Surface and Buried Explosions: An Explorative Review with Recent Advances

Abstract

Partially or fully buried explosive on detonation releases a large amount of kinetic energy a part of which gets dissipated during creation of crater and the rest gets converted into ground shock. Both phenomena are of complex nature with involvement of non-linearity in both loading and material characteristics. This review aims at providing an insight into mechanisms involved during an event of buried explosion with varying degree of confinement. Factors affecting crater formation and ground shock propagation in media are discussed in detail. An overview of various prediction methods developed over the years based on dimensional analysis and theory of similarity to estimate crater dimensions and magnitude of ground shock and ground motion along with their limitations is presented. Prediction models used to define and optimise rock fragmentation distribution in surface mining operations are additionally reviewed and discussed. Various state-of-the-art experimental and numerical techniques are discussed in brief. Finally, it discusses the challenges involved in both experimental and numerical analysis and thereby provides alternative solutions and suggestions for further investigations in specific areas of lacuna.

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Archives of Computational Methods in Engineering
https://doi.org/10.1007/s11831-021-09553-2
ORIGINAL PAPER
Surface andBuried Explosions: AnExplorative Review withRecent
Advances
JagritiMandal1 · M.D.Goel2 · A.K.Agarwal1
Received: 23 June 2020 / Accepted: 10 January 2021
© CIMNE, Barcelona, Spain 2021
Abstract
Partially or fully buried explosive on detonation releases a large amount of kinetic energy a part of which gets dissipated
during creation of crater and the rest gets converted into ground shock. Both phenomena are of complex nature with involve-
ment of non-linearity in both loading and material characteristics. This review aims at providing an insight into mechanisms
involved during an event of buried explosion with varying degree of confinement. Factors affecting crater formation and
ground shock propagation in media are discussed in detail. An overview of various prediction methods developed over the
years based on dimensional analysis and theory of similarity to estimate crater dimensions and magnitude of ground shock
and ground motion along with their limitations is presented. Prediction models used to define and optimise rock fragmentation
distribution in surface mining operations are additionally reviewed and discussed. Various state-of-the-art experimental and
numerical techniques are discussed in brief. Finally, it discusses the challenges involved in both experimental and numeri-
cal analysis and thereby provides alternative solutions and suggestions for further investigations in specific areas of lacuna.
1 Introduction
Explosives have been a huge part of mankind history ever
since the invention of gun powder in ninth century AD. Their
use in various sectors of industries further increased with the
invention of much stable explosive, dynamite, and detonator
by Albert Nobel in 1867 which enabled controlled explo-
sion along with safe storage and transportation of explosives.
These milestone inventions boosted the growth in mining
industries and various civil engineering projects. Most
common applications of explosive include fragmentation
of geo-material to extract minerals in open-cast and deep
underground mining, large earth excavation for construction
of railroads, dam, canal and harbours etc., and controlled
demolition of abandoned buildings. Though the invention of
explosive was made with good intents, it sooner found its use
in military sectors for warfare. Currently, intense research is
been done in developing low-yield earth-penetrating weap-
ons to destruct deeply buried structures with the least release
of radioactive and thermal radiation. An exponential rise in
explosives based terrorism has also been observed in the
last few decades posing a constant threat to civilians’ lives
and properties. Anti-personnel and anti-tank landmines
and vehicle-borne improvised explosive devices (VBIED)
have been frequently used in such acts. In both scenarios,
understanding the effect of explosion on surrounding media
is necessary, wherein, in the former case, to increase the
productivity and economic growth and in the latter case, to
minimise the damage incurred on the structures.
Research focusing on buried explosion, considering both
productive and destructive aspects of its application, is a
complex affair owing to many factors involved. The foremost
factor is the loading itself which is non-linear, dynamic and
transient with high magnitude and frequency, making it dis-
tinctly different from other loading types [1]. Secondly, the
naturally varying characteristics of geo-material invalidate
the application of findings and conclusions of one study on
other geotechnical problems universally.
A large number of nuclear and chemical explosion testing
were carried out before the signing of Partial Nuclear Test
Ban Treaty in 1963 which restricted all testing except under-
ground explosion and Comprehensive Nuclear Test Ban
Treaty in 1996 which covered banning of all kinds of nuclear
tests. Few of these surface and buried explosion testing are
mentioned in Figs.1 and 2 along with their yield charges,
* Jagriti Mandal
jagritimandal@students.vnit.ac.in
1 Department ofMining Engineering, Visvesvaraya National
Institute ofTechnology, Nagpur, Maharashtra440010, India
2 Department ofApplied Mechanics, Visvesvaraya National
Institute ofTechnology, Nagpur, Maharashtra440010, India

J.Mandal et al.
1 3
crater dimensions, height of mushroom cloud and depth of
burial in case of buried explosions. The yield ranged from
few tons to megatons. Where a similar gradual growth in
cloud height and crater dimensions with an increase in yield
can be observed in case of surface explosion, the same can-
not be expected for buried explosion as evident in Fig.2 due
to the effect of varying confinement. Based on these obser-
vations and obtained test results many theories, scaling laws
and empirical formulas governing the effect of explosion
on geo-materials were developed which are used till date.
Nordyke [2] was the first to report the earliest studies on the
effect of nuclear and chemical explosions on soil. The study
compared the results from four nuclear explosion testing in
Nevada, US and presented a preliminary theory on various
mechanisms involved during buried explosion. This theory
became the base for estimating the crater dimensions [3].
In the following years, many variations of scaling laws and
empirical relationships were made [4–9].
On detonation of a buried explosion large amount of
kinetic energy is released, a part of which gets dissipated
during the creation of crater and the rest gets converted
into ground shock. The paper focuses on these two buried
explosion phenomena in details, discussing the working
mechanisms behind them. Effect of various factors on these
mechanisms is discussed in detail in this review. It further
overviews on various prediction methods developed over the
years to estimate crater dimensions and magnitude of ground
shock and ground motion. Prediction models for optimisa-
tion of rock fragmentation distribution in mining operations
are additionally reviewed. Finally, the state-of-art develop-
ment made in the area of experimental and computational
research for the topic in concern is discussed in brief.
2 Mechanics Involved inBuried Explosion
When an explosive confined from all directions detonate
it produces gases as by-products of exothermic reaction
at very high temperature and pressure. A shock wavefront
is formed due to violent generation of these gases which
Fig. 1 Various surface burst testing events with their respective yield in TNT, crater dimensions and height of mushroom cloud

