characteristics of the unloading process of rocks under high initial stress

Computers and Geotechnics 45 (2012) 83–92

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Computers and Geotechnics

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Characteristics of the unloading process of rocks under high initial stress

Ming Tao a,b, Xibing Li a,⇑, Chengqing Wu b

a School of Resources and Safety Engineering, Central South University, Changsha, Hunan, PR Chinab School of Civil, Environmental and Mining Engineering, The University of Adelaide, SA, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 January 2012Received in revised form 3 May 2012Accepted 8 May 2012Available online 4 June 2012

Keywords:Initial stressUnloading processStress release rateImplicit and explicit methodsRock failure

0266-352X/$ - see front matter Crown Copyright � 2http://dx.doi.org/10.1016/j.compgeo.2012.05.002

⇑ Corresponding author. Tel./fax: +86 7318836450.E-mail address: [email protected] (X. Li).

The unloading process of rocks under high initial stress is complex, and verifying the mechanism ofthe unloading process in the field or in a laboratory is not straightforward. In this study, the unloadingprocess of rocks under high initial stress was characterised by a mathematical physics model, whichwas then implemented in the finite element program LS-DYNA for analysis. In particular, the implicitand explicit methods were performed in sequence in the finite element simulation of rocks with initialstresses. In the numerical simulation, the characteristics of the dynamic unloading process of rocks wereinvestigated for various peak initial stresses, initial stress release paths and initial stress release rates(ISRRs). The numerical results indicated that the rock failure could be induced by the release of the initialstress; furthermore, there is a relationship between the magnitude of the unloading failure and the peakinitial stresses, the initial stress release paths and the ISRRs. When the initial stresses were at the samelevel, the equivalent initial stress release rate (EISRR) was introduced to quantitatively describe thecharacteristics of the unloading process. Using the numerical results, the unloading failure process wascharacterised, and a method for the static stress initialisation-dynamic unloading of rock was developed.

Crown Copyright � 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Underground rocks or ores have initial stresses and deforma-tions. In civil tunnelling and underground mining excavation, rocksare removed using different excavation techniques, such as using atunnel boring machine and blasting excavation, which create adynamic unloading process during excavation. The undergrounddynamic unloading process is well understood by the spring mod-el, as shown in Fig. 1 [1]. The natural rocks refer to the state shownin Fig. 1a. Rocks with initial stresses before excavation can havecompression deformation, in which a quasi-static equilibriumexists between the core ahead of the tunnel face and the surround-ing ground. This ‘‘rock spring’’ state is shown in Fig. 1b. As the sup-port effect of the core disappears with its excavation, deformationsoccur around the working face accompanied by initial stress redis-tribution. The state is shown in Fig. 1c, wherein the unloadingprocess is similar to the spring back.

The magnitude of the initial deformation of the ‘‘rock spring’’mainly depends on the magnitude of the deformations producedby the initial stress. Under the high initial stress condition, theunloading-induced stress release or transfer and the related defor-mation relaxation during excavation can drastically influence thestability of the underground openings. For instance, during the min-ing excavation process, unloading might induce rock destruction

012 Published by Elsevier Ltd. All

[2–5] or even rock bursts [6,7]. Therefore, capturing the unloadingmechanism of the rocks is necessary and significant for tunnellingand mining engineering.

The initiation and the propagation of stress-releasing-induceddamage or failure is a necessary precursor of all of the undergroundexcavation unloading problems in mining and civil tunnelling engi-neering. In practice, stress-induced failure can be directly observedusing an electron microscope, such as the acoustic emission and themicroseismic electron microscopes [8–10]. Cai and Kaiser [11]presented a method to characterise the excavation failure or theexcavation disturbed zone based on microseismic monitoring data.Malmgren et al. [2] measured the excavation-disturbed zone using aseismic monitor. These studies are very useful for predictingunloading fractures and rock bursts. Alternatively, an importantunloading parameter that has been identified is the energy releaserate [12,13], which verified that the failure of the potential energyrock is related to the magnitude of the energy release rate. In addi-tion, the role of the stress release path has also been examined in thecontext of mining or tunnelling through hard rock [14,15]; thesestudies found that the stress history of the rocks and their responsesto excavation may be influenced by time effects. However, theseauthors did not quantitatively describe the unloading characteris-tics of a given unloading process.

Previous studies have investigated the unloading failure ofrocks in the laboratory [16–18]. The rock-burst process under truetriaxial unloading has also been recently investigated in the labo-ratory using the acoustic emission monitoring technique [19].

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(a) Natural state (b) Initial state (c) Unloading state

Fig. 1. Three rock mass states [1].

84 M. Tao et al. / Computers and Geotechnics 45 (2012) 83–92

These unloading experiments indicated that the rock failure couldbe due to changes in the initial stress conditions. However, theunloading process is difficult to control; achieving the unloadingfailure process of hard rocks in the laboratory is not straightfor-ward. Therefore, most of the rocks unloading experiments areconducted on soft rock or medium-hard rock.

