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Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach
Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
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Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach X.P. Zhou ⇑, H.Q. Yang State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, PR China School of Civil Engineering, Chongqing University, Chongqing 400045, PR China
a r t i c l e
i n f o
Article history: Received 14 March 2017 Revised 3 May 2017 Accepted 3 May 2017 Available online xxxx Keywords: Damage localization Crack-weakened rocks Pseudo traction method Bifurcation of crack growth pattern Strain energy density factor approach
a b s t r a c t The key failure mechanism of rock mass subjected to compressive loads is due to the damage localization. Conventional methods can not be applied to study the dynamic damage localization physical behavior of crack-weakened rock mass. The strain energy method has the capability to investigate the dynamic damage localization feature in crack-weakened rock mass. Therefore, in order to analyze the dynamic damage localization behavior, strain energy density factor approach is adopted in this paper. A two-dimensional model with periodic rectangular noncollinear array of cracks is established. Then, the onset condition of periodic distribution cracks in rock mass is obtained using strain energy density factor approach. By analyzing the bifurcation of crack growth pattern, the critical length and stress of damage localization of crack-weakened rock mass is determined as well as the location of damage localization. In addition, parameters sensitivity analysis is carried out, the effects of the length of crack, friction coefficient, fracture toughness, confining stress, velocity of crack growth, inclination and spacing between lines and rows on the onset condition for damage localization and bifurcation pattern of rocks are discussed. It can be concluded that onset condition of damage localization is mainly affected by its distribution of initial cracks. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction There are many discontinuities, such as bedding planes, joints, shear zones and faults, which are contained in rock mass. In many cases, the broken of rock mass is mainly controlled by the distribution of discontinuities in the rock mass. The key factor of rock failure is the damage localization process of microcracks in rock mass under compressive stress. Therefore, damage localization can serve as a precursor to the failure of rock mass. It is pointed out that the failure process of rock mass consists of the following three major stages, as shown in Fig. 1. Firstly, the steady state of stress accumulation appears, in which all cracks uniformly grow. Secondly, the alternative evolution of periodic microcracks occurs when they uniformly grow to some extent.
⇑ Corresponding author at: State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, PR China. E-mail addresses: [email protected], [email protected] (X.P. Zhou).
That is to say, the crack growth pattern bifurcation is characterized as only parts of cracks keeping propagation while others keeping still. Thirdly, some cracks coalesce with each other until rock mass fails. This process is called damage localization of rock mass. Based on these above observations, it can be concluded that damage localization of rock mass results from the bifurcation of crack growth pattern. Consider an infinite array of two-dimensional cracks of the same size in Fig. 1, all cracks grow simultaneously under far field applied loads when crack interaction is neglected. However, the bifurcation of crack growth pattern occurs when the effect of crack interaction is considered. Therefore, crack interaction is key factor for the occurrence of bifurcation and the onset condition for damage localization. That is to say, a bifurcation of the crack growth pattern emerges as soon as the critical condition is reached [2,3]. Various theoretical models were developed to analyze damage localization, these models can be simply classified into three kinds. The first kind of these models is based on the statistical microdamage theory. In order to reveal the underlying mesoscopic mechanism governing the experimentally observed failure in solids
http://dx.doi.org/10.1016/j.tafmec.2017.05.006 0167-8442/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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Increase loads
(1) Uniform growth state
2. Pseudo traction acting on the crack surface and stress intensity factor
(2) Bifurcation
2.1. The basic model
(3) Crack coalescence state Fig. 1. A typical observation of bifurcation of the growth pattern in experiments [1].
subjected to impact loads, some researchers [4,5] put forward a model of statistical microdamage evolution to analyze the macroscopic failure. The second kind of these models was derived from the strain-gradient-enhanced plastic theory. A strain-gradientenhanced damage model was proposed by Zhao et al. [6,7] to analyze the localization of rock-like or concrete-like brittle materials. The third kind of these models was known as the micromechanical damage model. Many efforts were done on this field. For example, considering the rapid stress drop and strain softening of material, the fracture process zone near the tips of a mode-I crack in a brittle damaged material was studied using the Dugdale-Barenblatt model [8]; the complete stress-strain relation for rock-like materials were investigated [9–16]. Although these three kinds of methods can reflect some characters of damage localization in crack-weakened rock mass, the mechanism of damage localization and the onset condition for damage localization still need to be investigated. In this paper, more general equation for noncollinear microcracks is derived based on a simplified analytical model for the bifurcation of collinear microcracks. Here, the possible bifurcation of crack growth pattern in twodimensional doubly periodic noncollinear crack arrays subjected to biaxial compression is investigated. By analyzing the bifurcation of crack growth pattern, the critical length and the critical stress for damage localization of crack-weakened rock mass are determined as well as the location of damage localization. Finally, parameters sensitivity analysis is carried out. This paper is organized as follows. In Section 2, the pseudo traction induced by crack interaction is determined. In Section 3, a dynamic fracture criterion is adopted. In Section 4, onset condition for damage localization is derived. In Section 5, parameters sensitivity analysis is carried out.
As shown in Fig. 2, a periodic rectangular noncollinear array of sliding cracks is used to analyze the damage localization of crackweakened rock mass under biaxial compression. Similar to the works by Horii and Nemat-Nasser [17], Kemeny [18], Deng et al. [19], Li [20] and Wang et al. [21], the sliding crack array can be simplified as an array of tensile cracks subjected to a pair of split1 ting forces F and the far field compressive stress r1 1 and r2 . For simplicity, Fig. 2 is further decomposed into configurations in Fig. 3a and b. The actual curvilinear wing crack in Fig. 2 is approximated by a straight opened crack growing parallel to the direction of the maximum principal compressive stress r1 1 . In Fig. 3a, an array of tensile cracks with the length of 2l, the horizontal spacing of H, and the vertical spacing of 2w is loaded by a pair of splitting forces T at its center. In Fig. 3b, an array of tensile cracks is subjected to axial compressive stress r1 1 and confining pressure r1 2 . Considering an elastic body containing an array of cracks, which is arranged as M rows and N columns, the periodic rectangular array of cracks in Fig. 3a is labeled by its location, such as the crack at the bottom left corner labeled by (1, 1), while the crack at the top right corner labeled by (M, N). The local Cartesian coordinate system of each cracks is (oij, x2ij, x1ij) in Fig. 3. As shown in Fig. 3, the spacing between adjacent rows of the same column is defined as d, while the spacing between adjacent columns of the same row is defined as b. The following three main assumptions are adopted in this model. Firstly, before bifurcation occurs, all the lengths of initial cracks are assumed to be equal to a, and all the lengths of wing cracks are 2l. Meanwhile, the dip angle of initial crack with respect to the maximum principal compressive stress r1 1 is h. Secondly, before bifurcation occurs, wing cracks always propagate along the direction of the maximum principal compressive stress r1 1 . Thirdly, it is assumed that the loading rate is not fast enough to consider the effects of cracks on the transmission of stress wave. These three assumptions were also adopted by the previous literatures [16,20].
Fig. 2. The periodic rectangular array of sliding cracks.
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
Fig. 3. The simplified periodic rectangular array of cracks.
During the phase of the activation of frictional sliding on the faces of the pre-existing cracks, the effect of the interaction among cracks on the deformation of rock mass can be neglected. The expression of the resolved stress acting on the initial crack surface is written as
(
1 ss ¼ cos h sin hðr1 1 r2 Þ 2 1 2 rn ¼ sin hr1 þ cos hr1 2
ð1Þ
It is assumed that the frictional contact on the initial crack surface is of the Mohr-Coulomb type. The effective shear stress driving the frictional slip on crack surface can be expressed as
s ¼ ss lrn
Based on the pseudo-traction methods, the original problem in Fig. 2 can be decomposed into two sub-problems. One subproblem is about crack array loaded by splitting forces, as shown in Fig. 3a. While the other sub-problem is about crack array loaded by far field compressive stress, as shown in Fig. 3b. In each sub-problem, each crack is loaded by unknown pseudo-tractions ðkÞ rðkÞ pq and spq , which are induced by the existence of the other
crack. The pseudo-tractions can be expanded into Taylor series as [19]
rpq ispq ¼
ð2Þ
k 1 X xpq ðkÞ ðkÞ ðrpq ispq Þ lpq k¼0
ð7Þ
where l is the coefficient of dry friction. The force T defines the effect of the pre-existing crack sliding on the tensile crack, which can be estimated by
where k is the terms of Taylor series, rpq and spq are n term components of the pseudo-tractions.
1 T ¼ aw1 r1 1 aw2 r2
2.2. Cracks loaded by splitting forces
ð3Þ 2
where w1 ¼ sin hðsin 2h 2l sin hÞ; w2 ¼ sin hðsin 2h þ 2l cos2 hÞ. Considering the effects of crack interaction, the equivalent point force, which drives the growth of initial cracks located at the pth row and qth column, can be written as 1 F pq ¼ a½w1 ðr1 1 r1pq Þ w2 ðr2pq þ r2 Þ
ð4Þ
where r2pq and r1pq are the pseudo traction on the crack (p, q) along X and Y direction, respectively. For the model plotted in Fig. 3, the distance between arbitrary two crack centers ðp; qÞ and ðr; sÞ can be expressed as
dpqrs
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ ðjs qjH þ jr pjdÞ þ ð2jr pjw þ js qjbÞ
ðkÞ
Considering the crack array plotted in Fig. 3a, cracks are only loaded by splitting forces. Only 2N term of pseudo traction is taken into account. Before bifurcation of crack growth pattern occurs, the pseudo traction acting on each crack is the same. Moreover, the pseudo traction acting on each crack is symmetric, it is easy to find that the odd term components of the pseudo-tractions are equal to zero [19]. Then, we have ðkÞ rpq ¼
ð6Þ where j:j represents the absolute value.
r¼1 s¼1
r¼1 s¼1
M X N M X N X T sin h X T cos h þ ðEpqrs ðF pqrs þ k Þ k Þ p l pl r¼1 s¼1 r¼1 s¼1
The angle of connection line between arbitrary crack ðp; qÞ and ðr; sÞ with respect to the initial crack surface can be estimated by
arspq
" # 1 X M X N M X N X X pqrs ðmÞ ðmÞ ðApqrs Þ r þ ðB Þ s rs rs km km m¼0
ð5Þ
8 i h > < arctan ððsqÞHþðrpÞdÞ ½2ðr pÞw þ ðs qÞb P 0 2ðrpÞwþðsqÞb i h ¼ ððsqÞHþðrpÞdÞ > : p arctan ½2ðr pÞw þ ðs qÞb < 0 2ðrpÞwþðsqÞb
ðkÞ
ðkÞ spq ¼
1 M X N M X N X X X ðmÞ ðmÞ ðC pqrs ðDpqrs km Þrrs þ km Þsrs m¼0
r¼1 s¼1
ð8Þ
!
r¼1 s¼1
M X N M X N X T sin h X T cos h ðGpqrs ðHpqrs þ þ k Þ k Þ p l pl r¼1 s¼1 r¼1 s¼1
ð9Þ
where k; m ¼ 0; 1; 2 . . . 1; ðp; qÞ – ðr; sÞ。
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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Apqrs k2m
¼ gm
k X 1
l pqrs
d
Apqrs k2m1 gm ¼
rð0Þ
¼ gm
l pqrs
d
2x hkx l apqrs pqrs k2x mþx d x¼1
k X 1
2xþ1 hkx ð2x þ kÞ l apqrs pqrs k2xþ1 2xðm þ xÞ d x¼1
ð2mÞ! 22mþ1 ðm!Þ2
\ ð10Þ
sð2Þ ¼
M X N X ðB20 rð0Þ þ D20 sð0Þ þ B22 rð2Þ þ D22 rð2Þ þ F 2 T sin h þ H2 T coshÞ r¼1 s¼1
ð18Þ
ð11Þ
When the components of the pseudo-tractions containing terms of
ð12Þ
is neglected, we have orders higher than ðl=d Þ A20 ¼ B20 ¼ D20 ¼ E2 ¼ F 2 ¼ H2 0, and other coefficients are determined by Eqs. (a8)(a16) in Appendix A. From Eqs. (15) to (18), rp0 , rp2 , sp0 and sp2 can be defined as
pqrs 2
P P P PN PM PN 2 PM PN M N 2 cos h M r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 A00 D00 ¼ T PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
ð19Þ
rð2Þ ¼ 0
sð0Þ
P P P PN PM PN 2 PM PN M N 2 sin h M r¼1 s¼1 B00 þ 2 cos h r¼1 s¼1 D00 þ r¼1 s¼1 B00 r¼1 s¼1 A00 D00 ¼ T PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
hkx ¼ ð1Þk
ðk þ 2x 1Þ! 22x1 k!½ðx 1Þ!2
ð13Þ
ð21Þ
sð2Þ ¼ 0
ð22Þ
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array can be expressed as
pqrs apqrs þ napqrs nm ¼ ðn þ 2Þ cos½ma
ðm þ nÞ cos½mapqrs þ ðn þ 2Þapqrs þ m cos½mapqrs þ napqrs ð14Þ pqrs pqrs In Eqs. (8) and (9), Bpqrs nm , C nm and Dnm can be calculated by replacing pqrs pqrs pqrs apqrs by b , c and d in Eqs. (10) and (11), the detailed expresnx nx nx nx
K pq Ia
ð20Þ
K pq Ia ¼
1 pffiffiffiffiffi T sin h X pl 2g k rpq þ 2k pl k¼0
!
