It is of great significance for the protective design of submarine to study the influences of coverings on the damage characteristics of single and double cylindrical shells subjected to underwater contact explosions. The SPH models of single and double cylindrical shells coated with foam silicone rubber are established to analyze shockwave propagation, damage characteristics, and elastoplastic responses, which provides reasonable parameters of covering position and thickness. The results can be concluded as follows: the superposition of multiple waves may cause the inhomogeneity and discontinuity; for the single cylindrical shell with inner or outer coverings, the damage mode is mainly tensile and shear failure is caused by detonation waves and detonation products; compared with out-covering approach, the in-covering approach has better antishock performance; the best protective effect comes out when the thickness of covering is close to that of the shell; as for the double cylindrical shell without interlayer water, the destruction of inner shell mainly results from the puncture of high-speed fragments from the outer shell, so for the outer shell, out-covering is a better choice; however, since the interlayer water is very effective in protecting the inner shell, in-covering will be better for the inner shell.
With the development of precision-guided weapons, the probability of naval structures being attacked/damaged by underwater contact explosion increases gradually. Therefore, special materials or structures are always adopted to resist the severe shock. As a submerged vehicle, submarine is always coated with multiple acoustic coverings [
Foam silicone rubber is chosen as the covering material since it has no significant influence on acoustic performance. SPH method is applied to study the influences of the covering on the damage characteristics of single and double cylindrical shells. Firstly, the feasibility of SPH method to solve strong discontinuity problems will be discussed in terms of dynamic continuity and motion continuity; and then the coated single and double cylindrical shells will be modeled with SPH method, the results of which will be compared with those of AUTODYN to verify the effectiveness of the present SPH model; on the basis of these, the parameters of covering position and covering thickness will be considered to discuss the influences on the structural damage of single and double cylindrical shells; more reasonable parameters of the coverings will be presented in terms of antishock performances to give a reference for the structural design of submarines.
In SPH method, the approximation of function
Thus, the conservation of mass, momentum and energy in SPH can be expressed as (without regard to body force) [
As for fluid, the stress
Parameters in Mie-Gruneisen EOS for water [
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1000 | 1480 | 0.5 | 0 | 2.56 | 1.986 | 1.2268 | 357.1 |
As for solid, the stress
The solutions of
Whether the stress should be updated or not is determined by Mises yield criterion [
In Johnson-Cook constitutive model of steel [
Parameters in Johnson-Cook constitutive model of steel [
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350 | 275 | 1 | 0.36 | 0.22 | 1.0 | 1811 | 288 | 452 | 77 |
Parameters in the dynamic constitutive equation of the foam silicone rubber [
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0.03813 | 11022 | 11.85 | 561 | −4528 | 16301 | −26044 | 15570 |
Parameters in Jones-Wilkins-Lee (JWL) EOS for explosive gas [
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1630 | 6930 |
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4.15 | 0.95 | 0.30 |
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Parameters in Mie-Gruneisen EOS for steel [
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7890 | 1.587 | 3075 | 1.294 | 0 | 0.3 |
Parameters in Murnaghan EOS for foam silicone rubber [
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1037 | 1.863 | 1.596 |
According to the requirement of continuum on wavefront, the particle displacement is continuous, but its derivatives, such as velocity and strain, may be discontinuous for shock problems. In this section, the essential conditions of stress and strain on discontinuity surface, namely, the conditions of dynamic continuity and motion continuity [
Given a finite region
Kinematic state of shock wave [
The function
With the propagation of shock wave, not only the function
Due to the mass conservation in any volume
As a result of the opposite directions between
Substituting the above equations into (
Owing to the arbitrariness of wavefront
Substituting (
Applying the particle approximation, the following equation can be drawn:
The discontinuity surface turns to be
Then during
If
In general, the propagation velocity
Discretizing the two equations above, subtract (
Replace
Therefore, restoring the discretized equation (
According to the Helmholtz velocity decomposing theorem [
In addition, we use a slight penalty force of Lennard-Jones model to solve interface problem, and the molecular force is so slight that it just prevents particles’ penetration. When particles on both sides of an interface tend to penetrate, in the case where
The model of a coated single-hull submarine subjected to the attack of torpedo can be simplified as Figure
Parameters of single cylindrical shell.
