Complex Variable Theory Research Articles

Antiplane effective properties of two‐phase micropolar elastic fiber‐reinforced composites with parallelogram‐like unit cells

AbstractIn this contribution, heterogeneous micropolar elastic fiber‐reinforced composites (FRCs) with a periodic structure are analyzed using the two‐scale asymptotic homogenization method (AHM). We focus on predicting the antiplane effective properties of micropolar two‐phase FRCs with parallelogram‐like unit cells. The periodic structure is defined by unidirectional, infinitely long, and concentric cylindrical fibers embedded in a homogeneous matrix. Constituent materials are assumed centro‐symmetric isotropic materials, and perfect interface conditions are considered. The AHM allows us to address the local problems on the periodic cell and determine the corresponding effective properties. This is achieved by employing two‐scale asymptotic expansions for the displacement and microrotation fields, which depend on both macro‐ and micro‐scales. The complex variable theory, combined with the complex‐potential method and doubly periodic Weierstrass elliptic functions, is applied to determine the solution of the antiplane local problems. Simple closed‐form formulas are provided for the antiplane stiffness and torque effective properties of two‐phase micropolar elastic FRCs, which depend on the physical properties and volume fractions of constituents. Finally, numerical examples are reported and discussed. Comparisons with other theoretical models are also presented, and good agreements are obtained.

Failure Prediction of an Airfoil-Type Crack in Thermoelectric Coupling Materials

Airfoil-type cracks with curved surfaces and sharp tips are frequently encountered in modern engineering applications and the aerospace industry. However, such cracks significantly threaten structures due to stress concentration and further cause damage. This study investigates the theoretical analysis and failure prediction of airfoil-type cracks within thermoelectric coupling materials under the interaction of remote mechanical loads, heat flux and electric current density. By employing complex variable theory, conformal mapping methods, and the analytic continuation theorem, the full-field stress and stress intensity factors of an airfoil-type crack were determined under three different loading conditions. The results indicate that when the airfoil-shaped crack tip is subjected to horizontal compression, vertical tensile mechanical load, and both heat flux and current density parallel to the crack, the crack tip would reach a dangerous state. Additionally, the full-field stress distribution reveals stress concentration around the crack tip, which explains variations in SIFs under different external forces and shape factors. Four critical situations suppressing crack propagation under mixed loading conditions are proposed. Furthermore, using an acrylic plate and bismuth telluride alloy as examples, the critical values of external loading for the mixed-mode fracture are determined based on the strain energy density criterion. These critical values can be used to predict failure initiation, thus reducing the risk of structural failure due to airfoil cracks within thermoelectric coupling materials.

Study on the effective drainage range of long horizontal borehole for goaf gas drainage

AbstractLaying long horizontal borehole (LHB) in mining‐induced strata fractures for coal‐bed gas drainage in the goaf is a pivotal method for eliminating safety hazards, preventing atmospheric pollution, and realizing gas utilization. A crucial premise of this method is to determine the effective drainage range of the borehole, a key parameter directly affecting boreholes spacing and drainage efficiency. In this study, a model of gas seepage during LHB drainage within a circular equipotential boundary was presented, and the complex variable function theory and the mirror principle were adopted for deriving the complex potential function of gas flow under the condition of LHB drainage and the drainage volume function of a single center borehole. Besides, the main indicator to determinate the effective drainage range of LHB was given. Specifically, the indicator refers to the distance between the turning point (the point at which the change rate of the gas flow velocity is −0.002 around the borehole) and the center of the borehole. Subsequently, numerical simulation was conducted to obtain the effective drainage range of LHB within mining‐induced fractures of overlying strata, that is, 4 m < r < 7 m. Furthermore, discussion of the effective drainage range of LHB was carried out by analyzing gas flow velocity distributions and LHB drainage volumes. A case study demonstrates that a double‐borehole layout with a 10 m spacing can effectively control pressure‐relief gas in the goaf. Meanwhile, such a layout can achieve well‐balanced drainage efficiency of each borehole with a minimal difference of 0.31 m3 min−1, which is only about 7.9% of the average drainage volume of the two boreholes. The above results verify that the determined effective drainage range of LHB is reasonable. The research results can provide references for determining the drainage range of LHB in multi‐borehole layout of coal mines, and then determining the spacing of LHBs arrangement, thereby improving the efficiency of gas extraction.