Substituting Eq. (19) and Eq. (20) into Eq. (23), we have
! pffiffiffiffiffi T sin h 2 cos hPM PN B00 þ 2 sin hðPM PN A00 þ PM PN B2 PM PN ðA00 D00 ÞÞ r¼1 s¼1 r¼1 s¼1 r¼1 s¼1 00 r¼1 s¼1 ¼ pl T PM PN PM PN 2 PM PN PM PN pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
sions are derived from Eqs. (a1) to (a7) in Appendix A. pqrs For periodic crack array in Fig. 3, we have bnm ¼ cpqrs nm , pqrs pqrs pqrs Bpqrs ¼ C and G ¼ F . 2n2m 2n 2n2m 2n For simplification, only 2 (N = 1) term components of the pseudo-tractions are considered in this paper. Combining Eq. (8) with Eq. (9), we have
rð0Þ ¼
ð15Þ \
rð2Þ ¼
M X N X ðA20 rð0Þ þ B20 sð0Þ þ A22 rð2Þ þ B22 rð2Þ þ E2 T sinh þ F 2 T coshÞ r¼1 s¼1
ð16Þ
\
sð0Þ ¼
M X N X
ðB00 rð0Þ þ D00 sð0Þ þ B02 rð2Þ þ D02 rð2Þ þ F 0 T sin h þ H0 T coshÞ
r¼1 s¼1
ð17Þ
ð24Þ
2.3. Noncollinear crack array loaded by compressive stress Only 2N term pseudo tractions are taken into account, we have
rð0Þ ¼
M X N X ðA00 rð0Þ þ B00 sð0Þ þ A02 rð2Þ þ B02 sð2Þ þ A00 r1 2 Þ
ð25Þ
r¼1 s¼1
M X N X ðA00 rð0Þ þB00 sð0Þ þA02 rð2Þ þB02 rð2Þ þE0 T sinhþF 0 T coshÞ r¼1 s¼1
ð23Þ
rð2Þ ¼
M X N X ðA20 rð0Þ þ B20 sð0Þ þ A22 rð2Þ þ B22 sð2Þ þ A20 r1 2 Þ
ð26Þ
r¼1 s¼1
sð0Þ ¼
M X N X ðB00 rð0Þ þ D00 sð0Þ þ B02 rð2Þ þ D02 sð2Þ þ B00 r1 2 Þ
ð27Þ
r¼1 s¼1
sð2Þ ¼
M X N X ðB20 rð0Þ þ D20 sð0Þ þ B22 rð2Þ þ D22 sð2Þ þ B20 r1 2 Þ
ð28Þ
r¼1 s¼1
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
From Eqs. (25) to (28),
rð0Þ
rp0 , rp2 , sp0 and sp2 can be defined as
P PN 1 1 M r¼1 s¼1 D00 r2 ¼ r1 PM PN 2 PM PN PM PN PM PN 2 r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
sð0Þ ¼ PM PN r¼1
s¼1 A00 þ
5
PM PN
r1 2
2 s¼1 B00
r¼1
ð29Þ
PM PN r¼1
1þ
s¼1 B00 PM PN r¼1
s¼1 D00
ð30Þ
PM PN
s¼1 A00 D00
r¼1
rð2Þ ¼ 0
ð31Þ
where K ID is the mode I dynamic SIF, K I is the mode I static SIF, kð_lÞ is a function of crack growth velocity _l. The mode II dynamic SIF has
sð2Þ ¼ 0
ð32Þ
the same pattern.
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array can be written as
K pq Ib ¼ rpqb
1 X pffiffiffiffiffi pffiffiffiffiffi pl ¼ plðr1 2g k rpq 2 þ 2k Þ
ð33Þ
k¼0
Substituting Eq. (29) and Eq. (31) into Eq. (33), we have
K pq Ib
As it is very difficult to determine function kð_lÞ precisely, several approximate function was presented [14]. The crack array depicted in Fig. 3a and b are mixed mode, both mode I and mode II SIF _ and k2 ðlÞ _ are approximate. should be considered. k1 ðlÞ It was pointed out that the relationship between the growth velocity of crack and stress intensity factor can be approximately given by [19]
1 P PN 1 1 M r¼1 s¼1 D00 r2 A PM PN 2 PM PN PM PN s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
0 pffiffiffiffiffi ¼ pl@PM PN r¼1
Combining Section 2.2 with Section 2.3, the stress intensity factor of crack array in Fig. 2 under static loads can be expressed as
pffiffiffiffi _l ¼ cR 1:5K Ia þ 1:75K Ib 1:25K IC X 1:5K Ia þ 1:75K Ib 0:75K IC
P P 1 0 PM PN PM PN 2 PM PN M N pffiffiffiffiffi T sin h 2 cos h r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ K I ¼ pl@ TA PM PN PM PN 2 PM PN PM PN pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00 0 1 P PN 1 1 M pffiffiffiffiffi r¼1 s¼1 D00 r2 A pl@PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
Under dynamic loads, crack growth velocities have a great influence on the dynamic SIF. The dynamic SIF can be deduced from static SIF by adding special items related to the growth velocity of crack. In most cases, the mode I dynamic SIF can be expressed as
K ID
ð37Þ
ð35Þ
X ¼ ½1:5K Ia þ 1:75K Ib 1:25K IC 2 4ðK Ia þ K Ib K IC Þð0:5K Ia
3. Dynamic stress intensity factor and fracture criterion
K ID ðtÞ ¼ kð_lÞK I
ð34Þ
ð36Þ
þ 0:75K Ib 0:375K IC Þ
ð38Þ
where K IC is mode I fracture toughness of rocks. For crack array in Fig. 2, from Eq. (35), the stress intensity factor can be described as
P P 1 0 P PN PM PN 2 PM PN M N 2 cos h M pffiffiffiffiffi r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ T sin h ¼ pl @ TA PM PN PM PN 2 PM PN PM PN pl cR 0:75_l r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00 0 1 P PN 1 1 M r¼1 s¼1 D00 r2 cR _l pffiffiffiffiffi@ A pl PM PN P PN 2 P PN P PN A00 þ M B 1þ M D00 M A00 D00 cR 0:5_l cR _l
r¼1
s¼1
r¼1
s¼1 00
r¼1
s¼1
r¼1
ð39Þ
s¼1
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
The strain energy density factor of rocks is chosen as the crack growth criterion. The strain energy density factor of rocks can be written as [22,23]
S ¼ a11 K 2ID þ 2a12 K ID K IID þ a22 K 2IID þ a33 K 2IIID
ð40Þ
where K ID , K IID and K IIID are the mode I and mode II and mode III dynamic stress intensity factors, respectively. The coefficients of could be found in Refs. [22,23]. The onset of rapid crack propagation is assumed to start when max the strain energy density Smin associated with ðdW=dVÞmin reaches a critical value, i.e.
Smin ¼ SDc
ð41Þ
where SDc ¼ rC
dW
energy density
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array in Fig. 2 can be written as
cR 0:75_l
cR _l
pffiffiffiffiffi F pq pffiffiffiffiffi ðr2pq þ r1 2 Þ pl ¼ K ID pl cR 0:5_l
ð42Þ
the principal stress can be obtained as
r2pq
r1pq
Eq. (44) can be rewritten as
r1pq ¼ r1 1
vÞ D 2 ¼ ð1þv2Þð12 ðK IC Þ is the critical dynamic strain pE dV c factor, K DIC is dynamic fracture toughness of rocks.
4. Onset condition for damage localization of crack-weakened rock mass
cR _l
Before bifurcation of the crack growth pattern occurs, the length of each wing crack is the same. Considering the variation of Eq. (44), we can obtain P PN 1 2 ½2 cosð2apqrs Þ cosð4apqrs Þ, where f 1 ¼ M r¼1 s¼1 dpqrs 4 P PN 2 1 ½sinð4apqrs Þ sinð2apqrs Þ , f2 ¼ M r¼1 s¼1 dpqrs 2 P PN 1 cosð4apqrs Þ, f3 ¼ M r¼1 s¼1 dpqrs PM PN 1 4 f 4 ¼ r¼1 s¼1 dpqrs ½2 cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ.
w2 ðr2pq þ r1 2 Þ w1 2
2
4
4
T sin h 2 cos hf 5 l þ 2 sin hðf 1 l þ f 2 l f 4 l Þ T 2 4 2 4 w1 a pl f 1l þ f 2l 1 þ f 3l f 4l
pl
!
ð47Þ where f 5 ¼
PM PN r¼1
s¼1
1 dpqrs
2
½sinð4apqrs Þ sinð2apqrs Þ.
Substituting Eq. (47) into Eq. (42) and Eq. (41), we have
" # pffiffiffiffiffi T sin h 2 cos hf 5 l2 þ 2 sin hðf 1 l2 þ f 2 l4 f 4 l4 Þ pl T 2 4 2 4 pl f 1l þ f 2l 1 þ f 3l f 4l pffiffiffiffiffi D k2 ð_lÞðr2pq þ r1 2 Þ pl ¼ K IC
k1 ð_lÞ
P PN 1 1 M r¼1 s¼1 D00 r2 ¼ PM PN r1 PM PN 2 PM PN PM PN 2 r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
P P 1 0 P PN PM PN 2 PM PN M N 2 cos h M r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ w2 p l T sin h 1 @ ¼ ðr2pq þ r1 TA PM PN PM PN 2 PM PN PM PN 2 Þ r1 þ w1 a w1 pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
ð48Þ
ð43Þ
ð44Þ
After bifurcation of crack growth pattern occurs, the stress field can be defined as
9
rð2Þ 2pq ¼ r2 þ dr2pq > > = rð2Þ 1pq ¼ r1 þ dr1pq > ð2Þ lpq
¼ l þ dlpq
ð45Þ
> ;
where d represents the variation of variable.
dr2pq
PM PN
2
( " 2 M X N X 1 ¼ dlrs 2l pqrs ½2 cosð2apqrs Þ cosð4apqrs Þ 2 4 2 2 4 d ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 r¼1 s¼1 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 #) 4 2 4 1 1 1 3 3 pqrs pqrs 2 pqrs pqrs pqrs pqrs ½sinð4a Þ sinð2a Þ þ 2l pqrs cosð4a Þ 4l ½2 cosð2a Þ cosð4a Þ cosð4a Þ þ4l pqrs pqrs d d d 2r1 2 l
r¼1
1 s¼1 ½dlrs dpqrs
cosð4apqrs Þ
r1 2 ð1 f 3 Þ
ð46Þ
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Considering the variation of Eq. (48), we can obtain ( " 2 M X N _ X K ICD dlrs k1 ðlÞTdl k1 ð_lÞT 1 rs sin h dl ½sinð4apqrs Þ sinð2apqrs Þ pffiffiffiffiffi rs 4l cos hdlrs pqrs 2 5 3 _ pl d 2k2 ðlÞl k2 ð_lÞpl k2 ð_lÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l r¼1 s¼1 " ##) 2 4 4 1 1 1 3 3 2 þ4sin h l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ þ 2l ½sinð4apqrs Þ sinð2apqrs Þ 2l ½2cosð2apqrs Þ cosð4apqrs Þcosð4apqrs Þ pqrs pqrs d d d ( " ! 4 4 2 2 4 4 M X N _ X k1 ðlÞT½2cos hl f 5 þ 2sin hðf 1 l þ f 2 l f 4 l Þ 1 1 3 pqrs pqrs 2 pqrs pqrs pqrs þ dl 4l ½sinð4 a Þ sinð2 a Þ ½2 cosð2 a Þ cosð4 a Þ cosð4 a Þ rs pqrs pqrs 2 5 3 d d r¼1 s¼1 k2 ð_lÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l !#) 2 2 1 1 ½2cosð2apqrs Þ cosð4apqrs Þ þ pqrs cosð4apqrs Þ ð49Þ þ2l pqrs d d
dr2pq ¼
Combining Eq. (46) with Eq. (48), after a series simplification, we can obtain following matrix form, the expanded expression is written as Eq. (a17) in Appendix A.