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
propagates through the medium in which the explosive
is buried. The entire phenomenon can be segmented into
three phases: shock energy, stress and gas energy phase.
Furthermore, the medium surrounding the explosive can
be divided into three deformation zones: crushed zone,
plastic deformation zone and elastic deformation zone
as illustrated in Fig.3. Initial phase involves generation
of shock wave which travels at very high velocity in all
the directions and pulverises the immediate surrounding
material given that magnitude of shock wave exceeds the
compressive strength of medium. This zone is termed
as crushed zone whose size is generally 1.5–4 times the
radius of explosive charge [10]. The pressure is so high
that it leads to large strain and the material behaves non-
linearly. Its compressibility is greatly affected and mate-
rial acts like a fluid. Enveloping this zone is the transition
zone where the shock wave attenuates into stress wave,
hence, initialising the second phase. Transition zone,
then, eases into plastic deformation zone. The stress wave
which is compressive in nature travels radially outwards
developing tensile stresses in direction tangential to the
wavefront. Compressive strength of the material is higher
than the compressive stress in this zone, however, its ten-
sile strength falls insufficient in comparison to generated
tensile stress leading to formation of radial cracks. As
the stress wave further propagates outward, its intensity
reduces exponentially. As a result, material in this zone
suffers smaller strain inadequate enough to cause irrevers-
ible damage. This zone is termed as elastic deformation
zone which extends infinitely. Final phase of buried explo-
sion phenomenon is gas energy phase. The gases generated
due to detonation expand and fill the cracks formed by
stress waves or existing fissures, further disintegrating the
medium. Some studies have also stated that the pressure
generated due to expansion of gases create a quasi-static
stress field which entails similar effect as of stress waves
resulting from attenuated shock wave on the medium
[10–12].
Fig. 2 Various buried explosion testing events with their respective yield in TNT, depth of burial, crater dimensions and height of mushroom
cloud

J.Mandal et al.
1 3
3 Crater Formation Mechanism
Cavity formation is usually associated with buried explo-
sion. Depending on the degree of confinement, which is
governed by depth of burial (DOB), the cavity could be a
surface crater or camouflet. When the explosive is near to the
ground surface the shock wave generated gets reflected back
on reaching the soil-air interface as tension or rarefaction
waves. Spalling of topsoil layer occurs when the magnitude
of rarefaction wave surpasses its tensile strength. Simultane-
ously, the cavity expands vertically upwards due to violent
production of gases which further encourages the spalling of
top surface resulting in formation of a dome shape structure.
The effect of material heaving caused due to rapid adiabatic
expansion of gas produced is called gas acceleration. This
dome continues to rise in height and disintegrates when it
can no longer withstand the gas pressure. On doing so a
large mass of soil is thrown out in the air leading to forma-
tion of crater in the surface. Some portion of the ejected soil
falls back into the crater due to gravitational force and rest
of them forms a ‘lip’ like structure surrounding the crater.
The former is termed as fallback while the latter is called
ejecta [13].
Figure4 illustrates the formation of crater in case of shal-
low buried explosion. The visible crater is termed as appar-
ent crater whereas the true crater, although difficult to make
the exact distinction, lies between the fall back material and
ruptured region of earth. True lip of crater is always covered
with the ejecta material which together form apparent lip.
3.1 Factors Affecting Crater Formation
3.1.1 Effect ofDepth ofBurial
As stated earlier depth of burial plays a prime role in cra-
ter formation. Deeper the explosive is buried larger will
be the size of crater formed. This theory proves to be true
till a certain depth of burial after which the size of crater
starts reducing. This depth of burial is termed as optimal
depth of burial at which radius and depth of crater become
approximately same [13]. Besides the size, the complete cra-
ter mechanism changes when the depth of burial is varied.
For a deeply buried burst, the mechanism primarily involves
heaving of overlying material. Gas venting into the atmos-
phere happens late in the phenomenon whereas larger por-
tion of blast energy couples with the ground. In shallow
buried burst, gas venting takes place early and considerable
part of blast energy dissipates in the atmosphere. Material
ejection occurs as a combined effect of upward movement
of top ground surface due to gas expansion and tensile wave
generated due to free surface [14].
Subsidence crater is always associated with deep buried
explosion which occurs when the raptured material in the
form of chimney collapses and compacts. Summarising the
Fig. 3 Pressure zones near a confined explosion

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
entire crater formation for both shallow and deep buried
burst, it can be stated that there are three prime mecha-
nisms involved in the cavity formation: spalling, gas accel-
eration and compaction and subsidence [2]. These three
mechanisms have varying degree of involvement in the
crater formation depending on the burial depth and are
depicted in Fig.5. Further, their involvement can be bet-
ter visualized in Fig.6. In surface burst, material above
the explosive gets thrown away as ejecta while the mate-
rial below experience crushing and compaction. Spalling
and gas acceleration tends to predominate as the depth of
burial increases. Further increase in burial depth increases
Fig. 4 Various stages in crater formation

J.Mandal et al.
1 3
the crater size, as discussed earlier, wherein gas accelera-
tion contributes primarily in cratering. However, most of
the ejected material falls back in crater due to gravity.
Subsidence with slight bulking is observed when the depth
exceeds optimum depth of burial. Camouflet is formed
when the explosive is buried at a greater depth resulting in
its complete confinement. Here, no venting of gases hap-
pens, instead, the gases get caged inside a hollow cavern-
like structure (Fig.6).
It is important to mention that depth of burial is often
expressed as scaled depth to study the effect of explosion
on geo-material, wherein, scaled depth is taken as,
According to a study conducted by Ambrosini and Luc-
cioni [15], scaled depth ranging from 0.2 to 0.8m/kg1/3
would result in a conventional crater, whereas partial cam-
ouflet would form for a scaled depth of 0.8m/kg1/3–1.39m/
kg1/3. A full camouflet is formed when the scaled depth is
more than 1.39m/kg1/3 [15].
Apart from depth of burial, other factors that influence
formation of crater or camouflet are gravitational force, size,
shape, and type of explosive and density, saturation degree
and the type of soil. Experimental and numerical studies
have been conducted in the past to understand the effect of
each of these factors on cavity formation [16–25]. Most of
the initial studies presented theories based on which empiri-
cal and semi-empirical formula to estimate crater dimen-
sions and ground shock parameters were developed and are
still used to validate the numerical tools.
3.1.2 Effect ofFactors Related toSoil
Variation of geological properties has been reported to have
a great influence on crater formation mechanism in many
studies. Dynamic nature of soil can drastically differ depend-
ing on the matter residing in the soil pores. Dry soil behaves
nearly independent of change in loading rate and is com-
pressible which makes it a good energy absorber leaning
more towards plastic deformation. On the other hand, water
(1)
Z
=
R
W
1∕3
Fig. 5 Contribution of primary mechanisms in formation of crater
with variation in DOB
Fig. 6 Varying shapes of crater depending on depth of burial

