# Calculation of Stress Intensity Factors at Both Ends of the Indenter under Eccentric Loads

Authors: Xuefeng Yang; Yili Duo
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

During rock breaking by drill, cone rolls on the rock surface and contact between gear teeth and surface is basically 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 Mode-I crack tip and with the same function distribution. In this paper, based on theJ2 integral conservation law, a new method is established for solving stress intensity factor of bias Mode-I plane indentation, which not only provides respective stress intensity factor of both ends of the indenter but also compares relations between eccentricity, partial load, and stress intensity factor. The study not only provides a method for establishment of new cone drill engineering design but also improves rock layer boring efficiency.


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## Body

## 1. 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 [1–3]. In 1952, the world’s first full-face rock tunnel boring machine was developed by American Robbins Company [4], opening a new era in lithostratigraphy full-face boring construction. With the rapid development of national economy, rock-breaking technology involves numerous engineering applications 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 drivage efficiency and reliability, whose concentrated expression is rock-breaking tool design and related theory. Scholars at home and abroad have undertaken a number of studies on the tool rock-breaking mechanism [5–10]. 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 the rock hardness, mechanical properties, and strength. Meanwhile, plus rapid increase in microcrack 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. Choi and Lee [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 decrease in joint angle and spacing. Zhang et al., by comparing the mechanical model of interaction between forward and positively installed disc cutter and the rock [13, 14], proved with plasticity theory that rock-breaking specific energy of the forward installed disk cutter is low. Moreover, the greater the radius of the disc cutter, the greater the relative reduction amount of rock-breaking specific energy of the forward 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 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 the common drill cone, tunneling shield cone hob and hole drilling, and other rock-breaking tools in the rock matrix can be generally divided into Hertz concentrated induction method of spherical indentation stress [15, 16], concentrated induction method of cylindrical indentation stress, and singular induction method of plane indentation [17–19]. Wherein, both Hertz spherical indentation method and plane indentation method have applications in rock-breaking engineering. Their common feature is that gear teeth force the rock surface 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, radius of the indenter will not be sufficiently small, so peak stress is after all finite. When rock strength is high, rock-breaking effect is limited, with high energy consumption and large damage area. Rock-breaking characteristic of cylindrical gear teeth is 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 the rock. Hence, the plane indentation method features prominent stress concentration, low load, and efficient rock breaking, which enjoy important engineering value. Therefore, in this paper, with two-dimensional plane indentation, for example, through the principle that the plane indentation method induces singular stress field on the rock surface, bias indentation stress intensity factor is given based on the J2 conservation integration method, and eccentricity effect on load size is analyzed to provide technical support for better results of the flat tooth cone drill in rock-breaking engineering applications.


## 2. Contact Pressure Problem

### 2.1. Plane Indentation

In recent years, plane indentation boundary cracking damage has been a major concern. As shown in Figure1, plane indentation-induced progressive stress field demonstrates r−1/2 singularity, with the same order as crack-tip singular stress field [20–23] and essentially the same mechanics. Existence of singular stress field will inevitably lead to rock damage and cracking. And theoretically, its peak stress can be infinite even under small load.Figure 1
A schematic representation of the contact between a two-dimensional rectangular punch and a substrate. (a) Contact by a rigid flat-end punch. (b) Related Mode-I crack.
(a)
(b)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, the cracking problem of the indentation boundary is actually the cracking problem without the initial crack surface or the 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 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 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 development of the new flat tooth cone drilling tool and shield flat hob and can form new and more perfect rock fragmentation theories.