The unloading failure mechanism of rock using the disturbedstate concept was studied by Wu and Zhang [20]. The localisationof deformation and the stress–strain relationship for brittle rockmaterials under the unloading process were identified by Zhou[21]. However, some researchers found that many special phenom-ena in the unloading process of deep mines cannot be interpretedby the classic mechanism, such as the phenomenon of zonal disin-tegration [22]. Therefore, there are still many uncertainties duringthe unloading process.

In civil and underground engineering, numerical analysis meth-ods are popular and powerful for modelling brittle materials, suchas rock and concrete. An extensive review of the numerical methodsused in rock mechanics is presented by Jing and Hudson [23] andJing [24]. The numerical analysis methods can be divided into theimplicit and the explicit methods. For a static or a quasi-static ini-tialisation-dynamic unloading problem, the implicit method iscommonly used to perform the static stress initialisation of an expli-cit dynamic calculation. Many current commercial numericalsoftware packages have the capability of simulating implicit orexplicit problems. Currently, the static or quasi-static initialisationproblem is commonly solved using one software package at thebeginning and then transferring the results to another softwarepackage to perform a dynamic analysis. This process is typicallyreferred to as ‘‘incorporate coupling’’ [25,26]. However, incorporatecoupling requires extra coding and increases the computationalcomplexity of the problem. The possibility of coupling differentnumerical tools is constrained by the amount of computationresources, even though the speed of computers has improvedsignificantly.

In this study, the finite element program LS-DYNA wasemployed to simulate the unloading process of rocks with initialstresses. The solver of LS-DYNA is based on explicit time integra-tion, but an implicit solver is also available. The static stress ini-tialisation and the dynamic unloading process can be performedin sequence in the same software package, drastically reducingthe computational complexity of the problem. In this paper,hard-rock unloading problems were studied using theoretical andnumerical methods. The dynamic unloading process of rock in anelastic state was described by an elastic dynamic equation usinga mathematical physics analysis method. The solution of the elasticdynamic equation identified the main factors dominating theunloading process. Based on these factors, a three-dimensional(3D) elastic–plastic model was then established by LS-DYNA. Inthe numerical model, the continuous surface cap model (CSCM)was employed for the rock material. The material model was vali-dated by simulating the performance of a standard rock sample

under uniaxial compression. This validated material model wasthen used to derive the initialisation-dynamic unloading processof the rock. More specifically, the implicit and the explicit finiteelement methods were performed in sequence to analyse this pro-cess. The main factors of the dynamic unloading process wereclearly identified; in particular, the results showed that thedynamic control of the equivalent initial stress release rate forinducing failure can minimise the excavation energy. Additionally,the equivalent initial stress release rate can be dynamically con-trolled to protect the stability of the surrounding rocks.

2. Problem layout and solution method

As discussed in the introduction, the unloading problem can berepresented by the spring-back model. However, as in anystatically indeterminate system, the magnitude of the spring backdepends on many factors. To simplify the problem, the unloadingprocess is assumed to occur only in one direction, and the initialspatial stress exists only in one direction, i.e., the circumferenceis stress-free. Therefore, the initialisation–unloading process isestimated using the sample shown in Fig. 2.

This sample is a cylindrical rod, and the left end of the rod in they–z plane is fixed in the x, y and z directions. The loading and unload-ing processes are performed at the right end in the direction of the x-axis. All of the surfaces are free except those at the two ends.

Based on the elastic dynamic theory, the elastic dynamicdifferential equations in the x, y and z domains can be describedin Cartesian coordinates as follows [27]:

@2u@t2 ¼ c2 @2u

@x2 þ@2u@y2 þ

@2u@z2

!þ lg ð1Þ

where u is the displacement, t is the time, c is the speed of sound, lis the friction coefficient, and g is the acceleration of gravity.

Assuming that the initialisation–unloading process only occursin the x-axial direction, ou/oy = 0 and ou/oz = 0. Therefore, Eq. (1)can be simplified as

@2u@t2 ¼ c2 @

2u@x2 þ lg ð2Þ

Alternatively, setting the x value in a regional context, i.e., 0 6 x 6 l,where l is the length of the sample, and assuming the equation ofthe initial stress is r(t),

@u@xjx¼l ¼

rðtÞE

ð3Þ

where E is Young’s modulus. Based on Hooke’s law, the boundarycondition is

ujx ¼ 0@u@x jx¼l ¼

rðtÞE

(ð4Þ

When t = 0,

ujt¼0 ¼ xe0 ¼xrð0Þ

Eð5Þ

Then, the initial condition is

ujt¼0 ¼ xe0 ¼ xrð0ÞE

@u@t jt¼0 ¼ 0

(ð6Þ

Therefore, the elastic, dynamic unloading problem is as follows:

@2u@t2 ¼ c2 @2u

@x2 þ lg

ujx¼0 ¼ 0; @u@x jx¼l ¼

rðtÞE

ujt¼0 ¼xrð0Þ

E ; @u@t jt¼0 ¼ 0

8>>><>>>:

ð7Þ

FixedUnloading

-Loading

Free

l

xΔ

z

xy

0Fixed

Unloading

-Loading

Free

l

xΔ

-

z

xy

0

Fig. 2. The rock loading and unloading model.

