Mohr Diagram Research Articles

Guest Editors' Introduction: Special Section on IEEE Visualization Applications

THIS issue contains extended versions of three of the bestrated application papers taken from the IEEE Visualization 2004 Conference. Application papers present the contribution of visualization techniques toward the understanding of application-specific data. In particular, application papers must explain the effectiveness of the visualization methods for the particular application domain. For the IEEE Visualization 2004 Conference, 24 out of 71 submitted applications papers were chosen for inclusion in the conference program. As paper cochairs of the IEEE Visualization applications track, we started from the opinion of the reviewers. From the 24 accepted papers, we have chosen three very high quality submissions and asked the authors to provide extended and revised versions of their original work for this issue. Each of the extended application papers was thoroughly reviewed by several experts and underwent revisions before it was recommended for acceptance. We are grateful to the reviewers for their detailed and timely reviews and to David Ebert, Editor-in-Chief of TVCG, for his strong support and assistance. The paper “Reconstruction and Visualization of Planetary Nebulae” by Marcus Magnor, Gordon Kindlmann, Charles Hansen, and Neb Duric exploits strong symmetry characteristics of planetary nebulae to recover spatial structures from 2D images. With GPU-based volume rendering and nonlinear optimization, the nebula’s emission density is recovered. Fascinating 3D visualizations of planetary nebulae are created, which further educational purposes and can support astrophysicists in better understanding the formation process of these phenomena. The paper “Advanced Virtual Endoscopic Pituitary Surgery” by Andre Neubauer, Stefan Wolfsberger, MarieTherese Forster, Lukas Mroz, Rainer Wegenkittl, and Katja Buhler investigates a minimally invasive endoscopic procedure for tumor removal. STEPS, a virtual endoscopy system, is presented to train surgeons on the transsphenoidal approach and assist experienced surgeons in planning a real endoscopic intervention. Interactive visualization is achieved through first-hit ray casting. The application provides various navigation and perception aids in the simulation of the surgical procedure. The paper “Visualization of Geologic Stress Perturbations Using Mohr Diagrams” by Patricia Crossno, David H. Rogers, Rebecca M. Brannon, David Coblentz, and Joanne T. Fredrich presents an inspection tool for finite element analyses of 3D real-valued second-order tensors. The Mohr diagram is a paper and pencil method from the material mechanics community. The authors originate the adaptation of this diagram type to visualize perturbed in-situ stress fields of geologic features. The Mohr diagram is used as an interactive glyph for probing as well as filtering through brushing and linking of principal stresses. Scientific visualization research is largely application driven. Many techniques are strongly motivated by the application scenario for which they were developed. Visualization research cannot be a goal by itself. It just acts as an interface tool to the domain expert to facilitate his understanding of his data and problems. Applications are therefore crucial in providing fruitful incitation for the further research of the visualization experts. The papers presented here are prime examples of how visualization helps to understand intricate application phenomena. We hope that the readers will find the papers inspiring and stimulating samples of work from our fascinating field.

Visualization of Geologic Stress Perturbations Using Mohr Diagrams

Huge salt formations, trapping large untapped oil and gas reservoirs, lie in the deepwater region of the Gulf of Mexico. Drilling in this region is high-risk and drilling failures have led to well abandonments, with each costing tens of millions of dollars. Salt tectonics plays a central role in these failures. To explore the geomechanical interactions between salt and the surrounding sand and shale formations, scientists have simulated the stresses in and around salt diapirs in the Gulf of Mexico using nonlinear finite element geomechanical modeling. In this paper, we describe novel techniques developed to visualize the simulated subsurface stress field. We present an adaptation of the Mohr diagram, a traditional paper-and-pencil graphical method long used by the material mechanics community for estimating coordinate transformations for stress tensors, as a new tensor glyph for dynamically exploring tensor variables within three-dimensional finite element models. This interactive glyph can be used as either a probe or a filter through brushing and linking.

