Deformation and failure of thin spherical shells under dynamic impact loading: Experiment and analytical model
Introduction
Shell structures are widely used in practical engineering applications such as fabricating pipelines and pressure vessels and in the fields of aerospace and ocean engineering [1]. As three-dimensional designs advance and civil engineering technology improves, the number of large curved shell structures applied in practical Scenarios has increased, e.g., the newly designed airport in Beijing (Fig. 1). For a long service life of shells, impact loading is one of the most important scenarios that must be considered. The deformation and perforation of an impacted shell would compromise the completeness of a structure, reducing its load-bearing capacity and resulting in safety problems. Typically, the impact loading and corresponding deformation are localized corresponding to the shell size. This localized response of the curved shell enables a spherical shell impacted by a flat nose projectile to be used to represent actual scenarios in civil engineering. Only key parameters in the deformation must be considered, such as the dimple radius, dimple depth, perforation plugging, and residual velocity of the projectile, to represent the deformed properties of actual curved shells applied in civil engineering and provide reliable and intuitive feedback for engineering design. Additionally, some structures, such as bunkers and tanks, are built directly using spherical shells (Fig. 2). Therefore, studying the dynamic response of a spherical shell loaded by an impact is crucial in engineering design.
Several investigations have been conducted on the dynamic responses of various shells, such as composite shells [[2], [3], [4], [5], [6], [7], [8]] and the composite structure of shells [[9], [10], [11], [12], [13], [14], [15], [16], [17], [18]], which are typically used to improve the properties of shells used in engineering. However, the prediction method of a loaded shell, particularly one under dynamic loads, is crucial to the application of shell structures. Numerical methods are frequently used to predict the response of various shells, e.g., finite element [[19], [20], [21], [22], [23], [24], [25]] and finite differential methods [[26], [27], [28]]. Although numerical methods are adaptive to complex geometric configurations, their parametric effects on the dynamic response of the shell are not as intuitive as those of theoretical methods. In addition, the simulated scale in numerical simulations must be sufficiently large to be suitable for engineering applications in which very fine meshes are required for a detailed description of the localized response of large-impacted thin shells. To accomplish this, the numerical model requires several elements; hence, the efficiency of the numerical method is reduced. In civil engineering, timely, efficient, and intuitive design methods are required; theoretical methods satisfy these requirements but numerical methods do not. Therefore, we used a theoretical model to investigate the dynamic deformation and perforation of an impacted spherical shell.
In engineering, theoretical methods provide an efficient analysis with acceptable accuracy. Several theoretical investigations of spherical shells have been conducted. Li et al. [[29], [30], [31]], Pang et al. [32] applied the energy method and first-order shear deformation theory to investigate the vibration of spherical composite shells. Several of the proposed theoretical methods are based on the energy method [33]. The most well-known theoretical method is the geometrical method proposed by Pogorelov [34], in which the primary part of the dimple was an isometric transformation of the initial shape. Furthermore, he formulated and solved the equations that describe the connection between the dimple and an undeformed part. His study provided a basis for many subsequent investigations. This geometrical approach was proven by Evkin [35] using an asymptotic method developed to investigate the large deflection [36,37] and buckling [[38], [39], [40]] of composite shells. The asymptotic method provides a simple formula for the deformation energy. However, these studies primarily considered elastic shells, and almost no plasticity was discussed in detail. In addition, only distributed pressures and concentrated forces were discussed without considering dynamic impact loading. MansoorBaghaei and Sadegh [41] provided a linearization of equations governing the impact of a thin-walled elastic spherical shell on an elastic barrier. The closed-form solution was verified through comparisons of the contact force and time duration using the finite element method. This closed-form solution is highly beneficial in engineering design as it enables impacts to be parametrically studied. However, plasticity, which is an important factor in the deformation and perforation of a dynamic impacted shell, was not considered in the solution. Based on the isometric transformation proposed by Pogorelov [34], Ning et al. [42], Ning and Song [43], performed theoretical analyses on the deformation and perforation of spherical shells using rigid–plastic and rigid–viscoplastic hardening strengthened models impacted by a cylindrical projectile. The analytical predictions were compared with experimental and numerical results. In these studies, a viscoplastic model was applied to the dimple deformation rather than to the perforation, which was not consistent with experimental observations. Moreover, the effects of the assumed parameters were not analyzed in detail. These investigations were insufficient; hence, the results could not be used in engineering applications.