M. Tao et al. / Computers and Geotechnics 45 (2012) 83–92 85

The principle of homogeneous separation can be used to solve thisproblem [28], resulting in the general solution shown below.

uðx; tÞ ¼ f ðl; c; E;l; g;rðtÞÞ ð8Þ

As in a determined condition, l, c, E, l and g are known. Therefore,the solution indicates that the unloading displacement is only re-lated to the initial stress equation r(t) and the unloading time t. Ifthe initial stress equation and the unloading time are determinedbased on a certain unloading path, the exact value of the unloadingdisplacement or the deformation can be obtained.

Additionally, the different excavation methods can be charac-terised by different initial stress release paths and different initialstress release rates. Therefore, in this study, r(t) is defined as theunloading path or the initial stress release path, and the ratio ofthe stress to the time (dr/dt) is defined as the initial stress releaserate (ISRR) to verify the characteristics of the unloading process. Inother words,

ISRRðtÞ ¼ drðtÞdt

ð9Þ

Please note that the ISRR is a constant for a linear unloading path.Therefore, this study first investigated linear unloading paths withconstant ISRRs. Nonlinear unloading paths are then studied to char-acterise the unloading process by a constant equivalent ISRR(EISRR), subsequently denoted by �k.

For hard rocks, it is difficult to capture the characteristics of anunloading failure following different paths in the field or labora-tory. To more directly and reasonably solve this problem, the mainfactors dominating the equation are employed in conjunction withthe numerical simulation method to study the unloading processof hard rocks in the following sections.

3. Material models for rock

Many material models, such as the geologic cap model, theTaylor–Chen–Kuszmaul model, the Johnson–Holmquist modeland the continuous surface cap model (CSCM) have been devel-oped recently to simulate the damage and the fracture progressof rock or rock-like material in LS-DYNA [29]. In general, thegeologic cap model deviates significantly from the failure and com-paction surfaces quickly, causing the model to produce erroneousresults or to stop due to an error condition [30,31]. The Taylor–Chen–Kuszmaul model is advantageous for considering tensiledamage in the destruction of the material but does not considercompression damage as extensively [32]. In contrast, the John-son–Holmquist model is advantageous for considering compres-sion damage but does not consider tensile damage as extensively[33]. The CSCM, which is widely used in LS-DYNA for brittle mate-rials such as concrete, was employed to model the rock in thisstudy. In this model, the shear failure and the compaction surfacesare ‘‘blended’’ together to form a ‘‘smooth’’ or ‘‘continuous’’ surface[30]. The material model includes an isotropic constitutiveequation, yield and hardening surfaces, a damage formulation tosimulate the softening and the modulus reduction, and a rateeffects formulation for the increased strength resulting from the

strain rate [34]. Here, the basic model features are described, andthe descriptions of other features can be found in the LS-DYNAkeyword user’s manual [29]. The yield equation is expressed as

f ðI1; J2; J3; kÞ ¼ J2 �R2FcF2f ð10Þ

where I1, J2 and J3 are the first, second and third invariants of thestress tensor, respectively, k is the cap hardening parameter, andR is the Rubin three-invariant reduction factor. Fc is the hardeningcap, and Ff is the shear failure surface, which is defined in terms of I1

as

Ff ðI1Þ ¼ a� kexp�bI1 þ hI1 ð11Þ

where the parameters of a, b, k and h are selected by fitting themodel surface to the strength measurements from tri-axialcompression tests conducted on plain rock cylinders.

The cap moves to simulate the change in the plastic volume,expanding to simulate plastic volume compaction and contractingto simulate plastic volume expansion, referred to as dilation.The motion of the cap is based on the hardening rule by

epv ¼W 1� exp�D1ðX�X0Þ�D2ðX�X0Þ2

� �ð12Þ

where epv is the plastic volume strain, W is the maximum plastic vol-

ume strain, X is the location on the I1-axis, D1 and D2 are the modelinput parameters, and X0 is the initial location of the cap whenk = k0.

Damage in the CSCM is governed by the fracture energy and thematerial softening in tension and regions of low confining pressure.Damage accumulation is based on two distinct formulations, whichare referred to as brittle damage and ductile damage. The damageparameter applied to the six stresses is equal to the current maxi-mum of the brittle or ductile damage parameters [29].

The strength of the model is increased with increasing strainrate. The CSCM applies the rate effects to the limit surface, theresidual surface and the fracture energy [33]. To limit the rateeffects at high strain rates, the user may input overstress limitsin tension and compression.

