Modification of Griffith–McClintock–Walsh model for crack growth under compression to incorporate stick-slip along the crack faces

International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318

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Modification of Griffith–McClintock–Walsh model for crack growth under compression to incorporate stick-slip along the crack faces Gaurav Singh n, Robert W. Zimmerman Department of Earth Science & Engineering, Imperial College London, London, SW7 2AZ, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 19 July 2014 Received in revised form 24 September 2014 Accepted 27 September 2014 Available online 29 October 2014

Griffith developed a now-classical model for crack growth under compressive loading, based on finding the location of the maximum tensile stress at some point on the crack perimeter. McClintock and Walsh extended Griffith's analysis by considering the effect of friction along the closed crack faces, using the assumption that the shear traction is equal to the normal traction multiplied by the friction coefficient, at all points on the crack face. Although this relation holds in regions of slip, it does not hold in regions where the crack faces are stuck and not sliding. In this paper, the problem is revisited, using a more accurate treatment of friction that accounts for both stick and slip regions. It is found that the more accurate treatment of friction does not alter the crack growth criterion obtained by McClintock and Walsh. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Crack Friction Stick-slip Contact mechanics

1. Introduction For nearly a century, many conceptual models for the brittle failure of materials such as rocks have been based on the concept that failure begins with the growth of small pre-existing cracks. Griffith's energy-based criterion for the initiation of crack growth in a body subjected to tensile loading is well known [1,2]. Griffith also proposed a model for failure under compressive loading, due to crack growth [3]. This model is not based on energy considerations, but rather on the assumption that a crack will begin to grow when the maximum local tensile stress at some point on its perimeter reaches a certain critical value. Although identifying rock failure with the growth of a single crack is known to be an oversimplification [4,2], Griffith's concept still plays an important role in developing models for rock failure. In Griffith's original analysis [3], no account was taken for the fact that, under compression, the two opposing crack faces would come into contact, and, if there was a shear component to the traction along the crack faces, frictional forces would be present. The effect of friction was incorporated into this problem by McClintock and Walsh [5], who assumed that the frictional stress σf acting on the contacting crack surfaces can be written as

σ f ¼ μσ n

ð1Þ

where μ is the friction coefficient, and σn is the normal contact stress. This provided an improvement to Griffith's model, in which n

Corresponding author. E-mail address: [email protected] (G. Singh).

http://dx.doi.org/10.1016/j.ijrmms.2014.09.020 1365-1609/& 2014 Elsevier Ltd. All rights reserved.

friction was ignored completely. However, it is known that Eq. (1) holds only when friction acts at its maximum static value, or when there is sliding (assuming then that the coefficient of static and sliding friction are the same) [6]. Nevertheless, this treatment of the crack-face contact has been used widely [7,8], without any explicit mention that it is an oversimplification of the laws of friction. Crack branching has been studied using the same idea [9]. Although more sophisticated and accurate treatments of friction between contacting deformable bodies have been developed over the past decades [10,11], this more accurate approach does not seem to have yet been used to analyze the problem of friction on crack faces under compressive loading. The purpose of this paper is to incorporate a more realistic friction model into this problem, so as to determine whether or not such a treatment will appreciably alter the conclusions reached by McClintock and Walsh [5]. A more realistic representation of friction (as an inequality) in compressional fracture had been first reported in [12] and later improved in [13]. The latter work rightly states that rock mechanicians have considered friction in rock fracture independently from the treatment given in other engineering fields. In these earlier works, the problem was formulated as a mixed complementarity problem, which was solved using numerical methods. It was in fact claimed [13] that the stress on a crack in such a situation cannot be obtained analytically. In the present work, an analytical method is developed to represent σf in a more accurate way using the standard contact mechanics approach for two elastic bodies in contact [14]. Specifically, the upper and the lower parts/surfaces of the body (i.e., the two crack faces) are denoted by the subscripts u and l, respectively. The shift (tangential relative displacement)

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between the two crack surfaces caused by a combined effect of elastic deformation and rigid body motion is [14] hðxÞ ¼ uxu ðx; 0Þ uxl ðx; 0Þ þ C

ð2Þ

where uxu and uxl are the displacements in the x-direction for the upper and lower crack faces, respectively, and the constant term C represents a rigid-body motion. A positive shift indicates a displacement of the upper crack face in the positive x-direction relative to the lower face. The stick between the two crack surfaces is defined as a state when there is no time-varying shift between _ the upper and lower crack faces, i.e., hðxÞ ¼ 0, with the overdot denoting the derivative with respect to time. The generally accepted friction law [6] for the static/sliding friction coefficient μ states that

