Numerical simulation of underwater explosion bulge test

Materials and Design 30 (2009) 4335–4341

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Numerical simulation of underwater explosion bulge test R. Rajendran * BARC Facilities, Bhabha Atomic Research Centre, Kalpakkam, Tamil Nadu 603 102, India

a r t i c l e

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Article history: Received 25 February 2009 Accepted 11 April 2009 Available online 18 April 2009 Keywords: Explosion bulge test Elastic response Depth of bulge Terminal strain to fracture

a b s t r a c t Ambient temperature underwater explosion bulge test is simulated using ANSYS/LS-DYNA. Elastic and inelastic numerical experiments are carried out. Inelastic prediction is validated with a physical experimental data. Terminal strain to fracture is established. A methodology is thus arrived at for carrying out the material qualification for explosive loading using numerical simulation. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The ability of a material to withstand large plastic deformation before it fractures is a major criterion in underwater structural applications [1,2]. Explosion bulge test (EBT) has been used as a final qualification test to verify the dynamic plasticity of structural materials. Hartbower and Pellini [3,4] developed explosion bulge test as a material qualification tool for defence structural materials. The principal objective of the EBT was to assess the material behaviour without establishing its relationship with the underwater shock forces. MIL-STD-2149A (SH) formulated by the US Navy [5] recommends air blast as the source of energy to evaluate the resistance of base materials and weldments to fracture under rapid loading conditions. It also recommends repetitive shock loading on the test plate with a reduction in thickness in each shot until final strain to fracture. ARE, UK [6,7] and DREA, Canada [8] independently developed underwater EBT to minimize the charge and the environmental noise nuisance. Hodgson and Boyd [9] gave a detailed account of behaviour of weldments subjected to explosive loading. Pellini [10] placed the test panel over a circular die which was chamfered to provide smooth entry of the bulge. Uniform loading of the test specimen was ensured by off-set of the explosive from the test plate. The unsupported area of the plate was approximately 28% of the total area. Failure criterion was either physical failure of the weld joint by the appearance of a crack or the required reduction in thickness. Pellini [11] demonstrated the effect of temperature on the plastic deformation of weld panels when subjected to explosive loading. The ductility of the plates decreased with decrease in temperature and a behaviour similar * Tel./fax: +91 44 27480282. E-mail addresses: [email protected], [email protected] 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.04.013

to the energy absorption of Charpy’s V-notch specimen as a function of temperature was observed. Monoblast EBT was developed by Porter and Morehouse [12] for the shock qualification of 51 mm thick steel plate. EBT was carried out by Rajendran and Narasimhan [13] on high strength low alloy steel plates that were rigidly clamped at their sharp edged boundary to study their deformation and fracture behaviour. Numerical simulation of air-backed circular and rectangular plates clamped at their periphery and subjected to underwater shock loading was presented by Rajendran [14]. Present investigation brings out a numerical simulation of ambient temperature explosion bulge test. This simulation differs from the work presented by Rajendran [14] by the way of application of boundary condition to the test plate because of its adherence to MIL-STD-2149A (SH) [5]. Here, smooth entry is provided to the deforming plate at its clamped boundary with chamfered edge of the anvil. Numerical model is validated with an experimental data. Terminal strain to fracture is established. 2. Explosive loading The peak pressure Pm of an exponentially decaying underwater shock wave that is generated at a stand off S which is greater than ten times the radius of the charge of equivalent TNT weight W is given in MPa as [15]

W 1=3 Pm ¼ 52:16 S

!1:13 ð1Þ

and the decay constant h in microseconds which is the time taken by the peak pressure to fall to 1/e times its initial value is given as [15]

