Blast Loaded Plates

Marine Structures 22 (2009) 99–127

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marstruc

Review

Blast loaded plates

R. Rajendran a,*, J.M. Lee b,1

a BARC Facilities, Kalpakkam 603 102, Tamil Nadu, Indiab Department of Naval Architecture and Ocean Engineering, Pusan National University, 30 Jangjeon-Dong, Geumjeong-Gu, Busan609-735, Republic of Korea

a r t i c l e i n f o

Article history:Received 21 August 2006Received in revised form 1 April 2008Accepted 22 April 2008

Keywords:DetonationShock wave propagationAir blastUnderwater explosionPlate damage

* Corresponding author. Tel./fax: þ91 44 274802E-mail addresses: [email protected] (R. R

1 Tel.: þ82 51 510 2342; fax: þ82 51 512 8836.

0951-8339/$ – see front matter � 2008 Elsevier Ltdoi:10.1016/j.marstruc.2008.04.001

a b s t r a c t

Plates form one of the basic elements of structures. Land-basedstructures may be subjected to air blast loads during combatenvironment or terrorist attack, while marine structures may besubjected to either air blast by the attack of a missile above the wa-ter surface or an underwater explosion by the attack of a torpedoor a mine or a depth charge and an aircraft structure may be sub-jected to an in-flight attack by on-board explosive devices. Further-more, gas explosion occurs in offshore installations and industries.This review focuses on the phenomenological evolution of blastdamage of plates.

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1. Introduction

Plated structures are important in a variety of aero, marine and land-based applications includingaircrafts, ships, offshore platforms, box girder bridges, power/chemical plants, bins, bunkers and boxgirder cranes. Internal explosion on-board commercial aircraft using explosive devices results in com-plete loss of aircraft. Land-based structures experience air blast loading during war or terrorist attack oraccidental gas explosion. Marine structures undergo air blast loading due to accidental gas explosionsand or the attack of rockets and missiles above the waterline and underwater explosion loading due tothe explosion of torpedoes, mines and depth charges below the waterline.

For an aircraft, frames (circumferential reinforcing members) and stringers (rows of longitudinal re-inforcements) are riveted or adhesively bonded to the thin aluminium fuselage skin. The curvature ofthe fuselage is small compared to the individual shell size. Therefore, the individual panels are

82.ajendran), [email protected] (J.M. Lee).

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Nomenclature

A area of the plate (m2)/explosive constantB, R1, R2, U explosive constantsac crack length (m)C structural damping matrixca ambient speed of sound in air (m/s)cp peak wind velocity behind the shock front (m/s)/plastic wave speed of the plate material

(m/s)csa velocity of shock wave in air (m/s)cw velocity of sound in water (m/s)D material constantE Young’s modulus of the plate materialEp shock energy transferred to the plate per unit area (J/m2)EqTNT TNT equivalent of the explosiveEs energy of the shock wave per unit area (J/m2)ETNT energy of TNT explosive (J)F peak load on the plate (N)F(t) time-dependent load on the plate (N)G matrix relating structural degrees of freedom to the fluidI impulse per unit area (N s/m2)It total impulse (N s)JWL Jones–Wilkins–LeeK structural stiffness matrixk stiffness of the plate (N/m)M structural mass matrixm mass per unit area of the plate (kg/m2)n material parameterPm peak pressure/peak overpressure (MPa)Po atmospheric pressure (MPa)p exponent of power-type strain law/pressure (MPa)pi incident pressure (MPa)q dynamic pressure (MPa)/material constantre radius of the explosive (m)S stand off (m)S0 scaled distance (m/m)T kinetic energy of the plate (J)t time (s)/thickness (m)tc cavitation time (m/s)td positive duration of the blast wave (s)tf thickness of the fluid element (m)U internal energy of the explosive per unit volume (J/m3)V volume (m3)/impact velocity (m/s)Vm maximum velocity attained by the plate (m/s)v velocity of the water particle behind the shock front (m/s)vi incident water particle velocity (m/s)vs scattered fluid particle velocity (m/s)

R. Rajendran, J.M. Lee / Marine Structures 22 (2009) 99–127100

W TNT equivalent of the explosive charge quantity (kg)/work imparted to the platestructure (J)

x lateral displacement of the plate (m)a waveform parameteraj Johnson’s damage numberb aspect ratiod central plastic deflection of the plate (m)3f uniaxial fracture strain (m/m)F dimensionless numberh coupling factorhm moment amplification factorn Poisson’s ratioq time constant of the underwater shock wave (s)/half angle of the petal (�)r density of air behind the shock front (kg/m3)ro ambient density of air (kg/m3)rp density of the plate material (kg/m3)rw density of water (kg/m3)so flow stress of the material (MPa)su ultimate stress (MPa)sy static yield stress (MPa)syd dynamic yield stress (MPa)s burn time (s)x shock factor (kg0.5/m)xe effective shock factor (kg0.515/m1.03)j inverse mass number (kg/kg)z knock-down factor

Subscriptsa air-backed platec circular platecr criticalm maximump plater rectangular platet totalw water-backed platey yield

R. Rajendran, J.M. Lee / Marine Structures 22 (2009) 99–127 101

approximated as flat plates [1]. The curvature of the ship plating is small and it is supported by weldedlongitudinal and transverse stiffeners at its edges. The plate between the stiffeners is therefore consid-ered as a flat plate [2].

The motivation for this review arises from the primary concern for the design of plated structuresagainst blast load. As elucidated in the preceding paragraph, plates form one of the basic elements ofthe structures. Therefore, studying the blast response of plates helps understanding and improvingtheir blast resistance. The topic can be broadly categorized into (1) the detonation process or the rapidchemical reaction of the explosive, (2) the shock wave propagation in the medium in which detonationtakes place, (3) the interaction of the shock wave with the plate and (4) the response of the plate to theinput shock loading. These four aspects are brought out in this review with an attempt to gain greaterinsight into the blast damage phenomenon.