### 2.2. Rectangular Indenter Contact Problem

Two typical rock-cutting saws commonly used in engineering are shown in Figure2. In drilling engineering, the interface of the rock and bit interface process is very complex. In order to describe the rock failure and mechanism, many scholars usually select a simplified indention test using the flat-end indenter [24, 25], and the indentation hardness is also usually used to assess the drillability of rocks [26–28]. Figure 2(a) shows a typical flat-tipped indenter, and the saw tooth of the drill is composed of a cylindrical shell periodic array flat-end diamond indenter. The mechanism of this type of drilling tool for rock breaking is that the rock is ground by diamond abrasive with the control of metal bond. The rock-breaking mechanism of the circular saw is shear failure, as shown in Figure 2(b). It is worth noting that both rock-breaking tools have a rectangular indenter, this is also the focus of this paper. Zhao et al. [29] took the brittle rock granite as the research object to test the performance of different tools. Guha et al. simulated flat punch indentation from the perspective of a moulding technique using a modified Fleck–Hutchinson plasticity model [30]. Campbell and Gill investigated the flat punch ISE across many orders of magnitude, and an analytic expression for the loading and unloading response is derived [31]. Nevertheless, there were almost no reports regarding the eccentric load problem.Figure 2
Two typical rock-cutting saws commonly used in engineering. (a) Flat-tipped indenters. (b) Circular saw.
(a)
(b)The common and yet unique character is that these rock-cutting tools can be modeled by periodic rigid and flat-tipped indenters, shown in Figure2. 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 microcracks on the contact surface even if it is free of any microcracks. This surface cracking mechanism can potentially be utilized to assist rock cutting, in additional to the diamond grits embedded on the cutter surface.


## 2.1. Plane Indentation

In recent years, plane indentation boundary cracking damage has been a major concern. As shown in Figure1, plane indentation-induced progressive stress field demonstrates r−1/2 singularity, with the same order as crack-tip singular stress field [20–23] and essentially the same mechanics. Existence of singular stress field will inevitably lead to rock damage and cracking. And theoretically, its peak stress can be infinite even under small load.Figure 1
A schematic representation of the contact between a two-dimensional rectangular punch and a substrate. (a) Contact by a rigid flat-end punch. (b) Related Mode-I crack.
(a)
(b)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, the cracking problem of the indentation boundary is actually the cracking problem without the initial crack surface or the 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 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 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 development of the new flat tooth cone drilling tool and shield flat hob and can form new and more perfect rock fragmentation theories.


## 2.2. Rectangular Indenter Contact Problem

Two typical rock-cutting saws commonly used in engineering are shown in Figure2. In drilling engineering, the interface of the rock and bit interface process is very complex. In order to describe the rock failure and mechanism, many scholars usually select a simplified indention test using the flat-end indenter [24, 25], and the indentation hardness is also usually used to assess the drillability of rocks [26–28]. Figure 2(a) shows a typical flat-tipped indenter, and the saw tooth of the drill is composed of a cylindrical shell periodic array flat-end diamond indenter. The mechanism of this type of drilling tool for rock breaking is that the rock is ground by diamond abrasive with the control of metal bond. The rock-breaking mechanism of the circular saw is shear failure, as shown in Figure 2(b). It is worth noting that both rock-breaking tools have a rectangular indenter, this is also the focus of this paper. Zhao et al. [29] took the brittle rock granite as the research object to test the performance of different tools. Guha et al. simulated flat punch indentation from the perspective of a moulding technique using a modified Fleck–Hutchinson plasticity model [30]. Campbell and Gill investigated the flat punch ISE across many orders of magnitude, and an analytic expression for the loading and unloading response is derived [31]. Nevertheless, there were almost no reports regarding the eccentric load problem.Figure 2
Two typical rock-cutting saws commonly used in engineering. (a) Flat-tipped indenters. (b) Circular saw.
(a)
(b)The common and yet unique character is that these rock-cutting tools can be modeled by periodic rigid and flat-tipped indenters, shown in Figure2. 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 microcracks on the contact surface even if it is free of any microcracks. This surface cracking mechanism can potentially be utilized to assist rock cutting, in additional to the diamond grits embedded on the cutter surface.


