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CHEMICAL ENGINEERING TRANSACTIONS
VOL. 66, 2018
A publication of
The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet
Guest Editors: Songying Zhao, Yougang Sun, Ye Zhou
Copyright © 2018, AIDIC Servizi S.r.l.
ISBN 978-88-95608-63-1; ISSN 2283-9216
Blasting Damage Model of Rock with Initial Damages and Its
Application
Pingyuan Yanga, Xiaoen Wu*b, c, Junhua Chenb
aInstitute of Mineral Resources, Chinese Academy of Geological Sciences, Beijing 100037, China
bSchool of Civil Engineering, Central South University, Changsha 410075, China
cChina Energy Engineering Group Equipment Co., Ltd, Beijing 100007, China
xewu@ceec.net.cn
Rock is a material subject to initial damage. Rock blasting-induced damage model is the most commonly used
constitutive model for numerical simulation of rock blasting. At present, however, most models neglect the
effects of initial damage on blasting. To this end, this paper explored the merits and demerits of typical blasting-
induced damage models. Then, a rock blasting-induced damage model was proposed by modifying the typical
blasting-induced damage models, with the effects of initial damage on damage evolution taken into account.
Numbers of parameter in the proposed model are low and all the parameters can be obtained easily from
laboratory tests under uniaxial dynamic and static loading. The criterion to determine the fracture state of rock
is also given, which is related to initial damage and used for confirming the blasting-induced fracture zone.
1. Introduction
At present, the most commonly used constitutive models for blasting are based on damage mechanics. Grady
and Kipp (1980) presented a blasting model (GK model) suggesting that tension plays a decisive role in dynamic
fracture of rock. Taylor et al., (1985, 1986) established an elastic-damage model (TCK model), regarding as
damage evolution is caused only by volume-tensile strain. Kuszmaul (1987) revised the formula in TCK model
for activated cracks to establish a blasting model (KUS model) considering effects of crack density on damage
evolution. Parameters in GK, TCK or KUS model must be obtained by high-strain-rate tensile tests which are
difficult to be conducted. Therefore, Liu and Katsabanisj (1997) proposed a simple damage evolution equation
and established the volume-tensile damage model (LK model) featuring easily obtained parameters. Yang et
al., (1996) recognizing that rock strength are different in compression and tension, established an equivalent
tensile damage model (YBK model) with simple damage evolution relation. In this model, parameters can be
obtained through uniaxial compression test at high strain rate. Overall, blasting-induced damage models have
been constantly improved.
The typical blasting-induced damage models suppose that damage is the evolution of internal defects, however,
in these models, damage evolution equations fail to reflect the influence of initial damage on damage calculation.
Yang et al., (1999) indicated that initial damage plays an important role in mechanical properties of rock. Gao
et al., (2000) also argued that rocks are a material subject to initial damage that acts as a key factor for rock
blasting. For large-scale projects, such as mine, hydropower station and long-distance mountain tunnel, the
types of rock mass are basically the same, but maybe there are differences in initial damage of rock mass. In
this case, initial damage is critical to designing blasting parameters as well as analyzing the security and stability
of surrounding rock (Chen et al., 2016). For example, granite and basalt are major rock mass in three Gorges
and Xiluodu hydropower stations respectively, both of which are large-scale underground excavation projects
in China. For the two large-scale projects, Initial damage or integrity is considered as the main factor influencing
blasting excavation.
In order to provide references for blasting excavation considering the influence of initial damage of rock, the
author proposed a blasting-induced damage model through analysis of features of typical blasting-induced
models, in which damage evolution relation can consider the influence of initial damage on damage evolution.
DOI: 10.3303/CET1866074
Please cite this article as: Yang P., Wu X., Chen J., 2018, Blasting damage model of rock with initial damages and its application, Chemical
Engineering Transactions, 66, 439-444 DOI:10.3303/CET1866074
439
2. Features and analysis of typical blasting damage models
2.1 Damage evolution law
In typical blasting-induced damage models, modes of rock failure mainly include fracture by damage
accumulation and plastic-flow fracture. Damage accumulation is induced by crack-density development. If rocks
are isotropic material, then damage variable can be formulated as:
(1)
where, D is a scalar for the damage variable of rock unit, representing the mechanical deterioration degree
relative to the initial state; Cd is the crack density under the dynamic loading, generally relying on tensile strain;
and f is an expression using crack density as the variable.
For typical blasting damage models, damage variables can be calculated according to Table 1. In this paper,
suppose tension equals positive and compression equals negative.