2. The explosion process

The explosion is a rapid chemical reaction in a substance, which converts the original material intoa gas at very high temperature and pressure evolving large amount of heat (4389 kJ/kg of trinitrotol-uene (TNT) explosive) [3]. The explosion process is divided into two parts: (1) the detonation processand (2) the interaction process between the product gases and the surrounding medium (air in atmo-sphere and water in underwater). During the detonation process, a detonation wave generates andpropagates in the explosive. The parameters that are used to assess the detonation performance ofan explosive are the Chapman–Jouguet (C–J) detonation pressure [4], the temperature of detonation[5] and the detonation velocity [6]. Typically for an explosive (TNT) with a density of 1650 kg/m3,the Chapman–Jouguet (C–J) detonation pressure [4] is 21,000 MPa, the detonation temperature [5] is3720 K and the detonation velocity [6] is 6950 m/s. Once the process of detonation is completed, theinteraction of the product gases with the surrounding medium takes place. The product gases withhigh pressure and temperature expand outward by generating a pressure wave. The gaseous productsare assumed to be inviscid at this high temperature and thus the viscous forces are not considered forthe explosive modeling. In the water medium, an instantaneous compression of the water surround-ing the gas emits a pressure pulse that propagates into water with a velocity that is three times higherthan the velocity of sound in water [7]. This higher velocity levels off rapidly and attains a velocity thatis only 20% higher than the sound velocity at a distance that is five times the charge radius after whichthe pressure wave falls to the sound velocity at around 20 times the charge radius and propagates atthat constant velocity. The gas bubble expands at a velocity that is much slower than the pressurepulse. In air explosion, the shock wave moves with the gas–air interface [8]. An equation of state(EOS) of the explosive relating energy, pressure and volume is essential for the numerical modelingof the detonation process. The most commonly used EOS to describe the state of detonation productsis Jones–Wilkins–Lee (JWL), which is given as [8]

pJWLðV ;UinÞ ¼ A�

1� U

R1V

�eð�R1VÞ þ B

�1� U

R2V

�eð�R2VÞ þ U

VEin (1)

where A, B, R1, R2and U are constants [8], pJWL is the pressure, V is relative the volume compared to theinitial volume of the explosive and Uin is the internal energy per unit volume. The first term in JWLequation known as the high-pressure term dominates first for V close to one, the second term is influ-ential for V close to 2 and last term corresponds to the expanded state.

3. The shock wave propagation

Significant contrast exists in the wave propagation phenomena between the air and the water me-dia due to (1) their different physical properties and (2) the interface phenomena between the explo-sive product gases and the surrounding medium [9]. The physical properties that matter for thepropagating medium are the velocity of sound, the density, the compressibility, the temperature andthe ambient pressure. While air is compressible water is considered as incompressible. Both air andwater are treated as inviscid. The velocity of sound in air at sea level is 340 m/s. The velocity of soundin water is 1483 m/s (approximately 4.36 times the velocity of sound in air at sea level). The sound ve-locity increases with temperature. The density of air at sea level is 1.25 kg/m3 and the density of wateris 1000 kg/m3 (approximately 800 times the density of air). The relations between the shock wave pa-rameters and the charge quantity and stand off for both air blast and underwater blast are empiricallyformulated and verified with a number of experiments.

3.1. Air blast

A schematic of the blast wave is shown in Fig. 1. The shock wave has an instantaneous rise and anexponential fall [10]. The parameters of interest for the damage process are the peak overpressure (thatis the pressure above the atmospheric pressure), the positive duration and impulse with respect to thescaled distance. The negative phase of the blast wave is generally ignored. An explosion of higher yield

p(t)

Positive phaseduration

Negative phaseduration

Ambientpressure

Peakoverpressure

Fig. 1. A schematic of the blast wave.


will arrive at a point sooner than an explosion of lower yield. The higher the overpressure at the shockfront the greater is the velocity of the shock wave. As the blast wave progresses outward, the pressureat the shock front decreases and the velocity falls of accordingly. At long ranges, when the overpressuredecreases to 7 kPa, the velocity of the blast wave approaches the ambient speed of sound. The durationof the overpressure phase increases with the energy of the explosive yield and the distance from theexplosion. The instantaneous pressure p(t) of the positive phase of an ideal air blast wave is given bythe Friedlander equation as [10]

pðtÞ ¼ Po þ Pm

�1�

�ttd

�e�at=td

�(2)

where Po is the ambient pressure, t is the instantaneous time, td is the positive duration of the pressurepulse and a is called waveform parameter that depends upon the peak overpressure Pm of the shockwave. The waveform parameter a is regarded as an adjustable parameter which is selected so thatthe overpressure–time relationships provide suitable values of the blast impulse. For chemical explo-sions, the peak overpressure is expressed as [10]

Pm=Po ¼808

h1þ

�S0

4:5

�2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S0

0:048

�2r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ

S00:32

�2r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ

S01:35

�2r (3a)

and for nuclear explosions [10]

Pm=Po ¼ 3:2� 106S0�3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�S0

87

�s �1þ S

0

800

�(3b)

The scaled distance S0 is given as [11–13]

S0 ¼ SW1=3

(4)

where S is the stand off from the explosion in m and W is the TNT equivalent of the explosive chargeweight in kg. The time of arrival of the shock wave ta for a radial distance r from an explosive radius ofradius re is given as [10]


ta ¼1ca

Z r

re

"1

1þ 6Pm7Po

#1=2

dr (5)

The duration of the shock pulse td for the chemical explosion in ms is [10]

td

W1=3¼

980h1þ

S0

0:54

�10ih1þ

S0

0:02

�3ih1þ

S0

0:74

�6i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S0

6:9

�2r (6a)

For a nuclear explosion [10]

td

W1=3¼

180h1þ

S0

100

�3iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S040

�r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S0

285

�5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S0

50;000

�6

r6

s (6b)

The shock wave velocity csa is expressed as [12]

csa ¼ ca

�1þ 6Pm

Po

�1=2

(7)

where ca is the ambient speed of sound. The particle velocity, cp (or peak wind velocity behind theshock front) is given as [12]

cp ¼5Pm

7Po

ca

ð1þ 6Pm=7PoÞ1=2(8)

The density, r, of the air behind the shock front is related to the ambient density ro as [12]

r=ro ¼7þ 6Pm=Po

7þ Pm=Po(9)

The dynamic pressure q, which is the kinetic energy per unit volume of air immediately behind theshock front, is given as [12]

q ¼ 52

P2m

7Po þ Pm(10)

The peak overpressure Pm in MPa is given as [14]

Pm ¼ 1:13S0ð�2:1Þ for 1 � S0 � 10 (11a)

Pm ¼ 0:183S0ð�1:16Þ for 10 � S0 � 200 (11b)

where S is in m.The impulse of the shock wave I in N s/m2 is given as [14]

I ¼ 203S0ð�0:91Þ for 1 � S0 � 10 (12a)

I ¼ 335S0ð�1:06Þ for 10 � S0 � 200 (12b)

and also as [10]


0:067 1þ�

S0

0:23

�42

I ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

S02ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

S0

1:55

�33

r (12c)

3.2. Gas explosion

A gas explosion is a process where combustion of a premixed gas cloud, that is, fuel–air or fuel/oxidizers causing rapid increase of pressure [15]. The pressure generated by the combustion wavewill depend on how fast the flame propagates and how the pressure expands away from the gas cloud(governed by confinement). The consequence of gas explosion ranges from no damage to totaldestruction.

