Coaxial Strain Research Articles

Ductility of garnet as an indicator of extremely high temperature deformation

Discoidal shaped garnets (Alm63 Prp33 Grs3 Sps1) occur in the foliation plane of lineated quartzofeldspathic mylonites of the Morin shear zone located along the eastern boundary of the Morin anorthosite complex (Grenville Province, Quebec). The flattened shape of garnets records a component of coaxial strain ( X = Y ⪢ Z), different from the dominant rotational shear strain ( X > Y > Z) in this mylonite. TEM observations indicate dislocation slip and recovery to be the main mechanism for the deformation of garnet. Extrapolation of available experimentally-determined flow laws to natural strain-rates shows an inversion of flow strength between quartz ± feldspar and garnet: garnet, much stronger than quartz and feldspar at temperatures below 700°C, becomes weaker than quartz and feldspar above 900°C. The coaxial ductile strain recorded by the garnets could be produced by a local, uncharacterized deformation, under extremely high temperature conditions at the roof of the rising anorthositic diapir (∼1155 Ma). Subsequent, lower temperature rotational deformation is related to strike-slip movement along the Morin shear zone (∼1020 Ma).

Archean uplift of a subprovince boundary in the Canadian Shield, revealed by magnetic fabrics

A high strain zone, reported elsewhere as a subprovince or tectonic-terrane boundary, shows evidence of uplift during a penetrative D 1 event. Some outcrops reveal this by conventional L-S petrofabric observations. However, all outcrops permit measurements of the anisotropy of low field magnetic susceptibility (AMS) which here records the preferred crystallographic orientation of chlorite, biotite and amphibole. Furthermore, anisotropy of anhysteretic magnetic remanence (AARM) reflects the preferred orientation of ferromagnetic grains (magnetite and pyrrhotite). All three types of fabric show that the rocks were shortened in a north-south direction, perpendicular to the terrane boundary, and stretched upwards in a nearly coaxial strain history. A few anomalous magnetic fabrics may indicate some late D 1 subhorizontal E-W extension, compatible with transpression along the boundary. We conclude that this boundary is a zone of high strain associated with the uplift of the high-grade plutonic rocks of the Winnipeg River subprovince, with only slight evidence of minor components of late transpression during the penetrative D 1 fabrics.

A graphical representation of coaxial plane strains and volume changes

Coaxial strain history, although not representing a common situation in rocks, can provide the opportunity to quantify very important parameters such as the amount of volume change occurring during compaction and tectonic deformation. In the case of coaxial superposition of compaction and tectonic plane strain, a co-ordinate system is defined with the x-y plane lying parallel to bedding and the z axis normal to it, and with the x-z plane corresponding to the plane in which tectonic strain occurs. Bulk strain data are represented on a simple deformation plot with axes 1 + e x , 1 + e x (where e x and e x are the finite extensions along the x and z axes of the co-ordinate system, respectively). With respect to the conventional Flinn diagram, the use of this plot for representing coaxial deformation paths brings several advantages, in that: (1) the magnitudes of the finite strains and their orientation with respect to the chosen co-ordinate system are represented; (2) the volume changes occurring during both compaction and tectonic strain are easily visualized for all the stages of progressive deformation; and (3) the transitions from the oblate to the prolate shape (and vice versa) of the finite strain ellipsoid during progressive deformation are shown continuously, without the artificial zig-zag pattern of the Flinn diagram.

Kinematic analyses of the Hsüehshan Range, Taiwan: A large‐scale pop‐up structure

The Hsüehshan Range, exposed in the northern and central Taiwan slate belt, is a fault‐bounded structural high cored by biotite grade slates and metasandstones. Syntectonic overgrowths in pyrite pressure shadows indicate that much of the eastern Hsüehshan Range experienced coaxial strain histories and that finite strain magnitudes generally increase toward the hinterland. Near the eastern boundary of the Hsüehshan Range, however, pressure shadows record noncoaxial strain histories consistent with a top‐to‐the‐east sense of shear along a steep NW dipping shear zone. This noncoaxiality is attributed to SE directed backthrusting on the Lishan fault, which separates the higher‐grade, Eo‐Oligocene rocks of the Hsüehshan Range from the lower‐grade Miocene rocks of the Backbone Range. Because it is bounded to the east by the SE‐vergent Lishan fault and to the west by a series of NW‐vergent thrusts (e.g., the Chüchih fault), the Hsüehshan Range is envisaged as a pop‐up structure. Strain magnitudes measured from pressure shadows in the coaxial part of the range are consistently lower than those predicted by steady state wedge models that assume all deformation is accommodated by penetrative strain. Departure from the model predictions is attributed primarily to strain localization along discrete fault surfaces (e.g., the Lishan fault). The Hsüehshan Range tapers in width to the south; thus the pop‐up may be buried or die out to the south where the collision is younger.

