Scale Invariance of Structural Responses of a Stiffened Cylindrical Shell Under Underwater Explosion Loads

Abstract

The dynamic response of the hull structure under transient strong impact loads exhibits nonlinear behavior influenced by multi-physical field coupling. This response is highly sensitive to coefficient parameters and boundary conditions. Even small disturbances can lead to bifurcations and abrupt changes in the response, resulting in chaotic behavior within the dynamic system. Therefore, accurately predicting the dynamic response of similar hull structures becomes challenging, limiting the identification of damage modes and the design of impact-resistant hull structures. In this study, an impact test of a stiffened cylindrical shell under underwater explosion loads is conducted. The deviation index of adjacent moving orbits of the shell’s dynamic response in the time domain is calculated, revealing the chaotic characteristics of the impact response system. The concept of a maximum predictable critical time for the response system is proposed. Range analysis indicates that the evolution of structural dynamic motion response exhibits scale invariance. Based on generalized Brownian motion, a mathematical framework for a dynamic response scaling theory for hull structures subjected to explosive loading is established. Finally, impact tests on model structures at different scales are performed. Test results show that the dynamic response of large-scale, similar model structures can be accurately predicted using the scaling equation derived from the impact response system, with an error margin controlled within 10%.

Keywords

Underwater explosion / Structural response / Chaotic characteristics / Fractal characteristics / Scale invariance

References

[1] BoeingG. Visual analysis of nonlinear dynamical systems: chaos, fractals, self-similarity and the limits of prediction. Systems, 2016, 4437.

[2] CapassoVBrownian Motion and Diffusions, 2011, Berlin Heidelberg. Springer. .

[3] CarpinteriA, AccorneroF. Dimensional analysis of critical phenomena: self-weight failure, turbulence, resonance, fracture. Physical Mesomechanics, 2021, 24(4): 459-463.

[4] ChengLL, HuangFL, WuHJ, DongH, TianSC. Comparative analysis of blast load transfer and structural damage in multicabin structures during air and underwater explosions. Ocean Engineering, 2024297

[5] DuJ, ZhangJL, YangZ, LiSL, YangY. Spatial ergodicity dimension: Ergodic self-similarity of trajectories in discrete chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 2025, 144108671.

[6] EmamzadehSS. Nonlinear dynamic response of a fixed offshore platform subjected to underwater explosion at different distances. Journal of marine science and application, 2022, 21(4): 168-176.

[7] FanY, TanZY, YangGD, CuiXZ, DengK, TianB. Analysis of dynamic behavior and damage mode of shallow buried tunnel due to underwater explosions. Ocean Engineering, 2024296

[8] FarmerME. Foundations of chaos and fractals. Application of Chaos and Fractals to Computer Vision, 2014.

[9] JinZ, YinC, ChenY, CuiYD, KhooBC, JiaoS, HuaH. Study on the cavitation effects induced by the interaction between underwater blast and various boundaries. Ocean Engineering, 2021222

[10] KenmogneF, NoahP, DongmoED, EbandaFB, BayihaBN, OuagniM, SimoH, KammogneA, WokwenmendamML, ElongE, NgapgueF. Effects of time delay on the dynamics of nonlinear beam on elastic foundation under harmonic moving load: chaotic detection and its control. Journal of Vibration Engineering & Technologies, 2022, 10(6): 2327-2346.

[11] KimTS, KimS. Singularity spectra of fractional Brownian motions as a multi-fractal. Chaos Solitons & Fractals, 2004, 19(3): 613-619.

[12] LiHL, ZhangJY, WangLK, LiB, ZhangZF, ZhangGY, ZongZ. Bubble dynamics and structural damage caused by underwater explosion. International Journal of Mechanical Sciences, 2025297

[13] LiJ, RongJ. Experimental and numerical investigation of the dynamic response of structures subjected to underwater explosion. European Journal of Mechanics - B/Fluids, 2012, 32: 59-69.

[14] Journal of Offshore Mechanics and Arctic Engineering, 2011, 1331

[15] LiSM, ZhangAM, CuiP, LiS, LiuYL. Vertically neutral collapse of a pulsating bubble at the corner of a free surface and a rigid wall. Journal of Fluid Mechanics, 2023962

[16] LiangLH, XunPP, WangBH. Shock response spectrum prediction of underwater explosion based on an ARMA model. Journal of Vibration and Shock, 2013, 32(13): 23-26

[17] LiuLT, YaoXL, ZhangAM, SunBS. Research on the estimate formulas for underwater explosion bubble jet parameters. Ocean Engineering, 2018, 164: 563-576.

