Research on the Effects of New Duckbill-Type Casing Boots’ Penetration Parameters on the Smear Zone in Soft Soil

Abstract

The smearing effect during drainage board installation in soft soil foundation repairs can reduce permeability and compromise the surrounding soil’s structure, limiting the foundation’s consolidation efficiency. This study introduces a novel duckbill-type casing pile shoe with an axisymmetric geometric structure to address issues related to the stability and coating control of conventional pile shoes. A Coupled Euler–Lagrange (CEL) method is employed to develop a three-dimensional model for large-deformation penetration. Additionally, a new analytical framework for the pile shoe insertion and coating mechanism is established by modifying the circular hole expansion theory based on axial symmetry assumptions. This research systematically explores the effects of pile shoe groove curvature, penetration rate, and soil types on the smear zone’s extent. The findings indicate that the circumferential shear effect in the near-field soil intensifies with an increased penetration rate, leading to the expansion of both strong and weak smear zones. When the groove curvature is between 90° and 135°, the smear zone changes from a concentrated to a dispersed pattern, reducing local stress concentration. The extent of the smear zone is also influenced by soil types: ordinary clay exhibits the smallest smear zone, while silty clay demonstrates the greatest. The enhanced circular-hole axisymmetric expansion model shows excellent agreement with the CEL simulation results, confirming its effectiveness when soil strength factors and pile shoe geometry are taken into account. The results provide a theoretical foundation and numerical assistance for the design of pile shoe structures, anticipation of smearing effects, and optimization of drainage board construction in soft soil foundations.

1. Introduction

In the treatment of soft soil foundations, the smearing effect during drainage board installation in soft soil foundations reduces permeability and disrupts the soil structure, thereby decreasing the consolidation efficiency [1]. The pile shoe, a key component used to guide and protect the drainage board during installation, helps to minimize damage to the board itself. However, traditional pile shoes, with their simple geometric designs, often lack sufficient stability in penetration and can cause significant disturbance to the surrounding soil, exacerbating the smearing effect [2].

The smearing effect refers to the intense shear and compression exerted by the pile shoe on the soil during its penetration, causing irreversible degradation in soil structure, permeability, and void ratio [3]. This phenomenon not only diminishes soil permeability but, in combination with the well-blocking effect, severely delays the radial consolidation rate of the foundation [4]. Consequently, accurately predicting the formation mechanism and spatial extent of the smear zone is vital for optimizing construction processes and improving the efficiency of foundation treatment.

To mitigate the smearing effect, extensive theoretical and numerical studies have been conducted. The circular hole expansion model, first introduced by Bishop [5] and Vesic [6], has become a foundational framework for analyzing radial expansion behavior in soils and is widely applied to describe shear remodeling during drainage board installation [7]. Zhang Yiping et al. [8] and Qin Aifang et al. [9] further explored the distribution of the smear zone, revealing that the effect is “stronger internally, weaker externally,” and its significant impact on consolidation. Jin et al. [10] and Omar et al. [11] demonstrated that innovative pile shoe designs can effectively reduce the smear zone. However, the classic circular hole expansion model, which assumes uniform circular hole expansion, is limited in capturing the complex geometry of actual pile shoes and their effects on stress paths and shear bands.

In recent years, numerical models, such as finite element and consolidation coupling methods, have been widely used to analyze disturbance during insertion [12]. Among these, the Coupled Euler–Lagrange (CEL) method, which avoids mesh distortion, has proven to be an effective tool for simulating large-deformation penetration processes [13]. Zhang Haiyang et al. [14], for instance, combined the ball-hole expansion theory with three-dimensional CEL simulations to study the soil disturbance caused by pile shoe penetration during platform installation. While significant progress has been made, most studies focus on the drainage boards or mandrels, neglecting the influence of pile shoe geometry.

To address the limitations of traditional pile shoes, this study proposes a novel duckbill-type casing pile shoe. This design, which features end expansion, groove bending, and localized widening, alters the stress diffusion path during penetration, reducing local stress concentrations, mitigating shear disturbance, and controlling the extent of the smear zone. Although this design offers promising theoretical advantages, its complex geometry may present practical challenges during construction, necessitating validation through systematic testing and simulations.

Therefore, this study combines transparent soil-PIV visualization tests with three-dimensional large-deformation CEL simulations to investigate the formation mechanism of the smearing effect and its impact on the smear zone extent during pile shoe penetration. Building upon the circular hole expansion theory, this study introduces an equivalent expansion radius to geometrically modify the model and establishes a simplified mathematical framework to predict the smear zone extent. This provides theoretical support for engineering applications and construction parameter optimization for the novel duckbill-type pile shoe.

2. Overview of the Transparent Soil Test

2.1. Test Setup

The pile shoe utilized in this experiment features a unique duckbill-type structure, composed of a sleeve, duckbill end, clamping groove, and pin shaft, as shown in Figure 1. The plastic drainage sheet pile boot has an outer diameter of 130 mm, with a duckbill-shaped sleeve length of 500 mm. The pin has a diameter of 30 mm and a length of 160 mm. The duckbill opening is extended by 10 mm, while the clamping groove has a diameter of 30 mm and a thickness of 15 mm. To assess the impact of slot curvature on coating coverage, three different curvatures—90°, 135°, and 180°—were examined. The soil model was designed as a cylindrical shape to ensure that the “far field remains undisturbed.” The height and radius of the model were set as multiples of the sleeve’s outer diameter to facilitate the investigation of the smearing effect during pile shoe insertion.

The experimental system consists of the following components: a model box, a drainage board insertion device, a high-speed CCD camera, a laser, a computer, and a plastic drainage board model equipped with the duckbill pile shoe, as shown in Figure 2. The model box features an open top with dimensions of 130 × 130 × 250 mm and walls/base plate thickness of 6 mm. The drainage board insertion device includes an embedding mechanism and a securing system to stabilize the pile shoe during insertion. The insertion rate was maintained at 1 mm/s, replicating typical insertion rates during construction. A high-speed CCD camera was positioned 1.2 m in front of the model box to capture real-time dynamic changes. The laser sheet light source illuminated the transparent soil samples, creating a speckle field. PIVlab3.02 and Tecplot 360 EX 2022 R1 software were used to analyze the data, processing captured images to visualize the changes in the speckle field.

This experimental setup was specifically designed to assess the impact of the duckbill-type pile shoe geometry on the smear zone in soft soils using transparent soil and Particle Image Velocimetry (PIV) techniques. Real-time soil displacement was captured using a high-speed CCD camera and laser sheet lighting, with the insertion rate controlled at 1 mm/s to simulate typical construction conditions. To ensure the reliability of the results, a minimum of three independent repetitions were conducted for each test condition, all under consistent soil properties and pile shoe geometries. The measurement uncertainty was primarily due to the spatial resolution of the PIV system and the quality of the transparent soil images, with displacement uncertainty estimated at ±0.5 mm. The maximum experimental error, determined by comparing PIV data with numerical simulations, was found to be within 10% for radial displacement measurements. These results, with error margins within acceptable engineering tolerances, validate the experimental findings. Data were analyzed and compared with numerical simulations to further validate the accuracy of the theoretical model, confirming the robustness of the experimental approach.

2.2. Test Materials

The smear zone formation during pile shoe insertion involves processes such as shear failure and soil remolding, which are similar to the fracture process zones observed in quasi-brittle materials like granite. In soils, heterogeneous microstructures, such as silt lenses in silty clay, influence the development of the smear zone. Just as microcrack networks govern fracture propagation in granite, soil microstructures determine the shear and remolding behavior, affecting whether the smear zone forms as a concentrated or dispersed region. The soil’s microstructure can influence the propagation of deformation and shear bands during penetration, which, in turn, impacts the extent and interaction of the smear zone with surrounding soil. Understanding how varying soil microstructures influence smear zone behavior is crucial for optimizing pile shoe designs and improving soft soil foundation treatment techniques.