Simultaneous pure and simple shear: the unifying deformation matrix

Any simultaneous combination of finite simple shear and finite pure shear is a linear transformation which can be expressed as a single transformation matrix. For two dimensions, the matrix is upper triangular with an off-diagonal term, Γ, called the effective shear strain. Γ is a simple function of the pure and simple shear components. For three dimensions, a simultaneous combination of thrusting in the x direction, thrusting in the y direction, and a wrench in the x direction, in addition to 3 orthogonal components of coaxial strain, can also be represented by a 3 × 3, upper triangular matrix. Here, three off-diagonal terms ( Γ xy , Γ x, y , and Γ y, z ) occur. Γ xy is a simple function of the horizontal coaxial strain values and thrusting in the x direction, Γ yz depends on the coaxial strain components in the y and z directions and the thrusting in the y direction, while Γ xz is related to all six strain components. The matrix also allows for volume change, either homogeneously or preferentially in a single direction. A method of decomposing the deformation matrix into a series of incremental deformation matrices, where each incremental deformation records the same kinematic vorticity number as the finite deformation is shown. The orientation and magnitude of the finite-strain ellipsoid (ellipse) is easily and accurately found at any increment during the deformation.

Inter-related plutonism and deformation in South Victoria Land, Antarctica

AbstractThe Lower Palaeozoic Bonney Pluton is a regionally extensive coarse-grained, variably megacrystic, monzodioritic to granitic body that crops out over 1000 km2 in South Victoria Land. It intruded upper amphibolite facies Koettlitz Group metasediments and interlayered orthogneisses. Magmatic fabrics are developed in the centre of the pluton by flow alignment of K-feldspars before the majority of phases had crystallized, whereas solid-state fabrics developed in the pluton margins by ductile–plastic deformation. Structures developed in the host-rocks vary around this elongate northwest–southeast-trending pluton. Upright, tight northwest–southeast-trending macroscopic folds are developed at the sides of the pluton, with axis-parallel stretching lineations and boudinage indicating strong northwest–southeast extension. Broad warps of tight macroscopic folds, and mesoscopic refolded folds, sheath folds and complicated interference patterns characterize areas at the ends of the pluton. Emplacement of the pluton involved radial expansion in a regional northeast–southwest compression, and growth predominantly in the northwest–southeast direction. Superposition of the radial expansion and regional compression resulted in an inhomogeneous strain field at a regional scale, with coaxial strain at the sides of the pluton and non-coaxial strain at the ends. Upright folds developed at the pluton's sides, and became tighter with continued coaxial deformation. Non-coaxial structures developed at the ends of the pluton and were pushed aside by the growing pluton into areas of coaxial deformation, resulting in complex folding, re-folding and sheath folds. Metamorphism of the host-rocks and migmatite development was more intense at the sides of the pluton than near the ends, possibly due to different P-T-t paths of host-rocks around syntectonic plutons.

Tectonic evolution of the northeast portion of the Archean Abitibi greenstone belt, Chibougamau area, Quebec