Comparative testing of ellipse-fitting algorithms: implications for analysis of strain and curvature

Several types of geological problem involve fitting an ellipse to sparse data in order to define a property such as strain or curvature. The sensitivity of ellipse-fitting algorithms to noise in input geological data is often poorly documented. Here we compare the performance of some well-known approaches to this problem in geology against each other and an algorithm developed for machine vision. The specific methods tested here are an analytical method, a Mohr diagram method, two least squares methods and a constrained ellipse approach. The algorithms were tested on artificial datasets of known elliptical and noise properties. These results allow the selection of an ellipse fitting method in a variety of geological applications and also allow an assessment of the absolute and relative accuracy of a chosen method for various combinations of sample numbers and noise levels. For the most accurate semi axis magnitude and orientation estimates with five or more input data, the ‘mean object’ least squares approach is recommended. However, the other least squares method also yields good results and is also suitable for three or four data points. Where curvature data is being assessed, the least squares method is preferred as it can handle negative principal curvature values.

Graphical methods of representing the stress components on different inclined planes at one point

The Mohr diagram is one of the most powerful and efficient graphical methods in stress/strain analysis and structural design. However, it cannot be used to determine the direction of the shear stress or shear strain for three-dimensional states. In this paper, a three-dimensional figure is given to describe the variation of normal and shear stress on different inclined planes at one point. Based on a mathematical formulation, the direction of the total shear stress is determined, and a graphical method is proposed to represent both the magnitude and the direction of the normal stress and shear stress together.

Reverse migration of seismicity on thrusts and normal faults

In this work, is analysed the control exerted by the stress axes orientation on the evolution of seismic sequences developing in compressive and extensional regime. According to the Anderson fault theory, the vertical stress is the minimum principal stress in compressional tectonic regimes, whereas it is the maximum principal stress in extensional regimes. Using Mohr diagrams and discussing the present knowledge about the distribution of vertical and horizontal stress with depth we show that, in absence of localised fluid overpressure, such changes imply that thrust and normal faults become more unstable at shallower and greater depths, respectively. These opposite mechanical behaviours predict, in a rather isotropic body, easier rupture at shallower level in compressional regimes later propagating downward. On the contrary, a first deep rupture propagating upward is expected in extensional regimes. This is consistent with observations from major earthquakes from different areas in the world. We show that the exceptions to downward migration along thrusts occur along steeply inclined faults and probably imply localised supra-hydrostatic fluid pressures. Moreover, we show that the inversion of the meaning of the lithostatic load has consequences also for the role of topography. High topography, increasing the vertical load, should inhibit earthquake development in compressional environments and should favour it in extensional settings. Although several factors, such as geodynamic processes, local tectonic features and rock rheology, are likely to control earthquake locations, stress distribution and tectonic regime, these model predictions are consistent with seismicity distribution in Italy, central Andes and Himalaya. In these areas, large to medium compressional earthquakes occur at the low elevation borders of compressional mountain belts, whereas large extensional earthquakes occur in correspondence to maximum elevations.

Structural overview of selected Group II kimberlite dyke arrays in South Africa: implications for kimberlite emplacement mechanisms

Group II kimberlite dykes occur in small, dominantly en-echelon dyke-fracture arrays, with individual dyke-fractures showing small angular variations from their array trends (5° to 15°). The analysed dyke systems are characterized by closely matching opposing dyke contacts, “ in-situ ” breccia, multiple kimberlite stringers within a dilated dyke-parallel fracture cleavage, wedge-shaped apophyses in bent bridges at dyke-fracture offsets/overlaps, kimberlite-free offset/overlap areas and calcite vein fibres orthogonal to dyke contacts. Commonly found microscopic structures include synemplacement/syncrystallization calcite veinlets, containing high aspect ratio stretched fibrous calcite, and elongate phlogopite phenocrysts and serpentinized olivine phenocrysts growing across the width of these veins. Both macro- and microscopic structures support a model of orthogonal host rock dilation during kimberlite emplacement. Terminations of dyke-fracture segments show minimal curvature or overlap, suggesting that remote horizontal stresses dominated during their emplacement (“passive” intrusion), as opposed to magma overpressured systems wherein dyke or dyke-fracture overlaps curve strongly towards each other (“active” intrusion). The application of Mohr diagrams suggests that low differential stresses, with no or only a very minor shear component, prevailed at the time of emplacement. The dominance of remote horizontal forces, imparting small differential stresses to the brittle portions of the crust, a closely-spaced, dilating dyke-parallel fracture cleavage ahead of the dyke tip (imparting a local suction) and the low-volume, low-viscosity, highly volatile nature of kimberlitic magmas may explain their empirically-constrained high emplacement velocities. This, in turn, explains the means by which such magmas may entrain significant volumes of high specific-gravity mantle material. Mobile hydrofracturing in the fringe zones around dilated craton-scale jointing is proposed to be a viable mechanism for kimberlite emplacement.