μσ n ðxÞ o σ f ðxÞ o  μσ n ðxÞ in stick region

ð3Þ

and _ σ f ðxÞ ¼  μσ n ðxÞ sgn ðhðxÞÞ in slip region

ð4Þ

where _ hðxÞ ¼ u_ xu  u_ xl þ C_

ð5Þ

After deriving expressions for σn and σf using the abovedescribed treatment of friction that follows from contact mechanics, the criterion for crack growth [3] will be invoked to study the effect that this may have on the conclusions that were reached by Griffith and McClintock and Walsh.

2. McClintock and Walsh's frictional crack model for compressive strength In this section, the analysis of McClintock and Walsh [5] will be reviewed to consider the effect that friction on the contacting crack faces will have on the condition for crack growth, using Griffith's assumption of crack growth under a condition of critical local tensile stress [3]. A crack, inclined at an angle η, which has closed under a major (σ1) and a minor (σ2) principal stress ðσ 1 4 σ 2 Þ, has a normal stress σn and a frictional shear stress σf acting along its faces. Fig. 1 shows a cracked body and its equivalent stress state. The figure and the subsequent problem formulation use the same compression is negative sign convention that was initially used by [5], and which is typically used in most of the contact mechanics literature. In this case, it must be noted that σ1 and σ2 will be negative for compressive loading. Using the standard coordinate transformation, the local stresses can be written as
σ þ σ 
σ  σ  σ yy ¼ 1 2 þ 1 2 cos 2η ð6Þ 2 2
σ  σ  σ xy ¼ 1 2 sin 2η ð7Þ 2

The stress system of Fig. 1 can be separated into superimposable stress systems, as shown in Fig. 2. It is worth noting again, that σ1 and σ2 will be negative for compressive loading. It is known that the maximum local tensile pffiffiffiffiffiffiffi stress exceeds the applied stress by a factor of the order of l=ρ, where l is the halflength of the crack and ρ is the radius of curvature at the tip [15]. Hence, for a crack of small aspect ratio, the local tensile stress is much greater than any other stress component. Therefore, the contributions of states (b)–(d) in Fig. 2 can be ignored, as the aim is to find the maximum stress on the crack. The stress due to the remaining two states (a) and (e) in Fig. 2 is given as

σ ββ ¼ ðσ xy  σ f Þ

2β for state ðaÞ α20 þ β2

ð8Þ

2α 0 for state ðeÞ α20 þ β2

ð9Þ

σ ββ ¼ ðσ yy  σ n Þ

where α0 ¼ ðρ=lÞ1=2 , l is the half length of the crack, ρ is the radius of curvature at the tip, and α0 cot β is the slope dy=dx along the crack perimeter. The crack is assumed to be open (and hence no friction acting) until the applied stress σyy reaches a value σc. The normal stress σn on the crack surface is assumed to increase linearly with σyy after the crack has closed, meaning that

σ n ¼ σ yy  σ c

ð10Þ

After substitutions from Eqs. (6) and (7), the total tensile stress σ ββ is found by summing Eqs. (8) and (9): 2 3 β σ þ ð μ ð σ þ σ  2 σ Þ þ ð σ  σ Þð sin 2 η þ G cos 2 η ÞÞ c c 1 2 1 2 7 26 α0 7 σ ββ ¼ 6 5 α0 4 1 þ ðβ=α0 Þ2 ð11Þ when

σ1 þ σ2 σ1  σ2 2

þ

2

Z σc

ð12Þ

and

σ xy 4 σ f

ð13Þ

The location at which this stress achieves its maximum value is found by differentiating σ ββ with respect to β =α0 , and setting the result equal to zero. Temporarily defining

σ n ¼ Gðσ 2 þ σ 1  2σ c Þ þ ðσ 1  σ 2 Þð sin 2η þ G cos 2ηÞ The point β=α0 at which σ ββ is maximum is found to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2σ σ n2 β=α0 ¼  c 1 7 1 þ 2 σn 4σ c

ð14Þ

ð15Þ

When Eq. (15) is substituted into Eq. (11), the maximum stress along the crack perimeter is found to be

α0 σ ββ ¼

σn

2ðβ =α0 Þ

¼

σ n2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!

