CN104653193A - Energy theory-based prediction method for stress of TMB (tunnel boring machine) disk hob - Google Patents

Abstract

The invention discloses an energy theory-based prediction method for stress of a TMB (tunnel boring machine) disk hob. The prediction method comprises the following steps: (1) determining the normal pushing vertical force FV, the tangential rolling force FR and the lateral force FS of the hob in a rock breaking process; (2) establishing a hob stress model in the rock breaking process; (3) establishing a rock strain model during the rock breaking process of the hob; (4) establishing an energy-balance equation for the hob and rock; (5) according to the established rock strain model, by setting the principal stress direction of the rock and the principal stress direction of the hob under a cylindrical coordinate to be corresponding, obtaining surface forces on a wedge-shaped surface when the hob breaks the rock, and predicting the stress of the TMB disk hob. Aiming at analysis and research status of all influence factors in the mechanical property of the existing TMB disk hob, the invention discloses the microcosmic energy theory-based novel prediction method for the stress of the hob; through the influence of the added rotation speed of the hob on the stress characteristics of the hob to in previous studies, a basis is provided for the design theory of a cutter with TBM boreability.

Description

TBM disk cutter based on energy principle is subject to force prediction method

Technical field

The present invention relates to the stressed Forecasting Methodology of TBM disk cutter, particularly relate to the theoretical prediction model that the hobboing cutter mechanical characteristic based on energy is analyzed.

Background technology

Tunneling boring rock tunnel development machine (TBM) be have integrate mechanical, electrical, gas, liquid, speed of application be fast, pollute the high automation tunnelling machinery of the various features such as low, safeguard work personal security, in underground engineering construction, occupy sizable ratio, now become the important symbol that three dimensional city is built.

TBM cutterhead belongs to equipment construction key component.Require that the TBM disk cutter carried out breaks rock force analysis and established the theoretical foundation of cutterhead design according to geological adaptability, many scholars have done a large amount of research work [1-7] in theoretical research and numerical simulation.Comprehensive analysis is known: in hobboing cutter rock break-off process, hobboing cutter characteristic is inseparable with rock properties; Undertaken analyzing by finite element emulation software in broken rock research process and become effective research means.Finite element software is based upon the mass computing simulation software that elastic plastic theory basis grows up, and the development of its model has certain hysteresis quality.Therefore, some researchers slowly start from elastic plastic theory basis from the new coupling relation analyzed between hobboing cutter and rock, Michael Alber [8] is based on stress analysis, adopt abrasion resistance index descriptive study geostatic stress and confined pressure on the impact of rock abrasivity, point out the dependence of CAI index to pressure, and under the stress state of different underground chambers, the wearing and tearing of development machine cutterhead are assessed.SeisukeOkubo [9] proposes and adopts based on elastic plastic theory, power-driving the linearity curve of Binding experiment data replaces conventional linear elastic constitutive model model, draw out engineering time and the load-time graph of development machine, the loading conditions in TBM tunneling process can be analyzed qualitatively, and be applied to the Finite Element (FEM) or other suitable computational methods that more real and complicated simulation combines.

In sum: the research based on elastic plastic theory basis is scarcely out of swaddling-clothes, the stress-strain analysis being entirety with excavation face cavern, progressively to launch, is also more and more subject to the attention of scholars simultaneously.This invention, based on the elastic plastic theory basis of finite element theory, sets up the stress model in hobboing cutter rock break-off process; According to the energy principle in elastic theory, theory analysis is carried out to the load change rule in hobboing cutter rock break-off process, introduce hobboing cutter driving speed parameter simultaneously, analyze and hobboing cutter mechanical characteristic is affected, for research cutterhead driving property is laid a good foundation, for the theoretical research of breaking rock from the hobboing cutter of microcosmic point provides certain reference frame.

Bibliography:

[1]CHIAIA B.Fracture mechanisms induced in a brittle material by a hard cuttingindenter[J].International Journal of Solids and Structures,2001,38:7747－7768。

[2]CARPINTERI A,CHIAIA B,INVERNIZZI S.Numerical analysis of indentationfracture in quasi-brittle materials[J].Engineering Fracture Mechanics,2004,71:567－577。

[3]GONG Q M,ZHAO J,JIAO Y Y.Numerical modeling of the effects of jointorientation on rock fragmentation by TBM cutters[J].Tunnelling and UndergroundSpace Technology,2005,20(2):183－191。

[4]GONG Q M,JIAO Y Y,ZHAO J.Numerical modeling of the effects of joint spacingon rock fragmentation by TBM cutters[J].Tunnelling and Underground SpaceTechnology,2006,21:46－55。

[5] Liao Zhiyi, Liang Zhengzhao, Yang Yuefeng, Wang Yan, Tang Chunan. the numerical simulation [J] of jointed rock mass destructive process under cutter dynamic action. geotechnical engineering journal, 2013, 6 (35): 1147-1155 (LIAO Zhi-yi, LIANG Zheng-zhao, YANGYue-feng, WANG Yan, TANG Chun-an.Numerical simulation of fragmentation processof jointed rock mass induced by a drill bit under dynamic loading [J] .Chinese Journalof Geotechnical Engineering, 2013, 6 (35): 1147-1155).

[6]Jung-Woo Cho,Seokwon Jeon,Sang-Hwa Yu,Soo-Ho Chang.Optimum spacing ofTBM disc cutters:A numerical simulation using the three-dimensional dynamicfracturing method[J].Tunnelling and Underground Space Technology,2010,25:230–244。

[7] Mo Zhenze, Li Haibo, Zhou Qingchun, He Enguang, Zou Fei, Zhu little Ming, Zhao Yu. the tunnel machine hobboing cutter based on UDEC breaks rock numerical simulation study [J]. rock mechanics, 2012, 4 (33): 1196-1209 (MO Zhen-ze, LI Hai-bo, ZHOUQing-chun, HE En-guang, ZOU Fei, ZHU Xiao-ming, ZHAO Yu.Research on numericalsimulation of rock breaking using TBM disc cutters based on UDEC method [J] .Rockand Soil Mechanics, 2012, 4 (33): 1196-1209).