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Stress Characteristics and Mechanical Behavior of Rock Masses with an Opening under Complex Deep Underground Stress Conditions

In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (λ) and the angle (β) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When λ < 1.0 and β = 45°, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.

Non‐iterative stress‐displacement solution for surrounding rock mass with multiple non‐circular deep tunnels

AbstractA non‐iterative stress‐displacement solution is proposed in this paper to investigate surrounding rock mass with multiple non‐circular deep tunnels. Compared with the Schwarz alternating method, the proposed method only spends one‐time matrix operation, showing considerable efficiency and accuracy of the matrix solution. First, a matrix solution is formulated to obtain the conformal mapping function for the non‐circular tunnel. Then, in consideration of multiple boundary conditions, the generalized complex variable theory and fast Fourier transform (FFT) algorithm are used to formulate a matrix solution of potential functions. The stress and displacement fields around multiple non‐circular tunnels can be further determined from the matrix solution. A series of numerical examples are conducted to verify the proposed method, perform the convergency study, and discuss the effects of tunnel geometries, distances, and arrangements. The convergency study shows that the more accurate conformal mapping function requires more sampling points in the FFT. In addition, the high stress zone would be amplified if the arrangement of multiple tunnels makes the high stress zone adjacent or overlapped.

Buckling optimization of variable-stiffness composite plates with two circular holes using discrete Ritz method and potential flow

Potential flow around two equal-radius cylinders is derived analytically and applied to generate the curvilinear fiber path of variable-stiffness composite (VSC) plates with two circular holes. As complex variable theory and conformal mapping are used to generate the potential flow around two equal-radius cylinders, the location and size of the two equal-radius circular holes are arbitrary. By changing the angle of incoming flow, the global fiber angle of a variable-stiffness lamina can be simulated and the local fiber orientation angle at any point is determined by the global potential flow field. Buckling performance of variously-shaped VSC plates with two circular holes are investigated via a novel numerical method − discrete Ritz method (DRM). DRM combines the extended interval integral, Gauss quadrature, and variable stiffness within a rectangular domain and builds a discrete energy system to simulate the plate, which allows the geometric boundaries of the plate to vary. The strain energy of a plate is modeled in the rectangular domain, discretized using Gauss points, and is characterized by variable stiffness, which characterizes both the material distribution and plate geometry simultaneously. After that, a three-dimensional sampling optimization (3DSO) method is adopted to optimize the curvilinear fiber configurations of VSC plates, and its buckling performances are compared with those of constant stiffness composites (CSC) with optimal straight fibers. Significant improvements on load-carrying capacity can be achieved compared to straight ones, demonstrating that using potential flow is one of the most efficient way to generate curvilinear optimal fiber path with maximum load-carrying capacity for VSC plates with material discontinuity. Moreover, DRM exhibits good precision and stability for buckling analysis of VSC plates with complex geometries.

A matrix-form complex variable method for multiple non-circular tunnels in layered media

This paper presents a matrix-form complex variable method that can predict the stress and displacement field around multiple tunnels in layered media. The proposed matrix-form method avoids complicated coefficient determinations and iterative algorithm, which effectively improves the computation efficiency. Firstly, the expressions for the stress and displacement field are given under the generalized complex variable theory. The conformal mapping technique and fast Fourier transform algorithm are used to formulate the matrix-form expressions of boundary conditions. Considering the boundary conditions of the tunnels and soil interfaces, the global matrix solution for multiple tunnels in layered media are obtained by assembling the matrix solutions of all single layers. The present theory is verified by comparison with the existing literature, finite element analysis and field data in engineering practices, and several numerical examples are given to reveal the effects of elastic properties, tunnels' position, equivalent layer and the layer thickness surrounding tunnels.

