Analytical modeling of glass composite and metal reinforcement adhesion with cements and concrete

Adhesion refers to the ability of concrete to resist slipping of reinforcement under loading in reinforced concrete products. The term "reinforced concrete" is considered as a uniform composite material as long as there is adhesion on the contact surface between the reinforcement and the concrete. In case of disruption of the reinforcement and concrete interaction, the construction is represented by separate elements. The aim of this work is to simulate the process of glass composite and metal reinforcement adhesion with cement concrete. To solve this problem, analytical modeling was performed in the ANSYS 19.0 Workbench software package, with the help of which the nature of damage accumulation in the rebar and the distribution of stresses in the concrete mass are determined.


Introduction
The problem of the reinforcement compliance and its mutual work with concrete under longitudinal load is one of the main problems of the reinforced concrete theory. Therefore, a huge number of domestic and foreign scientists took up this problem. The main theory describing the adhesion between a mutually displaced rebar and concrete is the theory by MM. Kholmyansky [1].
The objective of the current study is analysis of the stress-strain state (SSS) and determination of adhesion forces in the zone of interaction of reinforcement with concrete under the action of static axial pulling force. To simplify the computational apparatus without reducing the overall accuracy in the adhesion analysis in the considered admissible displacement boundaries at the crack joint, an elastoplastic concrete deformation diagram (Prandtl diagram), on the basis of which M.M. Kholmyansky and his students (Fig. 1a) approximated the normal adhesion law, is used: (1) * Corresponding author: kzis@nnn.tstu.ru where B and α are adhesion parameters; g x is the mutual displacement relative to the concrete mass. Under the elastoplastic law, adhesion is characterized by the parameters τ 0 and g * . However, Fig. 1b shows the formation of the section with plastic deformations at exceeding the maximum stress value. Therefore, to establish the parameters 0 и 0, the stress-strain state (SSS) in the area of anchoring the rebar in concrete should be considered for reinforced concrete structures [3].
To simplify this task, we consider a concrete prism centrally reinforced with a rebar with a symmetrically applied load from two sides (Fig. 1c). Consequently, the equilibrium equation for this problem takes the following form [1]: The equation of the mutual deformation of the concrete and the rebar: Using the previously obtained equation for shear stress τ (x), it takes the following form: To form a system of equations physically describing τ ad and g(х), the adhesion law is represented in the following form: The functionF(g(x)) represents the adhesion law, in which the shear stresses τ (x) are taken as a constant value along the entire length of the rebar anchoring. The dependence F(g (x)) is proposed in the form of an elastoplastic diagram τ − g [4]: ( ) = { ( ), при ( )/ * ≤ 1 * , при ( )/ * ≥ 1 (6) Having obtained the system of equations, the calculation at the elastic stage and later in the elastoplastic stage of work should be considered. If the elastic stage is g(x)/g * ≤ 1, with x = L, then the maximum displacement under loading is: The transition from the elastic stage to the elastoplastic one is characterized by the formation of sections with constant shear stresses τ 0 when the maximum displacement is reached g 0 > g * .
If the elastoplastic stage is g(x)/g * ≥ 1, with x = L, then the maximum displacement under loading is: * = 0 ℎ (8) The proposed system of equations allows to set the amount of the rebar displacement relatively to concrete. However, this dependence is applicable only for steel and composite reinforcement. In addition, it does not take into account elastoplastic deformations in concrete, which requires a deeper and more complete study of this problem.

