Theoretical Study on Flat Tooth Cone Fracture Type Rock Breaking based on Plane Indentation

During rock breaking by drill, cone rolls on the rock surface, and contact between gear teeth and the surface is positive Mode-I plane indentation and bias Mode-I plane indentation. During bias action, there will be an asymmetrical Mode-I singular stress field at both ends of the indenter. The stress field singularity is of the same order as the Mode-I crack tip and with the same function distribution. In this paper, based on J 2 integral conservation law, a new method is established for solving the stress intensity factor of bias Mode-I plane indentation, which not only provides respective stress intensity factor of both ends of indenter but also compares relations between eccentricity, partial load and stress intensity factor. The study not only provides a method for the establishment of a new cone drill engineering design but also improves rock layer boring efficiency.


Introduction
Rock breaking technology is widely used in petroleum, coal mining and other energy industries, underground urban space construction and tunneling shield and other military and civilian facilities, as well as environmental and architectural stone processing [1][2] [3].In 1952, the world's first full face rock tunnel boring machine was developed by the American Robbins Company [4], opening a new era in lithostratigraphy full face boring construction.With the rapid development of the national economy, rock-breaking technology involves numerous engineering such as resource development, transportation, roads, and urban construction.As one of the major basic issues affecting economic and social development, it enjoys particularly prominent importance.
Rock breaking is directly related to driving efficiency and reliability, whose concentrated expression is rock-breaking tool design and related theory.Scholars at home and abroad have undertaken many studies on tool rock breaking mechanism [5][6] [7][8] [9] [10].Tan Qing studied rock fracture characteristics of boring machine cutterhead under rock temperature variation [11].The results showed that the higher the rock temperature, the lower rock hardness, mechanical properties and strength.Meanwhile, plus a rapid increase in micro-crack and cracks, rock breaking efficiency is increased.But rock is a poor conductor of heat, so there are certain difficulties in this method to improve rock breaking efficiency.Sung-Oong [12] proposed disc cutter rock breaking numerical research under joint characteristics.The results showed that when disc cutter rock breaking direction was opposite to that of texture, rock breaking input energy would be increased constantly with a decrease in joint angle and spacing.Zhang Zhaohuang, by comparing the mechanical model of interaction between forwarding and positively installed disc cutter and the rock [13] [14], proved with plasticity theory that rock breaking specific energy of forwarding installed disk cutter is low.Moreover, the greater radius of the disc cutter, the greater the relative reduction amount of rock breaking specific energy of forwarding installed disc cutter.According to engineering statistics, disc cutter consumption is huge.Frequent replacement is caused due to tool life and fast wear, which not only delays the construction period, causing staff fatigue but also increases project cost.Thus, cone type tool becomes the major rock breaking tool for hard rock heading machine, shield machine and other equipment with wide application.
Stress inducing mode, rock breaking behavior and application of common drill cone, tunneling shield cone hob and hole drilling and other rock breaking tool in rock matrix can be generally divided into Hertz concentrated induction method of spherical indentation stress [15][16], concentrated induction method of cylindrical indentation stress, singular induction method of plane indentation [17][18] [19].Wherein, both Hertz spherical indentation method and plane indentation method have applications in rock breaking engineering.Their common feature is that gear teeth force rock surfaces to induce local stress concentration.Hertz spherical indentation method is mainly characterized by maximum stress in the indenter or central indentation.Despite high peak stress, the radius of the indenter will not be sufficiently small, so peak stress is after all finite.When rock strength is high, the rock-breaking effect is limited, with high energy consumption and large damage area.Rock breaking characteristics of cylindrical gear teeth are similar to that of the Hertz spherical indentation method, which will not be repeated here.However, even when the plane indentation method exerts a little load, high stress will be induced in rock.Hence, the plane indentation method features prominent stress concentration, low load and efficient rock breaking, which enjoys important engineering value.Therefore, in this paper, with a two-dimensional plane indentation for example, through the principle that plane indentation method induces singular stress field on the rock surface, bias indentation stress intensity factor is given based on J2 conservation integration method, eccentricity effect on load size is analyzed, to provide technical support for better results of flat tooth cone drill in rock breaking engineering applications.

