Contact interaction of deformable continuous media of various physical nature

. An approach is proposed for modeling the contact interaction processes of reinforcement bundles with concrete, as well as for underground elements of foundations and buildings reinforced foundations by compensatory injection with adjacent soil. A special contact finite element of a continuous medium with specific properties has been constructed which makes it possible to simulate the elastic interaction of two contacting surfaces their displacement relative to each other and separation. The developed algorithm allows within the framework of a single finite element method, to calculate the joint deformation of the elements of metal reinforcing elements, concrete and “weak” soils. The proposed calculation methods are implemented within the consistent equations framework of a geometrically nonlinear three-dimensional elasticity theory. On the basis of the proposed methods, the problem of "pulling out" a steel reinforcement bar from a concrete block was solved, a comparison was made with experimental data and with the results obtained in publicly available application software packages, a good agreement was noted.


Introduction
The current state of the problems arising in the finite element implementation is given in the books [1][2].At present, there are a number of approaches to solving the contact problem by the finite element method (FEM) [3][4][5][6][7][8].From an algorithmic point of view, the simplest method is based on calculating the coefficients of mutual influence of points of contacting bodies in the normal and tangential directions.In a number of works, contact mechanics was considered by analogy with plastic flow [9][10][11][12][13][14][15][16][17], where the analogy between the laws of plastic flow and the laws of motion of rigid or elastic blocks with dry friction was used.
Another way to solve FEM contact problems is opened with the use of special docking elements that model the force-displacement diagram on the interfaces of interacting bodies.
In a number of researches a solution of the contact problem is proposed without usage of any analogies and docking elements [19][20][21].In contrast to the previous approach, where contact elements combine interacting bodies into one system, in these approach elements are characterized by a separate consideration of contacting bodies.

The contact interaction modeling algorithm
The mechanism of interaction of elements of deformable media through contacting surfaces A and B can be described as follows.Let us assume that between the contacting surfaces there is a certain contact layer of finite thickness 0 H , and denote by H the current value of the gap between the contacting surfaces.
Initially, assume that the contact surfaces are not connected to each other, and the artificially introduced contact layer only allows us to identify various options for contact interaction.Let's consider several cases of interaction of contacting surfaces., and for stresses and strains in the contact layer, the relations can be written: , , , is not satisfied, then the following the relations should be used:

But in this case, an additional condition
There is a possibility of slippage.7. If the contacting surfaces are "glued" before the deformation process, then contact interaction options 1-3 will be performed not for limited thickness of the contact layer A similar situation develops when the contact surfaces "slip" by tangential load.In this case, the limiting shear stress is chosen not from the condition np H f   , but from the condition of its admissibility before shear failure.
If the detachment of the contacting surfaces has occurred, then all variants of deformation 1-6 are valid.
The resolving equation can be written based on the principle of virtual displacements in the variational form:

     
where the sum over m is the sum over the volumes of the finite elements of the deformable medium, the sum over k is the sum over the volumes of the finite elements of the contact layer; P -surface load acting on part of the boundary.To solve the formulated physically nonlinear problem, the iterative method was used, which is a variant of the initial stress method.The following variational equation is basic for determining the k-th iteration:

Contact Finite Elementb Problem Calculation Algorithm
To implement the mathematical model of the interaction of the overlays described in the previous paragraph within the framework of the FEM.It is convenient to define the socalled contact element.Approximations of front surfaces are introduced: , , , , where   , i N  are the known shape functions for the two-dimensional approximation.To approximate the displacement vector a similar representation was used: The iterative procedure of the "initial stress method" type used is the following sequence of steps.The first approximation was determined from the solution of the variational equation based on the principle of virtual displacements, assuming the validity of Hooke's law:

  
, where     ,  are the reduced stress and strain vectors on the k -th three-dimensional soil fragment;     , QP -vectors of mass and surface loads.This equation assumes that the kinematic connections between the fragments (continuity conditions for displacements) and the kinematic boundary conditions are satisfied a priori.Equilibrium equations for each fragment, static conjugation conditions and static boundary conditions are fulfilled automatically in the integral sense.
In this case, the resolving equation is linear and can be written as:
3 Numerical results

Modeling the pull-out process from a mass concrete for a reinforcement bar
Based on the proposed methodology, a computational experiment was carried out to study the adhesion of fiberglass reinforcement to concrete.
Fig. 1.The dependence of the pulling force F on the displacement of fiberglass reinforcement Δ, 1according to the test results from [22]; 2 -according to the numerical simulation from [22]; 3according to the method proposed in [23]; 4 -according to the method proposed from in the work.
The experiment carried out in [22] was numerically recreated.Figure 1 shows the diagrams pulling force -slippage of reinforcement relative to concrete, obtained according to the results of the experiment [22] and according to the method proposed in [23].
Figures 2 and 3 show a comparison of the solution the problem obtained in ANSYS and ABAQUS, taking into account cracking using the XFEM method.

Conclusion
It can be noted that in the problem of "pulling out" reinforcement from concrete, taking into account the nonlinear deformation of the reinforcement allows one to obtain an acceptable agreement with the experiment, as well as with the results obtained in ANSYS and ABAQUS, taking into account cracking using the XFEM method.

Acknowledgements
The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project No. FZSM-2023-0009.

. 3 . 4 .
true.This situation occurs in the presence of pre-compression, when 0 HH  If the contacting surfaces try to separate (and for the case of the absence of force), then the contacting surfaces move freely and the relations For the case of free slip, shear stresses do not arise.This variant of interaction occurs at 0 HH  , in this case shear stresses in the contact layer variant of elastic interaction of contacting surfaces with compression and shear without slippage is possible.This situation arises at 0 HH  be satisfied, where f is the linear coefficient of friction.6.If the condition HH f  


are respectively the volumes of the elements of the deformable medium and the contact layer;       ,, u  -stresses, deformations and displacements of the deformable medium;     , HH  -stresses and strains in the contact layer,   g -gravitational acceleration vector,   g  -gravity,   Conferences 431, 06029 (2023) ITSE-2023 https://doi.org/10.1051/e3sconf/202343106029