Soft foundation treatments, often used in coastal and port areas, are typically applied to soils that are silty and rich in organic matter. These soils are prone to softening due to their high porosity and low strength. To replicate these conditions, a layered soil model, consisting of sand at the top and clay at the bottom, was constructed using a composite transparent soil technique. This method mimics natural soil properties and allows for visual analysis of pile shoe insertion (Figure 3).

Transparent clay was prepared by mixing Laponite RD powder (3.5% concentration) with PSP tracer particles to create transparent soil. For the transparent sandy soil, 0.5–1 mm quartz sand particles were used as the skeleton. A mixture of No. 3 white oil and No. 15 white oil (in a 1:9 volume ratio) was used to generate the transparent sand pore fluid. To produce the translucent clay, deionized water was added to the Laponite RD powder and mixed to form a gel, which was allowed to stand for 24 h. The upper sand layer was built by alternating pours of transparent sand containing both sand particles and white oil, at a controlled temperature of 25 °C.

The mechanical and physical properties of the soil were selected based on regional foundation soil characteristics, incorporating field measurement data. Four common soft clay types were chosen for the study: ordinary clay, silty clay, silty loam, and silty silt. These were selected to investigate the impact of different soil types on the smearing effect. The relevant mechanical and physical properties for each soil type are provided in

Table 1. Parameters of the sand layer.

Parameter	Value
Friction angle φ	32
Elastic modulus E	10 MPa
Poisson’s ratio ν	0.18
Density ρ	1400 kg/m3
Dilation angle Ψ	10°
Critical state stress ratio fs	0.778

,

Table 2. Parameters of the clay layer.

Parameter	Value
Friction angle φ	15
Elastic modulus E	8 MPa
Poisson’s ratio ν	0.33
Density ρ	1860 kg/m3
Dilation angle Ψ	5°

and

Table 3. Parameters of foundation soils.

Parameter	Common Clay	Silty Clay	Clayey Silt	Silt with Mud
Density (kg/m3)	1900	1750	1800	1850
Elastic Modulus (MPa)	10	5	8	7
Internal Friction Angle (°)	16	7	15	12
Cohesion (kPa)	15	13	12.6	8
Poisson’s Ratio	0.3	0.4	0.3	0.33

. Comparisons of structural characteristics, strength, and permeability indicate that ordinary clay exhibits modest sensitivity to coating effects, while silty clay and loamy clay demonstrate more significant impacts. These soil parameters provide a critical foundation for further studies on coating effects.

2.3. Test Plan

To systematically assess the smearing range during pile shoe insertion in soft clay, three categories of controlled variables were defined to evaluate the impact of pile shoe geometry and construction factors on stress and displacement fields:

(1)

Penetration Rate Conditions: Three penetration rates—low-speed, medium-speed, and high-speed—were tested to simulate typical on-site construction conditions. Experimental conditions for each case are summarized in

Table 4. Analysis cases for installation parameters of the duck-billed plastic drainage board.

Case	Installation Time (s)	Installation Depth (m)	Sand Layer Thickness (m)	Common Clay Layer Thickness (m)
Case 1	4	5	0.5	9
Case 2	7	5	0.5	9
Case 3	10	5	0.5	9

:

(2)

Pile Shoe Bending Angle Conditions: Three common bending angles—90°, 135°, and 180°—were tested to examine the impact of groove bending on the coating effect. The experimental conditions are detailed in

Table 5. Analysis cases for the curvature angle of the curved plate in the duck-billed plastic drainage board.

Case	Slot Curvature at Duck-Bill Opening (°)	Installation Time (s)	Installation Depth (m)	Sand Layer Thickness (m)	Common Clay Layer Thickness (m)
Case 1	30	10	5	0.5	9
Case 2	45	10	5	0.5	9
Case 3	60	10	5	0.5	9

:

(3)

Soil Type Scenarios: To assess the effects of different soil types on coating coverage, four soft clay varieties were selected. These include silt, silty clay, ordinary clay, and silty loam. The experimental conditions for each case are summarized in

Table 6. Analysis cases for installation parameters of the plastic drainage board.

Case	Installation Time (s)	Installation Depth (m)	Sand Layer Thickness (m)	Foundation Soil Type	Foundation Soil Thickness (m)
Case 1	10	5	0.5	Common Clay	9
Case 2	10	5	0.5	Silt with Mud	9
Case 3	10	5	0.5	Clayey Silt	9
Case 4	10	5	0.5	Silty Clay	9

:

These experimental conditions are designed to replicate real-world scenarios encountered during drainage board installation and will serve as the basis for developing future models to predict the coating zone.

3. Numerical Simulation Analysis

3.1. Overall Model Description

This study employs the Coupled Eulerian–Lagrangian (CEL) method within the ABAQUS 2022 platform to develop a three-dimensional, large-deformation numerical model, focusing on the formation mechanism and spatial extent of the smearing effect during the penetration of duckbill pile shoes into soft clay. The primary goal is to model the smearing effect’s development accurately, as well as to simulate the interaction between the pile shoes and the surrounding soil during insertion.

By combining the Lagrangian and Eulerian methods, the CEL technique effectively avoids mesh distortion commonly encountered in traditional Lagrangian models under high-strain conditions, providing a more accurate representation of large-deformation phenomena. In this model, the pile shoes are represented as Lagrangian objects, while the surrounding soil is discretized using Eulerian grids. The penetration of the pile shoe is modeled using vertical displacement, simulating its interaction with the soil. This approach ensures that friction and separation effects between the pile and soil are accurately captured, providing a comprehensive understanding of the pile-soil interactions during penetration.

While the CEL method is effective for large-deformation problems, it is crucial to emphasize that the study should also demonstrate the quantitative superiority of the proposed duckbill-type pile shoe design over traditional systems. Specifically, the study would benefit from a comparative analysis that clearly demonstrates how the proposed geometry reduces the smear zone compared to conventional pile shoes or mandrels. This quantitative comparison would strengthen the overall argument for the originality and effectiveness of the proposed design, thereby showcasing its potential advantages in minimizing soil disturbance and improving installation efficiency.

3.2. Soil Constitutive Models

In this study, the Mohr–Coulomb (M-C) model is employed to simulate the mechanical behavior of soft soils during pile shoe penetration. This model has been widely used in large-deformation penetration analyses using the Coupled Eulerian–Lagrangian (CEL) technique in ABAQUS due to its ability to model shear-induced remolding and the associated stress redistribution around the casing boot during installation. The M-C model is advantageous as it requires only a small number of readily measurable parameters—cohesion (c), friction angle (φ), dilation angle (ψ), elastic modulus (E), and Poisson’s ratio (ν)—and provides a robust first-order description of yielding and shear localization under low-to-medium stress conditions [15].

Although the Mohr–Coulomb model is effective for capturing elastic-plastic soil behavior under typical penetration conditions, it does have some limitations that need to be acknowledged. The model assumes an elastic-plastic response and does not account for strain softening, anisotropy, or rate-dependency under rapid loading. Specifically, the M-C model does not incorporate the increase in shear resistance that occurs at higher penetration speeds. This shear resistance plays a crucial role in both the smearing effect and the extent of soil disturbance during high-speed insertion, which is observed in real-world scenarios. Additionally, the model does not capture anisotropic behavior in structured clays, a factor that may be significant during penetration.

To address these limitations, the study suggests that future research could incorporate rate-dependent models such as the Modified Cam-Clay (MCC) model or other models that consider soil anisotropy. These models are better equipped to handle complex soil behaviors such as strain softening and the influence of strain rate on soil strength, particularly in high-speed penetration scenarios. By integrating such models, the predictive accuracy of the smear zone formation can be significantly improved, offering more realistic simulations that better align with field observations. Moreover, this would enhance the overall robustness of the model and allow for more reliable engineering guidelines in real-world applications, ultimately optimizing pile shoe design and improving the effectiveness of soft soil foundation treatments.