[8]Michael Alber.Stress dependency of the Cerchar abrasivity index(CAI)andits effects on wear of selected rock cutting tools[J].Tunnell ing and UndergroundSpace Technology,23,(2008):351–359。

[9]Seisuke Okubo,Katsunori Fukui.Applicability of the variable-compliance-type constitutive equation to rock breakage by excavation machinery[J].Tunnellingand Underground Space Technology,26,(2011):29–37。

Summary of the invention

For in TBM hob rock break-off process, change unification mutually with the ess-strain of rock research field, by hobboing cutter macroforce characteristic, be converted to the model of microstress.On this basis, according to conservation of energy condition, hobboing cutter stress model is solved, finally obtain the stressed new Forecasting Methodology of hobboing cutter in rock break-off process under effect of stress.By with other two classical Forecasting Methodology comparative analyses, this by force prediction method Changing Pattern and energy variation rule and other models all consistent.According to the analysis of model self, add the analysis that radius and energy response stressed to hobboing cutter installed by hobboing cutter rotating speed and hobboing cutter in a model, the conclusion of " along with the increase of cutterhead rotating speed, 3 also increase to power thereupon " that obtain with utilizing finite element emulation software also matches.

A kind of TBM disk cutter based on energy principle of the present invention, by force prediction method, comprises the following steps:

Step 1, determine hobboing cutter in rock break-off process normal direction pushing vertical force F _v, tangential rolling force F _r, lateral force F _s;

Step 2, the hobboing cutter stress model set up in rock break-off process, comprising:

2-1, set up hobboing cutter and invade rock maximum cross-section upper stress equilibrium equation: in hobboing cutter driving rock break-off process, the movement locus of each hobboing cutter revolves round the sun around whole cutter head center around the rotation of hobboing cutter disc centre while, and the effect of hobboing cutter to rock comprises normal direction pushing vertical force F _v; , tangential rolling force F _r, lateral force F _s, under cylindrical coordinate, the power acted on hobboing cutter 3 directions is P _r, P _θ, P _x; Hobboing cutter is in rock break-off process, and entirety is disc, and tool point angle leads to for wedge shape sword, and the basil is 2 α, if certain on hobboing cutter wedge shape sword is some A point, the circular cylindrical coordinate at A point place is (r ₀, θ, x), if the unit plane power acting on 3 directions on the sword face of wedge shape sword is hobboing cutter invades rock maximum cross-section upper stress equilibrium equation:

$\left[\begin{array}{ccc}{\mathrm{\σ}}_r& {\mathrm{\τ}}_{\mathrm{\θr}}& {\mathrm{\τ}}_{\mathrm{xr}}\\ {\mathrm{\τ}}_{\mathrm{r\θ}}& {\mathrm{\σ}}_{\mathrm{\θ}}& {\mathrm{\τ}}_{\mathrm{x\θ}}\\ {\mathrm{\τ}}_{\mathrm{rx}}& {\mathrm{\τ}}_{\mathrm{\θx}}& {\mathrm{\σ}}_x\end{array}\right]\·\left[\begin{array}{c}\cos \mathrm{\α}\\ 0\\ \sin \mathrm{\α}\end{array}\right]=\left[\begin{array}{c}{p}_r\\ p_{\mathrm{\θ}}\\ p_x\end{array}\right]---\left(1\right)$

In formula (1): σ _r, σ _θ, σ _xthe main stress bar that-hobboing cutter is corresponding under cylindrical coordinate, Mpa;

τ _{θ r}, τ _xr, τ _{r θ}, τ _{x θ}, τ _rx, τ _{θ x}the shear stress of-all directions, Mpa;

2-2, set up the hobboing cutter stress deformation equation of comptability: choose the cylindrical-coordinate system centered by hobboing cutter initial point, when hobboing cutter breaks rock, hobboing cutter produces inertia force around disc-shaped center roller rotation and is by the balance differential equation formula that stress represents be then:

$\frac{\∂{\mathrm{\σ}}_r}{\∂r}+\frac{1}{r}\frac{\∂{\mathrm{\τ}}_{\mathrm{r\θ}}}{\∂\mathrm{\θ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂x}+\frac{{\mathrm{\σ}}_r-{\mathrm{\σ}}_{\mathrm{\θ}}}{r}+\mathrm{\ρ}{\mathrm{\ω}}_r^{2}r=0---\left(2\right)$

In formula (2): r-hobboing cutter radius, m; ρ-hobboing cutter density, Kg/m ³; ω _r-hobboing cutter rotating speed, rad/s;

If: hobboing cutter is arranged on cutterhead and cutterhead is whole relation, and the installation radius of hobboing cutter is R, A point is ((R+x) cos φ, (R+x) sin φ, Z) relative to the position of cutterhead, and the stress transmission relation in Two coordinate system is as follows:

σ _(R+x)＝σ _x,σ _φ＝σ _θ,τ _(R+x)φ＝τ _xθ,τ _(R+x)Z＝τ _rx

With the equilibrium equation that the hobboing cutter that stress represents revolves round the sun on cutterhead be then:

$\frac{\∂{\mathrm{\σ}}_x}{\∂(R+x)}+\frac{1}{(R+x)}\frac{{\∂\mathrm{\τ}}_{\mathrm{x\θ}}}{\∂\mathrm{\φ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂Z}+\frac{{\mathrm{\σ}}_x-{\mathrm{\σ}}_{\mathrm{\θ}}}{(R+x)}+\mathrm{\ρ}{\mathrm{\ω}}_{(R+x)}^2(R+x)=0---\left(3\right)$

In formula (3): the installation radius of hobboing cutter on R-cutterhead, m; ω _(R+x)-disk cutter revolution angular velocity, rad/s, ω _rr=ω _(R+x)(R+x);

If hobboing cutter material is perfect elastic body, ignore the load change that shear stress shearing strain produces, in conjunction with physical equation and geometric equation, obtaining the Coordinate deformation equation that stress method represents is:

$\frac{{\∂}^2{\mathrm{\σ}}_r}{{\∂r}^2}+\frac{3}{r}\frac{\∂{\mathrm{\σ}}_r}{\∂r}+(3+v)\mathrm{\ρ}{\mathrm{\ω}}_r^2=0---\left(4\right)$

In formula (4): ν-hobboing cutter material poisson's ratio;

2-3, according to formula (1) to formula (4), show that disk cutter breaks the rock stress components and is respectively:

${\mathrm{\σ}}_r=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2(r_0^2-r^2)---\left(5\right)$

