Study on the JH-2 Model Parameters for Metro Shield Cutting Reinforced Concrete Pile

In order to investigate the application of JH-2 (Johnson-Holmquist-2) constitutive model in concrete, a series of parameters study have been conducted to evaluate the JH-2 model in C30 concrete. The finite element analysis software ABAQUS is utilized to simulate the compression test of the three-dimensional standard concrete test based on the ALE adaptive mesh and the damage failure criterion of element deletion function. The numerical analysis of 3D concrete pile foundation and simplified shield machine cutter head to simulate shield cutting pile is established. Analyzing the numerical simulation and construction data, the key points of the shield machine cutting pile are illustrated. Research shows that: (1) JH-2 constitutive model can be used to simulate the concrete cutting. (2) The better damage parameters have been represented and verified against the results of experiments, which are D1 = 0.6 and D2 = 0.1 in the JH-2 model of C30 concrete. (3) In the simulation example of concrete compression and cutting pile, the failure form of concrete is brittle failure, and the non-linear characteristics of concrete has been observed. (4) The simulated torque vibration of the cutter head is relatively large than actual pile cutting, and the actual shield machine cutting pile construction should have a low rotation speed and low propulsion speed.


Introduction
The cutting is a very complex process, which includes elastoplastic mechanics, fracture mechanics, thermodynamics and tribology. The changes of internal stress, strain and temperature during the cutting cannot be accurately measured and detailed description due to the limitation of experimental technology. Therefore, numerical simulation can be used to conduct an effective analysis and research during the cutting (Yan and Chen 2011), and numerical simulation can also avoid too expensive tests, which fully follow the sustainable development in constructions.
Related materials have been used to investigate the cutting in the numerical simulation, which mainly includes JC model, D-P model, HJC model, JH-1 model, JHB model, JH-2 model, etc. The JC model is mainly used for metal cutting simulation. The D-P model is often used for rock and soil cutting (Zhu and Jia 2014; Li et al. 2018;Liu et al. 2019). The HJC model is mainly used for the simulation of large deformation and high strain rate of concrete and rock (Holmquist et al. 1993). However, the failure criterion of the HJC model is the equivalent plastic strain failure criterion. This criterion is mainly used to evaluate the compression failure, but it cannot effectively evaluate the failure caused by tension or shear. JH-1, JHB and JH-2 models are often used to simulate the model with large deformation and high strain rate of brittle materials, and can be used for ceramics, rock, concrete and other materials. JH-2 (Johnson and Holmquist 1994) is based on JH-1 (Johnson and Holmquist 1990) by adding continuous damage and deterioration effect of strength to describe the gradient failure process of the material. However, there are few literature represented that the JH-2 model is used to cut concrete. Oucif et al. (2020) applies the JH-2 model to concrete. But the corresponding parameters of the cited JH-2 model are referred to the original paper (Johnson and Holmquist 1994). This paper intends to analyze and summarize the acquisition method and JH-2 model application in C30 concrete based on various previous tests, and provides a new idea and the different simulation method for the concrete cutting.
The numerical analysis part of this paper is based on the actual shield cutting plain concrete pile foundation engineering. The shield tunnel is conflicted with the existing viaduct pile foundation with the planned route. And the construction company uses the method of pile foundation underpinning to build the new pile foundation of viaduct. However, the original pile foundation exists underground in the form of residual piles, which conflicts with the planned tunnel. The construction company originally planned to impact the residual pile below the planned shield tunnel through the method of downward impact. However, considering the complex geological conditions under the viaduct, numerous pipelines, and a complex location, the methods of impacting pile, drilling pile, and pulling pile are all too dangerous. Only the method of cutting pile by shield machine can be used. In order to prevent the shield machine from being unable to rotate on the shield machine cutter head with the main reinforcement bars twisted during cutting. They drilled the remaining pile to take reinforcement bars and changed the pile foundation into a ''beehive coal'' form. Then, mortar was poured into the drilled holes, and then the shield machine cutting work was performed. In theory, it is feasible to change reinforced concrete pile into plain concrete pile with reduced strength. The project is feasible.
2 JH-2 Model (Johnson and Holmquist 1994) The JH-2 model is suitable for simulating brittle materials, and can be used for simulation of ceramics, rocks, concrete, etc. The model analyses are mainly divided into two parts, the strength and the state equation.