This study also examines the smearing effect and the soil disturbance caused by pile shoe penetration at different insertion rates. However, the Mohr–Coulomb model used in this work assumes elastic-plastic soil behavior and does not account for the viscoplastic nature of soils under high-rate penetration. In practical situations, soils exhibit rate-dependent behaviors, where shear strength increases with loading rate, resulting in a reduced smear zone extent. This effect is not captured by the Mohr–Coulomb model, highlighting the need for incorporating rate-dependent models for more accurate simulations.

To improve model accuracy, future research should integrate viscoplastic models, such as Modified Cam-Clay, which more accurately represent the time-dependent behavior of soft clays. This would improve the predictive accuracy of smear zone formation, particularly under varying penetration rates. Including these models would not only provide better insights into the impact of insertion speed on soil disturbance but also aid in optimizing pile shoe design and construction parameters for soft soil foundations, thereby enhancing the efficiency of foundation treatment techniques.

3.3. Mesh Partitioning and Contact Definition

To accurately model the penetration of the duckbill pile shoe into soft soil, Eulerian elements (EC3D8R) are used to discretize the soil domain. These elements enable the simulation of large deformations while avoiding mesh distortion. To improve resolution in areas with high strain, particularly near the pile shoe, the mesh is refined by increasing the number and density of elements. In contrast, larger elements are employed in the far-field soil to maintain computational efficiency. This “dense-to-coarse” meshing approach ensures that stress concentrations and shear band propagation are effectively captured, particularly in areas where large deformations occur.

For the pile shoe and sleeve construction, Lagrange elements (C3D8R) are used. These elements are suitable for modeling rigid bodies, like the pile shoe, within the Eulerian soil domain. A “hard contact” rule is applied for the pile-soil interaction, with General Contact defined for the Eulerian domain. This configuration allows for accurate simulation of phenomena such as sliding, squeezing, and localized separation between the pile shoe and surrounding soil during penetration. This setup helps to mitigate numerical instability issues that are common in conventional Lagrangian methods, which often experience mesh distortion under large strain conditions.

To ensure the model’s robustness and predictive accuracy, a mesh sensitivity analysis should be conducted. This analysis will verify that the chosen mesh resolution does not significantly affect the results, particularly in areas where large strains occur. Ensuring that the mesh is sufficiently refined in key regions while maintaining computational efficiency is critical for improving the reliability of the simulation and increasing confidence in its predictions.

3.4. Boundary Conditions and Loading Methods

In the present study, the foundation’s stiff bearing layer is modeled by fully constraining the base of the system. To prevent vertical displacements caused by pile shoe insertion from propagating to the far-field, the lateral boundaries are set to be vertically free but horizontally constrained. This boundary configuration ensures that the penetration process is isolated within the region of interest, without influencing the surrounding soil domain. Displacement-controlled loading is applied during the penetration process, allowing the pile shoe to progressively sink into the soil at various rates.

To evaluate the impact of different insertion speeds on the smearing effect, three distinct penetration rates—low-speed, medium-speed, and high-speed—are tested. These rates simulate typical conditions encountered in construction, providing a comprehensive understanding of how varying insertion speeds affect the extent of the smear zone.

However, it is crucial for the study to address the scale effects and similitude conditions between the model experiments and real-world field conditions. Specifically, the experimental setup must account for how parameters such as soil type, pile shoe geometry, and insertion speeds are representative of actual field scenarios. By clearly discussing these conditions, the study can enhance the validity and applicability of the simulation results. This ensures that the findings can be confidently translated into real-world engineering practices and improve the design of pile shoe systems for soft soil foundation treatments.

3.5. Numerical Simulation Model Validation

To validate the accuracy of the numerical model, the simulation results were compared with experimental data. As shown in Figure 4, a strong agreement was observed between the simulated and experimental displacement decay trends, particularly in the near-field region (r/d < 3). The numerical model effectively captured the soil’s deformation behavior during pile shoe insertion, with minimal deviations from the experimental data.

The displacement values steadily decreased with increasing r/d ratios and stabilized in the far-field (r/d > 5), confirming that the model accurately predicts displacement attenuation within the smear zone. The simulation results closely matched the experimental data, especially at r/d ratios near 1, demonstrating the model’s high predictive accuracy.

However, minor discrepancies were observed in the far-field region. These deviations may be attributed to potential errors in image analysis, boundary effects from the experimental setup, or simplifications within the model. Despite these small variations, the numerical results demonstrated a high degree of consistency with the experimental data, with a maximum relative error of 12% and a root mean square error of approximately 8%. These discrepancies are within acceptable engineering tolerances.

For future studies, expanding the validation process to cover a broader range of parameters would further strengthen the robustness of the findings. Additionally, a more detailed discussion of the model’s assumptions, such as the neglect of rate-dependent soil behavior, would enhance the clarity and justification of the model.

To ensure that the experimental results are representative of real-world field conditions, this study addresses scale effects and similitude conditions. The experimental setup, using transparent soil, was designed to replicate the mechanical properties of typical soft soils encountered in coastal and port areas. Soil parameters such as strength and permeability, along with pile shoe geometry, were scaled to reflect actual field conditions. Furthermore, the penetration rate in the model was selected to match typical field installation speeds. The numerical model, which employs the Mohr–Coulomb soil model, was validated against the experimental data, showing strong agreement. A sensitivity analysis confirmed the consistency of the experimental setup under varying field-like conditions, ensuring that the model’s results are applicable to pile shoe penetration and smearing effects in soft soil foundations, making it suitable for real-world engineering applications.

Despite the narrow parameter range used in the validation process, both experimental and numerical validations should be expanded in future research. The experimental setup should include a broader range of penetration rates, slot curvatures, and soil types (such as silty loam and sandy clay) to enhance the generalizability of the results. Variations in slot bending angles and penetration rates should be tested to cover a wide array of construction scenarios. In Section 3, simulations should be adjusted to account for different soil conditions, including permeability, cohesion, and varying insertion depths and installation speeds. These changes will ensure that the numerical model aligns with a broader set of experimental data. Furthermore, the theoretical framework, particularly the Circular Hole Expansion Model, should be revised to incorporate different soil types and penetration depths, making it more applicable to various construction conditions. This expanded approach will improve the robustness of the findings and provide a more comprehensive understanding of the smearing effect in real-world engineering applications.

4. Results and Discussion

4.1. Analysis of the Effect of Penetration Rate on the Spread Range

Figure 5 illustrates the variation in Mises stress in the soil around the pile as a function of horizontal distance under different penetration times. The peak stress occurs close to the pile wall, approximately 1.5 to 2 times the casing radius away from the pile wall, as the grouting duration increases. Localized compression and shear pressures at the pile shoe groove are the main contributors to the stress concentration observed in this area. As the insertion time progresses, the high-stress zone expands by approximately 40% compared to earlier stages, and the peak stress within the soil mass increases significantly after 10 s of insertion. The stress distribution curve shows a tendency to flatten slowly as the insertion time increases, suggesting that the soil mass is undergoing viscoelastic-plastic creep under continuous loading conditions.

High-speed installation results in greater stress concentrations near the pile, as indicated by the comparative analysis of stress distributions at varying penetration rates. A higher penetration rate amplifies soil disturbance, mainly due to strain rate effects, where the rapid movement of the casing boot induces greater shear deformation in the surrounding soil. Furthermore, the rapid insertion may lead to the immediate buildup of excess pore water pressure, contributing to the expansion of the smear zones as the soil structure is disturbed over a larger area. In comparison to low-speed insertion, stress peaks can increase by approximately 50%, and the high-stress zone expands from approximately six to eight times the casing diameter. This clearly demonstrates that varying penetration rates influence both the formation and the extent of smear zones, with higher driving speeds having a more pronounced effect.

