Complex Variable Theory Research Articles

Analytical elastic stress solution and plastic zone estimation for a pressure-relief circular tunnel using complex variable methods

It is likely that a great number of excavation-induced failures, including rock burst, are activated in a highly anisotropic in situ stress field. Pressure-relief technologies are employed popularly as effective approaches of treating the induced failures, but there is a lack of study on their mechanisms in terms of the relief effect and distribution of the plastic zone. This paper proposes a pressure-relief method of grooving relief slots in the roof and floor of a circular opening considering the non-uniform stress distribution around a circular opening under anisotropic far-field stress. A conformal mapping method and complex variable theory are adopted to obtain a closed-form solution for released openings, stress ratios of the unreleased to released openings are used to describe the release effect, and effects of the lateral pressure coefficient and slot depth are discussed. Additionally, a negative exponential relation between the maximum stress ratio and the rock depth is found. On the basis of the elastic solution, this paper presents an approach of finding the distribution of the plastic zone around a non-circular opening for an anisotropic in situ stress state using the Mohr–Coulomb criterion, and investigates the release effect on the plastic zone for different lateral pressure coefficients and slot depths.

Stress intensity factors for cracks emanating from a circular hole in an infinite quasi‐orthotropic plane

AbstractAn infinite quasi‐orthotropic plane with a cracked circular hole under tensile loading at infinity is studied analytically. To this end, complex variable theory of Muskhelishvili is used. In addition, to obtain analytical functions, a new conformal mapping is proposed and expanded to series expressions. Stress intensity factors (SIFs) for two unequal cracks emanating from a circular hole are obtained. To validate the analytical SIFs in a quasi‐orthotropic plane, the results are compared with FEM and the results of isotropic plane. The SIFs for small cracks in a quasi‐orthotropic and an isotropic plane are different, because of difference between stress concentrations in points which cracks emanate from the hole. However, the results of quasi‐orthotropic plane converge to isotropic plane for the large cracks. Therefore, the SIFs of the large cracks in a quasi‐orthotropic plane can be replaced by the results of the center crack with equivalent length in an isotropic plane.

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACTAn analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.

An accessible, justifiable and transferable pedagogical approach for the differential equations of simple harmonic motion

ABSTRACTRecently, Gauthier introduced a method to construct solutions to the equations of motion associated with oscillating systems into the mathematics education research literature. In particular, Gauthier's approach involved certain manipulations of the differential equations; and drew on the theory of complex variables.Motivated by the work of Gauthier, we construct an alternative pedagogical approach for the learning and teaching of solution methods to these equations. The innovation lies in drawing on factorization techniques of differential equations and harmonizing them with Gauthier's approach of the theory of complex variables. When blended together to form a new approach, the significance lies in its accessibility, justifiability and transferrability to other problems.We pedagogically ground our approach in the educational development theory of Piaget, with the results informing the learning and teaching of solution methods to differential equations for lecturers, teachers and learners within universities, colleges, polytechnics and schools around the world.

Analytical stress and displacement around non-circular tunnels in semi-infinite ground

The construction of non-circular tunnels at shallow depths is becoming increasingly common to efficiently use excavation space. However, the stress concentration and ground movement are probably much greater around shallow non-circular tunnels than around deep ones; the greater stress and ground movement have significant negative effects on tunnel stability and the safety of nearby structures. This study provides a methodology for obtaining the analytical solutions of ground response induced by non-circular tunnel excavation. The surcharge loadings and gravity induced initial stresses, as well as the internal forces due to liners, are considered in the derivation. Using the Schwartz alternating method combined with complex variable theory, a series of two types of problems on simply connected domain, i.e., a non-circular hole in an infinite plane and a half-plane without holes, is in turn successfully solved. Convergent and accurate analytical solutions are finally achieved by superposing the solutions in all the alternating iterations. The analytical solutions exhibit a close agreement with the numerical results, except for the horizontal displacement along the line near the ground surface. Finally, a parametric study is performed for the elliptical/square tunnel subjected to infinite surcharge loadings. The proposed analytical solutions have great value for the conceptual understanding of the mechanical behavior of non-circular tunnels and for verifying the numerical models. Moreover, an alternative approach is provided for the preliminary designs of future shallow tunnels excavated in rock or medium/stiff clay.