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Nomenclature velocity of sound in water (m/s) natural frequency of the plate (Hz) mass per unit area of the plate (kg/m2) peak pressure (MPa) pressure acting on the air-backed plate (MPa) radius of the explosive (m) stand off (m) time (s) time to reach the maximum velocity of the air-backed plate (s)

cw fn m Pm PPa r S t tva

h ¼ 96:5W

1=3

W 1=3 S

!0:22 ð2Þ

The equation of motion of the free standing air-backed plate during underwater explosion is given as [15–21] 2

m

d xa dt

2

¼ 2P m et=h  qw cw

dxa dt

ð3Þ

where m is the mass per unit area of the plate, qw is the density of water, cw is the velocity of sound, t is the time and xa is the lateral displacement of the free standing air-backed plate. Applying initial conditions that the initial velocity and displacement of the plate are zero and introducing the dimensionless inverse mass number wa = qwcwh/m, the air-backed plate pressure Ppa(t) is [22]

Ppa ðtÞ ¼ 2P m et=h 

2P m wa t=h  ewa t=h  ½e ðwa  1Þ

ð4Þ

Instantaneous velocity of the plate is given as [16]

ma ðtÞ ¼

2P m h ½ewa t=h  et=h  mð1  wa Þ

ð5aÞ

V

va W xa

maximum velocity of the plate (m/s) velocity of the air-backed plate (m/s) TNT explosive charge quantity (kg) lateral displacement of the air-backed plate (m)

Greek symbols h decay constant of the underwater shock wave (ls) qw density of water (kg/m3) wa inverse mass number of the air-backed plate (kg/kg)

impulse for a depth of submergence of the target equal to or more than twice the stand off. This condition prevails for considerable depth of underwater explosion [25]. 3. Experimental details The quenched and tempered steel test plate had a dimension of 650 mm  650 mm  25 mm. The mechanical properties of the plate material are given in Table 1. Fixture or anvil was designed

Table 1 Material properties of the steel test plate. Property

Value

Young’s modulus (GPa) Poisson’s ratio Yield stress (MPa) Tangent modulus (MPa) Density (kg/m3) True fracture strain (m/m)

210 0.3 560 600 7800 0.1740

and the plate maximum velocity V is [17,18]

V¼

wa 2Pm h 1w wa a m

ð5bÞ

The time to reach the maximum velocity or the cavitation time, tva, is given as [2]

t ma ¼

h ln wa wa  1

ð5cÞ

Negative pressure after cavitation is insignificant and therefore ignored. Baker et al. [23] stated that dynamic loading can be considered as pure impulse loading as long as

fn h < 0:4

ð6Þ

where fn is the first mode of natural frequency of the plate. Modal analysis by block Lanczos mode extraction method of the plate clamped at its resting area yielded the first mode of natural frequency as 2981 Hz. The maximum time constant of the experiments on plate is 80 ls. Applying these values it is inferred that the loading under investigation is zero period impulsive loading [23]. Therefore, the maximum velocity of the plate is taken as the dynamic load on the plate. It is relevant to note here that the plates will have lower natural frequency than the computed values due to added mass effect of water. As observed in Eq. (6), the lowered natural frequency of the test plate keeps the underwater explosive loading in zero period impulsive loading regime. Rajendran [24] revealed that reloading component of underwater explosive impulse is as much as the primary shock pulse

Fig. 1. Experimental set up for the explosion bulge test.

R. Rajendran / Materials and Design 30 (2009) 4335–4341

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Fig. 2. Finite element model of the explosion bulge test.

Fig. 3. Photographic view of the deformed plate.

with the help of life buoys from the shock test site to the control centre. Every care was taken to adhere to the safety norms before going for firing. Test plate with fixture was recovered after the explosion to measure its bulge depth. The experimental set up is shown in Fig. 1. The plate velocity for the above explosion configuration is 56.5 m/s. Reloading component of the underwater explosion adds up an impulse equal to that of the primary pulse impulse [24]. Approximating the initial plane surface of the plate for the gas bubble reloading component, the primary shock pulse impulse is doubled (which essentially means that double the primary shock pulse plate velocity is employed for estimating the plate inelastic response. 4. Numerical simulation