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
present in saturated soil increases its incompressibility and
elastic nature, thereby, increasing the amount of impulse
transferred into the surrounding [26]. This occurrence has
been validated by many researchers during crater formation
[16, 18, 19, 27, 28]. Cratering efficiency of any explosive is
measured as ratio of volume of medium removed to explo-
sive yield which is higher for wet soil than dry soil. Further,
crater tends to take a shape of deep bowl for dry soil and
rock, wherein, the radius is, approximately, 2.5 times the
crater depth. In wet soil, the crater is much shallower with
dish shape profile with radius going as far as 15–20 times the
depth [29]. To further add, presence of water in soil pores
alters the cratering mechanism. Dry soil has relatively less
cohesion between the soil particles, hence, the heaved dome
structure formed during cavity formation cannot withstand
the tensile stress and breaks away soon. The energy released
in the air is primarily due to suddenly released detonation
gases. Alternatively, in the case of saturated soil which has
stronger cohesion the dome withholds longer compared to
dry soil. It results in transfer of large momentum from soil to
air with much bigger crater size. The rupturing of soil dome
occurs similar to that of a bubble popping, hence, the load-
ing mechanism for buried explosive in wet soil is termed as
bubble-type loading. This phenomenon has been observed in
numerical studies [30] and has been validated using experi-
mental results [31].
Influence of dry density of soil on cratering and pres-
sure generation has also been studied [19–21]. Dillon [20]
stated that similar to depth of burial there lies an optimum
soil density at which crater size for any type of soil would
be the largest. Lighter soil dissipates the blast energy due
to the presence of air pores, however, the energy released
in the air is larger than that of denser soil since majority of
the energy in denser soil gets expended during deformation
[21]. Similar behaviour is observed in the case of soil type
variation. Experimental tests conducted by Ehrgott etal.
[32] showed that peak impulse for dry sand was thrice the
peak impulse generated in the air by clayey and silty sand.
Among other observations made in the study, it was further
found that crater formed in dry sand was shallower with flat-
ter side slopes. Whereas, clayey and silty sands had deeper
crater with steeper side slopes. It can be concluded from
these observations that crater depth and slope of crater sides
decreases as air voids and shear strength of soil are increased
[32]. Unlike air-burst impulse, the impulse transferred into
the soil is larger for clay than saturated sand having similar
density and water content by a factor of three [33].
3.1.3 Effect ofFactors Related toExplosive
Charge weight The detrimental effect of increase in explo-
sive yield on crater formation or any other target is more
of common knowledge and has been proven by many
researchers. Increasing the charge weight of explosive
increases the blast energy, which is in the form of kinetic and
internal energies in case of chemical explosion, imparted
on the surrounding material on detonation. These energies
cause large strain in the material which on exceeding its
limiting strength results in plastic deformation.
Shape of explosive charge Effect of explosive charge
shape is most profound in the immediate medium sur-
rounding the charge. As the blast wave propagates further
away from the location of explosion the wavefront shape
and pressure profile becomes identical for explosives hav-
ing same charge weight and type but with different geo-
metrical shapes. Studies on effect of explosive charge shape
have been carried out in the past where characteristics of
air blast wave generated from spherical, tetrahedron, cube
and cylindrical charges, as well as, cylindrical charges with
varying height to diameter ratio have been compared using
experimental and numerical aids [22, 23, 34–37]. A gen-
eral conclusion that can be derived from all the studies is
that the shape variation influences the peak pressure of blast
wave, however, it has almost negligible effect on generated
impulse. Another notable observation made by research-
ers was that the exposed surface area of explosive charge
also affects the pressure build-up in the near surrounding.
The near region corresponding to largest surface area of
charge experiences highest blast pressure [37, 38]. Though
similar investigations have not been conducted for buried
explosives, same conclusions can be applied in this case too
assuming that the enveloping geo-material is homogenous
and isotropic. Considering that the above findings hold true
for buried explosive, the shape of explosive charge would,
therefore, govern the shape of the crater thus formed.
Explosive type Limited study is found on influence of
change in explosive type on crater mechanism since most
of the experimental tests and computational study has
been carried out using Tri-Nitro-Toulene (TNT) as it is an
accepted standard for explosives. However, for practical
purposes, explosives beside TNT are also used. Few of the
commercially used explosives and the ones used in military
are Nitro-Glycerin, Di-Nitro-Toulene (DNT), hexahydro-
trinitro-triazine (RDX), octahydro-tetranitro-tetrazocine
(HMX) and nitro-cellulose. Besides these explosives semtex,
dynamite and plastic explosive (C4) have been employed by
terrorist in many incidents. The factor ‘TNT equivalence’
has been in use for a longer time to determine the amount of
TNT required to produce the same amount of energy gener-
ated by unit weight of a different explosive [39]. Different
researchers have adopted different methods to calculate the
TNT equivalence factor for air blast based on either energy
released, peak pressure, impulse or detonation velocity
among which energy-based factor is most commonly used
[40–43]. However, they do not apply for buried explosion
considering the apparent difference in characteristics of blast

