Energy method in solving the problems of stability for a viscoelastic polymer rods

With the energy method in the form of Ritz-Timoshenko solved the stability problem for a polymer rod under axial compression for the clamping-free edge option. The proposed form of loss of stability is chosen as a sum of functions with indeterminate coefficients. The shape functions for various fastenings and the represented fastening of the rod in particular are considered. The result is obtained numerically using the MatLab complex. A study was made of the long critical loads. It is shown that if the compressive force does not exceed the long-term critical one, then stability loss does not occur.


Introduction
Analysis of the load-carrying capacity of the structure, in addition to strength calculation, should include the stability issues of the entire system and its individual elements.This is important for construction, mechanical engineering, since many structural elements work on compression and can lose stability at stresses much less than destructive [1][2][3][4] To reduce weight, cost and labor, and also to improve the quality of building structures and their elements, it is necessary to develop and introduce new types of light and lightweight structures.A significant role in the development of effective designs is performed by improving the calculation methods taking into account the actual work of the material [9][10][11][12].
One of the factors that have a significant effect on the stress-strain state and the deformability of the structures is creep.
The aim of the article is to solve the stability problems of rods, taking into account initial imperfections, various variants of fixing, and arbitrary creep laws.

Formulation of the problem
According to the Ritz-Timoshenko method, the assumed form of loss of stability is chosen as a sum of functions with indeterminate coefficients ( ) ( ) As i f we take functions from x , satisfying the kinematic boundary conditions of the problem, i.e. those that relate to the deflection and rotation angles, regardless of i a .
The resolving equations for determining the unknown coefficients of the series are obtained from the condition of the minimum of the total potential energy of the Э system: The total potential energy of the rod is written in the form: Where П -potential energy of deformation, А F = ∆ -work of external forces.The potential strain energy has the form: Here el ε is meant an elastic deformation that represents the difference between total deformation and creep strain:

The derivation of the resolving equations
Substituting (5) in (4), we obtain: Approximation of the ends of the elementary fragment dx of the rod (see Figure 1) during buckling can be determined by formula: To the determination of the approach of the ends of the rod during buckling.
Then the convergence of the ends of the whole rod will be written in the form: The work of external forces is defined as follows: The total potential energy of the system is written in the form: After substituting (1) in ( 6) and then minimizing it with unknown coefficients i a we obtain: Where { } Instant critical force can be determined from the condition that the determinant of the system (7): Thus, the problem of determining the critical force reduces to a generalized vector equation.In the case where the matrix [ ] B is not degenerate, system (7) can be multiplied by [ ] MATEC Web of Conferences 129, 05010 (2017) where [ ] E -unit matrix.The first critical force will correspond to the minimum of the eigenvalues of the

Method of solving the problem
Consider the function of the form ( ) i f x when fixing the rod embodiment "jamming -free edge" (see Figure 2).The initial deflection is defined as follows: The function takes the form of: The derivatives of ( 10) and ( 11) are written in the form: The integrals in the formulas of the coefficients of the system of equations are written in the form: In the case of constant stiffness of the rod ij k the coefficients take the form: , .32

Results and discussion
We consider a rod of rectangular constant cross-section from epoxy resin EDT-10.Rod sizes: 157 , The rod is acted upon by force 0.165 . This problem was solved by M. Y. Kozelskaya in [2], but by a different method.She received the critical time value -1 43 The Euler force for the problem under consideration:    The change in deflection as a function of x and t is shown in Fig. 4.
When solving the Ritz-Timoshenko method, the number of terms in the series n assumed to be equal to 4. The number of intervals for x and y is equal to 50.
From the table presented, it can be seen that the results are the same, which indicates their reliability, as well as the reliability of the calculation methodology developed by the authors.

Conclusions
The resolving equations are obtained and a method based on the finite difference method is developed, the Ritz-Timoshenko method for calculating the stability of creep rods of variable stiffness with allowance for initial imperfections, various fixing variants, and arbitrary creep laws.
For compressed polymer rods that obey the nonlinear Maxwell-Gurevich equation, a technique has been developed for determining long-term critical loads.It is established that if the compressive does not exceed a prolonged critical one, then stability loss does not occur.

Figure 3
Figure 3 shows a graph of deflection height.Critical time 1 41 кр t ч мин = .The rod loses stability over 130 hours.For a criterion of stability loss was taken condition: 10 l f 〉 .

Fig. 3 .
Fig. 3.The growth of the deflection at F=0.165 кНThis problem was solved both by the finite difference method and by the Ritz-Timoshenko method.

Fig. 4 .
Fig. 4. The growth of the deflection at F=0.165 кН The test problem was also solved for 80 F H = , which is less than дл F .Т the theoretical value of the deflection at the end of the creep process: 0.16 0.727 97.6 / 80 1 f ∞ = = − мм.The growth curve of the deflection when 80 F H = is shown in Figure 5.At 15000 t ч = , 0, 726 f = .

.
Comparison of the value of the deflection at different times is shown in table 1.Comparison of deflections of the rod, obtained by theRitz-Timoshenko method and using the finite difference method.