Shield machines are widely used in excavation projects such as metro and tunnel construction, featuring safety, high efficiency, strong geological adaptability, and low working noise (Zhu et al., 2008; Yang et al., 2024; Ding et al., 2022; Li et al., 2023). The cutterhead is one of the core components of the shield machine. When tunneling in composite strata, the cutterhead is subjected to eccentric loads due to the inhomogeneous distribution of geological conditions, probably causing the tunneling attitude to shift (Liu et al., 2025; Song et al., 2025).

The structural parameters of the cutterhead and the excavation parameters greatly affect the shield machine’s mechanical characteristics when the cutterhead excavates in composite strata (Göbl, 2010; Kong et al., 2024). Su et al. analyzed the structural characteristics of the panel-type cutterhead and spoke-type cutterhead, and compared the differences in their stiffness, strength, and weight of the two cutterheads (Su et al., 2011). Yang et al. investigated shield tunneling parameters and studied the influences of earth chamber pressure, thrust, and cutterhead rotational speed on tunneling speed and cutterhead torque (Yang et al., 2009). Sun et al. (2016) established a dynamic cutting force model based on the cavity expansion theory and a discretization method (Sun and Gao, 2023). They investigated the effects of geological conditions and operational parameters on the dynamic cutting forces and the vibration characteristics of the cutterhead.

During the tunneling process of the shield machine, due to the unevenness of the loads generated by the composite strata, the working conditions of the cutterhead in the composite strata are significantly different from those in a single stratum (Lin et al., 2021; Sun et al., 2016; Wu et al., 2024). Zhu et al. analyzed the leading causes and laws of the deviation of shield tunneling direction under different geological conditions through the measured deviation data of shield tunneling. They studied the orientation control technology of shield construction (Zhu, 2012). Li et al. conducted a failure analysis on the structural cracking of the cutterhead during construction through numerical simulations. They studied the rate of change of the stress intensity factor of cracks of different shapes (Li et al., 2021). Qian et al. performed a mechanical analysis of the coupling interaction between the cutterhead and the excavation face (Zhang et al., 2013). Zou et al. decoupled and solved the normal and tangential loads acting on the cutterhead, and established analytical expressions of thrust and torque under homogeneous geological conditions (Zou et al., 2024). Zhang et al. studied the dynamic loads received by the cutterhead during the tunneling process, established a thrust and torque prediction model considering the influences of overburden pressure, soil cutting, cabin pressure support, and shield shell friction (Zhang et al., 2014). Shang et al. investigated the free vibration characteristics of the cutterhead of the shield machine and proposed a meshless method based on radial basis functions (Shang et al., 2020). Yang et al. investigated the vibration characteristics of the cutterhead in soft-hard mixed strata through similarity model experiments, demonstrating the influence of the strength differences on the vibration acceleration and frequency distribution (Yang et al., 2020).

When a shield machine is tunneling in composite strata, due to the eccentric loads between the cutterhead and the composite strata, the tunneling posture of the cutterhead is prone to change, causing the tunneling direction to shift. When analyzing the mechanical characteristics of the cutterhead in the composite strata, the existing studies generally equate the interaction between the stratum and the cutterhead to the nonuniform loads acting on the cutterhead surface, ignoring the constraint effects of the rock mass on the cutterhead, which is different from the actual contact. The interaction between the cutterhead and the rock is generated through the disc cutter. Researchers have done investigations on the interaction and contact pressure distribution between the disc cutter and rock. Sun et al. (2016) developed a mechanics model for predicting the cutting forces (normal, rolling, and side forces) of constant cross-section disc cutters by solving the Boussinesq stress field problem (Sun et al., 2023). Zhang et al. employed the discrete element method (DEM) to investigate the contact pressure distribution between the disc cutter and rock surface, revealing that the maximum contact pressure depends solely on rock uniaxial compressive strength (Zhang et al., 2021). The pressure distribution within the rock-disc cutter interface was measured by strain gauges (Rostami, 2013) or digital image correlation experiments (Ashoor et al., 2025). Although studies on contact pressure can help researchers analyze the rock-breaking mechanism, contact pressure cannot be directly embedded into the dynamic model of the cutterhead system. Hertz contact theory translates the complex contact problem into computable spring stiffness via equivalent parameters. While this treatment sacrifices detailed local stress information, it enables the feasibility of dynamic studies for the cutterhead system. This represents the inevitable simplification required when scaling engineering problems down to micro-scale investigations.

In this work, the interaction between the stratum and the cutterhead is equivalent to mechanical spring constraints based on the Hertz contact theory. The static and dynamic characteristics of the cutterhead under different stratum constraints are investigated when the thrust is applied, and the influence of the rock-breaking loads of the disc cutter on the vibration response of the cutterhead is analyzed. The findings are helpful for the design of the cutterhead and the optimization of the tunneling parameters in the composite strata.