The Chibougamau area, occupying the northeastern part of the Abitibi greenstone belt is a large synclinorium of volcanic and sedimentary rocks enclosed within tonalitic gneisses. Several east–west–trending regional folds within this synclinorium are responsible for the vertical attitude of the strata. Synclinal structures, with youngest sediments within the core, possess axial-plane schistosity. Anticlines, on the other hand, either form domes with a core occupied by earlier tonalitic to dioritic plutons or are transected by a series of east–west-trending ductile faults (the Waconichi tectonic zone).An early deformation phase of low intensity (D1) generated broad, north–south folds without schistosity. The subsequent regional deformation, event D2, produced the large east–west folds. These deformations, in combination, produced the regional interference pattern of domes and basins. North–south horizontal shortening generated an east–west-trending schistosity associated with a vertical stretching lineation. Regional deformation at its climax produced a tightening of folds and rotation of fold axes parallel to the stretching lineation.Plutons deflected the regional east–west schistosity and formed concentric trajectories associated with "contact-strain aureoles." This produced small interaction zones or triple points characterized by strong vertical extension. These relations suggest an interference between a regional stress field, which produced north–south horizontal shortening, and local stress fields, controlled or deflected by granitoid plutons acting as competent bodies.East–west-trending ductile shear zones represent the final stage of the regional deformation. The observed northward and southward reverse movement along these east–west faults, their parallelism to the axial trace of folds, and the regional schistosity are probable evidence of a regime dominated by a coaxial strain.

Fabric and kinematic study of the Bíteš orthogneiss (southwestern Moravia): Result of large-scale northeastward shearing parallel to the Moldanubian/Moravian boundary

The body of the Bíteš orthogneiss (BO) in southwestern Moravia was affected by ductile shear deformation of probably late Variscan age. North to northeast-directed shearing is indicated by the orientation of sheath fold axes, S- C fabric, strong asymmetrical c-axis preferred orientation and NE-dipping shear bands. Deformational structures (stretching lineation, mylonitic foliation and sheath folds) associated with thrust-wrench tectonics are connected with lower amphibolite to green schist facies metamorphism. A strain study shows complex fabric development from the core to the boundaries of the BO thrust sheet. Fabrics in the core of the BO body are characterized by oblate strain ellipsoids, low strain intensity and quartz c-axis fabrics that can be approximated to small circle girdles centred around the Z-axis of finite strain. Strain ellipsoids near the boundaries of the BO thrust sheet are mostly prolate with high strain intensity and quartz diagrams show single or incomplete crossed girdles highly asymmetrical with respect to the structural framework. The coaxial strain history in the core, and the noncoaxial history which prevails at the boundaries of the BO thrust sheet, are interpreted as a result of different strain paths in the framework of a crustal-scale shear zone.

Experimental calcite fabrics in a synthetic weaker aggregate by coaxial and non-coaxial deformation

Calcite grain aggregates, supported by a weaker, macroscopically ductile, matrix of set Portland cement, were deformed triaxially at room temperature with a confining pressure of 150 MPa and at rates corresponding to a pure shear, natural strain rate of 10 −5 s −1. Extensive grain rotation accompanied the twinning of calcite. Stress orientations inside the specimens, determined from calcite twins, agree well with the stresses imposed upon the specimens. Thus calcite twins are sound kinematic indicators in this study. Various sample configurations were used to simulate pure shear, transpression and simple shear. In pure shear or coaxial strain, preferred dimensional orientations ( PDO) of calcite are produced more efficiently in the presence of high pore fluid pressure. Those grain alignments are stronger than the bulk strain would predict assuming homogeneous strain. Strain is overestimated by methods assuming continuum behaviour because they fail to take into account the extensive interparticle motions that occur in the presence of high pore fluid pressure. Fluid pressure suppresses the twinning of calcite by reducing the intergranular stresses. During deformation, small rapid variations in pore fluid pressure are believed to represent ephemeral dilatations accompanying particulate flow as groups of grains slide past one another. In dry specimens, pure shear is more effective than simple shear in producing a PDO of the calcite grains. Calcite grains are rotated rigidly and somewhat strained to produce L < S alignment fabrics in pure shear but give S-fabrics in transpression and perhaps also in simple shear. Thus grain-shape fabrics do not conform to the symmetry of bulk deformation in this study. Mean grain alignment is perpendicular to shortening in pure shear, initially inclined and later parallel to shear zone walls in transpression, and weak but statistically inclined to shear zone walls in simple shear. The mean orientation of grain-alignment fabrics is therefore a reliable kinematic indicator under the conditions investigated. At comparable strains in dry tests, e-twinning is best developed in transpression, next best developed in coaxial strain and least well shown by simple shear. High fluid pressure considerably reduces the incidence of twinning.