2 6 6 6 6 6 4
Q 11 11
Q 11 12
Q 12 11 .. . Q MN 11
Q 12 12 .. . Q MN 12
32
3 dl11 7 6 7 dl12 7 Q 12 7 MN 76 6 . 7¼0 .. .. 7 .. 7 76 5 4 . . 5 MN dlMN Q MN Q 11 MN
ð50Þ
As far as Eq. (50), the element Q pq rs represents the effects of the crack ðr; sÞ on the crack ðp; qÞ. The detailed expression can be described as Eq. (a18) in Appendix A. When Eq. (50) is singular, Eq. (50) has solutions which are not equal to zero. By solving Eq. (50), the critical length of wing crack lcr can be obtained when damage localization occurs. The matrix in Eq. (50) has a eigenvalue which is equal to zero. Moreover, the location of damage localization can be determined through the eigenvector of the matrix in Eq. (50) corresponding to the zero eigenvalue [3,16]. When the value in the eigenvector is plus, it is pointed out that the crack in the corresponding location keeps propagating after damage localization occurs. However, if the value in the eigenvector is minus, it is pointed out that the crack in the corresponding location stops propagating after damage localization occurs. When the confining pressure keeps constant, the critical stress for damage localization can be estimated from Eq. (50) as
r
1 1cr
fracture toughness, confining pressure and loading rate on the onset condition of damage localization and the bifurcation pattern of crack growth, Fig. 4 is plotted. In Fig. 4, the total number of cracks is 6. The length of preexisting crack is a, and each crack is the same value of length and inclination. 5.1. Dependence of the dimension of cracks on damage localization The effect of the length and inclination of cracks on damage localization is illustrated in this section. The following material parameters come from some experimental tests for granite: the confining pressure r2 = 1 MPa, the frictional coefficient of crack surface l = 0.5, dimension parameters for crack distribution H = 0.1 m, w = 0.18 m, d = 0.01 m, b = 0.005 m, fracture toughness KIC = 0.7 MPam1/2, the velocity of Rayleigh wave cR = 2000 m/s and the critical growth velocity for single crack l_ = 0.00001 m/s. Fig. 5 shows the effect of the length of crack on the critical axial stress. It is found from Fig. 5 that the critical stress of damage localization decreases as the crack length increases. When h = 40, 50 and 60, the critical length of damage localization is 0.5 mm, 0.6 mm and 0.72 mm, respectively. It can be observed from Fig. 6 that the critical length increases with increasing h. In addition, it is implied that the critical length doesn’t vary with the length of preexisting crack, when other conditions is fixed.
8 0 19 P PN 1 = 1 M r¼1 s¼1 D00 r2 1 < cR _l pffiffiffiffiffiffiffiffi@ A ¼ K ICD þ plcr PM PN PM PN 2 PM PN PM PN _ ; aw1 : cR 0:5l r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
P P 11 0 P PN PM PN 2 PM PN M N 2 cos h M _ r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ sin h w R 0:75l A pcffiffiffiffiffiffiffiffi þ 2 r1 @ PM PN PM PN 2 PM PN PM PN _ plcr w1 2 A þ B 1 þ D A D p l lÞ ðc 00 00 00 00 cr R r¼1 s¼1 r¼1 s¼1 00 r¼1 s¼1 r¼1 s¼1
where
r1 1cr is the onset stress for damage localization.
5. Sensitivity analysis In order to illustrate the dependence of the crack length, inclination, spacing between rows and columns, frictional coefficients,
ð51Þ
Fig. 7 shows the effect of h on the critical stress. In can bee concluded from Fig. 7 that when h is smaller than 45, the critical stress decreases as h increases. However, when h is bigger than 45, the critical stress increases as h increases. The critical stress is minimum when h is equal to 45. Fig. 8 shows the crack growth pattern after bifurcation occurs. It can be observed from Fig. 8 that when damage localization occurs, only some cracks keep propagating while others keep still. Three
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx 0.75 0.7
lcr /mm
0.65 0.6 0.55 0.5 0.45 0.4 0.35 20
30
40
50
60
70
θ /Deg. Fig. 6. Dependence of the critical length on h.
50 45
Fig. 4. Sketch for calculation model.
a=0.05
40
σ1 /MPa
35
50
σ1 /MPa
45
θ
30 25 a=0.03
20
a=0.05
15
60°
40
10
35
5
30
0 20
25
30
40
50
60
70
θ /Deg.
20 θ θ
15
Fig. 7. Dependence of the critical axial stress on h.
50° 40°
10 5 0 0
0.01
0.02
0.03
0.04
0.05
0.06
a /m Fig. 5. Dependence of the critical axial stress on the length of crack.
main conclusions are obtained from Fig. 8. Firstly, it is implied that the length of preexisting crack has no influence on the crack growth pattern after damage localization occurs. Secondly, the location of damage localization is related to the inclination of pre-existing cracks. Thirdly, when h is smaller than 45, the number of growing cracks decreases with increasing h after damage localization occurs. However, when h is bigger than 45, the number of growing cracks increases with increasing h after damage localization occurs. The critical stress is minimum when h is equal to 45.
5.2. Dependence of the distribution of crack array on damage localization The effect of the spacing between rows and columns of crack on damage localization is illustrated in this section. The following material parameters are used in computations: r2 = 1 MPa, a = 0.04 m, h = 45, l = 0.5, d = 0.01 m, b = 0.005 m, K = 0.7 MPam1/2, cR = 2000 m/s and l_ = 0.00001 m/s. IC
Figs. 9 and 10 respectively show the dependence of the critical axial stress on w and H. It can be concluded from Figs. 9 and 10 that when the spacing between cracks is small, the critical axial stress
increases with increasing the spacing. However, the critical axial stress keeps constant when the distance between cracks is far enough. Figs. 11 and 12 show the dependence of the critical axial stress on d and b, respectively. It can be concluded from Fig. 11 that when d is smaller than 0.025 m, the critical stress of damage localization decreases as d increases. However, d is bigger than 0.025 m, the critical stress of damage localization increases as d increases. The critical stress is minimum when d is equal to 0.025 m. It can be found from Fig. 12 that when b is smaller than 0.02 m, the critical stress of damage localization increases as b increases. However, b is bigger than 0.02 m, the critical stress of damage localization decreases as b increases. The critical stress is maximum when b is equal to 0.02 m. Figs. 13 and 14 show the dependence of the critical length on w and H, respectively. It can be concluded from Figs. 13 and 14 that when the spacing between cracks is small, the critical length increases with increasing the spacing. However, when the distance between cracks is far enough, the critical length keeps constant. Figs. 15 and 16 show the dependence of the critical length on d and b, respectively. It can be observed from Fig. 15 that when d is smaller than 0.025 m, the critical length of damage localization decreases as d increases. However, when d is bigger than 0.025 m, the critical length of damage localization increases with increasing d. The critical length is minimum when d is equal to 0.025 m. It can be found from Fig. 16 that when b is smaller than 0.02 m, the critical length of damage localization increases with increasing b. However, when b is bigger than 0.02 m, the critical length of
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30
30 , a=0.03 m
45
55 , a=0.03 m
Fig. 8. Sketch of the bifurcation growth model of cracks with different h and a.
16 17
14 15 13
10
σ1 /MPa
σ1/MPa
W=0.18 m
H=0.2 m
12
H=0.15 m
8
11 W=0.12 m
9 7
H=0.1 m
6
W=0.06 m
5
4 0.03
0.08
0.13
0.18
0.23
0.28
w /m
3 0.04
0.14
0.24
0.34
0.44
H /m
Fig. 9. Dependence of the critical axial stress on w.
Fig. 10. Dependence of the critical axial stress on H.
damage localization decreases as b increases. The critical length is maximum when b is equal to 0.02 m. Fig. 17 shows the crack growth pattern after bifurcation occurs. It can be seen from Fig. 17 that when w and H are small, the num-
ber of growing crack decreases with increasing the spacing after damage localization occurs. However, when w and H are far enough, the number of growing crack increases with increasing the spacing after damage localization occurs.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
9.45
12
9.4
10
9.35
8
lcr /mm
σ1 /MPa
10
9.3
W=0.18 m 6
9.25
4
9.2
2
W=0.12 m W=0.06 m
9.15
0
0.01
0.02
0.03
0.04
0.05
0 0.04
0.06
0.14
0.24
d /m
0.34
0.44
H /m
Fig. 11. Dependence of the critical axial stress on d.
Fig. 14. Dependence of the critical length on H.
1.92
9.36 9.34
1.88
9.32
lcr /mm
σ1/MPa
9.3 9.28 9.26
1.84
1.8
9.24 9.22
1.76
9.2
1.72
9.18 0
0.01
0.02
0.03
0.04
0
0.05
0.01
0.02
0.03
0.04
0.05
0.06
d /m
b /m Fig. 12. Dependence of the critical axial stress on b.
Fig. 15. Dependence of the critical length on d.
8
1.88
7 1.84
5
lcr /mm
lcr /mm
6
H=0.2 m
4
1.8
3 2 1 0
0.03
1.76
H=0.15 m H=0.1 m 0.08
0.13
0.18
0.23
0.28
w /m Fig. 13. Dependence of the critical length on w.
Fig. 18 shows the crack growth pattern at different b and d after bifurcation occurs when w = 0.12 m and H = 0.2 m. It can be found from Fig. 18 that the number of growing cracks doesn’t vary with b and d after damage localization occurs. However, the location of damage localization is related to b and d.
1.72
0
0.01
0.02
0.03
0.04
0.05
d /m Fig. 16. Dependence of the critical length on b.
5.3. Dependence of frictional coefficients and fracture toughness on damage localization The effect of the frictional coefficients l and fracture toughness KIC on damage localization is illustrated in this section.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
11
Fig. 17. Sketch of the bifurcation growth model of cracks with different w and H.
