ModelingofShearCrackPropagationinRockMassesUsing Mesh-FreeLRPIM · 2021. 3. 25. · LRPIM. ree-dimensional shear crack propagation with the LRPIM has not yet been studied. A three-dimensional

Research ArticleModeling of Shear Crack Propagation in Rock Masses UsingMesh-Free LRPIM

Qingbo Li 1 Nengxiong Xu 2 Weifeng Wan 1 and Yazhe Li 1

1Yellow River Engineering Consulting Co Ltd Zhengzhou China2School of Engineering and Technology China University of Geosciences (Beijing) Beijing 100083 China

Correspondence should be addressed to Yazhe Li liyazhecugbeducn

Received 16 October 2020 Revised 22 January 2021 Accepted 2 March 2021 Published 25 March 2021

Academic Editor Daniele Baraldi

Copyright copy 2021 Qingbo Li et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

emodeling of shear cracks in materials is critical in various engineering applications such as the safety analysis of concrete structuresand stability analysis of rock slopes Based on the idea of Goodman element the elastic-plastic constitutive model of the shear cracks isderived and the elastic-plastic analysis of shear crack propagation is realized in the local radial basis point interpolation method(LRPIM)is method avoids the loss of accuracy caused by themesh in the analysis of fracture propagation and the crack propagationof rock brittle material is simulated e investigation indicates that (1) the LRPIM results are close to the FDM results whichdemonstrates that it is feasible to analyze shear cracks in rockmasses (2) Compared with the results of the built-in oblique crack modelwhen the LRPIM is used to analyze crack propagation the results are close to the experimental results showing that the LRPIM canmodel shear crack propagation in a rock mass (3) e propagation path using the LRPIM is not sufficiently smooth which can beexplained as the crack tip stress and strain not being sufficiently accurate and still requiring further improvement

1 Introduction

Rock masses are discontinuous media that contain manykinds of defects such as cracks bedding and holes Existingshear cracks have a substantial influence on the mechanicalcharacteristics of rock masses A rock mass with manydefects is extremely sensitive to the action of forceserefore it is necessary to analyze the influence of shearcracks and their propagation under the action of externalforces

Some research results have been obtained throughphysical experiments e mechanism of crack propagationin concrete or rock specimens containing cracks under shearloading conditions is studied [1] and ultrasonic wavesrecorded during direct shear experiments on rock jointswere employed to investigate the shear failure processes [2]

Some methods that do not need too many criteria orspecial elements have been proposed [3 4] which have greatpotential in engineering applications Peridynamics (PD)describes the mechanical behavior of matter by solving thespatial integral equation which avoids the singularity and

complexity of the traditional model based on continuityassumption when facing discontinuity problems It has beensuccessfully applied to the simulation of various disconti-nuity problems with different scales and has become a re-search hotspot in the current international computationalmechanics and related fields In the simulation process ofnear-field dynamics model the phenomena of crack initi-ation propagation and bifurcation are naturally generatedwith the simulation process In addition to adding a single-bond fracture condition there is no need for too manyjudgment conditions or special elements [5 6]

Scholars have proposed various theories and calculationmethods to study the discontinuous characteristics of rockmasses e main numerical methods for simulating crackpropagation are the finite element method (FEM) [7] ex-tended finite element method (XFEM) [8] and meshlessmethod [9] e FEM was applied to study the mechanicalbehavior of rock masses and concrete in references [10 11]When the FEM is used to study crack propagation it isnecessary to remesh the elements which substantially in-creases the workload and leads to low calculation accuracy

HindawiAdvances in Civil EngineeringVolume 2021 Article ID 6654812 13 pageshttpsdoiorg10115520216654812

and efficiency Meshing and remeshing has significantlyimproved in the past years An alternative crack propagationalgorithm has been proposed which effectively circumventsthe variable transfer procedure adopted with classical meshadaptation algorithms [12] And neural networks can beused to compute the crack growth in fracture specimenswhich finally leads to a prediction of classical fracturetoughness parameters [13] ese shortcomings can also beavoided by the XFEM proposed by Belytschko and Moes[8 9 14] e XFEM rationally realizes accurate crack de-scriptions and has become a popular research topic amongnumerical methods for crack problems [15]

Moes et al [9] used the XFEM to simulate bond crackpropagation with the stress intensity factor as the crackindex Mariani et al [16] used the XFEM to simulate qua-sistatic bond crack propagation in quasibrittle materialsDeb and Das [17] combined the XFEM with cohesion tosimulate shear crack slip Zhou and Yang [18] proposed amultiscale XFEM to study crack propagation and rockbridge penetration under multidirectional compression andthe conclusions were in good agreement with the experi-mental results

Compared with the FEM and XFEM the meshlessmethod is based on field nodes that can eliminate or partiallyeliminate the difficulties caused by meshing e interpo-lation (approximation) function of the meshless methoddoes not depend on the elements so it has the advantages ofeasy higher order approximation formation and realizationof local node encryption bringing convenience to the nu-merical analysis of cracks

Belytschko [19] first used the EFGM method to analyzecrack propagation Krysl [20] further extended the EFGMmethod to crack propagation calculations in a three-di-mensional model Belytschko [21] applied the meshlessmethod to simulate dynamic crack propagation in concretestructures and they simulated the crack propagation di-rection and velocity in concrete structures very well ZhuangXiaoying et al [22] realized accurate fracture modeling in atwo-dimensional model using meshless methods the visi-bility criterion and level sets

e above applications all use the meshless EFGM butthis method requires the background grid for integrationand is not a true meshless method e LRPIM [23] methodis a local weak computing method that does not require thebackground grid and is a real meshless method Little re-search has been conducted on crack problems with the

LRPIM ree-dimensional shear crack propagation withthe LRPIM has not yet been studied A three-dimensionalshear crack propagation calculation is established based onelastic-plastic theory and fracture mechanics e reliabilityof the method is verified through numerical experimentsand the problems that need to be improved are discussed

is paper is organized as follows Section 2 brieflyintroduces the background of the LRPIM Section 3 mainlyconcentrates on the methods of modeling shear cracks withthe LRPIM Section 4 discusses the results Finally Section 5summarizes the conclusions

2 Background LRPIM

If only the static case is considered the BVP of three-di-mensional solid mechanics can be expressed as