3.3. Underwater explosion

The schematic of the blast wave in Fig. 1 is applicable for the underwater explosion too. The ambientpressure is the hydrostatic pressure. In air blast, the peak overpressure is typically of the order of kPa,which is comparable to the atmospheric pressure of 100 kPa, whereas in underwater explosion thepeak overpressure is of several orders of magnitude greater than the hydrostatic pressure. Therefore,the hydrostatic pressure is ignored and the peak overpressure is simply called as peak pressure. Theparameters that are of interest for an underwater explosion wave from the point of view of the platedamage are the peak pressure, Pm, the time constant, q, the free field impulse, I and the energy carriedby the shock wave Es. The pressure p(t) at a given point decays from its peak value Pm exponentiallywith time t as [3,16]

pðtÞ ¼ Pme�t=q (13)

where q is the time taken by the shock wave to decay to 1/e of peak value. The peak pressure, Pm, in MPais [3,16]

Pm ¼ 52:16

W1=3

S

!1:13

¼ 52:16S0�1:13 (14)

The velocity of the water particle behind the shock front is given as [3]

vðtÞ ¼ pðtÞ=rwcw (15)

The time constant q in s is given as [3,16]

q ¼ 96:5� 10�6

W1=3� W1=3

S

!�0:22

¼ 96:5� 10�6

W1=3�

S0ð0:22Þ (16)

The free field impulse per unit area, I, in N s/m2 is given as [3,16]

I ¼ 5760

W1=3� W1=3

S

!0:89

¼ 5760

W1=3�

S0ð�0:89Þ (17)

The energy carried by the shock wave per unit area in J/m2 is given as [3,16]

Es ¼ 98;000

W1=3� W1=3

S

!2:1

¼ 98;000

W1=3�

S0ð�2:1Þ (18)


These formulae are applicable for a stand off that is greater than 10 times the explosive charge radius.A variation of the ratio of the underwater explosion impulse, Iw, to the air blast impulse, Ia, as a functionof reduced stand off is shown in Fig. 2. The underwater explosion load carries an impulse that is at least22 times that of air blast for the same reduced stand off configuration. The dip in the curve is due to thechange in the formula for the air blast impulse prediction when the reduced stand off changes from 9 to10 m/kg(1/3).

4. Fluid–plate interaction

4.1. Air blast

When an air blast wave encounters a plate on which it impinges at zero angle of incidence, it getsnormally reflected. The peak plate overpressure (commonly known as reflected pressure), Ppm, isobtained from the Rankine Hugoniot relationship for an ideal gas as [17]

Ppm ¼ 2Pmð7Po þ 4PmÞ=ð7Po þ PmÞ (19)

The ratio of the plate impulse per unit area (commonly known as reflected impulse), Ip, to the inci-dent impulse per unit area, I, is approximated as [17]

Ip=I ¼ Ppm=Pm (20)

For a weak shock wave, Pm� Po. This leads from Eqs. (19) and (20) that the plate peak overpressure isdouble that of the incident peak pressure and hence the impulse imparted to the plate is double that ofthe incident impulse.

The requirement for simulating the maximum uniform lateral impulse on the plate area to studythe blast response of plates, taking into account the minimum use of the quantity of the explosivefor the reasons of safety and economy leads to a genre of experiments [18–34]. For numerical modeling

0 20 40 60 80 100

S' (m/kg1/3

)

10

20

30

40

Iw

/Ia

Fig. 2. The variation of the ratio of free field impulse for underwater explosion to air blast as a function of reduced standoff.


of the deformation phenomena, the impulse that is measured experimentally during the zero stand offfiring (that is, the explosive is separated from the test pate by a thin polystyrene sheet) is divided by theburn time of the explosive to arrive at the assumed uniform rectangular lateral pressure [35]. A blastpeak pressure that is 10 times or bigger than the corresponding static collapse pressure is assumed asa rectangular pressure pulse without loss of accuracy [36,37]. The step pressure, Pm, is estimated asa function of the measured total plate impulse, Itp, the exposed area of the plate, A, and the burntime s as [38]

Pm ¼ItpAs

(21)

Shock tubes are built to simulate the blast waves with required plate peak pressure and plate im-pulse [39–44]. Spherical shock wave fronts generated by the direct detonation process lead to a com-plex space–time evolution of the pressure distribution on a plane plate, which results in poorprediction of the plate deformation [40]. On the contrary, the plane shock wave fronts generated bythe shock tube allow precise modeling.


For the coupled fluid–structure interaction, the motion of the plate is given as [45–49]

M€xþ C _xþ kx ¼ FðtÞ (22)

where M is the structural mass matrix, C is the structural damping matrix, K is the structural stiffnessmatrix, x is the structural displacement and F(t) is the time-varying load applied to the structure. Bysuperimposing the imaginary fluid mesh on the fluid–plate boundary, the surface compatibility ofthe submerged plate is written as [46]

FðtÞ ¼ �GAf ðpi þ psÞ (23)

where G is the matrix relating the structural degrees of freedom to the fluid, Af is the matrix containingthe areas of the elements in the fluid mesh, pi is the incident pressure of the underwater explosion andps is the scatted pressure of the plate. Compatibility requirements dictate that the surface normal ve-locity of the plate and the fluid is equal, that is [46],

GT _x ¼ vi þ vs (24)

where vi is the incident water particle velocity from the underwater explosion, and vs is the scatteredwater particle velocity from the plate.

The fluid is assumed to be inviscid and incompressible. The scatted pressure and the scattered fluidparticle velocity are related by [46]

ps ¼ rwcwvs (25)

From Eqs. (24) and (25),

ps ¼ rc

GT _x� vi

�(26)

Eq. (26) is substituted into Eq. (23) to obtain the load–time history as [46]

FðtÞ ¼ �GAf

hpi þ rc

GT _x� vi

�i(27)

The force–time history is finally substituted into Eq. (22) to obtain the differential equation for theresponse of the plate as [46]

M€xþ

C þ GAf GTrwcw

�_xþ Kx ¼ �GAf ðpi þ rwcwviÞ (28)


The term rwcw represents the additional damping term to the plate due to the energy radiated awayfrom the plate into the fluid. The only unknown term in Eq. (28) is the plate displacement x whichis solved by finite element method. The response equation of the plate is valid until the fluid pressuregoes below the local atmospheric pressure or in other words, until cavitation occurs. Ignoring dampingand the nodal displacement for the duration the pressure pulse acts, for a one dimensional plate Eq.(28) reduces to [3,7,50–52]

m€xþ rwcw _x ¼ 2Pme�t=q (29)

where m is the mass per unit area of the plate. Applying initial conditions and introducing the dimen-sionless inverse mass number ja¼ rwcwq/m, the plate pressure, Pp(t), is [53]

PpðtÞ ¼ 2pme�t=q � 2Pmja

ðja � 1Þhe�t=q � e�jat=q

i(30)

from which the plate peak pressure is given as

Pp ¼ 2Pmj1

1�jaa (31)

and the plate maximum velocity is

Vma ¼2Pmq

mj

ja1�jaa (32)

The shock energy transferred to the air-backed plate Ep is

Epa ¼12

mV2ma ¼

2P2mmq2

mj

2ja1�jaa (33)