J.Mandal et al.
1 3
wave above ground and under ground [24, 25]. Nonetheless,
the factors are still employed due to lack thereof.
3.2 Estimation ofCrater Dimensions
Crater formation and its size can be used as a suitable tool
to measure the amount of destruction an explosive charge
can cause in the surrounding geomaterial. However, due to
restrictions put against performing physical tests, there is
a lack of experimental data, especially for large explosive
charges. In addition, the mechanics behind explosion is so
complex that analytical formula cannot be derived to predict
the parameters involved. Dimensional analysis and theory
of similarities have proven to be helpful in such cases and
it is based on these methods scaling laws were founded. It
involves using existing data from an experiment conducted
in similar environment and scaling the parameters to either
smaller or larger entity. Earlier studies showed that scal-
ing law developed individually by Bertman Hopkinson and
Carl Cranz during First World War gave satisfactory results
according to which crater dimensions were directly propor-
tional to cube root of explosive charge weight [44–47].
Henrych [48] presented semi-empirical formula which
was derived during initial years of development of theory of
explosion. Based on theory of model similarity, the formula
was derived as,
where R is the radius of crater, W is the weight of explosive
and k* is the coefficient of proportionality which depends on
the characteristics of soil medium.
Cooper Jr. [29] derived an empirical formula based on
high explosive cratering experimental data to approximately
calculate crater depth (D) and radius (R) once the volume of
crater (V) is known. Wherein, volume of crater is function
of geological and source characteristics, height of burst and
explosive yield. For dry soil, soft and hard rock and certain
wet soils, the radius and depth of crater are,
Although cube root law fit well for smaller charges,
great variability was observed for larger yields. This could
be due to the fact that Hopkinson scaling law neglects
the effect of gravity which plays a major role in crater
formation. Another study where gravity was assumed to
dominate the cratering mechanism, dimensional analysis
concluded that crater size varied proportionally to charge
yield with exponent of 1/4 [49, 50]. However, where expo-
nent of cube root law overestimated the crater depth, expo-
nent of quarter root law underestimated the crater depth
(2)
R=k
∗3
√W
(3)
1.1 V1∕3<R<1.4 V1∕3
(4)
0.35 V
1
∕
3
<D<
0.7
V
1
∕3
for larger yield [29, 51]. As depth increases the resistance
towards relative movement provided from soil increases in
vertically upward direction. This supports the observations
made in majority of large yield experiments wherein the
crater so formed were wider yet shallower in shape which
ran contradictory to prediction made using above two scal-
ing theories [5]. Experiments conducted in following years
with larger yield explosives confirmed the active involve-
ment of gravity and material strength in crater mechanism
[4, 6]. Based on a regression analysis, Chabai [6] reported
that collected data correlated relatively closer to explosive
yield with exponent of 1/3.4 instead of 1/3 and 1/4 when
both the factors were considered. Holsapple and Schmidt
[9] investigated further to determine the exact prerequisites
under which certain scaling laws could be applied. They
concluded that neither cube root law nor quarter root law
holds true simultaneously for same physical conditions.
The selection of scaling law depends strictly on which
variables were kept constant and not included among pi
variables for the dimensional analysis.
Baker etal. [51] formulated an empirical model, using
dimensional analysis, consisting of six parameters: charge
weight, W, crater radius, R, burial depth of explosive, d,
soil density, ρ, and two strength parameters characterizing
soil behaviour. First strength parameter associates with soil’s
constitutive parameters and has dimensions similar to that
of stress. The second parameter incorporates the effect of
gravitation and has dimensions similar to ratio of force and
cube root of length. Observations made by Chabai [6] were
used to further modify the model, thereby, presenting the
scaled crater radius as,
Deliberating on the right measure for K and σ based on
the fact that neither of their values can approach either zero
or infinity for any time of soil, it was concluded that specific
weight, ρg and ρc2 were the closest measures for the two
strength parameters, respectively. Substituting K and σ with
ρg and ρc2 where c is seismic velocity in soil, equation then
becomes
The above equation was verified with experimental data
over a TNT yield range of 45kg to 4.5 × 105kg by plotting
R/d against W 7/24/d. The plots presented scattering of data
in lower ranges corresponding to low yield or deeply buried
explosion where W 7/24/d < 0.3. This signifies sensitivity of
scaled radius to minute variation of independent parameter.
Experimental data conditioned well for large yield or shal-
low buried explosion for which W7/24/d > 0.3.
(5)
R
d
=f
(
W
7∕24
𝜎
1∕6
K
1∕8
d)
(6)
R
d
=f
(
W7∕24
𝜌
7∕24
g
1∕8
c
1∕3
d)

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
Most of the work carried out in developing scaling mod-
els for crater dimensions are based on theory of similarity
wherein empirical equations are formulated based on results
from series of experiments having similar physical attrib-
utes. Thus, direct application of scaling laws to non-similar
experiments is liable to discrepancies in results and still is
the topic of discussion and research.
4 Ground Shock andGround Motion
Ground shocks are a result of energy released in the medium
by underground burst and is the prime reason behind ground
motion experienced during explosion. There are two ways in
which ground shocks are generated: direct coupling and air
blast. Both types of shocks travel downwards in outward fash-
ion as illustrated in Fig.7. Wherein the former contributes
more towards crater formation and deformation of medium
surrounding the crater, the latter is experienced in region
further away from crater. Airburst induced shock is stronger
near the surface and attenuates with increase in depth. It is
short-lived in comparison to direct induced ground shocks.
The waveform of air blast-induced shock is in form of abrupt
spike with gradual declination, whereas, direct-coupled
ground shock wave has sinusoidal shape. In regions near to
the burst, the velocity of air blast-induced ground shock is
more than the velocity of sound in the medium. This region is
called as super-seismic region. Ground motion in this region
is downwards and radially outwards with point of explosion
as the centre. Moving away from the location of explosion,
airburst induced shock wave slows down and direct-coupled
induced shock wave outruns it. This region is termed as out-
running region and the ground shock waves here is a combi-
nation of both types. Differentiation between the two types of
shock waves can be established by determining their arrival
times. Direct-coupled induced ground shock has similar veloc-
ity as seismic waves thus making it easier to estimate its arrival
time. Ground motion in outrunning region consists of undula-
tion with gradual damping as the distance increases. In shallow
buried explosion, air blast-induced ground shock is dominant,
however, it has little to negligible participation in deeply bur-
ied explosion [52, 53].
4.1 Estimation ofGround Shock andGround Motion
Westine [8] developed empirical formulas using similitude
analysis and experimental data to predict peak particle dis-
placement and peak particle velocity,
(7)
X
R
po
𝜌c21∕2
=
0.04143

W
𝜌c2R3
1.105
tanh1.5

18.24

W
𝜌c2R3

0.2367

Fig. 7 Ground shock phenomenology for buried explosion

J.Mandal et al.
1 3
where, X is peak particle displacement and U is peak particle
velocity. ρ and c are soil density and soil seismic veloc-
ity, respectively, representing soil constitutive properties
wherein their product is a measure of compressibility of
media under shock wave propagation. W is explosive energy,
and R is standoff distance. po is atmospheric pressure, how-
ever, it does not indicate the involvement of atmospheric
pressure in the prediction model, instead, it introduces rela-
tive compressibility for varying geo-materials. Though the
above prediction model covers a large range of scaled dis-
tances it does not take into account the effect of coupling
between explosion and ground in present state.
To incorporate the coupling effect the energy released
from explosion is modified into parameter, Ea/Par3, where
Ea is actual energy released in Joules, Pa is atmospheric
pressure in N/m2 and r is radius of spherical cavity con-
taining the explosive. The corresponding value of equiva-
lent fully coupled energy release, Ee, can be determined
(8)
U
cpo
𝜌c21∕2
=
6.169 ×10−3

