Isotropic Curve Research Articles

Experimental study of compression behavior and its correlation with the microstructure of a deep-water sediment

Understanding the volume change behavior of deep-water sediments is essential for the safety design of deep-water engineering structures. In this study, the volume change behaviors of marine sediments from the South China Sea were studied through oedometer and isotropic compression tests. Scanning electron microscopy (SEM) and mercury intrusion porosimetry (MIP) tests have been conducted to investigate the microstructure evolution of two types of sediments under loads. The experimental results showed that the structural anisotropy of intact specimens is more pronounced in oedometer tests with the increase of stress, however, depolarization occurs in the isotropic consolidation test. The volume change after yield in the oedometer and isotropic consolidation tests comes from inter-aggregate pore variations associated with the adjustment of the soil fabric. The reconstituted specimen presents a more uniform distribution of pores than that of the intact specimen, and the macropores are more easily compressed for the reconstituted specimen than those of the intact specimen. With increasing stress, the oedometer compression and isotropic consolidation curves of intact specimens gradually approach those of the reconstituted specimen. The deformation mechanism under high stresses is that soil particles are reoriented and the variation of micropores.

Elastic strips along isotropic curves in complex 3-spaces

Elastic strips along isotropic curves in complex 3-spaces

A novel finite element approximation of anisotropic curve shortening flow

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H^1 -error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

Isotropic compression behavior for methane hydrate-bearing soil with varying temperature and pore pressure

Hydrate exploitation may trigger large-scale seafloor subsidence, especially when the depressurization method is used. The isotropic compression characteristics for methane hydrate-bearing soil (MHBS) need to be investigated to evaluate this deformation failure. Change in temperature and pore pressure (TP) condition can generate a considerable influence on the mechanical deformation and strength of MHBS. However, few studies focus on the isotropic compression deformation of MHBS. This paper carried out a series of isotropic compression tests on MHBS with different hydrate saturations and TP conditions. Experimental results show that the MHBS specimens exhibit a completely different isotropic compression curve at different hydrate saturations and TP conditions. The compressibility of MHBS is reduced remarkably with increased hydrate saturation, raised pore pressure, and decreased temperature. Hydrate saturation does not affect the compression index, but it explicitly reduces the swelling index. The yield stress is also enhanced with increasing hydrate saturation. The compression index of MHBS is independent of TP condition, while the swelling index is weakened and the yield stress is enhanced as temperature decreases and/or pore pressure increases. Through employing the phase state parameter as an indicator for TP condition, the influence of TP condition on the isotropic compression characteristics of MHBS is analyzed. Besides, the isotropic compression mechanism of MHBS at different hydrate saturation and TP condition is illustrated.

SANISAND-H: A sand bounding surface model for high pressures

This paper presents a new bounding surface constitutive model for sand, named SANISAND-H, capable of predicting the shear and volumetric behavior of the material under extra-high range of pressures, in addition to normal pressures. The model is an extension of the SANISAND08 model (Taiebat and Dafalias, 2008) with a closed cone-type yield surface that obeys rotational and isotropic hardening. The critical state line and the limiting isotropic compression curve of the SANISAND08 model are modified to accommodate a wide range of pressures. A new constitutive ingredient is the definition of a reference compression curve that serves as bounding surface for establishing the isotropic hardening rule of the yield surface. The kinematic hardening rule and stress-dilatancy relationship of the SANISAND08 model are significantly modified, to better capture the response of sands under high pressure. The value of the back stress ratio at the initiation of a new loading process is re-introduced, ensuring a smooth elastic-plastic transition necessary for accurate simulation of the response under loading reversals. The SANISAND-H model simulations are in very good agreement with test data for Toyoura and Cambria sands under various drained and undrained monotonic stress loading paths in a wide range of densities and pressures.