John Charlton Polkinghorne, 1930–2021

AbstractJohn Charlton Polkinghorne was very successful in two careers: as a theoretical physicist and as a theologian. As a physicist at Cambridge, he was one of the pioneers of applying complex variable theory to the study of properties of scattering amplitudes and contributed to the theory of strong (nuclear) interactions. In addition, he was a successful teacher, with many of his students gaining senior academic positions in universities. A postmaster's son from Weston‐super‐Mare, he married Ruth Martin, also a mathematics student at Cambridge. They had three children. John surprised many, apart from those who knew him closely, by resigning from his Chair at Cambridge to study for ordination in the Anglican Church. He became an influential theologian devoting much study to the interaction of science and religion. His position might be summed up in a phrase he was fond of, “God holds the Universe in being”, which was John's solution to what is called the fine‐tuning problem in physics: that the parameters of Nature turn out to be exquisitely adjusted to allow our universe to exist in its richness. John was a formidable administrator, became President of Queens' College, Cambridge, and chaired UK government committees on research ethics.

Stress distribution and failure characteristics around U-shaped caverns with different height-to-width ratios under biaxial compression

Understanding the stress distribution and failure characteristics around a U-shaped cavern is vital to the stability analysis of rock structures such as tunnels and roadways. In this study, the stress distribution around a U-shaped cavern with different height-to-width (h/w) ratios under biaxial compression was first analyzed based on the complex variable theory. Then, numerical simulations were conducted using PFC2D to study the failure process and energy evolution around a U-shaped cavern with different h/w ratios. Theoretical solution shows that the maximum stress concentration factor occurs at the bottom corner of the U-shaped cavern. With the h/w ratio increasing from 0.75 to 3.0, the stress concentration factor around the sidewall shows a declining trend, while the opposite trend is observed in the vault and floor. Numerical results show that when the maximum principal stress is parallel to the vertical direction, the thin slab buckling occurs in the sidewall of the cavern with low h/w ratios (h/w = 0.75 and 1.0), accompanied by the release of relatively high kinetic energy. Instead, the wedged-shaped slab spalling occurs in the sidewall of the cavern with high h/w ratios (h/w = 1.5, 2.0 and 3.0), which can be considered as a static process. Although the U-shaped cavern with low h/w ratios has higher integrity after failure, more stain energy is accumulated in the remaining surrounding rock, which is likely to cause rockbursts under a higher stress state or dynamic disturbance. When the maximum principal stress is horizontal, the rock slab separates from the cavern floor and presents a reverse trend with the increasing h/w ratio. The above results indicate that the h/w ratio and the direction of maximum principal stress should be considered comprehensively in the design of the support system of U-shaped caverns.

Investigation of the Application of Complex Function Theory in Underground Mine Design: A Case Study

This study, with the engineering background of the design of a stope involving a sublevel mining method in a certain underground metal mine, explored the application of stress-solving methods based on the complex variable function theory in actual engineering. Three mathematical calculation models based on the functions of a complex variable were established. Through triangle interpolation, mapping functions of a plane with a roadway section and a plane with the stope section were determined. An improved Schwarz alternating method was adopted to study the stability of the roadway and the influence of an adjacent roadway from the perspective of the stress field. In addition, in light of the distribution characteristics of a gangue in the stope, the design parameters of a pillar were optimized, with the pillar’s optimal dimensions determined. The results showed that when the magnitudes of two far-field principal stresses in the rock mass are relatively close, the distribution of the surrounding rock stress around the roadway is more uniform, and tensile stress is less likely to occur. The excavation of a neighboring roadway exacerbates to some extent the side stress of the other roadway, especially the compressive stress concentration on the side closer to the neighboring roadway. However, when the distance between the two roadways is significantly larger than the roadway size, this effect is not pronounced. In the engineering case studied in this research, the thickness of the pillar is approximately linearly positively correlated with the safety factor of the pillar approximately linearly negatively correlated with the recovery rate. Overall, this research explored the application of the complex variable function theory in an underground mine design, demonstrating its accuracy and practicality.

