Damage and fracture are the main causes of structural failure, which have a significant impact on engineering safety. Crack propagation problem is also a fundamental scientific challenge that needs to be solved urgently. In this paper, the relevant theoretical basis for simulating damage and fracture, such as a fracture mechanics model, damage evolution model and numerical calculation methods, such as the finite element method, boundary element method and peridynamics theory, are introduced in the form of literature review. This paper also reviews the commonly used CAE software for structural damage and fracture analysis, including general-purpose finite element programs such as the damage and fracture analysis module that comes with ABAQUS, as well as specialized fracture analysis software, damage tolerance tools, fatigue life analysis tools, etc. The development status of some autonomous CAE software is also discussed. Finally, this paper analyzes some challenges faced by CAE software for damage and fracture simulation, and looks forward to the future development direction of domestic CAE software.

A surrogate model is a new research idea direction of aerodynamic data generation. A traditional surrogate model relies on a large number of high-precision simulation sample points and their responses values to ensure the accuracy of the model. A multi-credibility surrogate model can reduce the computational cost while maintaining a certain accuracy by integrating multi-layer high and low credibility models, which is of great significance for reducing a missile development cycle. In this paper, the influence of different numbers of low-confidence sample points on the multi-confidence model and the optimal ratio of high-low-confidence sample points are studied, and a multi-confidence sampling method suitable for aerodynamics data is proposed. It is applied to the construction of three multi-credibility surrogate models in the prediction of missile aerodynamic data, among which the Co-Kriging model has the best comprehensive prediction effect. The recommended ratio of high and low confidence sample size is between 1: 4 and 1: 3.

In order to investigate the effect of cable damage on in-plane free vibration characteristics of cable-beam composite structures, three dimensionless parameters of cable damage intensity, extent and position are introduced in this paper to establish an in-plane dynamic model of single-cable cantilever beam composite structures with cable damage. The eigenvalue problem of in-plane free vibration of a single cable-cantilever beam model is solved by the method of separation of variables. At the same time, the finite element models under undamaged and damaged conditions of the cable are established for verification, and the results are in good agreement with the theoretical results. The results show that the frequency of the combined structure will decrease obviously only when the sag or damage intensity and extent of the cable are large. When the damage intensity and extent of the cable increase to a certain value, 1∶1 in-mode resonance phenomena tend to appear for high order frequencies. The asymmetric initial configuration caused by damage can increase the frequency of the combined structure, and some mixed modes of the combined structure change to local modes, while some local modes change to mixed modes.

Addressing the challenge of accurately solving unstable stick-slip vibration problems in non-smooth dynamics, this paper proposes a solution algorithm based on Physics-informed Neural Networks (PINN). Firstly, the classical stick-slip vibration problem is dynamically modeled using the linear complementarity theory under unilateral constraints. Then, the linear complementarity relationship is designed as a loss function to guide the training of the neural network, constructing a PINN algorithm for solving multi-point friction-induced stick-slip vibration problems. The accurate simulation of complex responses of multiple sliders'stick-slip vibrations in frictional systems is conducted. By comparing the numerical results with the Switching Model method that includes event detection and the traditional Time-Stepping method without event detection, the accuracy of the PINN algorithm is verified. The proposed PINN algorithm transforms the traditional optimization problem calculation into network training of the machine learning algorithm, making it suitable for stick-slip vibration analysis with multiple contact points. This method achieves accurate nonsmooth state transitions and provides a convenient and easy-to-use new approach for the accurate simulation of complex nonlinear vibration responses in multi-degree-of-freedom frictional systems.

The power battery pack of an electric vehicle is a complex system, whose efficient and high fidelity modeling is an urgent need for solving collision problems. This paper firstly proposes a hybrid reduced order battery pack model, with fine modeling in the central region and centralized mass in other parts through a simple connection contact. Compared with the refined full model, the maximum stress error is relatively small, and the computational efficiency is improved by about 3.5 times. Then, to consider the effect of the boundary further, a centroid module structural model with equivalent contact is proposed. Compared with the hybrid reduced order model, the collision displacement and the maximum stress of the centroid module structure model are much closer to those of the fine full model. Finally, an impact response analysis is conducted on the centroid module structural model and the fine full model, by comparing the behaviors of the collision center point and the impact object. It is found that both models have the similar trend in the response curves with small errors, which verifies the feasibility of the proposed model.