## 3. Plane Indentation Configuration and Singular Stress Field

As is shown in Figure1(a) of the plane indentation contact surface, width of the rectangular rigid indenter is 2l, load on the rigid indenter is P, acting on the smooth half-plane, Poisson’s ratio is ν, and elastic modulus is E. Indentation interfacial stress distribution according to classical analysis is as follows [32, 33]:(1)px1=−Pπl2−x2.When coordinate transformation is given, makex = r − l and expand with the binomial distribution method; then, pressure distribution between the indentation interface near the indenter is as follows:(2)px1=−Pπ12lr+⋅⋅⋅.Based on flat-end indenter (two-dimensional) induced Mode-I singular stress field, cracking behavior of the indentation boundary is studied. As shown in Figure1(a), when the rigid flat-end indenter acts on the half-plane frictionless surface, there is the following stress field in the indentation boundary [34]:(3)σrrσθθσrθ=−KΙ−ind2πrcosθ21+sin2θ2cos3θ2sinθ2cos2θ2.Compared with the singular stress field Mode-I crack, except the negative sign in equation (3), singular stress field in the indentation boundary is exactly the same as the 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(4)KΙ−ind=Pπl.As can be known from the above equation, for the crack shown in Figure1(b), stress distribution at x2 = 0 is consistent with the indentation problem in Figure 1(a). Thus, the indentation problem in Figure 1(a) is equivalent to the crack problem in Figure 1(b).


## 4. 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 boundaryS + 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 displacement u, given by a time-like parameter “t.” The result of energy release rate per unit time, ∂Π/∂t, is given, i.e.,(5)∂Π∂t=∫swvimids,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 in the same form as equation (5).Figure 3
Three-dimensional deformation fields and integration path.Letvi = ei = Δi/Δ, which corresponds to two components of unit boundary shift, so that e1 = cosα, e2 = sinα, for the two-dimensional solids, where α is the angle between shift Δ 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(6)G¯=−∂Π∂Δ=∫sweinids.Now, let all points on the boundarys move in the same direction, and combine this result with the conservation law for two-dimensional problem Ji. The energy release rate expressed by equation (6) can then be rearranged along the v-direction, i.e.,(7)G¯=Jiei=J1cosa+J2sina.From the geometrical point of view, boundary cracking or crack initiation regardless 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 limits ⟶0 taken, as shown in Figure 4. Then, the energy release rate of boundary cracking can be defined as(8)G=J1s⟶0cosa+J2s⟶0sina,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 x2-direction or the driving force of boundary cracking in x2-direction when the limits taken exist.Figure 4
Model of boundary cracking whens ⟶ 0 for two-dimensional solids. (a) Notch model for boundary cracking. (b) Boundary cracking.The geometrical model for crack-tip shattering initiated from a free boundary inn 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 equations (5)–(8), the energy release rate for crack-tip shattering is obtained, i.e.,(9)G=∑l=1nJ1sl⟶0cosal+J2sl⟶0sinal.Figure 5
Model of boundary scattered-cracking whens ⟶0 for two-dimensional solids. (a) Multinotch model for boundary scattered-cracking. (b) Boundary scattered-cracking.As expected, the boundary shift or cracking should be moving inwards within the solid, andJ2sl⟶0 must be positive indicating an energy release.


## 5. Conservation Law

### 5.1.J Integral

For closed curve, free of cracks or holes, as shown in Figure6, based on the J integral conservation law, the following zero integral expression can be obtained [35, 36]:(10)Jj=∫swnj−Tiui,jds,j=1,2.Figure 6
Circular integral path graph.Equation (10) contains two integral components (J1 and J2). When the crack plane is parallel to axis x1, J1 means crack growth energy release rate and J2 means crack opening energy release rate. Both can be used to calculate stress intensity factor of elastic body cracks, including calculation of 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 then compare eccentricity size.


