The deformation matrix and the deformation ellipsoid

doi:10.1016/0191-8141(79)90004-X

Journal of Structural Geology

Volume 1, Issue 4, 1979, Pages 299-307

https://doi.org/10.1016/0191-8141(79)90004-X Get rights and content

Abstract

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.

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Lineation and foliation are common passive markers that develop and then rotate with increasing deformation. Although their rotation during progressive deformation has been shown to be relatively complex, especially for transpression or transtension deformation (Flinn, 1979; Fossen and Tikoff, 1998; Lin et al., 1998; Jiang, 2014), there are some general assumptions that can be made regarding the orientation of lineation and foliation during progressive homogeneous deformation.2 For all types of oblate ellipsoid (flattening strain), poles to foliation on a stereonet should define a circular to ellipsoidal zone around the Z axis and lineations should always remain within the extensional field (Fig. 13C).
Separating the Parautochthonous and Allochthonous belts, the Allochthon Boundary Thrust (ABT) is a major lithotectonic boundary that generally runs parallel with the length of the Grenville Province, but with some transverse segments. In this contribution, we investigate one of these segments, the Manicouagan Imbricate Zone (MIZ), by adding new field-based data supplemented by new U–Pb deformation ages to a compilation of previous research conducted on both sides of the ABT in the region. Our results reveal striking differences between the footwall and hanging wall of the ABT in terms of strain and structures and concludes that the transverse orientation of the ABT is not related to folding. An updated strain analysis allows us to determine that deformation of the MIZ was dominated by a constrictive-type of strain. We build on a previously proposed conceptual model to document ductile extrusion over basement ramps driven by tectonic forcing and lateral density contrast in the upper crust at ca. 990 Ma. The insertion of the hot ductile nappe into partially molten metasediments of the surrounding Parautochthonous Belt was accommodated by constrictive strain within the former and flattening and transverse folding in the latter.
Shear zones – A review
2017, Earth-Science Reviews
Strain in the lithosphere localizes into tabular zones known as shear zones that grow from small outcrop-size individual zones to large composite structures. Nucleation is related to distributed microscale flaws or mesoscale structures such as fractures and dikes, and they soon establish displacement profiles similar to faults. Also similar to faults, they grow in width and length primarily by segment linkage as they accumulate strain and displacement, and this process typically results in shear zone networks. Consequently, mature shear zones are heterogeneous and composite zones characterized by anastomosing patterns and local variations in thickness and finite strain. Kinematic vorticity estimates suggest that most shear zones deviate from simple shear, and even if subsimple shear may be a useful reference model in many cases, finite strain data indicate that many shear zones involve three-dimensional combinations of coaxial and non-coaxial deformation, such as transpression and transtension. Strain geometry and kinematic vorticity can vary significantly within shear zone networks, which makes it difficult to estimate the bulk deformation type for a composite shear zone or shear zone network. However, perhaps the most challenging aspect is that of progressive deformation, i.e. to what extent and how flow parameters change during deformation (non-steady state deformation), which needs to be addressed by a combination of detailed field observations and numerical modeling.
Textures and melt-crystal-gas interactions in granites
2015, Geoscience Frontiers
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In between when L = S, it corresponds to plane strain. The distinction between oblate and prolate objects is the base of strain measurements and interpretation (Flinn, 1979). In igneous felsic intrusions, the average plane intersecting parallel to the K-feldspar, and/or to the biotite subfabric trace determines the foliation plane (Oertel, 1955).
Felsic intrusions present ubiquitous structures. They result from the differential interactions between the magma components (crystal, melt, gas phase) while it flows or when the flow is perturbed by a new magma injection. The most obvious structure consists in fabrics caused by the interactions of rotating grains in a flowing viscous melt. New magma inputs through dikes affect the buk massif flow, considered as global within each mineral facies. A review of the deformation and flow types developing in a magma chamber identifis the patterns that could be expected. It determines their controlling parameters and summarizes the tools for their quantification. Similarly, a brief review of the rheology of a complex multi-phase magma identifies and suggests interactions between the different components. The specific responses each component presents lead to instability development. In particular, the change in vorticity orientation, associated with the switch between monoclinic to triclinic flow is a cause of many instabilities. Those are preferentially local. Illustrations include fabric development, shear zones and flow banding. They depend of the underlying rheology of interacting magmas. Dikes, enclaves, schlieren and ladder dikes result from the interactions between the magma components and changing boundary conditions. Orbicules, pegmatites, unidirectional solidification textures and miarolitic cavities result from the interaction of the melt with a gaseous phase. The illustrations examine what is relevant to the bulk flow, local structures or boundary conditions. In each case a field observation illustrates the instability. The discussion reformulates instability observations, suggesting new trails for ther description and interpretation in terms of local departure to a bulk flow. A brief look at larger structures and at their evolution tries to relate these instabilities on a broader scale. The helical structures of the Říčany pluton, Czech Republic and by the multiple granitic intrusions of Dolbel, Niger illustrate such events.
