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.     

Volumetric strains in neutral surface folding

Neutral surface folding is a significant contributor to fold development. This mechanism produces contrasting strains in the inner and outer arcs of the folded layer that arise from principal stress orientations that are approximately parallel and perpendicular to the layer. We demonstrate that such stress patterns imply significant gradients of mean stress across the folding layer, being more tensional on the outer arc and more compressive in the inner arc. This could pump fluids towards the outer arc during folding and result in heterogeneous volume changes. We conclude that the neutral surface folding model should be adapted to accommodate volumetric strains, in order to explain dilatational structures (e.g. open fractures, veining) on the extrados and volume-loss structures (e.g. pressure solution seams, stylolitic cleavages) on the intrados. This dilatation has economic implications as it allows prediction of sites of mineralization and zones of secondary permeability in fold-related hydrocarbon traps.     

New Approach to Study Extended Evolution of Supergranular Flows and Their Advection of Magnetic Elements

Using velocity and magnetogram data extracted from the full-disk field of view of MDI during the 1999 Dynamics Program, we have studied the dynamics of small-scale magnetic elements (3–7 Mm in size) over time periods as long as six days while they are readily visible on the solar disk. By exploiting concurrent time series of magnetograms and Doppler images, we have compared the motion of magnetic flux elements with the supergranular velocity field inferred from the correlation tracking of mesogranular motions. Using this new method (which combines the results from correlation tracking of mesogranules with detailed analysis of simultaneous magnetograms), it is now possible to correlate the motions of the velocity field and magnetic flux for long periods of time and at high temporal resolution. This technique can be utilized to examine the long-term evolution of supergranulation and associated magnetic fields, for it can be applied to data that span far longer time durations than has been possible previously. As tests of its efficacy, we are able to use this method to verify many results of earlier investigations. We confirm that magnetic elements travel at approximately 350 m s ^–1 throughout the duration of their lifetime as they are transported by supergranular outflows. We also find that the positions of the magnetic flux elements coincide with the supergranular network boundaries and adjust as the supergranular network itself evolves over the six days of this data set. Thus we conclude that this new method permits us to study the extended evolution of the supergranular flow field and its advection of magnetic elements. Since small-scale magnetic elements are strongly advected by turbulent convection, their dynamics can give important insight into the properties of the subsurface convection.     

The representation and calculation of the deviatoric component of the geological stress tensor

A new diagram for the representation of stress states is proposed and compared with Nadai's stress diagram. The diagram is a graph whose axes are labelled as the differences of the principal stresses (σ₂-σ₃ as ordinate axis, σ₁-σ₂ as the abscissa; where σ₁ > σ₂ > σ₃ are the principal stresses). The design of the plot has been deliberately modelled on that of the ‘log Flinn’ diagram which is used to represent finite strain ellipsoids. The position of the plotted stress state on this diagram depends on the nature of the deviatoric (non-hydrostatic) component of the stress tensor. The distance of the plotted stress from the origin corresponds broadly to the departure of the stress from a hydrostatic state and the parameter R, defined as the gradient of the line joining the plotted state to the origin, expresses the type of symmetry possessed by the stress tensor.It is explained how the diagram can be used to represent calculated palaeostresses and, in particular, how the parameter R can be found directly from some existing methods of stress analysis currently in use.Besides its proposed function to represent the results of such analyses it is felt that the use of the diagram may make clear the essential elements of the definitions of well-known terms used to describe particular stress states (e.g. plane stress, triaxial stress, axial compression, etc.).     

The superellipsoidal form of coarse clastic sediment particles

An approximating form for coarse water-worn clasts based on the superellipsoid is proposed. The sectional outlines of pebbles from Gower (Wales) beaches approximate superellipses described by this equation in rectangular coordinates $$\frac{{x^p }}{{a^p }} + \frac{{x^p }}{{b^p }} = 1$$ wherea andb are principal semi-axes. For AB planes of the pebbles,p is close to 2, but in AC sections,p averages about 2.6. The measured volumes of pebbles are poor approximations to the previously proposed ellipsoidal model of pebble form. Instead, volumes are shown to accord with a three-dimensional form consisting of a superellipsoid of revolution, i.e., a solid of revolution produced by rotating a superellipse about one of its principal axes.