Strength
The normalized equivalent stress of the JH-2 model is where r Ã i is the normalized intact equivalent stress, r Ã f is the normalized fracture stress, D is the damage factor (0 D 1:0). The normalized equivalent stress (r Ã , r Ã i , r Ã f ) has following that: where r is the actual equivalent stress and r HEL is the equivalent stress at HEL. The normalized intact strength is given by The normalized breaking strength is determined by The material constants are A, B, C, m, n and SFMAX, where A and n are the complete strength parameters of the material, B and m are the residual strength parameters of the material, and C is the rate parameter of the material. The normalized pressure is P Ã ¼ P=P HEL , where P is the actual pressure and P HEL is the pressure at HEL. The normalized maximum tensile hydrostatic pressure is T Ã ¼ T=P HEL , where T is the maximum tensile hydrostatic pressure that the material can withstand. The dimensionless strain rate is e Ã ¼ e=e 0 , where e is the actual strain rate and e 0 ¼ 1:0s À1 is the reference strain rate. The strength model curve is shown in Fig. 1.

Damage
The cumulative way of fracture damage is expressed as where De p is the plastic strain during the integration cycle, and e p f ¼ f P ð Þ is the plastic strain at break under a constant pressure P. The specific expression is where D 1 and D 2 are the damage parameters of e p f . The physical explanation of damage and fracture is illustrated in Figs. 2 and 3. If the material is held under a constant pressure and then subjected to a straining deformation at a constant strain rate, the damage begins to accumulate when the material begins to flow plastically (at r¼r i ). Then the material begins to soften relative to its intact strength. This softening may be related to the transition of the material from a larger particle size to a smaller particle size under increased plastic strain. When the material is completely damaged (D ¼ 1:0), the strength doesn't decrease with the increase of plastic strain (r¼r f ).
But the material performance cannot be conducted under higher pressure in these tests. Therefore, the damage function and fracture strength must be inferred from other data.
Equation (7) is the equation of state in Fig. 3. When D ¼ 0, it represents the hydrostatic pressure before the crack starts. When 0\D 1, damage begins to accumulate and the volume may expand (pressure increase or volumetric strain increase). DP is the pressure increment. Where K 1 is the bulk modulus, K 2 and K 3 are constants; l is the volume strain, where l ¼ q=q 0 À 1, q is the current density and q 0 is the initial density.

JH-2 Model Parameter Acquisition
The determining parameters of the JH-2 model in concrete are extremely complicated in the test. The tests include plate impact test, dynamic triaxial test, SPHB test and other related tests. This section focuses on the data summary and fitting of the literature's data that has done corresponding concrete tests in the past, and obtains a set of JH-2 model parameters suitable for concrete.   (Johnson and Holmquist 1994) 3.1 Basic Physical Parameters (Table 1).

State Equation Parameters
The relationship between the volumetric strain of the material and the hydrostatic pressure in the JH-2 model is shown in Eq. (7). When the material is not broken (D = 0), the state equation is K 2 and K 3 need to be calculated by curve fitting according to the volume strain and hydrostatic pressure data obtained from one-dimensional strain plate impact test.
The data measured in literature (Wang et al. 2008) are shown in Table 2.