4σ c 1 7

1þ

σ n2 4σ 2c

ð16Þ

The material is assumed to be isotropic, so that the same criterion (of critical tensile stress on the crack) of crack growth applies to all points along the crack. Letting T0 be the critical local tensile stress at which the crack begins to grow, the maximum value of σ ββ [3] can be expressed as

σ ββ ¼

Fig. 1. Stress components referred to crack axes.

2T 0

α0

ð17Þ

The value of this maximum σ ββ is constant, and is independent of the loading mode. Thus, the required condition for fracture can be found by the substitution of Eq. (17) into Eq. (16). Further

G. Singh, R.W. Zimmerman / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318

313

Fig. 2. Superimposable stress states.

substitution of the value of σ n gives

rffiffiffiffiffiffiffiffiffiffiffiffiffi

μðσ 2 þ σ 1  2σ c Þ þ ðσ 1  σ 2 Þð sin 2η þ G cos 2ηÞ ¼ 4T 0 1 

σc

T0

ð18Þ

which is true for all crack orientations η. To find the specific value of η that gives the maximum σ ββ , Eq. (18) must be differentiated with respect to η, and the resulting expression equated to zero. The angle η that gives the maximum tensile stress σ ββ is found to be   1 1 ð19Þ η ¼ tan  1 2 μ substitution of which into Eq. (18) gives the condition for fracture as rffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi σ ð20Þ μðσ 2 þ σ 1  2σ c Þ þ ðσ 1  σ 2 Þ 1 þ μ2 ¼ 4T 0 1  c T0 This is the criterion for failure of a rock under far-field compressive loading, considering sliding along the crack faces, according to McClintock and Walsh [5]. If the coefficient of friction is set to zero, this expression reduces to the one derived by Griffith. It is worth pointing that this criterion has the form of a linear relation between stress σ1 and σ2, and so it coincides with, and provides some mechanistic justification for, the empirical failure criterion of Coulomb [2].

3. Crack faces as deformable bodies in contact In this section, concepts from the modern theory of contact mechanics will be used to study the contact between the two crack faces. To be consistent with the contact mechanics literature, compressive stresses will again be reckoned as negative. 3.1. Physical model Following the analyses of Griffith and McClintock and Walsh, the crack is assumed to have a shape in which the crack faces are nearly parallel to each other, and where the distance between the

Fig. 3. Zoomed-in view of the initial crack shape (solid curve) and the deformed crack faces (dashed curves) that are in contact at the mid-point; the þ sign denotes the centre of the crack, where the crack faces are assumed to come into initial contact.

crack faces is much less than the crack length. It is known that if the initial crack profile were perfectly elliptical, the two faces would come into complete and abrupt contact at some value of the normal load [16]. On the other hand, there are numerous crack shapes, not very different from elliptical, for which contact first occurs at a single point. In both cases, a shear stress component would not contribute to crack closure [17]. Hence, it seems reasonable to assume that, under a sufficiently large compressive normal stress, the crack faces would first come into contact at a single point. As contact occurs first at a single point, both crack tips will be located far from the point of contact, irrespective of its exact location, as long as the contact point is not too close to either tip. Hence, it is convenient to take this point of contact to be located the mid-point of the crack (Fig. 3). Although the mid-point of the crack has been assumed to be the initial point of contact, the full extent of the region of contact, as the load acts, will only emerge as a part of the solution to the problem. The following two concepts need to be employed in this treatment: (1) The contact force should not be tensile, meaning that σ n ðxÞ r 0 if the contact region extends from x ¼  a to x ¼a, as shown in Fig. 4. This will be naturally true for a crack under compression. (2) There should be no interaction between the crack faces outside the contact region, i.e., the crack tips are assumed to be far

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G. Singh, R.W. Zimmerman / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318

Fig. 6. Crack faces lightly in contact with an initial gap g0 ðxÞ. Fig. 4. A zoomed-in view of the two crack faces in contact, and the forces acting, with the origin located at the centre of the crack.

rock, but will automatically hold for the lower region, also. Fig. 6 shows the two crack surfaces in contact, with the initial gap between them taken to be some known function g 0 ðxÞ. The upper part of the cracked rock is given a vertical rigid-body translation C0 and a small clockwise rotation C1, such that the gap solely due to rigid body translation and rotation would become gðxÞ ¼ g 0 ðxÞ  C 0  C 1 x

ð21Þ

The normal tractions will cause elastic deformation, represented by the displacements in the lower and upper parts of the rock with respect to the crack. The gap will be further modified as gðxÞ ¼ g 0 ðxÞ  uyl ðx; 0Þ þ uyu ðx; 0Þ C 0  C 1 x