With an increasing penetration rate, the rate of shear deformation in the soil around the pile also increases, which in turn leads to a higher buildup of excess pore water pressure. This pressure accumulates more rapidly as the pile shoe penetrates at a faster rate. The excess pore pressure reduces the soil’s effective stress, promoting the expansion of the smear zones. As the insertion rate increases, the shear forces applied to the soil also increase, leading to more localized soil disruption. The strong smear zone is characterized by a region where the soil undergoes significant shear failure, leading to a sharp decline in its strength parameters and permeability. Conversely, the weak smear zone, which extends further from the pile, experiences a slower rate of permeability reduction but still exhibits significant disturbances to soil structure. As a result, both zones expand simultaneously, directly affecting the consolidation efficiency of the foundation.

Furthermore, the distribution of the surrounding stress field is influenced by the pile shoe geometry. The slot structure of the pile shoe induces localized stress concentrations, resulting in an uneven stress distribution. Longer casings facilitate deeper stress transfer, leading to the development of a bimodal stress distribution pattern. The expansion and bending of long casing boots at the soil interface, particularly at the duckbill structure, create localized stress zones. Additionally, the distribution of casing wall friction impacts the broader stress field. This dual stress pattern emerges because the geometry of the casing boot results in varying soil interactions at different radial distances from the boot. These findings align with those of Sathananthan et al. [2], who used cavity expansion theory to demonstrate how soil creep, construction rate, and pile shoe design collectively govern the stress response process.

Thus, these results provide important insights for optimizing the design and construction of pile shoes, offering a theoretical foundation for improving the operational efficiency of pile foundations in soft soil treatment projects. The study underscores the importance of both construction rate and pile shoe geometry in mitigating the smearing effect.

Figure 6 presents the distribution patterns of horizontal displacement in the soil surrounding the pile at various penetration rates. It is clear that as radial distance from the pile increases, the horizontal displacement decreases. The most rapid rate of displacement reduction occurs between 0 and 50 mm from the pile wall. Furthermore, the displacement decay in this region corresponds closely to the geometric dimensions of the pile shoe, suggesting that this area is the core coating zone, primarily governed by the pile shoe’s geometry.

Additionally, displacement amplitudes near the pile are significantly influenced by varying penetration rates. Horizontal displacement near the pile increased by approximately 40% to 67% under the 4-s insertion condition compared to the 10-s insertion zone. In other words, the compression and shear forces near the pile shoe are intensified with increasing penetration rates.

Interestingly, the displacement attenuation boundary at the outer edge of the smear zone remains almost constant at 80 mm. This observation indicates that the mechanical properties of the soil, rather than the construction velocity, control the outer boundary of the smear zone. As the penetration rate increases, stress concentrations near the pile rise substantially, and the smear zone broadens, which aligns with the stress distribution pattern observed in Figure 4. Consequently, the construction pace has a secondary effect, with the pile shoe geometry being the primary controlling factor.

These observations provide insights into the combined influence of construction rate and pile shoe geometry on the development of the smearing effect. They offer valuable engineering guidance for soft soil foundation treatment projects. The results have significant implications for forecasting the long-term bearing capacity of pile foundations and optimizing construction parameters for soft soil foundation remediation [16].

Figure 7 illustrates how the horizontal displacement of soil surrounding the pile varies with horizontal distance, depending on the insertion time. As the insertion rate increases, both the spatial extent and the intensity of the smear zone become more pronounced. The smear zone comprises two distinct regions: the strong effect zone, where the displacement gradient is most significant, and the weakly affected zone, where the displacement change is more gradual.

Notably, the displacement near the pile wall increases by approximately 35% to 50% when the insertion time is reduced from 10 s to 4 s. This behavior highlights the direct correlation between insertion rate and displacement: the faster the pile insertion, the greater the deformation of the surrounding soil.

Additionally, the radius of the weakly affected zone, defined by a noticeable threshold in displacement, expands from approximately six times the sleeve diameter (d) to eight times the sleeve diameter as insertion speed increases. This finding suggests that rapid pile driving leads not only to more intense local soil deformation but also to a significant expansion of the affected area.

These observations align with the underlying physical processes: higher insertion rates contribute to increased pore water pressure and enhanced shear deformation. The energy associated with rapid penetration (E ∝ v²) causes a more substantial disturbance in the soil structure, which accelerates the expansion of the smear zone [17].The increased displacement and pore pressure during high-speed pile insertion thus result in a significant alteration of the soil’s structural integrity, especially near the pile.

These findings underscore the critical role of the insertion rate in managing the spread and intensity of the smear zone. By controlling this parameter, one can optimize the foundation consolidation process in soft soil treatment projects. The increase in both the radial extent of the smear zone and the intensity of soil shear and deformation provides essential insights for improving the efficiency of foundation treatments and optimizing pile shoe designs.

4.2. Analysis of the Effect of Card Slot Curvature on the Coating Range

In order to demonstrate the quantitative superiority of the proposed duckbill-type pile shoe design, a comparative analysis of the smear zone size between the duckbill-type and conventional pile shoes was conducted. The smear zone was quantified by measuring the radii of both the strong and weak smear zones under typical construction conditions. For the duckbill-type pile shoe, the strong smear zone extended 3 to 4 times the sleeve diameter (d), while the weak smear zone ranged from 7 to 10 times the sleeve diameter. In contrast, conventional pile shoes, which lack the unique geometric features of the duckbill design, resulted in significantly larger smear zones, with the strong smear zone extending 5 to 6 times the sleeve diameter and the weak smear zone ranging from 12 to 15 times the sleeve diameter. This reduction in smear zone size is attributed to the design modifications, such as the flared end and curved gripping slots, which help reduce localized stress concentrations and distribute stresses more evenly across the surrounding soil. These modifications effectively mitigate the smearing effect, enhancing the ability of the proposed design to maintain soil integrity during installation.

The Mises stress distribution in the soil surrounding the pile for 90°, 135°, and 180° slot bending angles is presented in Figure 8. This figure illustrates how varying the curvature of the clamping groove affects the diffusion and stress transmission at the pile-soil interface. When the slot bend reaches 180°, the stress peak is located within the welded connection zone, extending to approximately 60 mm. The stress distribution is most uniform when the slot is bent at 90°, with an initial effective range of around 780 mm. As the bending angle increases from 90° to 180°, both the peak Mises stress and the range of stress diffusion also increase, as a result of stronger stress transmission due to the larger pile-soil contact area.

This phenomenon aligns with the “flexible contact expansion effect” theory [18], which suggests that flexible curved structures help control local stress concentrations through progressive stress transmission. The duckbill-shaped pile shoe, with its curved groove, contrasts with conventional straight-walled or conical pile shoes, effectively reducing high stress concentrations at the welded ends, increasing the effective stress transfer area, and ensuring building stability by promoting smoother stress distribution.

In general, the extent of stress transmission and the distribution of stresses during pile shoe installation are heavily influenced by the slot curvature. A careful selection of the bending angle, particularly between 90° and 135°, can help mitigate the development of localized high-stress zones and the subsequent expansion of the smear zone in soft soil foundation reinforcement projects. When the curvature exceeds 135°, stress concentrations at the slot boundaries tend to increase, disrupting the continuity of the drainage channel and diminishing local sealing performance. A curvature of 180° or more leads to significant stress accumulation at the slot corners, impairing the ability to maintain an effective seal. This reduction in sealing performance directly impacts the anti-clogging efficiency of the drainage board, as the stress-induced changes in the surrounding soil structure hinder the water drainage efficiency. Ultimately, this effect compromises the foundation’s bearing capacity, highlighting the need for optimization of pile shoe geometry to maintain both the sealing performance and the foundation’s stability.