W
𝜌c2R3
0.8521
tanh

26.03

W
𝜌c2R3

0.30

using the chart given in Fig.8a. Once the value of Ee is
known, the values of peak particle displacement and peak
particle velocity can be obtained using charts presented in
Fig.8b and Fig.8c [8, 54].
Drake and Little [55] formulated semi-empirical equa-
tions to predict soil stresses along with ground motion in
terms of peak particle velocity and peak acceleration using
seismic properties of soil which were later included in
the Army design manual, TM5-855–1, now superseded by
UFC 3–340-01 (2002). However, the latter improved docu-
ment is not yet accessible to civilian researchers. These
formulas are expressed as,
Similarly, Smith and Hetherington [54] provides equations
to predict soil pressure for three different types of soils,
(9)
x
W
1∕3=60 f
c(
2.5R
W
1∕3
)1−n
(10)
u
=48.8f
(
2.52R
W
1∕3
)−n
(11)
P
=160f𝜌c
(R
W
1∕3
)−n
Fig. 8 a Equivalent energy
release, b Coupled peak dis-
placement in geo-material and c
Coupled peak particle velocity
in geo-material ( modified from
Smith and Hetherington [54])

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
where P is peak pressure in soil (N/m2), u is peak particle
velocity (m/s), ρ is density of soil (kg.m3), R is distance from
explosive (m), W is weight of explosive (kg), n is attenuation
coefficient of soil, c is seismic velocity (m/s) and f is cou-
pling factor [54, 55]. Shock energy coupling is an important
factor inherent to buried explosion. It signifies the amount of
shock energy transmitted into the ground and is a function
of burial depth. Airburst involves limited coupling whereas
surface blast has larger degree of coupling. Direct coupling
with complete transmission of energy happens when the
explosive is fully confined in the soil with negligible air
void in between. To incorporate effect of ground coupling
while determining soil stress and ground motions coupling
factor ‘f’ has been introduced in their formula. It is a ratio of
magnitude of ground shock resulting from a partially buried
explosion to that of fully contained explosion. TM5-855–1
provides a graph, Fig.9, to determine the value of coupling
factor for soil at various value of scaled buried depth. For
near-surface burst, the value of coupling factor is 0.14 while
for a fully buried explosion the value is 1.
This graph is used universally for all soil types to deter-
mine the coupling factor at any scaled buried depth based
on the assumption that ground shock propagates similarly
in all soil types irrespective of the difference in their geo-
technical properties. Subsequently, many studies have been
(12)
P
=









𝜌uc for fully saturated clays
𝜌u0.6c+n+1
n−2u
for saturated clays
𝜌u

c+

n+1
n−2

u

for sand
published where the ground motion and soil pressure has
been predicted using varying versions of empirical formula
presented by TM5-855–1 design manual and coupling fac-
tor given above for validation purpose [56–65]. Some of the
prominent works are listed in Table1. Studies have reported
large discrepancies going as far as 800–1300% between
obtained results and values calculated using the empirical
formula. This necessitates in-depth research for revision of
these empirical equations or development of new models to
predict ground motion and soil pressure.
Propagation of ground shock through a medium is highly
sensitive to variations in geometry of explosive, depth of
burial and properties of the geomaterial, especially, water
content and porosity. For instance, degree of saturation
vastly affects the stress and ground motion caused by the
shock in the soils rich in clay. High density coarser soil is
relatively less influenced by water content. On the other
hand, low density coarser soil behaves similarly as cohe-
sive soils. These behaviours can very well be observed in
Fig.10, a graph presented in design manual, TM5-855–1.
The graph shows variation of peak stress in medium with
scaled distance for five different types of soils, namely, dry
loose sand, sandy loam or dry sand with medium density,
dense sand or wet sand, very wet sandy clay and saturated
clay or clay shale. Saturated clay may even generate ground
shock of magnitude higher than low-density dry soils by two
orders or more [55].
Significant effect of saturation and porosity is observed
in variation of ground shock and ground motion in rock
similar to soil mass. Saturated rock produces more ground
motion than dry rock with higher porosity under same load-
ing conditions [68]. Water content is capable of changing the
mechanical properties of geomaterial, hence, affecting the
wave propagation, however, studies have shown that porosity
has more influence than water content [68–70]. Sensitivity
study carried out by Kurtz [70] with porosity and water con-
tent as variables revealed that coupling factor is more sensi-
tive to porosity than water content which might be attributed
to the fact that higher porosity enables considerable attenu-
ation of shock energy. Their influence could further extend
to medium falling within two cavity radius [69].
Drake etal. [71] addressed these issues and modified the
empirical formula for coupling factor, however, the changes
were not made in the curve shown in Fig.9 which is still
followed till date. Many studies have objected to the use of
single curve defining the coupling factor versus depth of
burial for all types of soil [32, 72–74]. Besides, usage of
coupling factor to determine only the peak values of stress
and particle velocity is inadequate since both parameters are
time-variant and undergoes complex decay process over a
time period. Hence, extensive research is required to deter-
mine coupling factor considering various soil types and the
time-variant nature of blast loading.
Fig. 9 Coupling factor for soil varying with scaled depth of burial, d (
modified from TM5-855–1 [55])

