Submerged circular cylindrical shell subjected to two consecutive shock waves: Resonance-like phenomena

https://doi.org/10.1016/j.jfluidstructs.2013.03.010Get rights and content

Abstract

A submerged evacuated circular cylindrical shell subjected to a sequence of two external shock waves generated at the same source is considered. A semi-analytical model combining the classical methods of mathematical physics with the finite-difference methodology is developed and employed to simulate the interaction. Both the hydrodynamic and structural aspects of the problem are considered, and it is demonstrated that varying the delay between the first and second wavefronts has a very significant effect on the stress–strain state of the structure. In particular, it is shown that for certain values of the delay, the constructive superposition of the elastic waves travelling around the shell results in a ‘resonance-like’ increase of the structural stress in certain regions. The respective stress can be so high that it sometimes exceeds the overall maximum stress observed in the same structure but subjected to a single-front shock wave with the same parameters, in some cases by as much as 50%. A detailed parametric analysis of the observed phenomenon is carried out, and an easy-to-use diagram summarizing the finding is proposed to aim the pre-design analysis of engineering structures.

Introduction

The analysis of shell structures interacting with shock waves and/or acoustic pulses has been a subject of intense research attention for at least sixty years, and a considerable body of work now exists on the matter, from the simple modal analysis of the pioneering analytical works of the post-war period (e.g. Mindlin and Bleich, 1953) to the complete three-dimensional contemporary numerical studies (e.g. Gregson et al., 2006, Qiankun and Gangyi, 2011); a detailed review of the relevant research effort from 1950 to 2005 can be found in Iakovlev (2006). It appears, however, that the most favored and frequently addressed scenario was, rather universally, that of a structure (or several structures) responding to a single shock wave. This, of course, is easily explained if one recalls the historical reasons that drove the progress in the area, i.e. the development of naval warfare and subsea surveillance. At the same time, it is not difficult to envision a scenario where a submerged structure experiences the impact of more than one shock wave within a short time period, exposure to multiple near-simultaneous explosions being one example. Yet, the literature published on the topic of a multi-front loading is extremely limited (we offer an overview of the available relevant studies a little later), and it appears that a study exposing the effects of such a loading would be quite welcome.

Along with the outlined general considerations, we also had a much more specific reason to initiate the work we present here. Namely, it can now be considered well-established (e.g. Iakovlev, 2007, Iakovlev, 2004, Iakovlev, 2008a, Iakovlev, 2008b, Iakovlev, 2009; Iakovlev et al., 2010, Iakovlev et al., 2011a, Iakovlev et al., 2011b) that the dynamics of the stress–strain state of a shock-subjected shell is dominated by a variety of wave phenomena in both the shell and the fluid(s). The peak stress in a submerged shell, for example, can be attributed to either the direct impact of the incident wave (occurring, obviously, only once during the interaction) or to the multiple constructive superpositions of elastic waves travelling around the shell, the latter often causing higher stress than the former. This observation, naturally, leads to a question that appears to be of considerable practical significance: If there is more than one incident wavefront, is it possible that the interaction between the shell-borne elastic waves originated as a result of the initial impacts of the different incident fronts could, in some cases, lead to resonance-like phenomena that significantly alter the dynamics of the stress–strain state, and possibly result in stresses that are higher than those observed for a single incident wavefront? If the answer is ‘yes’, the practical implications of such a finding would be very significant—it seems reasonable to anticipate that if ‘resonance-like’ shock-induced stresses indeed are a possibility for a multiple-front loading, the existing standards would need to be adjusted in order to ensure that the conditions of operation where the peak stresses in question are possible are avoided, or, where that is not possible, adequate countermeasures are employed.

There is also another, somewhat more subtle, reason for undertaking this study. Namely, the scenario where a structure is subjected to multiple shock waves originated within a short time period is not limited to the cases where there exists an actual sequence of underwater explosions. It also occurs when there is only one explosion but it takes place in the proximity of one or more reflective surfaces, such as rigid walls, free surface, or other structures. In that case, the reflected waves follow the primary incident wave, and they all have an effect on the structure. The present study can therefore be seen as an attempt at the analysis aimed at determining the basic mechanisms of the interaction that is expected in multi-surface environments. If the above-mentioned resonance-like phenomena are indeed observed, studying them will tell us a lot about the least (or, in some cases, most) desirable geometrical configurations of the immediate surroundings of shock-sensitive engineering structures. The most practically interesting scenarios that can be analyzed using the proposed model include the reflection from the sea bottom and sea surface, the former generating the secondary wave of the same sign (the case we consider here), and the latter resulting in a negative secondary wave (the case we leave for future investigations), and we believe that the present study could be the first step towards a computationally efficient methodology of pre-design analysis of naval structures subjected to reflected waves.