The following material parameters are used in computations:
r2 = 1 MPa, H = 0.1 m, a = 0.04 m, h = 45, w = 0.16 m, d = 0.01 m,
b = 0.005 m, cR = 2000 m/s, _l = 0.00001 m/s. Fig. 19 shows the dependence of the critical stress for damage localization on the frictional coefficients l when KIC = 0.12 MPam1/2. It can be observed from Fig. 19 that the critical stress of damage localization increases with increasing the frictional coefficients l. Fig. 20 shows the dependence of the critical stress of damage localization on the fracture toughness KIC when l = 0.5. It can be observed from Fig. 20 that the critical stress of damage localization increases with increasing the fracture toughness KIC. Fig. 21 shows the dependence of the crack growth pattern on different frictional coefficients l and fracture toughness KIC after bifurcation occurs. It can be found from Fig. 21 that the number of growing cracks doesn’t vary with the frictional coefficients l and fracture toughness KIC after damage localization occurs. 5.4. Dependence of loading condition on damage localization The effect of the velocity of crack growth l_ and the confining pressure r2 on damage localization is illustrated in this section. The following material parameters are used in computations: l = 0.5, H = 0.1 m, a = 0.04 m, h = 45, w = 0.16 m, d = 0.01 m, b = 0.005 m, cR = 2000 m/s, KIC = 0.7 MPam1/2. Fig. 22 shows the dependence of the critical stress for damage localization on the confining pressure r2 when l_ = 105 m/s. It can be seen from Fig. 22 that the critical stress of damage localization increases with increasing the confining pressure r2. Fig. 23 shows the dependence of the critical stress of damage localization on the velocity of crack growth _l when
r2 = 1 MPa. It can be observed from Fig. 23 that the critical stress of damage localization increases with increasing the velocity of crack growth _l. Moreover, the numerical results show that the number of growing cracks doesn’t vary with the velocity of crack growth _l and the
confining pressure r2 after damage localization occurs. The crack growth pattern in this example after damage localization occurs is the same as Fig. 21. 6. Conclusions By analyzing the bifurcation of crack growth pattern, the critical length and stress of damage localization of crack-weakened rock mass are determined as well as the location of damage localization. In addition, parameters sensitivity analysis is carried out. The main conclusions are summarized as follows: (1) Damage localization is induced by the bifurcation of crack growth pattern. After damage localization occurs, only some cracks keep propagating while others keep still. (2) The critical stress of damage localization is related to the length of pre-existing crack, inclination, the spacing between rows and columns, frictional coefficient on the crack surface, fracture toughness of rocks, the confining pressure and the loading rate. (3) The crack growth pattern doesn’t concern with the inclination of preexisting crack, the spacing between rows and columns, length of preexisting crack, friction coefficients, fracture toughness, confining pressure and the loading rate after damage localization occurs.
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Fig. 18. Sketch of the bifurcation growth model of cracks with different b and d.
11
35
10
30
9 8
σ1 /MPa
σ1/MPa
25 20 15
7 6 5
10
4
5
3 2
0 0
0.2
0.4
0.6
0.8
μ Fig. 19. Dependence of the critical axial stress on l.
1
0
0.5
1
1.5
2
1/2
KIC /MPa•m
Fig. 20. Dependence of the critical axial stress on KIC.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
Appendix A
pqrs
bnm ¼ ðn þ 2Þ sin½mapqrs þ napqrs þ ðm þ nÞ sin½mapqrs þ ðn þ 2Þapqrs ðm 2Þ sin½mapqrs þ napqrs ða1Þ pqrs cpqrs þ napqrs þ ðm þ nÞ sin½mapqrs þ ðn þ 2Þapqrs nm ¼ n sin½ma
m sin½mapqrs þ napqrs
ða2Þ
pqrs
dnm ¼ n cos½mapqrs þ napqrs þ ðm þ nÞ cos½mapqrs þ ðn þ 2Þapqrs ðm 2Þ cos½mapqrs þ napqrs Epqrs ¼ n
d
2m hnm l apqrs pqrs n2m 2m 1 d m¼1
pqrs
F pqrs ¼ n
n X 1
l
n X 1
l
2m hnm l pqrs bn2m pqrs 2m 1 d m¼1
pqrs
d
ða3Þ
ða4Þ
ða5Þ
Fig. 21. Sketch of the bifurcation growth model of cracks.
¼ Gpqrs n
30
σ1 /MPa
2m hnm l cpqrs pqrs n2m 2m 1 d m¼1
pqrs
d
25
¼ Hpqrs n
20
n X 1
l
d
l
n X 1
2m hnm l pqrs dn2m pqrs 2m 1 d m¼1
pqrs
ða6Þ
ða7Þ
15 10
A00 ¼
2 1 l ½2 cosð2apqrs Þ cosð4apqrs Þ 2 dpqrs
ða8Þ
A02 ¼
2 1 l 1 ½2 cosð2apqrs Þ cosð4apqrs Þ ¼ A00 8 dpqrs 4
ða9Þ
B00 ¼
2 1 l ½sinð4apqrs Þ sinð2apqrs Þ 2 dpqrs
ða10Þ
B02 ¼
2 1 l 1 ½sinð4apqrs Þ sinð2apqrs Þ ¼ B00 8 dpqrs 4
ða11Þ
D00 ¼
2 1 l cosð4apqrs Þ 2 dpqrs
ða12Þ
D02 ¼
2 1 l 1 cosð4apqrs Þ ¼ D00 8 dpqrs 4
ða13Þ
5 0
0
2
4
6
8
10
σ2 /MPa Fig. 22. Dependence of the critical axial stress on r2.
18 16 cR=800 m/s 14
σ1/MPa
cR=1500 m/s 12 cR=2000 m/s 10 8 6 4 -6.00
-4.00
-2.00
0.00
2.00
4.00
lg(i)/(m/s) Fig. 23. Dependence of the critical axial stress on _l.
Acknowledgements This work was supported by project 973 (Grant no. 2014CB046903), the National Natural Science Foundation of China (Grant nos. 51325903, 51679017, 51409026), Natural Science Foundation Project of CQ CSTC (No. CSTC, cstc2015jcyjys30001, cstc2015jcyjys30006, cstc2013jcyjys30002, cstc2016jcyjys0005).
E0 ¼
l pqrs
d
F0 ¼
l d
l pqrs
d
½2 cosð2apqrs Þ cosð4apqrs Þ ¼ 2A00
ða14Þ
½sinð4apqrs Þ sinð2apqrs Þ ¼ 2B00
ða15Þ
2
pqrs
H0 ¼
2
2 cosð4apqrs Þ ¼ 2D00
ða16Þ
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
2r1 2 l
PM PN r¼1
s¼1
2 1 dlrs dpqrs cosð4apqrs Þ
r1 2 ð1 f 3 Þ
2 4 2 2 4 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ( " 2 M N XX 1 dlrs 2l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d r¼1 s¼1 4 2 1 1 3 2 ½sinð4apqrs Þ sinð2apqrs Þ þ 2l pqrs cosð4apqrs Þ þ4l pqrs d d #) 4 1 3 pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ cosð4a Þ 4l pqrs d _ k1 ðlÞTdlrs sin h k1 ð_lÞT þ þ 2 5 3 _ _ k2 ðlÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l ( k2 ðlÞpl " 2 M X N X 1 dlrs 4l coshdlrs pqrs ½sinð4apqrs Þ sinð2apqrs Þ d r¼1 s¼1 " 2 1 þ4sin h l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ2l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ##) 4 1 3 2l ½2cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ pqrs d 2 2 4 4 k1 ð_lÞT½2cos hl f 5 þ 2sin hðf 1 l þ f 2 l f 4 l Þ 2 5 3 k2 ð_lÞ½ðf ( " 2 f 4 Þl þ ðf 1 þ f 3 Þl l 4 M X N X 1 3 2 dlrs 4l ð pqrs Þ ½sinð4apqrs Þ sinð2apqrs Þ d r¼1 s¼1 ! 4 1 pqrs ½2cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ d !#) 2 2 1 1 pqrs pqrs pqrs þ2l ½2cosð2 a Þ cosð4 a Þ þ cosð4 a Þ ¼0 pqrs pqrs d d ða17Þ
2 1 pqrs 2r1 l cosð4 a Þ pqrs 2 d
r1 2 ð1 f 3 Þ 2 4 2 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ½ðf 1 þ f 3 Þl2 þ ðf 2 f 4 Þl4 1 (" 2 1 2l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ4l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d 2 4 1 1 3 þ2l pqrs cosð4apqrs Þ 4l ½2 cosð2apqrs Þ pqrs d d k1 ð_lÞT sin h cosð4apqrs Þcosð4apqrs Þ þ 2 k2 ð_lÞpl _ k1 ðlÞT þ _lÞ½ðf f Þl5 þ ðf þ f Þl3 l k ð 2 2 4 3 (" 2 1 1 4lcos h pqrs ½sinð4apqrs Þ sinð2apqrs Þ þ 4 sin h d " 2 1 l pqrs ½2 cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ2l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ##) 4 1 3 2l ½2cosð2apqrs Þ cosð4apqrs Þcosð4apqrs Þ pqrs d 2 2 4 4 _ k1 ðlÞT½2 coshl f 5 þ 2 sin hðf 1 l þ f 2 l f 4 l Þ 2 5 3 _ f Þl þ ðf 1 þ f 3 Þl l (" k2 ðlÞ½ðf 2 4 4 1 3 2 4l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ! 4 1 pqrs pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ cosð4a Þ d !#) 2 2 1 1 pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ þ pqrs cosð4a Þ þ2l pqrs d d ða18Þ
Q pq rs ¼
References [1] K.T. Chau, G.S. Wang, Condition for the onset of bifurcation in en echelon crack arrays, Int. J. Numer. Anal. Meth. Geomech. 25 (2001) 289–306. [2] H.B. Muhlhaus, K.T. Chau, A. Ord, Bifurcation of crack pattern in arrays of twodimensional cracks, Int. J. Fract. 77 (1996) 1–14. [3] K. Ikeda, K. Murota, M. Nakano, Echelon modes in uniform materials, Int. J. Solids Struct. 31 (19) (1994) 2709–2733. [4] Y.L. Bai, J. Bai, H.L. Li, F.J. Ke, M.F. Xia, Damage evolution, localization and failure of solids subjected to impact loading, Int. J. Impact Eng. 24 (2000) 685– 701. [5] Y.L. Bai, M.F. Xia, F.J. Ke, H.L. Li, Statistical microdamage mechanics and damage field evolution, Theoret. Appl. Fract. Mech. 37 (2001) 1–10. [6] J.D. Zhao, W.Y. Zhou, Y.G. Liu, On the strain-gradient-enhanced damage model and its application to the localization analysis, Acta. Mech. Sin. 24 (2002) 445– 452 (in Chinese). [7] R.H.J. Peerlings, R. De Borest, W.A.M. Brekelmans, J.H.P. De Vree, Gradient enhanced damage for quasi- brittle materials, Int. J. Numer. Anal. Methods Eng. 39 (1996) 3391–3404. [8] X.Q. Feng, S.W. Yu, Analyses of damage localization at crack tip in a brittle damage material, Eng. Fract. Mech. 53 (2) (1996) 169–177. [9] X.P. Zhou, J.H. Wang, Y.X. Zhang, Q.L. Ha, Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogenous rock under uniaxial tensile loading, Appl. Math. Mech. 25 (9) (2004) 1031–1038. [10] X.P. Zhou, One set of bifurcation of crack pattern in two-dimensional intermittent cracks under far field shear stress, Int. J. Nonlinear Sci. Numer. Simul. 8 (1) (2007) 55–58. [11] X.P. Zhou, Q.H. Qian, H.Q. Yang, Bifurcation condition of crack pattern in the periodic rectangular array, Theoret. Appl. Fract. Mech. 49 (2008) 206–212. [12] T. Heinze, B. Galvan, S.A. Miller, Modeling porous rock fracturing induced by fluid injection, Int. J. Rock Mech. Min. Sci. 77 (2015) 133–141. [13] S. Pietruszczak, E. Haghighat, Modeling of deformation and localized failure in anisotropic rocks, Int. J. Solids Struct. 67–68 (2015) 93–101. [14] J. Dautriat, M. Bornert, N. Gland, A. Dimanov, J. Raphanel, Localized deformation induced by heterogeneities in porous carbonate analysed by multi-scale digital image correlation, Tectonophysics 503 (2011) 100–116. [15] B. Pardoen, J. Talandier, F. Collin, Permeability evolution and water transfer in the excavation damaged zone of a ventilated gallery, Int. J. Rock Mech. Min. Sci. 85 (2016) 192–208. [16] H.Q. Yang, Y.Y. Zeng, Y.F. Lan, X.P. Zhou, Analysis of the excavation damaged zone around a tunnel accounting for geostress and unloading, Int. J. Rock Mech. Min. Sci. 69 (2014) 59–66. [17] H. Horii, S. Nemat-Nasser, Brittle failure in compression: splitting, faulting, and brittle- ductile transition, Philos. Trans. R. Soc. Lond. 319 (1986) 337–374. [18] J.M. Kemeny, A model for non-linear rock deformation under compression due to sub-critical crack growth, Int. J. Rock Mech. Min. Sci. 28 (6) (1991) 459–467. [19] H. Deng, S. Nemat-Nasser, Microcrack interaction and shear fault failure, Int. J. Damage Mech. 3 (1994) 3–37. [20] H.B. Li, J. Zhao, T.J. Li, J.X. Yuan, Analytical simulation of the dynamic compressive strength of a granite using the sliding crack model, Int. J. Numer. Anal. Meth. Geomech. 25 (2001) 853–869. [21] J. Wang, J. Fang, B.L. Karihaloo, Asymptotics of multiple crack interactions and prediction of effective modulus, Int. J. Solids Struct. 37 (2000) 4261–4273. [22] G.C. Sih, A special theory of crack propagation, methods of analysis and solutions of crack problems, in: G.C. Sih (Ed.), Mechanics of Fracture I, Noordhoof, Leyden, 1973. [23] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Netherlands, 1991.