J.Mandal et al.
1 3
5 Rock Fragmentation Models
In mining industries, blasting is employed to fragment
large rock masses into smaller units. The efficiency of blast
operation is governed by how optimally fragmented the
rocks are since it affects the transportation, hauling, and
crushing and milling processes which ultimately effects
the mine productivity. Rock fragmentation depends on
two sets of variables, characteristics of rock mass and
drilling-blasting parameters. The former set cannot be
controlled, however, the latter set can be altered to attain
optimised rock fragmentation [75]. These parameters are
illustrated in Fig.11. Many fragmentation prediction meth-
ods have been developed to define the fragmentation size
distribution curve and how it is influenced by the param-
eters mentioned in Fig.11. These methods are classified
as direct and indirect. Direct methods involve measuring
fragment sizes using sieve analysis which is accurate, how-
ever, is not cost and time effective. Cumulative distribution
function (CDF) or the fragment size distribution curve
is obtained by plotting the weights of fragments left on
each sieve against the mesh sizes. Indirect methods which
include empirical equations and visual analysis, although
not as accurate as direct method, gives satisfactory results
at comparatively less cost and time [76].
Rosin–Rammler model is often used in many prediction
models to define CDF. It is presented as,
where x is the sieve mesh size, xc is characteristic size taken
as 50% passing or mean fragment size, x50 and n is uniform-
ity index [77]. Empirical methods defining the mean frag-
ment size depends on the parameters listed in Fig.11, for
example [78],
(13)
PRR =
1
−e
−(x∕xc)
n
(14)
x50 =f1(exp losive )×f2(rock mass properties )×f3(geometry )
Table 1 List of researches using TM5-855–1 equations for validation
Source Loading
type
Attenuation
coefficient
Discrepancy (%) Remarks
P V a
Lu etal. [56] Buried 2 – 319 799 Magnitudes measured on front wall of buried structure
2.5 – 1344 799
Leong etal. [57] Buried 2.5 39–725 – – Used acoustic impedance suggested by TM5-855-1; Compared using
experimental data
Nagy etal. [66] Surface 2.5 30–100 – – Used acoustic impedance suggested by TM5-855-1
2.75 17–46 – –
Yang etal. [58] Surface 2.75 102–858 – – Used acoustic impedance suggested by TM5-855-1
Yankelevsky etal. [67] Buried 2.5 24.1–200 – – Used measured acoustic impedance
Koneshwaran etal. [60] Surface 2.5 30–70 – – Used acoustic impedance suggested by TM5-855-1
2.75 30–50 – –
Mobaraki etal. [61] Surface 2.75 0–82 – – Used acoustic impedance suggested by TM5-855-1
Mussa etal. [64] Surface 2.75 8–90 Used acoustic impedance suggested by TM5-855-1
Mussa etal. [63] Surface 2.75 1–60 – – Used acoustic impedance suggested by TM5-855-1; Altered radial
velocity factor and mesh remapping factor during discretization of
FE model
Mandal etal. [65]Surface 2.75 53–81 – – Used acoustic impedance suggested by TM5-855-1
Buried 0–13
Fig. 10 Variation in stress for different types of soil ( modified from
TM5-855–1 [55])

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
The most commonly adopted approach to predict mean
fragment size (in cm) is Kuznetsov’s equation, which presents,
where A is a constant depending on Protodyakonov hardness
of rock, wherein, for medium hard rock, A = 7; for heavily
fissured hard rock, A = 10; and for weakly fissured hard rock,
A = 13. E is relative explosive weight strength compared to
ANFO in percentage, q is powder factor in kg/m3 and Q is
the explosive charge weight in kg [79]. Uniformity index
depends on geometry of blast and is given as,
where B burden (m), d diameter of blast hole (mm), W stand-
ard deviation for accuracy in drilling (m), S spacing (m), H
bench height (m), L length of charge (m).
Table2 enlists few of the empirical models used to predict
fragmentation distribution. Among them, Kuz-Ram model is
the most popular model owning to its ease of application [75,
80, 81].
(15)
x
50 =AQ1
∕
6
q
0.8
(
115
E)
19∕
30
(16)
n
=
(
2.2 −
14B
d)(
1−
W
B)[
1+
(S
B
−1
)/
2
]L
H
6 State‑of‑art Development: Experimental
andNumerical Techniques
This section discusses various techniques available and
adopted to simulate buried explosion and the resultant events
involved such as cratering and ground shock propagation in
medium.
6.1 Experimental Techniques
Experimental test is without a doubt an effective approach
to investigate any physical event especially one as complex
as buried explosion. However, it bears certain disadvantages
which cannot be neglected. Firstly, it involves great risk and
the operation demands high level of skill and competency.
Secondly, due to high expenditure, it cannot be repeated
multiple times. Considering the above drawbacks research-
ers developed experimental models at reduced scale based
on the principles of theory of similarities and scaling laws.
Most commonly adopted methods involve inertial accelera-
tion by altering the gravitational force, g, such as centrifuge,
Fig. 11 Parameters affecting rock fragmentation distribution in blast operation

J.Mandal et al.
1 3
Table 2 Comparison of empirical fragmentation models
Fragmentation model Remarks Mean fragment size Fragment size distribution Source
Bond-Ram model Gives energy needed to crush a rock mass
into muck pile consisting of 80% frag-
ments of size less than 100 microns
Bond equation Rosin–Rammler equation Bond [82]
Swedish detonic research foundation
(SveDeFO) model
Considers rock mass properties and blast
pattern; Does not directly include the
effect of rock mass properties
Developed a separate equation Uses a constant uniformity index,
n = 1.35
Hjelmberg [83]
Kuz-Ram model Easy to parameterise; Provides direct
relation between fragmentation and
blast design parameters; Underestimates
quantity of fines; Ignores time delay and
interaction of blast waves released from
adjacent holes
Kuznetsov’s equation Rosin–Rammler equation Cunningham [79]
Kou-Rustan equation Gives 50% passing size; Considers char-
acteristics of discontinuities; Does not
predict shape of fragmentation curve
Developed a separate equation - Kou and Rustan [84]
Crushed Zone Model (CZM) and Two
Component Model (TCM)
Extension of Kuz-Ram model; Over-
comes the underestimation of fines by
Kuz-Ram model
Kuznetsov’s equation Rosin–Rammler equation Kanchibotla [80], Thornton
etal. [85] and Djordjevic
[86]
Kuznetsov-Cunningham-Ouchterlony
(KCO) model
Capable of predicting fines quantity Kuznetsov’s equation Swebrec function with curve undulation
factor
Ouchterlony [87]
Modified Kuz-Ram model Includes uniformity and blastability
indices thereby considering quantity of
fines; Does not consider upper limit of
fragment size
Modified Kuznetsov’s equation Rosin–Rammler equation Gheibie etal. [88]