ð22Þ

As the gap g(x) for two contacting crack faces in the contact region is zero, it can be written that uyl ðx; 0Þ uyu ðx; 0Þ ¼ g 0 ðxÞ  C 0  C 1 x; Fig. 5. Stress components without a crack to show direct proportionality to the forces F and T.

away from the region of contact. There is only one region of contact. However, once the present analysis is accepted, it is possible to extend the working to multiple points/regions of contact, representing a more realistic picture. 3.2. Mathematical analysis To develop a mathematical form of the physical model discussed above, the theory of contact mechanics of two deformable bodies [14] will be employed. As can be seen from Fig. 4, it is assumed that the two crack surfaces of the rock are being pressed together by a normal compressive concentrated force F, and then subjected to a monotonically increasing tangential concentrated force T. These concentrated forces are merely a simplified representation of the stress acting over the region of contact centered at the point of contact. The region of contact always spreads out from a point of contact, and so assuming that the stress over this region acts as a concentrated force seems to be a reasonable assumption. The normal compressive and the tangential friction stress on the crack surfaces are equal and opposite for equilibrium of the mid-point of contact region and the crack. The forces F and T are not new forces, and their values can be easily derived from the state of stress, ignoring the crack, as shown in Fig. 5. The sign convention followed in Fig. 5 is the same as that of contact mechanics (tension positive), with the origin at the point of contact (assumed at centre of crack) as shown in Fig. 4. The normal and tangential tractions at the contact plane will be separately found. 3.2.1. Normal tractions In the stage when only normal tractions are applied, there is no tendency for slip, and hence no tangential tractions are induced. In order to approximate the two contacting crack surfaces as halfplanes over the region of contact, it will be assumed that the radius of curvature at the contact point satisfies R b a. The mathematical statements below are written for the upper part of the cracked

aoxoa

ð23Þ

Treating the distributed load as the limit of a set of point loads of magnitude σ n ðξÞ dξ, the surface displacement uyl ðx; 0Þ of the lower body can be written as [14] uyl ðx; 0Þ ¼ 

ðκ þ 1Þ 4π G

Z

a a

σ n ðξÞ ln jx  ξj dξ 

ðk  1Þ 8G

Z

a a

σ f ðξÞ sgnðx  ξÞ dξ ð24Þ

where G and κ are the shear modulus and the Kolosov parameter, respectively. An alternative representation of the function sgn(x) is 2HðxÞ  1, where H(x) is the Heaviside step function. It follows that the derivative of sgn(x) is 2δðxÞ, and that the derivative of the Ra integral  a σ f ðξÞ sgnðx  ξÞ dξ is twice the value of σ f ðξÞ at the point ξ ¼ x. Differentiating Eq. (24) with respect to x gives Z duyl ðκ þ 1Þ a σ n ðξÞ dξ ðκ  1Þ ¼  σ ðxÞ ð25Þ 4π G  a ðx  ξÞ 4G f dx Following the same arguments as above, a similar expression for the tangential displacement uxl of the lower face is written as Z duxl ðκ þ 1Þ a σ f ðξÞ dξ ðκ  1Þ ¼ þ σ n ðxÞ ð26Þ 4π G  a ðx  ξÞ 4G dx Similar expressions for the upper face of the crack can be derived in the same way. The following can now be written as Z a d A σ n ðξÞ dξ ðuyl  uyu Þ ¼  ð27Þ dx 4π  a ðx  ξÞ d A ðu  uxu Þ ¼  dx xl 4π

Z

a a

σ f ðξ Þ dξ ðx  ξÞ

ð28Þ

where ðκ þ 1Þ A¼2 G

ð29Þ

Eq. (27) can now be substituted into the x-derivative of Eq. (23) to obtain Z a dg 0 ðxÞ A σ n ðξ Þ dξ  C1 ¼  ; aoxoa ð30Þ dx 4π  a ðx  ξÞ Hence, the problem reduces to a Cauchy singular integral equation for the unknown stress σ n ðξÞ. This equation can be solved by substituting x ¼ a cos ϕ and ξ ¼ a cos θ, followed by expansion of

G. Singh, R.W. Zimmerman / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318

the two sides of the equation in Fourier series: Z π 1 dg 0 ðϕÞ A σ n ðθÞ sin θ dθ ; 0oϕoπ  C1 ¼   4π 0 ð cos ϕ  cos θÞ a sin ϕ dϕ The trigonometric result Z π cos ðkθÞ dθ  π sin ðkϕÞ ¼ sin ϕ 0 cos ϕ  cos θ