As for the literature pertaining to the present topic, several studies exist where a structure, or several structures, are subjected to a loading that is different from a single-front shock wave with no reflected waves following it; even though none of them explicitly aim at addressing the structural effects of a multi-front loading, they are still quite useful for understanding the general mechanisms of shock–structure interaction when there is more than one incident wave. Namely, Jialing and Hongli (1997) considered two rigid cylinders inside a channel conveying a shock wave, and addressed the complex scattering patterns that developed as the shock wave interacted with the cylinders. Oakley et al. (2001) considered a system of three cylinders placed inside a large cylindrical confinement, and studied the response of such a system to an external shock wave, with the focus on the propagation and reflection of the shock waves inside the confinement. A system consisting of a large number of cylindrical structures placed inside the fluid domain confined by two significantly larger cylindrical shells was considered by Sigrist et al. (2007).

There also are several studies that deal with the analysis of shock wave propagation in various surroundings different from the ‘classical’ case of the infinite medium, but where the structure itself is not necessarily present; even though they are not fluid–structure interaction studies per se, they are very useful for gaining a basic understanding of how multi-front shock waves could form in reality. To that end, an explosion in the presence of a free surface was considered by Gitterman and Shapira (2000), and Jappinen and Vehvilainen (2006) addressed an explosion under ice.

Some of the studies where a single shell or a system of thin-walled structures is subjected to a single shock wave have proven to be useful as well in the present context because analyzing the multiple reflections that are taking place in such systems help to understand some of the aspects of the interaction with a multi-front loading. Namely, Huang, 1979a, Huang, 1979b considered two co-axial cylinders (Huang, 1979a) and two concentric spheres (Huang, 1979b) subjected to an external shock wave, with the focus on the structural dynamics, Wardlaw and Luton (2000) addressed an explosion inside a double-walled cylindrical structure with the fluid between the walls, and analyzed both the structural and fluid dynamics of the interaction, and Chambers et al. (2001) studied an explosion inside a cylindrical shell.

Finally, when a fluid-contacting shell is subjected to a multi-front loading, the structure of the resulting scattered-radiated field is expected to be more complex than that seen for a single-front loading, and this makes the understanding of and the ability to accurately simulate the shell-radiated field even more important. Thus, referring to the studies where the structure of the radiated field is analyzed in detail becomes as important here as ever. To that end, we mention the experimental studies by Neubauer, 1968a, Neubauer, 1968b, Neubauer and Dragonette (1970), Ahyi et al. (1998) and Derbesse et al. (2000) where the acoustic fields generated when an evacuated or fluid-filled cylindrical shell responded to an external acoustic pulse were photographed, Derbesse et al. (2000) and Ahyi et al. (2005) where the radiated fields were imaged for more complex shell systems such as a cylindrical shell with a spherical endcap, and Merlen et al. (1995) and Latard et al. (1999) where the images of the field radiated by an elastic sphere can be found. The numerical studies of the non-stationary fields radiated by elastic structures seem to be more limited, and we mention the one by Voinovich et al. (2001) where the radiation by a submerged evacuated shell is simulated using a finite element approach.

Section snippets

Mathematical formulation

We consider a circular cylindrical shell of radius r0 and thickness h0, Fig. 1. We assume that h0/r0⪯¡1 and that the deflections of the shell surface are small compared to its thickness; we also assume that the Love–Kirchhoff hypothesis holds true. The density, Poisson's ratio, and Young's modulus of the shell material are ρs, ν, and Es, respectively, and the sound speed in the shell is cs. The transverse and normal displacements of the middle surface of the shell are v and w, respectively.

Fluid dynamics

In order to obtain the hydrodynamic components, we first apply the Laplace transform with respect to the time to the dimensionless wave equation (4) written in the cylindrical coordinates to arrive at2Φ^r2+1rΦ^r+1r22Φ^θ2s2Φ^=0,where Φ^ is the Laplace transform of ϕ^, and s is the transform variable.

Then, we separate the spatial variables in order to obtain the general solution of (12) satisfying the zero condition for r at infinity asΦ^=AnKn(rs)cosnθ,n=0,1,,where Kn is a modified Bessel

Structural dynamics

In order to obtain the dimensionless displacements of the middle surface of the shell, v and w, we consider their matching series expansionsv=n=0vnsinnθandw=n=0wncosnθ,and rewrite the dimensionless shell equations (5), (6) in terms of the harmonics vnsinnθ and wncosnθ. This yields, for every n, a system of two integro-differential equationsγ2d2vndt2+cn11vn+cn12wn=0,γ2d2wndt2+cn21vn+cn22wn=χ^p^n0+p^nd0td2wn(η)dη2ξne(r,tη)dηr=1,where cn11=n2+k02n2,cn12=cn21=nk02n3,cn22=1+k02n4,γ=c^s1,and

Results and discussion

We consider a thin steel shell, h0/r0=0.01, cs=5000 m/s, ρs=7800kg/m3, and ν=0.3, submerged in water, cf=1400 m/s and ρf=1000kg/m3. Both of the waves of the incident sequence are chosen to be identical, with the rate of exponential decay λ and the pressure in front at the moment of the initial contact pα as 0.0001314 s and 250 kPa, respectively, and it is assumed that both waves are originated at the same source located at the distance of R0=5r0 from the axis of the shell (i.e., the standoff of SR=4

Conclusions

We have considered the interaction between an evacuated circular cylindrical shell and a sequence of two consecutive shock waves originated at the same source. We have modeled the interaction using a hybrid analytical–numerical approach combining the classical methods of mathematical physics with finite-difference methodology, and observed a number of interesting and practically important effects. Our focus was on the structural dynamics, and we have demonstrated that for certain values of the

Acknowledgments

S. Iakovlev and C. Seaton acknowledge the financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada (Discovery Grant 261949). S. Iakovlev also acknowledges the financial support of the Killam Trusts.