An early history of pure shear in the upper plate of the raft river metamorphic core complex: black pine mountains, southern Idaho

Although commonly obscured by simple shear, pure shear fabrics occur locally within many metamorphic core complexes. The cover rocks to the Raft River metamorphic core complex exposed within the Black Pine Mountains display an early coaxial strain history which developed prior to the formation of low-angle fault-bounded allochthons. At higher structural levels this is documented by pressure shadows with straight sutures, and oppositely-rotated antitaxial calcite veins. These rocks were sub-horizontally extended parallel to bedding, by up to 160%, in an ≈ E-W direction. The marbles of the lowest exposed levels exhibit well-developed crystallographic preferred orientation fabrics. Calcite c- and a-axis pole figures are symmetric to slightly asymmetric, with obliquities ranging from 0° to 8°, which indicate that the percent pure shear ranges between 100 and 76%. Initial constraints on the timing of this layer-parallel extension suggest that extension was occurring locally within the Sevier belt hinterland during the late Cretaceous. These data, coupled with previous studies, suggest that pure shear might be a much more common component of strain in rocks deformed in extensional environments than is generally acknowledged.

Calcite fabrics in the hinge zones of experimentally folded single layers of marble and limestone

The preferred orientations of poles to (0006) and (10 1 4) planes in calcite have been studied in the inner and outer arcs of experimentally produced single-layer buckle folds in Carrara marble and Solnhofen limestone. The textures produced in the medium-grained marble differ from that of the fine-grained limestone. Further, the pole figures from the inner fold arcs are characterized by a general orthorhombic symmetry and a strong preferred orientation of the crystallographic fabric elements, whereas that from the outer fold arcs generally exhibits a triclinic symmetry and a weak preferred orientation. The difference in the pole figures in medium-grained marble and fine-grained limestone is attributed to the mechanism of internal deformation in calcite. The orthorhombic symmetry of the pole figures from the inner fold arc suggests a coaxial strain history in this zone, whereas the triclinic symmetry of the fabric diagrams from the outer fold arcs is attributed to a non-coaxial strain history.

The Tarskavaig Nappe of Skye, northwest Scotland: a re-examination of the fabrics and their kinematic significance

AbstractEarly petrofabric studies of quartz c axis preferred orientation within the Tarskavaig Nappe, located at the southern end of the Moine thrust zone, have been interpreted as indicating that structures within the nappe have been produced by tectonic movements acting at right angles to those responsible for formation of structures within the overlying, but immediately adjacent, Moine Nappe. Such a dramatic contrast in inferred transport direction has not been recognized within the rest of the thrust zone.Re-examination of the microstructures and c axis fabrics within mylonitic metasediments from the western part of the Tarskavaig Nappe has revealed that these tectonites are characterized by maximum principal extension axes which trend sub-parallel to the WNW-trending Moine thrust zone transport direction. Later folding has reorientated and modified these structures within the eastern part of the nappe producing fabrics which have erroneously been taken to indicate a transport direction orientated sub-perpendicular to that of the rest of the Moine thrust zone.WNW-trending maximum principal extension axes and WNW-directed overthrust senses are indicated by microstructures within schists from the adjacent Moine Nappe, suggesting that structures within both the Tarskavaig and Moine nappes may be associated with a common west-northwest transport direction.Essentially coaxial (pure shear) strain paths (indicated by symmetrical c axis fabrics) dominate the internal parts of the Tarskavaig Nappe. Close to the base of the nappe, non-coaxial strain paths, originally associated with WNW-directed overthrusting, are indicated by asymmetrical c axis fabrics and oblique grain shape alignments. Strain compatibility considerations indicate that the lower kinematic domain (which must either be contemporaneous with, or later than, formation of the upper domain) must be characterized by a combination of pure and simple shear deformation.