The singularity at θ ¼ mπ ðx ¼ 7 aÞ in the above expression can be removed if ð31Þ

and the expansions

σ cos ðkθÞ σ n ðθÞ ¼ ∑ nk sin θ k¼0

ð33Þ

1 dg 0 ¼ ∑ g k sin ðkϕÞ dϕ k¼1

ð34Þ

are invoked to write Eq. (31) as 1

∑ g k sin ðkϕÞ þ C 1 a sin ϕ ¼ 

k¼1

Aa sin ϕ πσ nk sin ðkϕÞ Aa ∑ σ sin ðkϕÞ ∑ ¼ 4 k ¼ 1 nk 4π k ¼ 1 sin ϕ 1

1

ð35Þ Equating the Fourier coefficients yields

σ nk ¼ 

4g k Aa

σ n1 ¼ 

4ðC 1 a þ g 1 Þ Aa

forall k 4 1

ð36Þ ð37Þ

Hence, all the terms needed to find σn are now known, except for k ¼0,1 (noting that σn1 in Eq. (37) is in terms of the unknown C1). The coefficients σn0 and σn1 can be found by inserting Eq. (33) into the equilibrium conditions: Ra (1) Force balance:  a σ n ðξÞ dξ ¼  F Ra (2) Moment balance:  a σ n ðξÞξ dξ ¼ 0 as the ordinate of the crack system is along F as can be seen in Fig. 4. From the force balance, it follows that Z a Z π F ¼ σ n ðx0 Þ dx0 ¼ σ n ðθÞa sin θ dθ a

1

Z π

k¼0

0

¼ a ∑ σ nk

o

cos ðnθÞ dθ ¼ π aσ n0

ð38Þ

giving σ n0 ¼ F=π a and σ n1 ¼ 0. From geometric considerations, the following holds true: 2

d g0 1 ¼ R dx2

ð39Þ

which, after integrating twice and substituting x ¼ a cos ϕ, gives g0 ¼ C 0 þ

x2 a2 cos ð2ϕÞ a2 þ ¼ C0 þ 4R 2R 4R

Differentiation with respect to dg 0 a sin ð2ϕÞ ¼ 2R dϕ 2

ð40Þ

ϕ now gives ð41Þ

with the only non-zero coefficient in the Expansion (34) being g2 ¼ 

a2 2R

ð42Þ

leading to

σ n2 ¼

Ga Rðκ þ 1Þ

F

πa ð32Þ

1

315

ð43Þ

Therefore, the complete form of the Expansion (33) is written as   F Ga cos ð2θÞ = sin θ σ n ðθÞ ¼  þ ð44Þ π a Rðκ þ 1Þ

¼

Ga Rðκ þ 1Þ

which implies that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðκ þ 1ÞR a¼ πG

ð45Þ

ð46Þ

The value of a can be taken from Eq. (46), and the substitution 2G=ðκ þ 1Þ ¼ E=4 be made for plane stress, E being the Young's modulus of the rock, to write σ n as Fð1  cos ð2θÞÞ 2F sin θ ¼ πa π a sin θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2F a2  x2 σ n ðxÞ ¼  π a2

σ n ðθ Þ ¼ 

ð47Þ ð48Þ

giving the normal stress distribution for  a o x o a. 3.2.2. Tangential tractions The tangential tractions will now be considered for two crack faces in contact. It will be assumed that there will be a constant contact area 2a and a constant normal traction during this second phase of loading. Eq. (5) is differentiated with respect to x and substituted in Eq. (28) to find Z a σ_f ðξÞ dξ A  ¼ 0;  a o x o a ð49Þ 4π  a x  ξ which is another singular Cauchy integral equation, which can be solved through the previously-used method to obtain T_

ffi σ_f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi π a2  x2

ð50Þ

As a is constant during this phase of tangential loading, the above equation reduces to T