An inverse approach to constraining strain and vorticity using rigid clast shape preferred orientation data
2014, Journal of Structural Geology
We describe a three-part approach for modeling shape preferred orientation (SPO) data of spheroidal clasts. The first part consists of criteria to determine whether a given SPO and clast shape are compatible. The second part is an algorithm for randomly generating spheroid populations that match a prescribed SPO and clast shape. In the third part, numerical optimization software is used to infer deformation from spheroid populations, by finding the deformation that returns a set of post-deformation spheroids to a minimally anisotropic initial configuration. Two numerical experiments explore the strengths and weaknesses of this approach, while giving information about the sensitivity of the model to noise in data. In monoclinic transpression of oblate rigid spheroids, the model is found to constrain the shortening component but not the simple shear component. This modeling approach is applied to previously published SPO data from the western Idaho shear zone, a monoclinic transpressional zone that deformed a feldspar megacrystic gneiss. Results suggest at most 5 km of shortening, as well as pre-deformation SPO fabric. The shortening estimate is corroborated by a second model that assumes no pre-deformation fabric.
Evaluation of transtension and transpression within contractional fault steps: Comparing kinematic and mechanical models to field data
2014, Journal of Structural Geology
Citation Excerpt :
That the material line segment is infinitesimal and that homogeneity of deformation only applies in the infinitesimal neighborhood of the particle does not restrict the magnitudes of the components of F – they may describe finite deformation. In the structural geology literature, a special case of the deformation gradient tensor has been referred to as the “deformation matrix” and is given the symbol D (Flinn, 1979; Fossen and Tikoff, 1993; Tikoff and Fossen, 1993). The application of D departs from that of F in several ways.
Rock deformation often is investigated using kinematic and/or mechanical models. Here we provide a direct comparison of these modeling techniques in the context of a deformed dike within a meter-scale contractional fault step. The kinematic models consider two possible shear plane orientations and various modes of deformation (simple shear, transtension, transpression), while the mechanical model uses the finite element method and assumes elastoplastic constitutive behavior. The results for the kinematic and mechanical models are directly compared using the modeled maximum and minimum principal stretches. The kinematic analysis indicates that the contractional step may be classified as either transtensional or transpressional depending on the modeled shear plane orientation, suggesting that these terms may be inappropriate descriptors of step-related deformation. While the kinematic models do an acceptable job of depicting the change in dike shape and orientation, they are restricted to a prescribed homogeneous deformation. In contrast, the mechanical model allows for heterogeneous deformation within the step to accurately represent the deformation. The ability to characterize heterogeneous deformation and include fault slip – not as a prescription, but as a solution to the governing equations of motion – represents a significant advantage of the mechanical model over the kinematic models.
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2013, Journal of Structural Geology
The dynamic theory of deformable ellipsoidal inclusions in slow viscous flows was worked out by J.D. Eshelby in the 1950s, and further developed and applied by various authors. We describe three approaches to computing Eshelby's ellipsoid dynamics and other homogeneous deformations. The most sophisticated of our methods uses differential-geometric techniques on Lie groups. This Lie group method is faster and more precise than earlier methods, and perfectly preserves certain geometric properties of the ellipsoids, including volume. We apply our method to the analysis of naturally deformed clasts from the Gem Lake shear zone in the Sierra Nevada mountains of California, USA. This application demonstrates how, given three-dimensional strain data, we can solve simultaneously for best-fit bulk kinematics of the shear zone, as well as relative viscosities of clasts and matrix rocks.

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The deformation matrix and the deformation ellipsoid

Abstract

The construction and computation of three-dimensional progressive deformations

J. geol. Soc. Lond.

The computation of deformations

Geol. J.

Latent Roots and Latent Vectors