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
linear accelerator etc. The working principle behind any
inertial accelerator is that if the gravity of a reduced scaled
model can be increased artificially to a sufficient level it will
be capable of simulating its corresponding prototype. To
incorporate the strain-dependent behaviour of medium under
dynamic loading such as blast, energy, E, is also included.
Thus, if g is the terrestrial gravity, G is the elevated grav-
ity for reduced scale testing, then energy corresponding to
large-scaled prototype testing will be,
In other words, a large yield explosion and cratering can
be simulated in a laboratory using few grams of explosive if
the gravity is increased sufficiently [9]. Earliest experiments
using an inertial accelerator can be dated back to 1940′s
conducted in the Soviet Union. Viktorov and Stepenov [89]
was the first reported experiment which used rocket sled
as a linear accelerator to elevate the gravity. Although lin-
ear accelerators gave satisfactory results, they were sooner
replaced by centrifuge accelerators due to their limited range
of gravity increment. Pokrovskii, a Soviet Union researcher,
was among the first to use centrifuge modelling to simulate
effect of explosion on soil at smaller scale in the early 1930s.
Over the following years, improved versions of centrifuge
testing have been developed and employed to simulate bur-
ied explosion and cratering [90–99]. Centrifuge has quite
a larger range of gravity variation, thereby, making it most
suitable for reduced scaling testing considering its reliability
and simplicity. However, with higher gravity, which is neces-
sary to simulate large-scale explosion, Coriolis phenomenon
is observed [96]. Coriolis acceleration increases with rota-
tion rate which gets higher when ‘g’ is increased. This effect
makes the soil particles to incline in undesirable direction,
consequently, affecting the ejecta trajectory and soil flow
during crater formation [100].
Alternatively, there exists another approach which
employs scaling of material strength instead of changing the
gravity. Equivalent material method involves degrading the
strength of medium and explosive to simulate high strength
geo-material and high energy explosive, respectively, partic-
ipating in large yield blast event with unaltered gravity. Sad-
ovskii etal. [101] was the first documented research which
incorporated the concept of equivalent material method to
scale down large yield buried explosion without elevating
the gravity. They used a vacuum chamber half filled with dry
sand. To simulate buried explosion, they used a thin rubber
envelope bulb filled with air which ruptured on being elec-
trically heated releasing air at a very pressure. The release
of compressed air behaved similarly to detonation of buried
explosive exempting the lack of heat generation. Empiri-
cal relation developed based on results obtained agreed
well with large yield experiments. This method has been
(17)
e=g3E=(nG)3E
employed to simulate delayed blasting, large yield explosion
and buried explosion in multi-layered media so far [102,
103].
6.2 Numerical Techniques andComputer
Simulation
Though numerical analysis and computational methods
were already in use in the early 1900s by mathematicians
and astronomers, it was only during World War II that use
of numerical simulation with the help of computers gained
momentum. The first large-scale use was for Manhattan
Project to simulate nuclear explosion [104]. Since then a
lot of improvement has been made in numerical simulation
techniques considering this field. Blast analysis of structure
is divided into two stages: initial stage comprises of pre-
dicting the energy released due to detonation and the sec-
ond stage involves determining the effect of blast energy on
the surrounding medium. Computational methods adopted
for blast analysis are categorised as uncoupled and coupled
depending on whether these two stages are performed simul-
taneously or not. Uncoupled approach is simple and is less
time consuming, however, it involves the risk of over pre-
diction of blast load which can lead to conservative analysis
of structure. On the other hand, coupled method provides
realistic results.
Various blast modelling techniques are provided by dif-
ferent computer programs to simulate explosion. Each tech-
nique varies considering the type of element formulation
adopted for numerical modelling. Four types of element
formulations commonly in use are Lagrangian, Eulerian,
Arbitrary Lagrangian–Eulerian and Smooth Particle Hydro-
dynamics (SPH). In Lagrangian element formulation, the
material is fixed with the mesh, hence, on loading the mesh
deforms with the material. Principle drawback of the method
involves tangling and excessive distortion of mesh on appli-
cation of high strain loading leading to computational insta-
bility. This drawback can be overcome by adopting Eulerian
element formulation, wherein, the mesh is fixed in the space
and material moves through mesh on deformation. How-
ever, this method is computationally expensive since it has
to perform intensive material tracking through interfaces.
To overcome the drawbacks of these two methods, Arbitrary
Lagrangian–Eulerian method was developed. It is a combi-
nation of best features the above two methods can offer. The
mesh is either fixed to space similar to Eulerian method,
moves along with material like Lagrangian method or can
be somewhere in between as illustrated in Fig.12. Thereby,
it reduces the possibility of excessive distortion while allow-
ing better resolution of flow details of material through
interfaces at low computational expense. Smooth Particle
Hydrodynamics (SPH) is a modified version of Lagrangian
approach wherein the system is modelled as shaped clusters

J.Mandal et al.
1 3
of particles. Each particle is assigned with material proper-
ties, location and initial velocity before initiation of analysis.
Motion of these particles is governed by the external loading
applied and internal interaction properties defined. Since this
method is meshless it experiences the least distortion. Fur-
thermore, tracking of material is much easier, thus reducing
the computational efforts.
In a fully Lagrangian formulation based model, blast
loading can be simulated using either CONWEP function
or by defining the pressure–time history using a load curve.
However, both the approaches have certain drawbacks
which makes it unsuitable for simulating buried explosion.
CONWEP function is strictly applicable for a scaled dis-
tance range of 0.147m/kg1/3–40m/kg1/3, whereas, buried
explosion falls under contact explosion type [105]. In the
case of load curve method, although it doesn’t face simi-
lar issue as mentioned earlier, it oversimplifies the blast
loading and neglects three-dimensional characteristic of
blast wavefront. Contrarily, Eulerian, Arbitrary Lagran-
gian–Eulerian and SPH methods are complex and time-
consuming but they are capable of realistically simulating
buried explosion. Multi-Material Lagrangian–Eulerian
method is the most commonly adopted approach, wherein
materials having fluid characteristics like explosive, air
and soil in the immediate vicinity of explosive are mod-
elled using Eulerian formulation. Other materials are mod-
elled using Lagrangian formulation. Eulerian–Lagrangian
interface is defined using certain contact options to take
care of undesirable penetration problems which vary with
software to software.
Aside from obvious reasons why numerical simulation
is advantageous over experimental research such as ease of
operation, cost-effectiveness, safety and repeatability, the
former provides in-depth and visual insights into the com-
plex phenomena occurring during buried explosion. Often
difficulties are faced while recording data from any explo-
sion test. Firstly, measuring equipment cannot be stationed
in the immediate range of explosion. Secondly, equipment
required to measure output from explosion are very expen-
sive, any damage incurred on them, which is highly likely
even at far-field zone, it would result in huge monetary loss.
Therefore, knowledge on what happens in the soil immedi-
ately surrounding the explosive was very limited prior to
arrival of computational methods. Computational analysis
becomes convenient in such situations. It allows understand-
ing complex mechanical, thermodynamic and kinematic
characteristics of soil under blast loading with added visual
aids.
Fig. 12 Comparison between Lagrangian, Eulerian and Arbitrary Lagrangian–Eulerian element formulation

