Considering that the stress on the contact surface between the soil column and a rigid plane is σ , the stress increment d σ propagates along the soil column as a plastic wave at velocity d σ . For a micro-segment of length d x , within d t time, we have: c p = d x d t (1)

According to the law of conservation of momentum: d F d t = m d v (2) where F = A σ and m = ρ A d x substituted into Eq. 2, yield: d ( A σ ) d t = ρ A d x d v (3)

Subsequently, substituting Eq. 1 into Eq. 3 and simplifying, we obtain: d v = 1 ρ d σ c p (4)

The velocity of the plastic wave is: c p = d σ ρ d ε (5)

By substituting Eq. 5 into Eq. 4 and simplifying, the relationship between d σ 、 d ε 、 d 𝜐 can be derived as: d ε = ρ d σ d 𝜐 2 (6)

Integrating with the Duncan-Chang model, an explicit formula for the stress-strain relationship curve during the soil compression process is directly provided, as follows: σ 1 − σ 3 = ε 1 a + b ε 1 (7) where a 、 b represents an experimental constant, σ 1 is the major principal stress, and σ 3 is the confining pressure. In the context of this study, the confining pressure is denoted as σ 3 = 0 , which, when substituted into Eq. 7, and upon differentiation of both sides, yields the relationship between d σ 1 and ε 1 : d σ 1 = a a + b ε 1 2 d ε 1 (8)

By substituting Eq. 8 into Eq. 6, the relationship between the maximum strain of the soil column and the impact velocity can be obtained: 1 a + b ε 1 d ε 1 = ρ a d 𝜐 (9)

To determine the axial plastic strain produced in the soil column due to the impact from a rigid object, Eq. 9 is therefore integrated as Eq. 10: ∫ 0 ε 1 1 a + b ε 1 d ε 1 = ∫ v 1 0 ρ a d 𝜐 (10)

This yields: ln a + b ε 1 + b ρ a 𝜐 1 = ⁡ ln   a (11)

From Eq. 11, the theoretical expression for the axial plastic strain of the soil column under the effect of initial impact velocity can be derived as follows: ε 1 = e l n a − b ρ a 𝜐 1 − a b (12)

As shown in Figure 1, the ends of the soil column undergo lateral deformation when impacted by a rigid planar tamper. At the boundary between elastic and plastic deformation, the soil column changes from the initial cross-sectional area A 0 to A . From the condition of continuity, we have: A 0 c p + 𝜐 = A c p (13)

Thereby there is: d 0 d = A 0 A = c p c p + v (14) In Eq. 14, d 0 and d respectively represent the initial diameter and the deformed diameter of the soil column’s cross-section. Utilizing the relationship between the velocity of material points and the level of stress, we have: σ 1 = ρ c p 𝜐 (15)

By substituting Eq. 15 into Eq. 14, we can obtain: d 0 d = A 0 A = c p c p + 𝜐 = σ 1 σ 1 + ρ 𝜐 2 (16)

By substituting Eq. 7 into Eq. 16, we can obtain: d 0 d = σ 1 σ 1 + ρ 𝜐 2 = ε 1 ε 1 + ρ 𝜐 2 a + b ε 1 (17)

From Eq. 17, the relative magnitude of the lateral deformation at the ends of the soil column, denoted as d 0 d , can be deduced using the soil column’s maximum axial plastic strain value ε 1 , the soil column’s mass density ρ , and the soil column’s initial velocity 𝜐 .

Considering that the typical parameter values selected for soil columns should be compatible with the actual situation, let’s take a = 0.11 、 b = 0.17 , with the soil column’s mass density as ρ = 1.8 g / m m 3 , and the tamper’s initial velocities as 𝜐 0 = 1 m / s 、 𝜐 0 = 2 m / s 、 𝜐 0 = 3 m / s 、 𝜐 0 = 4 m / s 、 𝜐 0 = 5 m / s . Substituting these selected parameters into the derived Eqs 12, 17, the values of ε 1 and d 0 d for the soil column under the impact of the tamper at different initial velocities v 0 can be calculated. The theoretical values and their trends are shown in Table 1 and Figure 2.