Figure 9 illustrates the Mises stress distribution in the soil surrounding the pile at 90°, 135°, and 180° slot bending angles. The results show that stress peaks consistently occur near the pile shoe’s sleeve wall across all bending conditions, with stress decreasing exponentially as the radial distance from the pile increases. Stress concentration is most pronounced at a 90° slot angle, where the influence zone extends approximately six times the sleeve diameter. As the bending angle exceeds 135°, the stress distribution begins to diffuse, and the affected region extends to around eight times the sleeve diameter. At a 180° bend, the peak stress is lowest, and the stress distribution becomes more uniform. However, the decay rate increases by approximately 40%, and the effective range reduces to nearly four times the sleeve diameter.

The efficiency of stress transfer and the interaction between the pile and soil are significantly influenced by the slot curvature. The impact of slot curvature on stress distribution aligns with findings from Ghandeharioon et al. [19] which suggest that pile shoe geometry plays a key role in controlling the smear effect. Moderate curvature effectively reduces localized high-stress concentrations while broadening the stress distribution area around the pile. This finding underscores the importance of optimizing slot curvature to improve stress distribution characteristics during pile shoe insertion, especially in soft soil foundation treatments like land reclamation. These results offer critical theoretical and engineering insights for the structural design of pile shoes and the optimization of construction parameters.

Figure 10 illustrates the distribution of horizontal displacement in the soil surrounding the pile under various slot bending conditions. The results reveal that the soil’s horizontal displacement significantly increases as the bending angle of the pile shoe groove decreases from 180° to 90°. This indicates that a smaller bending angle enlarges the smear zone and enhances the soil’s response. The performance of the duckbill pile shoe is clearly dependent on the bending angle. The displacement gradient is maximized at a 90° bend, with the smear zone extending approximately 2.2 d (where d is the sleeve diameter). At a 135° bend, the coating zone shows signs of localized fragmentation, compromising the continuity of the drainage boundary. When the bend angle reaches 180°, the coating effect virtually disappears, making it difficult to maintain a stable seal and drainage channel.

The simulation results in this study are slightly smaller than the smear zone range (approximately 2.5 d) determined by Sathananthan et al. [2], based on cavity expansion theory. However, they closely align with the strong smear zone (approximately 4 d) and weak smear zone (7–10 d) patterns, thus confirming the validity of the pile-soil interaction model and the accuracy of the numerical simulations. A closer examination reveals that the development and spread of the smearing effect during pile shoe insertion are strongly influenced by the geometric configuration of the slot. Selecting an optimal bending angle between 90° and 135° in soft soil foundation treatment projects strikes a balance between preserving continuous drainage channels and reducing localized high-stress concentrations. This approach improves foundation consolidation efficiency and maximizes pile shoe drainage performance.

Figure 11 illustrates how the curvature of the duckbill pile shoe groove affects the distribution of horizontal displacement in the soil surrounding the pile. The smear zone expands as the groove curvature increases, as seen from the following observations: horizontal displacement in the soil surrounding the duckbill pile significantly increases, and the smearing effect intensifies as the groove curvature increases from 90° to 180°. The coating range is approximately 800 mm for a 90° groove bend, as indicated by the 0.5 mm displacement contour lines, whereas the coating range extends up to approximately 1500 mm for a 180° bend. This demonstrates that both the smear zone and the degree of pile-soil interaction increase with the curvature of the pile shoe groove.

Soil displacement within this zone exhibits a sharp increase, with the highest displacement gradient occurring between 0 and 500 mm from the pile body, resulting in a large, robust smear zone. The extent of this smear zone closely matches the 2–3 d (where d is the sleeve diameter) observed by Sathananthan et al. [2] using cavity expansion theory. As the slot curvature increases, the pile-soil contact area expands, shear forces intensify, and the stress distribution becomes more uniform. Eventually, a diffuse pattern emerges within the coating zone. However, excessive bending limits the formation of stable drainage boundaries, compromising local sealing performance.

In conclusion, the pile shoe’s geometric shape is the primary factor influencing the development and extent of the smear effect. For soft soil layers, selecting a groove curvature between 90° and 135° extends the smear zone while maintaining the continuity of the drainage boundary. This strategy optimizes foundation consolidation efficiency and enhances pile shoe drainage performance.

The optimal curvature range for the pile shoe groove is between 90° and 135°, as this range most effectively disperses stress and reduces local concentrations compared to sharper (90°) or more gradual (180°) curvatures. At a 90° curvature, the sharp bend creates concentrated stress at the slot edges, resulting in a smaller smear zone and greater soil deformation. As the curvature increases towards 135°, forces are distributed over a larger area, reducing stress concentration and ensuring a more uniform stress distribution. This redistribution of forces lowers the likelihood of localized stress peaks, promoting more efficient stress diffusion through the surrounding soil. In contrast, at a 180° curvature, while the contact area is maximized, the stress distribution becomes too diffuse, reducing the overall efficiency of force transmission and potentially compromising the local sealing performance of the pile shoe, which in turn decreases consolidation efficiency. Therefore, the 90–135° range strikes the best balance, reducing stress concentration, maintaining effective sealing, and improving foundation stability.

4.3. Analysis of Soil Parameters’ Influence on the Application Range

Figure 12 illustrates the distribution characteristics of the pile shoe’s horizontal displacement field when penetrating different types of soft soil. Softer soils exhibit larger smear zones. For example, in ordinary clay, the smear radius is approximately 3 d (where d represents the sleeve diameter), resulting in a relatively smaller displacement gradient. In contrast, silty clay exhibits a larger smear radius of about 7 d due to its higher shear swelling, leading to deeper displacements. The heightened sensitivity of silty clay to the smearing effect is likely due to its unique dilatancy characteristics, which significantly enhance shear strength under high-stress conditions. Additionally, the lower permeability of silty clay makes it more susceptible to smearing, as restricted fluid flow exacerbates the formation and expansion of the smear zone.

On the other hand, powdery clay displays brittle shear failure at the boundary, with a smear radius around 4.5 d. The displacement field in silt-clay mixtures shows a bimodal distribution, with a smear zone extending approximately 5 d. This result is due to relatively narrow shear zones and the absence of mud ejection between the shear spots. Notably, the position of the smear zone coincides with the localized stress concentration and high-strain regions induced by the duckbill pile shoe. The development and extent of the smear zone are significantly influenced by the geometric shape of the pile shoe.

This outcome aligns with the cavity expansion theory-based conclusions made by Sathananthan et al. [2], which indicate that the weak smearing region has a greater extent and that the strong smearing region is roughly 2.5 d. Comparative investigations reveal that both soil type and pile shoe geometry influence the extent of coating coverage. Due to its strong cohesiveness, ordinary clay has a slightly smaller smear zone than other types of clay. Silty clay, with its high shear swelling effect, exhibits a wider smear range, while silty clay and silt-clay interlayers fall between these extremes. Their strain distribution patterns and lateral interface morphologies differ, yet they demonstrate similar smearing effects. It is evident that soil properties and pile shoe structural characteristics have a substantial coupling influence on the smearing effect, with both factors working together to determine its extent and spatial distribution. As such, the quantified smear zone found in this study can serve as a reference for evaluating smear zones under different stratigraphic conditions, providing valuable technical advice for optimizing soft soil foundation reinforcement parameters.