The energy carried by the free field shock wave is

Es ¼1rc

Z N

0p2ðtÞ dt ¼ 1

rc

Z N

0

Pme�t=q

�2dt ¼ P2

mq2

2mja(34)

The ratio of the plate energy to the free field shock energy, ha, which is also known as coupling factor is

ha ¼Epa

Es¼

�2P2

mq2

m

�j

2ja1�jaa�

P2mq

2

2mja

� ¼ 4j1þja1�jaa (35)

The time to reach the maximum velocity or the cavitation time, tca, is given as

tca ¼q ln ja

ja � 1(36)

The impulse acting on the air-backed plate per unit area is given as

Ip ¼ 2Pmqjja

1�jaa (37a)

The maximum achievable impulse per unit area for an infinitely rigid plate from Eq. (37a) is [54]

Ipm ¼ 2Pmq (37b)

or

Ipa ¼ zIpm ¼ 2zaIf (37c)

where za is the knock-down factor for air-backed plate which is given as

za ¼ jja

1�jaa (37d)


For ja¼ 1/2, the impulse imparted to the plate, Ip, is equal to the free field impulse. Or in other words,Ip¼ I when 2rwcwq¼m. As the plate becomes more and more rigid, the impulse ratio approaches unity,meaning thereby, the underwater explosion impulse becomes double that of the free field impulse as inthe case of the impulse imparted by a weak air blast wave on a plate. The ratio of the variation of theplate impulse for air-backed underwater exploded plates, Ipw, to air blasted plates, Ipa, as a function ofreduced stand off is shown in Fig. 3.

For a water-backed plate, the equation of motion of the plate is modified as [7,50]

m€xþ 2rwcw _x ¼ 2Pme�t=q (38)

The maximum velocity of the water-backed plate is [7]

Vmw ¼Pmq

mj

jw1�jww (39)

where jw¼ 2rwcwq/m. The maximum velocity for the water-backed plate is reached when the platepressure equals the hydrostatic pressure of the water behind the plate. The energy imparted to the wa-ter-backed plate, Epw, is

Epw ¼12

mV2mw ¼

P2mq2

2mj

2jw1�jww (40)

The ratio of the energy of the water-backed plate to the shock wave energy, hw, is given as

hw ¼Epw

Es¼

P2

mqrc

�j

1þjw1�jww

P2mq

2rc

� ¼ 2j1þjw1�jww (41)

0 20 40 60 80 100

S'(m/kg1/3

)

0

10

20

30

Ip

w/Ip

a

Plate impulse ratiom=15.6kg/sq mm=31.2kg/sq mm=62.4kg/sq mm=289.5 kg/sq m

Fig. 3. The variation of the ratio of impulse for underwater explosion to air blast on a plate as a function of reduced standoff.


The ratio of ha to hw gives the ratio of the strain energy of the air-backed plate to the water-backed plateduring the elastic regime of deformation. This ratio from Eqs. (35) and (41) is given as

ha

hw¼�

Ep=Es

Epw=Es

�¼ 2

�4jað1�2ja Þj

�2jað1�ja Þð1�2ja Þa (42)

The time to reach the maximum velocity, tcw, is given as [7]

tcw ¼q ln jw

jw � 1(43)

The impulse acting on the water-backed plate per unit area is given as

Ipw ¼ 2Pmqjjw

1�jww ¼ 2If zw (44a)

where zw is the knock-down factor for water-backed plate which is given as

zw ¼ jjw

1�jww (44b)

The ratio of the primary pulse plate impulse for an air-backed plate to a water-backed plate gives theplastic damage ratio of these plates for the primary shock. From Eqs. (37) and (44),

Ipa

Ipw¼ 2

�2ja1�2ja j

�jað1�jaÞð1�2ja Þa (45)

For jw¼ 1/2, the impulse imparted to the plate, Ipw, is equal to half of the free field impulse. Or inother words, Ipw¼ If/2 when 4rwcwq¼m. As the plate becomes infinitely rigid, the maximum plate im-pulse equals the free field impulse. This is in contrast to the air-backed plate that undergoes twice thefree field impulse for identical conditions. The variation of plate impulse, Ip, and energy, Ep, as a fractionof the respective free field parameters for air- and water-backed plates as a function of inverse massnumber is shown in Fig. 4.

0 4 8 12 16

0

0.4

0.8

1.2

1.6

2

Ip/If

Air-backed plate impulseWater-backed plate impulseair-backed plate shock energy transferwater-backed plate energy transfer

0

0.2

0.4

0.6

0.8

1

Ep/E

s

Fig. 4. The variation of impulse and energy of air- and water-backed plates with the inverse mass number for underwater explosion.


The damage caused by the reloading component of the underwater shock wave on an air-backedplate is larger than the damage caused by the primary pulse itself. While reloading due to cavitationis absent for water-backed plates, reloading due to the gas bubble is present for both air- and water-backed plates. Reloading occurs when the depth of explosion is at least half the stand off and reachesit maximum when the depth of explosion becomes double the stand off [55].

5. Blast damage

5.1. Air blast

5.1.1. Uniform blast loadingSimple methods of structural dynamics were applied by Biggs [56] and Clough and Penzien [57] by

applying single degree of freedom system (SDOF) and idealizing the plate as the beam for obtaining theblast damage. The damage of plates that are subjected to non-contact air blast is conventionallyassessed by the simulated experimental methods or by taking the measured pressure–time historyon the plate for the simulated environment as the prescribed load and performing numerical analysis[1,13,58,59].

5.1.1.1. P–I diagrams. Elastic and plastic response of single degree of freedom (SDOF) systems subjectedto blast loading can be presented in the form of pressure–impulse (P–I) diagrams [60]. According to theP–I diagram of a specific structure or structural element, a certain load with the peak pressure and im-pulse above the critical value will result in the damage of the structures, vice versa, the structure is safeif the peak pressure and impulse combination is located below the curve.

In quasi-static loading realm, the deformation depends only on the peak load F and the structuralstiffness k. The response is independent of the duration of loading and the mass of the structure.The work done on the structure is equated to the strain energy for the quasi-static loading regime(loading period to the natural period of the structure is greater than 6.36). For a linear elastic system,the strain energy, UE, is given as [61]

UE ¼12

kx2max (46)

The maximum permissible work imparted, W, to the structure by a constant force whose amplitudedecreases insignificantly is [61]

W ¼ Fxmax (47)

where F is the force that is acting which is given by multiplying the area, A, with the pressure Pm. FromEqs. (46) and (47) [61],

xmax

ðF=kÞ ¼ 2:0 (48)

Eq. (48) is called quasi-static asymptote.In the impulsive realm (loading period to the natural period of the structure is less than 0.0636), the

deformation is directly proportional to the impulse. The kinetic energy imparted to the structure, T, isequated to the strain energy UE [61].