σijj + bi 0

σijnj ti in Γt

ui ui in Γu

u x t0( 1113857 u0(x) x isin Ω

(1)

where ui ti and u0 are the given displacement the givensurface force and the initial displacement I j (123)represents x y z

Considering the solid mechanics problem defined indomain Ω the local weighted residual method is used fornode I to satisfy the governing equation and the local weakequation of the node is obtained e form of the locallyweighted residuals is defined on the local integral domainΩq and the corresponding boundary Γq in the followingform

1113946Ωq

1113954WI σijj + bi1113872 1113873dΩ 0 (2)

where 1113954Wi is a weight function or test function centered atfield node I

e follow formula is obtained by integral operation

1113946Ωq

1113954WIσijjdΩ 1113946Γq

1113954WItidΓ minus 1113946Ωq

1113954WIjσijdΩ (3)

where ti is the component of surface force vector on theboundary

We bring formula (10) into formula (6)

1113946Γq

1113954WItiσijdΓ minus 1113946Ωq

1113954WIjσij minus 1113954WIbi1113960 1113961dΩ 0

1113946Γqi

1113954WItidΓ + 1113946Γqu

1113954WItidΓ + 1113946Γqt



(4)

where Γqi is the inner boundary of the integral domain Γqu isthe part of the natural boundary that intersects the integral

domain and Γqt is the part of the essential boundary thatintersects the integral domain

2 Advances in Civil Engineering

In the local weak meshless method based on RPIM theradial base point interpolation shape function is used toobtain the displacement of the calculated point and thestrain and stress vectors are further obtained

ε(6lowast1) B(6lowast3n)u(3nlowast1) (5)

σ(6lowast1) D(6lowast6)ε(6lowast1) D(6lowast6)B(6lowast3n)u(3nlowast1) (6)

B is a differential operatorD is the material constant matrixe system equation is discretized into matrix form and

the following formula is obtained

1113946Ωq

1113954VTI σdΩ minus 1113946

Γqi

1113954WItdΓ minus 1113946Γqu

1113954WItdΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(7)

where 1113954WI is the weight function matrix and 1113954VTI is the de-

rivative matrix of weight functionBring formulas (5) and (6) into formula (7)

1113946Ωq

1113954VTI DBudΩ minus 1113946

Γqi

1113954WT

I nDBudΓ minus 1113946Γqu

1113954WT

I nDBudΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(8)

e matrix form can be expressed as

KI( 1113857(3lowast3n)(u)(3nlowast1) fI( 1113857(3lowast1) (9)

where KI is the stiffness matrix of the field node I

KI 1113946Ωq

1113954VTI DBdΩ minus 1113946

Γqi

1113954WT

I nDBdΓ minus 1113946Γqu

1113954WT

I nDBdΓ

(10)

where fI is the nodal force vector including the physicalforce in the problem domain and the surface force on thenatural boundary

fI 1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ (11)

e standard gauss integral and the rectangular localintegral domain are used e final system equations can beobtained by assembling all these equations based on thesystem of population numbers

K3Nlowast3NU3Nlowast1 F3Nlowast1 (12)

For crack expansion the relationship between thenodes and Gauss integral points is constantly changingerefore the visual criterion is used to address the rangeof the influence domain If the line between the field nodeand the Gauss integral point (either a solid boundary orcrack) intersects the node is considered to be ldquomaskedrdquoAfter assembling the total rigid matrix a penalty func-tion is used to apply the displacement boundaryconditions

3 Methods

31 Overview A three-dimensional elastic-plastic local ra-dial basis point interpolation method for calculating crackpropagation is established e key techniques used in thissection are described in detail (1) According to the principleof the minimum potential energy the governing equation ofthe shear crack action is derived based on the idea of theGoodman element which is commonly used in finite ele-ment analysis (2) e elastic-plastic constitutive model ofthe shear cracks is derived based on the elastoplastic theory(3) According to crack mechanics theory the criterion ofshear crack propagation is obtained which can be used forshear crack propagation

32 Derivation of the Governing Equation of the Shear CrackFor the contact problem of shear cracks the materials onboth sides of the crack cannot be embedded into each otherHowever to calculate the interaction between cracks theidea of the Goodman element which is often used in theFEM is used to address this problem [19] It is assumed thatthere will be a small amount of overlap between thematerialson both sides of the crack and a relatively large normalstiffness is set When there is a small amount of overlapbetween the materials above and below the crack a largenormal force will occur At the same time the shear stiffnesscoefficient is set to reflect the tangential force generated bythe rubbing of the crack

e crack profile is shown in Figure 1 e red line is thecrack profile the hollow dot is the field node and the solidpoint is on the crack It should be pointed out that the pointson the crack are not directly involved in the calculation ofthe model Taking a point on the crack as the origin the localcoordinate system is establishede normal direction of thecrack is along the yprime axis and the two tangent directions arealong the xprime axis and zprime axis Two points are set up in thepositive and negative directions of the yprime axis and there is asmall distance from the origin which represent the materialabove and below the crack respectively

e energy equation considering crack interaction isshown in equation (13) which represents the work

Πc 12

1113946ΓΔuTσdΓ (13)

where Δu is the difference in the normal displacement of thematerials on both sides of the crack If Δu is negative there isan interaction between the two sides of the crack and it isnecessary to calculate the stress-strain effect of this actionΔv and Δw represent the difference in the displacementbetween the two shear directions in the local coordinatesystem

Δu

Δu

Δv

Δw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

uj

vj

wj

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (14)

where ui vi wi uj vj and wj represent the displacements ofpoints i and j in the local coordinate system e

Advances in Civil Engineering 3

displacement value can be obtained by interpolation fromthe displacement information of the field node In this paperthe radial base point interpolation method is used as follows

ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Φiu (15)

where u is the displacement vector of the field node in thesupport domain of the interpolation node and the vectorlength is 3n

u ui1 v

i1 w

i1 u

i2 v

i2 w

i2 u

in v

in w

in

1113872 1113873 (16)

and Φi is the displacement shape function of node i in thefollowing form

Φi

ϕi1 0 0 ϕi

2 0 0 ϕin 0 0

0 ϕi1 0 0 ϕi

2 0 0 ϕin 0

0 0 ϕi1 0 0 ϕi

2 0 0 ϕin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (17)

e shape function and displacement vector should beintegrated when solving the problem Here we should payattention to the corresponding point number We need toexpand the form as follows