and fabricated as per the guidelines given in MIL-STD-2149A (SH) [5]. The exposed diameter of the plate was 380 mm. To prevent water ingress into the air backed volume a rubber gasket was provided in the anvil. The fixture had a cavity with a depth of 200 mm to comfortably accommodate the plastic deformation of the deforming plate. Explosive with booster and detonator (having a TNT equivalent of 1 kg) was tied to a charge hanger at a distance of 40 cm from the front surface of the plate in line with the geometric centre of the plate. A stand off of 40 cm gives within 10% variation a uniform distance from the explosive charge to any point on the unsupported area. The farthest point on the clamped area from the point of explosion is 50% more than the stand off. This variation will have no effect on the deforming unsupported area as the explosive force only enhances the clamping force. Firing cable was connected to a detonator. The ratio of the radius, r, of the explosive to the stand off, S, is 7.6. The shock wave velocity is only 20% higher than the acoustic velocity at the charge radius to the stand off ratio of 5 [17]. Therefore the shock wave is approximated as a plane acoustic wave for which the theory of non-contact underwater explosion becomes applicable. The test assembly was tied to a pontoon and immersed in water to a depth of 4 m and towed to the test site from the bank. The firing cable was taken

The test fixture or anvil had a hole of 190 mm radius with a chamfer radius of 50 mm at its top. It had a thickness of 200 mm. The outer surface was 650 mm  650 mm. The fixture was modeled as a linear elastic material. The Young’s modulus is 210 GPa, Poisson’s ratio is 0.3 and density is 7800 kg/m3. The anvil was modeled using 12191 eight node tetrahedral elements. Translational and rotational degrees of freedom were arrested for the symmetric and external surfaces of the anvil. After a mesh refinement study, the plate was modeled using 1352 eight node hexahedral elements and bilinear isotropic material whose properties are given in Table 1. Strain rate effects were ignored. Symmetric boundary conditions were applied to one fourth model of the plate. Surface to surface automatic contact algorithm was applied between the anvil top surface and the plate bottom surface. The plate surface at its edges was fully clamped. Initial velocity calculated using Eq. (5(b)) was employed on the entire surface of the plate. The finite element model of the plate with anvil is shown in Fig. 2. A time gap of tens of milliseconds between the primary shock loading and the gas bubble loading make them appear as discrete loading in elastic regime [26]. Therefore, for studying the elastic response of plates, initial velocity is applied on the test plate. For

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studying the plastic response, as presented in the preceding sections, twice the initial velocity is applied to the plate. 5. Results and discussion 5.1. Model validation A photographic view of the deformed plate is shown in Fig. 3. The displacement contour of the deformed plate for 1 kg explosive at a stand off of 40 cm is shown in Fig. 4. The apex (centre) bulge depth (displacement) of the plate 68.86 mm. Measured value

through experiment is 68 mm. The agreement between numerical prediction and experimental observation is 99%. In view of the laborious exercise that has to be carried out for the generation of the experimental data, a good comparison that has been seen between prediction and measurement is taken as the adequate basis for the model validation. 5.2. Elastic response Von Mises elastic stress acting on the plate for 1 m/s impact velocity is shown in Fig. 5. Stress-time history at the geometric

Fig. 4. Deformation contour of the EBT plate for 1 kg TNT explosive and a stand off of 40 cm.

Fig. 5. Von Mises stress on the stress plate; plate velocity = 1 m/s.

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16

80

Displacement (mm)

Apex Von Mises stress (MPa)

R. Rajendran / Materials and Design 30 (2009) 4335–4341

60 40 20

12 8 4 0

0 0

0.5

1

1.5

2

2.5

0

2

4

6

8

10

12

Time (ms)

Time (ms)

Fig. 8. Displacement time history of the plate for an impact velocity of 30 m/s.

Apex Von Mises stress (MPa)

Fig. 6. Apex Von Mises stress at the centre of the rear surface of the plate; plate velocity = 1 m/s.

range of the plate from elastic to inelastic regime. Typical displacement time history of the plate for an impact velocity of 30 m/s is shown in Fig. 8.