Accurately constructing the nonlinear hysteresis loop model at the bolt connection is crucial for the vibration reduction and safety performance evaluation of a satellite load-carrying structure. Traditional time-domain analysis methods of computational models require substantial time costs, and typical data-driven models struggle to construct high-precision hysteresis models. To address these challenges, a novel Residual Improvement Deep Learning Algorithm (RIDLA) is proposed for constructing the hysteresis loop model of displacement and force at the bolt connection. The algorithm fully leverages the capacity of Long Short-Term Memory (LSTM) neural networks to fit nonlinear relationships in time series. It adopts an innovative approach by creating a multi-level residual improvement deep learning model that iteratively refines predictions based on measured responses, resulting in highly accurate modeling of hysteresis at bolt connections. The performance of the RIDLA method is validated using experimental data from cyclic loading of a subcomponent of a satellite load carrying structure. The findings demonstrate that RIDLA achieves highly accurate predictions of the displacement and force hysteresis loop at the bolt connection. Additionally, the RIDLA method could be applied to predict the dynamic responses of other complex non-linear systems.

Piezoelectric materials have advantages such as rapid actuation, ease of preparation, and low energy consumption. Using piezoelectric materials for vibration control can improve structural performance. Studies have shown that the distribution of piezoelectric materials can significantly impact control effectiveness. Many researchers use topology optimization techniques to optimize the layout of piezoelectric materials or control voltages. In the topology optimization of piezoelectric intelligent structures, introducing various control coefficients as design variables can achieve a larger design space and further enhance control efficiency. This paper studies the optimal distribution of control coefficients for piezoelectric layers under harmonic excitation based on the Discrete Material Optimization (DMO) method. Using a negative velocity feedback control strategy for active control, dynamic compliance is selected as the objective function to effectively measure the structural vibration level. The design variables are the negative velocity feedback control coefficients for each pair of piezoelectric sensors and actuators. Sensitivity analysis is conducted using the adjoint variable method. Finally, two numerical examples are provided to verify the correctness of the proposed method.

Considering the issue that low frequency and broadband of elastic wave metamaterials cannot coexist, this paper realizes a low-frequency band gap by rotating asymmetric mechanical metamaterials, and further widens the low-frequency bandgap by introducing multiple orders. By utilizing the node rotation and ligament bending deformation characteristics of anti-chiral materials, the vibration body size and ligament stiffness of the anti-tetra chiral unit cell diagonal are gradually adjusted through ligament folding, and the multi-order asymmetric unit cell design is realized. The generation and change mechanism of elastic wave bandgap are explained by analyzing the resonance mode and transmission characteristics of the upper and lower bounds of the bandgap. The study shows that: in the asymmetric mode, the node rotation and ligament bending deformation characteristics of chiral materials are utilized to realize the rotation resonance of the mass block and open the bandgap, and the band gap is widened by the resonant superposition between two pairs of different mass blocks arranged alternately. Finally, the proposed asymmetric mechanical metamaterial is verified by experiments to have to demonstrate improved broadband and low-frequency vibration isolation performance.

Proppant transport in fractures is essentially a dense granular flow in a slot-shaped space. Applying the two-fluid method in numerical simulations of a field-scale particle flow is promising, but existing solid stress models cannot accurately describe the process of proppant accumulation. In this paper, the morphological change of a proppant pack under flow erosion was analyzed experimentally, and the important influence of cohesion on the change of the pack state was pointed out. Then, combined with the simulated results of a proppant transport and the results of a suspension apparent viscosity test, the influence of the particle radial distribution function on the solid kinetic pressure and the change trend of total solid pressure were analyzed, and the change rate of the solid friction pressure with the particle volume fraction was determined. Based on the granular matter theory and results of a direct shear test, the cohesion of the proppant pack was considered in the frictional viscosity model. The results show that the improved solid friction stress model can capture larger angles of the accumulation and settlement profiles, and correctly simulate the process of proppant accumulation.