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Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach X.P. Zhou ⇑, H.Q. Yang State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, PR China School of Civil Engineering, Chongqing University, Chongqing 400045, PR China
a r t i c l e
i n f o
Article history: Received 14 March 2017 Revised 3 May 2017 Accepted 3 May 2017 Available online xxxx Keywords: Damage localization Crack-weakened rocks Pseudo traction method Bifurcation of crack growth pattern Strain energy density factor approach
a b s t r a c t The key failure mechanism of rock mass subjected to compressive loads is due to the damage localization. Conventional methods can not be applied to study the dynamic damage localization physical behavior of crack-weakened rock mass. The strain energy method has the capability to investigate the dynamic damage localization feature in crack-weakened rock mass. Therefore, in order to analyze the dynamic damage localization behavior, strain energy density factor approach is adopted in this paper. A two-dimensional model with periodic rectangular noncollinear array of cracks is established. Then, the onset condition of periodic distribution cracks in rock mass is obtained using strain energy density factor approach. By analyzing the bifurcation of crack growth pattern, the critical length and stress of damage localization of crack-weakened rock mass is determined as well as the location of damage localization. In addition, parameters sensitivity analysis is carried out, the effects of the length of crack, friction coefficient, fracture toughness, confining stress, velocity of crack growth, inclination and spacing between lines and rows on the onset condition for damage localization and bifurcation pattern of rocks are discussed. It can be concluded that onset condition of damage localization is mainly affected by its distribution of initial cracks. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction There are many discontinuities, such as bedding planes, joints, shear zones and faults, which are contained in rock mass. In many cases, the broken of rock mass is mainly controlled by the distribution of discontinuities in the rock mass. The key factor of rock failure is the damage localization process of microcracks in rock mass under compressive stress. Therefore, damage localization can serve as a precursor to the failure of rock mass. It is pointed out that the failure process of rock mass consists of the following three major stages, as shown in Fig. 1. Firstly, the steady state of stress accumulation appears, in which all cracks uniformly grow. Secondly, the alternative evolution of periodic microcracks occurs when they uniformly grow to some extent.
⇑ Corresponding author at: State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, PR China. E-mail addresses: [email protected], [email protected] (X.P. Zhou).
That is to say, the crack growth pattern bifurcation is characterized as only parts of cracks keeping propagation while others keeping still. Thirdly, some cracks coalesce with each other until rock mass fails. This process is called damage localization of rock mass. Based on these above observations, it can be concluded that damage localization of rock mass results from the bifurcation of crack growth pattern. Consider an infinite array of two-dimensional cracks of the same size in Fig. 1, all cracks grow simultaneously under far field applied loads when crack interaction is neglected. However, the bifurcation of crack growth pattern occurs when the effect of crack interaction is considered. Therefore, crack interaction is key factor for the occurrence of bifurcation and the onset condition for damage localization. That is to say, a bifurcation of the crack growth pattern emerges as soon as the critical condition is reached [2,3]. Various theoretical models were developed to analyze damage localization, these models can be simply classified into three kinds. The first kind of these models is based on the statistical microdamage theory. In order to reveal the underlying mesoscopic mechanism governing the experimentally observed failure in solids
http://dx.doi.org/10.1016/j.tafmec.2017.05.006 0167-8442/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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Increase loads
(1) Uniform growth state
2. Pseudo traction acting on the crack surface and stress intensity factor
(2) Bifurcation
2.1. The basic model
(3) Crack coalescence state Fig. 1. A typical observation of bifurcation of the growth pattern in experiments [1].
subjected to impact loads, some researchers [4,5] put forward a model of statistical microdamage evolution to analyze the macroscopic failure. The second kind of these models was derived from the strain-gradient-enhanced plastic theory. A strain-gradientenhanced damage model was proposed by Zhao et al. [6,7] to analyze the localization of rock-like or concrete-like brittle materials. The third kind of these models was known as the micromechanical damage model. Many efforts were done on this field. For example, considering the rapid stress drop and strain softening of material, the fracture process zone near the tips of a mode-I crack in a brittle damaged material was studied using the Dugdale-Barenblatt model [8]; the complete stress-strain relation for rock-like materials were investigated [9–16]. Although these three kinds of methods can reflect some characters of damage localization in crack-weakened rock mass, the mechanism of damage localization and the onset condition for damage localization still need to be investigated. In this paper, more general equation for noncollinear microcracks is derived based on a simplified analytical model for the bifurcation of collinear microcracks. Here, the possible bifurcation of crack growth pattern in twodimensional doubly periodic noncollinear crack arrays subjected to biaxial compression is investigated. By analyzing the bifurcation of crack growth pattern, the critical length and the critical stress for damage localization of crack-weakened rock mass are determined as well as the location of damage localization. Finally, parameters sensitivity analysis is carried out. This paper is organized as follows. In Section 2, the pseudo traction induced by crack interaction is determined. In Section 3, a dynamic fracture criterion is adopted. In Section 4, onset condition for damage localization is derived. In Section 5, parameters sensitivity analysis is carried out.
As shown in Fig. 2, a periodic rectangular noncollinear array of sliding cracks is used to analyze the damage localization of crackweakened rock mass under biaxial compression. Similar to the works by Horii and Nemat-Nasser [17], Kemeny [18], Deng et al. [19], Li [20] and Wang et al. [21], the sliding crack array can be simplified as an array of tensile cracks subjected to a pair of split1 ting forces F and the far field compressive stress r1 1 and r2 . For simplicity, Fig. 2 is further decomposed into configurations in Fig. 3a and b. The actual curvilinear wing crack in Fig. 2 is approximated by a straight opened crack growing parallel to the direction of the maximum principal compressive stress r1 1 . In Fig. 3a, an array of tensile cracks with the length of 2l, the horizontal spacing of H, and the vertical spacing of 2w is loaded by a pair of splitting forces T at its center. In Fig. 3b, an array of tensile cracks is subjected to axial compressive stress r1 1 and confining pressure r1 2 . Considering an elastic body containing an array of cracks, which is arranged as M rows and N columns, the periodic rectangular array of cracks in Fig. 3a is labeled by its location, such as the crack at the bottom left corner labeled by (1, 1), while the crack at the top right corner labeled by (M, N). The local Cartesian coordinate system of each cracks is (oij, x2ij, x1ij) in Fig. 3. As shown in Fig. 3, the spacing between adjacent rows of the same column is defined as d, while the spacing between adjacent columns of the same row is defined as b. The following three main assumptions are adopted in this model. Firstly, before bifurcation occurs, all the lengths of initial cracks are assumed to be equal to a, and all the lengths of wing cracks are 2l. Meanwhile, the dip angle of initial crack with respect to the maximum principal compressive stress r1 1 is h. Secondly, before bifurcation occurs, wing cracks always propagate along the direction of the maximum principal compressive stress r1 1 . Thirdly, it is assumed that the loading rate is not fast enough to consider the effects of cracks on the transmission of stress wave. These three assumptions were also adopted by the previous literatures [16,20].
Fig. 2. The periodic rectangular array of sliding cracks.
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
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Fig. 3. The simplified periodic rectangular array of cracks.
During the phase of the activation of frictional sliding on the faces of the pre-existing cracks, the effect of the interaction among cracks on the deformation of rock mass can be neglected. The expression of the resolved stress acting on the initial crack surface is written as
(
1 ss ¼ cos h sin hðr1 1 r2 Þ 2 1 2 rn ¼ sin hr1 þ cos hr1 2
ð1Þ
It is assumed that the frictional contact on the initial crack surface is of the Mohr-Coulomb type. The effective shear stress driving the frictional slip on crack surface can be expressed as
s ¼ ss lrn
Based on the pseudo-traction methods, the original problem in Fig. 2 can be decomposed into two sub-problems. One subproblem is about crack array loaded by splitting forces, as shown in Fig. 3a. While the other sub-problem is about crack array loaded by far field compressive stress, as shown in Fig. 3b. In each sub-problem, each crack is loaded by unknown pseudo-tractions ðkÞ rðkÞ pq and spq , which are induced by the existence of the other
crack. The pseudo-tractions can be expanded into Taylor series as [19]
rpq ispq ¼
ð2Þ
k 1 X xpq ðkÞ ðkÞ ðrpq ispq Þ lpq k¼0
ð7Þ
where l is the coefficient of dry friction. The force T defines the effect of the pre-existing crack sliding on the tensile crack, which can be estimated by
where k is the terms of Taylor series, rpq and spq are n term components of the pseudo-tractions.
1 T ¼ aw1 r1 1 aw2 r2
2.2. Cracks loaded by splitting forces
ð3Þ 2
where w1 ¼ sin hðsin 2h 2l sin hÞ; w2 ¼ sin hðsin 2h þ 2l cos2 hÞ. Considering the effects of crack interaction, the equivalent point force, which drives the growth of initial cracks located at the pth row and qth column, can be written as 1 F pq ¼ a½w1 ðr1 1 r1pq Þ w2 ðr2pq þ r2 Þ
ð4Þ
where r2pq and r1pq are the pseudo traction on the crack (p, q) along X and Y direction, respectively. For the model plotted in Fig. 3, the distance between arbitrary two crack centers ðp; qÞ and ðr; sÞ can be expressed as
dpqrs
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ ðjs qjH þ jr pjdÞ þ ð2jr pjw þ js qjbÞ
ðkÞ
Considering the crack array plotted in Fig. 3a, cracks are only loaded by splitting forces. Only 2N term of pseudo traction is taken into account. Before bifurcation of crack growth pattern occurs, the pseudo traction acting on each crack is the same. Moreover, the pseudo traction acting on each crack is symmetric, it is easy to find that the odd term components of the pseudo-tractions are equal to zero [19]. Then, we have ðkÞ rpq ¼
ð6Þ where j:j represents the absolute value.
r¼1 s¼1
r¼1 s¼1
M X N M X N X T sin h X T cos h þ ðEpqrs ðF pqrs þ k Þ k Þ p l pl r¼1 s¼1 r¼1 s¼1
The angle of connection line between arbitrary crack ðp; qÞ and ðr; sÞ with respect to the initial crack surface can be estimated by
arspq
" # 1 X M X N M X N X X pqrs ðmÞ ðmÞ ðApqrs Þ r þ ðB Þ s rs rs km km m¼0
ð5Þ
8 i h > < arctan ððsqÞHþðrpÞdÞ ½2ðr pÞw þ ðs qÞb P 0 2ðrpÞwþðsqÞb i h ¼ ððsqÞHþðrpÞdÞ > : p arctan ½2ðr pÞw þ ðs qÞb < 0 2ðrpÞwþðsqÞb
ðkÞ
ðkÞ spq ¼
1 M X N M X N X X X ðmÞ ðmÞ ðC pqrs ðDpqrs km Þrrs þ km Þsrs m¼0
r¼1 s¼1
ð8Þ
!
