## 4 Model Verification  
### Cutting Force of a Single Tool  
Before calculating the resultant force and moment caused by pile cutting, it is necessary to obtain the force of cutting concrete with a single tool. Many studies have been carried out on this issue and cutting force in different condition was investigated through numerical simulation and laboratory test [36; 37; 38]. The cutting force of a single tool cutting concrete is mainly affected by tool parameters (such as blade width, edge angle, rack angle and relief angle), strength of concrete and movement parameters (such as penetration and cutting speed). The selection of the geometric parameters of the shield tools usually takes into account the construction stratum conditions, cutting efficiency, wear resistance, etc., and it is hard to change once the tools were finished. The effect of the variation of the cutting speed under normal construction conditions on the cutting force is not obvious. Therefore, this paper focuses on the calculation of cutting force with different penetration. A numerical simulation model of a single tool cutting concrete was established based on finite element software LS-DYNA. The Holmquist-Johnson-Cook (HJC) model is introduced to simulate the failure of concrete material [39] and modified HJC model parameters are used according to the results of laboratory tests. The HJC model is a constitutive model widely applied to the numerical simulations of the dynamic responses of concrete-like materials. The comprehensively considers strain rate effect, damage evolution effect, confining pressure effect, crushing effect and pressing effect. The concrete cutting process by shield tool is a process of concrete damage, crushing and destruction caused by dynamic penetration. Therefore, HJC model is considered suitable in the numerical calculation of cutting force in concrete materials. The model mainly consists of three parts: strength model, damage evolution model and state equation. 1. The material strength model is described by the normalized equivalent stress, as in Equation (5) and Figure 5a\[\sigma_{\rm eq}*=[A(1-D)+B(p*)^{N}](1+\ln\dot{e}*) \tag{5}\]  
where \(\sigma_{\rm eq}*=\sigma_{\rm eq}/f_{c}\) is the normalized equivalent stress; \(\sigma_{\rm eq}\) denotes the actual equivalent stress; \(S_{\rm FMAX}\) is the normalized maximum strength that the concrete material can withstand and \(\sigma_{\rm eq}\leq S_{\rm FMAX}\); \(f_{c}\) is the unconfined uniaxial compressive strength; \(p*=p/f_{c}\) is the normalized pressure, where \(p\) is the actual pressure; \(\dot{e}*=\dot{\epsilon}/\epsilon_{0}\) is the dimensionless strain rate; \(T*=T/f_{c}\) is the normalized tensile strength, where \(T\) is the unconfined uniaxial tensile strength; \(A,B\) and \(N\) are material constants which represent the normalized cohesive strength, the normalized pressure hardening coefficient and the pressure hardening exponent, respectively; \(C\) is the strain rate coefficient; \(D\) (\(0\leq D\leq 1\)) is the accumulated damage. 2. Damage of the HJC model is accumulated from both the equivalent plastic strain and plastic volumetric strain, as in Equation (6) and Figure 5b: \[D=\frac{\Delta\varepsilon_{\rm p}+\Delta\mu_{\rm p}}{\varepsilon_{\rm p}^{ \rm f}+\mu_{\rm p}^{\rm f}}\] (6) \[\varepsilon_{\rm p}^{\rm f}+\mu_{\rm p}^{\rm f}=D_{1}(p*+T*)^{D_{2}}\geq E_{FMIN}\] (7) where \(\Delta\varepsilon_{\rm p}\) and \(\Delta\mu_{\rm p}\) are the effective plastic strain increment and plastic volumetric strain during a cycle of integration, respectively; \(\varepsilon_{\rm p}^{\rm f}+\mu_{\rm p}^{\rm f}\) is the total plastic strain under a constant pressure until fracture; \(D_{1}\) and \(D_{2}\) are the damage constants; \(E_{FMIN}\) is the minimum strain. 3. The relationship between the volume strain and actual pressure in the HJC model is expressed through equation of state and separated into three phases, as shown in Figure 5c.