${\mathrm{\σ}}_{\mathrm{\θ}}=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2-\left(\frac{1+3v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}^2---\left(6\right)$

${\mathrm{\σ}}_x=\frac{p_x}{\cos a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2(R_0^2-R^2)---\left(7\right)$

In formula (5), formula (6) and formula (7): r ₀-A point to the radius length of hobboing cutter center, m; R ₀-A point to the radius length of cutter head center position, m;

Step 3, set up rock strain model in hobboing cutter rock break-off process: A point is under Lagrange remainder, and the coordinate on hobboing cutter is:

$\left\{\begin{array}{c}{x}^{\′}=x\\ y^{\′}=r_0\sin \mathrm{\θ}\\ z^{\′}=r_0\cos \mathrm{\θ}\end{array}\right.---\left(8\right)$

The degree of depth that hobboing cutter is pushed down into rock in normal direction vertical force is u _r, suppose A point and rock planes overlapping, A point causes the displacement of rock respective change to be:

$\left\{\begin{array}{c}x=0\\ y=r_0\sin \mathrm{\θ}\\ z=u_r-(r-r_0\cos \mathrm{\θ})\end{array}\right.---\left(9\right)$

By r ₀use y respectively with θ, z is expressed as functional relation:

$\left\{\begin{array}{c}{r}_0=\sqrt{y^2+z^2}\\ \mathrm{\θ}=\arctan \frac{y}{z}\end{array}\right.---\left(10\right)$

According to the incremental model of the theory of plasticity, in hobboing cutter rock break-off process, the rock strain corresponding with A point is expressed as:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}\frac{\∂X}{\∂x}& \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{1}{2}(\frac{\∂X}{\∂z}+\frac{\∂Z}{\∂x})\\ \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{\∂Y}{\∂y}& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})\\ \frac{1}{2}(\frac{\∂X}{\∂x}z+\frac{\∂Z}{\∂x})& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})& \frac{\∂Z}{\∂z}\end{array}\right]---\left(11\right)$

In formula (11): consider u _rfor constant term, if shear stress and shearing strain are 0, on relative hobboing cutter wedge shape sword, the rock strain of A point is:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]---\left(12\right)$

In formula (12), ε _xx, ε _xy, ε _xz, ε _yx, ε _yy, ε _yz, ε _zx, ε _zy, ε _zz-rock relative strain;

Step 4, set up the energy-balance equation of hobboing cutter and rock: from the first law of thermodynamics and second law, TBM is in tunnel excavating process, and the energy transferring that hobboing cutter breaks rock comprises the input of energy, the gathering and dissipating, output 4 processes of energy of energy; When the external work that hobboing cutter provides is all for broken rock, and set rock system as an entirety, ignore the accumulation process of energy and the influence factor of temperature, the energy-balance equation setting up hobboing cutter and rock according to possibility merit principle is as follows:

${\∫}_S{p}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dS}+{\∫}_V{f}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dV}={\∫}_V{\mathrm{\σ}}_{\mathrm{ij}}^{\left(1\right)}{\mathrm{\ϵ}}_{\mathrm{ij}}^{\left(2\right)}\mathrm{dV},(i,j=\mathrm{1,2.3})---\left(13\right)$

The energy-balance equation that step 5 is set up according to step 4, meanwhile, if the principal direction of stress of rock is corresponding with the principal direction of stress under the cylindrical coordinates of hobboing cutter, face power when hobboing cutter breaks rock on lozenges is respectively:

$P_r=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2\}d{S}_r---\left(14\right)$

$P_{\mathrm{\θ}}=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2\}d{S}_{\mathrm{\θ}}---\left(15\right)$

$P_x=\∫\{\frac{{\mathrm{\σ}}_3}{2\cos a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{R}_0^2\}d{S}_x---\left(16\right)$

${\mathrm{dS}}_r=2u_r\sqrt{r^2-{(r-u_r)}^2}\tan a---\left(17\right)$

${\mathrm{dS}}_{\mathrm{\θ}}=h\sqrt{r^2-{(r-u_r)}^2}---\left(18\right)$

${\mathrm{dS}}_x=u_r^2\tan a---\left(19\right)$

Formula (14) is in formula (19): P _r, P _θ, P _xbe respectively the normal direction pushing vertical force F of hobboing cutter _v, tangential rolling force F _rwith lateral force F _s, KN; S _r, S _θ, S _xbe respectively the projected area in hobboing cutter normal direction pushing vertical force, tangentially rolling force and lateral force direction, m ²; σ ₁, σ ₃be respectively the main stress bar corresponding to rock element body, wherein, σ ₁=σ ₃, Mpa; u _rfor the displacement of A point incision rock, work as r ₀when position overlaps with position, Rock Cutting face, r ₀=r-u _r, m; R ₀for the position of A point on cutterhead, R ₀=R+u _rtan a, m.

Compared with being subject to force prediction method with existing many hobboing cutters, the advantage of this method is:

The present invention is by the derivation of model in force prediction method, for this theoretical present situation of microcosmic energy viewpoint shortcoming network analysis, in hobboing cutter rock break-off process, change unification mutually with the ess-strain of rock research field, by hobboing cutter macroforce characteristic, be converted to the model of microstress.On this basis, according to conservation of energy condition, hobboing cutter stress model is solved, finally obtain the stressed new Forecasting Methodology of hobboing cutter in rock break-off process under effect of stress, for hobboing cutter proposes new research direction by force prediction method, meanwhile, the method more in the past Forecasting Methodology is compared, add boring parameter-driving speed, can be and propose reference frame based on the cutterhead design theoretical foundation of performance evaluation driving property from now on.

Accompanying drawing explanation

Fig. 1 is the force analysis figure of disk cutter;

Fig. 2 is that disk cutter breaks rock analysis chart and knife edge stress envelope;

Fig. 3 is the stress envelope of disk cutter loading direction and blade cell cube;

Fig. 4 is the installation site figure of hobboing cutter on cutterhead;

Fig. 5 is hobboing cutter rock cutting system capacity transmittance process analysis chart;

Fig. 6 is normal direction pushing vertical force model contrast in hobboing cutter rock break-off process;

Fig. 7 is normal direction pushing vertical force model contrast in hobboing cutter rock break-off process;

Fig. 8 is the energy variation rule of tangential rolling force model in hobboing cutter rock break-off process;

Fig. 9 is the Changing Pattern of hobboing cutter energy with angular velocity;

Figure 10 is the energy variation rule of lateral force model in hobboing cutter rock break-off process;

Figure 11 is that lateral force energy is with the Changing Pattern installing radius and cutting speed.