### 5.2.J1 and J2 Integral Calculation of the Mode-I Crack

The integral path is a circle with a radius ofr, and the integral is shown in Figure 6. Mode-I crack stress field expression is(11)σ11=KΙ2πrcosθ21−sinθ2sin3θ2,σ22=KΙ2πrcosθ21+sinθ2sin3θ2,σ12=KΙ2πrcosθ2sinθ2cos3θ2.For homogeneous, isotropia material elastomers, the elastic strain energy density is(12)w=12σijεij.In equation (11), θ is the angle with the x1 axis, rad. Then, substituting equation (11) to equation (12), we find(13)w=1+νKΙ24Eπr145−cos2θ+4cosθ−2μ1+cosθ,where v is Poisson’s ratio and E is the modulus of elasticity, GPa. The expression of surface stress T is(14)T1=σ11n1+σ12n2,(15)T2=σ21n1+σ22n2.Substituting equation (11) to equations (14) and (15), the following result can be found:(16)T1=KΙ2πr34cos3θ2+14cosθ2,(17)T2=KΙ2πr32cosθ2sinθ.The expression of Mode-I crack tip displacement field is(18)u1=21+νKΙ4Er2π5−8νcosθ2−cos3θ2,u2=21+νKΙ4Er2π7−8νsinθ2−sin3θ2.However, there is a relation between the Cartesian and polar coordinates:(19)∂∂x1=cosθ∂∂r−sinθr∂∂θ,(20)∂∂x2=sinθ∂∂r+cosθr∂∂θ.Next, substituting equation (18) to equations (19) and (20),(21)∂u1∂x1=1+νKΙ4E2πr3−8νcosθ2+cos5θ2,(22)∂u2∂x1=1+νKΙ4E2πr8ν−9sinθ2+sin5θ2,(23)∂u1∂x2=1+νKΙ4E2πr7−8νsinθ2+sin5θ2,(24)∂u2∂x2=1+νKΙ4E2πr5−8νcosθ2−cos5θ2.From equations (16), (17), and (21)–(24), it will yield(25)T1∂u1∂x1+T2∂u2∂x1=1+νKΙ264πrE−2cos3θ+40−48νcos2θ+18−32νcosθ+16ν−24,(26)T1∂u1∂x2+T2∂u2∂x2=1+νKΙ264πrE−2sin3θ+40−48νsin2θ+22−32νsinθ.From equations (13), (25), and (26), according to equation (10), J1 and J can be expressed as follows:(27)J1=1−ν2KΙ22πE∫1−cos2θdθ,(28)J2=1−ν2KΙ22πE∫−sin2θdθ.


## 5.1.J Integral

For closed curve, free of cracks or holes, as shown in Figure6, based on the J integral conservation law, the following zero integral expression can be obtained [35, 36]:(10)Jj=∫swnj−Tiui,jds,j=1,2.Figure 6
Circular integral path graph.Equation (10) contains two integral components (J1 and J2). When the crack plane is parallel to axis x1, J1 means crack growth energy release rate and J2 means crack opening energy release rate. Both can be used to calculate stress intensity factor of elastic body cracks, including calculation of 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 then compare eccentricity size.