Combined Bending with Tension of Isotropic Plate with Crack Considering Crack Banks Contact and Plastic Zones at its Tops

Abstract Stress-strain state of isotropic plate with rectilinear through-crack at combined action of bending and tension, realized by applying distributed forces and bending moments at infinity, the vectors of which are parallel and perpendicular to the crack, is investigated. Under the influence of the internal stress the crack faces contacts on area of constant width near the upper base of plate, and plastic zones forms in its tips. Using methods of the theory of complex variables, complex potentials plane problem of elasticity theory and the classical theory of plates bending, solving of the problem is reduced to the set of linear conjugation problems and their analytical solution is built in a class of functions of limited plastic zones in the crack tips. The conditions of existence of the solution of the problem in these terms are determined. Using Treska plasticity conditions in the form of surface layer or the plastic hinge, the length of plastic zone and crack opening displacement are found analytically. Their numerical analysis for various parameters of the problem is conducted.

The Neumann Problem for the k-Cauchy–Fueter Complex over k-Pseudoconvex Domains in $$\mathbb {R}^4$$ R 4 and the $$L^2$$ L 2 Estimate

The k-Cauchy–Fueter operator and complex are quaternionic counterparts of the Cauchy–Riemann operator and the Dolbeault complex in the theory of several complex variables, respectively. To develop the function theory of several quaternionic variables, we need to solve the non-homogeneous k-Cauchy–Fueter equation over a domain under the compatibility condition, which naturally leads to a Neumann problem. The method of solving the $$\overline{\partial }$$ -Neumann problem in the theory of several complex variables is applied to this Neumann problem. We introduce notions of k-plurisubharmonic functions and k-pseudoconvex domains, establish the $$L^2$$ estimate and solve the Neumann problem over k-pseudoconvex domains in $$\mathbb {R}^4$$ . Namely, we get a vanishing theorem for the first cohomology group of the k-Cauchy–Fueter complex over such domains.

Analytical Solution for 2D Inter-Well Porous Flow in a Rectangular Reservoir

Inter-well fluid flows through porous media are commonly encountered in the production of groundwater, oil, and geothermal energy. In this paper, inter-well porous flow inside a rectangular reservoir is solved based on the complex variable function theory combined with the method of mirror images. In order to derive the solution analytically, the inter-well flow is modeled as a 2D flow in a homogenous and isotropic porous medium. The resulted exact analytical solution takes the form of an infinite series, but it can be truncated to give high accuracy approximation. In terms of nine cases of inter-well porous flow associated with enhanced geothermal systems, the applications of the obtained analytical solution are demonstrated, and the convergence properties of the truncated series are investigated. It is shown that the convergent rate of the truncated series increases with the symmetric level of well distribution inside the reservoir, and the adoption of Euler transform significantly accelerates the convergence of alternating series cases associated with asymmetric well distribution. In principle, the analytical solution proposed in this paper can be applied to other scientific and engineering fields, as long as the involved problem is governed by 2D Laplace equation in a rectangular domain and subject to similar source/sink and boundary conditions, i.e., isolated point sources/sinks and uniform Dirichlet or homogeneous Neumann boundary conditions.

Exact Solution of the Two-Dimensional Problem on an Impact Ideal-Liquid Jet

The two-dimensional problem on the collision of a potential ideal-liquid jet, outflowing from a reservoir through a nozzle, with an infinite plane obstacle was considered for the case where the distance between the nozzle exit section and the obstacle is finite. An exact solution of this problem has been found using methods of the complex-variable function theory. Simple analytical expressions for the complex velocity of the liquid, its flow rate, and the force of action of the jet on the obstacle have been obtained. The velocity distributions of the liquid at the nozzle exit section, in the region of spreading of the jet, and at the obstacle have been constructed for different distances between the nozzle exit section and the obstacle. Analytical expressions for the thickness of the boundary layer and the Nusselt number at the point of stagnation of the jet have been obtained. A number of distributions of the local friction coefficient and the Nusselt number of the indicated jet are presented.

