Analytical investigations of in situ stress inversion from borehole breakout geometries

This study aims to investigate the feasibility of deriving in situ horizontal stresses from the breakout width and depth using the analytical method.Twenty-three breakout data with different borehole sizes were collected and three failure criteria were studied.Based on the Kirsch equations,relatively accurate major horizontal stress(sH)estimations from known minor horizontal stress(sh)were achieved with percentage errors ranging from 0.33%to 44.08%using the breakout width.The Mogi-Coulomb failure criterion(average error:13.1%)outperformed modified Wiebols-Cook(average error:19.09%)and modified Lade(average error:18.09%)failure criteria.However,none of the tested constitutive models could yield reasonable sh predictions from known sH using the same approach due to the analytical expression of the redistributed stress and the nature of the constitutive models.In consideration of this issue,the horizontal stress ratio(sH/sh)is suggested as an alternative input,which could estimate both sH and sh with the same level of accuracy.Moreover,the estimation accuracies for both large-scale and laboratory-scale breakouts are comparable,suggesting the applicability of this approach across different breakout sizes.For breakout depth,conformal mapping and complex variable method were used to calculate the stress concentration around the breakout tip,allowing the expression of redistributed stresses using binomials composed of sH and sh.Nevertheless,analysis of the breakout depth stabilisation mechanism indicates that additional parameters are required to utilise normalised breakout depth for stress estimation compared to breakout width.These parameters are challenging to obtain,especially under field conditions,meaning utilising normalised breakout depth analytically in practical applications faces significant challenges and remains infeasible at this stage.Nonetheless,the normalised breakout depth should still be considered a critical input for any empirical and statistical stress estimation method given its significant correlation with horizontal stresses.The outcome of this paper is expected to contribute valuable insights into the breakout stabilisation mechanisms and estimation of in situ stress magnitudes based on borehole breakout geometries.

Phase estimation in bispectral domain based on conformal mapping and applications in seismic wavelet estimation

Seismic wavelet estimation is an important part of seismic data processing and interpretation, whose preciseness is directly related to the results of deconvolution and inversion. Wavelet estimation based on higher-order spectra is an important new method. However, the higher-order spectra often have phase wrapping problems, which lead to wavelet phase spectrum deviations and thereby affect mixed-phase wavelet estimation. To solve this problem, we propose a new phase spectral method based on conformal mapping in the bispectral domain. The method avoids the phase wrapping problems by narrowing the scope of the Fourier phase spectrum to eliminate the bispectral phase wrapping influence in the original phase spectral estimation. The method constitutes least-squares wavelet phase spectrum estimation based on conformal mapping which is applied to mixed-phase wavelet estimation with the least-squares wavelet amplitude spectrum estimation. Theoretical model and actual seismic data verify the validity of this method. We also extend the idea of conformal mapping in the bispectral wavelet phase spectrum estimation to trispectral wavelet phase spectrum estimation.

Conformal analysis of fundamental frequency of vibration of simple-supported elastic ellipse-plates with concentrated substance

Aimed at calculating the fundamental frequency of vibration of special-shaped, simple-supported elastic plates, Conformal Mapping theory is applied, and the mathematical method of trigonometric interpolation with interpolation points mutual iterative between odd and even sequences in boundary region is provided, as well as the conformal mapping function which can be described by real number region between complicated region and unit dish region is carried out. Furthermore, in the in-plane state of constant stress, vibrating function is completed by unit dish region method for simple-supported elastic plates with concentrated substance of complicated vibrating region, and the coefficient of fundamental frequency of the plate is analyzed. Meanwhile, taking simple-supported elastic ellipse-plates as an example, the effects on fundamental frequency caused by eccentric ratio, the coefficient of constant in-plane stress, as well as the concentrated substance mass and positions are analyzed respectively.

An analytical model for coplanar waveguide on silicon-on-insulator substrate with conformal mapping technique

In this paper, the authors present an analytical model for coplanar waveguide on silicon-on-insulator substrate. The four-element topological network and the conformal mapping technique are used to analyse the capacitance and the conductance of the sandwich substrate. The validity of the model is verified by the full-wave method and the experimental data. It is found that the inductance, the resistance, the capacitance and the conductance from the analytical model show they are in good agreement with the corresponding values extracted from experimental Sparameter until 10 GHz.