The CSCM for rock with appropriate parametric values is incor-porated into LS-DYNA in the following section to validate thematerial model by simulating the performance of rock samplesunder standard uniaxial compression loading.

4. Validation of the material model

To validate whether the capacity of the material model is suit-able for simulating the failure mechanism of rock, the materialmodel was used to simulate tests on samples under standard uni-axial compression. A standard sample with a cross-sectional diam-eter of 50 mm and a height of 100 mm under uniaxial compressionwas tested in the laboratory under displacement control conditionsuntil failure. The experimental results provided some rock proper-ties, which are shown in Table 1.

For the CSCM, the properties of the fracture energy (Gft intension, Gfs in shear, and Gfc in compression), the rate effect onthe fracture energy (repow) and the maximum principal strain at

Table 1Material properties of rock.

Poisson’s ratio Density (kg m�3) Internal friction angle (�) Young’s modulus (E) (GPa) Uni-axial compression strength (MPa) Uni-axial tensile strength (MPa)

0.16 2700 52 39.8 152.69 9.3


which erosion occurs should be determined. In general, the CSCMdeveloper recommends using the default properties for the tensilefracture energy Gft = Gfs, where Gfc = 100Gft. If more shear or com-pression-based damage is desired, then Gft = 0.5Gfs or Gfc = 50Gft

are reasonable reductions. If more overall damage is desired, thenrepow = 0.5 is a reasonable reduction [35].

Furthermore, a 3D model is developed using 3D solid elementsfor the uniaxial compression simulation, which is the same size asthe standard testing sample in the laboratory. The uniaxialcompressive loads controlled by the displacement are also appliedvertically. The final input values are adjusted based on the resultsof the calculation and the experiment. In Figs. 3 and 4, the numer-ical results are compared with those measured experimentally,including the failure patterns and the stress–strain relationships.

The results illustrate that there are only slight differences in thefailure patterns and stress–strain relationships, indicating that the

Fig. 3. Comparison of the simulated and the experimental wedges for uniaxialcompression (note: the red and green zones indicate plastic deformation, the bluezones indicate the elastic deformation). (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Comparison of the simulated results of the stress–strain relationship withthe test data for uniaxial compression.

material modelling is suitable for simulating the hard rock. Thematerial properties used in the CSCM for rock are shown in Table 2.

5. Numerical simulations and results

The numerical model validated above is used to conduct para-metric studies on the unloading process of rock with initial stres-ses. In this section, the numerical results from a static stressinitialisation-dynamic unloading case are described in detail toverify the mechanism of the unloading process.

5.1. Numerical simulation of the unloading

The numerical simulation of unloading is discussed briefly. First,the implicit finite element method is used to solve the static stressinitialisation of the rock before the dynamic unloading begins. Thedeformed shape, the pre-stress and the strain in the rock are thentransferred to the explicit code, which is accomplished by creatinga database file that updates the geometry and the stress history ofthe explicit element so that it matches the implicit static solution.An explicit file, which can be obtained by modifying the previousimplicit file that was used in the initialisation, is then prepared toperform the dynamic unloading process. An unloading curve withan initial displacement of zero and a command for calling theresults of the implicit initialisation are added while the node andelement components are deleted because they are not requiredduring the explicit stage. In particular, the unloading curvedescribes the initial stress release path. Various unloading curvesrepresent different initial stress release paths, and, if the unloadingcurve is not defined, an ‘‘instantaneous unloading’’ process isexpected. Finally, the explicit solver is utilised to execute the rockdynamic unloading problem in the static stress initialisation stage.

Rocks underground are in a 3D stress state, and the unloadingprocess of rocks under a 3D initial stress is very complex; therefore,it is complicated to use 3D modelling to investigate this problem.However, civil tunnelling and mining excavation only operate inone direction. To simplify the problem, it is assumed that thecircumference of the rock sample is stress-free, i.e., the confiningstress is zero, which simplifies this case to a 1D stress state problemto characterise the unloading process of rock under axial initialstress only.

In addition, although a 1D stress state is assumed for therock sample under an initial axial stress, i.e., the static stressinitialisation-dynamic unloading is only performed along thex-axial direction; it is solved using 3D finite element analysis in thisstudy. For a purely numerical technique, it is time consuming to usea 3D model to solve a 1D problem. However, this analysis demon-strated that a 3D finite element model is more realistic and can beemployed to solve this real problem for further research. Therefore,we used 3D modelling to solve the 1D stress state problem in thisstudy. The rock sample described in Fig. 2 is discredited as shownin Fig. 5 in the LS-DYNA program. The rod length l is 1000 mm,and the rod diameter D is 100 mm. The left end of the rod in they–z plane is fixed (x, y and z) at x = 0, and the initialisation–unload-ing process along the x axial is performed at x = l.

