General information 
Description 
The main idea of using thin shells for energy absorption is based on their nonlinear behavior by large deflections (Figure 1). Let us consider the path ABCDEF of a cycle of quasistatic loading. First, the load increases until the point B is reached, and then the snapthrough BD is realized, because the part BE of the equilibrium diagram is unstable. Also, the snapback EF occurs, when the load decreases. The dashed area BDEF corresponds to the energy absorbed by the loading cycle. If deformations of the structure remain elastic, the process can be repeated a number of times. 
Figure 1. Nonlinear loaddisplacement diagram of thin shell of revolution. 
In reality, the process is more complicated than in the above idealized quasistatic scheme. Various modes of highfrequency vibrations arise at the instance of buckling and energy is dissipated as a result. But it is clear that the idealized quasistatic scheme is valid when the period of the lowest frequency of these vibrations is much less than the time involved in the entire cycle.
One of the possible shock absorbers is sketched in the Figure 2. It consists of horizontal rigid element, where the load is applied, vertical spring with given stiffness and elastic reversing shell of revolution, whose deflection amplitude can be large compared to shell thickness. In addition to the spring, this device can also contain usual hydraulic shock absorber. 
Figure 2. Energy absorber with inversing shell of revolution. 
When designing energy absorber, it is important to be able to calculate the forcedisplacement characteristics and absorption energy, as well as to evaluate stresses and deformations of the structure for the given range of loading. 
Assumptions 




Methodology 
The asymptotic formulae derived from the Reissner’s equations are used in the calculation. The asymptotic analysis is based on the small parameter, which is propotional to the ratio of the shell thickness to its radius of curvature. It is important to note that the accuracy of obtained asymptotic results coincides with the accuracy of the original boundary value problem for a thin shell, which itself is based on the asymptotic limit of threedimensional equations of theory of elasticity with respect to the same small parameter.
An example of calculation is shown in the Figure 3 by curve 1. There are two branches of the curve. The first one corresponds to the deformation of the toroidal part of the shell of revolution. The second (almost linear) branch is related to the deformation of the conical shell. The relevant linear function, describing forcedisplacement diagram of the spring, is shown by dash dot line. 
Figure 3. Load parameter versus deflection amplitude of the shell. 
The finite element method was used for validation of the asymptotic solution and calculation. We calculated displacements and stresses for the structure with the following properties: radius of the rigid horizontal element r_{0}=1; small radius of the torus r_{1}=0.2; height of the shock absorber H=0.5; constant thickness of the shell h=0.000625; modulus of elasticity E=2048000 and Poisson’s ratio of material ν=0.5; the angle of the conical shell α_{0}=π/3. These properties correspond to the case shown in the Figure 3 by curve 1. There is a perfect agreement between asymptotic and numerical solutions in the whole range of deflection amplitude.
The comparison of maximum bending moments calculated for the same structure using both numerical and asymptotic methods is shown in the Figure 4. Here also the result of finite element analysis is marked by dots. Numerically obtained bending moments are slightly smaller than the asymptotic ones. The difference became smaller for larger deflections or thinner shells. 
Figure 4. Maximum bending moment versus deflection amplitude parameter. 
We supose that this calculation will be valuable for the investigation of the influence of geometrical and stiffness parameters of the structure on its behavior and optimal design.
For some comments or more information about calculation of nonlinear behavior of thin shells of different shapes and boundary conditions, geometrical and stiffness properties, and under different types of load please contact the author of the asymptotic method and present calculation Alex Yevkin by email and take a look at his personal web site. 
References 
