A novel electron backscatter diffraction (EBSD) ‐based finite‐element (FE) wave propagation simulation is presented and applied to investigate seismic anisotropy of peridotite samples. The FE model ...simulates the dynamic propagation of seismic waves along any chosen direction through representative 2D EBSD sections. The numerical model allows separation of the effects of crystallographic preferred orientation (CPO) and shape preferred orientation (SPO). The obtained seismic velocities with respect to specimen orientation are compared with Voigt‐Reuss‐Hill estimates and with laboratory measurements. The results of these three independent methods testify that CPO is the dominant factor controlling seismic anisotropy. Fracture fillings and minor minerals like hornblende only influence the seismic anisotropy if their volume proportion is sufficiently large (up to 23%). The SPO influence is minor compared to the other factors. The presented FE model is discussed with regard to its potential in simulating seismic wave propagation using EBSD data representing natural rock petrofabrics.
Key Points
Our combined approach allows separating different causes for seismic anisotropySeismic anisotropy is dominated by crystallographic preferred orientationThe novel EBSD‐based FE simulation has several advantages discussed in the paper
Geological folds in transpression are inherently 3D structures; hence their growth and rotation behavior is studied using 3D numerical finite-element simulations. Upright single-layer buckle folds in ...Newtonian materials are considered, which grow from an initial point-like perturbation due to a combination of in-plane shortening and shearing (i.e., transpression). The resulting fold growth exhibits three components: (1) fold amplification (vertical), (2) fold elongation (parallel to fold axis), and (3) sequential fold growth (perpendicular to axial plane) of new anti- and synforms adjacent to the initial fold. Generally, the fold growth rates are smaller for shearing-dominated than for shortening-dominated transpression. In spite of the growth rate, the folding behavior is very similar for the different convergence angles. The two lateral directions always exhibit similar growth rates implying that the bulk fold structure occupies an increasing roughly circular area. Fold axes are always parallel to the major horizontal principal strain axis (λ→max, i.e., long axis of the horizontal finite strain ellipse), which is initially also parallel to the major horizontal instantaneous stretching axis (ISA→max). After initiation, the fold axes rotate together with λ→max. Sequential folds appearing later do not initiate parallel to ISA→max, but parallel to λ→max, i.e. parallel to the already existing folds, and also rotate with λ→max. Therefore, fold axes do not correspond to passive material lines and hinge migration takes place as a consequence. The fold axis orientation parallel to λ→max is independent of convergence angle and viscosity ratio. Therefore, a triangular relationship between convergence angle, amount of shortening, and fold axis orientation exists. If two of these values are known, the third can be determined. This relationship is applied to the Zagros fold-and-thrust-belt to estimate the degree of strain partitioning between the Simply Folded Belt and the bordering strike-slip fault-system.
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•3D fold growth comprises amplification, elongation, and sequential growth.•Fold structures maintain a lateral aspect ratio of ≈1, occupying a circular area.•Fold axes initiate and actively rotate with the major principal strain axis, λ→max.•Hinge migration depends on convergence angle α; it is more marked for simple-shear.•The relation between α, strain, and fold orientation is independent of viscosity.
We use the finite element method to simulate slow viscous (Newtonian) flow in two dimensions without gravity and to model asymmetric (S- and Z-shaped) and symmetric (M-shaped) parasitic folds during ...multilayer folding. During multilayer folding, the matrix between stiffer layers shows a deformation close to pure shear in the hinge area and a combination of pure and simple shear in the limb areas. Thinner layers placed between thicker layers develop symmetric parasitic folds in the hinge, and eventually asymmetric parasitic folds in the limbs of the larger fold. Our results verify numerically the theory that asymmetric parasitic folds develop from symmetric buckle-folds that are sheared by the hingeward relative displacement of the thick layers in the limbs of the first-order fold. To develop asymmetric shapes, the amplitudes of the parasitic folds must exceed a critical value before the first-order fold begins to amplify. Otherwise, the parasitic folds are unfolded during flattening that takes place in the limb area between the thick layers. More than five thin layers are necessary to generate distinct asymmetric parasitic folds for the applied model setting. More layers generate higher amplification rates in the thin layers and, hence, higher amplitudes.