Detailed description of the invention

Below in conjunction with accompanying drawing and research, the present invention is described in further detail.

The present invention is based on the TBM disk cutter of energy principle by force prediction method, mainly by with other two classical forecast model comparative analyses, according to the analysis of model self, add the analysis that radius and energy response stressed to hobboing cutter installed by hobboing cutter rotating speed and hobboing cutter in a model, the conclusion of " along with the increase of cutterhead rotating speed, 3 also increase to power thereupon " that obtain with utilizing finite element emulation software also matches.And utilizing the experimental data and model calculating data that record in C.Balci document, compared with the calculating data proposing model with this method, as seen along with the raising of rock strength and the increase of cutter pile penetration, the result of calculation of this method is closer to experimental data.

This TBM disk cutter, by force prediction method, comprises the following steps:

Step 1, determine hobboing cutter in rock break-off process normal direction pushing vertical force F _v, tangential rolling force F _r, lateral force F _s;

Not only tunnel in rock break-off process at TBM, the movement locus of each hobboing cutter revolves round the sun around the rotation of hobboing cutter disc centre but also around whole cutter head center, and therefore, the destruction of disk cutter to rock comprises normal direction pushing vertical force F _v; Tangential rolling force F _r; Lateral force F _s, the stress model of hobboing cutter as shown in Figure 1.

Step 2, the hobboing cutter stress model set up in rock break-off process, comprising:

2-1, set up hobboing cutter and invade rock maximum cross-section upper stress equilibrium equation:

Set up hobboing cutter and invade rock maximum cross-section upper stress equilibrium equation: in elastic plastic theory, the power acted on hobboing cutter 3 directions belongs to external force, under cylindrical coordinate, is respectively P _r, P _θ, P _x.As shown in Figure 2: TBM hob is in rock break-off process, and entirety is disc, tool point angle is generally wedge shape sword (basil is 2 α), postulated point A be on disk cutter wedge shape sword certain a bit, then the circular cylindrical coordinate putting A place can be expressed as (r ₀, θ, x), if act on the unit plane power in 3 directions on the sword face of wedge shape sword set up hobboing cutter and invade rock maximum cross-section upper stress equilibrium equation:

$\left[\begin{array}{ccc}{\mathrm{\σ}}_r& {\mathrm{\τ}}_{\mathrm{\θr}}& {\mathrm{\τ}}_{\mathrm{xr}}\\ {\mathrm{\τ}}_{\mathrm{r\θ}}& {\mathrm{\σ}}_{\mathrm{\θ}}& {\mathrm{\τ}}_{\mathrm{x\θ}}\\ {\mathrm{\τ}}_{\mathrm{rx}}& {\mathrm{\τ}}_{\mathrm{\θx}}& {\mathrm{\σ}}_x\end{array}\right]\·\left[\begin{array}{c}\cos \mathrm{\α}\\ 0\\ \sin \mathrm{\α}\end{array}\right]=\left[\begin{array}{c}{p}_r\\ p_{\mathrm{\θ}}\\ p_x\end{array}\right]---\left(1\right)$

In formula (1), σ _r, σ _θ, σ _xthe main stress bar that-hobboing cutter is corresponding under cylindrical coordinate, Mpa;

τ _{θ r}, τ _xr, τ _{r θ}, τ _{x θ}, τ _rx, τ _{θ x}the shear stress of-all directions, Mpa;

2-2, set up the hobboing cutter stress deformation equation of comptability:

Analyze the effect of hobboing cutter rotation load, choose the cylindrical-coordinate system centered by disk cutter initial point, when hobboing cutter rock cutting, as shown in Figure 3, hobboing cutter produces inertia force around disc-shaped center roller rotation to stress direction figure the balance differential equation formula that foundation represents with stress:

$\frac{\∂{\mathrm{\σ}}_r}{\∂r}+\frac{1}{r}\frac{\∂{\mathrm{\τ}}_{\mathrm{r\θ}}}{\∂\mathrm{\θ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂x}+\frac{{\mathrm{\σ}}_r-{\mathrm{\σ}}_{\mathrm{\θ}}}{r}+\mathrm{\ρ}{\mathrm{\ω}}_r^{2}r=0---\left(2\right)$

In formula (2), in formula (2): r-hobboing cutter radius, m; ρ-hobboing cutter density, Kg/m ³; ω _r-hobboing cutter rotating speed, rad/s;

Disk cutter also around cutter head center revolution, does not consider hobboing cutter spinning motion, as shown in Figure 4.Suppose that hobboing cutter is arranged on cutterhead and cutterhead is whole relation, if the installation radius of hobboing cutter is R, some A is ((R+x) cos φ, (R+x) sin φ relative to the position of cutterhead, Z), the inertia force that hobboing cutter produces around cutter head center revolution is considered through hobboing cutter rotation and revolution loading analysis, the stress transmission relation in Two coordinate system is as follows:

σ _(R+x)＝σ _x,σ _φ＝σ _θ,τ _(R+x)φ＝τ _xθ,τ _(R+x)Z＝τ _rx

With the equilibrium equation that the hobboing cutter that stress represents revolves round the sun on cutterhead be then:

$\frac{\∂{\mathrm{\σ}}_x}{\∂(R+x)}+\frac{1}{(R+x)}\frac{{\∂\mathrm{\τ}}_{\mathrm{x\θ}}}{\∂\mathrm{\φ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂Z}+\frac{{\mathrm{\σ}}_x-{\mathrm{\σ}}_{\mathrm{\θ}}}{(R+x)}+\mathrm{\ρ}{\mathrm{\ω}}_{(R+x)}^2(R+x)=0---\left(3\right)$

In formula (3), the installation radius of hobboing cutter on R-cutterhead, m; ω _(R+x)-disk cutter revolution angular velocity, rad/s, ω _rr=ω _(R+x)(R+x);

Due to self symmetry of disk cutter and cutterhead installation site, suppose that hobboing cutter material is perfect elastic body, generation of not wearing and tearing.For convenience of calculating, ignoring the load change that shear stress shearing strain produces, only to consider above main stress bar distortion upwards and displacement, in conjunction with physical equation and geometric equation, trying to achieve through abbreviation the Coordinate deformation equation that stress method represents is:

$\frac{{\∂}^2{\mathrm{\σ}}_r}{{\∂r}^2}+\frac{3}{r}\frac{\∂{\mathrm{\σ}}_r}{\∂r}+(3+v)\mathrm{\ρ}{\mathrm{\ω}}_r^2=0---\left(4\right)$

In formula (4): ν-hobboing cutter material poisson's ratio.