Ductile strain and metamorphism in an extensional tectonic setting: a case study from the northern Snake Range, Nevada, USA

Summary In recent years considerable attention has focused on metamorphic core complexes of the Basin and Range Province of the western US Cordillera. These highly extended areas are characterized by an upper plate that has been brittlely attenuated by normal faults separated by a sub-horizontal detachment surface from a lower plate that has been ductilely thinned and stretched. A study of mesoscopic structures, finite strain, microstructures, quartz c -axis fabrics and 40 Ar/ 39 Ar geochronology was undertaken in order to characterize the nature, geometry, kinematic history and timing of ductile extension in the lower plate of the northern Snake Range metamorphic core complex in E-central Nevada. These data provide new insights into the processes of deep-seated ductile strain beneath supracrustal normal fault mosaics in highly extended regions. Mesoscopic structures and finite strain analyses indicate that the lower plate underwent plane strain with a sub-vertical Z-axis and a sub-horizontal WNW-ESE X -axis. The magnitude of strain increases dramatically from W-to-E, and is constant vertically. Bedding and foliation are everywhere parallel and bedding is thinned 30–90%. The nature and geometry of microstructures and c -axis fabrics changes progressively from W-to-E. Lower strain rocks on the W flank of the range are characterized by a single bedding-parallel foliation, defined by flattened detrital grains, globular grains which have their c -axes parallel to the Z -axis and symmetrical cross-girdle c -axis fabrics. Further E to slightly higher strains, the rocks are characterized by asymmetrical cross-girdle c -axis fabrics, although globular grains with their c -axes parallel to Z are still present. The high-strain rocks on the E flank of the range are characterized by C -planes, asymmetrical mica ‘fish’ and oblique quartz foliations, and asymmetrical single-girdle c -axis fabrics. These data suggest that the lower plate has not deformed entirely by either coaxial or non-coaxial strain, and we propose an evolutionary model whereby the lower plate deformed during an early period of coaxial strain followed by a later component of non-coaxial strain accompanied by coaxial strain on the E flank of the range. 40 Ar/ 39 Ar geochronology on lower-plate hornblendes and micas indicates that temperatures of deformation were &gt;280°C, but &lt;530°C, and increased with depth. Geochronological data suggest that lower-plate ductile strain is post-latest Cretaceous, and biotite and muscovite data indicate that ductile deformation was probably ongoing 22–26 Ma. Our data and regional geological relations suggest that ductile extensional deformation in the northern Snake Range occurred because of localized heat input to shallow levels of the crust, and together with seismic reflection data, indicate that lower-plate rocks probably represent the top of a regional metamorphic terrane of Tertiary age.

Relationships between strain and quartz crystallographic fabrics in the Roche Maurice quartzites of Plougastel, western Brittany

An integrated microstructural and petrofabric study of the plastically deformed and partially recrystallized Roche Maurice quartzites of Plougastel, western Brittany, has revealed a clear correlation between the pattern of c-axis fabrics displayed by detrital quartz grains and the symmetry of the calculated strain ellipsoid. In specimens with flattening ( k = 0) strains, c axes lie on a small circle girdle (opening angle 28–42°) centred about the principal finite shortening direction (Z). For specimens that exhibit approximate plane strain ( k = 1), cross-girdle c-axis fabrics consisting of a small circle girdle centred about Z and connected through the intermediate principal extension direction ( Y) were detected. Within individual specimens c-axis fabrics of syntectonically recrystallized new quartz grains within the matrix are similar to those of detrital quartz grains. c axes of new grains located within the relatively undeformed sections of the host detrital grains are commonly orientated at angles between 10 and 40° to the host c axis and are, in addition, statistically orientated at a higher angle to Z than their host c axes. These relationships are interpreted as indicating that both host grain control and the local strain (and/or stress) field may have influenced the process of recrystallization; the relative influence of these factors is, however, unknown. Microstructural and petrofabric studies indicate that the Roche Maurice quartzites have been subjected to essentially coaxial strain histories. The role of syntectonic recrystallization in facilitating continued plastic deformation in quartzites subjected to such strain histories is considered.

Plastic behavior of dual phase steel following plane-strain deformation

Dual-phase, high-strength steel sheet has been prestrained in plane-strain tension. Residual hardening and ductility properties were evaluated by performing subsequent uniaxial tensile tests in either co-axial or noncoaxial principal strain axis orientations. In contrast to similar work on aluminum-killed 1008/1010 steel sheet, only minor changes were found in the subsequent flow behavior of dual-phase steel, and no significant difference was found between the two orientations. The small effect of an abrupt strain path change observed in this study is consistent with the low work hardening rate of this alloy.