Φ

ϕi1 0 0 ϕi

2 0 0 ϕin 0 0 minusϕj

1 0 0 minusϕj2 0 0 minusϕj

n 0 0

0 ϕi1 0 0 ϕi



n 0

0 0 ϕi1 0 0 ϕi


1 0 0 minusϕj2 0 minusϕj

n 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18)

Here σ is set as the node stress vector and D is thematerial matrix then σ

σ

τs

σn

τn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ D times Δu (19)

σn is the normal force and τs and are the component forcesin the two tangent directions

kn and ks are the normal and tangential stiffnessesrespectively

D

ks 0 0

0 kn 0

0 0 ks

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20)

It is necessary to transform the displacement of thenodes from the global coordinate system to the local co-ordinate system R is a transformation matrix that convertsglobal coordinates to local coordinates

R

l1 m1 n1

l2 m2 n2

l3 m3 n3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)

e displacement in the local coordinate system is ob-tained as follows

uprime RΦu (22)

A functional considering the crack energy is establishedbased on the principle of the minimum potential energy Itsform is shown in the following formula

Πc 12

1113946ΓΔuprimeTσdΓ

Πc 12

1113946ΓΔuprimeTDΔudΓ

(23)

By variational calculation

δΠc δuT1113946ΓΦTRTDRΦdΓ1113874 1113875u 0

KC 1113946ΓΦTRTDRΦdΓ

(24)

KC is the stiffness matrix reflecting the action of the crackWhen there are no repeated calculation points in the fieldnode of the crack the action of the crack is reflected by thesquare matrix of 3(m + n) middot 3(m + n) e stiffness matrixcan be combined into the stiffness matrix established by thelocal weak meshless method In this process we should payattention to the corresponding relationship between thenodes

e matrix form of the crack surface energy functional isobtained as follows

jzprime

xprime

i

yprime

Figure 1 Schematic diagram of the local coordinate system


KC 1113946ΓΦTRTDRΦdΓ AiΦ

TRTDRΦ (25)

33 Elastoplastic 4eory of the Shear Crack Limited by thestrength of the crack when the shear stress of the crackexceeds the maximum value the shear stress of the crackshould not continue to increase It is assumed that the crackis a smooth surface which accords with the deformationcharacteristics of ideal plasticity

e stress state of the point on the crack can bedecomposed into three directions of stress as shown inFigure 2 A local coordinate system is established at a nodeon the crack and the normal stress of the crack is σn If theresultant force of the tangential force is expressed as τs thenthe value of the shear stress in the direction perpendicular toσn and is zero It is assumed here that the direction of τs is thesame as that of the relative displacement of the crack

According to the MohrndashCoulomb strength criterion thegeneral form of the yield function is as follows

f τs

11138681113868111386811138681113868111386811138681113868 minus c minus σntanφ (26)

where c and ϕ are the cohesion and internal friction of thecrack τs is the shear stress of the crack and σn is the normaldirection of the crack

zf

zσ1113896 1113897

zf

zτs

zf

zσn

zf

zτn

1113890 1113891

T

1 minustanϕ 01113858 1113859T (27)

e elastic matrix of the stress displacement relationship

can be written as follows D

ks 0 00 kn 00 0 ks

⎛⎜⎝ ⎞⎟⎠

In elastoplastic theory the deformation matrix includesthe elastic modulus and Poissonrsquos ratio in kPa e stress isobtained by multiplying the matrix and the strain vectorHere the stiffness coefficient is in the deformationmatrix andits unit is kPam which corresponds to the displacementdifference between the upper and lower nodes of the crackthe unit ism that is the strain is replaced by the displacement

e plastic matrix is as follows

Dp D zgzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zgzσ1113864 1113865 (28)

Under the ideal elastic-plastic condition A 0 Anadaptive flow criterion is adopted as g f e elastic-plasticdeformation matrix of the shear crack surface is as follows

Dp D zfzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zfzσ1113864 1113865

1S0

k2s ksS1 0

ksS1 S21 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S0 ks + kntan2ϕ

S1 minuskntanϕ

(29)

e elastic-plastic matrix after yielding is as follows

Dep D minus Dp

Dep

ks minus1S0

k2s minus

1S0

ksS1 0

minus1S0

ksS1 kn minus1S0

S21 0

0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

e deformation matrix under ideal elastoplastic con-ditions is only related to the stiffness coefficient and internalfriction angle

From the above derivation we can obtain the consti-tutive relation of the shear cracks under ideal plastic con-ditions Before yielding the deformation matrix is D andafter yielding the elastic-plastic deformation matrix is DepIn this way the yield criterion and elastic-plastic constitutiverelation of the cracks are established

In the calculation when the crack is open the normalstress is zero the crack is a free surface and the contact effectbetween the cracks is not considered When the normalstress of the crack is not zero it indicates that the cracks arein contact with each other and that there is a force effect Inthe solution the stress state of the point on the crack isreflected by modifying the deformation matrix

34 Crack Propagation Criterion In this paper the maxi-mum circumferential tensile stress theory is selected toanalyze crack propagation For the I-II tension shearcomposite mode crack [24] under the plane strain state thecritical conditions are as follows

kIC cosθ02

kIcos2θ02

minus32kIIsin θ01113890 1113891 (31)

e expansion step can be set to a fixed value which is acommon setting method After determining the crack cri-terion we can judge whether the crack tip is cracked anddetermine the direction of the crack According to previouswork this paper simulates three-dimensional crack propa-gation by expanding a finite number of crack front points intheir respective crack front normal planes

e crack is simulated by a group of small triangles anda group of line segments are used to simulate the crack frontWhen themodel is balanced the current stress-strain state of

n

n

s

Figure 2 Local coordinate system


the nodes in the calculation domain can be obtained and thepropagation of the crack can be calculated As shown inFigure 3 the stress states of points B C and E in the crackfront edge in Figure 3(a) meet the expansion criteria andexpand in a certain direction and a new crack tip endpoint isestablished in its propagation direction In diagram 3(b) twonew triangular planes are formed by the new tip point andtwo points A and C which realize the first step expansion ofthe crack e extensions of points C and E are realized bythe same method and the new crack is reconstructed asshown in Figure 3(d) With the expansion of the crack therelationship between the calculation point and the field nodewill constantly change It is necessary to recalculate thestiffness matrix solve the equilibrium equation and thencarry out the next step of the expansion calculation