Mohr diagram interpretation of anisotropic diffusion indices in MRI

Descriptions of diffusion anisotropy in MR Diffusion Tensor Imaging are often based on scalar indices such as surface-to-volume ratio, volume ratio, fractional anisotropy and rotational anisotropy. Recently, Mohr diagram was introduced to visualize the anisotropic diffusion information, but in a graphical form. Even though both scalar indices and the Mohr diagram are derived from the elements of the diffusion tensor, so far, the relationships between these measures and the key aspects of the Mohr diagram have not been established. This paper demonstrates these relationships both qualitatively and quantitatively for two microscopically heterogeneous biologic tissues, kidney and spinal cord.

Easy handling of tectonic data: the programs TectonicVB for Mac and TectonicsFP for Windows™

TectonicVB for Macintosh and TectonicsFP for Windows TM operating systems are two menu–driven computer programs which allow the shared use of data on these environments. The programs can produce stereographic plots of orientation data (great circles, poles, lineations). Frequently used statistical procedures like calculation of eigenvalues and eigenvectors, calculation of mean vector with concentration parameters and confidence cone can be easily performed. Fault data can be plotted in stereographic projection (Angelier and Hoeppener plots). Sorting of datasets into homogeneous subsets and rotation of tectonic data can be performed in interactive two-diagram windows. The paleostress tensor can be calculated from fault data sets using graphical (calculation of kinematic axes and right dihedra method) or mathematical methods (direct inversion or numerical dynamical analysis). The calculations can be checked in dimensionless Mohr diagrams and fluctuation histograms.

Mohr diagram representation of anisotropic diffusion tensor in MRI.

With current approaches, it is difficult to visually comprehend the complete information contained in a diffusion tensor (DT) measured from a microscopically heterogeneous biological tissue. Therefore, in this work the Mohr diagram is introduced to graphically display the key aspects of DTs and to interpret the underlying anisotropy, both qualitatively and quantitatively. Specifically, the mathematical basis for the construction of the Mohr diagram from the elements of DTs is described, the merits of the approach are illustrated with examples of DTs reported for various biological tissues, and the results are discussed in the context of relating the diffusion anisotropy to the tissue structures.

Polar Mohr diagram method and its application in calculating the shear displacements of general shear zones with volume loss

The main problem, in determining the shear displacement of a general shear zone with volume change using the available formula, is that it is hard to know the initial angle between the planes (or lines) in the plane of shear. A planar deformation analysis of this kind of ductile shear zone is carried out with the polar Mohr diagram. If the volume change is induced by homogeneous contraction in the Z direction of the shear zone, there are sufficient conditions for constructing a polar Mohr diagram regardless of sequence of the simple shear and volume change. Therefore, the angle between a line and the shear direction before and after the deformation can be measured. Making use of these lines the shear strain and the volume change can be calculated and the shear displacement can be determined.