Outside radius of water (m) | Inside radius of water (m) | Inside radius of shell (m) | Thickness of shell (m) | Radius of TNT (m) |
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3.0 | 1.22 | 1.2 | 0.02 | 0.05 |
Different cases of single cylindrical shell and numbers of particles (different positions are listed as Cases 1–3, in the case where the rubber thickness is 0.020 m; different thicknesses are listed as Cases 4–7, in the case where the inner surface is coated with rubber).
Cases | Name | Numbers of particles | ||||
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Water | Steel | TNT | Rubber | Total | ||
Case 1 | No-covering | 364507 | 9300 | 121 | 0 | 373928 |
Case 2 | In-covering | 364507 | 9300 | 121 | 9153 | 383081 |
Case 3 | Out-covering | 355250 | 9300 | 121 | 9257 | 373928 |
Case 4 | 0.010 m | 364507 | 9300 | 121 | 13680 | 387608 |
Case 5 | 0.015 m | 364507 | 9300 | 121 | 11406 | 385334 |
Case 6 | 0.025 m | 364507 | 9300 | 121 | 6886 | 380814 |
Case 7 | 0.030 m | 364507 | 9300 | 121 | 4575 | 378503 |
Models of single-hull.
In order to verify the effectiveness of the present SPH method, the no-covering model of underwater contact explosion is selected, the SPH result will be compared with that of AUTODYN.
The propagations of shockwave obtained from different methods are shown in Figure
Pressure nephograms of different methods: (a) SPH method, (b) AUTODYN.
The pressure-time curves of test point A.
The pressure of test point A in the cases of different initial particle spacing is shown in Figure
The pressure-time curves of test point A in the cases with different initial particle spacing;
After TNT detonation, the shockwaves propagates and overlaps in multilayer media, which will cause the inhomogeneity in time and space. According to the principle of impedance matching, the results above are in good accordance with the physics law, and the numerical results are fairing and smooth, which verifies that the dynamic and motion continuity conditions are satisfied in the SPH method.
For the single cylindrical shell, the pressure of point A is shown in Figure
The pressure of test point A; Cases
The pressure of the in-covering with different covering thickness is shown in Figure
The pressure of test point A in the cases of in-covering with different covering thicknesses;
Mode 1: plastic large-deformation; Mode 2: tensile tearing in outer fibers at the support; Mode 3: transverse shear failure at the support.
The equivalent plastic strain of coated shell is shown in Figure
The equivalent plastic strain of coated shell; (a), (b), and (c) correspond to no-covering, in-covering, and out-covering, respectively.
The equivalent plastic strain of different covering thicknesses at about 2 ms; (a)–(f) correspond to the covering thickness of 0, 1.0 cm, 1.5 cm, 2.0 cm, 2.5 cm, and 3.0 cm, respectively.
The crevasse radius of different coated shells; Cases
The crevasse radius of different covering thicknesses is shown in Figure
The crevasse radius of different covering thicknesses;
Based on the researches of single cylindrical shell, the influences of covering position and interlayer water on the damage characteristics of the double cylindrical shell are studied. The basic model is shown in Figure
Parameters of double cylindrical shell.
Outside radius of water (m) | Inside radius of water (m) | Outside radius of shell (m) | Thickness of shell (m) | Radius of TNT (m) | ||
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Inner shell | Outer shell | Inner shell | Outer shell | |||
3.0 | 1.22 | 0.865 | 1.22 | 0.02 | 0.015 | 0.05 |
Different cases of double cylindrical shell (Cases 1–5 belong to the case of double-hull without interlayer water; Cases 6–10 belong to the case of double-hull full of interlayer water).