Accurately simulating the impact characteristics of a dam break is of paramount significance for the prediction and mitigation of dam-break flow disasters. The B-spline material point method (BSMPM), as an improved algorithm of the material point method (MPM), effectively enhances computational accuracy and improves convergence. However, the BSMPM solves the governing equations based on a tensor grid rather than an Eulerian background grid. Moreover, its interpolation shape functions have a larger influence domain. Consequently, when solving problems involving fluid-structure coupling and contact, issues such as premature contact, difficulty in capturing contact interfaces, and challenges in calculating contact forces arise. Within the same tensor grid space of the BSMPM, accurate capture of contact interfaces is achieved based on the relative velocities and unit outward normals of the same nodes; employing the Greville Abscissa enables precise contact of contacting objects, thereby avoiding premature or spurious contact; through the Lagrange multiplier method, interface contact forces are accurately determined, thus constructing a high-precision contact algorithm for fluid-structure strongly coupled problems, facilitating research on the simulation of dam-break fluid impact with rigid and elastic obstacles, and enabling a comparison with existing experimental or simulated results. The results demonstrate that the simulated impact loads and structural deformation evolution patterns correspond well with existing experimental/simulated results. For rigid obstacles, the peak impact pressure exhibits concave parabolic growth and positive correlation exponential function growth with increasing water level and dam-break slope, respectively. For elastic obstacles, the peak impact pressure decreases exponentially with the increase in the height of the probing point. The feasibility and effectiveness of simulating dam-break flow impact problems using the BSMPM contact algorithm are validated, providing a new perspective for simulating dam-break flow impact problems.

The Euler equation is one of the fundamental equations describing fluid motion in Computational Fluid Dynamics, and the existence of discontinuous solutions poses challenges in constructing numerical algorithms for solving this type of equation. To achieve high-resolution numerical results for the Riemann problem of the two-dimensional Euler equation, this paper constructs a pressure-difference adaptive rotating entropy stable scheme. Utilizing the rotating invariance of the equations, the normal vector outside the boundary is decomposed into two orthogonal components, and an entropy stable scheme is implemented in each directions. The determination of the components of the two components relies on the rotation angle. In this paper, a pressure function is introduced to adaptively adjust the rotation angle of the scheme based on local pressure variations. The resolution of the entropy stable scheme is enhanced by introducing the adaptive rotation angle. Numerical examples show that the numerical results obtained by this scheme exhibit good symmetry and high resolution.

Research on acoustic propagation in multiple fluids has important application values in naval architecture and ocean engineering, such as sound propagation in pipelines filled with water and air, and the detection of buried objects. There are two difficulties in solving such problems with the use of the classical finite element method: one is the serious numerical dispersion error in the finite element solutions under medium and high wave numbers; the other is the need to use refined mesh grids to discretize the fluids near the coupling interface. These difficulties lead to a large computational cost for the finite element method, and the manual intervention to generate refined grids. Compared with the finite element method, the weak-form meshfree method does not require traditional grids, and the dispersion error effect in its solution is much weaker, ensuring good computational accuracy and efficiency. However, the meshfree shape functions are usually discontinuous in the problem domain, resulting in the inability of the continuity condition of the acoustic particle velocity to be naturally satisfied on the interface. Therefore, this paper uses the penalty function method to reconstruct the continuity condition of the acoustic particle velocity on the interface, and proposes a Galerkin weak form suitable for meshfree methods for sound propagation in multiple fluids. Numerical analysis shows that the meshfree solutions is consistent with the reference solutions, and the computational accuracy and efficiency of the meshfree method can be higher than the finite element solutions.

A simple numerical implementation method is proposed for the Chaboche-type viscoplastic constitutive model coupled with Lemaitre anisotropic damage theory. Using the decoupled algorithm, the damage tensor is updated based on the forward difference format at the beginning of each incremental step. The damage tensor is considered as a constant in the discretization process of the constitutive equations. Based on the hypothesis of strain equivalence, the formulations containing only partial tensors are constructed in the effective deviatoric stress space, and the radial return process is simplified to solve a nonlinear scalar equation concerning the accumulated plastic strain increment. The numerical implementation method and the derivation of consistent tangent operator are provided based on the Voigt notation scheme. The comparison between the experimental data and the simulation results of isotropic scalar damage model under uniaxial and multiaxial stress states validates the effectiveness and high computational efficiency of this method. Numerical results under different time step sizes also indicate the good accuracy and stability.