2-3, according to formula (1) to formula (4), show that disk cutter breaks the rock stress components and is respectively:

${\mathrm{\σ}}_r=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2(r_0^2-r^2)---\left(5\right)$

${\mathrm{\σ}}_{\mathrm{\θ}}=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2-\left(\frac{1+3v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}^2---\left(6\right)$

${\mathrm{\σ}}_x=\frac{p_x}{\cos a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2(R_0^2-R^2)---\left(7\right)$

In formula (5), formula (6) and formula (7), r ₀-A point to the radius length of hobboing cutter center, m; R ₀-A point to the radius length of cutter head center position, m;

Step 3, set up rock strain model in hobboing cutter rock break-off process:

A point is under Lagrange remainder, and the coordinate on hobboing cutter is:

$\left\{\begin{array}{c}{x}^{\′}=x\\ y^{\′}=r_0\sin \mathrm{\θ}\\ z^{\′}=r_0\cos \mathrm{\θ}\end{array}\right.---\left(8\right)$

Disk cutter, under the effect of normal direction vertical force, is pressed into the degree of depth u that rock is certain _rin hobboing cutter rock cutting process, rock crackle forming generally occurs in immediately below hobboing cutter, due to the effect of rolling force, the displacement produced in the Y direction is with to connect thrum relevant, when ignoring hobboing cutter and installing the affecting of radius, do not produce the change of displacement in X-direction, therefore on hobboing cutter wedge shape sword, A point causes the displacement of rock respective change to be:

$\left\{\begin{array}{c}x=0\\ y=r_0\sin \mathrm{\θ}\\ z=u_r-(r-r_0\cos \mathrm{\θ})\end{array}\right.---\left(9\right)$

By r ₀use y respectively with θ, z is expressed as functional relation:

$\left\{\begin{array}{c}{r}_0=\sqrt{y^2+z^2}\\ \mathrm{\θ}=\arctan \frac{y}{z}\end{array}\right.---\left(10\right)$

According to the incremental model of the theory of plasticity, in hobboing cutter rock break-off process, the rock strain corresponding with A point is expressed as:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}\frac{\∂X}{\∂x}& \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{1}{2}(\frac{\∂X}{\∂z}+\frac{\∂Z}{\∂x})\\ \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{\∂Y}{\∂y}& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})\\ \frac{1}{2}(\frac{\∂X}{\∂x}z+\frac{\∂Z}{\∂x})& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})& \frac{\∂Z}{\∂z}\end{array}\right]---\left(11\right)$

In formula (11): ${\mathrm{\ϵ}}_{\mathrm{xx}}=\frac{\∂X}{\∂x}=\frac{\∂X}{\∂r_0}\frac{\∂r_0}{\∂x}+\frac{\∂X}{\∂\mathrm{\θ}}\frac{\∂\mathrm{\θ}}{\∂x};$

Consider u _rfor constant term, be convenience of calculation, ignore shear stress and shearing strain and break on hobboing cutter the impact that rock produces, its value is 0, and only analyze the load change under main stress bar strained condition, the rock strain obtaining A point on relative hobboing cutter wedge shape sword is simultaneously:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]---\left(12\right)$

In formula (12), ε _xx, ε _xy, ε _xz, ε _yx, ε _yy, ε _yz, ε _zx, ε _zy, ε _zz-rock relative strain.

Step 4, set up the energy-balance equation of hobboing cutter and rock:

From the first law of thermodynamics and second law, TBM is in tunnel excavating process, and the energy transferring of hobboing cutter rock cutting is a dynamic process, comprises the input of energy, the gathering and dissipating, output 4 processes of energy of energy, as shown in Figure 5.

When the cutting energy (external work) that hobboing cutter provides, time all for rock crushing, stock-removing efficiency now is also the highest.Suppose that rock system is an entirety, ignore the process of rock mass energy accumulation, because hobboing cutter cuts in rock mass process, speed is slower, the influence factor of temperature can be ignored, based on conservation of energy principle, hobboing cutter breaks rock external work will all be used for the broken Dissipated energy of rock, according to merit principle setting up energy-balance equation:

${\∫}_S{p}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dS}+{\∫}_V{f}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dV}={\∫}_V{\mathrm{\σ}}_{\mathrm{ij}}^{\left(1\right)}{\mathrm{\ϵ}}_{\mathrm{ij}}^{\left(2\right)}\mathrm{dV},(i,j=\mathrm{1,2.3})---\left(13\right)$

The energy-balance equation that step 5 is set up according to step 4, meanwhile, if the principal direction of stress of rock is corresponding with the principal direction of stress under the cylindrical coordinates of hobboing cutter, face power when hobboing cutter breaks rock on lozenges is respectively:

$P_r=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2\}d{S}_r---\left(14\right)$

$P_{\mathrm{\θ}}=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{r}_0^2\}d{S}_{\mathrm{\θ}}---\left(15\right)$

$P_x=\∫\{\frac{{\mathrm{\σ}}_3}{2\cos a}+\left(\frac{3+v}{8}\right)\mathrm{\ρ}{\mathrm{\ω}}_r^2{R}_0^2\}d{S}_x---\left(16\right)$

${\mathrm{dS}}_r=2u_r\sqrt{r^2-{(r-u_r)}^2}\tan a---\left(17\right)$

${\mathrm{dS}}_{\mathrm{\θ}}=h\sqrt{r^2-{(r-u_r)}^2}---\left(18\right)$

${\mathrm{dS}}_x=u_r^2\tan a---\left(19\right)$

Formula (14) is in formula (19): P _r, P _θ, P _xbe respectively the normal direction pushing vertical force F of hobboing cutter _v, tangential rolling force F _rwith lateral force F _s, KN; S _r, S _θ, S _xbe respectively the projected area in hobboing cutter normal direction pushing vertical force, tangentially rolling force and lateral force direction, m ²; σ ₁, σ ₃be respectively the main stress bar corresponding to rock element body, wherein, σ ₁=σ ₃, Mpa; u _rfor the displacement of A point incision rock, work as r ₀when position overlaps with position, Rock Cutting face, r ₀=r-u _r, m; R ₀for the position of A point on cutterhead, R ₀=R+u _rtan a, m.

