Snapping of a viscoelastic cylindrical panel under loading with small volume compressed gas

The paper considers the dynamics of the process of a flat rectangular elastic panel having an initial deflection equal to the critical deflection of the static problem of panel stability. The panel is loaded with compressed gas enclosed in a small volume container. Using this value, the load acting on the panel during the snapping process will depend on the deflection of the panel as a result of the adiabatic expansion of the compressed gas. The main goal of the work is to investigate the stability region of a cylindrical panel when loaded with compressed gas. The problem is solved using a developed methodology based on the method of separating variable equations of mathematical physics. Based on calculations, it was found that when the panel is loaded with compressed gas enclosed in a small-volume container, a stabilizing effect appears, i.e. a certain deflection interval appears where the panel does not lose stability. The presence of external pressure makes it possible to expand this area, i.e. the external environment increases the “stabilizing” effect. Also, taking into account the viscoelastic properties of the material expands this area to 10%.


Introduction
Polymer and composite materials are widely used in the field of modern aircraft construction and aircraft [1].Such materials have energy dissipation properties and a high level of rheological properties.A significant number of scientists have dealt with such problems.In particular, the problems of snapping and prediction with viscoelastic properties are discussed in [2][3][4].When loading a cylindrical panel with compressed gas (or liquid), the solution will depend on the physical properties of the gas (specific gravity, viscosity, etc.) and the ratio of the volumes of compressed gas and liquid [5].If the connection is with a double-sided loading panel, then the clicking time is an order of magnitude longer than in the case of loading with compressed gas.If the connection is one-sided, then the loading panel is subject to separation at certain values of panel curvature.
Thus, the problem is posed about the dynamics of the process of snapping panels depending on the method of loading the panel.The mechanical behavior of materials while ensuring strength has led to the use of viscoelastic polymer materials with high energy dissipation.And solving the problem of damping vibrations of structural elements under stationary and non-stationary conditions for various purposes depends on the composition of the composite material [6,7].To study the damping of resonant vibrations of a panel during the snapping process, viscoelastic components of polymer particles with high energy losses are introduced into its structure [8,9].Depending on the type of anisotropy of the material and inclusions, as well as the geometric configuration of the inclusions, the panel material can be viscoelastic isotropic, nano-transversally isotropic or orthotropic.So, for example, if an isotropic matrix contains chaotically located spherical or cylindrical inclusions of deformable material with and without rheological properties, then the panel material will be isotropic with the given characteristics presented, for example, in [10].

Problem statement and solution method
Consider an annular plate with an internal pinched towards the free outer edges.At point D, a concentrated force P normal to the middle plane is applied to the plate.The limit equilibrium method is used to solve the problem of loss of load-bearing capacity of a plate under the assumption that there is no hardening of the material.It is assumed that when the force P reaches a certain critical value Ркр two cylindrical plastic hinges arise along the segments O1A and O1B, the area inside the corner AO1B is deformed into a conical surface with the vertex at point O1, the area outside the corner remains undeformed.The relationship between stress and strain is taken as [11,12]  к =  ̃ к к   + 2 ̃к к .-respectively, the cosine and sine are the Fourier images of the relaxation kernel of the material.As an example of a viscoelastic material, we take the three-parametric relaxation kernel () =  − / 1− .The influence function R(t-τ) is subject to the usual requirements of integrability, continuity (except for t=τ), definite sign and monotonicity: ⃗ -displacement vector of the j-th layer environment.
In a polar coordinate system with a pole at point O1, is proposed an approximation of a curved surface in the form where w -is the deflection, is a dimensionless constant,  -is the coordinate angle,  -calculated from the horizontal radius of the OS.If the deflection w receives a small increment then the dihedral angle  at edge O1A increases where n -is the normal to edge O1A.Bending moment arising in a plastic hinge , where   is the yield strength, h -is the thickness of the plate, which does work when incrementing where where ddistance from pole O1 to point D of force application.
The principle of possible displacements when neglecting the work of chain forces and bending moments on elastic deformations gives А  = 2  , where After introducing dimensionless parameters takes the form The critical value of the dimensionless force  is determined by passing expression (4), using the parameter ,   ,  ; the value realized during the deformation process is determined from the transcendental equation   = 0. (5)

Results and discussion
Table 1 shows the values и  depending on the parameter  0 .A rectangular plate is considered in terms of an elastic cylindrical flexible panel having an initial deflection equal to the upper critical deflection of the static problem of panel stability.The viscoelastic panel is loaded with a small volume of compressed gas.In this case, the value of the load acting on the panel during the snapping process will depend on the deflection of the panel, as a result of the adiabatic expansion of the compressed gas.To obtain numerical results, the parameters of the relaxation kernel are taken in the form  () =  − / 1−  = 0,048;  = 0,05;  = 0,1.The calculation results are presented in Table 1.Table 1 shows that the critical speeds vary depending on the parameters   and .

Conclusions
Thus, a solution method and algorithm for solving the problem of snapping a viscoelastic cylindrical panel when loaded with a small volume of compressed gas has developed.It has been established that the presence of external pressure makes it possible to expand the range of stability of the panels, i.e. the external environment increases the "stabilizing" effect.Also, taking into account the viscoelastic properties of the material expands this area to 10%.

Table 1 .
Change in critical load.