Analytical solution and numerical verification for the pressure-relief method of a circular tunnel

This paper presents an elastic analytical solution to a circular tunnel with releasing slots at high stress areas near the hole by using a conformal mapping method and the complex variable theory. Compared to the original stress distribution around the circular hole, the releasing effect on elastic stresses is evaluated. After grooving slots, low stress area is generated where the high stress concentration is located. This is agreeable with what was predicted by the finite difference FLAC2D. Besides, displacements are obtained along the periphery of the released hole and are in accordance with those of FLAC2D. In addition to the intersection of the mapping contour, the influences of the sampling points distribution, series number in mapping function, and slot shape are discussed. It is inevitable that the mapping accuracies for the slot and the circle cannot be satisfied at the same time The mapping effect on the circle has to be considered primarily since the stress distribution around the circle is much more significant than the tunnel stability. The analytical solution can be available and fast method of estimating the releasing effect of the application on the tunnel without rock parameters.

Problem on extremal decomposition of the complex plane

The paper is devoted to one extremal problem in geometric function theory of complex variables associated with estimates of functionals defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the power of the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper we generalized it to the case of the more general system of points and obtained a solution of this problem for some concrete values of n and y.

The Cauchy-Riemann equation in Clifford analysis of several variables

The aim of this article is to extend the theory of several complex variables to the non-commutativerealm. In this paper,we solve the non-homogeneous Cauchy-Riemann equation under the differential compatibility condition, find the explicit integral representation of the solution and derive the Hartogs theorem in Clifford analysis of several variables. This will generalize the corresponding results in several complex variables.

Frost heaving and frost cracking of elliptical cavities (fractures) in low-permeability rock

A 9% expansion of freezing water in the elliptical cavities (fractures) of rock leads to frost heaving pressure and thermal stress, which are the main cause of the freeze-thaw damage of a fractured rock mass. Considering the influence of thermal stress and water flow in elliptical cavities (fractures), a mathematical model for predicting frost heaving pressure in the elliptical cavities (fractures) of rock is presented, assuming that the rock matrix is impermeable. The maximum tensile stress and frost cracking angle on the elliptical cavity are deduced and the critical condition for simplifying the elliptical cavity to planar fracture is suggested based on the method of complex variable function and elastic mechanics theory. Moreover, the effects of the main parameters on frost heaving pressure and frost-cracking conditions are determined to investigate the frost cracking of water-filled elliptical cavities (fractures). The flow flux of water, the aspect ratios of cavities (fractures) and the elastic moduli of rock have a great direct influence on the frost heaving pressure and cracking angle, whereas the dip angle of the cavity (fracture) and the thermal expansion coefficient of rock indirectly affect them by changing the thermal stress. It is suggested that the stress intensity factor is adopted to predict the frost propagation of planar elliptical cavities only if the aspect ratio exceeds fifteen.

Analytical model of cracking due to rebar corrosion expansion in concrete considering the structure internal force

Based on the assumptions of uniform corrosion and linear elastic expansion, an analytical model of cracking due to rebar corrosion expansion in concrete was established, which is able to consider the structure internal force. And then, by means of the complex variable function theory and series expansion technology established by Muskhelishvili, the corresponding stress component functions of concrete around the reinforcement were obtained. Also, a comparative analysis was conducted between the numerical simulation model and present model in this paper. The results show that the calculation results of both methods were consistent with each other, and the numerical deviation was less than 10%, proving that the analytical model established in this paper is reliable.

Derivation and application of an analytical rock displacement solution on rectangular cavern wall using the inverse mapping method.

Rectangular caverns are increasingly used in underground engineering projects, the failure mechanism of rectangular cavern wall rock is significantly different as a result of the cross-sectional shape and variations in wall stress distributions. However, the conventional computational method always results in a long-winded computational process and multiple displacement solutions of internal rectangular wall rock. This paper uses a Laurent series complex method to obtain a mapping function expression based on complex variable function theory and conformal transformation. This method is combined with the Schwarz-Christoffel method to calculate the mapping function coefficient and to determine the rectangular cavern wall rock deformation. With regard to the inverse mapping concept, the mapping relation between the polar coordinate system within plane ς and a corresponding unique plane coordinate point inside the cavern wall rock is discussed. The disadvantage of multiple solutions when mapping from the plane to the polar coordinate system is addressed. This theoretical formula is used to calculate wall rock boundary deformation and displacement field nephograms inside the wall rock for a given cavern height and width. A comparison with ANSYS numerical software results suggests that the theoretical solution and numerical solution exhibit identical trends, thereby demonstrating the method’s validity. This method greatly improves the computing accuracy and reduces the difficulty in solving for cavern boundary and internal wall rock displacements. The proposed method provides a theoretical guide for controlling cavern wall rock deformation failure.