ffi σ f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi π a2  x2

ð51Þ

The overdot in the above equations indicates a derivative with respect to time. The constant of integration that appears in the above equation must vanish, because the tangential force T is not acting initially and acts only after the normal force F has been acting. This point had been made earlier in Section 3.2. Hence, if stick is assumed throughout the region of contact, it leads to a singularity in σ f ðxÞ at the edges x ¼ 7 a. This would violate Inequality (3) for all values of μ, since σ n ðxÞ is bounded. Therefore, some slip must occur near the edges of the contact region for any T a 0. Hence, 0 r c o a must hold at all times. The problem of the slip zone was originally solved by [10] and [11] independently. In the slip zone, the tractions satisfy the condition σ f ¼  μσ n . Thus, the solution for the tangential tractions is considered as the summation of two components: (1) A frictional traction pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μF a2  x2 σ f ðxÞ ¼  μσ n ðxÞ ¼ ð52Þ π a2 from Eq. (48) throughout the contact region  a o x o a. (2) A correction traction σ nf that must be zero in the slip zones, and which is sufficient to restore the Inequality (3) in the stick zone. To find this correction shear, the shift due to distribution (Eq. (52)) is found to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z a 2 2μF a2  ξ dξ dðuxl  uxu Þ A 0 h ðxÞ ¼ ¼ dx 4π  a π a2 ðx  ξÞ Z π 2 μFA sin θ dθ μFAx ¼ ¼ ð53Þ 2π a2 0 cos ϕ  cos θ 2π a2

316

G. Singh, R.W. Zimmerman / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318 0

In the stick region, the shift is independent of x (ie. h ðxÞ ¼ 0), and hence the correction shear must be able to cancel the RHS of Eq. (53). Through an analogy with Eqs. (52) and (53), this can be achieved through pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μF c2  x2 σ nx ¼  ; coxoc ð54Þ π a2 which is a traction similar in form to that given by Eq. (52), but distributed over a smaller centrally located zone of width 2c (stick zone). The condition 0 rc o a always holds. The complete friction distribution can therefore be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μF pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ f ðxÞ ¼ 2 ð a2  x2  Hðc2  x2 Þ c2  x2 Þ;  a o x o a ð55Þ πa The stick-zone semi-width c can be determined by fulfilling the condition Z a
c 2 σ f ðxÞ dx ¼ μF  μF ð56Þ T¼ a a from Eq. (55), thereby arriving at sffiffiffiffiffiffiffiffiffiffiffiffiffiffi T c ¼ a 1 μF

ð57Þ

indicating that the stick zone shrinks to zero when μF-T, as would be expected. At this stage, the substitutions T=F ¼ σ xy =σ yy can be made in Eqs. (46) and (57) to arrive at 2ðκ þ 1ÞRσ yy πG   σ xy c 2 ¼ a2 1  a¼

ð58Þ ð59Þ

μσ yy

Thus, expressions for σf and σn have been derived for crack surfaces in contact. They must be substituted into the developments reviewed in Section 2. The combined σ ββ from Eqs. (8) and (9) can be written as

σ ββ ¼ X 1

α0 α20 þ β

2

þ X2

β α20 þ β

2

ð60Þ

where X 1 ¼ 2ðσ yy  σ n Þ and X 2 ¼ 2ðσ xy  σ f Þ, both do not depend on β. The assignment ζ ¼ β =α0 is now made, and the above equation can be written as

α0 σ ββ ¼

1 1þζ

2

½X 1 þ ζ X 2 

ð61Þ

The first and the second derivative tests are done to find the value of ζ that would give the maximum value of α0 σ ββ . The result is ffi
 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4X 21 1  2X 2 þ5 X2 þ 1 7 X2 ð62Þ ζ¼ 2 This value of ζ is substituted into Eq. (61), along with the use of Griffith's criterion [3] for crack growth α0 σ ββ ¼ 2T 0 , where T0 is the fracture load in simple tension, to find 0
 rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B X1 þ X2B @

2X 1 X2 þ 1

0
2X 1 B  X2 B 1þ@

7 2

4X 21 þ5 C X 22

C A

ffi 12 ¼ 2T 0  rffiffiffiffiffiffiffiffiffiffiffiffi 4X 21 7 þ 5 C X 22 C 2 A

ð63Þ

þ1

It can be easily verified that this equation gives the same result as of Eq. (18) for Griffith criterion in the absence of friction. For the sake of comparison with Eq. (18), the dependence on x; a can be eliminated by assuming x-a. This derived condition is valid for all orientations η of the crack. To find the maximum of this maximum stress for a certain orientation η, Eq. (63) must be maximized with respect to η. The expressions of σn and σf as varying functions, including both stick and slip in the contact region, will be substituted into Eq. (63) through X1 and X2, with the objective of finding a relationship similar to Eq. (19) corresponding to the state of maximum stress at the crack. This is extremely difficult to do analytically, and so this maximum will be found numerically, by substituting the whole range of values of β, η and x and picking the one that yields a maximum.