Precise method to control elastic waves by conformal mapping

The transformation method to control waves has received widespread attention in electromagnetism and acoustics. However, this machinery is not directly applicable to the control of elastic waves, because it has been shown that the Navier＇s equation does not usually retain its form under coordinate transformation. In this letter, we prove the form invariance of the Navier＇s equation under the conformal mapping based on the Helmholtz decomposition method. The needed material parameters are provided to manipulate elastic waves. The validity of this approach is confirmed by an active stealth device which can disguise the signal source by changing its position. Experimental verifications and potential applications may be expected in nondestructive testing, structural seismic design and other fields.

Some Extension on the Kellogg Theorem

In this paper,by using the tools of second order smooth module,we discusses the conformal mapping from a unit circle onto a simply connected domain enclosed by a smooth Jordan curve L ,and further improves the Kellogg theorem.

A Dugdale-Barenblatt model for a strip with a semi-infinite crack embedded in decagonal quasicrystals

The present study is to determine the solution of a strip with a semi-infinite crack embedded in decagonal quasicrystals,which transforms a physically and mathematically daunting problem.Then cohesive forces are incorporated into a plastic strip in the elastic body for nonlinear deformation.By superposing the two linear elastic fields,one is evaluated with internal loadings and the other with cohesive forces,the problem is treated in Dugdale-Barenblatt manner.A simple but yet rigorous version of the complex analysis theory is employed here,which involves a conformal mapping technique.The analytical approach leads to the establishment of a few equations,which allows the exact calculation of the size of cohesive force zone and the most important physical quantity in crack theory：stress intensity factor.The analytical results of the present study may be used as the basis of fracture theory of decagonal quasicrystals.

An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors(SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole(a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.

Interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional hexagonal quasicrystal with piezoelectric effect

The interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional(1 D) hexagonal quasicrystal with piezoelectric effect is considered. A general formula of the generalized stress field, the field intensity factor, and the image force is derived, and the special cases are discussed. Several numerical examples are given to show the effects of the material properties and the dislocation position on the field intensity factors and the image forces.

INTEGRAL EQUATION METHOD'S APPLICATION IN HOLE-EDGE STRESS OF COMPOSITE MATERIAL PLATE WITH DIFFERENT SHAPED HOLES

The strength of composite plate with different hole-shapes is always one of the most important but complicated issues in the application of the composite material. The holes will lead to mutations and discontinuity to the structure. So the hole-edge stress concentration is always a serious phenomenon. And the phenomenon makes the structure strength decrease very quickly to form dangerous weak points. Most partial damage begins from these weak points. According to the complex variable functions theory, the accurate boundary condition of composite plate with different hole-shapes is founded by conformal mapping method to settle the boundary condition problem of complex hole-shapes. Composite plate with commonly hole-shapes in engineering is studied by several complex variable stress fimction. The boundary integral equations are founded based on exact boundary conditions. Then the exact hole-edge stress analytic solution of composite plate with rectangle holes and wing manholes is resolved. Both of offset axis loadings and its influences on the stress concentration coefficient of the hole-edge are discussed. And comparisons of different loads along various offset axis on the hole-edge stress distribution of orthotropic plate with rectangle hole or wing manhole are made. It can be concluded that hole-edge with continuous variable curvatures might help to decrease the stress concentration coefficient; and smaller angle of outer load and fiber can decrease the stress peak value.

GREEN FUNCTIONS FOR A DECAGONAL QUASICRYSTALLINE MATERIAL WITH A PARABOLIC BOUNDARY

This investigation presents the Green functions for a decagonal quasicrystalline ma- terial with a parabolic boundary subject to a line force and a line dislocation by means of the complex variable method. The surface Green functions are treated as a special case, and the explicit expressions of displacements and hoop stress at the parabolic boundary are also given. Finally, the stresses and displacements induced by a phonon line force acting at the origin of the lower half-space are presented.

Analytic solutions to a finite width strip with a single edge crack of two-dimensional quasicrystals

In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modesⅠand Ⅱ using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_Ⅰ and K_Ⅱ, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to analyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With the method of complex variable function, a series' of conformal mapping functions are obtained from different cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And, the general expressions for the equations of a cylindrical shell, including the solutions of stress concentrations meeting the boundary conditions of the large openings' edges, are given in the mapping plane. Furthermore, by applying the orthogonal function expansion technique, the problem can be summarized into the solution of infinite algebraic equation series. Finally, numerical results are obtained for stress concentration factors at the cutout's edge with various opening's ratios and different loading conditions. This new method, at the same time, gives a possibility to the research of cylindrical shells with large non-circular openings or with nozzles.