Mathematical modeling for describing the rebar adhesion with concrete
This paper presents the results of the development of a mathematical model which takes into account the contact interaction of reinforcement with concrete [5]. The model of Menetrey-Willam [6] based on the Willam-Warnke yield surface [9], including the dependence on three independent invariants of the stress tensor, was adopted as a concrete model. The Willam-Warnke yield surface differs from the Mohr-Coulomb yield surface by the absence of sharp edges, which may cause difficulties in solving the Mohr-Coulomb surface stress. It also has some characteristics of the Drucker-Prager model [8] and can simulate similar materials.
The Menetrey-Willam model is best suited for modeling the behavior of bound inert materials, since it allows to take into account the redistribution of the strength characteristics inside the material. The hardening and softening parameter is defined as follows [7]: where is a material parameter;Ω andΩ are compression and hardening/softening functions, respectively, which depend on compression or hardening/softening parameters of и ; is the strength limit under uniaxial tension; is the strength limit under uniaxial compression; is the strength limit under biaxial compression. To describe the work of the reinforcement material, a model of the generalized Hooke's law was adopted with the following assumptions:  The material is considered to possess orthotropic properties when specifying physicomechanical characteristics of the glass composite reinforcement (GCR). For this purpose the Orthotropic Elasticity material model [5] is used to describe the GCR operation. This model is also based on Hooke's law, but it has a material stiffness matrix in which the deformation characteristics in the YX and YZ planes are symmetric.  As it was previously established, the primary reason of adhesion failure of GCR with concrete is the detachment of the reinforcement profile from the main rebar. A cohesive zone model (CZM) [9] is used to simulate this process, where the destruction of the binder matrix is regarded as a gradual separation of the reinforcement periodic profile surface from the main rebar with the subsequent delamination. In this model, the destruction of the contact surface occurs gradually, taking into account the accumulation of defects in the adhesive matrix inside the contact layer. The CZM is also used to simulate a bonding cement gel in the contact layer between the reinforcement and the concrete mass. The CZM model is described by the following equation [9]: where is stress; is contact stiffness; is tangential displacement in the contact area; is a violation parameter of the adhesive matrix.
The violation parameter of the adhesive matrix: where ̅̅̅ is displacement (slip length) in the contact area at the maximum stress; is displacement in the contact area with the destruction of the adhesive matrix. At ≤ 1, at In this case, the critical destruction energy is calculated as: The GCR strength can be evaluated by various criteria, which can directly identify the probable pattern or mechanism of destruction, for example, indicate whether matrix or fiber failure will occur. Therefore, each strength criterion can be assigned a coefficient, which will show the priority in the analysis of the material destruction. For example, when stretching the GCR, the rebar external fibers are more likely to be destructed, and this criterion should be assigned a higher fiber destruction strength coefficient. In this paper, the strength criterion for the GCR is the Puck criterion [10], which is appropriately suitable for recording materials with an orthotropic fibrous structure and highlights the destruction of the reinforcement fibers, delamination of the material and interfiber destruction inside the material. The destruction of a separate GCR fiber is described in terms of maximum values of permissible stresses or strains: If we consider the destruction of the matrix material, the dependence will take the following form [10]: where , are material strength limits for uniaxial tension or compression; , are material strength limits on shear.
As in many criteria, the destruction is considered to be the moment when the relative strength reaches 1. Therefore, the function takes the form [10]: Due to the twisting of the GCR fibers, an additional reduction coefficient should be introduced to calculate the delamination (for the rebar it is = 0,945), which eventually leads to the following formula [10] Steel is an isotropic material by structure that has identical mechanical properties in all directions. However, steel reinforcement has a yield point, therefore, to specify plastic deformations, a bilinear isotropic model was adopted, according to which the yield surface expands evenly in all directions. The initial slope of the curve (σ-ε) is equivalent to the elastic modulus of the material. Beyond the limit, an increase in plastic deformation is observed. To describe this dependence, the concept of "tangent modulus", which cannot be less than zero or greater than the elastic modulus, was introduced. The maximum equivalent stresses according to Mises are used as the destruction criterion. For steel reinforcement used in our research it was 610 MPa.

Analysis of the results of mathematical modeling
The numerical simulation of the SSS in the contact area was carried out in the ANSYS 19.0 Workbench package based on the results of mechanical testing of reinforcement and concrete according to GOST 24452-80 and GOST 12004-81. The obtained mechanical characteristics of concrete and reinforcement are given in [12].
The adhesion of reinforcement samples with concrete B25 was determined using a pullout test made on concrete cubes with a 100 mm edge according to GOST 31938-2012. All the samples hardened and gained strength in normal conditions (90 days, t = 20 ± 2 °C, w = 65%) [11]. The depth of the samples in concrete was l/d = 5 mm. The movement of the rebar relative to concrete was fixed by an indicator with an accuracy of 0.001 mm installed on the loaded end of the rebar.

Fig. 2. The scheme of the sample for pull-out test
For convenience, the shear stress τad was used as a result of measuring the bearing capacity: where N is the force applied; l is the anchoring depth; P is a circle perimeter. During testing, the load applied to each of the 10 samples was recorded using a forcemeasuring sensor. Simultaneously, the displacement of the rebar was measured (Δ) relative to the upper end of the concrete. Subsequently, a generalized curve "shear stressesdisplacements" was built.
The mutual displacement of the reinforcement relative to concrete was determined by the formula: where is the fixed value of the rebar displacement; is the reinforcement elongation between fasteners and the reference end of the sample, which was defined as: where l is the initial length of the reinforcement between the fastener and the reference end of the sample; A is a rebar area; E a is an elastic modulus of the reinforcement. Fig. 3 shows the convergence of the experimental data with the results of numerical simulation. It can be seen that the model describes the rebar displacement in the concrete mass. However, the maximum value of the bearing capacity was 5% lower than during the experiments. This is due to the fact that a smaller amount of plastic deformations in the concrete mass was put into the model. It should also be noted that the nature of the shifts initially represented a linear relationship, which was reflected by the actual shift schedule. For a more accurate reflection of the first solution stage, we had to add two additional conditions. Firstly, at the first loading stage, there is a gluing effect of the cement gel which takes the form presented in the formula 6. Secondly, this is the reinforcement compression by the concrete mass due to shrinkage and hardening of the concrete (Fig. 4). Furthermore, these conditions are not used in the calculation of the inelastic stage of the material, which allows to obtain a more accurate match of the calculated and experimental data for the entire range of materials.

