-deformation bands. Both mechanisms are intensified with progressive strain, decreasing the deformation-band spacing and increasing the R- to R′-angles. The tendency of young Riedel structures to organize in dense elongated networks is related to strain localization during the shear-zone evolution. We suggest a kinematic explanation for the evolution of Riedel-structure networks, which relates the network geometry to the progressive accumulation and localization of shear strain.

Controls on strain localization in a two‐dimensional elastoplastic layer: Insights into size‐frequency scaling of extensional fault populations

We use a two‐dimensional (2‐D) finite element code to investigate how mechanical properties and boundary conditions influence progressive strain localization during extension of a 2‐D elastoplastic layer. The deforming medium is modeled using a strain softening Von Mises rheology, with a Gaussian heterogeneity in yield strength distributed randomly in space. Specifically, we examine the effects of changing (1) the thickness of the deforming layer, (2) the basal boundary condition, (3) the range of strengths in the deforming layer, (4) the strength loss on failure, and (5) the total accommodated strain. Discrete zones of plastic shear strain are observed to nucleate and grow progressively during each experiment. In order to quantify and thus discriminate between the deformation patterns produced under differing conditions, we calculate the size‐frequency distributions of the shear band populations. Size is defined as the total plastic strain represented by each discrete structure. A wide range of size‐frequency distributions is observed, including both power law and exponential as well as distributions that show breaks in scaling with a transition from power law (at small sizes) to exponential (at large sizes). We show that these variations in population statistics are directly reflecting a change in strain localization in space and time caused by changing model parameters. Furthermore, by making an analogy between the model shear bands and tectonic faults, we provide insights into key features of fault size‐frequency distributions that have been measured in continental and oceanic extensional settings and also observed in analogue experiments.

Geomechanics in Reservoir Simulation: Overview of Coupling Methods and Field Case Study

The paper addresses the modeling of geomechanical effects induced by reservoir production and their influence on fluid flow in the reservoir. Geomechanical effects induced by reservoir production can be particularly pronounced in stress sensitive reservoirs, such as poorly compacted reservoirs and fractured reservoirs. The authors review the main coupled mechanisms associated with the production of these reservoirs, and describe the different approaches that can be used to solve the coupling between fluid flow and geomechanical problems. A field case study is then presented. A stress dependent reservoir simulator-ATH2VIS-was used to quantify effects associated with the production of a highly heterogeneous and compartmentalized limestone reservoir. This simulator relies on a partial coupling approach with different time steps for reservoir and geomechanical simulations and manages data exchanges at given time intervals between the ATHOSTM reservoir simulator developed at IFP and the VISAGETM geomechanical simulator (VIPS Ltd. , 2001). The results of the coupled reservoir geomechanical simulations indicate that perturbation of the reservoir mechanical equilibrium specifically leads to progressive strain localization on a limited number of faults. Only specific parts of these faults are critically stressed, depending on pore pressure variations in their vicinity, temperature variations, and fault strikes compared with stress orientation. The normal strains resulting from geomechanical computations are interpreted in terms of permeability variations using a fracture and fault permeability model to improve the dynamic description of fluid flow and history matching.

Geometric controls on the evolution of normal fault systems

Abstract The growth of normal fault arrays is examined in basins where sedimentation rates were higher than fault displacement rates and where fault growth histories are recorded by thickness and displacement variations within syn-faulting sequences. Progressive strain localization is the principal feature of the growth history of normal faults for study areas from the Inner Moray Firth, a sub-basin of the North Sea, and from the Timor Sea, offshore Australia. The kinematics of faulting are similar in both study areas. Fault displacement rates correlate with fault size, where size is measured in terms of either displacement or length. Small faults have higher mortality rates than larger faults throughout the growth of the fault system. Displacement and strain are progressively localized onto the larger faults at the expense of smaller faults at progressively larger scales. Strain localization and the preferential growth of larger faults are attributed to geometric factors, such as size and location, rather than to the mechanical properties of fault rock in individual faults. This conclusion is supported by numerical models that reproduce the main characteristics of fault system growth established from both study areas.

Strain Localization and Undrained Steady State of Sand

Experimental results are presented that characterize the behavior of a loose, fine-grained, water-saturated sand tested under globally undrained conditions is a plane strain apparatus. Together with local measurements of boundary forces and deformations, stereophotogrammetry is used to track the progressive localization of strain. The constitutive behaviors in the tests prior to localization can be characterized as either undrained local softening or undrained load softening and subsequent rehardening. In both cases, a consistent pattern of onset of the formation of the persistent shear band, mobilization of the maximum effective friction, and the complete formation of the band was observed. Conditions approximating steady state only were observed in the shear bands. The effect of strain localization on the conventionally derived steady state line was found to be minimal when undrained load softening behavior was observed, but significantly different steady-state lines were found from results of plane strain and axisymmetric tests. The differences between tests conducted under these two stress conditions assessed to evaluate the practical implications of the observed differences.

Deformation partitioning and foliation reactivation during transpressional orogenesis, an example from the Central Longmen Shan, China

Progressive localization of strain within the Songpan—Garzê Fold Belt, at the eastern margin of the Tibetan Plateau, has resulted in the formation of the Wenchuan—Maowen Shear Zone and preservation of excellent overprinting relationships between three major phases of ductile and semi-ductile deformation. These relationships show that deformation in the eastern Songpan—Garzê Fold Belt evolved from crustal thickening during SW-directed compression ( D 1) to localized sinistral transcurrent shear ( D 2) and finally to increasingly localized high temperature ( D 3 p ) and retrograde ( D 3 r ) reverse shear in response to SE-directed compression. Both D 2 and D 3 deformations were co-planar and reactivation of the S 2 foliation led to the formation of a composite S 2–3 fabric. However, inclusion trails in garnet porphyroblasts preserve evidence of a transition between these strike-slip and reverse shear end members, indicating that D 2 and D 3 were part of an episode of progressive transpressional shear. Metamorphic grade increases from chlorite to kyanite zone within the Wenchuan—Maowen Shear Zone, reaching syn- D 3 p peak conditions of ∼600°C and 10 kbar. The D 3 p , and D 3 r reverse shears were responsible for tectonic exhumation of the shear zone and resulted in the formation of retrograde D 3 r high strain zones along the margins of the Xuelongbao Granite and adjacent to the Wenchuan—Maowen Fault.

A few methods of analytical calculation of forming-limit curves

The main aim of the present work is the modification of analytical calculation methods for forming limit curves. The calculated curves were compared with experimental data for steel and aluminium sheets and the best modifications were chosen. The following modifications were analysed: combination of the Swift diffuse instability criterion developed by Lian with the localized necking M-K theory developed by Jalinier and co-workers. The initial calculation, to some degree of deformation, was performed using: the Swift model and then the Jalinier localized theory until fully localization of the strain was applied; the application of Hill's non-quadratic yield criteria for material with in-plane isotropy to the Jalinier theory; the application of the heterogeneity coefficient described by surface roughness to the Jalinier theory; and the development of the Gillis-Jones theory, which eliminates the necessity of the determination of the initial heterogeneity coefficient. The latter theory is based on the assumption that instability develops during successive cycles of deformation: uniform deformation, progressive strain localization, force drop and fully localized necking. Good correlation was found between theoretically calculated and experimentally determined curves for all modified methods, but the best agreement was for the Jalinier theory with the heterogeneity coefficient described by surface roughness and the Gills-Jones theory. The theories can be used successfully for the CAD of sheet forming processes.