## 1 Introduction
In terms of theoretical analysis methods, semi-analytical methods, transfer matrix methods, and continuum-mechanics-based approaches have been widely applied to study the mechanical performance of pile foundations under lateral loads [32; 33]. For example, [PERSON] et al. obtained a semi-analytical solution for pile performance under coupled longitudinal and lateral loads using the transfer matrix method, providing an efficient tool for mechanical analysis under complex loads [34]; [PERSON] and [PERSON] adopted the finite difference method and elastic foundation beam finite element method to study the lateral deformation and internal force distribution of piles in layered foundations [35]. However, these studies mostly assume a perfect pile-soil contact interface, neglecting the potential influence of local detachment at the pile-soil interface, which may occur in practical engineering scenarios. In reality, laterally loaded long piles, especially those with large slenderness ratios, may undergo significant deformation under loading, leading to local detachment at the pile-soil interface. This phenomenon not only alters the internal force distribution of the pile but also significantly impacts its load-bearing path [36; 37]. For instance, [PERSON] found that local contact detachment could increase pile head negative shear forces by up to three times [38]; [PERSON] further pointed out that under local scour conditions, con tact detachment significantly amplifies pile bending moments and deformations, thereby reducing pile stability, which is particularly pronounced in post-seismic liquefied soil layers [39]. Moreover, introducing dynamic soil-structure interaction models enables a more comprehensive description of the effects of local contact detachment on the mechanical performance of pile foundations [40; 41]. Experimental studies by [PERSON] et al. revealed that increasing the stiffness of the soil layers around the pile head and foundation can significantly enhance the lateral load-bearing capacity of pile foundations, offering new strategies for pile foundation design under complex geological conditions [42]. However, existing research has primarily focused on the overall load-bearing behavior of piles under lateral loads, with insufficient attention to the impact of local interface detachment. This oversight may lead to an overestimation of pile mechanical performance in practical engineering, particularly under post-seismic and complex geological conditions, posing potential safety risks. Based on the above analysis, further exploration of the effects of local pile-soil contact detachment on the lateral performance of piles, especially the mechanical response in complex multilayered foundations, is of significant theoretical and engineering value. This research can deepen the understanding of pile deformation and internal force distribution while providing a scientific basis for post-seismic repairs and optimized designs. This study investigates the mechanical behavior of laterally loaded long piles with local pile-soil detachment in layered foundations, emphasizing the influence of partial detachment on pile performance--an aspect often overlooked in conventional analyses. By employing the modified Vlasov foundation model and the separation of variables method, this research systematically derives analytical solutions for pile deformation and internal forces in both complete contact and local detachment scenarios. Unlike traditional models assuming full pile-soil contact, this study introduces a more realistic consideration of local detachment, providing a novel approach to quantifying its impact on lateral deformation, bending moments, and shear forces. The findings offer new insights and quantitative guidance for optimizing pile foundation design in complex layered soils, contributing to safer and more reliable geotechnical engineering practices.