r¼1 s¼1
M X N M X N X T sin h X T cos h ðGpqrs ðHpqrs þ þ k Þ k Þ p l pl r¼1 s¼1 r¼1 s¼1
ð9Þ
where k; m ¼ 0; 1; 2 . . . 1; ðp; qÞ – ðr; sÞ。
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Apqrs k2m
¼ gm
k X 1
l pqrs
d
Apqrs k2m1 gm ¼
rð0Þ
¼ gm
l pqrs
d
2x hkx l apqrs pqrs k2x mþx d x¼1
k X 1
2xþ1 hkx ð2x þ kÞ l apqrs pqrs k2xþ1 2xðm þ xÞ d x¼1
ð2mÞ! 22mþ1 ðm!Þ2
\ ð10Þ
sð2Þ ¼
M X N X ðB20 rð0Þ þ D20 sð0Þ þ B22 rð2Þ þ D22 rð2Þ þ F 2 T sin h þ H2 T coshÞ r¼1 s¼1
ð18Þ
ð11Þ
When the components of the pseudo-tractions containing terms of
ð12Þ
is neglected, we have orders higher than ðl=d Þ A20 ¼ B20 ¼ D20 ¼ E2 ¼ F 2 ¼ H2 0, and other coefficients are determined by Eqs. (a8)(a16) in Appendix A. From Eqs. (15) to (18), rp0 , rp2 , sp0 and sp2 can be defined as
pqrs 2
P P P PN PM PN 2 PM PN M N 2 cos h M r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 A00 D00 ¼ T PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
ð19Þ
rð2Þ ¼ 0
sð0Þ
P P P PN PM PN 2 PM PN M N 2 sin h M r¼1 s¼1 B00 þ 2 cos h r¼1 s¼1 D00 þ r¼1 s¼1 B00 r¼1 s¼1 A00 D00 ¼ T PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
hkx ¼ ð1Þk
ðk þ 2x 1Þ! 22x1 k!½ðx 1Þ!2
ð13Þ
ð21Þ
sð2Þ ¼ 0
ð22Þ
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array can be expressed as
pqrs apqrs þ napqrs nm ¼ ðn þ 2Þ cos½ma
ðm þ nÞ cos½mapqrs þ ðn þ 2Þapqrs þ m cos½mapqrs þ napqrs ð14Þ pqrs pqrs In Eqs. (8) and (9), Bpqrs nm , C nm and Dnm can be calculated by replacing pqrs pqrs pqrs apqrs by b , c and d in Eqs. (10) and (11), the detailed expresnx nx nx nx
K pq Ia
ð20Þ
K pq Ia ¼
1 pffiffiffiffiffi T sin h X pl 2g k rpq þ 2k pl k¼0
!
Substituting Eq. (19) and Eq. (20) into Eq. (23), we have
! pffiffiffiffiffi T sin h 2 cos hPM PN B00 þ 2 sin hðPM PN A00 þ PM PN B2 PM PN ðA00 D00 ÞÞ r¼1 s¼1 r¼1 s¼1 r¼1 s¼1 00 r¼1 s¼1 ¼ pl T PM PN PM PN 2 PM PN PM PN pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
sions are derived from Eqs. (a1) to (a7) in Appendix A. pqrs For periodic crack array in Fig. 3, we have bnm ¼ cpqrs nm , pqrs pqrs pqrs Bpqrs ¼ C and G ¼ F . 2n2m 2n 2n2m 2n For simplification, only 2 (N = 1) term components of the pseudo-tractions are considered in this paper. Combining Eq. (8) with Eq. (9), we have
rð0Þ ¼
ð15Þ \
rð2Þ ¼
M X N X ðA20 rð0Þ þ B20 sð0Þ þ A22 rð2Þ þ B22 rð2Þ þ E2 T sinh þ F 2 T coshÞ r¼1 s¼1
ð16Þ
\
sð0Þ ¼
M X N X
ðB00 rð0Þ þ D00 sð0Þ þ B02 rð2Þ þ D02 rð2Þ þ F 0 T sin h þ H0 T coshÞ
r¼1 s¼1
ð17Þ
ð24Þ
2.3. Noncollinear crack array loaded by compressive stress Only 2N term pseudo tractions are taken into account, we have
rð0Þ ¼
M X N X ðA00 rð0Þ þ B00 sð0Þ þ A02 rð2Þ þ B02 sð2Þ þ A00 r1 2 Þ
ð25Þ
r¼1 s¼1
M X N X ðA00 rð0Þ þB00 sð0Þ þA02 rð2Þ þB02 rð2Þ þE0 T sinhþF 0 T coshÞ r¼1 s¼1
ð23Þ
rð2Þ ¼
M X N X ðA20 rð0Þ þ B20 sð0Þ þ A22 rð2Þ þ B22 sð2Þ þ A20 r1 2 Þ
ð26Þ
r¼1 s¼1
sð0Þ ¼
M X N X ðB00 rð0Þ þ D00 sð0Þ þ B02 rð2Þ þ D02 sð2Þ þ B00 r1 2 Þ
ð27Þ
r¼1 s¼1
sð2Þ ¼
M X N X ðB20 rð0Þ þ D20 sð0Þ þ B22 rð2Þ þ D22 sð2Þ þ B20 r1 2 Þ
ð28Þ
r¼1 s¼1
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
From Eqs. (25) to (28),
rð0Þ
rp0 , rp2 , sp0 and sp2 can be defined as
P PN 1 1 M r¼1 s¼1 D00 r2 ¼ r1 PM PN 2 PM PN PM PN PM PN 2 r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
sð0Þ ¼ PM PN r¼1
s¼1 A00 þ
5
PM PN
r1 2
2 s¼1 B00
r¼1
ð29Þ
PM PN r¼1
1þ
s¼1 B00 PM PN r¼1
s¼1 D00
ð30Þ
PM PN
s¼1 A00 D00
r¼1
rð2Þ ¼ 0
ð31Þ
where K ID is the mode I dynamic SIF, K I is the mode I static SIF, kð_lÞ is a function of crack growth velocity _l. The mode II dynamic SIF has
sð2Þ ¼ 0
ð32Þ
the same pattern.
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array can be written as
K pq Ib ¼ rpqb
1 X pffiffiffiffiffi pffiffiffiffiffi pl ¼ plðr1 2g k rpq 2 þ 2k Þ
ð33Þ
k¼0
Substituting Eq. (29) and Eq. (31) into Eq. (33), we have
K pq Ib
As it is very difficult to determine function kð_lÞ precisely, several approximate function was presented [14]. The crack array depicted in Fig. 3a and b are mixed mode, both mode I and mode II SIF _ and k2 ðlÞ _ are approximate. should be considered. k1 ðlÞ It was pointed out that the relationship between the growth velocity of crack and stress intensity factor can be approximately given by [19]
1 P PN 1 1 M r¼1 s¼1 D00 r2 A PM PN 2 PM PN PM PN s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
0 pffiffiffiffiffi ¼ pl@PM PN r¼1
Combining Section 2.2 with Section 2.3, the stress intensity factor of crack array in Fig. 2 under static loads can be expressed as
pffiffiffiffi _l ¼ cR 1:5K Ia þ 1:75K Ib 1:25K IC X 1:5K Ia þ 1:75K Ib 0:75K IC
P P 1 0 PM PN PM PN 2 PM PN M N pffiffiffiffiffi T sin h 2 cos h r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ K I ¼ pl@ TA PM PN PM PN 2 PM PN PM PN pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00 0 1 P PN 1 1 M pffiffiffiffiffi r¼1 s¼1 D00 r2 A pl@PM PN PM PN 2 PM PN PM PN r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
Under dynamic loads, crack growth velocities have a great influence on the dynamic SIF. The dynamic SIF can be deduced from static SIF by adding special items related to the growth velocity of crack. In most cases, the mode I dynamic SIF can be expressed as
K ID
ð37Þ
ð35Þ
X ¼ ½1:5K Ia þ 1:75K Ib 1:25K IC 2 4ðK Ia þ K Ib K IC Þð0:5K Ia
3. Dynamic stress intensity factor and fracture criterion
K ID ðtÞ ¼ kð_lÞK I
ð34Þ
ð36Þ
þ 0:75K Ib 0:375K IC Þ
ð38Þ
where K IC is mode I fracture toughness of rocks. For crack array in Fig. 2, from Eq. (35), the stress intensity factor can be described as
P P 1 0 P PN PM PN 2 PM PN M N 2 cos h M pffiffiffiffiffi r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ T sin h ¼ pl @ TA PM PN PM PN 2 PM PN PM PN pl cR 0:75_l r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00 0 1 P PN 1 1 M r¼1 s¼1 D00 r2 cR _l pffiffiffiffiffi@ A pl PM PN P PN 2 P PN P PN A00 þ M B 1þ M D00 M A00 D00 cR 0:5_l cR _l
r¼1
s¼1
r¼1
s¼1 00
r¼1
s¼1
r¼1
ð39Þ
s¼1
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The strain energy density factor of rocks is chosen as the crack growth criterion. The strain energy density factor of rocks can be written as [22,23]
S ¼ a11 K 2ID þ 2a12 K ID K IID þ a22 K 2IID þ a33 K 2IIID
ð40Þ
where K ID , K IID and K IIID are the mode I and mode II and mode III dynamic stress intensity factors, respectively. The coefficients of could be found in Refs. [22,23]. The onset of rapid crack propagation is assumed to start when max the strain energy density Smin associated with ðdW=dVÞmin reaches a critical value, i.e.
Smin ¼ SDc
ð41Þ
where SDc ¼ rC
dW
energy density
Before bifurcation of crack growth pattern occurs, the stress intensity factor of crack array in Fig. 2 can be written as
cR 0:75_l
cR _l
pffiffiffiffiffi F pq pffiffiffiffiffi ðr2pq þ r1 2 Þ pl ¼ K ID pl cR 0:5_l
ð42Þ
the principal stress can be obtained as
r2pq
r1pq
Eq. (44) can be rewritten as
r1pq ¼ r1 1
vÞ D 2 ¼ ð1þv2Þð12 ðK IC Þ is the critical dynamic strain pE dV c factor, K DIC is dynamic fracture toughness of rocks.
4. Onset condition for damage localization of crack-weakened rock mass
cR _l
Before bifurcation of the crack growth pattern occurs, the length of each wing crack is the same. Considering the variation of Eq. (44), we can obtain P PN 1 2 ½2 cosð2apqrs Þ cosð4apqrs Þ, where f 1 ¼ M r¼1 s¼1 dpqrs 4 P PN 2 1 ½sinð4apqrs Þ sinð2apqrs Þ , f2 ¼ M r¼1 s¼1 dpqrs 2 P PN 1 cosð4apqrs Þ, f3 ¼ M r¼1 s¼1 dpqrs PM PN 1 4 f 4 ¼ r¼1 s¼1 dpqrs ½2 cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ.
w2 ðr2pq þ r1 2 Þ w1 2
2
4
4
T sin h 2 cos hf 5 l þ 2 sin hðf 1 l þ f 2 l f 4 l Þ T 2 4 2 4 w1 a pl f 1l þ f 2l 1 þ f 3l f 4l
pl
!
ð47Þ where f 5 ¼
PM PN r¼1
s¼1
1 dpqrs
2
½sinð4apqrs Þ sinð2apqrs Þ.