## 5.2.J1 and J2 Integral Calculation of the Mode-I Crack

The integral path is a circle with a radius ofr, and the integral is shown in Figure 6. Mode-I crack stress field expression is(11)σ11=KΙ2πrcosθ21−sinθ2sin3θ2,σ22=KΙ2πrcosθ21+sinθ2sin3θ2,σ12=KΙ2πrcosθ2sinθ2cos3θ2.For homogeneous, isotropia material elastomers, the elastic strain energy density is(12)w=12σijεij.In equation (11), θ is the angle with the x1 axis, rad. Then, substituting equation (11) to equation (12), we find(13)w=1+νKΙ24Eπr145−cos2θ+4cosθ−2μ1+cosθ,where v is Poisson’s ratio and E is the modulus of elasticity, GPa. The expression of surface stress T is(14)T1=σ11n1+σ12n2,(15)T2=σ21n1+σ22n2.Substituting equation (11) to equations (14) and (15), the following result can be found:(16)T1=KΙ2πr34cos3θ2+14cosθ2,(17)T2=KΙ2πr32cosθ2sinθ.The expression of Mode-I crack tip displacement field is(18)u1=21+νKΙ4Er2π5−8νcosθ2−cos3θ2,u2=21+νKΙ4Er2π7−8νsinθ2−sin3θ2.However, there is a relation between the Cartesian and polar coordinates:(19)∂∂x1=cosθ∂∂r−sinθr∂∂θ,(20)∂∂x2=sinθ∂∂r+cosθr∂∂θ.Next, substituting equation (18) to equations (19) and (20),(21)∂u1∂x1=1+νKΙ4E2πr3−8νcosθ2+cos5θ2,(22)∂u2∂x1=1+νKΙ4E2πr8ν−9sinθ2+sin5θ2,(23)∂u1∂x2=1+νKΙ4E2πr7−8νsinθ2+sin5θ2,(24)∂u2∂x2=1+νKΙ4E2πr5−8νcosθ2−cos5θ2.From equations (16), (17), and (21)–(24), it will yield(25)T1∂u1∂x1+T2∂u2∂x1=1+νKΙ264πrE−2cos3θ+40−48νcos2θ+18−32νcosθ+16ν−24,(26)T1∂u1∂x2+T2∂u2∂x2=1+νKΙ264πrE−2sin3θ+40−48νsin2θ+22−32νsinθ.From equations (13), (25), and (26), according to equation (10), J1 and J can be expressed as follows:(27)J1=1−ν2KΙ22πE∫1−cos2θdθ,(28)J2=1−ν2KΙ22πE∫−sin2θdθ.