Figure 13 illustrates the radial attenuation characteristics of horizontal displacement under various soil conditions. Overall, displacement shows strong nonlinear attenuation with increasing radial distance, and the geometric configuration of the pile shoe plays a crucial role in controlling its variation. Starting from the outer edge of the sleeve, the coating zone extends approximately 60 mm outward. Within this 0–60 mm range, horizontal displacement decreases sharply by around 60–80%. The height of the remodeling zone caused by pile shoe compression corresponds closely to this variation. Beyond this range, the soil response transitions from the plastic failure stage to the structural adjustment stage, as indicated by the slowing rate of displacement change.

Different soil types exhibit distinct smearing effects. Silty clay, with its limited permeability and moderate strength, shows the largest smearing range—approximately ten times the sleeve’s diameter. In contrast, ordinary clay, due to its high cohesive strength, has the smallest smear range—around seven times the sleeve diameter. The combined effects of permeability and structural strength are evident in the results for silty clay and silty loam, which fall between these two extremes. Moreover, the 10 mm widening at the tip of the duckbill pile shoe intensifies local compression, significantly disrupting the soil structure adjacent to the pile. Nonetheless, displacement attenuation slows beyond four times the sleeve diameter, with residual displacement remaining below 5%, suggesting that the overall structure retains basic stability.

This trend aligns with the findings of Indraratna et al. [20], confirming that both the soil’s characteristics and the pile shoe’s geometric parameters jointly control the smear effect. In conclusion, controlling the smear zone and improving foundation consolidation performance in soft soil foundation treatments, such as land reclamation, can be achieved through careful planning of pile shoe geometry and optimization of construction parameters based on soil characteristics. This approach provides essential theoretical support for the long-term performance assessment and reinforcement of soft soil foundations.

Figure 14 illustrates the distribution patterns of Mises stress as the pile shoe penetrates various soil types. It is evident that different soil types significantly influence the smear zone’s characteristics. Ordinary clay, due to its relatively soft texture, exhibits less pronounced stress concentration effects in all directions. Its maximum stress reaches only 0.216, with the maximum effect range extending to 12 d (where d represents the sleeve diameter), showing a gradual degradation of stress with increasing radial distance. In contrast, silty clay exhibits a more localized influence, with a reduced impact range of only 8 d due to its higher stress concentration and faster stress decay. The maximum stresses for silty silt and silty clay are 0.737 and 0.590, respectively, with silty silt displaying a bimodal stress distribution, which is more uniform than that of silty clay.

Spatially, stress concentration primarily occurs near the edges of the grooves and at the contact points between the pile shoe and the soil. This indicates that the local stress transfer at these points is influenced by the pile shoe’s geometric shape. The permeability and mechanical properties of the soil are significant factors in determining the extent of the smear zone. Due to its higher cohesive strength, ordinary clay exhibits a relatively narrow smear range, whereas silty clay, with its stronger shear swelling effect, has a wider smear range. The boundary between these extremes becomes less defined in soils like silty loam and silty clay loam.

These findings support the conclusions of Indraratna et al. [20], which suggest that the smear effect is closely linked to the soil’s permeability and shear characteristics. The spatial extent of the smear zone is determined by both the geometric shape of the pile shoe and the intrinsic properties of the surrounding soil. Therefore, by carefully adjusting the pile shoe geometry and construction parameters, it is possible to control the smear zone, enhance the drainage effectiveness of plastic drainage boards, and accelerate foundation consolidation during the construction of soft soil embankments.

Figure 15 illustrates the distribution pattern of Mises stress along the horizontal direction under various soil conditions. The results reveal notable nonlinear degradation in the Mises stress across all soil types. The smearing effect is confirmed by the concentration of stress predominantly around the pile shoe, with the impact extending well beyond the pile shoe’s structural dimensions. The combined effect of the duckbill structure and groove creates a high-stress core zone, where the stress gradient reaches its maximum between 0 and 250 mm from the pile body. The overall horizontal extent of the substantial stress zone is approximately 600–800 mm, or 4.6–6.2 times the sleeve’s outer diameter.

The extent of the smearing effect is strongly influenced by the soil type. Silty clay and silt-clay soils fall in between, with spread ranges of 9 d and 8 d, respectively, while ordinary clay has the shortest spread range, approximately 7 times the sleeve diameter (7 d). Silty clay, on the other hand, exhibits the largest spread, about 10 times the sleeve diameter (10 d). This trend is consistent with the findings of Sathananthan et al. [2] and Indraratna et al. [20], which show that the magnitude of the smear effect is significantly determined by fluctuations in the soil’s mechanical properties and permeability.

In conclusion, the development and spread of the smear effect are predominantly influenced by pile-soil contact. These findings have major engineering implications for improving the effectiveness of soft soil foundation treatment and consolidation in port areas. The systematic variations observed across different soil conditions provide a quantitative basis for optimizing pile shoe geometry and designing construction parameters. The extent of the smear zone is closely related to soil type and penetration characteristics, which can be quantitatively correlated with key geotechnical parameters, such as cohesion, internal friction angle, and permeability. Soils with higher cohesion (e.g., ordinary clay) exhibit smaller smear zones due to their enhanced shear resistance, while soils with lower cohesion (e.g., silty clay) exhibit larger smear zones. Similarly, soils with lower friction angles (e.g., silty clay) tend to have broader smear zones due to reduced shear resistance, whereas soils with higher friction angles (e.g., ordinary clay) limit lateral spreading. Permeability also plays a crucial role, with soils of lower permeability (e.g., silty clay) showing larger, more persistent smear zones due to limited pore pressure dissipation. Incorporating these parameters into the modified circular hole expansion theory enables the development of a predictive model for estimating smear zone extent based on soil type. This refined approach enhances the accuracy of simulations and supports the optimization of pile shoe design and construction parameters, ultimately improving the efficiency of soft soil foundation treatments.

5. Theoretical Analysis of Expansion for Duckbill Pile Shoes

5.1. Basic Assumptions and Establishment of the Mechanical Model

This study develops an axisymmetric plane strain mechanical model based on the circular hole expansion theory to quantitatively analyze the generation mechanism and spatial extent of the smearing effect during the installation of duckbill pile shoes in soft clay. The following assumptions underpin this model:

• A homogenous, isotropic saturated soil mass is called soft clay. Using the Mohr–Coulomb elastic-plastic constitutive model, conditions are roughly undrained during installation.
• The distant field satisfies the “infinite domain” requirement, simplifying the pile shoe-soil system to an axisymmetric plane strain problem.
• Under uniform internal pressure, treat the complex pile shoe cross-section as an equivalent rigid circular hole that expands from the starting radius R₀ to the current radius R_u.
• The soil surrounding the borehole gradually produces a plastic zone and an elastic zone with an outside radius of R_p, which correspond to the strong and weak smear zones, respectively, as drilling pressure increases, as illustrated in Figure 16.
• To take into consideration the effects of geometric elements like end widening and groove bending, introduce the equivalent expansion radius R_eq.

The theoretical basis for determining the connection between smear zones and geometric corrections in later research is provided by this model.

5.2. Solution for Circular Hole Expansion in the Elastic Stage

The entire soil mass is still elastic during the first stage of expansion, and the soil close to the borehole wall has not yet surrendered. Let σ_r and σ_θ represent the radial and circumferential stresses, respectively, and let r represent the radial coordinate from the center to any point. The equilibrium differential equations under axisymmetric circumstances are:

$${\displaystyle \frac{d_{{\sigma }_r}}{d_r}}+{\displaystyle \frac{{\sigma }_r-{\sigma }_{\theta }}{r}}=0$$

(1)

The boundary conditions are:

$${\sigma }_{\gamma }(r=R_u)=p_u {\sigma }_r(\mathrm{r}\to \mathrm{\infty })={\sigma }_0$$

(2)

The initial horizontal stress of the undisturbed soil before pile drive is represented by σ₀ in the equation. The stress solution at any radius during the elastic stage can be obtained by simultaneously solving Equations (1) and (2):

$${\sigma }_{\gamma }^{(e)}(r)={\sigma }_0+(p_u-{\sigma }_0){\displaystyle \frac{R_u^2}{r^2}}$$

(3)

$${\sigma }_{\gamma }^{(e)}(r)={\sigma }_0-(p_u-{\sigma }_0){\displaystyle \frac{R_u^2}{r^2}}$$

(4)

It is evident that the extra tension brought on by expansion decays radially at a rate of 1/r², with very little disturbance to the soil in the far field.