Plane Indentation
In recent years, plane indentation boundary cracking damage has been a major concern.As shown in [Figure 1], plane indentation induced progressive stress field demonstrates r -1/2 singularity, with the same order as crack tip singular stress field [20][21] [22][23] and essentially the same mechanics.The existence of a singular stress field will inevitably lead to rock damage and cracking.And theoretically, its peak stress can be infinite even under a small load.
It is worth noting that, unlike research objects of conventional fracture mechanics, indentation boundary is general without initial cracks.Under the act of plane indentation singular stress field, cracking problem of indentation boundary is cracking problem without initial crack surface, or crack initiation problem, which carries certain challenges, with difficulty much higher than the cracking problem.However, this kind of indentation-induced singular stress field is near surface stress field, with stress singular point and K control area on the surface of the rock mass.The field intensity of the field stress is under the sole control of indentation stress intensity factor Kind.Even if ambient boundary conditions change, singularity and distribution of stress field will not be undermined, with the only impact on the size of indentation stress intensity factor.This characteristic provides an ideal fracture mechanics model for research on flat tooth drilling tool fracture type rock breaking technology and applications, which is especially suitable for the development of new flat tooth cone drilling tool and shield flat hob and can form new and more perfect rock fragmentation theories.

Rectangular rigid head contact problem
Two typical rock-cutting saws commonly used in engineering are shown in [Figure 2].The common and yet unique character is that these rock-cutting tools can be modeled by periodic rigid and flat tipped indenters.It has been pointed out from the previous work that a singular stress field exists in an incompressible substrate at the sliding contact edge of a rigid flat-ended indenter pressing down onto the substrate.
It should be emphasized that such a singular indentation stress field is sufficient to generate micro-cracks on the contact surface even if it is free of any micro-cracks.This surface cracking mechanism can potentially be utilized to assist rock-cutting, in addition to the diamond grits embedded on the cutter surface.

Plane indentation configuration and singular stress field
As is shown in [Figure 1 (a)] of the plane indentation contact surface, the width of rectangular rigid indenter is 2l, load on rigid indenter is P, acting on smooth half-plane, Poisson's ratio is  , the elastic modulus is E. Indentation interfacial stress distribution according to classical analysis is as follows [24][25]:  When the coordinate transformation is given, make x=r-l, expand with binomial distribution method, the pressure distribution between indentation interface near the indenter is as follows:  1 (a)], when rigid flat end indenter acts on a half-plane frictionless surface, there is the following stress field in the indentation boundary [26]: Compared with singular stress field Mode-I crack, except negative sign in equation ( 3), singular stress field in indentation boundary is the same as Mode-I crack problem, demonstrating Mode-I singular stress field.Therefore, KI-ind is defined as "indentation stress intensity factor" which can be expressed as: As can be known from the above equation, for crack shown in [Figure 1