Table 1: Calculating variables for rock blasting-induced damage
Damage model Damage evolution equation Calculation of crack density
TCK d
16(1 )
9(1 2 )
D C
2
vIC
d 2
vmaxr
5
2( )
m
kK
C
c
KUS d
16(1 )
9(1 2 )
D C
2 1
vIC v
d 2
vmaxr
5 (1 )
2( )
m
kmK D
C
c
LK dexp( )V CD 0=1- -
L
d v vcr
( )
B
C A ε ε t
L= -
YBK
2
d
exp( CD )=1- -
Y
d Y cr
( )
B
C A = -
Notes: As shown in Table 1, μ is the Poisson's ratio of damage, KIC is the fracture toughness of rock mass, 𝜀�̇�𝑚𝑎𝑧 is tensile
strain rate at limit volume, �̇�𝑑 is the growth rate of crack density, �̅�𝑟 and 𝑐̅ are density and longitudinal wave velocity of
undamaged rocks respectively, k,m,AL,BL,AY and BY are material parameters, V0 is the volume of rock unit, εv is volumetric
strain, θ is equivalent tensile deformation (generally εv≥θ), εvcr is the threshold value of volumetric strain, θcr is the threshold
value of equivalent tension strain and t is time. When εv>θ, TCK and KUS models can be established. When εv>εvcr>0 and
θ>θcr>0, LK and YBK models can be built.
In Table 1, TCK, KUS, and LK models are tensile damage models under volumetric tensile stress. Unlike TCK
and KUS models, damage evolution in LK and YHB models is simply formulated and easy to calculate. If
damage, in TCK and KUS models, occurs from the beginning of loading, then in LK and YBK models damage
occurs only when strain exceeds the critical value εvcr and θcr respectively. On the basis of YBK model, whether
the volumetric stress is tensile or compressive, damage may accumulate, which has been verified. Compared
with TCK, KUS and LK models, YBK model features broader concept of damage, simple damage evolution
equation and easily obtained parameters, making it better describe the changes in mechanical property.
Therefore, damage variables defined in YBK model are easily to be used.
2.2 Damage constitutive relations
2.2.1 Constitutive relation of elastic damages
The relation of damage to undamaged modulus can be written as:
(2)
(3)
(4)
In Formulas (2) and (3), 𝐾,�̅� and �̅� means bulk, shear and elasticity modulus of undamaged rocks respectively,
while K,G and E are that of damaged rocks.
In LK and YBK models, damage is supposed not to affect the Poisson's ratio, and the formulas are given as:
(5)
(6)
(7)
d
( )D f C
(1 )K K D
(1 )G G D
(1 )E E D
2 (1 ) 3 (1 )E G K
2 (1 ) 3 (1 )E G K
440
http://dict.youdao.com/w/mode%20of%20failure/#keyfrom=E2Ctranslation
where 𝜇 = �̅� is the Poisson's ratio of undamaged rocks.
When stress is resolved into spheric and deviator stresses, constitutive relation of elastic damage in rock can
be expressed as:
(8)
(9)
where σm is the component of spheric stress, σ1, σ2 and σ3 (σ1≥σ2≥σ3) are the maximum, intermediate and
minimum principal stresses, sij is the components of deviator stress, 𝜀𝑣
𝑒 and 𝑒𝑖𝑗
𝑒 are elastic deformation and
elastic deviator strain, and is deviator strain (i, j=1, 2, 3). Accordingly, elastic damage relations are 𝜀𝑣
𝑒 = 𝜀𝑣 and
𝑒𝑖𝑗
𝑒 = 𝑒𝑖𝑗.
2.2.2 Plastic constitutive relation
In TCK, KUS, LK and YBK models, in case of D=0 and yield in rock units, the stress can be represented as:
(10)
2 2 2
1 3 2 3 3 12
[( ) ( ) ( ) ] / 6J (11)
where ψ is yield function, J2 is the second invariant of stress deviator and σY is the yield strength under uniaxial
loading.
Incremental constitutive relation for yield in rock can be expressed as:
e p
v v v m m
d d d d / ( / ) / 3K (12)
e p
d d d d / (2 ) ( / )
ij ij ij ij ij
e e e s G s (13)
where 𝜀v
𝑝
and 𝑒𝑖𝑗
𝑝
are plastic deformation of volume and deviator stress, and λ is the plastic factor (λ>0).
Formulas (1)-(9) describe the constitutive relations of typical blasting damage models. When a certain damage
variable value is reached, unloading occurs. The deformation modulus under unloading is the damage modulus
calculated from Formulas (2)-(4) from the damage value.