Erosion Mechanics of Soils with an Impermeable Subsurface Layer

More than 50% of annual soil loss in a number of temperate regions of the world occurs when frozen soils are thawing. Such losses occur as a consequence of relatively high surface soil water contents due to the presence of a subsurface impermeable layer. This laboratory study addressed the soil mechanical principles governing erosion of soils with an impermeable subsurface layer (frozen or compacted, etc.). We examined (i) the effect of a freeze–thaw cycle on soil cohesion and (ii) the effect of a subsurface impermeable layer on soil detachment during raindrop impact for two soils: a Galva loess (silty clay loam, mixed, mesic Typic Hapludoll) and a Nicollet glacial till (loam mixed, superactive mesic Aquic Hapludoll). A Mohr diagram was constructed, based on a series of triaxial tests at three matric potentials. Soil cohesion for each treatment was determined from the Mohr diagrams. Soil water content treatments of 0.15 and 0.25 g g−1 were imposed before freezing and thawing. The freeze treatment was 90 min at −12°C, followed by a 30‐min thaw period. Immediately thereafter, shear strength and detachment were measured. Soil cohesion values were <50 kg m−2 and were not affected by freezing or by water content‐at‐freezing. The single‐water‐drop detachment values for soils with an impermeable layer vs. without an impermeable layer increased from 0.016 ± 0.0065 g to 0.054 ± 0.0265 g for loess and 0.036 ± 0.0071 g to 0.145 ± 0.0635 g for till. With an impermeable subsurface layer, soil matric potential is driven towards 0 Pa, resulting in extremely weakened soils prone to high soil‐detachment values and quite likely high soil‐erosion losses.

A graphical technique to predict slip along a pre-existing plane of weakness

A graphical technique is proposed to determine whether a pre-existing plane of weakness will be reactivated by slip under a stress field. This technique is based on Coulomb-Navier criteria and the method of Yin and Ranalli (Yin, Z., Ranalli, G., 1992. Critical stress difference, fault orientation and slip direction in anisotropic rocks under non-Andersonian stress systems. J. Struct. Geol. 14, 237–244). It consists of calculating which mechanism, rupture or sliding, needs the smaller stress difference to liberate the deformation. Using the results of calculations over a wide range of plane orientations, we plotted, in an equal-area net, the line which separates the orientation fields where rupture needs less stress difference from the fields where slip on pre-existing plane is favored. We named these plots slip-rupture graphs. For the three Andersonian fault regimes, the graphs are presented as dendrograms. These dendrograms show the variation of the range of orientations favorable for reactivation as a function of cohesion and friction of the plane of weakness, depth, pore fluid pressure and the stress ratio. The slip-rupture graphs are compared with the Mohr diagram and slip-tendency graphs (Morris et al., 1996). Relative to Mohr diagrams, our graphs have the advantage that it is possible to work with geographic orientations of planes and principal stresses, and it is not necessary to transform the field data to a stress space. The slip-rupture graphs are similar to slip-tendency graphs; however, the former can lead to estimate physical parameters that make reactivation possible along planes with unfavorable orientations.

Polar Mohr constructions for strain analysis in general shear zones

The polar Mohr diagram is a useful tool for strain analysis in general shear zones. Polar Mohr diagrams for general shear zones can be constructed with the following measured data: (a) the stretches of two line markers which were originally perpendicular to each other; (b) the ratio of their stretches and the principal directions; (c) the stretches in the shear direction and in the principal direction; and (d) the stretches in any two directions. With these diagrams, the angle between eigenvectors, v, can be obtained, and the kinematic vorticity number W k in the shear zone and the ratio of pure shear rate ε to simple shear rate γ during deformation can be computed with the formulae W k = cos v and W k = cos[cot(2ε/γ)].

Superposed deformations by Mohr construction

The Mohr diagram can provide a simple constructional method for superposing various kinds of deformation. Examples are given for superposed pure shears, and various combinations of pure shear, simple shear and area change. Construction methods are developed on a Mohr diagram for reciprocal stretch vs rotation, which is actually a representation of the ‘backwards deformation’ tensor, d. The superposition of apure shear or a simple shear on an earlier deformation follows elementary principles in Mohr space, enabling the total deformation to be determined simply. More general deformations can be considered as a combination of dilation, pure shear and simple shear (if body rotations arc omitted), which are all amenable to the Mohr construction method. Although confined to problems of two-dimensional superposed deformation, this new method provides a useful alternative to mathematical or computer-based methods to determine the effects of oblique superposition of several deformations The visual appeal of such graphical constructions should aid in the teaching and understanding of deformation processes.