Damage to fracture in offshore engineering materials under several stress states: Blowout preventer valve application

This contribution proposes a study of the mechanical behavior from damage to fracture of normalized and annealed samples of AISI 4340 steel. Besides, a numerical study in a security device of offshore industry is conducted. Experimental tests were initially performed on smooth and notched cylindrical specimens subjected to monotonic tension as well as on rectangular specimens subjected to pure shear and a combination of shear and tension loads. Such experimental tests were selected to observe different combinations of the stress triaxiality and the normalized third invariant concerning the stress state. The two types of heat treatments were considered to achieve different levels of ductility and verify their influence on the fracture mode of the alloy. The approach based on the second invariant of the deviatoric stress tensor ( J 2), with nonlinear isotropic hardening, is assumed to describe the mechanical behavior of the material in axisymmetric and three-dimensional numerical simulations. Finite elements simulations were carried out to analyze the performance of a simplified version of a mechanism of a blow-out preventer-BOP valve. The experimental and numerical curves of force versus displacement were compared, highlighting the effect of the stress triaxiality, the third invariant and calibration condition on the behavior of the material. The evolution of the equivalent plastic strain was plotted, and the fracture onset compared with its maximum value. The results show that the AISI 4340 alloy is highly dependent on the stress triaxiality and third invariant. Moreover, it can be concluded that different calibration conditions for the isotropic hardening curve can result in different levels of forces for the correct performance of the BOP in a critical situation.

Conformal geometry of isotropic curves in the complex quadric

Let [Formula: see text] be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in [Formula: see text]. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into [Formula: see text], null with respect to the conformal structure of [Formula: see text]. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

Predicting the Isotropic Volumetric Compression Response of Hydrating Cemented Paste Backfill (CPB)

Deep and high-stress mining results in stress transfers onto the previously placed backfill, and mines have recorded several MPa induced backfill stresses. Understanding the backfill-rock mass interaction is therefore critical. Previous work considered tabular ore bodies undergoing primarily one-dimensional compression and showed how the backfill reaction curves could be estimated from oedometer laboratory test results. This work considers massive orebodies and develops a similar approach based on isotropic compression curves. Isotropic compression tests exceeding 6 MPa are carried out on samples with 3.0 to 11.1% binder content, tested at 1-day cure time to 28-day cure time. The compression curve is characterized in three stages: initial elastic compression up to a yield point, followed by a transition stage to the start of a final stage with a linear post-yield compression line in $$\varepsilon_{v} - \log \left( {p^{\prime } } \right)$$ space. Because these isotropic compression tests are rare (the reported results are the first for Cemented Paste Backfill), attempts are made to relate the isotropic compression test parameters to parameters from the more commonly used Unconfined Compression Strength (UCS) tests. Unifying equations as functions of binder content and cure time are found to determine the initial yield stress and the peak strength from UCS tests. These are then related to the corresponding parameters in isotropic compression. Finally, the slope of the post-yield compression line is found as a function of UCS of similar CPB with the same binder content and cure time. Then the isotropic compression behavior of CPB is reconstructed as a function of binder content and curing time using UCS values of similar CPB. Although the calibrated parameters are specific to the studied mine’s materials, the framework is general and applicable to other mines’ CPBs.

Representations of Rectifying Isotropic Curves and Their Centrodes in Complex 3-Space

In this work, the rectifying isotropic curves are investigated in three-dimensional complex space C3. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. Meanwhile, the rectifying isotropic curves are expressed by the Bessel functions explicitly. Last but not least, the centrodes of rectifying isotropic curves are explored in detail.

Spatial orientations of angular momentum vectors of galaxies in Supercluster S [173+014+0082

The spin vector orientation of 1302 SDSS (Sloan Digital Sky Survey) galaxies in Supercluster S[173+014+0082] having redshift 0.076 to 0.091 has been analysed. The positions, position angles and inclination angles of galaxies are used to convert two-dimensional observed parameters into three-dimensional angular momentum vectors of the galaxy using the `position angle-inclination' method. The expected isotropic distribution curves are determined performing numerical simulation by generating 107 virtual galaxies. The observed distribution is compared with the expected isotropic distribution curves using three statistical tools namely Chi-square test, auto-correlation test and Fourier test. Redshift map is studied and found that the distributions fit with the Gaussian. No preferred alignment of angular momentum vectors is noticed, supporting Hierarchy model of galaxy formation.
 BIBECHANA 18 (2021) 26-32