35 Calculation of the Shear Crack Propagation in a RockMass Combining the solution of the local weak meshlessmethod crack propagation and shear crack interaction theprogram is compiled according to the discrete rectangularform e calculation process is as follows

(1) e initial conditions are set including the ar-rangement of the nodes the definition of the ma-terial parameters the application of the boundaryconditions and the application of the loads

(2) Applying the nth-order load the substiffness matrixof the joint and the stiffness matrix of the shear crackare calculated e stress-strain state of the wholemodel is obtained by solving the equilibrium con-vergence of several iterative steps

(3) After entering the expansion step we can judgewhether the crack is expanding and determine thedirection of the crack propagation according to thestress state at the crack tip If the crack tip meets thecrack criterion crack propagation is realized and theiterative calculation is carried out again in step 2 If thecrack does not expand then proceed to the next step

(4) Judge whether level N load has been applied If soterminate the calculation otherwise return to step②

e calculation module of shear crack propagation isintegrated with the meshless LRPIM program

4 Results Case Studies

e correctness and stability of the calculation method areverified by the following three examples Example 1 verifiesthe calculation method and realizes the force transfer effectof cracks and case 2 verifies the influence of oblique shearcracks on the deformation characteristics of themodel underdifferent tangential stiffnesses to verify the correctness of thecalculation results In case 3 the propagation path of thebuilt-in inclined crack under axial compression is obtained

41 Model Information e information for the examplemodel is as follows the dimensions are 50mm times 50mmtimes

100mm the deformation parameters include an elastic

modulus of 16GPa and a Poissonrsquos ratio of 025 the upper andlower surfaces of the model are fixed and a vertical dis-placement is applied on the upper surface and the displace-ment value is 1mm

According to whether the interaction between the cracksis considered the experimental calculation is carried out intwo groups e first experiment does not use the developedcalculation module and the other test uses the developedcalculation module e normal stiffness is set to 20 timesthe elastic modulus and the tangential stiffness is set to 01times the elastic modulus e node distribution of themodel is shown in Figure 4

If the interaction between cracks is not considered thecalculated displacement nephogram is shown in Figure 5(a)e upper part of the model has a consistent vertical dis-placement value of 1mm Because the interaction betweenthe cracks is not considered the upper part of the modelcannot transfer the force to the lower part so the dis-placement value of the lower part is 0mm because there is noforce on the lower part is calculation result is consistentwith the expected results

If the interaction of cracks is considered the verticaldisplacement nephogram of the model is obtained as shownin Figure 5(b) e displacement of the model presents aregular ladder shape and the force between the cracks iseffectively transferred is shows that the local weakmeshless method considering the interaction between cracksurfaces is effective and the established calculation methodcan effectively calculate the interaction between the two sidesof the crack

42 Oblique Crack Model

421 Elastic CrackModel ere is an oblique through crackin the middle of the model in Figure 6 which divides themodel into two parts e parameters and boundary con-ditions of the model are consistent with those of test 1 butthe crack is oblique e normal stiffness is 20 times theelastic modulus and the tangential stiffness changes ecorrectness of the algorithm is verified by setting differenttangential stiffness coefficients Five test models are set andthe tangential stiffness is set to 0 times 005 times 01 times015 times and 02 times the elastic modulus

It should be noted that the crack is implicit in this al-gorithm e results do not include crack information onlyfield node information e field node at the crack of thedisplacement nephogram is not on the crack surface Inaddition the value shown in the legend is not the maximumand minimum value of the cloud map but the segmentedvalue e maximum values of the calculated results arecompared in Table 1

e horizontal displacement nephograms of the five testresults are shown in Figure 7 e maximum horizontaldisplacements are approximately 7069e-2mm 6829e-2mm 6604e-2mm 6393e-2mm and 6195e-2mm It canbe understood that due to the increasing tangential stiffnessthe deformation of the model on both sides of the crack isconstrained resulting in the decrease in the lateral


a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)

Figure 3 Geometric sketch of crack propagation [25]

Figure 4 Model and field node distribution diagram

minus6250E minus 05minus1250E minus 04minus1875E minus 04minus2500E minus 04minus3125E minus 04minus3750E minus 04minus4375E minus 04minus5000E minus 04minus5625E minus 04minus6250E minus 04minus6875E minus 04minus7500E minus 04minus8125E minus 04minus8750E minus 04minus9375E minus 04

(a)


(b)

Figure 5 Vertical displacement nephogram (m)


Table 1 Horizontal displacement of the model

Tangential stiffness (Pa) 00E 005E 01E 015E 02E

Maximum horizontal displacement (mm) LRPIM 7069eminus 2 6829eminus 2 6604eminus 2 6393eminus 2 6195eminus 2FDM 6997eminus 2 6887eminus 2 6782eminus 2 6682eminus 2 6585eminus 2

Error () 10 08 26 43 59

FLAC3D

69967e minus 0560000e minus 0550000e minus 0540000e minus 0530000e minus 0520000e minus 0510000e minus 0500000e + 00minus10000e minus 05minus20000e minus 05minus30000e minus 05minus40000e minus 05minus50000e minus 05minus60000e minus 05minus69932e minus 05

(a)

61890e minus 0553087e minus 0544283e minus 0535479e minus 0526675e minus 0517871e minus 0590671e minus 0626315e minus 07minus85408e minus 06minus17345e minus 05minus26149e minus 05minus34952e minus 05minus43756e minus 05minus52560e minus 05minus61364e minus 05

LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm

Figure 6 Schematic diagram of oblique through crack


FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued


deformation which is consistent with the expected calcu-lation results

As shown in Table 1 the maximum horizontal dis-placement decreased with increasing tangential stiffness andthe calculation results obtained by the two algorithms werevery close

e above derivation and calculation results assume thatunder elastic conditions the material and crack have no yieldcriterion or constitutive relation under elastic-plastic condi-tions e model considering the plastic deformation of crackswill be calculated and analyzed in the following section