Intensity-related Parameters
HEL is the Hugoniot elastic limit. Determining the HEL of concrete is more complicated and requires a plate impact test. HEL is based on the concept of static yield under one-dimensional strain. However, the yield stress of materials with one-dimensional strain is very strictly to the experimental equipment and conditions. There are few literature about the determination of HEL in concrete. Rosenberg (1993) proposed the relationship between the spalling strength of brittle materials and HEL as shown in Eq. (10).
where r spall is the spall strength, and literature (Kipp et al. 1998) took the spall strength of concrete as 30 MPa. By formula (10), the concrete HEL = 1481 MPa can be calculated.
To determine the parameters A and n, a triaxial test of concrete is required. Without considering the strain rate coefficient C, the formula (3) can be changed to The normalized hydrostatic pressure is P Ã ¼ P=P HEL , where the hydrostatic pressure is  . According to the concrete tensile strength formula T ¼ 0:62 ffiffiffi ffi f c p proposed by the American Concrete Association, the tensile strength of C30 concrete can be calculated as 3.4 MPa, which is T Ã ¼ T=P HEL ¼ 4:192 Â 10 À3 . The normalized equivalent stress is r Ã ¼ r=P HEL and the equivalent stress is By sorting out the concrete dynamic triaxial data in the literature (Yan et al. 2007;Li et al. 2011;Yi et al. 2020), the relationship between the equivalent stress and the hydrostatic pressure under different confining pressures is calculated, and through the nonlinear fitting of Eq. (13), A = 0.6304, n = 0.8437, R 2 = 0.9909. The fitting curve is shown in Fig. 5. Banadaki and Mohanty (2012) believes that due to the lack of information on the residual strength after destruction, parameter B is considered to be 1 /3 of parameter A, and parameter m is assumed to be the same as parameter n, then B = 0.2101 and m = 0.8437 can be taken.

Rate Parameter
The strain rate parameter C is equal to the parameter C in the HJC model, and the parameter C needs to be determined based on the SHPB test. Literature(Genmao et al. 2016) makes a linear fitting of the equivalent stress after eliminating the influence of hydrostatic pressure at different natural logarithmic strain rates of various kinds of concrete, and obtain C = 0.006.

Damage Parameters
The initial values of the damage parameters D 1 and D 2 are taken as D 1 = 0.04 and D 2 = 1.0 provided in the literature (Johnson and Holmquist 1994). The transition from intact strength to complete failure strength is described by the damage factor D. The plastic strain of brittle materials with complete failure is very small and impossible to measure directly, leading to the failure of damage parameters D 1 and D 2 . On the contrary, reliable values of D 1 and D 2 can be obtained by numerical simulation trial adjustment to achieve acceptable damage effect. In this paper, by simulating the compressive strength test of standard concrete test blocks to obtain the parameters. The concrete test block of 150 mm 9 150 mm 9 150 mm is established. D 1 is 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, and D 2 is 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. Combining the above values to simulate the compression test of 100 sets of concrete standard test blocks, the optimal damage effect data are obtained: D 1 = 0.6,   Figs. 6 and 7, and the failure cracks are ''X'' shape.
In summary, the data simulated in this paper are shown in Table 3:

Numerical Analysis of Cut Piles
In order to investigate the application of the JH-2 model in concrete, a numerical analysis case of cutting pile foundation by the shield machine cutter head is established according to the actual engineering. The pile foundation concrete is C30 concrete with a diameter of 1.5 m and a length of 18 m. The cutter head for cutting pile foundation is composite cutter head. The cutter for shield machine cutter head includes common cutter, disc cutter, advance cutter, profiling cutter, etc. And their excavation methods are cut, scratch and crush. The working method of the cutter is mainly by rotation of the cutter head, the blade exerts pressure on the soil, when the soil reaches the ultimate failure stress, shear deformation occurs, and the chips are discharged. The disc cutter is suitable for rock, and its working method is that the blade directly faces the rock. When the blade cuts into the rock, a compression core is demonstrated in the crushing zone. The compaction core further forms are radioactive cracks on the rock and gradually form rock fragments. Therefore, the disc cutter is also suitable for crushing brittle materials such as concrete, but the disc cutter is not suitable for cutting steel bars in reinforced concrete pile. When the disc cutter works in soft soil, the disc cutter will be collapsed due to the jam blocked. The advance cutter is to disturb the soil before cutting, which is more conducive to the work of the cutting tool. The profiling cutter is to create the required space for over excavation.   Appropriate simplifications were made in the numerical simulation modeling due to the limitation of computer performance. Therefore, the disc cutters, advance cutters and profiling cutters are all removed, and corbel of the cutter head is also removed. Keep the common cutter and center cutter of the cutter head, and set it as a rigid body for calculation. The diameter of the cutter head is 6200 mm. The width of common cutter is 180 mm, which is 150 mm higher than the cutter head. And the front and rear angles are 10°. The length of center cutter is 830 mm and the width of center cutter is 100 mm, which is 330 mm higher than the cutter head. The models of common cutter, center cutter and overall cutter head are shown in Figs 8, 9 and 10. The concrete of the pile foundation is C30, and the relevant parameters of the JH-2 model are distributed. The parameters are shown in Tables 2 and 3. The cutter material is tungsten carbide cemented carbide, the elastic modulus is 652 GPa, the Poisson's ratio is 0.22, and the density is 15700 kg/ m 3 .
The assembly is based on the actual project, and the position of the shield machine is located in the center of the pile. The mesh of the pile foundation is set to a hexahedral mesh, and the middle part of the cut pile is closed to ensure the accuracy of the calculation. Due to the complex structure of the cutter head, the cutter head mesh can only be set to a tetrahedron. Assembly and meshing are shown in Fig. 11. Set the non-cutting parts of the upper and lower sides of the pile body to be encastre. The ALE adaptive mesh is used for the pile, and the damage failure criterion of the element deletion function is applied to delete the damaged element.

Calculation Results and Analysis
The calculation results are as shown in Fig. 12. The damage nephogram of the pile in the process of cutting is shown in Fig. 12a. When the damage reaches to 1, the element reaches the maximum load bearing capacity and will be deleted. The Mises stress state during pile cutting is shown in Fig. 12b. The ''red circle'' in Figs. 12c is the large strain of the pile when the pile is cut, which explains the cracking trend of the component. The crack development process during pile cutting is shown in the ''red circles'' in Fig. 12c and d. When cutting to 1100 mm, cracks occurred on the upper and lower sides of the pile and same to the center of the pile. The cracks on the upper and lower sides are generated during the propulsion of the cutter head and are mainly affected by the thrust of the cutter head. The middle crack is caused by the rotation of the center cutter, which is mainly caused by the rotation of the cutter head when the cutter head is thrust. As shown in Fig. 12e, the crack penetrated the pile body when it was cut to 1270 mm, and the overall crack occurred. The final fracture result is shown in Fig. 12e. The fragments are divided into two large fragments under the action of the rotation of the cutter head, and the moving direction of the fragments is consistent with the linear velocity direction of the rotation direction of the cutter head. Figure 12f shows the density state of the pile when the pile is cut. This method can more intuitively show the tension, compression, and shear states of the pile. In this paper, the concrete density of the numerical simulation is set to Fig.11 Assembly and mesh 2400 kg/m 3 . As is shown in Fig. 12f, the concrete in contact zone with the cutter head is less dense in the last fragment, while the other side that is not in contact is larger. This is mainly due to the cutter head damage to the concrete pile for shear failure. Under the action of shear, the concrete density of this part is lower than the extruded part outside. It also proved that the compressive strength of concrete is stronger than the shear strength. This is mainly because the damage of concrete pile by the cutter head is shear damage.
Under the action of shear, the density of this part of the concrete is lower than that of the outer squeezed part. It is demonstrated that the compressive capacity of concrete is stronger than the shear capacity. Figure 13 is the broken line diagram of the cutter head torque when cutting pile. When cutting starts, the center cutter of the cutter head contacts the surface of the pile foundation at first. Because the center cutter is partly higher than the common cutter, the initial cutting torque is lower. When the cutter head continues to thrust and the common cutters cut the pile, the cutter head torque has a significant increase. Due to the advance cutter is removed in the process of modeling. Some of the common cutters are no longer     Figure 13 shows a state where the torque vibrates violently. When the crack starts to develop, the torque peak of the cutter head is lower than the previous part, and the amplitude is reduced. When the cutter head thrusts to the penetration crack of the pile, the torque of the cutter head decreases again and gradually tends to a relatively stable state. The broken concrete pieces are shown in Fig. 12e.