Convergence tests are conducted to investigate how manyelements are needed to achieve a reliable estimation by decreasingthe size of the elements until the difference between the results for

Table 2Input parameter used for the CSCM.

Young’smodulus (E)

Shearmodulus (G)

Bulkmodulus (K)

Density (q) Poisson’ratio (v)

Uni-axialCompressionStrength (fc)

Uni-axialTensilestrength (ft)

3.98E+10 1.72E+10 1.951E+10 2.7E+3 0.16 151 9.1

Compressing a h k b4.060E+08 0.07510 3.907E+08 1.0E�09

Shearing a1 h1 k1 b1

0.761 2.0E�05 0.0000 0.0000

Tension a2 h2 k2 b2

0.6800 2.290E�05 0.0000 0.07057

Cap Cap shape (R) Cap location (X0) Maximum plasticvolume change (W)

Linear hardening (D1) Quadratichardening (D2)

4.00 6.0E+08 0.003987 6.0E�10 0.0000

Fracture energy Gfc Gft Gfs

9000 180.00 90.00

Rate dependence Rate effects parameterfor uni-axial compressivestress (g0c)

Rate effectspower for uni-compressivestress (Nc)

Rate effects parameterfor uni-tensile stress (g0t)

Rate effects powerfor uni-axialtensilestress (Nt)

1.93E�04 0.604 1.760E�05 0.64


two consecutive element sizes is less than 5%. The convergencetests resulted of the element number that was employed in thesimulation. Eight-node solid elements were used for the model,and the total number of elements for this model was 20,000. Thesize of the finite elements varies as shown in the cross section ofFig. 5.

5.2. Characteristics of the different unloading processes

According to the mathematical physics analysis in Section 2 andthe finite element model outlined in Section 3, the dynamicunloading path follows different initial stress r(t) equations. Theinitial stress release time is controlled by the different unloadperiods.

Based on previously published results, if the rock mass isunloaded, the unloading forces may dominate over the tensilestrength of the rock mass, resulting in failure [15]. In addition, ahigher stress release rate leads to a higher possibility of rock failure[36,37]. In the instantaneous unloading process, the initial stressdoes not follow a path and is released in a very short time. Conse-quently, the instantaneous unloading represents both the highestinitial stress release rate (ISRR) and the shortest unloading period.Therefore, the peak initial stresses were set as 4 MPa, 8 MPa,15 MPa, 20 MPa and 30 MPa, and an instantaneous unloading isapplied to obtain the highest possibility of rock failure.

Furthermore, the dynamic unloading process can produce a ten-sile pulse, i.e., a tensile wave. In general, the velocity of the wave ofthe rock sample in Table 1 is found from

ct ¼

ffiffiffiffiEq

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3:98� 1010

2:7� 103

s¼ 3839 m=s ð13Þ

where ct is the velocity of the tensile wave, E is Young’s modulus ofthe rock and q is the density of the rock sample. The actual propa-gation time t of the wave from right to left is calculated by

z

xy

0

1000

F

Fixed

Fig. 5. Numerical model of

t ¼ lct¼ 1

3839¼ 0:26 ms ð14Þ

To give the wave a sufficient amount of time to propagate in therock, the simulation time is set to 10 � t, i.e., 2.6 ms, and the resultsof the instantaneous unloading test are shown in Fig. 6.

Fig. 6 illustrates the failure zones of the rock generated bydifferent initial stresses. The magnitude of the failure is very obvi-ous when the peak initial stress is high. The unloading tensile wavepropagates along the rock sample from right to left. When thewave arrives at the fixed left end, the wave reflects as a highermagnitude tensile wave. The reflected tensile wave is superim-posed on the incoming tensile wave, and the superposition has amagnitude higher than that of the dynamic tensile strength ofthe rock sample. This superposition will induce rock failure. Themaximum superposition magnitude is two times that of the peakinitial stress. In addition, the CSCM considers the rate dependence,which increases the dynamic tensile strength to a value higherthan the static tensile strength (9.1 MPa). Therefore, the 4 MPapeak initial stresses do not generate failure zone, but 8 MPa,20 MPa or 30 MPa peak initial stresses do. These numerical testsclearly indicate that the initial stress release process can inducerock failure zones. The characteristics of these different unloadingprocesses are discussed below.

In general, when the peak initial stress is at the same level, thekey factor of the unloading process is the ISRR. The linear path rep-resents a uniform unloading path based on the previous definitionof the ISRR; the slope of the linear unloading path is the ISRR.When the peak initial stress is defined, it is easier to change the dif-ferent magnitudes of the ISRR by changing the unloading period.The time of the initial stress release in Fig. 7 indicates that thestress release time is very short (0.004 ms) during the instanta-neous unloading process of the 30 MPa peak initial stresses. There-fore, based on the 30 MPa static initial stress state, several linearpaths are employed to investigate the influence of the ISRR. These

Cross section

=100 mm mm φ

ree

Loading-

Unloading

the unloading process.