Respectively the hobboing cutter 3 that the present invention derives will be discussed to the energy variation rule of power model below, postulated point A when the extreme higher position of wedge shape sword injection amount, r ₀=r-u _r, x=0.5t+u _rtan a, the rotary speed ω of hobboing cutter _r=(ω _r× R)/r, choosing 17 cun of hobboing cutters, to carry out cutting granite be that example is analyzed, and hobboing cutter parameter is in table 1, and granite parameter is in table 2.

Table 1.17 cun hobboing cutter geometric parameter

Cutter diameter (m)	The hobboing cutter basil (°)	Hobboing cutter density (Kg/m 3)	Blade width (m)
				0.432	121°	7850	0.014

Table 2. granite property parameters

Uniaxial compressive strength (UCS) (MPa)	Brazil tensile strength (BTS) (MPa)
		209	9.2

According to hobboing cutter geometric parameter and rock geology parameter, by Evans normal direction pushing vertical force model, and CSM normal direction pushing vertical force model, and the model that this method is derived contrasts, cutting depth is from 2mm to 16mm, and the stressed Changing Pattern of stress model as shown in Figure 6.As can be seen from the figure, the mechanical characteristic of the stress model of deriving in this method between Evans model and CSM model, and change of mechanical property all thereupon pile penetration increase and increase.Now be also noted that normal direction vertical force model that the present invention derives also installs radius with cutterhead rotating speed and hobboing cutter relevant, if cutting depth is 8mm, cutterhead rotating speed is from 0 to 10rad/min, hobboing cutter installs radius from 0 to 5000mm, its change of mechanical property rule as shown in Figure 7, the change of mechanical characteristic under normal direction vertical force as can be seen from Figure 7, installs the increase of radius with hobboing cutter and increases, increasing with the increase of cutterhead rotating speed.

This method is derived tangential rolling force model and E.F.Roxborough tangential rolling force model and the tangential rolling force model of CSM and is analyzed, and cutting depth is from 2mm to 16mm, and the change of mechanical property rule of stress model as shown in Figure 8.As can be seen from the figure, the mechanical characteristic that the tangential rolling force model that this method is derived consumes when broken rock is between CSM model and E.F.Roxborough model, and models fitting result closer to CSM model, the same all with CSM model and E.F.Roxborough model of its energy variation rule, increases along with the increase of injection amount.

It can also be seen that from the tangential rolling force model derived, tangential rolling force is also relevant with the rotating speed of cutterhead, suppose that the installation radius of hobboing cutter on cutterhead is 2m, under identical cutting-in (cutting depth is 8mm) hobboing cutter energy with angular velocity (cutterhead rotating speed is from 0 to 10rad/min) Changing Pattern as shown in Figure 9.As can be seen from the figure the tangential rolling force of hobboing cutter cuts the angular velocity breaking rock mechanical characteristic and cutterhead substantial connection, and that is, the raising of cutterhead rotating speed, the cutting mechanics characteristic of the tangential rolling force of hobboing cutter increases.

The lateral force model that this method is derived and E.F.Roxborough lateral force model Akimi Tomiro lateral force model are analyzed, and cutting depth is from 2mm to 16mm, and the change of mechanical property rule of stress model as shown in Figure 10.As can be seen from the figure, hobboing cutter lateral force in rock break-off process mechanical characteristic lower than between E.F.Roxborough lateral force model AkimiTomiro lateral force model, closer to E.F.Roxborough lateral force model, the energy consumed is very little, but the Changing Pattern that it can also be seen that lateral force mechanical characteristic from derivation formula is also with cutting speed with to install radius relevant, suppose that cutting injection amount is 8mm, lateral force energy is with installing the Changing Pattern of radius (cutterhead installs radius 0 to 5m) and cutting speed (cutterhead rotating speed is from 0 to 10rad/min) as shown in figure 11.As can be seen from the figure, lateral force change of mechanical property reduces with the increase of installing radius, increases with the increase of cutterhead rotating speed simultaneously.

In sum: it is 3 all identical with the change of mechanical property rule that the hobboing cutter utilizing LS-DYNA to simulate obtains under rotating speed different situations to power model that this method is derived, and provable this method model has certain reference significance.

In order to analyze the reasonability of the model derived according to elastic plastic theory further, to the hobboing cutter stress model experiment Analysis based on energy principle, this method adopt C.Balci study V-tool on different rock texture character can cutting impact this chapter in extract experimental data, carry out models fitting.In model, the selection of each parameter is all consistent with the experiment parameter of C.Balci, and hobboing cutter parameter and geologic parameter are as shown in table 3, table 4.

Table 3.19 cun hobboing cutter parameter

Cutter diameter (m)	The hobboing cutter basil (°)	Hobboing cutter density (Kg/m 3)	Blade width (m)
				0.483	150°	7850	0.020

Table 4. rock properties parameter

It is as shown in table 5 that C.Balci adopts the experimental data of linear incision record and the calculating data of model of the present invention to carry out contrast to 19 cun of V-type hobboing cutters.

Table 5 experimental data and gross data contrast

Gross data (V-type)-1, the theoretical value of reference data document; Gross data (V-type)-2, the present invention derives the theoretical value of model.

As can be seen from Table 5, the V-type hobboing cutter set up based on elastic plastic theory energy principle breaks the theoretical model mentioned in the relative C.Balci document of rock stress model, under the effect of normal direction vertical force, when the uniaxial compressive strength of rock is larger, the data obtained relative to hobboing cutter cutting experiment are more close; Under tangential rolling force effect, tangential rolling force stress model is closer to actual experiment data.As can be seen from rock behavio(u)r, hard rock, the result calculated of this model and actual result more close, see energy point of view analysis better can reflect that actual hobboing cutter breaks the mechanical characteristic of rock from thin as seen.