## 6. Influencing Factors of Flat Tooth Cone Bias

### 6.1. Bias Indentation Stress Intensity Factor

Positive stress intensity factor problem of the ideal smooth surface has been discussed in literature [37]. Bias Mode-I plane indentation is shown in Figure 7, and unbalance loading indentation stress intensity factor can be expressed as superposition of positive P and bending moment M stress intensity factor.Figure 7
Bias Mode-I plane indentation.
(a)
(b)In this paper, finite width rigid indenter with width at W and thickness at B (whereinB = 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 equation (10) based on J2 [38–40]:(29)J2−P=∮Sobcdefghijown2−Tiui,2ds=0.Figure 8
Contact model for elastic solids with finite boundaries and the crack model.
(a)
(b)On pathSbc and Sde, there exist Ti = 0 and n2 = 0. And the following calculation results can be obtained:(30)J2−P=∫Sbcwn2−Tiui,2ds=∫Sdewn2−Tiui,2ds=0.PathSgh and Sij are straight lines, while path Sfg and Sjo are quadrant. 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:(31)J2−P=∫Sghwn2−Tiui,2ds=∫Sijwn2−Tiui,2ds=0,(32)J2−P=∫Sfgwn2−Tiui,2ds=∫Sjown2−Tiui,2ds=1−ν2KΙ−ind22πEplane strain.For relatively far away indentation contact area and indentation contact area near the surface, the following approximate expression can be obtained:(33)J2−P=∫Scdwn2−Tiui,2ds=w¯−+Pu˜2,2−,(34)J2−P=∫Shiwn2−Tiui,2ds=−w¯++Pu˜2,2+,wherein u˜i represents the strain of the neutral axis, which can be obtained by bending theory in material mechanics. The symbol “−” indicates far-field cross section, and 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(35)J2−P=1−ν2KΙ−ind−P2πE+2∫SobwPds=w¯−+Pu˜2,2−−w¯++Pu˜2,2+=P2u˜2,2−−u˜2,2+,wherein u˜2,2− and u˜2,2+ may be expressed as follows [41, 42]:(36)u˜2,2−=−PEA,(37)u˜2,2+=−PEA∫01dξ1−2a/W1−ξ2.In the formula,a represents the crack length, A = WB shows the cross-sectional area at the far field Scd. Due to movement and cracking of the indentation border and that Sob and Sef are not in K-control area, its integral value is very small which can be ignored. Then, substitute equations (36) and (37) into equation (35) and obtain(38)J2−P=1−ν2KΙ−ind−P2πE=P2u˜2,2−−u˜2,2+=P2PEW∫01dξ1−2a/W1−ξ2−PEW.According to the above formula, stress intensity factor under concentrated force can be expressed as(39)KΙ−ind−P=πP2W1−ν2∫01dξ1−2a/W1−ξ2−11/2.Under concentrated moment, closed integration pathSobcdefghijo is still chosen. As shown in Figure 8(a), it can be obtained according to the integral conservation law and by referring to the above calculation method that(40)J2−M=1−ν2KΙ−ind−M2πE+2∫SabwMds=w¯−−M−ϕ˜−−w¯+−M+ϕ˜+=M2ϕ˜+−ϕ˜−.In the formula,φ˜+ and φ˜− are the curvature of cracked cross section and crack-free cross section after deformation [43]. I represents the inertia moment of the neutral axis of cross section (wherein, B = 1):(41)ϕ˜−=MEI,I=BW312,(42)ϕ˜+=MEI∫01dξ1−2a/W1−ξ23.Then, substitute equations (41) and (42) into equation (40). In equation (40), M = nP, where P is equal to the eccentric loads P and n is the distance from the center line (Figure 7(a)).(43)KI−ind−M=Pπ2W1−ν2∫0112×n/W21−2a/W1−ξ23dξ−12×nW21/2.Due to two nonsymmetrical Mode-I singular stress fields at the indenter corner under eccentric load, as shown in Figure9, two different stress intensity factors, in this paper, singular stress field at both ends of the 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 the indenter obtained via equations (39) and (43) is(44)KΙ−ind−R=Pπ2W1−ν2∫0111−2a/W1−ξ2+12×n/W21−2a/W1−ξ23dξ−1−12×nW21/2,(45)KΙ−ind−L=Pπ2W1−ν2∫0111−2a/W1−ξ2−12×n/W21−2a/W1−ξ23dξ−1+12×nW21/2.Figure 9
Nonsymmetrical Mode-I singular stress field at the indenter corner under eccentric load.As can be known from equations (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.In above sections, based on the conservation law, a direct and simple method is proposed to estimate the indentation stress intensity factor for singular stress field induced by eccentric loads. For various crack configurations, as shown in Figure8(b), the stress intensity factors have been found and collected in the handbooks [39]:(46)KΙ=FPP+FM3Mlπa2lW/2,(47)FP=1.1221−aW−0.06aW2+0.728aW3,(48)FM=43π1+lW+1.5lW2+2.5lW3−7.52lW4+21.216lW5.Here, a hypothesis is given byn/l = 0.3, so the following is obtained:(49)nW=0.30.5−aW,and then equation (44) is taken as an example to compare with equation (46), as displayed in Figure 10.Figure 10
Scheme of eccentric load stress intensity factor (n/l = 0.3 and µ = 0).The results are in good agreement with the superposition of stress intensity factors for the bending moment and the normal load, and Figure10 shows that equation (44) is correct. Traditionally, “Handbook of Stress Intensity Factors” is needed when stress intensity factors are calculated. However, through the study of this paper, equations (46)–(48) can be replaced by equation (44) to obtain stress intensity factor. This method not only has more accurate results but also saves calculation costs. What are especially worth noticing are equations (44) and (45) also indicate that the boundary cracking of Mode-I plane indentation with positive loads and eccentric loads has different critical cracking, and they are different from the crack growth problem, but they all have the same asymptotic singular stress field, the same mechanical essence, and the homologous cracking mechanism.