Some Practical Applications of Discretely Tunable Couplers’ Matrix Model

The findings of a research of the practical application of a previously developed model of a decametric waves discrete coupler are presented. Due to the high accuracy achieved in describing the transforming properties of Coupler in the operating frequency range, together with the apparatus of the theory of complex variables, in particular with the method of conformal maps of fractional linear functions, it was possible to solve a number of applied problems. A method for calculating the area of consistent loads of discretely tunable power circuits of arbitrary structure has been developed. For completeness of the result, the case of matching a generator with a complex value of internal resistance with a given quality (with an acceptable value of the traveling wave coefficient) has been considered. The proposed approach to matching assumes that for any possible antenna load there must be appropriate combinations of discretely tunable elements of the Coupler power circuit that ensure matching with a given quality. The design ratio for images of arbitrary circles and straight lines conformally mapped by a matching quadrilateral to a given complex domain (i.e., expressions for calculating the coordinates of the circle center and its radius, as well as coefficients of the equation of straight lines) has been obtained. Studies of the matching properties of various circuits implemented on discrete elements have been carried out. The results of this study are important when choosing the structural design of the Coupler power circuit. Calculation formulas are obtained that allow us to estimate the relative losses of active power in the Coupler power circuit. This contributes to the reasonable choice of the best configuration option from several, similar in quality matching, thereby avoiding complicated temperature settings of the circuit elements. An exhaustive search method of tuning the discrete contour of the Coupler has been proposed, which is extremely simple and does not require heavy calculations with matrices and complex numbers.

Stress redistribution in a multilayer chamber for compressed air energy storage in abandoned coalmine: Elastic analytical insights and material choice

AbstractCompressed air energy storage (CAES) is attracting attention as one of large‐scale renewable energy storage systems. Its gas storage chamber is one of key components for its success. A successful utilization of an abandoned coalmine roadway depends on the stability of the gas storage chamber. The chamber is a multilayer structure and the redistribution of the stress and displacement in each layer is critical to the chamber stability. So far, this redistribution mechanism under any lateral pressure coefficient and internal air pressure has been unclear and thus a quick and easy evaluation on the CAES chamber stability becomes difficult. In this study, the redistributions of stress and displacement in each layer are analytically solved based on complex variable function theory. First, the stress and displacement in three CAES multilayer structures are obtained under any lateral pressure coefficient and internal air pressure. Then, a comprehensive parameter b1 is obtained to describe the redistribution of stress and displacement in rock mass, lining layer, and grout layer. Its linkage with lateral pressure coefficient, internal air pressure, and material properties is analytically expressed. Thirdly, an analytical expression is obtained for the influence range of internal air pressure on chamber stress. Finally, the role and selection of lining and grouting layers are explored at different lateral pressure coefficient and internal air pressure. It is found that the CAES chamber stability can be effectively and quickly evaluated by these analytical solutions and comprehensive parameters and enhanced by a material with low bulk modulus but high shear modulus.

A novel complex variable solution for the stress and displacement fields around a shallow non-circular tunnel

A novel complex variable solution is derived in this paper for the stress and displacement fields around a shallow non-circular tunnel. The proposed method allows explicit computations in matrix form and has advantages of considerable efficiency and high accuracy. Firstly, the stress and displacement boundary conditions for the shallow non-circular tunnel are obtained using the generalized complex variable theory and conformal mapping technique. Then, the fast Fourier transform algorithm is employed to effectively perform the Fourier series fitting for these boundary conditions, and matrix equations for the potential functions in the frequency domain are established. Later, a matrix solution for the potential functions is formulated to determine the stress and displacement fields around the shallow non-circular tunnel. Finally, select numerical analyses are conducted to verify the proposed method, discuss the key parameters in the algorithm, and reveal the effect of the acceleration strategy using FFT as well as the effect of non-circular geometries on the stress and displacement fields.