600 5.3. Inelastic response

400

200

0

0

2

4

6

8

Plate velocity (m/s) Fig. 7. Variation of apex Von Mises stress at the centre of the rear surface of the plate.

centre of the rear surface of the plate for an impact velocity of 1 m/ s is shown in Fig. 6. A range of combination of the quantity of explosive and the stand off is possible to generate the plate velocity of 1 m/s. The peak Von Mises stress that is generated is 79.3 MPa. The variation of peak Von Mises stress with the plate velocity is shown in Fig. 7, which depicts a linear trend. The response of the plate for the impact velocities varying from 9 to 39 m/s showed the apex Von Mises stress exceeding the yield stress with a gradually discernable inelastic deformation. This implies that the explosive charge weight increasing from 0.160 to 3.242 kg of TNT at a fixed stand off of 1 m is the near transition

Displacement time history of the plate for the impact velocities varying from 40 to 120 m/s is shown in Fig. 9. The displacement of the plate for a plate velocity 40 m/s is 13.27 mm where as for a plate velocity 80 m/s, it is 41.88 mm. A simple multiplication of the primary shock pulse impulse by 2 will introduce sizeable amount of error as double that of the displacement for the plate velocity of 40 m/s is only 26.54 mm whereas the displacement for the plate velocity of 80 m/s is 41.88 mm. Considerable time gap between the primary shock pulse and the gas bubble pulse (tens of milliseconds) [26] makes the plate feel the gas bubble reloading as the second exclusive shock loading. The elastic energy that has dissipated after the deformation due to the primary shock loading is reabsorbed during the gas bubble reloading. The displacement of the plate for a plate velocity 60 m/s is 29.74 mm where as for a plate velocity 120 m/s, it is 68.12 mm. Double that of the displacement for the plate velocity of 60 m/s is 59.48 mm which is comparable to the displacement for the plate velocity of 120 m/s. Therefore at very large deformation, the dissipation of elastic energy becomes an insignificant fraction of the total deformation energy. 5.4. Terminal strain to fracture The Von Mises strain contour for the plate that was subjected to an impact velocity of 113 m/s is shown in Fig. 10. It is assumed that

V=40 m/s

Displacement (mm)

80

V=50 m/s V=60 m/s V=70 m/s V=80 m/s V=90 m/s

60

40

V=100 m/s V=110 m/s V=120 m/s

20

0 0

4

8

12

Time (ms) Fig. 9. Displacement time history of the bulge plate.

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R. Rajendran / Materials and Design 30 (2009) 4335–4341

Fig. 10. Von Mises plastic strain contour of the EBT plate for 1 kg TNT explosive and a stand off of 40 cm. Impact velocity = 113 m/s.

Fig. 11. Von Mises strain contour of the plate; impact velocity = 160 m/s.

the fracture in the plate initiates when the Von Mises strain at a location exceed the uniaxial true fracture strain of the plate material. The apex of the plate at its rear surface has reached the value of 0.1746, which is the terminal strain to fracture for the plate material. But the strain at the front face of the plate is at a lower value. Increasing the plate velocity gradually to 160 m/s spreads the terminal strain to fracture to the plate top surface from its rear

surface as shown in Fig. 11. In physical sense, this means increasing the TNT explosive charge quantity from 6.332 to 13.02 kg at a stand off of 1 m (r/S = 10.3–8.1). The depth of bulge for the onset of fracture is 99.63 mm as shown in Fig. 12. The depth of bulge of the plate increased by 44.68% from the moment the fracture strain is attained locally to the propagation of it across the thickness.

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Fig. 12. Displacement contour of the plate; impact velocity = 160 m/s.