422 Elastoplastic Crack Model e interface elements inFLAC3D also adopt the MohrndashCoulomb strength criterionand ideal plastic constitutive model e contact surfaceparameters include the normal stiffness tangential stiffnesscohesion internal friction angle tensile strength and ex-pansion angle e tensile strength and expansion angle ofthe contact surface are not considered in this paper

e crack is set as elastic-plastic the cohesion is 0 kPa andthe friction coefficient is 001 A normal stiffness of 20 andtangential stiffness of 01 times the elastic modulus are set

e calculation results of the LRPIM and FLAC3D areshown in Figure 8e results of the two numerical exampleshave the same displacement nephogram As shown inTable 2 under ideal plastic conditions the maximumhorizontal displacement of the model using the LRPIM is7051eminus 2mme results of the two methods are close andthe error is only 08

43 Built-In Inclined Crack Model For shear cracks thestress intensity factor is KI 0 because there is no relativedisplacement in the normal direction According to themaximum circumferential stress theory θ 705deg

Under the action of axial pressure the cracks of brittlerock with built-in inclined cracks will expand and the crackpropagation pattern is shown in Figure 9 [26]

e model parameters are the same as the parameters ofcommon marble e model information is set as followsdimensions of 62mmtimes 25mmtimes 110mm elastic modulus of17GPa Poissonrsquos ratio of 025 internal friction angle of 30degand fixed displacement of the upper and lower surfaces ofthe model e angle between the crack and the horizontalplane is 45deg and the crack length is 20mm e cohesiveforce of the model material is 209 kPa the cohesive force ofthe crack is 2 kPa the friction coefficient is 001 the normalstiffness is twice the elastic modulus and the tangentialstiffness is 001 times the elastic modulus e force diagramof the model is shown in Figure 10

A vertical displacement of 02mm is applied on the uppersurface of the model to simulate the stress state of compressionand promote the expansion of cracks Figure 11 shows thevertical displacement nephogram and stress nephogram in theinitial state ere is discontinuity in the model while thevertical stress at the crack tip is obviously larger

Figure 12 shows the crack morphology calculated by theLRPIM and the two ends of the crack are curved and extendupward which demonstrates the propagation process of thewing crack

5 Discussion

51 Simulation Analysis of Shear Cracks Using the LRPIMUnder certain assumptions the governing equation of theshear cracks under three-dimensional conditions is derivedand discretized into a programmable mode to realize thesolution and expansion of shear crack propagation in theLRPIM


FLAC3D

(i)


LRPIM (Ks = 02e)

(j)

Figure 7 Horizontal displacement nephogram (m)


e constitutive relation of the shear cracks is establishedaccording to elastic-plastic theory e elastic-plastic stress-strain relationship and yield criterion of the cracks arederived according to the MohrndashCoulomb strength criterionwhich is commonly used in rock mechanics Based on theabove theory the incremental iteration method is used tosolve the problem

e results of several numerical examples are basicallyconsistent with the expected results which verifies the cor-rectness of the relevant algorithms e horizontal throughcrack model verifies that the crack realizes force transfer Amodel of the oblique through cracks is established and thedeformation of the cracks in the elastic and elastic-plasticconditions is calculated Compared with the FDM results thesource of the difference is analyzed Compared with the cal-culation results of the classical model with built-in inclinedcracks the ability of this method to simulate the propagation ofshear composite cracks is verified

e results of the numerical model calculation show thatthe developed method can effectively simulate the shearcrack and its propagation in the rock mass and can accu-rately calculate the force between the two sides of the crack

52 Comparative Analysis In the simulation of crackpropagation compared with the results of classical physicalexperiments the crack propagation is not sufficientlysmooth which indicates that the calculation accuracy needsto be improved In meshless computing there are many

69949E minus 0560000E minus 0550000E minus 0540000E minus 0530000E minus 0520000E minus 0510000E minus 0500000E + 00minus10000E minus 05minus20000E minus 05minus30000E minus 05minus40000E minus 05minus50000E minus 05minus60000E minus 05minus69914E minus 05

(a)

60472E minus 0551870E minus 0543268E minus 0534666E minus 0526065E minus 0517463E minus 0588606E minus 0625870E minus 07minus83432E minus 06minus16945E minus 05minus25547E minus 05minus34149E minus 05minus42751E minus 05minus51353E minus 05minus59955E minus 05

(b)

Figure 8 Horizontal displacement nephogram (m) (a) FLAC3D (b) LRPIM

Table 2 Horizontal displacement comparison

Method LRPIM FLAC3D

Maximum horizontal displacement (mm) 7051eminus 2 6995eminus 2Error () 08

Shear fracture

θ

Figure 9 Crack propagation under uniaxial compression [26]

25mm

20mm

62mm

110mm

4793mm

45deg

Figure 10 Schematic diagram of cracks subjected to uniaxialcompression


factors that affect the accuracy of the model and the in-fluence law is complex Different distributions of modelnodes may result in different calculation results which mayaffect the crack propagation path of the model Whensimulating the crack propagation of a rock mass the crackrepresentation methods used are all implicit However thisrepresentation is more flexible than other methods Becausethere are no nodes on the crack the stress-strain state of thecrack cannot be accurately obtained In the future thedisplay method can be used to characterize cracks

6 Conclusion

In this paper the governing equation of the shear cracksunder three-dimensional conditions is derived e con-stitutive relation of the shear cracks is established accordingto elastic-plastic theory e stress-strain relationship of themodel material and crack is unified and the solution issolved by the incremental iteration method which realizesthe solution and expansion of shear crack propagation in theLRPIM e second development in the LRPIM program is

minus20000E minus 05minus28421E minus 05minus36842E minus 05minus45263E minus 05minus53684E minus 05minus62105E minus 05minus70526E minus 05minus78947E minus 05minus87368E minus 05minus95789E minus 05minus10421E minus 04minus11263E minus 04minus12105E minus 04

minus13789E minus 04minus14632E minus 04minus15474E minus 04minus16316E minus 04minus17158E minus 04minus18000E minus 04

minus12947E minus 04

(a)

minus54335E + 06minus78874E + 06minus10341E + 07minus12795E + 07minus15249E + 07minus17703E + 07minus20157E + 07minus22611E + 07minus25065E + 07minus27518E + 07minus29972E + 07minus32426E + 07minus34880E + 07minus37334E + 07minus39788E + 07minus42242E + 07minus44696E + 07minus47150E + 07minus49603E + 07minus52057E + 07

(b)

Figure 11 Nephogram of the model subjected to unidirectional compression (a) Vertical displacement (m) (b) Vertical stress (Pa)

