Dynamic stability of anisotropic fiber-reinforced plate

The dynamic stability problem of an anisotropic fiberreinforced plate under increasing compressing load is considered in a geometrically nonlinear formulation using the Kirchhoff-Love’s shell theory. The problem is solved using the Bubnov-Galerkin method based on a polynomial approximation of the deflections in combination with a numerical method based on quadrature formulas. For a wide range of variations of physical, mechanical, and geometrical parameters, the dynamic behavior of the plate is studied.


Introduction
During the intense development of the modern industry, a reduction in the materials consumption of machine structures is one of the main problems of mechanical and civil engineering. For material saving, the need arises to manufacture thin-walled structures. The thinner the element, and the more flexible it is, the more strongly its susceptibility to buckling and loss of stability is manifested. The latter is accompanied by a catastrophic development of deformations and, as a rule, by a structural failure. From this standpoint, in the production of lightweight, durable and reliable structures, it is reasonable to use the materials that make it possible not only to improve their operating characteristics but also to create the structures unfeasible with traditional materials. Here, the calculation procedure and structural design involving the consideration of their actual properties are rather complicated. Today, efficient solution algorithms for nonlinear problems of dynamic stability of shells, panels, and plates are the most pressing issue. The problems with a similar mathematical formulation were considered in [1][2][3][4][5][6][7][8][9][10][11][12].

Materials and methods
To construct the mathematical model of the problem of dynamic stability of a plate made of a material having anisotropic properties in a geometrically nonlinear formulation, we use the classical Kirchhoff-Love's shell theory. In this case, the normal and tangential forces , , , as well as the bending moments and torques , , , have the form [13][14][15][16][17]: where the 's are extensional stiffnesses, the 's are bending-extension coupling stiffnesses, and the 's are bending stiffnesses having the following form: Here is the number of plate layers, 1 , 2 are the elastic modulus, 12 is the shear modulus, 12 and 21 are the Poisson ratios, is the angle characterizing the direction of the fibers relative to the axis .
The relations between the deformations in the median surface , , , , , and displacements , , in directions , , have the form [18]: Substituting (1) and (2) into the equations of motion: we obtain a system of nonlinear differential equations in partial derivatives that satisfies the boundary conditions of the problem (the edges are simply supported). The solution of this system is sought in the form: where ( ), ( ), ( ), are the unknown functions of time. Substituting the approximating functions (3) into the resulting system of equations and performing the procedure of the Bubnov-Galerkin method, we obtain a system of nonlinear ordinary differential equations that, in turn, is integrated using the numerical method based on the use of quadrature formulas [19][20].

Results and Discussion
Let's consider the problem of dynamic stability of anisotropic fiber-reinforced rectangular plate of thickness ℎ with the sides and , subjected to dynamic compression along one of the sides by force ( ) = ( is the loading rate).
In the calculations, the following parameters of the plastic (KAST-V) rectangular plate have been used: 1 = 25.5 , 2 = 14.91 , 12 = 4.41 , 12 = 0.2, = 1900 / 3 , = = 0.5 ., ℎ = 0. 5 ., = 45 , = 2 / . As a criterion determining the critical time, we assume that the sag of the deflection should not exceed a value equal to the thickness of the plate. In shell structures, the greater the critical time, the more stable it is to dynamic loads. The following graphs correspond to the results obtained for the midpoint of the hinged plate. On the graphs, (meter) is the dimension for the deflection, and (second) is for time. Figure 1 shows a graph of the changes in the deflections of the midpoints of the plates of various thicknesses. The results show that an increase in plate rigidity due to an increase in plate thickness leads to a proportional increase in the critical time value. The various curves in Fig. 2 correspond to cases of changes in the deflections of the midpoint of a reinforced rectangular plate at different loading speeds. It should be noted here that in all cases, at the initial moments of time, the changes in the deflections are oscillations that are harmonic in shape, which begin to increase rapidly at certain points in time. The influence of changes in the direction of the fibers of the reinforced plate on the dynamic process is shown in (Figure 3). As the angle of direction of the fibers increases from 0 to 45 degrees, an increase in the critical time is observed. The difference between the critical time values for single-layer plates with fiber directions of 0 and 45 degrees is 20.7%. Modern reinforced composites are a set (composition) of several reinforced layers, each of which has its own mechanical properties. Thus, by changing the composite structure, it is possible to create constructions, the behavior of which can be predicted in advance. Their behavior depends on various factors such as loads, temperatures, and humidity. In this regard, the study of the behavior of laminated reinforced plates with different directions of fibers is of particular interest. Fig.4 shows the changes in the deflections of the midpoints of laminated reinforced plates made of KAST-V. Moreover, although all these plates have different fiber directions, however, their thickness is the same. The results show that for two-layer plates with fibers located at an angle of -45 degrees relative to the axis in one layer and 45 degrees in another, the critical time values are higher than the others. The layered fiber plate, which is parallel and perpendicular to the axis, has a lower critical time (i.e., it is less stable) than other plates with similar mechanical properties. The difference between the critical time values for the above two-layer plates is 21.8%. The results of studies of the behavior of reinforced plates for a wide range of changes in their mechanical, physical, and geometric parameters under dynamic compression of one of their sides are shown in Table 1.

Conclusion
The study of the problems of the dynamic stability of anisotropic reinforced plates shows that when subjected to dynamic compression along one of their sides, the critical time values mainly depend on the direction of the reinforced fibers in each layer. In single-layer and double-layer plates, the difference in the critical time values depending on the direction of the reinforced fibers in places is 20.7% and 21.8%, respectively. An analysis of the results shows that the most resistant to these types of loads are double-layer plates with fibers located at an angle of -45 degrees relative to the axis in one layer and 45 degrees in another.