In this work, the Hertz contact model of parallel contact between a plane and a cylinder (axis parallel to the plane) is used to describe the interaction between the disc cutter and rock mass, as shown in Figure 1a. The elastic contact theory calculates the contact stiffness between the disc cutter and the rock mass, and the contact stiffness is equivalent to the spring constraint between the cutterhead and the rock mass. The finite element model of the interaction between the shield cutterhead and the rock mass is further established. The finite element numerical simulation is used to analyze the dynamic characteristics of the cutterhead in different strata, study the static and dynamic responses of the cutterhead under the maximum thrust, and calculate the dynamic response of the shield cutterhead under the rock-breaking load of the disc cutter in different strata, to provide theoretical support for the dynamic control of the cutterhead driving parameters and driving attitude.

Assuming the contact between the cylinder and the elastic half-space is frictionless, and both the deformations of the cylinder and the half-space are small. According to Hertz contact theory, for the line contact between a cylinder and a half-space, the contact region is a rectangle of width 2b, and the contact pressure distribution is semi-elliptical (Figure 1b) and given by Equation 1 (Johnson, 1985; Hills et al., 1993) p x = p 0 1 − x / b 2 (1) where p ₀ is the maximum contact pressure. The load per unit thickness of the disc cutter P=F/L is the integral of the pressure distribution, as shown in Equation 2 P = ∫ − b b p x d x = ∫ − b b p 0 1 − x / b 2 d x = π b 2 p 0 (2)

This yields the maximum contact pressure p ₀ = 2P/πb. The pressure distribution within the contact zone should meet displacement compatibility, which requires the displacements within the contact zone to match the shape of the cylinder. The Hertz elastic contact solution gives the contact half-width by Equation 3 (Johnson, 1985) b = 4 PR π E * (3) where R is the radius of the cylinder, E ^* is the reduced elastic modulus, which is expressed in Equation 4 1 E * = 1 − ν 1 2 E 1 + 1 − ν 2 2 E 2 (4) where E ₁ and E ₂, ν ₁ and ν ₂ represent the elastic moduli and Poisson’s ratios of the cylinder and the half-space, respectively.

The indentation depth δ is defined to be the distance of normal approach between the two elastic bodies. For the contact between a cylinder and a half-space under plane-strain conditions, Hertz elastic contact theory gives the indentation depth, as shown in Equation 5 (Johnson, 1985) δ = P π E * ln 4 R b − 1 2 (5)

The contact stiffness is defined as the derivative of the load with respect to the indentation depth (Figure 1c) by Equation 6 K = d P d δ = π E * ln 4 R / b − 1 (6)

Substituting the expression of half-width b into the above equation, the final expression for contact stiffness is obtained, as shown in Equation 7 K = d P d δ = 2 π E * ln 4 π E * R P − 2 (7)

The parameters used for the calculation of disc cutter-rock contact stiffness are listed in Table 1. By substituting the parameters into the contact stiffness formula, the reduced elastic modulus for the moderately weathered limestone and the disc cutter is 24.14 GPa, and the contact stiffness is 2.94 × 10⁷ kN/m. The reduced elastic modulus of hard rock and the disc cutter is 59.28 GPa, and the contact stiffness is 6.15 × 10⁷ kN/m.

In the finite element modeling, both the cutterhead and disc cutter employ a linear elastic constitutive model. The material density is 7.85 × 10³ kg/m³, the elastic modulus is 210 GPa, and Poisson’s ratio is 0.3. The cutterhead is meshed using the SOLID187 element type. A fully fixed displacement constraint is applied to the flange end face, while the side surface of the cutterhead is defined with a frictionless support constraint. The maximum thrust applied to the shield machine cutterhead is 40,700 kN. For different ground strata, the constraint between the cutterhead and the rock mass is defined as a normal spring constraint, and a total of 100 spring constraints are applied. The two ends of the spring are connected with the cutterhead and the rock stratum to substitute for the contact stiffness. In the analysis of composite strata, the upper half of the stratum is set as moderately weathered limestone, and the lower half is set as hard rock. The schematic diagram of the contact between the cutterhead and rock stratum and the corresponding equivalent contact model are shown in Figure 2a and Figure 2b, respectively.

In this work, the displacement, modal characteristics, and vibration response of a shield cutterhead operating in composite strata are investigated, utilizing Hertz contact theory to model the cutterhead-rock interaction. The contact between the disc cutter and rock is equivalently represented by mechanical spring constraints, with stiffness derived from elastic contact theory for the cylinder-plane configuration. In composite strata, the displacement distribution of the cutterhead under maximum thrust exhibits significant asymmetry. The displacement of the upper half of the cutterhead is much larger than the lower half due to stiffness contrast. The natural frequencies depend primarily on the equivalent contact stiffness at the cutterhead-stratum interface, increasing progressively from moderately weathered limestone to composite strata, and finally to hard rock. Composite strata induce unique asymmetric modal shapes, resulting from heterogeneous contact stiffness and structural asymmetry. The vibration responses under rock-breaking loads in uniform strata are rotationally symmetric but asymmetric in composite strata. The maximum displacement of the cutterhead in hard rock stratum is reduced by 34.78% compared to moderately weathered limestone stratum. The displacement of the cutterhead in composite strata falls between the other two strata. This work provides a Hertz contact-based framework for analyzing shield cutterhead dynamics in heterogeneous strata.