### 6.2. Impact Analysis of Eccentricity

In this paper, through equations (44) and (45), effect of different eccentricity values on eccentric load is analyzed. Figure 11(a) shows that, in the case of increasing crack expansion, with increase in eccentricity, 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 11(b) shows that when eccentricity changes from 0.1 to 0.3, the situation is opposite to the case of Figure 11(a), and value change from 0.1 to 0.3 causes little effect which can be ignored. At the same time, research shows that when eccentricity is >0.3, imaginary number will appear in 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 service life of the indenter. This finding not only enlightens our drilling efficiency improvement but also means great significance for dimensional design of drill and solution of the engineering application problem.Figure 11
Effect of eccentricity onK/P. (a) R-end of the indenter. (b) L-end of the indenter.
(a)
(b)


## 6.1. Bias Indentation Stress Intensity Factor

Positive stress intensity factor problem of the ideal smooth surface has been discussed in literature [37]. Bias Mode-I plane indentation is shown in Figure 7, and unbalance loading indentation stress intensity factor can be expressed as superposition of positive P and bending moment M stress intensity factor.Figure 7
Bias Mode-I plane indentation.
(a)
(b)In this paper, finite width rigid indenter with width at W and thickness at B (whereinB = 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 equation (10) based on J2 [38–40]:(29)J2−P=∮Sobcdefghijown2−Tiui,2ds=0.Figure 8
Contact model for elastic solids with finite boundaries and the crack model.
(a)
(b)On pathSbc and Sde, there exist Ti = 0 and n2 = 0. And the following calculation results can be obtained:(30)J2−P=∫Sbcwn2−Tiui,2ds=∫Sdewn2−Tiui,2ds=0.PathSgh and Sij are straight lines, while path Sfg and Sjo are quadrant. 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:(31)J2−P=∫Sghwn2−Tiui,2ds=∫Sijwn2−Tiui,2ds=0,(32)J2−P=∫Sfgwn2−Tiui,2ds=∫Sjown2−Tiui,2ds=1−ν2KΙ−ind22πEplane strain.For relatively far away indentation contact area and indentation contact area near the surface, the following approximate expression can be obtained:(33)J2−P=∫Scdwn2−Tiui,2ds=w¯−+Pu˜2,2−,(34)J2−P=∫Shiwn2−Tiui,2ds=−w¯++Pu˜2,2+,wherein u˜i represents the strain of the neutral axis, which can be obtained by bending theory in material mechanics. The symbol “−” indicates far-field cross section, and 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(35)J2−P=1−ν2KΙ−ind−P2πE+2∫SobwPds=w¯−+Pu˜2,2−−w¯++Pu˜2,2+=P2u˜2,2−−u˜2,2+,wherein u˜2,2− and u˜2,2+ may be expressed as follows [41, 42]:(36)u˜2,2−=−PEA,(37)u˜2,2+=−PEA∫01dξ1−2a/W1−ξ2.In the formula,a represents the crack length, A = WB shows the cross-sectional area at the far field Scd. Due to movement and cracking of the indentation border and that Sob and Sef are not in K-control area, its integral value is very small which can be ignored. Then, substitute equations (36) and (37) into equation (35) and obtain(38)J2−P=1−ν2KΙ−ind−P2πE=P2u˜2,2−−u˜2,2+=P2PEW∫01dξ1−2a/W1−ξ2−PEW.According to the above formula, stress intensity factor under concentrated force can be expressed as(39)KΙ−ind−P=πP2W1−ν2∫01dξ1−2a/W1−ξ2−11/2.Under concentrated moment, closed integration pathSobcdefghijo is still chosen. As shown in Figure 8(a), it can be obtained according to the integral conservation law and by referring to the above calculation method that(40)J2−M=1−ν2KΙ−ind−M2πE+2∫SabwMds=w¯−−M−ϕ˜−−w¯+−M+ϕ˜+=M2ϕ˜+−ϕ˜−.In the formula,φ˜+ and φ˜− are the curvature of cracked cross section and crack-free cross section after deformation [43]. I represents the inertia moment of the neutral axis of cross section (wherein, B = 1):(41)ϕ˜−=MEI,I=BW312,(42)ϕ˜+=MEI∫01dξ1−2a/W1−ξ23.Then, substitute equations (41) and (42) into equation (40). In equation (40), M = nP, where P is equal to the eccentric loads P and n is the distance from the center line (Figure 7(a)).(43)KI−ind−M=Pπ2W1−ν2∫0112×n/W21−2a/W1−ξ23dξ−12×nW21/2.Due to two nonsymmetrical Mode-I singular stress fields at the indenter corner under eccentric load, as shown in Figure9, two different stress intensity factors, in this paper, singular stress field at both ends of the 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 the indenter obtained via equations (39) and (43) is(44)KΙ−ind−R=Pπ2W1−ν2∫0111−2a/W1−ξ2+12×n/W21−2a/W1−ξ23dξ−1−12×nW21/2,(45)KΙ−ind−L=Pπ2W1−ν2∫0111−2a/W1−ξ2−12×n/W21−2a/W1−ξ23dξ−1+12×nW21/2.Figure 9
Nonsymmetrical Mode-I singular stress field at the indenter corner under eccentric load.As can be known from equations (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.In above sections, based on the conservation law, a direct and simple method is proposed to estimate the indentation stress intensity factor for singular stress field induced by eccentric loads. For various crack configurations, as shown in Figure8(b), the stress intensity factors have been found and collected in the handbooks [39]:(46)KΙ=FPP+FM3Mlπa2lW/2,(47)FP=1.1221−aW−0.06aW2+0.728aW3,(48)FM=43π1+lW+1.5lW2+2.5lW3−7.52lW4+21.216lW5.Here, a hypothesis is given byn/l = 0.3, so the following is obtained:(49)nW=0.30.5−aW,and then equation (44) is taken as an example to compare with equation (46), as displayed in Figure 10.Figure 10
Scheme of eccentric load stress intensity factor (n/l = 0.3 and µ = 0).The results are in good agreement with the superposition of stress intensity factors for the bending moment and the normal load, and Figure10 shows that equation (44) is correct. Traditionally, “Handbook of Stress Intensity Factors” is needed when stress intensity factors are calculated. However, through the study of this paper, equations (46)–(48) can be replaced by equation (44) to obtain stress intensity factor. This method not only has more accurate results but also saves calculation costs. What are especially worth noticing are equations (44) and (45) also indicate that the boundary cracking of Mode-I plane indentation with positive loads and eccentric loads has different critical cracking, and they are different from the crack growth problem, but they all have the same asymptotic singular stress field, the same mechanical essence, and the homologous cracking mechanism.