MPa8 MPa8

MPa4 MPa4

MPa20 MPa20

MPa30 MPa30

Fig. 6. Failures zone under different peak initial stresses (note: the fissure zonesindicate fracturing, the green zones indicate plastic deformation, and the blue zonesindicate the elastic zone). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 7. Stress–time curve unloading surface at the right end.

Fig. 8. Different linear unloading paths.

(a) ISRR=1500 MPa/ms (T = 0.02 ms)

(b) ISRR=300 MPa/ms (T = 0.1 ms)

(c) ISRR=100 MPa/ms (T = 0.3 ms)

(d) ISRR=60 MPa/ms (T = 0.5 ms)

ms)0.8( MPa/ms37.5ISRR == T(f)

ms)0.9( MPa/ms33.3ISRR == T(g)

==

==

ms)0.7( MPa/ms42.9ISRR == T(e) 0.7( =

Fig. 9. Plastic strain zones of the rock samples under different ISRRs.


different linear paths representing different ISRRs are presented inFig. 8.

The simulated results of the unloading process are shown inFig. 9, where T was defined as the unloading period; these resultsillustrate that different initial stress release rates generate differentdynamic responses. If an obvious plastic strain zone is defined as afailure zone, the unloading failure can be characterised. First, whenthe unloading period ranges from 0.02 ms to 0.5 ms, multiple fail-ure zones are produced. In contrast, for unloading periods rangingfrom 0.5 ms to 0.9 ms, less ‘‘failure’’ is observed. For an unloadingperiod larger than 0.9 ms, there is no failure. Second, when theunloading period is very short, such as 0.02 ms and 0.05 ms, thefailure type is similar to that for instantaneous unloading. The rea-sons that the different unloading processes induce such differentresponses are discussed below.

The unloading process produces a tensile stress wave, and thewaveform is not well defined. However, if the unloading periodis very short compared to l/ct, as is the case for instantaneousunloading, the generated tensile wave can be approximated byan equation with only one peak zone, and the waveform can beconsidered equivalent to a rectangular wave. When this rectangu-lar wave arrives at the fixed left end, the reflection and the super-position double the stress magnitude. When the superposed stressis higher than the dynamic strength of the rock, a serious failurezone is generated. Instantaneous unloading only generates oneserious failure zone. Furthermore, when the unloading period isvery short compared to l/ct, the responses are similar to those forinstantaneous unloading, so the failure zones only appear near

the fixed left end. The reflection of the rectangular wave issketched in Fig. 10.

However, when the unloading period is increased, the rise timeof the tensile wave is not significantly shorter than l/ct. The tensilestresses produced have multiple peak zones because the superpo-sition of the reflected and the incoming waves is not concentratedat the left boundary. When the local tensile stress becomes higherthan the dynamic tensile strength of the rock, rock failure will oc-cur. Thus, multiple peak tensile stresses will generate multiple

waveTensile

Fixed

waveTensile

Fixed

waveTensile

Fixed

(a) Tensile wave produced by unloading

waveTensile

Fixed

waveTensile

Fixed

(b) Tensile wave arriving the left end

Reflection wave tensileionSuperposit

Fixed

(c) Reflection and superposition

Reflection wave tensileionSuperposit

Fixed

(d) Reflection and superposition

Fig. 10. Reflection and superposition process of the rectangular tensile wave.

Fig. 11. Different unloading paths.

Table 3Unloading paths.

Type Unloading function r(t)

Linear rl ¼ �rmax 1� tT

� �Cos rcos ¼ �rmax cos p

2T t� �

Exp rexp ¼ �rmaxe�2pT t

D tΔ

I

B

σ

ot

C

A

It

i

( )dt

tσ

D tΔ

I

B

σ

t

C

A

It

i

( )dt

σ

D

σΔ IB

σ

ot

C

A

I i

σ

( )dt

σD

σΔ IB

σ

tC

A

I i

σ

( )dt

tσ

(a) Differential time (b) Differential stress

d

d

Fig. 12. Differential sketch map of the unloading path.


failure zones, as shown in Fig. 9b–f. The magnitude of the peak ofthe wave, the wavelength and the wave period all depend on theISRR. Therefore, when the initial stress values are the same, thedominant factor in the unloading process is the release time (andthus the ISRR).

Until now, the ISRR was constant, which is not typically thecase. For example, the nonlinear unloading paths composed ofcosine and exponential functions that are shown in Fig. 11 illus-trate different unload histories for which the ISRR is time-depen-dent. Therefore, the characteristics of the unloading process wereused to quantitatively describe the magnitude of the ISRR.

The equations for these paths are presented in Table 3.