Cases | Name | Numbers of particles | ||||
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Water | Steel | TNT | Rubber | Total | ||
Case 1 | No-covering | 648852 | 36298 | 212 | 0 | 685362 |
Case 2 | In-covering on inner shell | 648852 | 36298 | 212 | 11632 | 696994 |
Case 3 | Out-covering on inner shell | 648852 | 36298 | 212 | 11795 | 697157 |
Case 4 | In-covering on outer shell | 648852 | 36298 | 212 | 16269 | 701631 |
Case 5 | Out-covering on outer shell | 644625 | 36298 | 212 | 16821 | 697956 |
Case 6 | No-covering | 709479 | 36298 | 212 | 0 | 745989 |
Case 7 | In-covering on inner shell | 709479 | 36298 | 212 | 11632 | 757621 |
Case 8 | Out-covering on inner shell | 706401 | 36298 | 212 | 11795 | 754706 |
Case 9 | In-covering on outer shell | 705143 | 36298 | 212 | 16269 | 757922 |
Case 10 | Out-covering on outer shell | 705127 | 36298 | 212 | 16821 | 758458 |
Models of double-hull.
The propagation of shockwaves in the case of in-covering on inner surface is shown in Figure
Pressure nephograms of double-hull.
The pressure at test point A is shown in Figure
The pressure of test point A; 1 and 2 mean no interlayer water and full interlayer water; cases are shown as Table
The pressure at test point B in the case of full interlayer water is shown in Figure
The pressure of test point B; 2 means full interlayer water; cases are shown in Table
Failure modes of different coated shells at about 2 ms; (a)–(d) mean no-covering (no interlayer water), no-covering (full interlayer water), in-covering of inner shell (full interlayer water), and out-covering of inner shell (full interlayer water).
Failure modes of different coated shells; Cases 1–10 are shown in Table
As for the outer shell, the damage process experiences plug failure, dent, and rolling. As for the inner shell, in the case of no interlayer water and no-covering, the failure of inner shell is caused by high-speed fragments from outer shell, which is called “3a,” as shown in Figure
The crevasse variation of outer shell without interlayer water; cases are shown in Table
The crevasse variation of the outer shell with full interlayer water is shown in Figure
The crevasse variation of outer shell with full interlayer water; cases are shown in Table
The crevasse variation of the inner shell without interlayer water is shown in Figure
The crevasse variation of inner shell without interlayer water; cases are shown in Table
The crevasse variation of the inner shell with full interlayer water is shown in Figure
The crevasse variation of inner shell with full interlayer water; cases are shown in Table
An SPH method with mesh-free and Lagrange properties is applied to solve extremely dynamic problems of cylindrical shell subjected to underwater contact explosion in this paper. The influences of coverings on damage characteristics of cylindrical shells are investigated. Through the parametric analyses of covering position and covering thickness, the following conclusions can be drawn. The propagation laws of shockwaves in multilayer media correspond with the theory of impedance matching, and the continuity conditions of discontinuity surface are verified by the smooth SPH results; the results of SPH agree well with those of AUTODYN, which verifies the feasibility and effectiveness of the present SPH method. For single cylindrical shell, the failure mode will be large-deformation and tensile failure when the covering is thinner than the shell, and tension failure and shear failure otherwise. The antishock performance does not necessarily increase with covering thickness; better protective effects appear when the covering thickness is close to that of the shell. As for double cylindrical shell, the obvious protective effect is found when there is out-covering on the outer shell or in-covering on the inner shell. However, considering the operability, the out-covering on inner shell is also a good choice when the interlayer is full of water. The damage of single shell is mainly caused by detonation wave and detonation products; as for double shell without interlayer water, the inner shell is mainly disrupted by high-speed fragments. Moreover, a certain protective effect will be provided by the interlayer water.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Excellent Young Scientists Fund (51222904), the National Defense Basic Scientific Research (B2420133001), and the National Security Major Basic Research Program of China (613157).