Figure 12 Crack propagation under uniaxial compression


carried out to establish the related computing ability Severalnumerical examples are used to verify the reliability andaccuracy of the algorithm Compared with the calculationresults the source of the difference is analyzed

e research results show that the LRPIM as a realmeshless method can simulate the shear crack of a rockmass and avoid the inconvenience of meshing so it has greatapplication potential e experimental results show thatfurther work is needed to improve the calculation accuracy

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Natural Key RampD Programof China (Grant no 2018YFC1508703) and level I RampDproject for Yellow River Engineering Consulting Co Ltd(Grant no 2017-ky(02))

References

[1] H Haeri K Shahriar and M Fatehi Marji ldquoParviz moarefvandsimulating the bluntness of tbm disc cutters in rocks usingdisplacement discontinuity methodrdquo in Proceedings of the In-ternational Conference on Fracture Beijing China June 2013

[2] A Hedayat H Haeri J Hinton et al ldquoGeophysical signatures ofshear-induced damage and frictional processes on rock jointsrdquoJournal of Geophysical Research Solid Earth vol 112 2018

[3] T Rabczuk and T Belytschko ldquoCracking particles a sim-plified meshfree method for arbitrary evolving cracksrdquo In-ternational Journal for Numerical Methods in Engineeringvol 61 no 13 pp 2316ndash2343 2010

[4] T Rabczuk G Zi S Bordas et al ldquoA simple and robust three-dimensional cracking-particle method without enrichmentrdquoComputer Methods in Applied Mechanics amp Engineeringvol 199 no 37ndash40 pp 2437ndash2455 2010

[5] H Ren X Zhuang Y Cai and T Rabczuk ldquoDual-horizonperidynamicsrdquo International Journal for Numerical Methodsin Engineering vol 108 no 12 pp 1451ndash1476 2016

[6] H Ren X Zhuang and T Rabczuk ldquoDual-horizon peridy-namics a stable solution to varying horizonsrdquo ComputerMethods in Applied Mechanics and Engineering vol 318pp 762ndash782 2017

[7] D V Swenson and A R Ingraffea ldquoModeling mixed-modedynamic crack propagation nsing finite elements theory andapplicationsrdquo Computational Mechanics vol 3 no 6pp 381ndash397 1988

[8] T Belytschko and T Black ldquoElastic crack growth in finite ele-ments with minimal remeshingrdquo International Journal for Nu-merical Methods in Engineering vol 45 no 5 pp 601ndash620 1999

[9] N Moes J Dolbow and T Belytschko ldquoA finite elementmethod for crack growth without remeshingrdquo InternationalJournal for Numerical Methods in Engineering vol 46 no 1pp 131ndash150 2015

[10] Y Jianping C Weizhong Y Diansen and Y JingQiangldquoNumerical determination of strength and deformability of

fractured rock mass by FEM modelingrdquo Computers andGeotechnics vol 64 pp 20ndash31 2015

[11] J Tejchman and J Bobinski ldquoContinuous and discontinuousmodelling of fracture in plain concrete under monotonicloadingrdquo 2013

[12] P Areias T Rabczuk J C De Sa and J Cesar ldquoA novel two-stage discrete crack method based on the screened Poissonequation and local mesh refinementrdquo Computational Me-chanics vol 58 no 6 pp 1003ndash1018 2016

[13] M Abendroth andM Kuna ldquoIdentification of ductile damageand fracture parameters from the small punch test usingneural networksrdquo Engineering Fracture Mechanics vol 73no 6 pp 710ndash725 2006

[14] T Belytschko Y Krongauz D Organ M Fleming andP Krysl ldquoMeshless methods an overview and recent devel-opmentsrdquo Computer Methods in Applied Mechanics andEngineering vol 139 no 1ndash4 pp 3ndash47 1996

[15] R Tian and L Wen ldquoImproved XFEM-An extra-dof freewell-conditioning and interpolating XFEMrdquo ComputerMethods in Applied Mechanics and Engineering vol 285no 3 pp 639ndash658 2015

[16] S Mariani and U Perego ldquoExtended finite element methodfor quasi-brittle fracturerdquo International Journal for NumericalMethods in Engineering vol 58 no 1 pp 103ndash126 2003

[17] D Deb and K C Das ldquoExtended finite element method forthe analysis of discontinuities in rock massesrdquo Geotechnicaland Geological Engineering vol 28 no 5 pp 643ndash659 2010

[18] X P Zhou and H Q Yang ldquoMultiscale numerical modelingof propagation and coalescence of multiple cracks in rockmassesrdquo International Journal of Rock Mechanics and MiningSciences vol 55 pp 15ndash27 2012

[19] T Belytschko L Gu and Y Y Lu ldquoFracture and crack growthby element free Galerkin methodsrdquoModelling and Simulationin Materials Science and Engineering vol 2 no 3App 519ndash534 1994

[20] P Krysl and T Belytschko ldquoe Element Free Galerkinmethod for dynamic propagation of arbitrary 3-D cracksrdquoInternational Journal for Numerical Methods in Engineeringvol 44 no 6 pp 767ndash800 2015

[21] T Belytschko D Organ and C Gerlach ldquoElement-freegalerkin methods for dynamic fracture in concreterdquo Com-puter Methods in Applied Mechanics and Engineering vol 187no 3-4 pp 385ndash399 2000

[22] X Zhuang C Augarde and S Bordas ldquoAccurate fracturemodelling using meshless methods the visibility criterion andlevel sets formulation and 2D modellingrdquo InternationalJournal for Numerical Methods in Engineering vol 86 no 2pp 249ndash268 2011

[23] G R Liu G Y Zhang Y T Gu and Y Y Wang ldquoAmeshfreeradial point interpolation method (RPIM) for three-dimen-sional solidsrdquo Computational Mechanics vol 36 no 6pp 421ndash430 2005

[24] R J Nuismer ldquoAn energy release rate criterion for mixedmode fracturerdquo International Journal of Fracture vol 11no 2 pp 245ndash250 1975

[25] Y Li N Xu J Tu and G Mei ldquoComparative modelling ofcrack propagation in elasticndashplastic materials using themeshfree local radial basis point interpolation method andeXtended finite-element methodRrdquo Royal Society Open Sci-ence vol 6 no 1 Article ID 6190543 2019