T ¼�

mA2

�I2

ðmAÞ2¼ I2

2mA(49)

Equating the kinetic energy, T, to the strain energy UE,

ffiffiffiffiffiffiffiffiffiffikmAp

xmax

I¼ 1:0 (50)


Eq. (50) is called impulsive asymptote. A combination of peak loads and durations with the same im-pulse will result in the same maximum deformation. The deflection is influenced by both structuralstiffness and structural mass. A dynamic load factor of 2 is conservative in this regime.

Between quasi-static realm and impulsive realm a transition realm exists which is known as dy-namic loading realm. The deformation here depends on the entire loading history. Here, the motionof the structure depends on pressure and impulse as well as structural stiffness and mass. Computa-tion of the quasi-static and impulsive asymptotes yields an approximation to the entire shockresponse.

For a rigid-plastic system loaded in the pressure realm, the strain energy, UP, is given as [61]

UP ¼ Rxmax (51)

where R is the resistance. Equating the work done, W, to the strain energy [61]

F=R ¼ 1:0 (52)

Eq. (52) is the quasi-static asymptote for the rigid-plastic structure. For the impulsive loading realm,equating the kinetic energy to the strain energy UP,

IffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixmaxmARp ¼

ffiffiffi2p

(53)

Eq. (53) is called impulsive asymptote for the rigid-plastic structure.The loading on a plate cause by gas or dust explosions is characterized finite rise time and a non-

exponential fall. The quasi-static loading asymptote for a finite rise time is equivalent to a staticloading. In the range 1:15 < I=xmax

ffiffiffiffiffiffiffiffiffiffikmAp

< 5:5, the loading with finite rise time is more severe thana loading with zero rise time. This behaviour is produced by resonance between the loading rateand the structural frequency. A typical P–I diagram for a blast loaded SDOF system is shown in Fig. 5.

5.1.1.2. Plastic deformation. The central deflection of plastically deformed plates is taken as the indica-tion of the measure of blast damage. Assumed modes method in which a shape function is used to rep-resent the global displacement function [62] was used in conjunction with rigid-plastic materialbehaviour and energy methods were applied by Schleyer et al. [63] and Langdon and Schleyer [64].For the contact air blast, where a rectangular pressure pulse can be assumed [18,65] analytical [36]and empirical [28] predictions are available based on the impulse imparted to the plate.

Johnson [66] proposed a guideline for assessing the behaviour of metals subjected to impact loadingusing a dimensionless number that is defined as

aj ¼rpV2

sd(54a)

where V is the impact velocity, rp is the plate material density and sd is the damage stress which istaken as equal to the plate material yield stress sy. Johnson’s damage number is applicable onlywhen plates have similar dimensions. The damage number can be written in terms of impulse as [67]

aj ¼I2tp

A2t2rpsy(54b)

where t is the thickness of the plate.A modified damage parameter F was introduced by Nurick and Martin [28] that incorporated plate

dimensions and loading. For circular plates [29]

Fc ¼Ipt

pRt2

rpsy

�1=2(55)

where R is the radius of the loaded portion of the circular plate. For rectangular plates [67]

Fig. 5. P–I diagram for SDOF systems undergoing blast load [61]. (a) Elastic response due to air blast, (b) plastic response due to airblast, (c) elastic response due to gas explosion and (d) plastic response due to gas explosion [61].


Fr ¼Ipt

2t2

4abrpsy

�1=2(56)

where 2a and 2b are the length and breadth of the plate.For large plastic deformation (mode I failure) of clamped circular plates [28] the deflection-thick-

ness ratio is empirically given as

�d

t

�c¼ 0:425Fc þ 0:227 (57)

and for clamped rectangular plates [28] the empirical relationship is

�d

t

�r¼ 0:471Fr þ 0:001 (58)

where d is the central deflection of the plate.Jones [36] predicted analytically the deflection-thickness ratio for fully clamped circular plates

without strain rate effects as


�d�¼ 0:817Fc (59a)

t c

Taking strain rate effect into account Eq. (59a) is modified as [36]

�d

t

�c¼ 0:817Fcffiffiffi

np (59b)

n ¼ 1þ

I2p

3r2Pt2DR

�rp

3sy

�1=2!1=q

(59c)

For fully clamped rectangular plates without strain rate effects, the deflection-thickness ratio asgiven by the analytical method of Jones is [36]

�d

t

�r¼ð3� 20Þ

nð1þ GÞ1=2�1

o2f1þ ð20 � 1Þð20 � 2Þg (60a)

G ¼2rpV2a2b2

3syt2 ð3� 220Þ�

1� 20 þ1

2� 20

�(60b)

20 ¼ b

3þ b2

�1=2�b

�(60c)

b ¼ ba

(60d)

Taking strain rate effect into account Eq. (60a) is modified as

�d

t

�r¼ð3� 20Þ

nð1þ G=nÞ1=2�1

o2f1þ ð20 � 1Þð20 � 2Þg (60e)

n ¼ 1þ"

Vtð3� 20ÞG1=2

6ffiffiffi2p

Db2f1þ ð20 � 1Þð20 � 2Þg

#1=q

(60f)

5.1.2. Localized blast loadingLocalized impact is the explosion or impact process that occurs over a localized region of the plate

(in contrast to the uniform loading that occurs over the whole area of the unsupported plate). Whena localized explosion takes place on a plate petalling occurs [68]. For small amplitudes of impulsethe plate undergoes dishing. Dishing occurs until tensile necking and fracture takes over the criticalvelocity, Vcr, of the plate which is given by

Vcr ¼ 2:83cpffiffiffiffi3fp

(61a)

where cp is the plastic wave speed which is given as

cp ¼ffiffiffiffiffiso

rp

s(61b)

where so is the flow stress which is given as


so ¼ffiffiffiffiffiffiffiffiffiffiffiffisysu

1þ p

r(61c)

where su is the ultimate stress and p is the exponent in the power-type stress–strain law. The detona-tion blows out a central cap of the radius rp (where rp is the radius of the explosive). This occurs at a cen-tral deflection d, which is given as

d ¼ 2:47rpffiffiffiffi3fp

(62)

When the impulse is above this value, the reminder of the initial kinetic energy goes into the petallingprocess. For n1 radial cracks that develop from a point in an infinite plate dividing it into n1 symmetricpetals, the central 2q angle of the petal equals 2p/n1. Taking the instantaneous crack length as ac, theperpendicular distance l, which can be considered as process parameter is given by ac cos q. By energybalance [68],

�Vcr

c

�2"�

VVcr

�2

�1

#¼ 5:2h0:6

m

�tre

��l

re� 1

�1:4

(63a)

where V is the impact velocity and hm is the moment amplification factor which takes into account thelarger bending resistance of the curved plate is given as [68]

hm ¼ 1þ 2ffiffiffiffiffiffiffi23f

pq2re

t(63b)

Normally 3–5 petals form in the plate [68]. The amount of impulse imparted to the plate is propor-tional to the weight of the explosive W.�

VVcr

�¼�

IpIpc

�¼�

WWcr

�(63c)

where Wcr is the critical weight of the explosive to generate the critical impulse, Ipc, that is required toblow out the central plot. Lee and Wierzbicki [69,70] reported analytical and numerical modeling onthe dishing, discing and petalling of plates subjected to localized impulse. Analytical and experimentalwork was reported by Wierzbicki and Nurick [71] to determine the location of tearing failure and thecritical impulse to failure. Experimental work on clamped circular plates subjected to localized impulsewas reported by Nurick and Radford [72] to study the formation of petalling failure. Jacob et al. [73]described the effect of varying both the loading conditions and the plate geometries on the deforma-tion of the plate and predicted numerically the plate response.