To investigate the geometrical relationships between folding and thrust faulting, we built a 3D geological model of the Helvetic fold-and-thrust belt in eastern Switzerland from several existing and ...two newly drawn cross-sections in the Säntis area. We partly redrew existing cross-sections and validated them by checking for line length balance; to fill areas with no data we drew additional cross-sections. The model was built based on surface interpolation of the formation interfaces and thrusts between the cross-sections, which allowed generating eight main surfaces. In addition, we used cave data to validate the final model in depth. The main structural elements in the Säntis area, the Säntis Thrust and the Sax-Schwende Fault, are also implemented in the model. The result is a 3D structural model of the area, which provides an intuitive way for examining a portion of a complex structural nappe. The 3D model highlights the shapes of the main anticline-syncline pairs and how these fold trains vary laterally in amplitude and wavelength. It shows how lateral variations in fold style correlate with regional shortening gradients as determined from line-length balancing. The model also clearly shows the lateral extension, the trend, and the variation in displacement along the principal faults. The reconstruction of horizons in 3D allows the investigation of cross-sections in any given direction. The 3D model is useful for developing and understanding how the internal nappe structures, namely folds and thrust faults, change along strike due to palaeogeographic and stratigraphic variations. Lateral stratigraphy variations correlate with different deformation responses of the nappe. Changes can occur either abruptly across transverse faults or in a more gradual manner.
The neutral line in a buckle fold, dividing areas of outer-arc extension from areas of inner-arc shortening, is a fundamental concept in structural geology. In the past, folds have been constructed ...kinematically from a given neutral line geometry using the tangential longitudinal strain pattern. In this study, a mechanical finite element model is used to numerically buckle single-layer folds with Newtonian and power-law viscous rheology. Two neutral lines can be distinguished, the incremental neutral line (zero layer-parallel strain rate) and the finite neutral line (zero finite layer-parallel strain). The former develops first and migrates through the layer from the outer towards the inner arc ahead of the latter. Both neutral lines are discontinuous along the fold and terminate either at the bottom or top interface of the layer. For decreasing viscosity ratio between layer and matrix and for decreasing initial amplitude, the neutral lines develop later during folding and, for some cases, no neutral line develops. The dynamical behaviour of the neutral lines is similar for Newtonian and power-law viscous rheology if the viscosity ratio is large, but substantially different for small viscosity ratios. The results are discussed in light of fold-related structures, such as outer-arc-extension structures and inner-arc-shortening structures.
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► The neutral lines in Newtonian and power-law viscous single-layer folds are studied. ► Two neutral lines can be defined: the finite and the incremental neutral line. ► During folding, both neutral lines migrate from the outer towards the inner arc. ► None of the two neutral lines is continuous along the fold. ► For small viscosity ratios the neutral lines do not develop.
Two-dimensional scattering of elastic waves in a medium containing a circular heterogeneity is investigated with an analytical solution and numerical wave propagation simulations. Different ...combinations of finite difference methods (FDM) and finite element methods (FEM) are used to numerically solve the elastodynamic wave equations. Finite difference and finite element techniques are applied to approximate both the time and space derivatives and are combined in various ways to provide different numerical algorithms for modeling elastic wave propagation. The results of the different numerical algorithms are compared for simulations of an incident plane P-wave that is scattered by a mechanically weak circular inclusion whereby the diameter of the inclusion is of the same order than the P-wave's wavelength. For this scattering problem an analytical solution is available and used as the reference solution in the comparison of the different numerical algorithms. Staircase-like spatial discretization of the inclusion's circular shape with the finite difference method using a rectangular grid provides accurate velocity and displacement fields close to the inclusion boundary only for very high spatial resolutions. Implicit time integration based on either finite differences or finite elements does not provide computational advantages compared to explicit schemes. The best numerical algorithm in terms of accuracy and computation time for the investigated scattering problem consists of a finite element method in space using an unstructured mesh combined with an explicit finite difference method in time. The computational advantages and disadvantages of the different numerical algorithms are discussed.