Multi-objective Optimal Design of Vibration Absorber with Simple Cell Mapping Algorithm

In this paper, we investigate a multi-objective optimal design of a tuned vibration absorber for a two degrees-of-freedom linear system, in which we take white Gaussian noise as the excitation. We use the simple cell mapping method to obtain solutions to the multi-objective optimization problem. We optimize tuned vibration absorbers in two different configurations. As our objective, we take various features of the response of the power spectrum density function. Our results show that the primary mass vibration with the optimally designed absorber has a lower peak power spectrum value. In addition, the two natural frequencies of the optimized system are further apart than in the non-optimized system, which implies that the absorber has a wider effective region in the frequency domain. © 2017, Tianjin University and Springer-Verlag GmbH Germany.

A NEW SOLUTION FOR THERMOPIEZOELECTRIC SOLID WITH AN INSULATED ELLIPTIC HOLE

A new solution is obtained for thermal analysis of insulated elliptic hole embedded in an infinite thermopiezoelectric plate. In contrast to our previous re- sults, the present formulation is based on the use of exact electric boundary conditions at the rim of the hole, thus avoiding the common assumption of electric impermeabil- ity. Using Lekhnitskii's formulation and conformal mapping, the elastic and electric fields can be expressed in a closed form in terms of complex potentials. The solutions for the crack problem are obtained by setting the minor axis of the ellipse approach to zero. As a consequence, the stress and electric displacement (SED) intensity factors and strain energy release rate can be derived analytically. One numerical example is considered to illustrate the application of the proposed formulation and compare with those obtained from impermeable model.

SCATTERING OF PLANE SH-WAVE BY A CYLINDRICAL HILL OF ARBITRARY SHAPE

The problems of scattering of plane SH-wave by a cylindrical hill of arbitrary shape is studied based on the methods of conjunction and division of solution zone. The scattering wave function is given by using the complex variable and conformal mapping methods. The conjunction boundary conditions are satisfied. Furthermore appling orthogonal function expanding technique, the problems can finally be summarized into the solution of a series of infinite algebraic equations. At last, numerical results of surface displacements of a cylindrical arc hill and of a semi-ellipse hill are obtained. And those computational results are compared with the results of finite element method (FEM).

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping, the diffraction of flexural waves and dynamic stress concentrations in thick plates with a cavity have been studied. A general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of an arbitrary cavity is obtained. By employing the orthogonal function expansion technique, the dynamic stress problem can be reduced to the solution of an infinite algebraic equation series. As an example, the numerical results for the dynamic stress concentration factor in thick plates with a circular, elliptic cavity are graphically presented. The numerical results are discussed.

Parabolic Wave Propagation Modelling in Orthogonal Coordinate Systems

This paper presents a numerical model study of the propagation of water waves using the parabolic approximation of the mild slope equation in the orthogonal coordinate system. Two types of coordinate systems are studied: (a) a general form of orthogonal coordinate system and (b) the conformal system, a special form of orthogonal coordinate system. Two typical examples, namely, expanded breakwaters and a circular channel, are studied to validate the model. First, the examples are studied by use of the general orthogonal coordinates. Then the same examples are computed by use of the conformal system. The computational results show that the conformal coordinate system generally gives better predictions than the general orthogonal system. A numerical technique for generating the conformal grid is combined with the numerical model to improve the practicability of the model. The comparison between the result from the numerical grid system and that from the analytical grid system shows that reliable computational results can be obtained by use of the numerical conformal grid system.

Mapping analysis of vibrating fundamental frequency for simple-supported elastic rectangle-plates with concentrated mass

By conformal mapping theory, a trigonometric interpolation method between odd and even sequences in rectangle boundary region was provided, and the conformal mapping function of rectangle-plate with arc radius between complicated region and unite dish region was carried out. Aiming at calculating the vibrating fundamental frequency of special-shaped, elastic simple-supported rectangle-plates, in the in-plane state of constant stress, the vibration function of this complicated plate was depicted by unit dish region. The coefficient of ftmdamental frequency was calculated. Whilst, taking simple-supported elastic rectangle-plates with arc radius as an example, the effects on fundamental frequency caused by the concentrated mass and position, the ratio of the length to width of rectangle, as well as the coefficient of constant in-plane stress were analyzed respectively.

Some Geometric Properties of the m-Möbius Transformations

Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations fm mapping onto . Even for the simplest entity, the pre-image by fm of a unique point, there is no way of visualization. Pre-images by fm of figures from C are like ghost figures in Cm. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in Cm are well known and vector calculus in Cm is familiar, yet the pre-image by fm of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in Cm. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.