6. Conclusions Numerical modeling of underwater explosion bulge test showed respectable correlation with the experiment. The transition from elastic to inelastic deformation for a thick plate takes place in a wide range of impact velocity. Incorporating reloading impulse in the primary shock pulse impulse introduces significant error at lower level of inelastic deformation which is due to the dissipated and reacquired elastic energy. With larger deformation, the error vanishes due to the fact that the elastic energy fraction becomes insignificant. Plate can undergoes considerable amount of plastic deformation from the incipient fracture state to through the thickness fracture. References [1] Sumpter JDG. Design against fracture in welded joints. In: Smith CS, Clarke JD, editors. Advances in marine structures. Elsevier Applied Science Publishers; 1986. p. 326–46. [2] Porter JF. Response of SMA and Narrow Gap HY80 weldments to explosive shock. Technical memorandum 88/ 206, DREA, Atlantic, Canada; 1988. [3] Hartbower CE, Pellini WS. Explosion bulge test studies of deformation of weldments. Weld J 1951:307s–18s. [4] Hartbower CE, Pellini WS. Investigation of factors which determine the performance of weldments. Weld J 1951:499s–511s. [5] US Navy. Standard procedure for explosion bulge testing of ferrous and nonferrous metallic materials and weldments, MIL-STD-2149A (SH); 1990. [6] Sumpter JDG. Prediction of critical crack size in plastically strained welded panels, ASTM-STP-995; 1989. p. 415–32. [7] Sumpter JDG. Design against fracture in welded structures. J Naval Sci 1987;13(4):258–70. [8] Porter JF, Morehouse DO. Development of the DREA underwater single shot explosion bulge procedure. In: Canadian fracture conference, Halifax, Nova Scotia, vol. 21; 1990. [9] Hodgson J, Boyd GM. Brittle fracture in welded ships: an empirical approach from recent experience. Inst Naval Archit 1958;100(3):141–80.

[10] Pellini WS. Use and interpretation of NRL explosion bulge test. Washington, DC: NRL; 1952. [11] Pellini WS. Evaluation of engineering principles for fracture safe design in steel structures. Washington, DC: NRL; 1969. [12] Porter JF, Morehouse DO. A monoblast explosion bulge procedure for two inch thick submarine pressure hull materials. DREA/TC, Defence Research Establishment Atlantic, Dockyard Laboratory, FMO Halifax, Nova Scotia, Canada B3K 2X0; 1993. [13] Rajendran R, Narasimhan K. Performance evaluation of HSLA steel subjected to underwater explosion. J Mater Eng Perform, ASM 2001;10:66–74. [14] Rajendran R. Numerical simulation of response of plane plates subjected to primary shock loading of non-contact underwater explosion. Mater Des 2009;30:1000–7. [15] Bjorno L, Levin P. Underwater explosion research using small amount of explosives. Ultrasonics 1976;14(6):263–7. [16] Cole RH. Underwater explosions. NJ, USA: Princeton University Press; 1948. [17] Keil AH. Introduction to underwater explosion research. UERD, Norfolk Naval Ship Yard, Portsmouth, Virginia; 1956. [18] Keil AH. The response of ships to underwater explosions. Trans Soc Naval Archit Mar Eng 1961;69:366–410. [19] Taylor GI. The pressure and impulse of submarine explosion waves on plates. Compendium of underwater explosion research, vol. 1. ONR; 1941. p. 1155– 74. [20] Temperley HNV. Theoretical investigation of cavitation phenomena occurring when an underwater pressure pulse is incident on a yielding surface. Underwater explosion research, vol. 3. ONR; 1950. p. 260–8. [21] Kennard AH. The effect of pressure wave on a plate or diaphragm. Compendium of underwater explosion research, vol. 3. ONR; 1944. p. 11–64. [22] Deshpande VS, Fleck NA. One-dimensional response of sandwich plates to underwater shock loading. J Mech Phys Solids 2005;53:2347–83. [23] Baker WE, Cox PA, Westine PS, Kulesz JJ, Strehlow RA. Explosion hazards and evaluation. New York: Elsevier Scientific Publishing Company; 1983. [24] Rajendran R. Reloading effects on plane plates subjected to non-contact underwater explosion. J Mater Process Technol 2008;206(1–3):275–81. [25] Sulfredge CD, Morris RH, Sanders RL. Calculating the effect of surface or underwater explosions on submerged equipment and structures. Oak Ridge, Tennesse, USA: Oak Ridge National Laboratory; 2001. . [26] Rajendran R, Narasimhan K. A shock factor based approach for the damage assessment of plane plates subjected to underwater explosion. J Strain Anal Eng Des 2006;41(6):417–25.