[26] H Horii and S Nemat-Nasser ldquoCompression-inducedmicrocrack growth in brittle solids axial splitting and shearfailurerdquo Journal of Geophysical Research vol 90 no B4p 3105 1985


and efficiency Meshing and remeshing has significantlyimproved in the past years An alternative crack propagationalgorithm has been proposed which effectively circumventsthe variable transfer procedure adopted with classical meshadaptation algorithms [12] And neural networks can beused to compute the crack growth in fracture specimenswhich finally leads to a prediction of classical fracturetoughness parameters [13] ese shortcomings can also beavoided by the XFEM proposed by Belytschko and Moes[8 9 14] e XFEM rationally realizes accurate crack de-scriptions and has become a popular research topic amongnumerical methods for crack problems [15]

Moes et al [9] used the XFEM to simulate bond crackpropagation with the stress intensity factor as the crackindex Mariani et al [16] used the XFEM to simulate qua-sistatic bond crack propagation in quasibrittle materialsDeb and Das [17] combined the XFEM with cohesion tosimulate shear crack slip Zhou and Yang [18] proposed amultiscale XFEM to study crack propagation and rockbridge penetration under multidirectional compression andthe conclusions were in good agreement with the experi-mental results

Compared with the FEM and XFEM the meshlessmethod is based on field nodes that can eliminate or partiallyeliminate the difficulties caused by meshing e interpo-lation (approximation) function of the meshless methoddoes not depend on the elements so it has the advantages ofeasy higher order approximation formation and realizationof local node encryption bringing convenience to the nu-merical analysis of cracks

Belytschko [19] first used the EFGM method to analyzecrack propagation Krysl [20] further extended the EFGMmethod to crack propagation calculations in a three-di-mensional model Belytschko [21] applied the meshlessmethod to simulate dynamic crack propagation in concretestructures and they simulated the crack propagation di-rection and velocity in concrete structures very well ZhuangXiaoying et al [22] realized accurate fracture modeling in atwo-dimensional model using meshless methods the visi-bility criterion and level sets

e above applications all use the meshless EFGM butthis method requires the background grid for integrationand is not a true meshless method e LRPIM [23] methodis a local weak computing method that does not require thebackground grid and is a real meshless method Little re-search has been conducted on crack problems with the

LRPIM ree-dimensional shear crack propagation withthe LRPIM has not yet been studied A three-dimensionalshear crack propagation calculation is established based onelastic-plastic theory and fracture mechanics e reliabilityof the method is verified through numerical experimentsand the problems that need to be improved are discussed

is paper is organized as follows Section 2 brieflyintroduces the background of the LRPIM Section 3 mainlyconcentrates on the methods of modeling shear cracks withthe LRPIM Section 4 discusses the results Finally Section 5summarizes the conclusions

2 Background LRPIM

If only the static case is considered the BVP of three-di-mensional solid mechanics can be expressed as

σijj + bi 0

σijnj ti in Γt

ui ui in Γu

u x t0( 1113857 u0(x) x isin Ω

(1)

where ui ti and u0 are the given displacement the givensurface force and the initial displacement I j (123)represents x y z

Considering the solid mechanics problem defined indomain Ω the local weighted residual method is used fornode I to satisfy the governing equation and the local weakequation of the node is obtained e form of the locallyweighted residuals is defined on the local integral domainΩq and the corresponding boundary Γq in the followingform

1113946Ωq

1113954WI σijj + bi1113872 1113873dΩ 0 (2)

where 1113954Wi is a weight function or test function centered atfield node I

e follow formula is obtained by integral operation

1113946Ωq

1113954WIσijjdΩ 1113946Γq


1113954WIjσijdΩ (3)

where ti is the component of surface force vector on theboundary

We bring formula (10) into formula (6)

1113946Γq

1113954WItiσijdΓ minus 1113946Ωq


1113946Γqi

1113954WItidΓ + 1113946Γqu

1113954WItidΓ + 1113946Γqt



(4)

where Γqi is the inner boundary of the integral domain Γqu isthe part of the natural boundary that intersects the integral

domain and Γqt is the part of the essential boundary thatintersects the integral domain







1113946Ωq

1113954VTI σdΩ minus 1113946

Γqi


1113954WItdΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(7)



1113946Ωq


Γqi

1113954WT


1113954WT

I nDBudΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(8)




KI 1113946Ωq


Γqi

1113954WT


1113954WT

I nDBdΓ

(10)


fI 1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ (11)




3 Methods





Πc 12

1113946ΓΔuTσdΓ (13)


Δu

Δu

Δv

Δw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

uj

vj

wj

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (14)




ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Φiu (15)


u ui1 v

i1 w

i1 u

i2 v

i2 w

i2 u

in v

in w

in

1113872 1113873 (16)


Φi

ϕi1 0 0 ϕi

2 0 0 ϕin 0 0

0 ϕi1 0 0 ϕi

2 0 0 ϕin 0

0 0 ϕi1 0 0 ϕi

2 0 0 ϕin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (17)


Φ

ϕi1 0 0 ϕi



n 0 0

0 ϕi1 0 0 ϕi



n 0

0 0 ϕi1 0 0 ϕi



n 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18)


σ

τs

σn

τn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ D times Δu (19)



D

ks 0 0

0 kn 0

0 0 ks

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20)


R

l1 m1 n1

l2 m2 n2

l3 m3 n3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)


uprime RΦu (22)


Πc 12


Πc 12


(23)




(24)



jzprime

xprime

i

yprime




TRTDRΦ (25)




f τs



zf

zσ1113896 1113897

zf

zτs

zf

zσn

zf

zτn

1113890 1113891

T

1 minustanϕ 01113858 1113859T (27)



ks 0 00 kn 00 0 ks

⎛⎜⎝ ⎞⎟⎠



Dp D zgzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zgzσ1113864 1113865 (28)


Dp D zfzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zfzσ1113864 1113865

1S0

k2s ksS1 0

ksS1 S21 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S0 ks + kntan2ϕ

S1 minuskntanϕ

(29)


Dep D minus Dp

Dep

ks minus1S0

k2s minus

1S0

ksS1 0

minus1S0

ksS1 kn minus1S0

S21 0

0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)





kIC cosθ02

kIcos2θ02

minus32kIIsin θ01113890 1113891 (31)



n

n

s























a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References


































1113946Ωq

1113954VTI σdΩ minus 1113946

Γqi


1113954WItdΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(7)



