Coefficients Calculation of Laurent Series in the Neighborhood of Complex Variable Function Poles

The theory of complex function is a key part of mathematics, which can solve the complex problems in production and life. It is of great significance to extend the research field of complex function theory. In this paper, taking a complex variable function as the research object, a calculation method of Laurent series coefficient of complex function pole neighborhood expansion was proposed to determine the complex variable function pole, determine the order of complex variable function pole, calculate the residue of high-order pole in complex variable function, thus judging the attribute of complex variable function. In this regard, the coefficient formula was used to calculate the coefficients of Laurent series in the neighborhood of the complex variable function poles.

Study on the Influence of Tectonic Stress on Stability of Horseshoe-Shaped Tunnels

While the tunnel is in the high tectonic stress environment and surrounding rock of tunnel has the characteristics of soft texture and stronger expansion, the preference of tunnel shape is horseshoe. An elastic-plastic model is analyzed by complex function theory in accordance with the deformation characteristics of a horseshoe-shaped tunnel in an engineering site. The numerical model of the tunnel is built by FLAC3D, and the influence of the magnitude and direction of structural stress on the horseshoe-shaped tunnel is studied in detail. Finally, the security support of the tunnel is discussed. Results show that the stress concentration phenomenon is easily focused on the left, right, and bottom sides of the tunnel; these places should therefore be the focus of attention of tunnel stability analysis. The magnitude and direction of tectonic stress greatly affect the stability of the horseshoe-shaped tunnel. Similarly, the magnitude of tectonic stress can significantly affect the deformation state of the tunnel. The direction of tectonic stress mainly reflects the orientation of the tunnel. In addition, the orientation of the tunnel should be arranged along the maximum direction of principal stress.

Coupling of Complex Function Theory and Finite Element Method for Crack Propagation Through Energetic Formulation: Conformal Mapping Approach and Reduction to a Riemann–Hilbert Problem

In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann–Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.

Study on mechanical response and crack development law of tunnel

Considering the linear elastic fracture mechanics and complex function theory, considering the non hydrostatic pressure field, the stress solutions of lining and surrounding rock of deep buried circular underground cavern with lining under the action of external internal pressure are studied. The fracture mechanics model of underground cavern supporting structure is established, and the numerical verification is carried out. The analytical solution of circular pressure cavern with lining shows that when the internal pressure is 0, it can degenerate into the existing classical solution. The results of finite element calculation and analytical calculation show that the tensile stress of lining is very large at the vault and arch bottom under the action of strong external supporting force. The calculation shows that the cracks of lining will expand and open under the action of tensile stress, and the lining is in the shape of “flat duck egg”, However, if there is no through rupture, there will be great stress concentration at the top and bottom of the arch, and the tensile stress value is much larger than that calculated by elastic mechanics. Therefore, it is a good method to configure a certain amount of reinforcement to enhance the stiffness and tensile strength of the lining.

Representation of Maximal Surfaces in a 3-Dimensional Lightlike Cone

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone Q3 is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via the representation formula.