# Transient Response of Dynamic Stress Concentration around a Circular Opening: Incident SH Wave
## 1 Introduction  
The study of elastic wave propagation in continuous media holds significant importance in elastic dynamics. Understanding stress concentration under dynamic conditions is crucial for ensuring the safety and reliability of engineering applications. It plays a pivotal role in seismic design, ore exploration, non-destructive testing, blasting, and the application and advancement of advanced composite materials. Defects such as openings, inclusions, and cracks are often present when manufacturing and processing artificial materials. Consequently, investigating dynamic stress concentration around scattering waves and defects has become a focal point of research [1]. As early as 1973, [PERSON] et al. [2] researched dynamic stress concentration problems associated with various defects. Since then, numerous numerical examples have been provided, and the mechanical properties of materials with defects have emerged as a prominent research area. In recent decades, the theoretical analysis of elastic wave propagation in continuous media has focused on three key aspects: model structure, defect morphology, and constitutive relationships. Scholars have extensively researched these problem categories [3, 4, 5, 6, 7]. For instance, [PERSON] et al. [3] utilized the wave function expansion method based on magneto-acoustic coupling dynamics theory to address the issues of acoustic wave scattering and dynamic stress concentration in e-type piezoelectric composite materials. [PERSON] et al. [4] investigated the dynamic response of a cylindrical shell surrounded by homogeneous,isotropic, and linear elastic media under plane harmonic excitation. [PERSON] et al. [5] studied the scattering of SH waves by two circular cavities in an elastic plate. They derived the incident and scattered conductive waves that meet the stress-free conditions of the upper and lower surfaces within the band domain. This was achieved through a combination of wave function expansion, complex functions, image method, and multipolar coordinate translation techniques. [PERSON] et al. [6] conducted an analytical study on the scattering problem of horizontal waves in two-dimensional approximately linear inhomogeneous media using the method of complex variables. They proposed a novel transformation method to normalize the governing equations of two-dimensional inhomogeneous media and employed complex variables to represent the wave field and stress. [PERSON] et al. [7] proposed a series solution for cylindrical SH-wave scattering in a shallow asymmetrical V-shaped canyon and studied the effect of source location on terrain amplification. By comparing the data of two published cases (cylindrical SH-wave symmetrical V-shaped canyon and planar SH-wave asymmetric shallow V-shaped canyon), they found that the difference of displacement amplitude between the asymmetric canyon and the symmetric canyon under the action of cylindrical SH wave can reach 270.2%, that is, the asymmetric effect of the canyon has a significant influence on the topography amplification. With the emergence of functionally graded materials, research on the mechanics of such materials has become indispensable. [PERSON] et al. [8] proposed a method to evaluate stress concentration near a circular opening at a permeable interface crack tip in exponential-gradient piezoelectric biomaterials. They established a mechanical model of the crack using the crack connection method and derived the dynamic stress intensity factors at the crack tip. [PERSON] et al. [9] developed a theoretical approach to investigate magnetoelastic coupling waves and dynamic stress intensity around a cylindrical opening in exponential gradient piezoelectric materials. They provided numerical examples of dynamic stress intensity factors near the opening. [PERSON] and [PERSON] [10] developed an experimental and numerical model to understand the thermal stress distribution of functionally graded plates with only the coefficient of thermal expansion.