Therefore, by with other two classical forecast model comparative analyses, the stressed Changing Pattern of forecast model and energy variation rule that the present invention proposes and other models all consistent.According to the analysis of model self, add the analysis that radius and energy response stressed to hobboing cutter installed by hobboing cutter rotating speed and hobboing cutter in a model, the conclusion of " along with the increase of cutterhead rotating speed; 3 also increase to power thereupon " that obtain with utilizing finite element emulation software also matches, and this model has certain theory value.Utilize the experimental data and model calculating data that record in C.Balci document, compared with the calculating data proposing model with this method, the increase of the visible raising along with rock strength and cutter injection amount, the result of calculation of this model is closer to experimental data, and this model has certain engineer applied and is worth.

Claims (1)

1. the TBM disk cutter based on energy principle is subject to a force prediction method, it is characterized in that, comprises the following steps:

Step 1, determine hobboing cutter in rock break-off process normal direction pushing vertical force F _v, tangential rolling force F _r, lateral force F _s;

Step 2, the hobboing cutter stress model set up in rock break-off process, comprising:

2-1, set up hobboing cutter and invade rock maximum cross-section upper stress equilibrium equation:

In hobboing cutter driving rock break-off process, the movement locus of each hobboing cutter revolves round the sun around whole cutter head center around the rotation of hobboing cutter disc centre while, and the effect of hobboing cutter to rock comprises normal direction pushing vertical force F _v; , tangential rolling force F _r, lateral force F _s, under cylindrical coordinate, the power acted on hobboing cutter 3 directions is P _r, P _θ, P _x;

Hobboing cutter is in rock break-off process, and entirety is disc, and tool point angle leads to for wedge shape sword, and the basil is 2 α, if certain on hobboing cutter wedge shape sword is some A point, the circular cylindrical coordinate at A point place is (r ₀, θ, x), if the unit plane power acting on 3 directions on the sword face of wedge shape sword is hobboing cutter invades rock maximum cross-section upper stress equilibrium equation:

$\left[\begin{array}{ccc}{\mathrm{\σ}}_r& {\mathrm{\τ}}_{\mathrm{\θr}}& {\mathrm{\τ}}_{\mathrm{xr}}\\ {\mathrm{\τ}}_{\mathrm{r\θ}}& {\mathrm{\σ}}_{\mathrm{\θ}}& {\mathrm{\τ}}_{\mathrm{x\θ}}\\ {\mathrm{\τ}}_{\mathrm{rx}}& {\mathrm{\τ}}_{\mathrm{\θx}}& {\mathrm{\σ}}_x\end{array}\right]\·\left[\begin{array}{c}\cos a\\ 0\\ \sin a\end{array}\right]=\left[\begin{array}{c}{p}_r\\ p_{\mathrm{\θ}}\\ p_x\end{array}\right]---\left(1\right)$

In formula (1):

σ _r, σ _θ, σ _xthe main stress bar that-hobboing cutter is corresponding under cylindrical coordinate, Mpa;

τ _{θ r}, τ _xr, τ _{r θ}, τ _{x θ}, τ _rx, τ _{θ x}the shear stress of-all directions, Mpa;

2-2, set up the hobboing cutter stress deformation equation of comptability:

Choose the cylindrical-coordinate system centered by hobboing cutter initial point, when hobboing cutter breaks rock, hobboing cutter produces inertia force around disc-shaped center roller rotation and is by the balance differential equation formula that stress represents be then:

$\frac{{\∂\mathrm{\σ}}_r}{\∂r}+\frac{1}{r}\frac{{\∂\mathrm{\τ}}_{\mathrm{r\θ}}}{\∂\mathrm{\θ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂x}+\frac{{\mathrm{\σ}}_r-{\mathrm{\σ}}_{\mathrm{\θ}}}{r}+{\mathrm{\ρ\ω}}_r^{2}r=0---\left(2\right)$

In formula (2):

R-hobboing cutter radius, m;

ρ-hobboing cutter density, Kg/m ³;

ω _r-hobboing cutter rotating speed, rad/s;

If: hobboing cutter is arranged on cutterhead and cutterhead is whole relation, and the installation radius of hobboing cutter is R, A point is ((R+x) cos φ, (R+x) sin φ, Z) relative to the position of cutterhead, and the stress transmission relation in Two coordinate system is as follows:

σ _(R+x)＝σ _x,σ _φ＝σ _θ,τ _(R+x)φ＝τ _xθ,τ _(R+x)Z＝τ _rx

With the equilibrium equation that the hobboing cutter that stress represents revolves round the sun on cutterhead be then:

$\frac{{\∂\mathrm{\σ}}_x}{\∂(R+x)}+\frac{1}{(R+x)}\frac{{\∂\mathrm{\τ}}_{\mathrm{x\θ}}}{\∂\mathrm{\φ}}+\frac{{\∂\mathrm{\τ}}_{\mathrm{rx}}}{\∂Z}+\frac{{\mathrm{\σ}}_x-{\mathrm{\σ}}_{\mathrm{\θ}}}{(R+x)}+{\mathrm{\ρ\ω}}_{(R+x)}^2(R+x)=0---\left(3\right)$

In formula (3):

The installation radius of hobboing cutter on R-cutterhead, m;

ω _(R+x)-disk cutter revolution angular velocity, rad/s, ω _rr=ω _(R+x)(R+x);

If hobboing cutter material is perfect elastic body, ignore the load change that shear stress shearing strain produces, in conjunction with physical equation and geometric equation, obtaining the Coordinate deformation equation that stress method represents is:

$\frac{{\∂}^2{\mathrm{\σ}}_r}{{\∂r}^2}+\frac{3}{r}\frac{{\∂\mathrm{\σ}}_r}{\∂r}+(3+v){\mathrm{\ρ\ω}}_r^2=0---\left(4\right)$

In formula (4): ν-hobboing cutter material poisson's ratio;

2-3, according to formula (1) to formula (4), show that disk cutter breaks the rock stress components and is respectively:

${\mathrm{\σ}}_r=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2(r_0^2-r^2)---\left(5\right)$

${\mathrm{\σ}}_{\mathrm{\θ}}=\frac{p_r}{\sin a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2{r}_0^2-\left(\frac{1+3v}{8}\right){\mathrm{\ρ\ω}}_r^2{r}^2---\left(6\right)$