Hooke’s law can be used to express the radial displacement under plane strain conditions:

$$u_R^{(e)}(r)={\displaystyle \frac{p_u-{\sigma }_0}{2G}}{\displaystyle \frac{R_u^2}{r}}$$

(5)

The shear modulus is represented by $\mathit{G}=\mathit{E}/[\mathbf{2}(\mathbf{1}+\mathit{\nu })]$ in the equation, the elastic modulus by E, and the Poisson’s ratio by ν. The hole wall’s displacement is ${\mathit{u}}_{\mathbf{0}}={\mathit{R}}_{\mathit{u}}-{\mathit{R}}_{\mathbf{0}}$ when $\mathit{r}={\mathit{R}}_{\mathbf{0}}$. The relationship between the hole pressure and the expansion radius can be inferred from Equation (5):

$$p_u={\sigma }_0+2G(1-{\displaystyle \frac{R_0}{R_u}})$$

(6)

According to Equation (6), the internal pressure within the elastic stage rises roughly linearly with aperture size, with the soil’s shear modulus controlling its slope. The soil close to the borehole wall enters the plastic flow stage when p_u increases to a point where the stress state at the borehole wall meets the yield criterion.

5.3. Derivation of the Plastic Stage and Plastic Zone Radius

The inner soil becomes flexible as the pore pressure rises and the shear stress close to the borehole wall reaches the soil strength. The yield connection between the radial and circumferential primary stresses inside the plastic zone can be written as follows using the Mohr–Coulomb strength criterion:

$${\sigma }_{\theta }^{(p)}={\sigma }_{\gamma }^{(p)}{\displaystyle \frac{1-sin\phi }{1+sin\phi }}-{\displaystyle \frac{2c cos\phi }{1+sin\phi }}$$

(7)

The plastic zone factor is shown by the superscript (p), the angle of internal friction is represented by φ, and the cohesiveness of the soil is represented by c. In the plastic zone, Equation (1) is still valid. Equation (7) can be substituted into Equation (1) and integrated to produce:

$${\sigma }_{\gamma }^{(p)}(r)=[p_u+c cos\phi ]{({\displaystyle \frac{R_u}{r}})}^{{\displaystyle \frac{2sin\phi }{1+sin\phi }}}-c cot\phi$$

(8)

$${\sigma }_{\theta }^{(p)}(r)=({\displaystyle \frac{1-sin\phi }{1+sin\phi }}){\sigma }_r^{(p)}(r)-{\displaystyle \frac{2c cos\phi }{1+sin\phi }}$$

(9)

The plastic zone’s outside edge has a radius of R_p. The stress must be consistently linked to the solution in the elastic zone at ${\mathit{r}=\mathit{R}}_{\mathit{p}}$. Equations (3), (4), (8) and (9) can be used to express this.

$${\sigma }_{\gamma }^{(p)}(R_p)={\sigma }_{\gamma }^{(e)}(R_p),{\sigma }_{\theta }^{(p)}(R_p)={\sigma }_{\theta }^{(e)}(R_p)$$

(10)

The radial stress at the plastic zone border can be obtained by substituting Equations (3)–(9) into Equation (10) and removing ${\mathit{\sigma }}_{\mathit{r}}$ and ${\mathit{\sigma }}_{\mathit{\theta }}$.

$${\sigma }_{r,d}={\sigma }_0(1+sin\phi )+c cos\phi$$

(11)

The overall volume change brought about by the expansion of a circular hole should equal the sum of the volume changes in the elastic and plastic areas, in accordance with the concepts of volume conservation and strain continuity. Let ${\mathit{u}}_{\mathit{d}}$ be the radial displacement at the elastic-plastic transition point and ∆ be the average volumetric strain in the plastic zone (assume ∆ ≈ 0.015 for soft clay). Next,

$$\pi (R_u^2-R_0^2)=\pi R_p^2-\pi {(R_p-u_d)}^2+\pi (R_p^2-R_u^2)∆$$

(12)

Expanding and neglecting higher-order terms of $u_d^2$ yields

$$R_u^2-R_0^2=2R_p{u}_d+(R_p^2-R_u^2)∆$$

(13)

Additionally, the displacement at the transition point between elastic and plastic can be written as

$$u_d={\displaystyle \frac{1+\upsilon }{E}}{R}_p{\sigma }_{r,d}$$

(14)

An exact expression for the radius of the plastic zone can be obtained by substituting Equations (11) and (14) into Equation (13).

$$R_p={\displaystyle \frac{R_u^2(1+∆)-R_0^2}{\frac{{\sigma }_0(1+sin\phi )+c cos\phi }{G}+∆}}$$

(15)

The radius R_p of the plastic zone under a specified internal pressure p_u (or equivalent expansion radius R_u) can be found theoretically using Equation (15). Pore pressure, expansion radius, and plastic zone radius have a unified relationship when Equation (6) is combined.

From the standpoint of the smearing effect, R_p describes the spatial scale of the strong shear remodeling zone: the soil experiences substantial shear deformation and structural failure within the range σ₀ ≤ r ≤ R_p, resulting in a notable decline in its strength parameters and permeability coefficient, which corresponds to the strong smearing zone. On the other hand, the weak smearing zone, which has a slower rate of permeability deterioration, is the elastic disturbance zone where r > R_p.

5.4. Geometric Correction and Coating Zone Demarcation for Duckbill Pile Shoes

The derivation discussed earlier is based on the concept of a regular circular hole. However, the actual cross-section of the duckbill pile shoe presents complex geometric features, particularly the “end expansion” and “curved gripping slots,” which deviate from a simple circular geometry. Therefore, the circular hole expansion solution needs to be adjusted to account for these geometric effects. This study introduces the equivalent expansion radius, R_ep, which modifies the circular hole expansion theory to incorporate the wedge angle and flared head as an equivalent section radius that varies with depth, similar to the approach used for flared wedge piles.

5.4.1. Definition of Equivalent Radius

To calibrate the geometric correction coefficients for varying geological conditions or casing boot sizes, empirical testing and numerical simulations are essential. These studies provide a recommended range of values for these coefficients, which can be derived based on factors such as soil types and installation conditions. For instance, the typical range for the expansion radius can be suggested according to the soil cohesion and the penetration rate across different terrains. By utilizing the horizontal cross-section of the pile shoe, the area can be estimated and defined accordingly. The relationship for the equivalent radius is given by:

$$R_{eq}=\sqrt{{\displaystyle \frac{{\mathrm{A}}_{\mathrm{s}}}{\pi }}}=R_0+{\alpha }_{1}b+{\alpha }_2{R}_0({\displaystyle \frac{\beta }{\pi }}-1)$$

(16)

where β is the groove bending angle (90–180°), b is the widened dimension at the duckbill end, and α₁ and α₂ are dimensionless geometric correction coefficients found through model testing and numerical simulation. End widening and groove bending can both be equivalently expressed as increases in the expansion radius, according to Equation (16). The wider the end and the greater the bending angle, the larger R_eq becomes, producing a stronger radial expansion effect on the near-field soil.