4. Results Purely for the sake of generating results for analysis of this work, the material properties of marble will be considered. All quantities have been non-dimensionalized when producing the final results, so as to generalize the observations. This choice of rock type is arbitrary. The material properties of marble used are the following: Poisson's ratio ν ¼0.166, and Young's modulus E¼ 70 GPa. It has been assumed during the analytical derivation that R b a. For l ¼1, R¼10 has been found to be sufficiently large so as to not make any difference to the generality of the results

Fig. 7. Permutations of μ and η for σ 1 =σ 2 ¼ 10; 100 which give imaginary half-stick region c.

G. Singh, R.W. Zimmerman / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 311–318

summarized below. Tests are conducted for different ratios of the major and minor principal stresses σ 1 =σ 2 ¼ ð1; 137, ð137; 1Þ with σ 2 =E ¼ 0:01=70; the significance of the value 137 will become apparent soon. The results can be studied separately, as each one of them offers different insights about the stick/slip contact mechanics model in Griffith's criterion of crack growth. It is worth mentioning again the assumptions under which the stick/slip contact mechanics model is built, keeping Inequality (3) intact: (i) a and c must be both real. For a general range of values μ and η, the parameters pairs that lie within the shaded areas of Fig. 7 do not satisfy this condition. (ii) 0 r c o a must hold to allow some slipping near the crack edges. (iii) a 5 l theoretically but 0 ra ol has been found to give generally sensible results. The variation of a=l with the principal stress ratio σ 1 =σ 2 follows a linear relationship as can be seen from Fig. 8. Eq. (46) is recalled here and simplified further by substituting σyy from Eq. (6) as 
σ σ 2 1 2ðκ þ 1ÞR ð1 þ cos 2ηÞ þ ð1  cos 2ηÞ 2 σ2 ð64Þ a¼ πG

317

modulus E and dimension R. It has been verified that σ 1 =σ 2 ¼ 1:1 shows a typical behavior of this domain, an example of which is discussed below. The case of σ 1 =σ 2 ¼ 10 has been considered here as an example. All points in Figs. 9–11 correspond to a point of maximum tensile stress along the crack (always occurring at the crack tip, as the assumption σ c ¼ 0 is upheld from the predecessor [5]) – the crack is supposed to grow at all these points. It is worth mentioning that the plots compared to the earlier known theory [5] are made for Eq. (19), as Eq. (20) would introduce an additional independent parameter T0 which would influence (but not disturb) the generality of the results. The results are divided into three zones, for assessment. In the first zone 0 o μ r 0:5, for maximum tensile stress on the crack, the size c of the stick region turns out to be imaginary (due to Eq. (57), which is physically not sensible. Even though the friction coefficient of rocks have usually been found to be higher than 0.6 [18], for the sake of physical generality it will be assumed that the coefficient of friction can take any positive value; hence, the need arises to seek a solution in this domain 0 o μ r0:5. A correction is made to find a meaningful relationship in this zone,

which after substitutions κ ¼ ð3  νÞ=ð1 þ νÞ and G ¼ E=2ð1 þ νÞ becomes   8Rσ 2 σ 1 ð1 þ cos 2ηÞ þ ð1  cos 2ηÞ ð65Þ a¼ πE σ2 which can be re-written as   Eπ a 1 σ 1 =σ 2 ¼ þ ð cos 2η 1Þ 8Rσ 2 1 þ cos 2η

ð66Þ

Let the ratio ðσ 1 =σ 2 Þc denote the critical state when the halfcontact region just becomes equal to the half-crack length,  i.e. a ¼ l . The above equation can now be written as    Eπ l 1 þ ð cos 2η 1Þ ð67Þ ðσ 1 = σ 2 Þc ¼ 1 þ cos 2η 8Rσ 2 

where E=σ 2 ¼ 70=0:01 and l =R ¼ 1=10. For 0 r η o π =2, the minimum value of ðσ 1 =σ 2 Þc , through first and second derivative tests has been found to occur at η ¼0. At η ¼0, ðσ 1 =σ 2 Þc ¼ 137:375 which serves as a cutoff point for the assumed material properties and dimensions. (1) When 1 o σ 1 =σ 2 r 137, Eqs. (6) and (7) cannot be simplified any further. The upper bound of 137 will depend on the Young's

Fig. 8. Variation of normalized half-region of contact a=l with the principal stress ratio σ 1 =σ 2 .