Structural condition assessment is crucial for ensuring the safe services of structures, with structural damage detection (SDD) being a core component. In this paper, a novel SDD method is proposed based on the adaptive grasshopper algorithm and sparse regularization. It aims to tackle accuracy decline of SDD results and instability involving uncertainties and incomplete measurement, thereby achieving sparse-regularization-based structural condition assessment. Firstly, adaptive Lévy flight and elite opposition-based learning strategies are incorporated into the adaptive grasshopper algorithm to prevent the SDD process from falling into local optima and to enhance the stability of SDD results. Secondly, a modal parameter-based objective function with sparse regularization is formulated to increase the sparsity of SDD results, thereby improving SDD accuracy and robustness. The optimization results of competition-based evolutionary computation benchmark functions show that the adaptive grasshopper algorithm exhibits better global convergence and identification stability compared with its standard version. Numerical and experimental results for simply-supported beams indicate that the proposed method can ensure reliable SDD accuracy even in the case of incomplete measurements, and it possesses good noise robustness as well.

The formation of porous media is influenced by a number of factors, including the deposition and fragmentation of particles, which result in the formation of interlayers with varying structures. These interlayers exert a significant influence on the mechanical behavior of porous media. This paper presents a systematic investigation into the influence of the inclination angle and thickness of the interlayer on the mechanical behavior of porous media, employing the discrete element method. The results demonstrate that the stress intensity of porous media containing interlayers is between those of the two homogeneous porous media and varies with changes in the inclination angle and thickness of the interlayers. The average coordination number between grains is found to be significantly affected by the thickness of the interlayer at the beginning of loading, but stabilized at the end of loading. The variation of the coordination number affects the distribution of strong and weak force chains, while the inclination angle and thickness of the interlayer determine the magnitude and direction of stress transfer in the force chains. Furthermore, the contact unit normal force and normal contact force are deflected with the increase of the inclination angle of the interlayer, demonstrating significant anisotropy. This study advances our understanding of the intricate mechanical behavior of porous media containing interlayers in strata, offering invaluable insights for optimization and practical application in geological engineering.

This paper presents an analytical method, namely interface stiffness transfer method, for evaluating the responses of multilayered elastic structures. Based on the Love function and general solutions, the stiffness matrix relationship of the displacement-stress state vectors is introduced to obtain the interface stiffness transfer matrix equation between adjacent layers, which satisfies an algebraic Riccati matrix equation. When the elastic layer is a half-space, an explicit solution is obtained directly for the interface stiffness matrix. The interface stiffness transfer matrix method starts from the bottom layer with a known stiffness, and then deals with one layer at a time until the uppermost layer is reached, obtaining the interface stiffness of the multilayered structure. Finally, by solving the symmetric equilibrium equations of the boundary conditions, the displacement-stress state vector of an arbitrary layer is obtained. This method keeps the advantages of the classical transfer matrix method, but naturally excludes its exponential growth terms. In particular, the proposed method is a powerful candidate for efficiently solving the algebraic Riccati equation for the optimal control problems. Numerical examples show the properties of the interface stiffness transfer method.

This paper presents a Bezier triangle meshing method that considers both clipped and non-clipped forms for a single NURBS surface. The proposed method is applied to analyze isogeometric Kirchhoff-Love shell structures. The process begins by interpolating NURBS surfaces into Bezier surfaces. Subsequently, the topological relationship between the clipping curve and each parameter node is calculated within the parameter domain. A Bezier contour curve set is then generated in the parameter domain by selecting points along the clipping curve. Utilizing this contour curve set, a triangular mesh is generated in the parameter domain. Finally, the Bezier triangle mesh in the physical domain is created through a mapping method. The adaptability and robustness of the algorithm are verified through three models, and the mesh quality is assessed. The results demonstrate favorable overall mesh quality. Building upon this foundation, the paper illustrates the application of a rotation constraint between Kirchhoff-Love shell elements, using the penalty function method with Scordelis-Lo's Roof shell model as an example. The accuracy of Kirchhoff-Love shell elements based on Bezier triangles is subsequently validated.

Small cross-sections of boundary elements easily induce the “internal tension” phenomenon of steel plate shear walls, making it difficult to fully utilize the seismic performance of buckling-restrained steel plate shear walls. The design of cross-sections of boundary elements is related to their internal force requirements, and analyzing the internal force requirements of boundary columns is meaningful. Based on the proposed buckling-restrained steel plate shear wall with multi-concrete panels (MBRSPSW), the analytical expressions for the internal force of the boundary column of the MBRSPSW were theoretically derived in this paper. Combined with the experimental research on the buckling-restrained steel plate shear wall horizontally assembled multi-concrete panels (H-MBRSPSW), its numerical model was established and verified. The internal force distributions of the boundary column obtained from numerical analysis and analytical calculation were further compared. The research results indicate that the inner steel plates in the MBRSPSW are divided into constrained regions and unconstrained regions. The axial force, shear force, and bending moment distributions of the boundary column from the analytical calculation results agree with those from the numerical analysis results, and the internal force calculation expressions of the boundary column are correct. The research results can provide a reference for the design of this category of steel plate shear walls.