5.2.1. Uniform explosion loading

5.2.1.1. Elastic response. A near miss underwater explosion in a war scenario may result in the plate re-sponse varying from elastic to plastic and in an extreme case fracture. The designer is therefore inter-ested in predicting the range of responses of the ship plates. Both warships and merchant ships haveliquid filled side shells. Therefore it is of interest to know what happens to these side shells during anunderwater explosion environment. Although there is no permanent damage to the plate that un-dergoes elastic deformation, it is the interest to the designer to know in advance the transient stateof stress it develops. For a small intensity of explosion, the stresses developed in the plate are in its elas-tic range. During elastic deformation the air-backed plate undergoes tens of thousands of ‘g’s (acceler-ation due to gravity) [74]. Water-backed plates, however, suffer relatively less damage. This is becausea considerable proportion of the shock wave transmits through the water at the rear of the plate. Inother words, water-backed plates simply transmit the maximum part of the shock wave energy.


5.2.1.1.1. Circular geometry. For air-backed plates [75–77] for the primary shock wave, the semi-an-alytical von Mises stress, sa, at the apex (center) of the plate is given as

sa ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ErpP2

mx2=ð1�xaÞa

r2wc2

wð1� nÞ

vuut (64a)

where E is the Young’s modulus of the plate material. In terms of effective shock factor [77,78]

sa ¼ 1179xea

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

ð1� nÞt

s(64b)

where E is Young’s modulus of the material, n is the Poisson’s ratio, xea is the effective shock factor forthe air-backed plates.

xea ¼ x1:03 ffiffiffihp

a (64c)

where x is the normal shock factor which is given as [7]

x ¼ 0:445

ffiffiffiffiffiffiWp

S(64d)

Traditionally, the shock factor is used to classify the severity of the attack. For x less than 0.15, the shockdamage is considered to be negligible; for x greater than 0.7, the shock damage is considered to be theseverest [7].

For water-backed plates, assuming a strain distribution pattern of the target plate similar to that ofair-backed plate [76–78] the semi-analytical von Mises stress is given as

sa ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ErpP2

mx2=ð1�xwÞw

2r2wc2

wð1� nÞ

vuut (65a)

or in terms of effective shock factor [77,78]

sa ¼ 1179xew

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

ð1� nÞt

s(65b)

where xew is the effective shock factor for a water-backed plate

xew ¼ x1:03 ffiffiffiffiffiffiffihwp

(65c)

A comparison of the apex (plate center) von Mises stress developed by the circular plate for variousshock factors is shown in Fig. 6. As expected, water-backed plates undergo less stress. Furthermore, theshock wave parameters based and the shock factor based stresses are in good agreement.

5.2.1.1.2. Rectangular geometry. For air-backed plates [76] and a Poisson’s ratio of 0.3 the semi-analytical von Mises stress is given as

sa ¼ 0:867

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14ErpP2

mx2=ð1�xaÞa

r2wc2

w

vuut (66a)

In terms of effective shock factor [77,78]

sa ¼ 1584x1:03 ffiffiffiffiffihap

ffiffiffiEt

r(66b)

Numerical and experimental underwater explosion simulations carried out [65] on aluminium plateof 1 m� 1 m� 0.01 m are compared well with Eq. (66).

0 0.02 0.04 0.06 0.08 0.1

Shock factor (kg1/2

/m)

0

200

400

600

Ap

ex V

on

M

ises

stress (M

Pa)

ABPSWPBABPSFBWBPSWPBWBPSFBABPE

Fig. 6. A comparison of the elastic response of air- and water-backed circular plates as a function of shock factor. ABPSWPB: air-backed plate shock wave parameters based; WBPSWPB: water-backed plate shock wave parameters based; ABPSFB: air-backed plateshock factor based; WBPSFB: water-backed plate shock factor based; and ABPE: air-backed plate experiment [76].


For water-backed plates, assuming a strain distribution pattern of the target plate similar to that ofair-backed plates [76,77]

sa ¼ 0:867

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7ErpP2

mx2=ð1�xwÞw

2r2wc2

w

vuut (67a)

or in terms of effective shock factor [77,78]

sa ¼ 1584x1:03 ffiffiffiffiffiffiffihwp

ffiffiffiEt

r(67b)

A comparison of the apex (plate center) von Mises stress developed by the rectangular plate for var-ious shock factors is shown in Fig. 7. A similar trend as for the circular plates is observed.

5.2.1.2. Limiting elastic range. As the intensity of explosion gradually increases, the plate reaches thelimit of the elastic range beyond which it undergoes permanent deformation. For all practical discus-sions here, elastic range limit is assumed to merge with the yield point without appreciable error. Mer-chant cargo vessels and warships are normally made up of mild steels whereas mine sweepers aremade up of austenitic steel (non-magnetic) and aluminium alloys. For strain rate sensitive materials,there are static and dynamic yield points, the dynamic yield stress, syd, is higher than the static yieldstress sy, and is typically related by the Cowper–Symonds relation:

syd ¼ sy

"1þ

�_3

D

�1=q#

(68)

where _3 is the average strain rate, and D and q are material constants which are given by Jones [36].

0 0.02 0.04 0.06 0.08

Shock factor (kg1/2

/m)

0

100

200

300

400

500

600

Ap

ex V

on

M

ises stress (M

Pa)

ABPSWPBABPSFBWBPSWPBWBPSFBABPE

Fig. 7. A comparison of the elastic response of air- and water-backed rectangular plates as a function of shock factor. ABPSWPB: air-backed plate shock wave parameters based; WBPSWPB: water-backed plate shock wave parameters based; ABPSFB: air-backed plateshock factor based; WBPSFB: water-backed plate shock factor based; and ABPE: air-backed plate experiment [76].