References (33)

  • J. Qiankun et al.

    A finite element analysis of ship sections subjected to underwater explosion

    International Journal of Impact Engineering

    (2011)
  • A.C. Ahyi et al.

    Experimental demonstration of the pseudo-Rayleigh (A0) wave

    Journal of the Acoustical Society of America

    (1998)
  • A.C. Ahyi et al.

    Re-radiation of acoustic waves from the A0 wave on a submerged elastic shell

    Journal of the Acoustical Society of America

    (2005)
  • G. Chambers et al.

    Pressure measurements on a deforming surface in response to an underwater explosion in a water-filled aluminum tube

    Shock and Vibration

    (2001)
  • Gitterman, Y., Shapira, A., 2000. Audio-visual and hydroacoustic observations of the Dead Sea calibration experiment....
  • Gregson, J., Link, R., Lee, J., 2006. Coupled simulation of the response of targets to close proximity underwater...
  • Cited by (10)

    • Semi-analytical models for deriving shock signals and response spectra on immersed and fluid-filled hulls subjected to shock waves

      2022, Ocean Engineering
      Citation Excerpt :

      The present paper aims at proposing additional validation test cases for the model initially developed for immersed structures and, more generally, at extending the semi-analytical approach and the parametric study to fluid-filled structures – a configuration which may be encountered in various applications of industrial interest. The recent literature indeed shows that semi-analytical models for elastic fluid-filled shells subjected to shock waves have been studied in very few papers, furthermore almost exclusively by Iakovlev et al., 2013, 2014. Their analysis is focused on the physics of the pressure field, while the present paper is concerned with the influence on submergence conditions on the dynamic responses of the hull and the equipment.

    • Interaction between a submerged cylindrical shell and a shock wave in the presence of a rigid wall

      2022, Journal of Fluids and Structures
      Citation Excerpt :

      The dynamics of the stress state is, however, a rather different matter — although it also exhibits a number of general phenomenological similarities to the simplified two-wave scenario, its understanding does require a sufficiently detailed graphical presentation. Prior to addressing the present scenario, however, it is helpful to summarize the overall physics of the process for the classical scenario of a single-wave, no-reflective-surface loading, Fig. 6 (the more detailed discussion of the matter can be found in Iakovlev et al., 2013a). If one now considers the possibility of arrival of a second incident wave, it is rather apparent that such timing of the arrival exists that the elastic waves it induces will constructively superpose with those induced by the primary incident wave, thus producing high stress not observed in the single-wave scenario.

    • Structural analysis of a submerged cylindrical shell subjected to two consecutive spherical shock waves

      2018, Journal of Fluids and Structures
      Citation Excerpt :

      At the same time, it is the highest stress that is the practitioner’s first priority, thus our focus in what follows will be on the stress state. The single-front scenario images illustrate the dynamics of the stress wave most clearly: After its inception at the instant of the initial contact, it starts to propagate downstream, (a) and (b), until its two branches circumnavigating the shell meet at the tail point and superpose constructively there, resulting in the well-known (Iakovlev, 2008; Iakovlev et al., 2013b) stress peak there, (c). After that, the two branches separate again and propagate upstream, (d) and (e), and eventually meet at the head point where they undergo another constructive superposition, (f), while the total stress is diminishing.

    • Resonance-like phenomena in a submerged cylindrical shell subjected to two consecutive shock waves: The effect of the inner fluid

      2014, Journal of Fluids and Structures
      Citation Excerpt :

      To that end, the ratios observed in the present case are 1.00:0.84:0.68, while they are 1.00:0.76:0.63 for the evacuated shell, which further emphasizes the relatively higher effect that multiple shock loading has on a fluid-filled structure. In particular, we note that, as was elaborated in Iakovlev et al. (2013), the present study, along with its direct use for the assessment of the effect of the addressed multi-front loading, can also be considered as the first attempt at analyzing the effects of shock loading on structures in the presence of reflective surfaces. And when such analysis is aimed at the pre-design structural optimization of geometrically complex systems in the context of improving their shock resistance, the knowledge of the existence of the most disadvantageous ranges of Δt is particularly valuable in that a relatively minor change in the geometry of the system can lead to a very significant reduction of the peak stresses experienced by its constituent structures.

    • Semi-analytical modeling of non-stationary fluid-structure interaction

      2020, Lecture Notes in Applied and Computational Mechanics
    View all citing articles on Scopus
    View full text