Surface andBuried Explosions: AnExplorative Review withRecent Advances
1 3
When soil is subjected to high-strain loading such as blast
it behaves more or less like a fluid, especially in the region
immediately surrounding the detonation. The behaviour
of soil changes in radially outward directions with explo-
sion as the centre from plastic flow to elastic deformation.
Further, considering the complexity and dynamic nature of
blast loading it becomes extensively difficult to replicate the
behaviour of soil under explosion. Soil is a multi-phase geo
material consisting of a skeleton made of soil solids filled
with air and water in the pores. Each material follows differ-
ent physical laws when subjected to loading. Soils undergo
two primary types of deformation when subjected to blast
loading. First mechanism involves failure of contact inter-
face between soil solids leading to deformation of soil skel-
eton which could be elastic or plastic in nature depending
on the intensity of blast pressure. In the second mechanism,
deformation occurs in all three soil phases. It depends on
the type and composition of soil and loading rate which
mechanism dominates over the other. For dry soil, first
mechanism occurs initially wherein the soil skeletons bears
the load completely since the voids are filled with highly
compressible air. As the loading continues the soil skeleton
deforms and soil acts like fluid marking the initiation of
second mechanism. Deformation mechanism of saturated
soil depends on loading rate. At high loading rate, which
is the case of explosion, second mechanism prevails [106].
Blast-induced liquefaction is another complex phe-
nomenon wherein excess porewater pressure generated
due to blast wave triggers liquefaction of saturated soil.
Though experimental works have been conducted on this
phenomenon it was inadequate to plot out the underlying
mechanism involved, which is where numerical simulation
becomes useful.
Wang etal. [107] developed a three-phase elastoplastic
soil model based on a conceptual model theory explained
in Henrych [48]. Researchers have used this model to
numerically simulate the propagation of blast wave
through soil at various saturation degree to gain insight
on blast-induced liquefaction. Viscoplastic model and its
modified versions have also been developed by research-
ers which is capable of representing fluid-compressibility
behaviour of soil under blast much better along with its
strain-rate dependency [108–114]. Equation of state is
additionally used to model the fluid flow and pressure-
dependent compressibility of soil. Some of the commonly
used equations of state for soil under explosion are Lyak-
hov model [56, 59, 106, 107, 112, 115–118], Mie-Grü-
neisen [21, 112, 119], Shock-Hugoniot equations [26, 30,
110, 120–125]. Along with strength model and equation of
state, failure model is additionally used to define the yield-
ing of soil. Drucker-Prager model [107, 126], Mohr–Cou-
lomb failure model [127, 128] and plastic cap model [112,
114, 129–131] are mostly adopted failure models for soil.
Table3 enlist various soil models adopted to simulate
behaviour of soil under explosion.
Table 3 Comparison of soil models used for blast analysis
Source Strength model Failure model Equation of state/material stiffness
Laine and Sandvik [120] Prandtl-Reuss yield model with
density-dependent shear modulus
Hydro tensile limit Shock-Hugoniot/Three-phase equa-
tion of state
Wang and Lu [106], Wang etal.
[115, 116, 132]
Modified yield strength model
adopted from Prapaharan etal.
[133]
Drucker-Prager Lyakhov
Lu etal. [56] (2005) Implicitly considered in the three-
phase soil model
Modified Drucker-Prager Implicitly considered in the three-
phase soil model
Fiserova [24] (2006) Granular model Hydro tensile limit Piecewise plastic compaction curves
Lee [127], Busch etal. [128] Duvaut-Lions viscoplasticity Modified Mohr–Coulomb Modified soil stiffness
Grujicic etal. [17, 26, 30, 121, 134] Implicitly defined Piecewise yield stress curves Mie-Grüneisen
Tong and Tuan [109] Perzyna viscoplasticity Drucker-Prager Three-phase equation of state
Yankelevsky etal. [122], Karinski
etal. [123, 124]
Implicitly defined Lundborg Shock-Hugoniot
Feldgun etal. [59, 117, 135] Implicitly defined Lundborg Lyakhov
Luccioni etal. [136], Desmet etal.
[119]
Drucker Prager strength criterion
with piecewise hardening
Mie-Grüneisen
Ghassemi etal. [131] - Plastic cap model Modified soil stiffness
An [137], An etal. [112] Perzyna viscoplasticity Cap model Mie-Grüneisen
Higgins etal. [113] Perzyna viscoplasticity SANISAND Modified soil stiffness
Roger [21], Roger etal. [138] Perzyna viscoplasticity Geologic cap model Mie-Grüneisen
Lu and Fall [139], Lu etal. [114] Perzyna viscoplasticity cap Plastic cap Mie-Grüneisen

J.Mandal et al.
1 3
7 Summary andObservations
The paper presents a comprehensive review of literature
available on surface and buried explosions. Two main
phenomena inherent to such events, crater formation and
generation of ground shock, are described in details. The
paper discussed various factors controlling the underlying
mechanisms involved in the two phenomena. These factors
include characteristics of explosive such as depth of burial,
size, shape and type, geotechnical parameters such as den-
sity, saturation degree and type and finally gravitational
force. Due to high degree of complexity and non-linearity
involved, formulation of analytical models is not possible
easily, thereby, empirical and semi-empirical formula has
been developed over the years based on experimental data.
These formulas are presented and briefly analysed here
along with their limitations. Based on this review, it is
observed that there exists large difference in prediction of
various parameters and those observed in real blast tests.
Hence, there exists an immediate need to explore other
methods, as discussed in this review, to arrive at accept-
able solutions. Further, prediction models to determine
rock fragmentation size distribution and mean fragment
size to optimise mining productivity are additionally dis-
cussed to complete the review. Lastly, the paper covers
various state-of-art development made in experimental
and numerical methods used in this field which need to be
explored further for the research development.
Funding None.
Availability of Data and Materials Not applicable.
Code Availability Not applicable.
Compliance with Ethical Standards
Conflict of interest On behalf of all authors, the corresponding author
states that there is no conflict of interest.
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