Horizontal cleavage in southeastern Sinai: the case for a coaxial strain history

A horizontal cleavage and associated lineation are developed in a low-pressure igneous-related metamorphic terrain in southestern Sinai. The cleavage is axial planar to recumbent folds, which are never large, and varies from a smooth slaty cleavage to a discrete crenulation cleavage. Structural evidence on the macro, meso and microscales suggests that cleavage and lineation were formed during irrotational extension and not by simple shear. Estimates of strain using the March method indicate 50–70% vertical shortening. This structural evidence when combined with metamorphic and age data strongly suggests that the cleavage was formed by the forceful emplacement of a pluton at depth.

Heterogeneous strain of a phyllite as revealed by porphyroblast-matrix relationships

Garnet porphyroblasts in a homogeneous phyllite specimen from the central Norwegian Caledonides provide insight into the distribution and type of strain which followed the garnet growth. Matrix schistosity is traceable through the porphyroblasts in the form of inclusions which allow the measurement of two parameters: the flattening of the matrix schistosity around the porphyroblasts, and an angle of rotation of the porphyroblasts relative to the matrix schistosity. Both parameters vary considerably as studied in a large thin section cut parallel to mineral lineation and perpendicular to schistosity. Accepting some simplifications and assumptions, it seems necessary to consider the strain as composed of two components: a noncoaxial strain component of a simple-shear type, and an approximately coaxial strain component, both of which are heterogeneously developed on the scale of the thin section. The shear planes of the simple-shear-type strain are likely to lie parallel, or at a very small angle, to the matrix schistosity. The linear fabric of the phyllite seems to be a material expression of the coaxial strain component. The λ 1 λ 2 plane of the finite strain ellipsoid most probably varies in orientation across the thin section and is only, by coincidence, parallel to the almost constantly oriented matrix schistosity.

The deformation matrix and the deformation ellipsoid

Homogeneous strain can be computed most easily by the methods of matrix algebra. Lines, planes and ellipsoids represented in matrix form can be homogeneously deformed by simple matrix multiplication by linear transformation matrices, the elements of which are the coefficients of the transformation equations. Deformation matrices or linear transformation matrices which cause geological-type homogeneous strain are divided into four classes based on the presence or absence of symmetry and/or orthogonality. The nature of the homogeneous strain caused by each class of deformation matrix is examined. Orthogonal-symmetrical and orthogonal matrices cause rotation. Symmetrical matrices cause irrotational strain with co-axial strain as a special case. Matrices which are neither orthogonal nor symmetrical cause many different types of rotational strain, some of which are examined.

Transected folds: A study illustrated with examples from Canada and Scotland

Transected folds have contemporaneous cleavage that is not parallel to their axial surface, but cuts through the axial surface and both limbs with the same sense. Such folds can develop with perfectly synchronous formation of folds and cleavage. This may occur in rocks undergoing a bulk coaxial strain history or in an approximately coaxial phase of a more complex strain history in which the strain axes do rotate relative to the deforming rocks. In a noncoaxial strain history, transected folds may also arise by a different mechanism, in which cleavage formation is slightly delayed relative to fold formation. This may be achieved where the grain-shaping (cleavage-forming), intragranular deformation mechanisms are initially suppressed because rock flow is more easily accommodated by intergranular movement. The temporary local volume increases required by this “grain-boundary sliding” may occur during de-watering. Transected folds may readily occur and have already been described in many areas. Therefore, field mapping techniques assuming a special geometrical relationship between folds and cleavage should be used only where the degree, or absence, of transection of folds by their contemporaneous cleavage has been established.

Distinction Between Coaxial and Noncoaxial Strains in Sheet Metal and Its Implications on the Assessment of Material Formability

Abstract The distinction between coaxial and noncoaxial strains is explained. It is shown that with noncoaxial strains deformation does not necessarily produce a change of shape, if circles and ellipses are used for strain measurements. If materials are tested in exisymmetrical processes, in which the strains are coaxial, and used in unsymmetrical ones, in which the strains are noncoaxial, then the formability will appear to be overestimated. A test for coaxiality, based on the use of squares and parallelograms for the strain measurements, is presented in the text and the formula used is derived in the Appendix.