Application of generalised functions and Fourier’s integral transformation in modelling the effect of a moving load on a beam resting on an elastic foundation

The paper is dedicated to modelling the behaviour of an infinitely long beam resting on a solid elastic foundation under the concentrated moving load effect. The modelling is carried out in the context of determining the high-speed vehicle stock impact on the track structure. When developing the method of problem solving, the known elasticity theory factors, wave-propagation theory, complex variable theory, as well as integral transformations of generalised functions were used. The mobile concentrated force is represented by the Dirac delta function. The Fourier’s integral transformation is used to solve the equation of motion and determine the beam defect. Contour integration using Cauchy residue theorem is applied to calculate the defect values in the original space. The general dependence form the beam defect on the load velocity is determined. It is shown that the defect values in the beam increase with moving load increasing speed. It is shown how the moving load velocity affects the character of the beam defect time dependence. Solutions are given for Euler-Bernoulli, Rayleigh and Timoshenko models of beams on an elastic base. The considered mathematical models behaviour is compared. It is shown that the character of beam interaction with the moving load, the defect values, as well as the reaching critical velocities character do not differ significantly for the considered beam models. The character of reaching the maximum defect in the application area of the mobile load is analysed. It is shown that the maximum defect values of beams on an elastic base occur after the moving load. The presented methodology can be used to evaluate the impact of rolling stock on the track structure, including the design of high-speed lines.

An analytical approximate solution for stress and displacement around a shallow circular tunnel excavated under concave slope topography

For shallow buried tunnels under slope, overloading on the ground surface and unloading of excavation both significantly influence the stability of the tunnel surrounding rock and slope. This study proposes an analytical approximate solution for the stress and displacement around shallow buried circular tunnel under slope boundary. In derivation process, the original problem is divided into two models: Model A is a slope plane without any holes subjected to gravity and surcharge load along the ground surface, and Model B is a tunnel contained in a slope plane loaded by released stresses along the tunnel boundary. Furthermore, Model B is decomposed into another two models: Model B1 represents a slope without holes subjected to virtual surface force along the ground surface, and Model B2 is an infinite domain containing a tunnel loaded by another virtual surface force along its boundary. Two sets of virtual surface forces are then achieved by an iterative calculation. The solution, which strictly meets all stress boundary and compatibility conditions of the original problem, are ultimately solved by superposing the solution of Models A, B1 and B2, respectively obtained by the method proposed by the authors (Models A, B1) and complex variable theory (Model B2). The convergence of iteration process and good agreement between the derived and numerical results under the same assumptions verify the correctness of the proposed solution. Furthermore, a parametric investigation is performed to investigate the influence of surcharge load position, slope topography and tunnel position on the stress and displacement around the tunnel and near the ground. The proposed solution is of great value for the conceptual understanding of mechanical behavior of shallow tunneling in slope terrain and for verifying the numerical models. Moreover, the solution is useful for the preliminary design of tunnels at shallow depth under slope topography.

Creep monitoring and parameters inversion methods for rock salt in extremely deep formation

When underground salt caverns are used for natural gas storage, the pressure difference between the inside and outside of the cavern will cause continuous creep deformation of the surrounding rock. Understanding the creep characteristics and constitutive model parameters of rock salt in actual formations is of great significance for guiding the construction and safe operation of gas storage facilities. In this paper, a creep contraction calculation model for slender cylindrical rock salt caverns was established firstly, based on the theory of complex variables and viscoelasticity. This model can calculate the radial displacement of the surrounding rock at any moment on the cross-section of the rock salt wellbore. Subsequently, two indirect monitoring methods for deep rock salt creep using the open hole section of the well were proposed based on this model, which involves using the cavern pressure change after wellhead closure and the liquid overflow rate after wellhead opening to indirectly reflect the deformation of underground rock salt. These two sets of field experiments can be used together to determine the values of the constitutive model parameters of rock salt creep under actual stress conditions. This theoretical framework was applied in the Ningjin Salt Cavern Gas Storage Project in China. Pressure monitoring was conducted for 150 days, and wellbore fluid overflow was measured for 45 h in a rock salt well at a depth close to 3000 m. The calculated results showed good agreement with the field monitoring data, and the parameters of the Burgers creep model for rock salt under actual conditions were determined. This demonstrates the strong feasibility of the monitoring method and the validity of the computational model. Finally, using the obtained creep parameters, a quantitative analysis of the creep contraction of the wellbore was performed considering factors such as internal pressure, in situ stress, and Poisson's ratio of the rock salt.