Furthermore, for any unloading process, the time integral of thestress is the stress impulse I. Different unloading paths have thesame magnitude of stress impulses, but the shape of the stress im-pulse is different. Both the magnitude of the stress impulse and theshape of the unloading path can affect the unloading process. Toinvestigate how the stress impulse affects the initial stress releaserate, the parameters kt and kr are introduced, and it is assumedthat kt and kr are defined by the weighted averages of the ISRRfor an arbitrary path.

kt ¼Pn

i¼1Iti

IdrðtÞ

dt¼R T

0 rðtÞr0ðtÞdtR T0 rðtÞdt

ð15Þ

kr ¼Pn

i¼1IriI

drðtÞdt¼R rmax

0 tðrÞr0ðtÞdrR T0 rðtÞdt

ð16Þ

where r(t) is the unloading path as a function of time, and t(r) andr0(t) are the inverse and the derivative of r(t), respectively. Themathematical meanings of Eqs. (15) and (16) are sketched inFig. 12. These equations are weighted averages of the ISRR, whichis a function of time.

Therefore, kt represents the contribution of the ISRR of the finitestress impulse I divided by the finite time, and kr represents thecontribution of the ISRR of the finite stress impulse I divided bythe finite stress. Because neither kt nor kr fully characterises theunloading process, suitable combinations of kt and kr are investi-gated. After testing many alternatives, the equation that providesthe best average of the ISRR is

�k ¼ kr

kr þ ktkr þ

kt

kr þ ktkt ð17Þ

Herein, �k is referred to as the ‘‘equivalent initial stress release rate(EISRR)’’.

Therefore, for the three different paths illustrated in Fig. 11, thecalculated values of the EISRR are as follows:

For the linear path: kt ¼ kr ¼ �k ¼ ISRR ¼ 33:3 MPa/ms.For the cosine unloading path:

kt ¼R 0:9

0 rcosðtÞr0cosðtÞdtR 0:90 rcosðtÞdt

¼ 45:4 MPa=ms

kr ¼R 30

0 tðrÞr0cosðtÞdrR 0:90 rcosðtÞdt

¼ 26:2 MPa=ms

and

kcos ¼kr

kr þ ktkr þ

kt

kr þ ktkt ¼ 38:4 MPa=ms

For the exponential path:

ISRR=87.6 MPa/ms (Linear Path)

ISRR=38.4 MPa/ms (Linear Path)

(a) Linear path

EISRR=38.4 MPa/ms (Cos Path)

EISRR=87.6 MPa/ms (Exp Path)

(b) Nonlinear path

Fig. 13. Plastic strain zones of the rock samples under linear and nonlinearunloading paths.

Actual time t=0.20 ms




Fig. 14. Plastic strain zones of the rock samples under the instantaneous unloadingprocess (note: the failure near the right end is induced by direct tension).

Fig. 15. Different linear unloading paths.


kt ¼R 0:9

0 rexpðtÞr0expðtÞdtR 0:90 rexpðtÞdt

¼ 53:1 MPa=ms

kr ¼R 30

0 tðrÞr0expðtÞdrR 0:90 rexpðtÞdt

¼ 105:0 MPa=ms

and

�kexp ¼kr

kr þ ktkr þ

kt

kr þ ktkt ¼ 87:6 MPa=ms

Thus, if Eq. (17) is approximately correct, the calculated value of�kcos ¼ 38:4 MPa/ms indicates that the failure under the cosineunloading path should be in between the responses to thelinear unloadings in Fig. 9e and f, where ISRR = 42.9 MPa/msand 37.5 MPa/ms, respectively. For the exponential path,�kexp ¼ 87:6 MPa/ms, so the failure should be in between the re-sponses to the linear unloadings in Fig. 9c and d, whereISRR = 100.0 MPa/ms and 60.0 MPa/ms, respectively. The resultsfor the nonlinear paths are shown in Fig. 13 with the correspondinglinear paths; as shown in the figure, the unloading failures are qual-itatively the same.

Many other nonlinear unloading paths were also tested, and theresults indicated that the ‘‘EISRR’’ can quantitatively describe thecharacteristics of the failure behaviour.

All of the failures generated in the previous simulations are in-duced by the reflection and the superposition of tensile waves be-cause the initial wave does not have a sufficient amount of energyto induce direct failure, i.e., the rock failure is not induced beforethe initial wave arrives at the fixed left end. Therefore, to achievedirect unloading failure, the initial stress rmax is gradually in-creased. The initial stress produces direct failure at approximately90 MPa (where the uniaxial compression strength is 151 MPa) forinstantaneous unloading (ISRR is the highest). The results areshown in Fig. 14.

Fig. 14 clearly illustrates that the failure zones are induced di-rectly at t = 0.0 ms at the right end. The arrival time of the unload-ing wave at the fixed left end is approximately 0.26 ms, but directfailure is also induced along the rod at an earlier time (such as0.20 ms in Fig. 14) after the passing of the first wave front.