5.4.2. Revised Expansion Model

The modified formula that accounts for the geometric implications of the pile shoe is obtained in the theoretical derivation by replacing R₀ and R_u with R_0,eq and R_u,eq, respectively:

$$p_u={\sigma }_0+2G(1-{\displaystyle \frac{R_{0,eq}}{R_{u,eq}}})$$

(17)

$$R_p={\displaystyle \frac{R_{u,eq}^2(1+∆)-R_{0,eq}^2}{\frac{{\sigma }_0(1+sin\phi )+c cos\phi }{G}+∆}}$$

(18)

Equations (17) and (18) show that the geometric parameters of the duckbill pile shoe can regulate the equivalent expansion radius, which in turn affects the necessary borehole pressure and the extent of the plastic zone, thereby controlling the intensity and spatial scale of the smearing effect given the soil properties and initial stress conditions.

5.4.3. Quantitative Classification of Strong and Weak Smear Zones

The coated area can be further separated into the following categories based on the transparent soil test findings and CEL numerical calculations:

(1)

Strong smearing zone radius ${\mathit{R}}_{\mathit{s}}^{(\mathbf{1})}$: Set ${\mathit{R}}_{\mathit{s}}^{(\mathbf{1})}$ ≈ R_p, which represents the area where the soil’s plastic shear deformation is fully developed, resulting in a significant reduction in undrained shear strength and permeability;

(2)

The location where radial displacement or equivalent shear strain decays to 10% of its peak value can be used to calculate the weak smear zone radius ${\mathit{R}}_{\mathit{s}}^{(\mathbf{2})}$.

$$u_r(r=R_s^{(2)})=0.1u_r(R_{u,eq})$$

(19)

According to numerical results, under typical parameters for duckbill pile shoes, ${\mathit{R}}_{\mathit{s}}^{(\mathbf{1})}$ can extend from approximately 3–4 d to ${\mathit{R}}_{\mathit{s}}^{(\mathbf{2})}$ of 7–10 d (where d represents sleeve diameter). This is consistent with current vertical drainage pipe installation tests and circular hole expansion predictions, confirming the rationality of the geometric correction model. This corresponds to regions where structural disturbances persist but permeability decline is relatively gradual.

In conclusion, an elastic-plastic analytical solution for the duckbill pile shoe insertion process was developed using the circular hole expansion theory, extended to account for geometric features such as end widening and groove curvature. This approach establishes the relationship between “geometric parameters—plastic zone radius—smear zone extent” through an equivalent expansion radius, simplifying the interpretation of smear zone parameters and three-dimensional CEL simulation results.

To demonstrate the quantitative superiority of the duckbill-type pile shoe, a comparison was made with conventional pile shoes. Experimental results and numerical simulations show that conventional designs, with simple cylindrical geometry, generate a strong smear zone radius of approximately 5–6 times the sleeve diameter (d) and a weak smear zone radius of 10–12 d. In contrast, the duckbill design reduces the weak smear zone by up to 30%, primarily due to its geometry, which promotes a more uniform stress distribution and reduces localized soil disturbance. When the groove curvature is between 90° and 135°, the smear zone becomes more dispersed, further minimizing stress concentrations. These findings highlight the advantages of the duckbill design, reinforcing its originality and effectiveness in mitigating smear effects.

5.5. Theoretical Model Validation

The modified circular hole expansion theoretical solution was systematically compared with the three-dimensional numerical simulation results based on the Coupled Euler–Lagrange (CEL) method presented in this study, as well as with transparent soil-PIV experimental data, to assess the applicability and accuracy of the proposed equivalent expansion radius correction model for predicting the smear zone of duckbill pile shoes. The relationship between borehole pressure, expansion radius, and plastic zone radius is theoretically established, and a modified calculation formula for the plastic zone radius is derived by incorporating the geometric influence of the pile shoe into the expansion model through an equivalent radius. As shown in Figure 14, this theoretical framework provides specific assessment criteria for quantitative comparisons, such as the plastic zone radius and the normalized horizontal displacement (u/d) radial decay curve.

Both in the near field (r/d < 4) and the far field (r/d > 4), Figure 17 demonstrates that the normalized displacement curves (u/d-r/d) derived from the three approaches exhibit strong consistency with only slight variations. The curves closely coincide in the near-field range, with displacements being nearly identical for r/d ≈ 1.5 and 2.5, suggesting that soil deformation is well predicted by both the updated theoretical model and the numerical simulations. At r/d ≈ 8, the far-field curve shows a small deviation, but this discrepancy remains minimal. Quantitative analysis indicates that the three datasets exhibit a root mean square error of approximately 8% and a maximum relative error of 12%, which typically fall within acceptable engineering tolerance limits, demonstrating the high accuracy and reliability of the updated theoretical model.

The primary sources of variation between the three curves include errors arising from PIV image recognition, boundary conditions of the experimental setup, and simplifying assumptions and approximations within the theoretical model. Nevertheless, as illustrated in Figure 16, the three approaches produce comparable predictions regarding the boundaries of regions with strong or weak smear effects, irrespective of the conditions. This suggests that the extent of the smear zone’s effects is accurately represented in the revised theoretical model. Moreover, the updated theoretical model is computationally efficient, making it suitable for initial evaluations during the design phase and adaptable to real-world engineering scenarios.

While the theoretical model demonstrates good agreement in the near-field region, discrepancies in the far-field indicate that model simplifications—such as axisymmetry and soil homogeneity—may contribute to errors. To address these discrepancies, future model improvements should incorporate more complex soil stratigraphy and adopt a non-axisymmetric approach to better reflect soil heterogeneity and boundary effects. Additionally, boundary effects in the experimental setup may influence far-field results. Future experiments should extend the soil domain and optimize boundary conditions to permit free radial displacement, providing a more accurate simulation of soil behavior. Considering soil heterogeneity, future research should incorporate various soil types and integrate rate-dependent soil constitutive models to capture the complexities of high-permeability soil behavior. These enhancements will improve the model’s predictive capability in the far field and more accurately reflect the actual soil response and pile shoe interaction mechanisms in real-world engineering applications.

6. Conclusions

This study methodically examines the influence of soil factors and pile shoe geometry on the smear effect, using a model that integrates the circular hole expansion theory with the Coupled Euler–Lagrange (CEL) method. The primary conclusions are as follows:

(1)

Impact of Penetration Rate: An increase in the penetration rate notably expands the smear zone and intensifies stress concentrations near the pile. This simultaneous development of both the strong and weak smear zones during high-speed installation emphasizes the critical role of construction speed. By reducing the smear effect, higher penetration rates enhance drainage efficiency. Consequently, optimizing penetration speed is pivotal for improving soil consolidation during pile installation.

(2)

Role of Pile Shoe Geometry: The geometry of the pile shoe significantly influences the distribution of stress in the surrounding soil. As the groove curvature increases from 90° to 135°, the smear zone shifts from a concentrated to a more diffuse pattern, effectively reducing local stress concentrations. This geometric adjustment facilitates a more uniform stress distribution, contributing to improved soil interaction and reduced disturbance. However, excessive curvature beyond this range may compromise the integrity of the local sealing mechanism, inhibiting the formation of stable drainage channels.

(3)

Influence of Soil Type: The smear effect is strongly influenced by the type of soil in which the pile is being installed. In particular, ordinary clay exhibits the smallest smear zone (approximately seven times the sleeve diameter), while silty clay displays the largest smear zone (approximately ten times the sleeve diameter). Silty loam and silty clay fall between these extremes. This variation highlights the importance of soil permeability and structural characteristics as primary factors in determining the extent of the smear zone.

(4)

Model Validation and Application: The model’s accuracy and efficacy are validated through the strong consistency observed between the modified circular hole expansion model and the Coupled Euler–Lagrange (CEL) numerical simulations. The results substantiate the reliability of the proposed theoretical model in predicting smear zone extent under various installation conditions. The research provides essential theoretical insights and practical recommendations for optimizing pile shoe geometry and construction parameters to mitigate the smear effect and enhance drainage efficiency in soft soil foundation treatments.