Analytical stress and displacement of twin noncircular tunnels in elastic semi-infinite ground

In general, twin noncircular tunnels may cause high stress concentrations and large settlements because of the strong interaction between tunnels and their noncircular shape. In this study, new analytical solutions are presented to efficiently predict the elastic stresses and displacements around shallow twin noncircular tunnels in rocks. The key factors, i.e., the real interactions between tunnels, arbitrary tunnel shapes, arrangements and pressures exerted at the internal tunnel boundaries due to liners or water pressures, are fully taken into account.The Schwartz alternating method is used to reduce the twin noncircular tunnels problem to a series of problems where only one noncircular tunnel is contained in the half-plane. Then, the solutions to the problem of a noncircular tunnel in the half-plane are obtained by complex variable theory combined with conformal mapping. Finally, convergent and highly accurate analytical solutions of twin noncircular tunnels are achieved by superposing the solutions of the reduced single tunnel problems. The analytical solutions are verified by the good agreement between the analytical and FEM results under the same assumptions. The proposed analytical solution can help to reveal the mechanical mechanisms of ground responses due to twin tunnelling in an elastic semi-infinite ground.

A 3D method called mdMRS for post-processing Magnetic Resonance Spectroscopy data

The data obtained from a scanner for Magnetic Resonance Spectroscopy (MRS) is three-dimensional (3D) since the FID data from the scanner has spectrum values which are 2D complex numbers and 1D time values. This paper describes an MRS frequency-based post-processing method that has the advantage that it quickly determines the values needed for metabolite intensities and concentrations, even for most overlapping spectral terms.The method of this paper does not depend on area under a curve or a user-chosen basis set. As few as three valid points corresponding to a peak on the spectral curve are all that is needed, other than similar information on a reference metabolite, for concentration determination of the associated metabolite. All four of the key values of peak location, amplitude, phase angle, and damping constant are determined simultaneously with high accuracy, dependent upon noise, and with computational simplicity from only three complex-valued constants. The theory behind mdMRS uses the knowledge that the projection of the 3D spectrum to the complex plane is a simple circle. Several concepts from complex variables theory are important, such as the Linear Fractional Transformation (LFT) that maps the frequency axis to that circle. The duality linking an LFT with a 2 × 2 Moebius matrix enables a fast iteration process that sharpens the four key value estimates on each iteration. The iteration removes all other terms when considering one of them. Applied to initial estimates, this leads to increasingly accurate output.To be useful to clinicians and to researchers developing treatments based on metabolite concentrations, the goal for post-processing of the data include both fast and accurate computations, with speed sufficient to provide results “on console” and output provided as concentrations. Besides the number of protons associated with a metabolite, concentrations are determined using both amplitude and damping constant values. Since the methods of mdMRS provide both of those characteristics simultaneously, the time from data collection to metabolite concentration output is minimized. The name of this new post-processing method was chosen since the attributes of the method help bring that goal closer to reality for Medical Doctors.

Analytical Predictions on the Ground Responses Induced by Shallow Tunneling Adjacent to a Pile Group

Prevalent buildings are supported by pile foundations in urban areas, and the importance of nearby excavation prediction is indisputable due to various engineering accidents caused by the density of urban buildings and the complexity of the underground environment. Recently, a case of tunneling adjacent to a pile group has received a lot of attention from the research community and engineers. In this study, a mechanical model of a shallow tunnel adjacent to a pile group is established. The proposed stress-release function is taken as the stress boundary condition of the tunnel periphery. Considering the pile group, the elastic stresses are calculated by complex variable theory, combined with the Mindlin’s solution. Then, the new analytical solutions to stress are obtained to predict the stratum responses induced by tunneling adjacent to the existing pile group loads inside the stratum in a gravity field. Ultimately, this study provides parameters to analyze their influence on ground stress and potential plastic zone, such as the stress release coefficient, pile group locations, and soil parameters. This research provides a theoretical basis for stratum stresses’ prediction in shallow tunneling engineering fields when tunneling adjacent to a pile group, and it can be applied to the construction of resilient cities.