Three-Dimensional Mohr's Circle for Shear Stress Components

The Mohr diagram is a graphical technique that is well known and widely used in stress analysis. Mohr's graphical construction enables the determination of the magnitude of the normal stress component for a given oblique plane. On the other hand, it only gives the magnitude of the total shearing stress, thus failing to provide its components. This paper is an extension of Mohr's work to overcome this shortcoming. A proposed method, based on the utilization of Cauchy's formula, along with vector analysis of both the stress vector and the normal stress component, is suggested. It allows full representation of the shearing stress components in Mohr space. This present technique is mathematically derived and graphically depicted.

The sliding-scale Mohr diagram

Mohr diagrams for the position gradient tensor are a useful tool in problems of finite deformation, for example to determine the angle between deformed lines and the angle of rotation of finite strain axes. The conventional Mohr diagram is less suitable, however, to visualise progressive deformation or deformation paths. A sliding-scale Mohr diagram is introduced which serves this purpose better. It consists of a Mohr circle of fixed diameter and a mobile reference frame origin that maps the deformation path. The sliding-scale Mohr diagram can be used as a visualisation tool for teaching purposes and for research on kinematics of progressive deformation.

Least squares best-fit circles (with applications to Mohr's diagram)

Procedures are outlined for the selection of a least squares best-fit circle to data points defined by rectangular Cartesian coordinates. Equations are derived to allow fitting of circles centered on the x-axis as well as off-axis Mohr circles. These procedures are applicable to the estimation of second-order tensors such as stress and strain by means of Mohr's diagram.

An interactive program for the graphical representation of striated faults and applied normal and tangential stresses

Accurate analysis of striated faults may lead to the calculation of the associated reduced stress tensor ?; this is defined by four parameters, namely the three principal stress orientations Si and a stress ellipsoid shape ratio i.e. ? = (?2 - ?3)/(?1 - ?3). For a given fault plane, the basic hypothesis is that the direction and sense of displacement St correspond to the tangential stress applied upon this plane.The program we propose is written in FORTRAN 77; for each striated fault, the tangential and normal stress vectors are computed with respect to the reduced stress tensor ?. The input data are the file of fault measurements and the four parameters of ?. The outputs are composed of: (1) the stereographic projection of fault planes with direction/sense of measured striation St and applied tangential stress. (2) A histogram of the angular differences between calculated tangential stresses and directions of striations. (3) Mohr circle for stress. An optional output is proposed for the case where only the principal stress directions Si are known; then Figures 1 and 3 are drawn for incremental variations of ? in the range 0,1. The resolved shear stress values along the directions of measured striations are also computed and plotted in the Mohr diagram. A synthetic fault set is used to illustrate the application of this program.

Mohr diagram for three-dimensional reciprocal stretch vs rotation : Treagus, S H J Struct GeolV12, N3, 1990, P383–395

Mohr diagram for three-dimensional reciprocal stretch vs rotation : Treagus, S H J Struct GeolV12, N3, 1990, P383–395

The Mohr diagram for three-dimensional reciprocal stretch vs rotation

Mohr diagrams for stretch provide a useful alternative to the more familiar λ′,γ′ Mohr diagrams, for solving two- and three-dimensional strain problems. Examples show how the polar graphs of stretch or reciprocal stretch vs rotation can be used to deform or undeform planes in three-dimensional strain. The strain ellipsoid can be represented by a three-circle reciprocal stretch diagram which represents the symmetric three-dimensional reciprocal stretch tensor. The diagram can also be applied to the representation of asymmetric three-dimensional deformation tensors. An example is given for a simple-shear deformation, which is represented by a distinctive ‘off-axis’ type of three-dimensional reciprocal stretch diagram.