Stress-strain state in a concrete mass
The paper deals with the SSS in a concrete mass with a force equal 0.85 of the destruction load. At higher loads, an avalanche-like increase in stresses and plastic deformations is observed, which is not interesting for us. It should be noted that the magnitude of adhesion, and, as a result, the SSS in the embedment area consists of a number of factors shown in Fig.4. The fundamental factor is the mechanical mesh of the rebar periodic profile with the concrete consoles. When considering the SSS, it is important to pay special attention to the concrete consoles formed between reinforcement profiles. Fig. 4 shows consoles from the two sides which in parallel take the distribution of compressive stress arising from the load transfer by the reinforcement profile [11][12][13][14][15][16][17][18]. Besides, to fulfill the equilibrium condition, tensile stresses are formed in the upper face. For the composite reinforcement, due to the lower stiffness of the profile, the concrete consoles fully work in compression, and since the profile is threaded at an angle of 30 degrees, the compression platform is formed at the same angle between the profile windings for equilibrium (Fig. 5b). The rest of the mass works on longitudinal tensile stresses in the Y plane. Kholmyansky M.M. noted that the classic concept of engineering strength of the concrete for solving the contact problem is incorrect to use. Therefore, to solve the problem, it is required to use the strength characteristics of concrete on a scale of 1 -0.01 mm. It is known [1] that increasing the sample sizes leads to increasing the probability of microdefects in the material structure. Therefore, when decreasing the sample sizes, the strength of concrete usually increases. In this case, it is confirmed by the fact that the maximum compressive stresses formed under the consoles reach 100 MPa, while tensile stresses are up to 2 MPa and more.
The results for comparing mutual shear stresses along the XY plane are presented in Fig. 6. The dependence of stresses takes an exponential form for both reinforcements; however, the magnitude has different values, since there are leaps in the place of the periodic profile, which are about 4 times lower for GCR than for steel. This is due to the higher elastic modulus and the homogeneous structure of the metal reinforcement, which in turn leads to pronounced leaps. When the embedment depth increases, the shear stresses of both reinforcements smoothly fade out (Fig. 7.). When considering plastic deformations, it can be seen that they are formed in those places where later the formation of cracks and the subsequent destruction of the adhesion will occur. For example, for the samples with metal reinforcement plastic deformations are formed in the cut-off areas of concrete consoles (Fig. 8a), while for GCR deformations these deformations are pronounced only along the main rebar in the planes of compressive stresses (Fig. 8b).

Damage accumulation in reinforcement
Experimental studies with steel reinforcement showed that the destruction of the rebar did not occur. So the maximum stress from the destruction load reached 0.783. At the same time, the value from the destruction load in the rebar periodic profile in the embedment area was 0.25. This testifies to the fact that during the pull-out testing the steel reinforcement profile did not exhaust its reserve, and the decisive reason was the destruction of concrete in the contact area. When considering a numerical model with GCR, the adhesion failure occurred between the main rebar and the periodic profile (CZM). In addition, the reinforcement showed the damage from tensile and compressive stresses in the GCR periodic profile. Their values were 1.579 and 1.065 from the Puck destruction load (Fig. 11 b, c), which was confirmed experimentally (Fig. 10). periodic profile (Fig.11d). The same destruction was observed when testing the GCR with the embedment depth of l/d≥6, since the destruction value of the matrix increased to 0.97 by Puck, and the strain intensity value was below the maximum by 15%.

Conclusion
1. The analysis of existing models of concrete deformation showed that the Menetrey-Willam model should be used to consider the distribution of stresses in the contact layer of concrete, taking into account hardening and softening of the concrete mass.
2. The developed numerical model of reinforcement adhesion with concrete has expanded the existing ideas about the SSS in the contact area of concrete with reinforcement. This allowed to establish the nonlinear character of the distribution of shear and normal stresses over the entire embedment depth. Thus, leaps of stresses are observed in the contact area of concrete and reinforcement periodic profile, which become the cause of major cracks. 3. The numerical simulation of the GCR embedment in the concrete mass based on the ANSYS 19.0 Workbench software package according to the previously determined mechanical characteristics of materials allowed to determine the nature of the damage accumulation in the reinforcement and to reveal that the destruction of composite reinforcement occurs from tensile stresses by breaking the fibers or the GSR matrix.