Construction Analysis
The shield machine used for pile cutting is Herrenknecht S-716, and the cutter head of the shield machine is a composite cutter head, as shown in Fig. 14. The rated torque of the main drive of the shield machine is 5835 kNÁm, and the relief torque is 6807 kNÁm. Figure 15 shows the working parameters of Herrenknecht S-716. Drilling core construction is shown in Fig. 16, and the core sample is shown in Fig. 17.
The tunneling torque of shield machine includes cutting torque, rotary resistance torque of the cutter head, counterforce torque generated by thrust load on cutter head, friction torque generated by the sealing device, friction torque on the front face of the cutter head, friction torque behind the cutter head, shear torque of the cutter head opening, and stirring torque in the soil silo (Guan and Gao 2008). Because of muck conditioning technology, the torque of the cutter head can be greatly reduced in soft soil excavation. The total torque obtained is an approximate theory without considering the muck conditioning technology. However, considering the muck conditioning technology and the large amount of foam in the actual pile cutting construction, most of the shield machine torque is provided by the cutter head cutting pile. The cutter head torque of the shield machine collected on the site is shown in Fig. 18.
During pile cutting, the maximum torque is 3.7 MNÁm, which shows that the cutter head is stuck. Due to incomplete cleaning of the steel bars, the steel bars could not be cut by the cutter, but the cutter head penetration is large, and the steel bar is wound around the cutter head. If the cutter head continues to rotate without stopping, it may cause excessive torque and shut down the shield machine. The site is successfully freed by reversing the cutter head. And the discharged steel bar is shown in Fig. 19. Table 4 shows the mathematical statistics of the simulation cutting torque and actual construction cutting torque.
Depending on the data comparison in Table 4, the numerical simulation torque results are relatively small compared to the construction torque, but the standard deviation of the simulation results is large. In other words, the cutting torque is scattered and unstable. The main reason is that the simulated cutter head only retains the common cutter and the center cutter, and only simulates the working conditions of the cutter head for pile cutting, ignoring the interaction between the cutter head and the soil, and other torque. However, instability of cutting can easily cause damage to the cutter head. In the actual construction,  .19 Residual rebar the shield machine should reduce the rotation speed and the propulsion speed to further decrease the damage of the shield machine cutter.

Conclusion
Based on the collection and summary of concrete impact damage related tests in previous literature. The parameters of JH-2 model of C30 concrete are calculated in this paper. The finite element software ABAQUS is utilized to simulate the failure of the three-dimensional standard concrete test blocks and simulate the pile cutting test by shield machine cutter head.
The conclusions are as follows: (1) JH-2 constitutive model can be used to simulate the concrete cutting.
(2) The better damage parameters have been represented and verified against the results of simulations, which are D 1 = 0.6 and D 2 = 0.1 in the JH-2 model of C30 concrete. (3) In the simulation example of concrete compression and cut pile, the failure form of concrete is brittle failure, and the non-linear characteristics of concrete have been observed. (4) The simulated torque vibration of the cutter head is relatively larger than actual pile cutting, and the actual shield machine cutting pile construction should have a low rotation speed and low propulsion speed.
Author contribution Ping Xu: Investigation-editing and supervision. Sixian Zuo: Writing, editing -data analysis and original drafts.
Data availability The data supporting the results of this study can be obtained from the corresponding author upon reasonable request.

Declarations
Conflict of interest The authors do not have any possible conflicts of interest.
Yi L, Hua C, Liangliang Z (2020) Energy analysis of C60 concrete under triaxial compression under different confining pressures. Chinese J Appl Mech 37:2086-2093 Zhu X, Jia YJ (2014) 3D mechanical modeling of soil orthogonal cutting under a single reamer cutter based on Drucker-Prager criterion. Tunn Undergr Space Technol 41:255-262 Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.