[18] LiuT, ZhangW, WuMQ, ZhengY, ZhangYF. Metastable nonlinear vibrations: Third chaos of bistable asymmetric composite laminated square shallow shell under foundation excitation. Composite Structures, 2021255

[19] Physics of Fluids, 2025, 376

[20] LiuYZ, RenSF, ZhaoPF. Application of the deep neural network to predict dynamic responses of stiffened plates subjected to near-field underwater explosion. Ocean Engineering, 2022247

[21] NikolicD, NesicL, AngelopoulosA, FildisisT. Brownian motion, fractal structure and verification of A. Einstein’s Formula. American Institute of Physics, 201012921297

[22] PanahiB, GhavanlooE, DaneshmandF. Transient response of a submerged cylindrical foam core sandwich panel subjected to shock loading. Materials & Design, 2011, 32(5): 2611-2620.

[23] PengT, WangGH, LuWB, GaoZ, PanXH, ShuYZ. Blast resistance performance and failure modes of prestressed thin-walled aqueducts subjected to underwater contact explosion. Thin-Walled Structures, 2024201

[24] PizarroA, AcuñaP, DimitriadisP, IliopoulouT, KoutsoyiannisD. Past, present, and future of the Hurst-Kolmogorov dynamics in Stochastics: A bibliometric analysis of the last 50 years in water resources. Journal of Hydrology, 2024643

[25] PrabaR, RamajeyathilagamK. Numerical investigations on the large deformation behaviour of ring stiffened cylindrical shell subjected to underwater explosion. Applied Ocean Research, 2020101

[26] Journal of Engineering Materials and Technology, 2017, 1392

[27] Symmetry-Basel, 2019, 1112

[28] SigristJF, Marie-Dit-DinardM, LeblondC. Semi-analytical models for deriving shock signals and response spectra on immersed and fluid-filled hulls subjected to shock waves. Ocean Engineering, 2022263

[29] SoninAA. A generalization of the II-theorem and dimensional analysis. Proceedings of the National Academy of Sciences of the United States of America, 2004, 101(23): 8525-8526.

[30] SuyalV, PrasadA, SinghHP. Nonlinear time series analysis of sunspot data. Solar Physics, 2009, 260(2): 441-449.

[31] TangY, SunSL, RenHL. Numerical investigation on a container ship navigating in irregular waves by a fully nonlinear time domain method. Ocean Engineering, 2021223

[32] TaoZ. Time-frequency characteristics of shock responses of underwater target to underwater explosion. Explosion and Shock Waves, 2012

[33] TelescaL, LapennaV, MacchiatoM. Mono- and multi-fractal investigation of scaling properties in temporal patterns of seismic sequences. Chaos Solitons & Fractals, 2004, 19(1): 1-15.

[34] VermaG. Self similarity in fractals. Research Journal of Engineering and Technology, 2016, 7: 75-78.

[35] Physical Review Fluids, 2021, 61

[36] Physics Of Fluids, 2020, 321

[37] XiaoF, ChenY, WangY, HuaHX, ZhuDW. Experimental research on the dynamic response of floating structures with coatings subjected to underwater explosion. Shock and Vibration, 2014, 20141705256

[38] YaoXL, HuangRJ, ZhaoK, YinYR. Data orbits similarity conversion law of scaled-down model tests of ship structures under strong impact loads. Thin-Walled Structures, 2025212

[39] YaoXL, ZhaoK, HuangRJ, YinYR. Research on the similarity of ship structure’s shock response under underwater explosion based on scaling theory. International Journal of Impact Engineering, 2025203

[40] YaoXL, ZhaoK, ShiDY. Study on the effectiveness of impact factor in underwater explosion model test. International Journal of Impact Engineering, 2024185

[41] YaparO, BasuPK. Fluid-structure interaction simulation of the effects of underwater explosion on submerged structures. Finite Elements in Analysis and Design, 2022199

[42] YuanJH, ZhuX. Dynamic response of a ring-stiffened cylindrical shell subjected to underwater explosive loading. Applied Mechanics & Materials, 2011, 105-107: 931-936.

[43] Physics of Fluids, 2023, 353

[44] ZhangWQ, YaoXL, WangYN, WangZK. Experimental study and numerical model adequacy assessment of hull structure dynamic response subject to underwater explosion. Ships and Offshore Structures, 2020, 15: 1023-1036.

[45] ZhangXF, FanJL. A new piecewise nonlinear chaotic map and its performance. Acta Physica Sinica, 2010, 59(4): 2298-2304.

[46] Buildings, 2024, 1410

[47] ZhuZY, LiXB, KeZY, XiXY, XieWM, YeZZ. The effect of interfacial imperfection on the nonlinear dynamic behavior of composite laminates under low-velocity impact. Nonlinear Dynamics, 2024, 11214: 11869-11886.

[48] ZverevV, RubinsteinB. Self-similarity symmetry and fractal distributions in iterative dynamics of dissipative mappings. Physics, 2007

RIGHTS & PERMISSIONS

Harbin Engineering University and Springer-Verlag GmbH Germany, part of Springer Nature

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.