The effective shock factor xe remains the same for air- and water-backed plates for generating a spec-ified stress level. The effective shock factor, xeyc, required for generating the static yield stress for cir-cular plates is given as [77,78]

xeyc ¼ 848� 10�6sys

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� nÞt

E

r(69)

and for rectangular plates as [77,78]

xeyr ¼ 631� 10�6sys

ffiffiffiEt

r(70)

It was established from experiments [79] that the dynamic yielding of mild steel during an underwaterexplosion occurs for a stress that is 1.4 times the static yield stress. The average strain rate for defor-mation during the elastic range was 0.5 s�1 [70]. Therefore, the effective shock factor for circularand rectangular plates for dynamic yielding of various hull materials is given by multiplying the rightside of Eqs. (69) and (70) by scaling factors (sd/sy). The Cowper–Symonds equations should be obtainedfrom dynamic tensile tests on the specific material being considered. Guideline values are derived byapplying the nominal material constants [36] and the average strain rate for the elastic range limit. Thescaling factors thus obtained are as follows: for mild steel 1.42, for high tensile steel 1.17, for aluminiumalloy 1.09, for a-titanium (Ti 50A) 1.54 and for AISI 304 stainless steel 1.59.

A look at Eq. (64d) shows that a range of charge quantity and stand off combination is possible forgenerating the desired shock factor. The charge quantity in kg and stand off in m required to generatethe shock factor for the given time constant are obtained from Eqs. (16) and (64d) as [80]

W ¼�

x

0:445

�0:5946� q

96:5� 10�6

�2:7026

(71)


S ¼�

0:445x

�0:7027� q

96:5� 10�6

�1:3513

(72)

5.2.1.3. Plastic deformation. When the effective shock factor exceeds the dynamic yield value, the plateundergoes permanent deformation. The extent of permanent deformation is indicated by the centraldeflection of the exploded plates and is proportional to the impulse imparted to the plate [3]. Forthe underwater explosion of the air-backed plates based on the free field impulse [81,82] the empiricalrelationship is given as [81,82]

�d

t

�c¼ 0:541Fc � 0:433 (73)

The variation of the deflection-thickness ratio as a function of the dimensionless parameter Fc for cir-cular plates is shown in Fig. 8. A comparison of Eqs. (57), (59a), (59b) and (73) with experimental datashowed good agreement [80]. Eq. (59a) over predicts because strain rate effects are not accounted for.For air-backed rectangular plates the empirical prediction is given as [80]

�d

t

�r¼ 0:553Fr þ 0:741 (74)

The variation of the deflection-thickness ratio as a function of the dimensionless parameter, Fr, for rect-angular plates is shown in Fig. 9. A comparison of Eqs. (58), (60a), (60e) and (74) with experimental datashowed good agreement [80]. Eq. (60a) over predicts because strain rate effects are not accounted for.

Eqs. (73) and (74) assume that the total deflection caused by the impulse due to primary and thereloading shock waves on the plate is equal to the deflection caused by the free field impulse.

0 10 20 30 40

0

10

20

30

Deflectio

n-th

ickn

ess ratio

Circular platesNurick & Martin Equation (57)Jones Equation (59a)Jones Equation (59b)Rajendran & Narsimhan Equation (73)Experimental (Ref [81])

Fig. 8. Variation of deflection-thickness ratio of circular plates with the dimensionless parameter F.

0 10 20 30 40

0

5

10

15

20

25

Deflectio

n-th

ickn

ess ratio

Rectangular platesNurick & Martin Equation (58)Jones Equation (60a)Jones Equation (60e)Rajendran & NarasimhanEquation (74)Experimental (Ref [81])

Fig. 9. Variation of deflection-thickness ratio of rectangular plates (b¼ 5/6) with the dimensionless parameter F.


For water-backed plates cavitation does not occur [7]. However, there will be shock loading on thewater-backed plate due to the gas bubble pulse. The total impulse for water-backed plates for the in-teraction of the primary shock pulse is given as

Ipt ¼ VmwAtrp (75)

from which dimensionless parameters are obtained for water-backed plates to calculate the central de-flection. A photographic view of a rectangular steel plate that underwent mode I (inelastic deforma-tion) is shown in Fig. 10.

Fig. 10. Photographic view of an underwater exploded plate that underwent mode I failure.


5.2.2. Contact underwater explosionIn underwater explosion scenario, contact underwater explosion where the explosive comes in di-

rect contact with the plate during the detonation process is equivalent to localized blast loading. Thephysics of the contact underwater explosion phenomena is not yet well understood. However, it is ofgreat relevance to the naval warfare because underwater weapons for attacking the subsurface vesselsare so designed that they detonate on impact with the target. Keil [50] reported that the early work oncontact explosion damage of ships was carried out by the Japanese imperial navy on the discarded ship.Experiments carried out by Keil [7] ruled out the depth of submergence on the damage of plates. Keil[7] described that there is a definite relation between the radius, R, of the hole that is being bored by anexplosive quantity, W, during contact with a plate of thickness t.

R ¼ 0:0704

ffiffiffiffiffiffiWt

r(76)

where R and t are in m and W is in kg. The boundary conditions were, however, not specified for thisempirical relationship. This relation is valid only above a certain charge quantity since a minimumquantity of explosive is required for making a hole in the plate of specified thickness. The critical chargeweight, Wcri, above which Eq. (76) is valid is given by [7]

Wcri ¼ 2:72t (77)

An analytical prediction for the radius of the hole that is bored on a clamped circular plate is given byRajendran and Narasimhan [83]

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hWEqTNTETNTJ

ptsy3f

s(78)

A comparison of the radius of the crack by Wierzibicki [68], Keil [7] and Rajendran et al. [82] is pre-sented in Fig. 11. The model proposed by Wierzbicki [68] for localized air blast impact under predictsfor smaller charges and over predicts as the charge quantity increases beyond 20 g for localized under-water explosion in comparison with the prediction methodologies by Keil [7] and Rajendran and Nar-asimhan [83]. A photographic view of a circular plate that underwent contact underwater explosion isshown in Fig. 12.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

700 10 20 30 40 50 60TNT explosive quantity (g)

Crack len

gth

(m

)

Equation (63)

Equation (76)

Equation (78)

Fig. 11. The variation of the plate crack length that is subjected to localized blast as a function of explosive charge quantity. Platethickness¼ 1.6 mm; strain to rupture¼ 0.3; flow stress¼ 330 MPa.

Fig. 12. Photographic view of petalling of a steel plate during contact underwater explosion.


6. Fracture

Mode I deformation of the plates under explosive loading continues with the increasing intensity ofthe explosion. The moment the total strain at any point at the edge of the plate attains its rupturestrain, tensile tearing or fracture (mode II failure) occurs. A photographic view of a steel plate that failedin tensile tearing is shown in Fig. 13.

Lee and Wierzbicki [69] proposed that facture initiates at the critical point of a structure when theaccumulated equivalent plastic strain with suitable weighting function reaches a critical value. Rajen-dran and Narasimhan [83] equated equivalent plastic strain of thin plates to their uniaxial fracturestrain as fracture criterion. Balden and Nurick [84] proposed that fracture occurs when the summationof the ratio of the incremental effective plastic strain to the failure strain becomes equal to one. Thefailure strain is a function of mean stress, strain rate and temperature. Langdon and Schleyer [64]equated the total strain to cause tearing at the outer fibres of a rectangular cross-section beam tothe sum of the membrane strain and the strain due to the curvature. The interaction of mode II fractureand mode III (shear) fracture is brought out by Rudhrapatna et al. [58,59].