2.3 Problems in typical rock blasting-induced model
As can be seen from Formulas (2)-(8) and Table 1, typical rock blasting models consider rock damages are
caused on the basis of intact state.
Take TCK, LK and YBK as an instance, suppose initial damage is D0, then, the formulas are given as follows:
, (TCK model) (14)
, (LK model) (15)
, (YBK model) (16)
where D0 is the damage variable at initial state.
When D0=0, bulk and shear modulus under initial loading are represented as K0 and G0. The following formulas
can be derived based on Formulas (2)-(4).
(17)
(18)
(19)
As can be seen from Formulas (17)-(19), bulk or shear modulus at initial state is the same with that at intact
state. In typical modes, the initial damage variable is supposed to be 0, namely D0=0. That is to say, damage
evolution is independent of damage history or initial damage has no effect on damage evolution.
e
1 2 3m v v
( ) / 3 (1 )K K D
e
2 2 (1 )
ij ij ij
s Ge G D e
2 Y
3
0
3
J
0
0D D
v
0ε <
0
0D D
v vcr
ε < ε
0
0D D
cr
<
0
K K
0
G G
0
E E
441
3. Blasting damage model considering initial damage
3.1 Damage evolution equation and constitutive relation
It can be seen from the above analysis that damage variable defined in YBK model is relatively reasonable. To
this end, this paper modified the YBK model to establish a model considering the effects of initial damage.
Damage evolution undergoes on the basis of initial damage. Therefore, damage evolution equation in YBK
model shown in Table 1 was modified into:
(20)
where D0 is the parameter for initial damage. If D0=0, Formula (20) is degraded as the damage evolution
equation of YBK model in Table 1. The evolution equation of YBK model is a special form of Formula (20).
Like equation for YBK model, crack density parameter Cd can be formulated as:
(21)
Equivalent tensile strain θ can be calculated as:
(22)
where εi is the principal strain in i direction.
At initial damage state, Cd=0. Then, the following formula can be derived from Formula (20) and (2)-(4).
(23)
(24)
(25)
(26)
where μ0 is the initial Poisson's ratio.
If bulk, shear and elastic modulus are constant for intact rocks, the modulus, according to Formulas (23)-(25),
depends on the initial damage variable at initial sate.
Combining Formulas (8)-(13), (20) and (23)-(25) can derive the constitutive relation between elastic and plastic
damages, with initial damage taken into account.
3.2 Blasting damage criterion considering initial damage
Damage variable increases with loading. When the damage variable reaches a certain value between 0 and 1,
relatively serious damages will occur to rock units exactly seeing the emergence of inner macro-cracks. Based
on Formula (20), the critical damage variable and crack density can be expressed as:
(27)
Where Dlim is the critical damage variable, showing the damage state in which macro-cracks are generated; and
Cdlim is the critical crack density, which can indicate the critical state of damage.
When damage variable satisfies the Formula below, rock unit is within the range of fracture or the area of macro-
cracks. Criterion for fracture can be described as:
(28)
For a certain rock, Cdlim is generally a constant. As can be seen from Formula (24), Dlim is a function in which D0
is the variable. Dlim increases as D0 increases. In this paper, the proposed blasting damage criterion is a function
of damage history. However, Dlim is a constant, i.e. 0.2, 0.2 and 0.22 in TCK, KUS and YBK models, which
neglect the effects of initial damage history.
3.3 Confirmation of calculating parameters
3.3.1 Confirming θcr, E0, and μ0 (or θcr, G0 and K0)
As the threshold value for equivalent extensile strain, θcr can be calculated as:
2
0 d
1 (1 ) exp( )D D C
Y
d Y cr
0
( ) d
t
B
C A t -
3
=1
[( ) 2]
i i
i
0 0
(1 )K K D
0 0
(1 )G G D
0 0
(1 )E E D
0
2 2
lim 0 dlim dlim
exp( ) [1 exp( )]D D C C
lim
D D
442
e e e e
cr t t 0 0 c 0 c 0
/ 2 2 /E E
(29)
where 𝜎𝑡
𝑒 and 𝜎𝑐
𝑒 are the stress of linear elastic limit under uniaxial tension and compression; and 𝜀𝑡
𝑒 and 𝜀𝑐
𝑒 are
the strain of linear elastic limit under static uniaxial tension and compression. “-” means compression is negative
but tension is positive.