Substituting Eq. (47) into Eq. (42) and Eq. (41), we have
" # pffiffiffiffiffi T sin h 2 cos hf 5 l2 þ 2 sin hðf 1 l2 þ f 2 l4 f 4 l4 Þ pl T 2 4 2 4 pl f 1l þ f 2l 1 þ f 3l f 4l pffiffiffiffiffi D k2 ð_lÞðr2pq þ r1 2 Þ pl ¼ K IC
k1 ð_lÞ
P PN 1 1 M r¼1 s¼1 D00 r2 ¼ PM PN r1 PM PN 2 PM PN PM PN 2 r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
P P 1 0 P PN PM PN 2 PM PN M N 2 cos h M r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ w2 p l T sin h 1 @ ¼ ðr2pq þ r1 TA PM PN PM PN 2 PM PN PM PN 2 Þ r1 þ w1 a w1 pl r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
ð48Þ
ð43Þ
ð44Þ
After bifurcation of crack growth pattern occurs, the stress field can be defined as
9
rð2Þ 2pq ¼ r2 þ dr2pq > > = rð2Þ 1pq ¼ r1 þ dr1pq > ð2Þ lpq
¼ l þ dlpq
ð45Þ
> ;
where d represents the variation of variable.
dr2pq
PM PN
2
( " 2 M X N X 1 ¼ dlrs 2l pqrs ½2 cosð2apqrs Þ cosð4apqrs Þ 2 4 2 2 4 d ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 r¼1 s¼1 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 #) 4 2 4 1 1 1 3 3 pqrs pqrs 2 pqrs pqrs pqrs pqrs ½sinð4a Þ sinð2a Þ þ 2l pqrs cosð4a Þ 4l ½2 cosð2a Þ cosð4a Þ cosð4a Þ þ4l pqrs pqrs d d d 2r1 2 l
r¼1
1 s¼1 ½dlrs dpqrs
cosð4apqrs Þ
r1 2 ð1 f 3 Þ
ð46Þ
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
Considering the variation of Eq. (48), we can obtain ( " 2 M X N _ X K ICD dlrs k1 ðlÞTdl k1 ð_lÞT 1 rs sin h dl ½sinð4apqrs Þ sinð2apqrs Þ pffiffiffiffiffi rs 4l cos hdlrs pqrs 2 5 3 _ pl d 2k2 ðlÞl k2 ð_lÞpl k2 ð_lÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l r¼1 s¼1 " ##) 2 4 4 1 1 1 3 3 2 þ4sin h l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ þ 2l ½sinð4apqrs Þ sinð2apqrs Þ 2l ½2cosð2apqrs Þ cosð4apqrs Þcosð4apqrs Þ pqrs pqrs d d d ( " ! 4 4 2 2 4 4 M X N _ X k1 ðlÞT½2cos hl f 5 þ 2sin hðf 1 l þ f 2 l f 4 l Þ 1 1 3 pqrs pqrs 2 pqrs pqrs pqrs þ dl 4l ½sinð4 a Þ sinð2 a Þ ½2 cosð2 a Þ cosð4 a Þ cosð4 a Þ rs pqrs pqrs 2 5 3 d d r¼1 s¼1 k2 ð_lÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l !#) 2 2 1 1 ½2cosð2apqrs Þ cosð4apqrs Þ þ pqrs cosð4apqrs Þ ð49Þ þ2l pqrs d d
dr2pq ¼
Combining Eq. (46) with Eq. (48), after a series simplification, we can obtain following matrix form, the expanded expression is written as Eq. (a17) in Appendix A.
2 6 6 6 6 6 4
Q 11 11
Q 11 12
Q 12 11 .. . Q MN 11
Q 12 12 .. . Q MN 12
32
3 dl11 7 6 7 dl12 7 Q 12 7 MN 76 6 . 7¼0 .. .. 7 .. 7 76 5 4 . . 5 MN dlMN Q MN Q 11 MN
ð50Þ
As far as Eq. (50), the element Q pq rs represents the effects of the crack ðr; sÞ on the crack ðp; qÞ. The detailed expression can be described as Eq. (a18) in Appendix A. When Eq. (50) is singular, Eq. (50) has solutions which are not equal to zero. By solving Eq. (50), the critical length of wing crack lcr can be obtained when damage localization occurs. The matrix in Eq. (50) has a eigenvalue which is equal to zero. Moreover, the location of damage localization can be determined through the eigenvector of the matrix in Eq. (50) corresponding to the zero eigenvalue [3,16]. When the value in the eigenvector is plus, it is pointed out that the crack in the corresponding location keeps propagating after damage localization occurs. However, if the value in the eigenvector is minus, it is pointed out that the crack in the corresponding location stops propagating after damage localization occurs. When the confining pressure keeps constant, the critical stress for damage localization can be estimated from Eq. (50) as
r
1 1cr
fracture toughness, confining pressure and loading rate on the onset condition of damage localization and the bifurcation pattern of crack growth, Fig. 4 is plotted. In Fig. 4, the total number of cracks is 6. The length of preexisting crack is a, and each crack is the same value of length and inclination. 5.1. Dependence of the dimension of cracks on damage localization The effect of the length and inclination of cracks on damage localization is illustrated in this section. The following material parameters come from some experimental tests for granite: the confining pressure r2 = 1 MPa, the frictional coefficient of crack surface l = 0.5, dimension parameters for crack distribution H = 0.1 m, w = 0.18 m, d = 0.01 m, b = 0.005 m, fracture toughness KIC = 0.7 MPam1/2, the velocity of Rayleigh wave cR = 2000 m/s and the critical growth velocity for single crack l_ = 0.00001 m/s. Fig. 5 shows the effect of the length of crack on the critical axial stress. It is found from Fig. 5 that the critical stress of damage localization decreases as the crack length increases. When h = 40, 50 and 60, the critical length of damage localization is 0.5 mm, 0.6 mm and 0.72 mm, respectively. It can be observed from Fig. 6 that the critical length increases with increasing h. In addition, it is implied that the critical length doesn’t vary with the length of preexisting crack, when other conditions is fixed.
8 0 19 P PN 1 = 1 M r¼1 s¼1 D00 r2 1 < cR _l pffiffiffiffiffiffiffiffi@ A ¼ K ICD þ plcr PM PN PM PN 2 PM PN PM PN _ ; aw1 : cR 0:5l r¼1 s¼1 A00 þ r¼1 s¼1 B00 1 þ r¼1 s¼1 D00 r¼1 s¼1 A00 D00
P P 11 0 P PN PM PN 2 PM PN M N 2 cos h M _ r¼1 s¼1 B00 þ 2 sin h r¼1 s¼1 A00 þ r¼1 s¼1 B00 r¼1 s¼1 ðA00 D00 Þ sin h w R 0:75l A pcffiffiffiffiffiffiffiffi þ 2 r1 @ PM PN PM PN 2 PM PN PM PN _ plcr w1 2 A þ B 1 þ D A D p l lÞ ðc 00 00 00 00 cr R r¼1 s¼1 r¼1 s¼1 00 r¼1 s¼1 r¼1 s¼1
where
r1 1cr is the onset stress for damage localization.
5. Sensitivity analysis In order to illustrate the dependence of the crack length, inclination, spacing between rows and columns, frictional coefficients,
ð51Þ
Fig. 7 shows the effect of h on the critical stress. In can bee concluded from Fig. 7 that when h is smaller than 45, the critical stress decreases as h increases. However, when h is bigger than 45, the critical stress increases as h increases. The critical stress is minimum when h is equal to 45. Fig. 8 shows the crack growth pattern after bifurcation occurs. It can be observed from Fig. 8 that when damage localization occurs, only some cracks keep propagating while others keep still. Three
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx 0.75 0.7
lcr /mm
0.65 0.6 0.55 0.5 0.45 0.4 0.35 20
30
40
50
60
70
θ /Deg. Fig. 6. Dependence of the critical length on h.
50 45
Fig. 4. Sketch for calculation model.
a=0.05
40
σ1 /MPa
35
50
σ1 /MPa
45
θ
30 25 a=0.03
20
a=0.05
15
60°
40
10
35
5
30
0 20
25
30
40
50
60
70
θ /Deg.
20 θ θ
15
Fig. 7. Dependence of the critical axial stress on h.
50° 40°
10 5 0 0
0.01
0.02
0.03
0.04
0.05
0.06
a /m Fig. 5. Dependence of the critical axial stress on the length of crack.
main conclusions are obtained from Fig. 8. Firstly, it is implied that the length of preexisting crack has no influence on the crack growth pattern after damage localization occurs. Secondly, the location of damage localization is related to the inclination of pre-existing cracks. Thirdly, when h is smaller than 45, the number of growing cracks decreases with increasing h after damage localization occurs. However, when h is bigger than 45, the number of growing cracks increases with increasing h after damage localization occurs. The critical stress is minimum when h is equal to 45.
5.2. Dependence of the distribution of crack array on damage localization The effect of the spacing between rows and columns of crack on damage localization is illustrated in this section. The following material parameters are used in computations: r2 = 1 MPa, a = 0.04 m, h = 45, l = 0.5, d = 0.01 m, b = 0.005 m, K = 0.7 MPam1/2, cR = 2000 m/s and l_ = 0.00001 m/s. IC
Figs. 9 and 10 respectively show the dependence of the critical axial stress on w and H. It can be concluded from Figs. 9 and 10 that when the spacing between cracks is small, the critical axial stress
increases with increasing the spacing. However, the critical axial stress keeps constant when the distance between cracks is far enough. Figs. 11 and 12 show the dependence of the critical axial stress on d and b, respectively. It can be concluded from Fig. 11 that when d is smaller than 0.025 m, the critical stress of damage localization decreases as d increases. However, d is bigger than 0.025 m, the critical stress of damage localization increases as d increases. The critical stress is minimum when d is equal to 0.025 m. It can be found from Fig. 12 that when b is smaller than 0.02 m, the critical stress of damage localization increases as b increases. However, b is bigger than 0.02 m, the critical stress of damage localization decreases as b increases. The critical stress is maximum when b is equal to 0.02 m. Figs. 13 and 14 show the dependence of the critical length on w and H, respectively. It can be concluded from Figs. 13 and 14 that when the spacing between cracks is small, the critical length increases with increasing the spacing. However, when the distance between cracks is far enough, the critical length keeps constant. Figs. 15 and 16 show the dependence of the critical length on d and b, respectively. It can be observed from Fig. 15 that when d is smaller than 0.025 m, the critical length of damage localization decreases as d increases. However, when d is bigger than 0.025 m, the critical length of damage localization increases with increasing d. The critical length is minimum when d is equal to 0.025 m. It can be found from Fig. 16 that when b is smaller than 0.02 m, the critical length of damage localization increases with increasing b. However, when b is bigger than 0.02 m, the critical length of
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30
30 , a=0.03 m
45
55 , a=0.03 m
Fig. 8. Sketch of the bifurcation growth model of cracks with different h and a.
16 17
14 15 13
10
σ1 /MPa
σ1/MPa
W=0.18 m
H=0.2 m
12
H=0.15 m
8
11 W=0.12 m
9 7
H=0.1 m
6
W=0.06 m
5
4 0.03
0.08
0.13
0.18
0.23
0.28
w /m
3 0.04
0.14
0.24
0.34
0.44
H /m
Fig. 9. Dependence of the critical axial stress on w.
Fig. 10. Dependence of the critical axial stress on H.
damage localization decreases as b increases. The critical length is maximum when b is equal to 0.02 m. Fig. 17 shows the crack growth pattern after bifurcation occurs. It can be seen from Fig. 17 that when w and H are small, the num-
ber of growing crack decreases with increasing the spacing after damage localization occurs. However, when w and H are far enough, the number of growing crack increases with increasing the spacing after damage localization occurs.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
9.45
12
9.4
10
9.35
8
lcr /mm
σ1 /MPa
10
9.3
W=0.18 m 6
9.25
4
9.2
2
W=0.12 m W=0.06 m
9.15
0
0.01
0.02
0.03
0.04
0.05
0 0.04
0.06
0.14
0.24
d /m
0.34
0.44
H /m
Fig. 11. Dependence of the critical axial stress on d.