Fig. 9. Variation of crack orientation η with friction coefficient μ for maximum tensile stress on the crack; σ 1 =σ 2 ¼ 10.

Fig. 10. Variation of the stick (c) and slip (a) region with friction coefficient μ for maximum tensile stress on the crack; σ 1 =σ 2 ¼ 10.

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5. Conclusion

Fig. 11. Variation in the ratio of stick and slip region c=a with friction coefficient μ for maximum tensile stress on the crack; σ 1 =σ 2 ¼ 10.

within the theory that has been used. Iterations are carried on along the maximum tensile stress until a real value of c is found. The values of η plotted in Fig. 9 as a function of μ correspond to the maximum tensile stress on the crack with a real value of c. Having made this alteration to get meaningful results, it is found that the results differ from those of [5]. The results of [5] are inconsistent with the stick/slip theory used in the present work as can be seen in Fig. 7. In the second zone 0:5 o μ r 2:7, stick and slip regions are both real, and c=a o1. The results are close to the predictions obtained under the assumption of slip-only condition of the entire crack [5]. The oscillations which are observed are numerical artifacts of the simple method which has been used to solve for the maxima of Eq. (62). There is no stick region at μ ¼0.5, and the stick region just begins to cover the entire region of contact just after μ ¼ 2.7. In the third zone, 2:7 o μ o 1, the stick region completely covers the slip region, and so c=a ¼ 1 (violating the assumption 0 rc o a) meaning σ f ðxÞ-1. This domain can be rejected for its lack of any physical meaning. (2) When 137 o σ 1 =σ 2 o1 ðσ 1 b σ 2 Þ, Eqs. (6) and (7) gives 1 2

ð68Þ

1 2

ð69Þ

σ yy ¼ σ 1 ½ cos ð2ηÞ þ 1 σ xy ¼ σ 1 sin ð2ηÞ

This automatically means that T=F ¼ tan η and thus, through Eq. (57), the half-stick region c is real (and physically meaningful) only when 1  tan η=μ Z0. A more accurate range of such values is shown in the inverse of data in Fig. 7. The case of σ 1 =σ 2 ¼ 1000 has been considered here as an example. The permissible domain of (η, μ) is physically sensible when the region of contact lies within the crack length as an extreme case, meaning 0 r a o l. It is easy to conclude that for 0 r η o π =2, σ 1 =2 r σ yy o σ 1 will follow. Substituting F ¼ 2aσ yy into Eq. (46) gives a p σ yy p σ 1 . Large σ1 for this linear proportionality gives a large value of the region of contact a ¼7.23 for the present case, which does not make sense, as it violates the initial assumption of 0 r a o l, where l ¼1 is the half-crack length. It has  þ been verified that a ¼ l for σ 1 =σ 2 ¼ 137 and a ¼ l for σ 1 =σ 2 ¼ 138. Therefore, this range of the loading ratio of principal stresses falls outside the scope of the stick-slip model of rock fracture.

There is a two-tier generalized conclusion of the present work. Firstly, the stick/slip model gives physically meaningful results only for a ratio 1 o σ 1 =σ 2 r γ where γ is a positive real number dependent on the ratios E=σ 2 and l=R at the contact point, but independent of the Poisson's ratio. In the present study, γ ¼137 has been found. Secondly, within 1 o σ 1 =σ 2 r γ , there is a range of κ o μ r ω where results of stick/slip theory are consistent with the results of the sliding crack model, meaning that the assumption that there is only slipping between the crack faces or that friction acts at its maximum static value is valid in this zone. The range of ω o μ o 1 gives unrealistic infinite σf, and thus is not physically relevant. The maximum tensile stress in the range 0 r μ o κ corresponds to imaginary stick regions. Thus, an alteration has been made in the present work to consider only those maximum tensile stress values that correspond to real stick regions. For the case of σ 1 =σ 2 ¼ 10, κ ¼0.5 and ω ¼2.7 have been observed in the present study. Overall, the stick/slip contact mechanics model, when applied to Griffith's crack growth criterion, gives meaningful results only for certain ranges of principal stress ratios and friction coefficients, as opposed to the entire real range of the previously known sliding crack model. A physical interpretation of what happens in reality for the entire real domain of parameters has been proposed.

Acknowledgments This work was funded by the Rio Tinto Centre for Advanced Mineral Recovery at Imperial College London.

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