An experimental investigation on deformation shape of a cylindrical shell with internal medium subjected to lateral contact explosion was carried out briefly. Deformation shapes at different covered width of lateral explosive were recovered experimentally. Based on the experimental results, a corresponding analytical approach has been undertaken with rigid plastic hinge theory. In the analytical model, the cylindrical shell is divided into end-to-end rigid square bars. Deformation process of the cylindrical shell is described by using the translations and rotations of all rigid square bars. Expressions of the spring force, buckling moment, and deflection angle between adjacent rigid square bars are conducted theoretically. Given the structure parameters of the cylinder and the type of the lateral explosive charge, deformation processes and shapes are reported and discussed using the analytical approach. A good agreement has been obtained between calculated and experimental results, and thus the analytical approach can be considered as a valuable tool in understanding the deformation mechanism and predicting the deformation shapes of the cylindrical shell with internal medium subjected to lateral contact explosion. Finally, parametric studies are carried out to analyze the effects of deformation shape, including the covered width of the lateral explosive, explosive charge material, and distribution of initial velocity.

Cylindrical shells are used in a wide variety of engineering applications, from the containment pressure vessels of nuclear reactors to the bracing elements of aerospace structures. Such structures may be subjected to a wide variety of short duration transient loads throughout the course of their working life, such as air blasts, underwater explosions, and high velocity impact. Accurate prediction of the dynamic plastic deformation and rupture of the cylindrical shell subjected to high intensity transient loading is of great importance in many industrial applications.

Early researches on the dynamic buckling and failure of cylinders were restricted to axisymmetric external radial pulse loading [

Depending on the load intensity and the special distribution of contact pressures, various forms of damage may result ranging from large amplitude lateral deflections to punch-through penetration, fracture initiation at the base plate, progression of tearing fracture, and finally massive structural damage. Yakupov [

In recent years, increasing attention of both engineering communities and government agencies has turned to the dynamic response of the cylindrical shell subjected to underwater explosion. Pédron and Combescure [

To investigate the behavior of the cylindrical shell with internal medium loaded by lateral contact explosion, several experiments have been conducted, and the experimental results are presented and discussed in detail in this paper. Based on the experiments, a corresponding analytical approach was conducted with rigid plastic hinge theory. Deformation processes of cylindrical shells are described using the translations and rotations of all rigid square bars. In this study, an infinitely long cylindrical shell filled with medium subject to lateral contact explosion is performed. Because the cylindrical shell has the unique deformation shape in the symmetric axis direction, we take a ring representing the cylinder. Due to solving the inertia moment of the ring, a unit height ring represents deformation of the infinite cylinder. Because the ring has the same value of thickness and height in the radial and axial direction, the cross section of the ring in the circumferential direction is square, so we call it “square bar.”

Given the structural parameters of the ring and the type of the explosive charge, deformation processes and shapes are reported using the analytical approach. A good agreement has been obtained between calculated and experimental results. Finally, parametric studies are carried out to analyze the effects of deformation shapes, depending on the covered width of the lateral explosive, explosive charge material, and distribution of initial velocity.

In order to investigate the dynamic response of the cylindrical shell with internal medium subjected to lateral contact explosion loading, some experiments were carried out with respect to the different covered widths of the lateral explosive charge. The photography and assembly schematics of experimental setups are shown in Figures

Photography of the experimental setups.

Assembly schematics of experimental setups: (a) section drawing and (b) profile drawing.

Section drawing

Profile drawing

The experimental setup consists of a cylindrical shell with internal medium (sand), a lateral explosive charge, a connecting rod, and two endplates. The thickness ^{3}. Two endplates, made from LY12 Aluminium, having a thickness of 10 mm, are fixed by the connecting rod so that fully closed condition will be simulated. Material properties of 1020 Steel, LY12 Aluminum, and sand are listed in Table

Material properties of 1020 Steel, LY12 Aluminium, and sand.

Material | Density (g/cm^{3}) |
Yield stress (MPa) | Young’s modulus (GPa) | Poisson’s ratio |
---|---|---|---|---|

1020 Steel | 7.85 | 275 | 210 | 0.29 |

LY12 Aluminium | 2.78 | 230 | 70 | 0.29 |

Sand | 1.75 | 4.23 | 0.01 | 0.26 |

The lateral charge is an emulsion explosive (DL103-80), which is made from 75% PETN, 20% emulsion, and 5% Pb_{3}O_{4}, and the density is 0.95 g/cm^{3}. The emulsion explosive DL103-80 is a sort of mild and flexible material that can be easily shaped. The inner radius, the thickness, and the covered width of the lateral explosive charge are 100 mm, 5 mm, and

Deformation shapes of the cylindrical shell with internal medium at three different covered widths of the lateral explosive charge were recovered. Deformation shapes after tests are shown in Figure

Experimental result of deformation shapes after tests at three different covered widths of the lateral explosive charge.

45°

90°

135°

Deformation processes of the cylindrical shell with internal medium subjected to lateral contact explosion are a high nonlinear problem. Due to complexities introduced by unsymmetric loading, large displacements, and rotations of the cylinder amplified by material nonlinearities, the problem does not lend itself easily to an analytical treatment. However, by introducing a suitable set of assumptions, a simple and realistic model can be established to describe the deformation processes of the cylindrical shell with internal medium.