Compression test is easier to be conducted than tension test. Therefore, a laboratory test for static uniaxial
compression can be conducted to determine the stress of linear elastic limit 𝜎𝑐
𝑒, the strain of linear elastic limit
𝜀𝑐
𝑒, the initial elastic modulus E0 and initial Poisson's ratio μ0. Then, Formula (29) can be used to deermine the
threshold value for equivalent extensile strain θcr, E0 and μ0. Based on Formulas (6)-(7) and (23)-(25), G0 and
K0 can be confirmed.
The relation 𝜀𝑐
𝑒 to D0 can be described as:
(30)
where 𝜀�̅�
𝑒 is the strain of elastic limit for intact rocks under uniaxial compression, and n is a material parameter.
𝜀�̅�
𝑒 and n can be determined through the uniaxial compression test for intact rocks under static loading,
3.3.2 Confirming D0, Cdlim and Dlim
According to the one-dimensional theory of longitudinal wave, the following formula can be given.
(31)
where c is the longitudinal wave velocity in a damaged rock.
In light of Formula (31), the initial damage variable can be calculated as:
(32)
where c0 is the longitudinal wave velocity in a rock at initial damage state.
D0 can be calculated according to Formula (32). In nature, there are almost no intact rocks. In this case, sampling
can be used for statistical analysis. As a result, sample with the fastest longitudinal wave velocity can be used
as intact material, of which the bulk, shear and elastic modulus as well as longitudinal wave velocity are deemed
as parameters for intact material.
Descent rate of the longitudinal wave velocity can be formulated as:
(33)
where represents the descent rate of the longitudinal wave velocity relative to the initial state.
Existing research has shown that when achieves some extent that the damage in rock unit will be in a critical
state. Then, the following formula works:
(34)
where 𝜛𝑙𝑖𝑚 is the critical value for the descent rate of the longitudinal wave velocity, clim is the critical value for
the longitudinal wave velocity in rock damage, and 𝜛𝑙𝑖𝑚and clim represent the critical state of damage.
Combining Formulas (31)-(34) can obtain the formula below:
(35)
The following equation is given by comparing Formula (35) and (27).
(36)
According to Construction Technical Specifications on Rock-Foundation Evacuating Engineering of Hydraulic
Structures (DL/T5389-2007, CHN), Clim=0.57 can be calculated by substituting 𝜛𝑙𝑖𝑚 = 0.15 in Formula (36).
When 𝜛𝑙𝑖𝑚 = 0.15, Formula (35) can be changed into:
lim 0
0.28 0.72D D (37)
e e
c c 0
(1 )
n
D
2
1 ( / )D c c
2
0 0
1 ( / )D c c
0 0
( ) /c c c
lim 0 lim 0
( ) /c c c
2 2
lim lim 0 lim
1 (1 ) (1 )D D
2
dlim lim
2ln(1 )C
443
3.3.3 Confirming AY and BY
Under the loading of normal stress rate, Formula (21) can be rewritten as:
Y Y 1
d Y cr con Y cr con Y
0
( ) d / ( ) / [ ( 1)]
B B
C A A B
- - (38)
where �̇�𝑐𝑜𝑛 represents a constant extensile strain rate of equivalence.
In case of critical damage, the following equation can be derived according to Formula (38).
Y 1
dlim Y lim cr con Y
( ) / [ ( 1)]
B
C A B
- (39)
where θlim means the equivalent extensile strain in the critical state of damage.
In case of constant strain rate under uniaxial compression, �̇�𝑐𝑜𝑛 and θlim can be represented as:
con 0 con
2 (40)
lim 0 clim
2 (41)
Where 𝜀�̇�𝑜𝑛 is the constant strain rate under the dynamic loading of uniaxial compression, εclim is the critical
strain at the direction of principal stress on uniaxial compression test.
At last, AY and BY are confirmed based on Formulas (39)-(41) and the results from uniaxial compression tests
at different strain rates. Overall, when rock unit is at an initial damage state, there are only 5 numbers of
parameter in the proposed model, including θcr, E0, μ0, D0, AY and BY (or θcr, K0, G0, D0, AY and BY).
4. Conclusion
(1) Typical blasting damage models in fact neglect the effects of initial state of rock on damage, so elastic, shear
and bulk modulus in the initial state are identical with that in the intact state.
(2) The proposed model improved by YBK model still retains features of the old one broader. It has a wide
concept of rock damage, simple damage evolution equation and easily obtained parameters through
compression tests (or uniaxial tension test) and sound wave velocity test. The numbers of parameter are so low
(only 5 parameters) that the proposed model is easy to be used.
(3) The damage evolution relation considers the effects of initial damage on damage evolution, so the proposed
model can consider the effects of initial damage on the blasting-induced damage in rocks.
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