Fuse -Replacement Noise-Filter Development for Automotive Audio–Video Navigation Systems Based on Loudness Isotropy Curves

With the development of advanced driver assistance systems, the more reliable the autonomous driving technology is, the more the rest and entertainment times of the driver of the car increases. Hence, the importance of the entertainment function of automotive audio-video navigation (AVN) systems is increasing. Currently, the AVN system of automobiles has a monitoring function for fault diagnosis and a combination of functions. Applying these technologies is challenging for drivers who want to tune the audio quality to their musical taste. In this study, a method for upgrading the sound quality using a power supply noise filter without deforming the AVN system was developed. The low-pass attenuation that appeared as a side effect was solved by applying a filter using the loudness isotropic curve. In the installation method of the filter, the method of using a fuse holder minimized the inconvenience of AVN detachment and wiring. Based on the results obtained in this study, further research and improvement of the filter are required for audio tuning of various models.

Узагальнений спосіб формування плоских ізотропних кривих

The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position. It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve. Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.

КОНСТРУЮВАННЯ ДИСКРЕТНИХ СІТОК ТА ПОВЕРХОНЬ НА ОСНОВІ ІЗОТРОПНИХ В-СПЛАЙНІВ

Modeling geometric objects in isotropic space allows us to simplify a number of calculations related to the calculation of differential characteristics. To create the ability to control the curved contours of objects, it is advisable to apply parametric curves based on characteristic polygons. For each type of such curves, there are limitations to the calculation of coordinates and lengths in isotropic space. Studies on modeling isotropic B-splines and based on them grids and surfaces are local in nature and require more detailed consideration. The paper proposes to model isotropic B-splines based on the recursive Cox de Boer formulas. The coordinates of the points for the spline are calculated based on the equal to zero side lengths of the characteristic polygons. Examples of constructed splines for different types of nodal vectors are given. It is shown that some types of nodal vectors do not allow the curve to pass through the start and end points of polygons. In this case, it is necessary to carry out adjustments to the nodal vector. The authors of the work developed an algorithm for modeling discrete grids and surfaces based on B-splines. All calculations are made in a complex space. When displaying geometric objects, the selection of the real part and the transition to the real space are carried out. As an example, visualization of the grid and surface based on an open uniform nodal vector is given. To research isotropic discrete curves, grids and surfaces based on B-splines, C # software was created for the .Net Core platform. Further research is related to the study of the influence of nodal vectors on the basis of an isotropic B-spline and surfaces. Keywords: isotropic curve, B-spline, discrete grids, discrete surfaces based on isotropic splines.

МОДЕЛЮВАННЯ ПРОСТОРОВИХ ІЗОТРОПНИХ КРИВИХ БЕЗЬЄ НА ОСНОВІ КРИВИХ ЗА ГОДОГРАФОМ ПІФАГОРА

Minimal surfaces are essential for geometric modeling, computer science, and for various branches of engineering, and the determination of the conditions for their construction has been studied in many works and includes a large number of methods. However, this topic has not yet been exhausted when using isotropic curves to construct minimal surfaces. These studies focus on methods for determining zero-length curves. For interactive control of such curves, it is advisable to use curves constructed on the basis of characteristic polygons, therefore, studies are based on Bézier curves. The paper highlights the experience of previous studies in the field of constructing spatial isotropic curves and plane Pythagorean-Hodograph curves. The modeling of spatial Pythagorean-Hodograph curves is based on the theory of quaternions. The authors of the paper present a different approach, namely, the construction of an isotropic spatial curve based on a plain Bézier Pythagorean-Hodograph curve. The transition to the spatial curve is carried out on the basis of determining the third coordinate from the condition that the length is equal to zero. In this case, a plain curve is built on the basis of real values, and the third coordinate will be purely imaginary. The authors carry out studies based on fifth-order curves. In this case, it is advisable to choose quadratic functions as basic polynomials. Three options are proposed for setting the initial values ​​of the coordinates for determining the Pythagorean-Hodograph curve. The problem of the boundary points of the curve and the determination of the intermediate points of the Bezier curve based on the given dependences have the most practical importance. To prove the reliability of the proposed provisions, the calculation of the points of the curve is performed and examples of simulated curves are given. Further research involves the use of the constructed curves for modeling minimal surfaces and isotropic portions of Bézier. Key words: isotropic curve, Bézier curve, Pythagorean-Hodograph curve.