To investigate how the unloading paths affect the response ofthe rock samples when rmax = 90 MPa, several linear paths (shownin Fig. 15) are tested, and their corresponding failure zones are pre-sented in Figs. 16 and 17. When the unloading period increases, i.e.,the ISRR decreases, the failure type gradually changes from directtension failure to indirect failure, resulting from reflection andsuperposition.

In addition, two nonlinear paths are investigated forrmax = 90 MPa and T = 0.012 ms, and the EISRR is used to quantita-tively describe the failure. The cosine and exponential equationsare employed again, as shown in Fig. 11 and Table 3. For the expo-nential path, the EISRR is �kexp ¼ 1:97� 104 MPa/ms, so the re-sponse should be in between the linear cases shown in Fig. 16aand b, where the ISRR values are 2.25 � 104 MPa/ms and1.5 � 104 MPa/ms, respectively. The exponential path generates di-rect failure. For the cosine path, �kcos ¼ 8:63� 103 MPa/ms, so thefailure is similar to those in Fig. 17a and b, whereISRR = 9.0 � 103 MPa/ms and 7.5 � 103 MPa/ms, respectively, anddirect failure does not occur. The results for both nonlinear pathswith the corresponding linear paths are shown in Fig. 18. The di-rect and the indirect failure behaviours are similar.

From the analysis above, it can be concluded that releasing theinitial stress of the rock can induce rock damage and eventual fail-ure. The newly defined ‘‘equivalent initial stress release rate(EISRR)’’ is the key factor dominating the unloading process. Thisfinding is significant because, in underground excavation

engineering, different excavation methods can be characterisedby different unloading processes. Based on this finding, in practicalengineering, by dynamically controlling the release of the initial

(a) ISRR=2.25×104 MPa/ms (T=0.04 ms)

(b) ISRR=1.50×104 MPa/ms (T=0.06 ms)

(c) ISRR=1.125 104 MPa/ms (T=0.08 ms)

(a) ISRR=2.25×104 MPa/ms (T=0.04 ms)

(b) ISRR=1.50×104 MPa/ms (T=0.06 ms)

(c) ISRR=1.125×104 MPa/ms (T=0.08 ms)

Fig. 16. Direct unloading failure, then reflection and superposition unloadingfailure (note: some of the failure zones at the right end are generated at thebeginning).

(a) ISRR=9.0×103 MPa/ms (T=0.1 ms)

(b) ISRR=7.5×103 MPa/ms (T=0.12 ms)

(a) ISRR=9.0×103 MPa/ms (T=0.1 ms)

(b) ISRR=7.5×103 MPa/ms (T=0.12 ms)

Fig. 17. Reflection and superposition unloading failure.

ISRR=8.63×103 MPa/ms




(a) Linear path

EISRR=8.63×103 MPa/ms (Cos Path)

EISRR=1.97×104 MPa/ms (Exp Path)

EISRR=8.63×103 MPa/ms (Cos Path)

EISRR=1.97×104 MPa/ms (Exp Path)

(b) Nonlinear path

Fig. 18. Plastic strain zones of the rock samples under the linear and the nonlinearpaths (note: some of the failure zones at the right end are generated in thebeginning).


stress to induce rock failure, an appropriate method can be devel-oped to minimise the excavation energy. Furthermore, employingan appropriate excavation method protects the stability of the sur-rounding rocks by decreasing the initial stress release rate. Forexample, during the blasting excavation process, the optimal EISRR

can be achieved by manipulating the spacing network geometry,the charge configuration and the delay time.

6. Conclusions

The mathematical physics analysis method was developed inconjunction with numerical simulations performed by LS-DYNAto study the unloading process of rocks, particularly to investigatehow the initial stress release paths and rates influence the unload-ing and the failure of rocks during the unloading process. Thenumerical results clearly indicate that releasing the initial stressof rocks can induce rock failure. The ‘‘equivalent initial stress re-lease rate’’ (EISRR) was then introduced to quantitatively describethe unloading process. Many unloading paths were employed andtested to confirm that the ‘‘equivalent initial stress release rate’’dominates the unloading process and the corresponding failurebehaviour. The results show that with the same initial stress level,the unloading characteristics of rocks are mainly dependent on themagnitude of the EISRR. This finding contributes to the develop-ment of an optional excavation method to generate stress-releas-ing rock failure to minimise the excavation energy and protectthe stability of the surrounding rocks by dynamically controllingthe equivalent initial stress release rate.

Acknowledgments

The research presented in this paper was jointly supported bythe 973 Program of China (Grant No. 2010CB732004), the NationalNatural Science Foundation of China (Grant No. 50934006 and10872218) and the Ph.D. Programs Foundation of Central SouthUniversity, China (Grant No. 2010bsxt10). The first author wouldlike to thank the Chinese Scholarship Council for financial supportto the joint Ph.D. at University of Adelaide, Australia and expressthe acknowledgment to Dr. Arris ST at Eindhoven University ofTechnology and Dr. Liang Huang at Hunan University, China.

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characteristics of the unloading process of rocks under high initial stress

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