Previous studies focused on the deformation of a dynamic loaded shell. Only Ning and Song [43] considered the failure of a shell impacted by a cylindrical projectile. However, they did not consider radial displacement, which is important for the displacement mode of the deformed spherical shell. In addition, only a rigid-plastic strengthened model was used to describe the stress evolution in the shell. A strain gradient was evident in the region around the contact edge between the projectile and shell, signifying that the strain rate should be considered in the shell perforation. The constitutive model has been proven to significantly affect the dynamic response of a loaded shell, particularly the strain rate effect [44]. The viscoplastic model must be considered for the part at which the strain rate is prominent in the strengthened model.
Considering the brevity of the aforementioned investigations, a theoretical analysis was conducted in this study to investigate the perforation of a viscoplastic spherical shell subjected to the impact of a cylindrical projectile at various velocities. Only the normal impact of a flat-nosed projectile on a thin shell was considered in this theoretical analysis. An isometric method combined with the assumed deformation of the edge region was used to obtain the deformation mode of the impacted spherical shell. Hamilton's principle was the dominant method used in a previous theoretical analysis [45]; this method was also used in this study to derive the governing equations of the dynamic response of a spherical shell impacted by a projectile. Aluminum is an excellent lightweight material with extensive applications in civil engineering. Thus, we selected aluminum as the material for our shell in this study. The governing equations of aluminum spherical shells were solved via Runge-Kutta integration to obtain the time evolution of the perforation and deformation of the impacted spherical shell. A series of theoretical predictions were conducted for different impact velocities, and the deformation was compared with the experimental results. The comparison revealed a good agreement, thereby validating the theoretical model proposed herein. Parametric analyses were performed to investigate the effects of the parameters set in the theoretical model on the deformation and perforation properties of the impacted spherical shell. The results obtained are meaningful for engineering design and can facilitate further investigation of localized impacted shell structures.
Section snippets
Deformation mode
According to the isometric transformation reported in Refs. [34,42,43], the deformation of a spherical shell impacted along the radial direction can be depicted as shown in Fig. 3. An isometric transformation is a transformation in which the distances between points are preserved, e.g., translations and rotations. The shape of the mean surface of the shell under such a deformation is inevitably similar to one of the shapes of its isometric transformation. Any isometric transformation of a shell
Experimental verification
As described earlier, aluminum was used as the shell material. The corresponding material parameters are listed in Table 1 with reference to Refs. [42,43].
Analysis and discussion
In this study, we assumed a constant initial edge size for region and a constant shear deformation region size for region I. The corresponding values of these two parameters were obtained from the experimental measurements. The results were validated, implying that our assumption was reasonable and reliable. However, the effects of the constant values on the deformation and perforation as well as the interaction between these two constant parameters are important and require further
Conclusion
In this study, the dynamic perforation of an aluminum spherical shell impacted by a cylindrical projectile was experimentally and theoretically investigated. Shear failure was considered to determine the perforation process of the impact. A viscoplastic strengthened model was adopted to calculate the perforation, and a rigid plastic model was used to describe the deformation of the impacted shell. A shear region with a constant size was assumed to calculate the shear strain and strain rate used
CRediT author statement
Jianqiao Li: Conceptualization, Methodology, Software, Investigation, Writing Original Draft. Huilan Ren: Formal analysis, Validation, Data Curation, Writing Reviewing and Editing. Jianguo Ning: Conceptualization, Resources, Supervision, Writing Reviewing and Editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This study was supported by the National Natural Science Foundation of China (Grant No. 11802026 and 11532012).
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