On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations

We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N.Q. Le.Since our results are shown to apply to the convex functions |x|p with p≥2 and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot–Carathéodory metric in the study of certain subelliptic PDEs.

On the weighted L2 estimate for the k-Cauchy–Fueter operator and the weighted k-Bergman kernel

The k-Cauchy–Fueter operators, k=0,1,…, are quaternionic counterparts of the Cauchy–Riemann operator in the theory of several complex variables. The weighted L2 method to solve Cauchy–Riemann equation is applied to find the canonical solution to the non-homogeneous k-Cauchy–Fueter equation in a weighted L2-space, by establishing the weighted L2 estimate. The weighted k-Bergman space is the space of weighted L2 integrable functions annihilated by the k-Cauchy–Fueter operator, as the counterpart of the Fock space of weighted L2-holomorphic functions on Cn. We introduce the k-Bergman orthogonal projection to this closed subspace, which can be nicely expressed in terms of the canonical solution operator, and its matrix kernel function. We also find the asymptotic decay for this matrix kernel function.

Fluid Mechanics and Complex Variable Theory: Getting Past the 19th Century

The subject of fluid mechanics is a rich, vibrant, and rapidly developing branch of applied mathematics. Historically, it has developed hand-in-hand with the elegant subject of complex variable theory. The Westmont College NSF-sponsored workshop on the revitalization of complex variable theory in the undergraduate curriculum focused partly on identifying scientific areas where complex variables plays an active role. The author presented a lecture on this topic and here we highlight parts of that presentation and follow-up discussion.

Estimates of products of inner radii of non-overlapping domains in the complex plane

This paper is devoted to one open extremal problem of Dubinin in geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the power of the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper, we obtained a solution of this problem for some concrete values of n and .

Robust and Adaptive Model Predictive Control of Nonlinear Systems [Bookshelf

This book provides a comprehensive study of nonlinear adaptive robust model predictive control (MPC). Chapters 2–5 present a framework for the analysis and synthesis of nonlinear robust MPC. This framework includes the treatment of robustness, computation methods, and performance improvement. Chapters 6–7 show how to develop the basic ideas for the design and analysis of the nonlinear adaptive robust MPC. One of the key techniques is the set-based approach, in which the internal model identifier allows the MPC to compensate for future changes in the parameter estimates and uncertainty associated with the unknown model parameters. Chapters 8–12 illustrate how to implement the synthesis approaches for nonlinear adaptive robust MPC, and a robust adaptive economic MPC is also proposed. This text also gives a finite-time identification method, which can be used to estimate the unknown parameters in finite time, provided a persistence of excitation (PE) condition is satisfied. This identification method is particularly effective in the online implementation of MPC. The early chapters study continuous-time systems, and Chapters 13–14 extend the set-based estimation and robust adaptive MPC to discrete-time problems. While adaptive robust MPC is an improvement on robust MPC, this book shows that feedback MPC can be used to improve the open-loop MPC. At each sampling instant, a sequence of parameter estimates can be performed/invoked to improve the control performance. Economic MPC is also incorporated so as to improve the control performance in a broader way. This book is intended for someone learning functions of a complex variable and who enjoys using Matlab. It will enhance the experience of learning complex-variable theory and will strengthen the knowledge of someone already trained in this branch of advanced calculus. Supplying students with a bridge between the functions of complex-variable theory and Matlab, this supplemental text enables instructors to easily add a Matlab component to their complex-variables courses. The book shows students how Matlab can be a powerful learning aid in such staples of complex-variable theory as conformal mapping, infinite series, contour integration, and Laplace and Fourier transforms. In addition to Matlab programming problems, the text includes many examples in each chapter along with Matlab code.