## 6.2. Impact Analysis of Eccentricity

In this paper, through equations (44) and (45), effect of different eccentricity values on eccentric load is analyzed. Figure 11(a) shows that, in the case of increasing crack expansion, with increase in eccentricity, 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 11(b) shows that when eccentricity changes from 0.1 to 0.3, the situation is opposite to the case of Figure 11(a), and value change from 0.1 to 0.3 causes little effect which can be ignored. At the same time, research shows that when eccentricity is >0.3, imaginary number will appear in 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 service life of the indenter. This finding not only enlightens our drilling efficiency improvement but also means great significance for dimensional design of drill and solution of the engineering application problem.Figure 11
Effect of eccentricity onK/P. (a) R-end of the indenter. (b) L-end of the indenter.
(a)
(b)


## 7. Conclusion

(1)
Different stress intensity factors at both ends of the indenter under bias are given based on theJ2 integral conservation law, no longer equates in positive stress intensity factor. The calculation of the stress intensity factor can be simplified from 3 formulas into one formula based on this methodology. Therefore, the calculation process is not only simple with good accuracy but also can provide a reference for engineering application.
(2)
When stress intensity factor remains constant, eccentricity growth can reduce input load, but when eccentricity is ≥0.3, imaginary number will appear in stress intensity factor of the left indenter; when 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.



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*Source: 5174680-2020-04-10.xml*