To study the influence of steel truss web shear deformation on the deflection of steel truss web composite box beams, the beams were first decomposed into a laminated structure composed of top and bottom flanges and a steel truss. A steel truss web shear deformation angle function was introduced to establish an analytical model, and the flexural deformation of a simply supported beam was analyzed as an example. The effective stiffness of the cross-section was determined by combining Euler beam theory and the analytical solution, and the mid-span deflection was calculated using this effective stiffness. The flexural characteristics under different load conditions were analyzed and compared with the Euler beam theory. The influence of structural parameters such as steel truss web diameter, steel truss web wall thickness, and steel truss web inclination angle on the effective stiffness was also examined. The results show that considering steel truss web shear deformation provides an analytical solution closer to the finite element results, with a maximum error of 6.24%. Using the effective stiffness can effectively predict the mid-span deflection, with a maximum error of 3.64% compared with the analytical solution. Among the structural parameters affecting the effective stiffness, steel truss web wall thickness has the greatest influence, followed by steel truss web diameter and steel truss web inclination angle. Additionally, the effective stiffness is positively correlated with steel truss web diameter and steel truss wall thickness but negatively correlated with steel truss web inclination angle.

The calculation of the probability distribution of performance functions is a core issue in uncertainty quantification and reliability design, and the recently proposed equivalent expectation method (EEM) is an effective way to solve this problem. This paper proposes an improved EEM. Putting forward an empirical calculation formula for the standard deviation coefficient of auxiliary random variablesand obtaining a more accurate probability distribution of the auxiliary function. Meanwhile, aiming at the accuracy issue in calculating the probability distribution of theperformance function is proposed, the calculation formula for the PDF of the performance function is derived using only one auxiliary function. In the process of calculating the PDF, proposing an exact theoretical transformation of probability distribution from auxiliary functions to performance functions is proposed, resulting in a more accurate PDF of the performance function. Finally, the effectiveness and accuracy of the method are verified through three numerical examples. The results indicate that this method is suitable for computing the probability distribution of high-dimensional nonlinear or implicit performance functions.

To investigate the evolution of dimpling and the mechanism of interface separation in bimetal clad pipes under external mechanical loading, a stress model was established. The study analyzed the effects of the ratio of diameter to thickness for the inner and outer pipes, forming pressure, initial forming clearance, and operational internal pressure on dent formation and interface separation. Results indicate that interface separation distance and rebound rate correlate positively with the ratio of diameter to thickness for the inner pipes, forming pressure, and initial forming clearance, and negatively with the ratio of diameter to thickness for the outer pipes. Higher operational internal pressure reduces interface separation but increases rebound rate. Internal pressurization of dented pipes decreases interface separation; for instance, under 2-MPa operating pressure, interface separation is 5% less compared with conditions under 2-MPa pressurization. Additionally, the difference in separation between these conditions decreases with increasing pressure. Adjacent dimpling results in increased interface separation in intermediate pipe segments, causing a broader interface separation area compared with isolated dimpling.

The non-isothermal complex flow caused by fluid impacting obstacles is very important to the industrial processes such as nuclear energy utilization. Through coupling various numerical techniques such as the density diffusive term, artificial viscous term, particle shifting technique, a stable and accurate non-isothermal smoothed particle hydrodynamics (SPH) scheme is established, and accurate simulation of non-isothermal complex flow caused by fluid impacting obstacles is realized. Based on the simulation for the non-isothermal flow past a heated cylinder, the non-isothermal dam break past single/multiple obstacles, it is demonstrated that: (1) the developed non-isothermal SPH scheme can not only compute a smooth pressure field and avoid the spurious oscillation of numerical solutions, but also predict accurately the temperature field and the key physical quantities; (2) this SPH scheme can also accurately show the interaction between the heat conduction process and the complex free-surface evolution, and has the capability to simulate non-isothermal complex flows past multiple obstacles.