Basic assumptions are as follows:

During deformation processes of the cylindrical shell, some parameters of square bars may vary at different moments, such as translational displacements, translational velocities, rotational displacements, rotational angles, and area surrounded by square bars, which affect the value of the spring force, the bending moment, and the deflection angle intensively.

Based on the above-mentioned assumptions, the analytical model is established, shown in Figure

Analytical model.

The spring force between adjacent square bars assumed as the perfect elastic-plastic is described by a changeable spring force (Figure

Spring force between two adjacent square bars.

The relative displacements between the end of the current bar and the head of the next bar are obtained by using the end displacement of the current square bar subtracting the head displacement of the next square bar. The relative displacements of adjacent square bars are described by the equation

Bending moment and corresponding deflection angle between two adjacent square bars are shown in Figure

Bending moment and deflection angle between bars.

The relationship between the bending moment and the deflection angle is expressed by the equation

Moment of each square bar consists of two aspects: the moment generated by the spring force and the moment resulting from the bar bending. Moments of square bars are described by the equation

The internal medium is compressed because of the translation and rotation of square bars. During movement processes of all bars, interactions between square bars and internal medium are shown in Figure

Interactions between square bars and internal medium.

It is assumed that the processes of the internal medium compressed undergo two stages. The first stage is that voids of internal medium are compacted, and the resistance force of each square bar is equal to

The resistance forces of bar generated by internal medium during the two stages are expressed by the equation

By calculating the spring forces between adjacent square bars and the resistance forces generated by the internal medium, translational accelerations of all square bars are obtained at corresponding time. By calculating the bending moments of all square bars, the rotational accelerations are obtained at corresponding time. The translational and rotational accelerations of square bars are described by

Utilizing (

Initial translational acceleration, rotational acceleration, and velocities of all square bars are set to 0, and the distributions of initial velocities are as follows:

According to the Gurney equations on contact explosion, the ring velocity can be obtained by using the equation [

According to specific structure parameters and the type of the explosive charge, the velocity of the ring can be obtained with (

Based on the analytical approach, calculated results are reported and discussed. Deformation processes of the cylindrical shell consist of two stages: stage I, velocities of the cylindrical shell obtained by the lateral contact explosion loading, and stage II, interactions between the cylindrical shell and internal medium. Figure

Distributions of positions and velocities of the cylindrical shell at different moments (

Distributions of positions and velocities of the cylindrical shell at different moments (

Distributions of positions and velocities of the cylindrical shell at different moments (

From the results of deformation shapes, a good agreement has been obtained between calculated and experimental results, and thus the analytical approach can be considered as a valuable tool in understanding the deformation mechanism and predicting the deformation shape of the cylindrical shell under lateral contact explosion loading.

Deformation shapes of the ring have a significant relationship with the covered width of lateral explosive, explosive materials, and initial velocities distribution. In order to better understand the deformation mechanism, parametric studies are carried out for the deformation shapes and corresponding results were discussed.

From the calculated and experimental results, it is obvious that the covered width of the lateral explosive charge is a key factor to the deformation shapes. Deformation shapes of various covered widths of the lateral charge are shown in Figure

Deformation shapes of the cylindrical shell at various widths of lateral explosive charge (

In order to investigate the effect of lateral charge, a series of calculated results are obtained by adjusting various initial velocities of the cylindrical shell, because higher initial velocities of cylindrical shell represent greater power of charge.

Figure

Deformation shapes of the cylindrical shell at various initial velocities (

In general, the lateral explosive charge has a uniform thickness in the circumferential direction within the central angle, and the ring close to the lateral explosive has the same velocity value. The velocity values varies with thickness of the circumferential explosive. The simplest assumption is that the thickness of the lateral explosive is a linear distribution from one side to middle position, and the thickness of one side is a half of middle position.

Due to the explosive mass reduction, the initial velocities close to the lateral charge are assumed to have a cosine distribution. Figure

Distributions of positions and velocities of the cylindrical shell at different moments (

The plotted data in Figures

This paper presents brief results of an experimental investigation on the deformation process of the cylindrical shell with internal medium under lateral contact explosion, and the deformation shapes were obtained. Based on the experiments, a corresponding analytical approach has been undertaken using the rigid plastic hinge theory.

Given the structural parameters and explosive charge, deformation processes and shapes are reported using the analytical approach. A good agreement has been obtained between calculated and experimental results, and thus the analytical approach can be considered as a valuable tool in understanding the deformation mechanism and predicting the deformation shapes of the cylindrical shell with internal medium subjected to lateral contact explosion. Finally, a parametric study is carried out to analyze the effects of deformation shapes, depending on the covered width of the lateral explosive, explosive materials, and distribution of initial velocities. Therefore, an optimal deformation shape can be achieved by adjusting the covered width of lateral and initial velocities distribution.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to acknowledge, with thanks, the financial support from the China National Natural Science Funding under Grants nos. 11202237 and 11132012.