3D fold growth rates Frehner, Marcel
Terra nova (Oxford, England),
October 2014, Letnik:
26, Številka:
5
Journal Article
Recenzirano
Odprti dostop
Geological folds are inherently 3D structures; therefore, they also grow in three dimensions. Here, fold growth in all three dimensions is quantified by numerically simulating upright single‐layer ...folds in 3D Newtonian media. Horizontal uniaxial shortening leads to a buckling instability, which grows from a point‐like initial perturbation in all three dimensions by fold amplification (vertical), fold elongation (parallel to fold axis) and sequential fold growth (parallel to shortening direction) of secondary (and further) folds adjacent to the initial isolated fold. The two lateral directions exhibit similar averaged growth rates, leading to bulk fold structures with aspect ratios in map view close to 1. However, fold elongation is continuous with increasing bulk shortening, while sequential fold growth exhibits jumps whenever a new sequential fold appears and the bulk fold structure therefore suddenly occupies more space. Compared with the two lateral growth directions, fold amplification exhibits a slightly higher growth rate.
The study of seismic anisotropy has benefited from the wide application of the electron backscatter diffraction (EBSD) technique that provides complete information on the crystallographic and shape ...preferred orientations in 2D sections. Classical effective medium theory statistically approximates the seismic anisotropy based on the crystallographic preferred orientation, but the shape preferred orientation is often idealized as e.g. parallel layering or oriented inclusions. Due to higher demands in precisely quantifying seismic anisotropy in natural rocks and taking full advantage of the EBSD technique, dynamic wave propagation methods have received broad attention. This paper presents the MATLAB program E-Wave based on a novel approach to directly use EBSD data for 2D numerical wave propagation simulation. The complete mechanical formulation and numerical benchmarks with simple model setups are presented. The E-Wave program allows straightforward EBSD data import, finite-difference simulations with one-button click, and automatic result analysis. The E-Wave program can be a helpful and independent tool in future works to shed light on the relationship between microstructures and seismic anisotropy, and contribute from the modelling perspective to studies in seismology, geodynamics and rock physics.
•E-Wave program is provided to perform dynamic wave propagation model.•Effects of SPO and CPO can be separated using E-Wave program.•Effects of elastic scattering and frequency influence on seismic anisotropy can be studied.
Parasitic folds are typical structures in geological multilayer folds; they are characterized by a small wavelength and are situated within folds with larger wavelength. Parasitic folds exhibit a ...characteristic asymmetry (or vergence) reflecting their structural relationship to the larger-scale fold. Here we investigate if a pre-existing geometrical asymmetry (e.g., from sedimentary structures or folds from a previous tectonic event) can be inherited during buckle folding to form parasitic folds with wrong vergence. We conduct 2D finite-element simulations of multilayer folding using Newtonian materials. The applied model setup comprises a thin layer exhibiting the pre-existing geometrical asymmetry sandwiched between two thicker layers, all intercalated with a lower-viscosity matrix and subjected to layer-parallel shortening. When the two outer thick layers buckle and amplify, two processes work against the asymmetry: layer-perpendicular flattening between the two thick layers and the rotational component of flexural flow folding. Both processes promote de-amplification and unfolding of the pre-existing asymmetry. We discuss how the efficiency of de-amplification is controlled by the larger-scale fold amplification and conclude that pre-existing asymmetries that are open and/or exhibit low amplitude are prone to de-amplification and may disappear during buckling of the multilayer system. Large-amplitude and/or tight to isoclinal folds may be inherited and develop type 3 fold interference patterns.
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•We model multilayer buckle folding with a predefined geometrical asymmetry.•If inherited, the asymmetry resembles a parasitic fold with wrong vergence.•In the larger-scale fold, layer-perpendicular flattening and flexural flow occurs.•Both work against the asymmetry, leading to de-amplification and unfolding.•Hence, a pre-existing wrong vergence is difficult to inherit during buckling.
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