The value of the work done by an isotropic vector force field along an isotropic curve

In the present paper we consider a 3-dimensional differentiable manifold M equipped with a Riemannian metric g and an endomorphism Q, whose third power is the identity and Q acts as an isometry on g. Both structures g and Q determine an associated metric f on (M,g,Q). The metric f is necessary indefinite and it defines isotropic vectors in the tangent space TpM at an arbitrary point p on M.The physical forces are represented by vector fields. We investigate physical forces whose vectors are in TpM on (M,g,Q). Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

Applying Isotropic Fractional-Rational Curves Toward Surface Modeling: New Insights from the Applied Mathematics Perspective

Given curves or points, one of the problems that have been documented in the recent past entails building smooth surfaces. The problem is pronounced due to computer technology and industry developments. Initially, zero Gaussian curvature shells and minimal surfaces were used. These surfaces relied on isotropic analytic curves. However, consumer properties restrict these curves. Hence, there is a growing need to steer expansions in surface shaping. In response to this trend, this paper proposed a technique for surface construction based on the role of isotropic fractional-rational curves. In the framework, the building of surfaces involved the use of flat orthogonal and isothermal grids, a process governed by the Weierstrass technique. Indeed, the latter approach was adopted due to the associated trend of demanding minimal surfaces.

Isotropic hardening curve characterization by the resultant profile of ball indentation tests

Traditionally, the material’s stress–strain relationship is acquired from uniaxial testing, which is widely used to describe its behavior under plastic deformation. When it comes to traditional hardness tests, such as in the Brinell, Knoop, Rockwell, and Vickers, they have been mostly used as a way to assess the capability a material has to resist plastic deformation. The technique developed and presented here has gone beyond that by determining other material properties in addition to hardness. This is performed by relating the applied load to the resultant indentation profile impressed in the material. Similar approaches have been devised aided by instrumented indentation devices. However, the methodology presented here does not require the use of such a device. It considers that the use of three configurations of the Brinell hardness test is enough to represent the material behavior within the boundaries of the plastic deformation experienced by it. Therefore, it is simpler than the usual approaches in this aspect. On the other hand, the indentation profile is also assessed along with the load–depth information obtained for each of those configurations. For that, a multi-image analysis of the indention impressions is performed in a confocal laser microscope. The indentation profile informs the amount of pile-up/sink-in experienced by the material. These phenomena are directly related to the strain hardening of the material and therefore influence the description of the plastic stress–strain curve. The extracted profiles are used as a reference with which the predicted output of repeated FEM modeling is compared. In this process, several trial stress–strain curves are provided to minimize the discrepancy between numerical and experimental data in an iterative FEM modeling of the indentation process. The process runs until reaching the established tolerance and thus providing the hardening parameters that best fit the experimental data. The SAE 1524 is analyzed in the present work employing the Kleinermann–Ponthot constitutive hardening model, which is composed of four parameters. It is shown that the determined parameters conform closely to the experimental data. Therefore, the procedure is viable for the material characterization task.

Surfaces Modelling Using Isotropic Fractional-Rational Curves

The problem of building a smooth surface containing given points or curves is actual due to development of industry and computer technology. Previously used for those purposes, shells of zero Gaussian curvature and minimal surfaces based on isotropic analytic curves are restricted in their consumer properties. To expand the possibilities regarding the shaping of surfaces we propose the method of constructing surfaces based on isotropic fractional-rational curves. The surfaces are built using flat isothermal and orthogonal grids and on the basis of the Weierstrass method. In the latter case, the surfaces are minimal. Examples of surfaces that were built according to the proposed method are given.

Isotropic Curves and Their Characterizations in Complex Space C4

Isotropic Curves and Their Characterizations in Complex Space C4