Fig. 14. Dependence of the critical length on H.
1.92
9.36 9.34
1.88
9.32
lcr /mm
σ1/MPa
9.3 9.28 9.26
1.84
1.8
9.24 9.22
1.76
9.2
1.72
9.18 0
0.01
0.02
0.03
0.04
0
0.05
0.01
0.02
0.03
0.04
0.05
0.06
d /m
b /m Fig. 12. Dependence of the critical axial stress on b.
Fig. 15. Dependence of the critical length on d.
8
1.88
7 1.84
5
lcr /mm
lcr /mm
6
H=0.2 m
4
1.8
3 2 1 0
0.03
1.76
H=0.15 m H=0.1 m 0.08
0.13
0.18
0.23
0.28
w /m Fig. 13. Dependence of the critical length on w.
Fig. 18 shows the crack growth pattern at different b and d after bifurcation occurs when w = 0.12 m and H = 0.2 m. It can be found from Fig. 18 that the number of growing cracks doesn’t vary with b and d after damage localization occurs. However, the location of damage localization is related to b and d.
1.72
0
0.01
0.02
0.03
0.04
0.05
d /m Fig. 16. Dependence of the critical length on b.
5.3. Dependence of frictional coefficients and fracture toughness on damage localization The effect of the frictional coefficients l and fracture toughness KIC on damage localization is illustrated in this section.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
11
Fig. 17. Sketch of the bifurcation growth model of cracks with different w and H.
The following material parameters are used in computations:
r2 = 1 MPa, H = 0.1 m, a = 0.04 m, h = 45, w = 0.16 m, d = 0.01 m,
b = 0.005 m, cR = 2000 m/s, _l = 0.00001 m/s. Fig. 19 shows the dependence of the critical stress for damage localization on the frictional coefficients l when KIC = 0.12 MPam1/2. It can be observed from Fig. 19 that the critical stress of damage localization increases with increasing the frictional coefficients l. Fig. 20 shows the dependence of the critical stress of damage localization on the fracture toughness KIC when l = 0.5. It can be observed from Fig. 20 that the critical stress of damage localization increases with increasing the fracture toughness KIC. Fig. 21 shows the dependence of the crack growth pattern on different frictional coefficients l and fracture toughness KIC after bifurcation occurs. It can be found from Fig. 21 that the number of growing cracks doesn’t vary with the frictional coefficients l and fracture toughness KIC after damage localization occurs. 5.4. Dependence of loading condition on damage localization The effect of the velocity of crack growth l_ and the confining pressure r2 on damage localization is illustrated in this section. The following material parameters are used in computations: l = 0.5, H = 0.1 m, a = 0.04 m, h = 45, w = 0.16 m, d = 0.01 m, b = 0.005 m, cR = 2000 m/s, KIC = 0.7 MPam1/2. Fig. 22 shows the dependence of the critical stress for damage localization on the confining pressure r2 when l_ = 105 m/s. It can be seen from Fig. 22 that the critical stress of damage localization increases with increasing the confining pressure r2. Fig. 23 shows the dependence of the critical stress of damage localization on the velocity of crack growth _l when
r2 = 1 MPa. It can be observed from Fig. 23 that the critical stress of damage localization increases with increasing the velocity of crack growth _l. Moreover, the numerical results show that the number of growing cracks doesn’t vary with the velocity of crack growth _l and the
confining pressure r2 after damage localization occurs. The crack growth pattern in this example after damage localization occurs is the same as Fig. 21. 6. Conclusions By analyzing the bifurcation of crack growth pattern, the critical length and stress of damage localization of crack-weakened rock mass are determined as well as the location of damage localization. In addition, parameters sensitivity analysis is carried out. The main conclusions are summarized as follows: (1) Damage localization is induced by the bifurcation of crack growth pattern. After damage localization occurs, only some cracks keep propagating while others keep still. (2) The critical stress of damage localization is related to the length of pre-existing crack, inclination, the spacing between rows and columns, frictional coefficient on the crack surface, fracture toughness of rocks, the confining pressure and the loading rate. (3) The crack growth pattern doesn’t concern with the inclination of preexisting crack, the spacing between rows and columns, length of preexisting crack, friction coefficients, fracture toughness, confining pressure and the loading rate after damage localization occurs.
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Fig. 18. Sketch of the bifurcation growth model of cracks with different b and d.
11
35
10
30
9 8
σ1 /MPa
σ1/MPa
25 20 15
7 6 5
10
4
5
3 2
0 0
0.2
0.4
0.6
0.8
μ Fig. 19. Dependence of the critical axial stress on l.
1
0
0.5
1
1.5
2
1/2
KIC /MPa•m
Fig. 20. Dependence of the critical axial stress on KIC.
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X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
Appendix A
pqrs
bnm ¼ ðn þ 2Þ sin½mapqrs þ napqrs þ ðm þ nÞ sin½mapqrs þ ðn þ 2Þapqrs ðm 2Þ sin½mapqrs þ napqrs ða1Þ pqrs cpqrs þ napqrs þ ðm þ nÞ sin½mapqrs þ ðn þ 2Þapqrs nm ¼ n sin½ma
m sin½mapqrs þ napqrs
ða2Þ
pqrs
dnm ¼ n cos½mapqrs þ napqrs þ ðm þ nÞ cos½mapqrs þ ðn þ 2Þapqrs ðm 2Þ cos½mapqrs þ napqrs Epqrs ¼ n
d
2m hnm l apqrs pqrs n2m 2m 1 d m¼1
pqrs
F pqrs ¼ n
n X 1
l
n X 1
l
2m hnm l pqrs bn2m pqrs 2m 1 d m¼1
pqrs
d
ða3Þ
ða4Þ
ða5Þ
Fig. 21. Sketch of the bifurcation growth model of cracks.
¼ Gpqrs n
30
σ1 /MPa
2m hnm l cpqrs pqrs n2m 2m 1 d m¼1
pqrs
d
25
¼ Hpqrs n
20
n X 1
l
d
l
n X 1
2m hnm l pqrs dn2m pqrs 2m 1 d m¼1
pqrs
ða6Þ
ða7Þ
15 10
A00 ¼
2 1 l ½2 cosð2apqrs Þ cosð4apqrs Þ 2 dpqrs
ða8Þ
A02 ¼
2 1 l 1 ½2 cosð2apqrs Þ cosð4apqrs Þ ¼ A00 8 dpqrs 4
ða9Þ
B00 ¼
2 1 l ½sinð4apqrs Þ sinð2apqrs Þ 2 dpqrs
ða10Þ
B02 ¼
2 1 l 1 ½sinð4apqrs Þ sinð2apqrs Þ ¼ B00 8 dpqrs 4
ða11Þ
D00 ¼
2 1 l cosð4apqrs Þ 2 dpqrs
ða12Þ
D02 ¼
2 1 l 1 cosð4apqrs Þ ¼ D00 8 dpqrs 4
ða13Þ
5 0
0
2
4
6
8
10
σ2 /MPa Fig. 22. Dependence of the critical axial stress on r2.
18 16 cR=800 m/s 14
σ1/MPa
cR=1500 m/s 12 cR=2000 m/s 10 8 6 4 -6.00
-4.00
-2.00
0.00
2.00
4.00
lg(i)/(m/s) Fig. 23. Dependence of the critical axial stress on _l.
Acknowledgements This work was supported by project 973 (Grant no. 2014CB046903), the National Natural Science Foundation of China (Grant nos. 51325903, 51679017, 51409026), Natural Science Foundation Project of CQ CSTC (No. CSTC, cstc2015jcyjys30001, cstc2015jcyjys30006, cstc2013jcyjys30002, cstc2016jcyjys0005).
E0 ¼
l pqrs
d
F0 ¼
l d
l pqrs
d
½2 cosð2apqrs Þ cosð4apqrs Þ ¼ 2A00
ða14Þ
½sinð4apqrs Þ sinð2apqrs Þ ¼ 2B00
ða15Þ
2
pqrs
H0 ¼
2
2 cosð4apqrs Þ ¼ 2D00
ða16Þ
Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006
14
X.P. Zhou, H.Q. Yang / Theoretical and Applied Fracture Mechanics xxx (2017) xxx–xxx
2r1 2 l
PM PN r¼1
s¼1
2 1 dlrs dpqrs cosð4apqrs Þ
r1 2 ð1 f 3 Þ
2 4 2 2 4 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ( " 2 M N XX 1 dlrs 2l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d r¼1 s¼1 4 2 1 1 3 2 ½sinð4apqrs Þ sinð2apqrs Þ þ 2l pqrs cosð4apqrs Þ þ4l pqrs d d #) 4 1 3 pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ cosð4a Þ 4l pqrs d _ k1 ðlÞTdlrs sin h k1 ð_lÞT þ þ 2 5 3 _ _ k2 ðlÞ½ðf 2 f 4 Þl þ ðf 1 þ f 3 Þl l ( k2 ðlÞpl " 2 M X N X 1 dlrs 4l coshdlrs pqrs ½sinð4apqrs Þ sinð2apqrs Þ d r¼1 s¼1 " 2 1 þ4sin h l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ2l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ##) 4 1 3 2l ½2cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ pqrs d 2 2 4 4 k1 ð_lÞT½2cos hl f 5 þ 2sin hðf 1 l þ f 2 l f 4 l Þ 2 5 3 k2 ð_lÞ½ðf ( " 2 f 4 Þl þ ðf 1 þ f 3 Þl l 4 M X N X 1 3 2 dlrs 4l ð pqrs Þ ½sinð4apqrs Þ sinð2apqrs Þ d r¼1 s¼1 ! 4 1 pqrs ½2cosð2apqrs Þ cosð4apqrs Þ cosð4apqrs Þ d !#) 2 2 1 1 pqrs pqrs pqrs þ2l ½2cosð2 a Þ cosð4 a Þ þ cosð4 a Þ ¼0 pqrs pqrs d d ða17Þ
2 1 pqrs 2r1 l cosð4 a Þ pqrs 2 d
r1 2 ð1 f 3 Þ 2 4 2 ½ðf 1 þ f 3 Þl þ ðf 2 f 4 Þl 1 ½ðf 1 þ f 3 Þl2 þ ðf 2 f 4 Þl4 1 (" 2 1 2l pqrs ½2cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ4l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d 2 4 1 1 3 þ2l pqrs cosð4apqrs Þ 4l ½2 cosð2apqrs Þ pqrs d d k1 ð_lÞT sin h cosð4apqrs Þcosð4apqrs Þ þ 2 k2 ð_lÞpl _ k1 ðlÞT þ _lÞ½ðf f Þl5 þ ðf þ f Þl3 l k ð 2 2 4 3 (" 2 1 1 4lcos h pqrs ½sinð4apqrs Þ sinð2apqrs Þ þ 4 sin h d " 2 1 l pqrs ½2 cosð2apqrs Þ cosð4apqrs Þ d 4 1 3 2 þ2l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ##) 4 1 3 2l ½2cosð2apqrs Þ cosð4apqrs Þcosð4apqrs Þ pqrs d 2 2 4 4 _ k1 ðlÞT½2 coshl f 5 þ 2 sin hðf 1 l þ f 2 l f 4 l Þ 2 5 3 _ f Þl þ ðf 1 þ f 3 Þl l (" k2 ðlÞ½ðf 2 4 4 1 3 2 4l ½sinð4apqrs Þ sinð2apqrs Þ pqrs d ! 4 1 pqrs pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ cosð4a Þ d !#) 2 2 1 1 pqrs pqrs pqrs ½2cosð2a Þ cosð4a Þ þ pqrs cosð4a Þ þ2l pqrs d d ða18Þ
Q pq rs ¼
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Please cite this article in press as: X.P. Zhou, H.Q. Yang, Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach, Theor. Appl. Fract. Mech. (2017), http://dx.doi.org/10.1016/j.tafmec.2017.05.006