Energy release rate for boundary cracking
This section describes a method for calculating the energy release rate associated with the initiation of a crack, starting from the free boundary.Consider a three-dimensional elastostatic boundary problem for a material contained within the boundary S+s illustrated in [Figure 3], where the portion s of the boundary is traction-free, and external loading only imposes tractions on S. Without changing the boundary conditions on S, impose a continuously varying sequence of static solutions, related to the displacements u, given by a time-like parameter "t".The result of energy release rate per unit time, t    / , is given, i.e., Figure 3. Three-dimensional deformation fields and integration path ds m wv t where vi denotes the "velocity" of the points on s and mi is the current outward normal to s.In the case of two-dimensional deformation fields, relevant to the present problem, the energy release rate remains the same form as Eq.( 5).
Let vi=ei=△i/△, which corresponds to two components of unit boundary shift, so that e1=cosα, e2=sinα, for the two-dimensional solids, whereαis the angle between shift △ and e1 and represents the direction of the boundary shift.Let ni=-mi be the unit inward normal on boundary s, which means that the boundary s moves inward, e.g., the case of crack growth.Thus, the energy release rate of boundary shift is given by Now, let all points on the boundary s move in the same direction, and combine this result with the conservation law for two-dimensional problem Ji.The energy release rate expressed by Eq.( 6) can then be rearranged along the v-direction, i.e., From the geometrical point of view, boundary cracking or crack initiation regardless of whether it occurs at a crack tip, or a notch, or from a smooth boundary, can be considered as a boundary shift, or a change in the boundary condition, with the limit s→0 taken, as shown in [Figure 4].Then, the energy release rate of boundary cracking can be defined as Where (J1)|s→0 denotes the energy release rate of the boundary s shifting in x1-direction or the driving force of boundary cracking in x1-direction; and (J2)|s→0 denotes the energy release rate of the boundary s shifting in x2direction or the driving force of boundary cracking in x2direction when the limits have taken exist.
The geometrical model for crack-tip shattering initiated from a free boundary in n directions with s=s1+s2+…….+sn→0 is illustrated in Figure 5 for the case of n=3 and s=s1+s2+ s3→0.By following the same procedure from Eqs. ( 5)-( 8), the energy release rate for crack-tip shattering is obtained, i.e., ( ) ( ) As expected, the boundary shift or cracking should be moving inwards within the solid, and (Jj)sl→0 must be positive indicating an energy release.

J Integral
For a closed curve free of cracks or holes, as shown in Figure 6, based on J integral conservation law, the following zero integral expression can be obtained [27][28]: Eq. ( 10) contains two integral components (J1 and J2).When the crack plane is parallel to axis x1, J1 means crack growth energy release rate, J2 means crack opening energy release rate.Both can be used to calculate the stress intensity factor of elastic body cracks, including the calculation of the plane indentation stress intensity factor.This paper will provide the main calculation process for calculating bias Mode-I plane indentation stress intensity factor based on integral J2, and will compare eccentricity size.

J1 and J2 integral calculation of Mode-I crack
The integral path is a circle with a radius of r, the integral is shown in [Figure 6].Mode-I crack stress field expression is: For homogeneous, isotropia material elastomers, the elastic strain energy density: In Eq. ( 11) θ is the angle with the x1 axis, rad.Then, substituting Eq. ( 11) to Eq.( 12), we find where v is Poisson's ratio, E is Modulus of elasticity, GPA.The expression of surface stress T is; Substituting Eq. ( 11) to Eqs. ( 14) and ( 15), and the following result can be found The expression of Mode-I crack tip displacement field is: However, there is a relation between the Cartesian and polar coordinates: Next, we substituting Eq. ( 18) to Eqs. (19) and (20), can be found as follows: From Eqs. ( 16)-( 17) and Eqs ( 21)-( 24 From Eqs. ( 13), ( 25) and ( 26), according to Eq. ( 10), J1 and J can be expressed as follows:

Bias Indentation Stress Intensity Factor
The positive stress intensity factor problem of the ideally smooth surface has been discussed in the literature [29].Bias Mode-I plane indentation is shown in [Figure 7], unbalance loading indentation stress intensity factor can be expressed as the superposition of positive P and bending moment M stress intensity factor.
In this paper, finite width rigid indenter with width at W and thickness at B (wherein B = 1) is taken as an example to study stress intensity factor under bias, as shown in Figure 8.In the case of concentrated force, choose a closed path of integration Sobcdefghijo as shown in Figure 8 (a).And the following results can be obtained according to Eq. ( 10), based on J2 [30] On path Sbc and Sde, there is Ti=0 and n2=0.And the following calculation results can be obtained: Path Sgh and Sij are straight lines, while path Sfg and Sjo are quadrants.When Sijo and Sfgh are in K-control area at the corner of the indenter, it is not difficult to obtain the following calculation results: For relatively far away indentation contact area and indentation contact area near surface, the following approximate expression can be obtained: Wherein i u ~represents a strain of neutral axis, which can be obtained by bending theory in material mechanics.The symbol "-" indicates far field cross-section, symbol "+" indicates near field cross-section.w represents strain energy density per unit length.Then, substitute equations ( 30) -( 34) into equation (29), and obtain that: ũ may be expressed as follows [33,34]: In the formula, a represents crack length, A = WB shows cross-sectional area at the far field Scd.Due to movement and cracking of indentation border, and that Sob and Sef are not in the K control area, its integral value is very small which can be ignored.Then substitute Eqs. ( 36) and (37) into Eq.( 35) and obtain: According to the above formula, stress intensity factor under concentrated force can be expressed as: Yili Duo, Yujun Xie, and Jun Hai Under concentrated moment, closed integration path Sobcdefghijo is still chosen.As shown in [Figure8 (a)], it can be obtained according to integral conservation law and by referring to the above calculation method that: In the formula, +  ~and −  ~are curvature of cracked cross-section and crack-free cross- section after deformation [35].I represent the inertia moment of the neutral axis of the crosssection (wherein, B=1).
Then, substitute equation ( 41) and (42) into equation ( 40), M = nP [Figure 8 (a)] and stress intensity factor under concentrated moment can be obtained after calculation: Due to two non-symmetrical Mode-I singular stress field at the indenter corner under eccentric load, as show in [Figure 9], two different stress intensity factors, in this paper, singular stress field at both ends of indenter is defined as L-end Mode-I singular stress field and R-end Mode-I singular stress field.Stress intensity factor at both ends of indenter obtained via Eq.(39) and Eq. ( 43) is: As can be known from Eqs. (44) and (45), this method innovatively provides different stress intensity factors at both ends of the indenter under bias, and takes into account the impact of eccentricity on stress intensity factor.

Impact Analysis of Eccentricity
In this paper, through equations ( 44) and ( 45), the effect of different eccentricity values on eccentric load is analyzed.[Figure 10 (a)] shows that in the case of increasing crack expansion, with an increase in eccentricity, the eccentric load will gradually become smaller under constant stress intensity factor; stress intensity factor will gradually become large under constant load.The difference is that, although [Figure 10 (b)] shows that when eccentricity changes from 0.1 to 0.3, the situation is opposite to the case of [Figure10 (a)], value change from 0.1 to 0.3 causes a little effect which can be ignored.At the same time, research shows that when eccentricity is> 0.3, an imaginary number will appear in the stress intensity factor of the left indenter, and with indefinitely growing eccentricity, the left indenter will leave the base.This is not conducive to improving drilling efficiency, which will wear the right indenter and reduce the service life of the indenter.This finding not only enlightens our drilling efficiency improvement but also means great significance for the dimensional design of drill and solution of engineering application problem.

Conclusion
(1) Different stress intensity factors at both ends of the indenter under bias are given based on J2 integral conservation law, and the calculation process is simple with good accuracy, which can provide a reference for engineering application.
(2) When the stress intensity factor remains constant, eccentricity growth can reduce input load, but when eccentricity is ≥0.3, the imaginary number will appear in the stress intensity factor of the left indenter; when the stress intensity factor is constant, indenter width reduction can also lower input load.
(3) Study on the relationship between stress intensity factor, eccentricity and indenter width can enable us to optimize flat tooth cone design, reduce energy consumption and improve rock breaking efficiency.

Figure 1 .
Figure 1.A schematic representation of the contact between a two-dimensional rectangular punch and a substrate

Figure 2 .
Figure 2. Two typical rock cutting saws commonly used in engineering (b)], stress distribution at x2=0 is consistent with the indentation problem in [Figure 1 (a)].Thus, indentation problem in [Figure1 (a)] is equivalent to crack problem in [Figure 1 (b)].

Figure 4 .
Figure 4. Model of boundary cracking when s → 0 for two-dimensional solids 5. Conservation law

Figure 9 .
Figure 9. Non-symmetrical Mode-I singular stress field at indenter corner under eccentric load

Figure 10 .
Figure 10.Effect of eccentricity on K/P