Fig. 13. Photographic view of an underwater exploded plate that underwent mode II failure (tensile tearing).


The ability of the hull structure to withstand large plastic deformation before fracture is a major cri-terion in naval structural design [85]. Explosion Bulge Test (EBT) has been used as the final qualificationtest to verify the dynamic plasticity of defence structural materials [86]. Explosive loading promotesbrittle fracture due to high strain rate influence of material flow properties. EBT has been developedby Hartbower and Pellini [87,88] to investigate the response of steel weldments to air blast. MIL-STD-2149A [89] formulated by the U.S. Navy recommends air blast as the source of energy to evaluatethe resistance of base materials and weldments to fracture under shock loading. It also recommendsrepetitive loading on the test plate with a reduction in thickness in each shot until final strain to frac-ture. Underwater EBT was developed by Sumpter [90] and Porter et al. [91] to minimize the chargequantity and environmental noise nuisance. Sumpter [92] formulated pass/fail criterion for crackedplates subjected to shock load. Fracture resistance of metal plates loaded into plastic regime by non-contact underwater explosion was reported by Gifford et al. [93,94].

7. Numerical methods

Elastic response of blast loaded plates was carried out by Veldman et al. [1] using ANSYS. The non-linear finite element analysis accounted for large deformation effects but neglected strain rate effects.Pressure–time history was modeled as decaying exponential function based on experimental data. Theslight negative phase of the shock wave and round reflection peak that were present during the exper-iment were ignored. Numerical simulation and experiments showed good correlation to within 5.5%for the plate central displacement. Jacinto et al. [13] performed numerical elastic analysis using ABA-QUS/standard 5.7. The plate was modeled using shell element. The boundary conditions were consid-ered as perfectly clamped. Dynamic analysis was performed using modal superposition and directintegration method. Pressure–time history was input from the experiment. It was brought out thatthe number of vibration modes was important because blast load excites high frequencies. The elementsize of the model should agree with the quantity of modes. More refined mesh captured high frequencywith less error.

Circular plate clamped at its edge and subjected to uniform blast load was simulated by Balden andNurick [84] using ABAQUS. A frictional contact was prescribed between upper and lower plate surfacesand corresponding plate surfaces. The friction co-efficient was assumed to be 0.3. The bolt array wassimulated with single elastic spring having spring stiffness based on the axial elastic response of thebolt array, acting between upper and lower flanges. The bolt array preload was calculated using the as-sumed bolt tightening torque and applied as load during the analysis. Uniform blast pressure was dis-tributed over the entire exposed surface of the plate. The blast pressure was obtained by dividing theimpulse with the burn time of the explosive. von Mises plasticity with isotropic hardening/softeningbehaviour together with strain rate sensitivity using Cowper–Symonds relationship was applied.The hardening curve was linearly changed in magnitude based on the change in yield stress at varioustemperatures. The variation of elastic response with reference to temperature was accounted for byproviding the variation of Young’s modulus of the material with temperature. Element failure criterionwas fixed as 200% failure strain and on nodal temperature. Comparison of experimental and numericalpredictions of input energy for uniform blast loading was good. Veldman et al. [1] used ANSYS/LS_DYNA–Release 7.1 for modeling the inelastic deformation of rectangular plates. The plate was mod-eled as four-node quadrilateral explicit thin shell elements ignoring strain rate effects. Bi-linear isotro-pic plasticity was assumed. Pressure–time history was modeled as decaying exponential functionbased on experimental data. The negative phase of the pressure pulse and the small ground reflectionwere neglected. An agreement to within 5.3% was seen between numerical and experimental results.

Explicit finite element code DYNA3D was adopted by Pan and Louca [95] to model the response ofa plate to gas explosion. Reaction-time history of the support assembly was compared with experimen-tally obtained and ABAQUS/Explicit results to simulate the exact boundary conditions. Four noded thinshell elements were used to model the frame–plate assembly. Pinned boundary condition was appliedto the outside part of the lower side of the specimen frame. To simulate the slip in bolt connections andpossible in-plane movement of the test rig, beam elements were used. Good comparison was seen be-tween predicted and measured peak displacement.


A detailed coupled fluid–structure interaction that is applied for underwater shock loading is pre-sented in Section 4.2. LS-DYNA and Underwater Shock Analysis (USA) codes were used by Shin [45] fornon-linear structural analysis of a ship model that is subjected to an underwater explosion. Doubly as-ymptotic approximation (DAA) that is used for the fluid–structure interaction eliminates the need formodeling the surrounding fluid volume by covering the wet surface of the structure with DAA boundaryelements. The fluid element thickness, tf, in the direction normal to the wetted surface is defined as [45]

2rwtfrpt

� 5 (79)

Velocity and acceleration response prediction by Shin [45] of a ship model using LS-DYNA and USAgave good comparison with shock test data

8. Conclusions

This paper brought out a detailed review of the phenomena of air and underwater explosions andtheir effects on plane plates. The process of detonation of the explosive is marked by the generation oflarge amount of heat with the associated pressure at a short interval. The shock wave parameters thatare significant for an air blast are the peak overpressure and the impulse. For an underwater explosion,the peak overpressure, time constant, free field impulse and energy are the four vital parameters thatare considered for the damage process.

By and large, interest is shown on the plastic damage of plates that are subjected to an air blast. Fora generated pressure–time history on a plate, a simple single degree of freedom (SDOF) system, or as-sumed modes method or numerical method is applied to derive the plate response.

When rectangular air blast pressure–time history is imparted, with the magnitude of the pressuremore than 10 times that of static collapse pressure of the plate, analytical and empirical methods areemployed to obtain the central plastic deflection with the measured impulse on the plate as input. Forperforming numerical analysis, the impulse is divided by the burn time of the explosive to get the pres-sure magnitude.

For underwater explosion, from the threat perception and survivability point of view, both elasticand plastic responses are of interest. Methods of evaluating the elastic stresses and the limiting elasticrange are presented for a mild intensity explosion. Empirical methods are presented for predicting theplastic central deflection for a severe explosion.

Localized blast leads to petalling of the plate. Analytical methods are available to compute the cracklength. Methods developed by Keil [7] and Rajendran and Narasimhan [83] for underwater explosionare compared against the methodology developed by Wierzbicki [68] for air blast.

Various methods of numerical simulation of a plate that undergoes air blast, gas explosion and un-derwater explosion are brought out comparing their predictive accuracy with experimental results.

Acknowledgement

This work was supported by the Advanced Ship Engineering Research Centre (ASERC) of PusanNational University, Republic of Korea, as part of its research program.

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Blast Loaded Plates

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