${\mathrm{\σ}}_x=\frac{p_x}{\cos a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2(R_0^2-R^2)---\left(7\right)$

In formula (5), formula (6) and formula (7):

R ₀-A point to the radius length of hobboing cutter center, m;

R ₀-A point to the radius length of cutter head center position, m;

Step 3, set up rock strain model in hobboing cutter rock break-off process:

A point is under Lagrange remainder, and the coordinate on hobboing cutter is:

$\left\{\begin{array}{c}{x}^{\′}=x\\ y^{\′}=r_0\sin \mathrm{\θ}\\ z^{\′}=r_0\cos \mathrm{\θ}\end{array}\right.---\left(8\right)$

The degree of depth that hobboing cutter is pushed down into rock in normal direction vertical force is u _r, suppose A point and rock planes overlapping, A point causes the displacement of rock respective change to be:

$\left\{\begin{array}{c}x=0\\ y=r_0\sin \mathrm{\θ}\\ z=u_r-(r-r_0\cos \mathrm{\θ})\end{array}\right.---\left(9\right)$

By r ₀use y respectively with θ, z is expressed as functional relation:

$\left\{\begin{array}{c}{r}_0=\sqrt{y^2+z^2}\\ \mathrm{\θ}=\arctan \frac{y}{z}\end{array}\right.---\left(10\right)$

According to the incremental model of the theory of plasticity, in hobboing cutter rock break-off process, the rock strain corresponding with A point is expressed as:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}\frac{\∂X}{\∂x}& \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{1}{2}(\frac{\∂X}{\∂z}+\frac{\∂Z}{\∂x})\\ \frac{1}{2}(\frac{\∂X}{\∂y}+\frac{\∂Y}{\∂x})& \frac{\∂Y}{\∂y}& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})\\ \frac{1}{2}(\frac{\∂X}{\∂x}z+\frac{\∂Z}{\∂x})& \frac{1}{2}(\frac{\∂Y}{\∂z}+\frac{\∂Z}{\∂y})& \frac{\∂Z}{\∂z}\end{array}\right]---\left(11\right)$

In formula (11): ${\mathrm{\ϵ}}_{\mathrm{xx}}=\frac{\∂X}{\∂x}=\frac{\∂X}{{\∂r}_0}\frac{{\∂r}_0}{\∂x}+\frac{\∂X}{\∂\mathrm{\θ}}\frac{\∂\mathrm{\θ}}{\∂x};$

Consider u _rfor constant term, if shear stress and shearing strain are 0, on relative hobboing cutter wedge shape sword, the rock strain of A point is:

$\left[\begin{array}{ccc}{\mathrm{\ϵ}}_{\mathrm{xx}}& {\mathrm{\ϵ}}_{\mathrm{xy}}& {\mathrm{\ϵ}}_{\mathrm{xz}}\\ {\mathrm{\ϵ}}_{\mathrm{yx}}& {\mathrm{\ϵ}}_{\mathrm{yy}}& {\mathrm{\ϵ}}_{\mathrm{yz}}\\ {\mathrm{\ϵ}}_{\mathrm{zx}}& {\mathrm{\ϵ}}_{\mathrm{zy}}& {\mathrm{\ϵ}}_{\mathrm{zz}}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]---\left(12\right)$

In formula (12), ε _xx, ε _xy, ε _xz, ε _yx, ε _yy, ε _yz, ε _zx, ε _zy, ε _zz-rock relative strain;

Step 4, set up the energy-balance equation of hobboing cutter and rock:

From the first law of thermodynamics and second law, TBM is in tunnel excavating process, and the energy transferring that hobboing cutter breaks rock comprises the input of energy, the gathering and dissipating, output 4 processes of energy of energy;

When the external work that hobboing cutter provides is all for broken rock, and set rock system as an entirety, ignore the accumulation process of energy and the influence factor of temperature, the energy-balance equation setting up hobboing cutter and rock according to possibility merit principle is as follows:

${\∫}_S{p}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dS}+{\∫}_V{f}_i^{\left(2\right)}{u}_i^{\left(1\right)}\mathrm{dV}={\∫}_V{\mathrm{\σ}}_{\mathrm{ij}}^{\left(1\right)}{\mathrm{\ϵ}}_{\mathrm{ij}}^{\left(2\right)}\mathrm{dV}---(i,j=\mathrm{1,2.3})---\left(13\right)$

The energy-balance equation that step 5 is set up according to step 4, meanwhile, if the principal direction of stress of rock is corresponding with the principal direction of stress under the cylindrical coordinates of hobboing cutter, face power when hobboing cutter breaks rock on lozenges is respectively:

$P_r=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2{r}_0^2\}{\mathrm{dS}}_r---\left(14\right)$

$P_{\mathrm{\θ}}=\∫\{\frac{{\mathrm{\σ}}_1}{2\sin a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2{r}_0^2\}{\mathrm{dS}}_{\mathrm{\θ}}---\left(15\right)$

$P_x=\∫\{\frac{{\mathrm{\σ}}_3}{2\cos a}+\left(\frac{3+v}{8}\right){\mathrm{\ρ\ω}}_r^2{R}_0^2\}{\mathrm{dS}}_x---\left(16\right)$

${\mathrm{dS}}_r=2u_r\sqrt{r^2-{(r-u_r)}^2}\tan a---\left(17\right)$

${\mathrm{dS}}_{\mathrm{\θ}}=h\sqrt{r^2-{(r-u_r)}^2}---\left(18\right)$

${\mathrm{dS}}_x=u_r^2\tan a---\left(19\right)$

Formula (14) is in formula (19):

P _r, P _θ, P _xbe respectively the normal direction pushing vertical force F of hobboing cutter _v, tangential rolling force F _rwith lateral force F _s, KN;

S _r, S _θ, S _xbe respectively the projected area in hobboing cutter normal direction pushing vertical force, tangentially rolling force and lateral force direction, m ²;

σ ₁, σ ₃be respectively the main stress bar corresponding to rock element body, wherein, σ ₁=σ ₃, Mpa;

U _rfor the displacement of A point incision rock, work as r ₀when position overlaps with position, Rock Cutting face, r ₀=r-u _r, m;

R ₀for the position of A point on cutterhead, R ₀=R+u _rtana, m.