1113946Ωq


Γqi

1113954WT


1113954WT

I nDBudΓ

1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ

(8)




KI 1113946Ωq


Γqi

1113954WT


1113954WT

I nDBdΓ

(10)


fI 1113946Γqt

1113954WItdΓ + 1113946Ωq

1113954WT

I bdΩ (11)




3 Methods





Πc 12

1113946ΓΔuTσdΓ (13)


Δu

Δu

Δv

Δw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

uj

vj

wj

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (14)




ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Φiu (15)


u ui1 v

i1 w

i1 u

i2 v

i2 w

i2 u

in v

in w

in

1113872 1113873 (16)


Φi

ϕi1 0 0 ϕi

2 0 0 ϕin 0 0

0 ϕi1 0 0 ϕi

2 0 0 ϕin 0

0 0 ϕi1 0 0 ϕi

2 0 0 ϕin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (17)


Φ

ϕi1 0 0 ϕi



n 0 0

0 ϕi1 0 0 ϕi



n 0

0 0 ϕi1 0 0 ϕi



n 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18)


σ

τs

σn

τn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ D times Δu (19)



D

ks 0 0

0 kn 0

0 0 ks

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20)


R

l1 m1 n1

l2 m2 n2

l3 m3 n3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)


uprime RΦu (22)


Πc 12


Πc 12


(23)




(24)



jzprime

xprime

i

yprime




TRTDRΦ (25)




f τs



zf

zσ1113896 1113897

zf

zτs

zf

zσn

zf

zτn

1113890 1113891

T

1 minustanϕ 01113858 1113859T (27)



ks 0 00 kn 00 0 ks

⎛⎜⎝ ⎞⎟⎠



Dp D zgzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zgzσ1113864 1113865 (28)


Dp D zfzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zfzσ1113864 1113865

1S0

k2s ksS1 0

ksS1 S21 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S0 ks + kntan2ϕ

S1 minuskntanϕ

(29)


Dep D minus Dp

Dep

ks minus1S0

k2s minus

1S0

ksS1 0

minus1S0

ksS1 kn minus1S0

S21 0

0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)





kIC cosθ02

kIcos2θ02

minus32kIIsin θ01113890 1113891 (31)



n

n

s























a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References






























ui

vi

wi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Φiu (15)


u ui1 v

i1 w

i1 u

i2 v

i2 w

i2 u

in v

in w

in

1113872 1113873 (16)


Φi

ϕi1 0 0 ϕi

2 0 0 ϕin 0 0

0 ϕi1 0 0 ϕi

2 0 0 ϕin 0

0 0 ϕi1 0 0 ϕi

2 0 0 ϕin

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (17)


Φ

ϕi1 0 0 ϕi



n 0 0

0 ϕi1 0 0 ϕi



n 0

0 0 ϕi1 0 0 ϕi



n 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18)


σ

τs

σn

τn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ D times Δu (19)



D

ks 0 0

0 kn 0

0 0 ks

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20)


R

l1 m1 n1

l2 m2 n2

l3 m3 n3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)


uprime RΦu (22)


Πc 12


Πc 12


(23)




(24)



jzprime

xprime

i

yprime




TRTDRΦ (25)




f τs



zf

zσ1113896 1113897

zf

zτs

zf

zσn

zf

zτn

1113890 1113891

T

1 minustanϕ 01113858 1113859T (27)



ks 0 00 kn 00 0 ks

⎛⎜⎝ ⎞⎟⎠



Dp D zgzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zgzσ1113864 1113865 (28)


Dp D zfzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zfzσ1113864 1113865

1S0

k2s ksS1 0

ksS1 S21 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S0 ks + kntan2ϕ

S1 minuskntanϕ

(29)


Dep D minus Dp

Dep

ks minus1S0

k2s minus

1S0

ksS1 0

minus1S0

ksS1 kn minus1S0

S21 0

0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)





kIC cosθ02

kIcos2θ02

minus32kIIsin θ01113890 1113891 (31)



n

n

s























a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References






























TRTDRΦ (25)




f τs



zf

zσ1113896 1113897

zf

zτs

zf

zσn

zf

zτn

1113890 1113891

T

1 minustanϕ 01113858 1113859T (27)



ks 0 00 kn 00 0 ks

⎛⎜⎝ ⎞⎟⎠



Dp D zgzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zgzσ1113864 1113865 (28)


Dp D zfzσ1113864 1113865 zfzσ1113864 1113865

TDA + zfzσ1113864 1113865

TD zfzσ1113864 1113865

1S0

k2s ksS1 0

ksS1 S21 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S0 ks + kntan2ϕ

S1 minuskntanϕ

(29)


Dep D minus Dp

Dep

ks minus1S0

k2s minus

1S0

ksS1 0

minus1S0

ksS1 kn minus1S0

S21 0

0 0 ks

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)





kIC cosθ02

kIcos2θ02

minus32kIIsin θ01113890 1113891 (31)



n

n

s























a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References

















































a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References





























a

bc

de

f

xprime

yprime

(a)

a

b bprimec

de

f

xprime

yprime

(b)

a

b bprime

cprimecd

ef

xprime

yprime

(c)

a

b bprime

cprime

eprime

cd

e

f

xprime

yprime

(d)




(a)


(b)






Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References
































Error () 10 08 26 43 59

FLAC3D


(a)


LRPIM (Ks = 00e)

(b)

Figure 7 Continued

50mm

50mm

100mm

40mm60mm



FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References





























FLAC3D


(c)


LRPIM (Ks = 005e)

(d)

FLAC3D


(e)


LRPIM (Ks = 01e)

(f )


FLAC3D

(g)


LRPIM (Ks = 015e)

(h)

Figure 7 Continued













5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References








































5 Discussion



FLAC3D

(i)


LRPIM (Ks = 02e)

(j)








(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References


































(a)


(b)



Method LRPIM FLAC3D


Shear fracture

θ


25mm

20mm

62mm

110mm

4793mm

45deg




6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References






























6 Conclusion





(a)


(b)






Data Availability




Acknowledgments


References































Data Availability




Acknowledgments


References





























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ModelingofShearCrackPropagationinRockMassesUsing Mesh-FreeLRPIM · 2021. 3. 25. · LRPIM. ree-dimensional shear crack propagation with the LRPIM has not yet been studied. A three-dimensional