Dynamic Buckling Criterion Of Composite Structures Using Lyapunov Exponent


The respected Comrade Kim Jong Un said:

"In scientific research, it should focus on satisfactorily resolving the theoretical and practical, scientific and technological problems arising in building a powerful socialist country, developing the sector of basic sciences and pushing back the frontiers of science and technology."

Recently, the composite mechanics research group in the faculty of mechanics has developed a new criterion for buckling analysis of composite structures subjected to pulsed dynamic loading and has achieved the success of establishing an analytical method based on it.

Generally, the buckling of a structure under dynamic loading occurs in various forms depending on the type of structure and the applied load, and thus the stability criteria are different.

Although there are several known criteria for determining the dynamic buckling load in which a structure under impulsive dynamic loading is transformed from steady to unsteady, the most widely used criterion is the Budansky-Hutchinson criterion (BH criterion).

This criterion, combined with finite element software, is an effective method to determine dynamic buckling loads regardless of load type, application time, geometry and boundary conditions of the structure, which has been used predominantly in the literature dealing with dynamic buckling.

This criterion has the advantage of using finite element software instead of directly handling the differential equation, while there are some disadvantages associated with the choice of observation points and the analysis of the displacement diagram.

Especially in the case of composite structures subjected to impulse loading, the Budansky-Hutchinson criterion does not give satisfactory results owing to local and global buckling or the ambiguity of the diagram due to the complex interaction between delamination and matrix cracks.

The main limitation of this criterion is that the generated displacement diagram is ambiguous.

According to this criterion, the critical load is the load that causes a sudden increase in the load-displacement relationship diagram.

However, as some cases show, the diagram is not always obtained in the standard form shown in the literature, and it is not always monotonically increasing or randomly fluctuating with the choice of the observation point or initial defect, and even in the unstable state, it is not a sudden change in the shape of the curve, but only an increase in the amplitude, which makes the diagram obscure to be judged.

In this case, it is clearly impossible to judge dynamic buckling by the BH criterion based on the sudden change in the diagram.

This is because this criterion is not completely quantitative and there is a qualitative aspect related to visual observation of the shape of the diagram.

We improved the BH criterion using the Lyapunov exponent.

The basic idea for improving the BH criterion is to use the Lyapunov exponent originally developed to identify the occurrence of chaotic motion in nonlinear dynamical systems theory, mainly in time series analysis.

Considering the nonlinear dynamical system theory, a more quantitative evaluation of dynamic buckling can be obtained through the Lyapunov exponent of the dynamical system based on the displacement time series.

The newly proposed improved criterion is called the Budansky-Hutchinson-Lyapunov criterion or simply the BHL criterion.

The definition of the BHL criterion is as follows.

The loss of stability of the structure under impulse loading occurs when the Lyapunov exponent of the displacement time series obtained at the point of maximum buckling displacement is not negative. In the improved criterion, the observation point is chosen as the point that produces the maximum buckling displacement, rather than as an empirical one.

Also, the Lyapunov exponent for the displacement time series at the observation point is found and its sign is evaluated to determine the dynamic buckling load.

According to the nonlinear dynamical system theory, the negative Lyapunov exponent means that the structure undergoes finite displacement vibration under a given impulse load and the structure is dynamic stable.

On the contrary, the positive Lyapunov exponent indicates a state in which the structure undergoes chaotic motion and a relatively large displacement or irregular motion with a rapidly increasing amplitude, whereas the zero Lyapunov exponent indicates a dynamic bifurcation.

Based on the proposed dynamic buckling criterion, the nonlinear dynamic buckling problem of multilayer composite plates with distributed delamination and matrix cracks is analyzed and compared with the simulation results by the engineering analysis software ANSYS, to evaluate the applicability of the BHL criterion.

The results of the study were published in the journal International Journal of "Structural Stability and Dynamics"(22(3), 2022) by Elsevier under the title of "Dynamic buckling of composite structures subjected to impulse loads the Lyapunov exponent"(https://doi.org/10.1142/s0219455422500869).