Quasistatic elastic-plastic loading and unloading of rods with coulomb dry friction

Loading of elastic-plastic rods with account on Prandtl’s scheme is considered in the paper, and nonlinear interactions are taken into account under the law of Coulomb friction. Exact analytical solutions of quasi-stationary problems of interaction of elastic-plastic rod and undeformable media surrounding it are given.


Introduction
In works [1][2][3][4][5][6][7][8][9][10][11], the problem of quasistatic loading of structural elements in the form of an elastic rod interacting with its environment according to the Coulomb dry friction law was formulated and solved. It is known that in the problems of longitudinal vibrations of rods interacting with soil media, attention is drawn to the fact that in certain situations, it is possible to neglect the inertial terms in the equations of motion (works [12,16,20,21], etc.) in comparison with the arising shear stresses on the surfaces of interaction between the structure and the environment. In this case, quasistatic formulations of nonlinear problems arise.
The paper considers the loading of an elastic-plastic rod according to the Prandtl scheme, and nonlinear interactions are taken into account according to the Coulomb dry friction law. Exact analytical solutions of quasi-stationary problems on the interaction of an elastic-plastic rod and the non-deformable soil environment surrounding it are given. The loading of the rod particles along two loaded sections of the Prandtl diagram and the unloading of the elastoplastic rod is taken into account. Rod structures are assumed to be long enough, which are found in deep drilling structures but loaded in the longitudinal direction.

Statement of the problem of dynamic loading of a bar structure and a method for its solution
Let a semi-infinite elastoplastic rod, which is in the initial state in contact with its surface with a non-deformable medium, be loaded from the end section x = 0 (the x-axis is directed along the rod axis and x> 0). Compressive loading at x = 0 is reduced smoothly from zero, and it is set according to the following scheme ( Figure 1). Figure 2, the diagram of elastic-plastic loading and unloading of the rod particles according to the Prandtl scheme is shown. The loading given at x = 0 can be determined by the following analytical representation: Obviously, smooth loading and then unloading of the rod particles at x = 0 leads to the need to use different equilibrium equations in the loading and unloading zones. Suppose the loading and unloading process is considered only within the limits of elasticity, then in the areas of loading and unloading of a semi-infinite bar. In that case, the boundaries of these areas, which change with time, must be determined. Solutions to problems in these areas must be built under satisfying conditions for the continuity of displacements and stresses at the boundaries. Suppose there is a monotonic loading of the section x = 0 according to Fig.  1, then with the passage beyond the elastic limit. In that case, two loading zones are formed: the area of plastic and elastic deformations of the bar. Solutions in areas with movable boundaries, boundaries, and conditions on them must be determined in the process of solving problems. The analytical construction of problems becomes more complicated if the particles of the rod are unloaded from the states corresponding to points 2 ( Fig. 2).
In the area for points 1 in Figure 1, when there is a loading process, which does not exceed s  at the end section, we have: These formulas were also obtained in [2][3][4]. In the region of loading beyond the elastic limit, when s We integrate the second equation (2) for a fixed time t from zero to x: Here t = t1 + t '(t'> 0) and for times t> t1 we have: Based on the first relation (2), we have: From here, we get: Solutions (5) are valid on the interval 0 <x <ls2 (t), where the boundary relations for elastically and plastically deformable regions can be used to determine ls2 (t): Hence we have: In the region ls2 (t) <x <lsl (t), where the rod is deformed within the elasticity, we again have: Integrating it from ls2 (t) to x, we have: where is the border Considering that here x u E     and integrating, we find: Relation (7) makes it possible to determine u (ls2, t) from the condition u (lsl, t) = 0. Given this circumstance, we get: If we substitute the values of the above expressions for ls1 and ls2, then after some calculations, we get: from (5) and (9), we get: equating the right-hand sides (10) and (11) we get: we substitute (12) into (5), and after substituting the expressions for ls1 and ls2 we get: Thus, under monotonic loading of a semi-infinite bar in a region plastically deformed with a movable boundary, stresses and displacements are calculated by the formulas: where 2 0 s l x   . We add here the solutions obtained above in the elastically deformable region: It is not difficult to make sure that stresses and displacements are continuous functions at the moving boundary between the regions In the unloading area At the moment t = t 2 from 2 0 s l x   , we have: When unloading, we have: From (16), we have: Thus, for each fixed * x , we find   3.77 -to the elasticplastic zone, 3.77 <x <11.78 -to the elastic zone, for different moments of time. Figure 4 shows that in the unloading region, the deformation increases in absolute value, i.e., the structure is stretched and then abruptly passes into the plastic zone, decreasing along the length of the rod. In the unloading zone, the stress increases until the time t3, then gradually decreases along the length of the rod.

Conclusions
1. The problem of quasistatic loading and unloading of long bar structures in an elastic medium and interacting with it according to the Coulomb dry friction law has been solved. Analytical solutions are given for quasi-stationary problems for an elastic-plastic rod and its surrounding soil medium.