When constructing the numerical model based on PFC, we first set the geometric boundary and determine the relevant properties of the particle material, and then determine the interaction law of particle-particle and particle-boundary. As shown in Figure 3, in this numerical simulation, the cylindrical soil particle units are represented using standard spherical particles as the basic discrete elements. The rigid impactor is simulated with a clump, composed of multiple particles rigidly bound together. The parameters in the following table are chosen with due consideration of the actual situation of fine-grained soils and the fact that the deformation of soils under ramming is studied in this paper. The basic parameters of the materials for both are presented in Table 2.

To achieve a uniformly dense soil column, a stratified under-pressure method was employed, generating particles in five layers to simulate the soil body. In order to prevent the occurrence of uneven initial internal stress within the soil column, once the entire column reached its designed height, the “solve aratio” command in PFC3D was utilized to allow the particles to self-balance under the influence of gravity. This ensures that the unbalanced force on each soil particle is less than 10⁻⁵N. Additionally, with a focus on computational efficiency of the model, 4,319 overlapping particles were collectively bound to simulate a rigid impact object, and 5,225 ball particles were used to model the soil column. Considering the mechanical characteristics of impact loading, a hysteresis damping model was adopted for the interactions between particles and between particles and the impact object. The model parameters were determined through extensive research and a comprehensive review of literature (as shown in Tables 2, 3).

To test the applicability of the model, impact experiments were conducted using the established discrete element numerical model at five different initial velocities (respectively 1 m/s, 2 m/s, 3 m/s, 4 m/s, and 5 m/s). At the beginning of the experiment, the generated soil column was called upon and its lower end was fixed. The rigid impact object was positioned 5 cm above the top of the soil column and assigned a downward initial impact velocity. It's important to note that during the impact process, the acceleration due to gravity was zero, meaning that the effects of gravity were not considered. Throughout the impact, data such as the bulging at the ends of the soil column, changes in the length of the soil column, and other deformation data were recorded until the velocity of the rigid impact object reduced to zero, marking the completion of the impact process.

As shown in Figure 4, the deformation characteristics at the ends of the soil column, when subjected to impacts at various initial velocities, exhibit common features: The lateral deformation at the ends is greatest at the contact surface and gradually decreases further away. The deformed ends still maintain a circular shape, and the overall plastic deformation of the soil column presents a “mushroom shape” that is larger at the top and smaller at the bottom. More notably, there are differences in the deformation of the soil column ends under different impact velocities. Firstly, the axial plastic deformation of the soil column increases with the increase in impact velocity. In terms of lateral deformation, the deformation at the ends increases with increasing impact velocity. However, at excessively high velocities (v ₀ = 4.0 m/s, v ₀ = 5.0 m/s), a few particles at the impacted end of the soil column are observed to disintegrate, a phenomenon similar to the slight particle splashing observed in actual soil columns under impact.

To verify the applicability of the theoretical model, a comparative analysis was conducted from two perspectives: the axial plastic strain of the soil column and the lateral deformation at its ends. Figure 5 illustrates the axial plastic deformation of the soil after impacts at different velocities. It is observed that as the impact velocity increases, the axial deformation also increases, albeit at a decreasing rate. Comparing theoretical and experimental values reveals good consistency across the five impact velocities, with the largest discrepancy occurring at an impact velocity of 2 m/s, where the difference between the experimental and theoretical values of axial deformation is only 4.2 mm. Regarding the lateral deformation at the ends of the soil column (Figure 6), the theoretical and experimental values under different impact velocities still show good agreement. The greatest difference occurs at an impact velocity of 5 m/s, which can be attributed to the disintegration and farther scattering of individual particles at the ends of the soil column under high-velocity impacts. Considering both axial and lateral deformations, it can be concluded that the theoretical values aptly describe the deformation characteristics of the soil column under impact. This finding has significant implications for further research into the deformation characteristics of foundations and subgrades under impact compaction.