The discontinuous Galerkin (DG) method has been widely adopted due to its excellent properties such as high accuracy and ease of parallelization. The adaptive mesh refinement (AMR) technique has been widely adopted to improve computational efficiency with much less computational cost compared with uniform global refinement to the same level with AMR. This paper combines the advantages of DG and AMR, and a new hybrid limiter is applied to the DG method on adaptive Cartesian grid based on p4est, an open-source library. The limiter exhibits advantages of high precision, compactness, robustness, and ease of implementation. The shock wave is captured with a shock indictor and the performance of the new hybrid limiter is compared with that of the total variational bounded (TVB) limiter in this paper. The result shows that the performance of the former is significantly better than that of the latter. A series of numerical examples for Euler equations and Navier-Stokes equations are used to verify the feasibility and efficiency of the proposed method. The results show that the new hybrid limiter performs very well in the AMRDG method, it has lower dissipation and great shock capture ability, and the computational efficiency is greatly improved while the accuracy is guaranteed.

In computational fluid dynamics, mesh quality greatly affects the accuracy and computational efficiency of numerical simulation results. The Bubble does not require the consideration of intersection judgments and has a relatively simple data structure, which has significant advantages in mesh generation efficiency and quality. The process of improving the mesh quality by moving nodes based on the traditional Bubble is optimized in this article, and we define it as the Bubble-Opt method. In this method, a bubble radius selection method combined with neural networks is used to generate the initial bubbles, and an improved bubble dynamic movement technique is used to adjust the bubbles to the appropriate position. The Delaunay method is used to connect the center of bubbles to form the final optimized mesh. Then, the optimization effects of different bubble radius selection methods and Bubble-Opt methods are compared under different process parameters. Taking the flow around a 2D cylinder as an example, the geometric quality and transition ratio of the mesh before and after optimization are tested. For this example, there is a set of optimal parameters and a radius selection method that achieve the best mesh quality optimization effect. The average transition ratio can be improved by about 17.37%, the average mesh quality can be improved by about 13.60%, and the minimum transition ratio and minimum mesh quality can be significantly improved. Finally, under the radius selection method and process parameters, taking two-dimensional cylindrical flow and NACA0012 airfoil flow as examples, the numerical simulation results are compared with experimental data from both qualitative and quantitative perspectives, indicating a significant improvement in the overall grid quality.

The computation mechanism of the calculation method for the completed state of existing suspension bridges is unclear, and the target state is unreasonable. A reasonable numerical analysis algorithm is proposed for bridge formation state. The cable theory consisting of the initial end angle and horizontal cable force is validated based on the relationship between the initial end angle and cable force in the theory of catenary equations. A system of bridge state analytical equations are constructed based on the optimization principle of the target parameters of each component of the suspension bridge. The calculation equation for the main cable configuration based on the geometric closure conditions of the suspension bridge's main cable. The mechanical equilibrium equations are constructed for each component based on the mechanical equilibrium conditions of the suspension cables and stiffening beams. Based on the principle of minimizing the bending moment of the stiffening beam components and the principle of uniform cable force of the suspension cable components in the completed state of the suspension bridge, a calculation equation system for the stiffening beam and suspension cable is established. The intelligent algorithm GRG is used to optimize the numerical solution of the objective function of the completed state of a suspension bridge. A case study of a kilometer-long level suspension bridge project. The derived analytical algorithm is compared with the calculation results of the finite element model and rigid supported continuous beam algorithm. The results show that the difference between the analytical algorithm and the finite element model calculation is relatively small in terms of force of main cable, shape-finding of main cable, and the bending moment of the stiffening beam. Compared with the rigid support continuous beam algorithm, the analytical algorithm has computational advantages in ensuring the uniformity of cable forces in bridge suspension cables and the extreme bending moment of stiffening beams.

Using fractional derivatives to modify the Zener standard rheological solid model and considering the instantaneous rheological effect of the soil around the pile, a vertical coupled vibration model of the pile-soil system is constructed. The frequency-domain analytical solution of the system dynamic control equation is derived using Laplace transform and potential function decomposition methods. The time domain response under instantaneous excitation at the pile top is obtained through numerical Laplace inversion. Then, numerical examples are used to analyze the frequency domain characteristics of displacement and dynamic stiffness, and dynamic damping of end-bearing pile vertical vibration in a rheological clay layer, as well as the wave response under instantaneous excitation at the pile top. Research has found that the rheological effect of soil reduces the amplitude of pile top displacement and dynamic stiffness and the rheological effect of soil causes a decrease in the amplitude of the pile top response and a weakening of the reflected wave signal under instantaneous excitation.