Circles of Curvature at Points of Parabola in Isotropic Plane

The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.

On Solving Problems of Thermal Conductivity on an Anisotropic Plane with a Weakly Permeable Film

The problem of thermal conductivity on an anisotropic plane (x; y) divided into two halfplanes D1(1 < x < 0; y 2 R) and D2(0 < x < 1; y 2 R) by a weakly permeable film x = 0 is considered at given heat sources and a given initial temperature. The anisotropy ellipses are arbitrary (in magnitude and direction) and are the same at all points of the plane. Using the method of convolution of Fourier expansions, the solution of the problem is expressed in single quadratures through the well-known solution of the classical Cauchy problem on an isotropic plane without a film. The results obtained are of practical interest in the problems of heat propagation and conservation in materials with anisotropic properties (crystalline, fibrous materials), in the presence of a thermal insulation film.

Parabolas in the isotropic plane

In this paper we study the properties of a parabola in an isotropic plane and compare the results obtained with their Euclidean analogues.

Gob’s Circles of a Triangle in the Isotropic Plane

In this paper, we consider Gob’s circles of an allowable triangle in the isotropic plane and derive their equations in the case of a standard triangle. We prove that the potential axes of the circumscribed circle and Gob’s circles are the angle bisectors of the given triangle. We investigate some other interesting properties of Gob’s circles and some other circles related to the given triangle.

Determining the transverse isotropic rock’s static elastic moduli with cylindrical plugs: shortfalls, challenges and expected outcomes

Cylindrical shaped plugs can be tested using a Hoek-Cell like apparatus that allows for efficient and inexpensive measurements of a rock’s static elastic properties. However, when it comes to Transverse Isotropic material, this approach has a natural limitation due to the isotropic radial stresses; particular attention to the boundary conditions and the proper design of pressurization steps is warranted. Typical attempts to constrain the complete set of compliances ( S), using multiple plugs of different orientations, are impeded by the heterogeneity and pressure-dependent elasticities inherent to sedimentary rocks. Through stepwise pressure increases, we can constrain four normal compliances S 11, S 12, S 13, S 33 , describing two Young’s moduli and three Poison’s ratios using a single horizontal plug drilled parallel to the rock’s isotropic plane, contrary to the common assumption that at both horizontal and vertical plugs are needed. The measurement of the shear modulus S 44 needs to be obtained using a plug that is drilled oblique to the isotropic plane; replicating the in-situ stress environment is not possible using this approach. Lastly, the specimen’s anisotropic plane’s geometry is elliptical under isotropic radial stress; this causes a discrepancy between the strain gauge’s contraction and the actual strain. We propose an iterative inversion approach to account for this issue and calculate the exact strains useful for inferring S ij from measurements reported by strain gauges. The example included in this writing shows that without correction, inferred values of S ij may suffer errors of 20%.

On Diagonal Triangle of Non Cyclic Quadrangle in Isotropic Plane

Geometry of the non cyclic quadrangle in the isotropic plane was introduced in [2] and [6]. Herein, its diagonal triangle is studied and some nice properties of it are given.

Effects of randomly orienting penny-shaped cracks on the elastic properties of transversely isotropic rocks

Fractured reservoirs, as one kind of unconventional reservoirs, have great potential for oil and gas development, and their accurate characterization requires the development of rock-physics models that better simulate real fractured rocks. However, current models focus mainly on the elastic properties of rocks with aligned cracks, while the effects of randomly orienting cracks in transversely isotropic (TI) rocks are poorly studied even though such conditions are frequently encountered in the earth. To address this problem, we have derived models for the elastic properties of rocks with a TI background permeated by 3D inclined cracks and randomly orienting cracks. Then, based on the developed models, we comprehensively study the effects of the two inclination angles (i.e., the dip angle between the cracks and the isotropic plane and the rotation angle between the cracks and the plane normal to the isotropic plane, respectively) of 3D inclined cracks on the elastic properties of TI rocks. We determine that the two angles have significant influences on the elastic coefficients and hence the elastic velocities, and that their influences on the elastic properties are varying in different directions. We further investigate the effects of crack density and aspect ratio of randomly orienting cracks on the elastic properties of the fractured rocks with a TI background. The results show that the increasing crack density and crack aspect ratio reduce the elastic coefficients and velocities for rocks with randomly orienting cracks, in which the relations between compressional-wave velocities and the crack properties (i.e., crack density and crack aspect ratio) are obtained to aid the interpretation of the acquired acoustic exploration data. The proposed new models can greatly improve the modeling capability for the elastic properties of rocks with a TI background permeated by inclined and randomly orienting cracks.

Bound state in a model of electromagnetic field interacting with a material plan

In the model with Chern-Simons potential describing the coupling of electromagnetic field with a two-dimensional material, the possibility of the appearance of bound field states, vanishing at sufficiently large distances from interacting with its macro-objects, is considered. As an example of such two-dimensional material object we consider a homogeneous isotropic plane. Its interaction with electromagnetic field is described by a modified Maxwell equation with singular potential. The analysis of their solution shows that the bound state of field cannot arise without external charges and currents. In the model with currents and charges the Chern-Simons potential in the modified Maxwell equations creates bound state in the form of the electromagnetic wave propagating along the material plane with exponentially decreasing amplitude in the orthogonal to its direction.