A Cracking Model for Reinforced Concrete Cover, Taking Account Of

materials

Article A Cracking Model for Reinforced Concrete Cover, Taking Account of the Accumulation of Corrosion Products in the ITZ Layer, and Including Computational and Experimental Veriﬁcation

Tomasz Krykowski 1,* , Tomasz Ja´sniok 2 , Faustyn Recha 1 and Michał Karolak 3

1 Department of Mechanics and Bridges, Faculty of Civil Engineering, Silesian University of Technology, Akademicka 5, 44-100 Gliwice, Poland; [email protected] 2 Department of Building Structures, Faculty of Civil Engineering, Silesian University of Technology, Akademicka 5, 44-100 Gliwice, Poland; [email protected] 3 3D Team Oddział Wrocław, Wrocławski Park Biznesu, Bierutowska 57-59, 51-317 Wrocław, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-603-912-594

Received: 11 November 2020; Accepted: 23 November 2020; Published: 26 November 2020

Abstract: The paper presents the finite element method model (FEM) which allows the forecasting of the evolution of damage in a concrete cover together with experimental verification of the model. The objective of the model is to define the corrosive volume strain tensor rate effected by corrosion, which comprises the accumulation of corrosion products in pore spaces as well as in micro-cracks which develop at the initial stage of cover degradation. The propagation of damage in the contact zone was captured by taking into account the function describing the degradation of the interface transition zone depending on the cover tightening time–critical time. The method of determining the critical time along with the method of taking into account the effective electrochemical equivalent of iron was also analyzed in this paper. The work presents the experimental verification of the model using an accelerated corrosion test of reinforcement in concrete and strain measurements with optical methods. The conducted tests demonstrate satisfactory compliance of the model with the test results.

Keywords: reinforcement corrosion; cover cracking; accelerated corrosion test; optical measurements; FEM; elastic-plastic-fracturing model; corrosion strain tensor

1. Introduction The damage to reinforced concrete elements caused by reinforcement corrosion is a very important problem pertaining to the operation of reinforced concrete structures. The reasons for electrochemical corrosion of reinforcing steel used in reinforced concrete elements stem principally from the carbonization process of concrete cover resulting in the decrease in concrete pH value or/and the diffusion process of chloride ions to the surface of reinforcing bars or direct current leaks from electric tractions [1]. The regulations used in the design of structural elements enforce the obligation to use covers of appropriate thickness [2,3], but in reality this thickness may be insufficient and can initiate corrosive processes resulting in a progressive decrease in the load capacity of reinforced concrete elements. Good knowledge of corrosion process mechanisms allows the determination of both the effects of corrosion (e.g., cross-sectional reduction of rebars or the place of developed cracks), and the estimation of the time at which such effects can occur [4]. Reinforcement corrosion processes are complex problems, and two major phases related to the life time of the structure texp can be basically distinguished in them. These are the phases of electrochemical

Materials 2020, 13, 5375; doi:10.3390/ma13235375 www.mdpi.com/journal/materials Materials 2020, 13, x 2 of 18

Reinforcement corrosion processes are complex problems, and two major phases related to the

life time of the structure t can be basically distinguished in them. These are the phases of electrochemical corrosion initiation (in) and the degradation phase (D), which comprises the work of the corrosion cell and the propagation of mechanical damage in the cover. Schematically the process of degradation can be simplified according to the scheme shown in Figure 1. This figure describes the characteristic stages of the corrosion process causing degradation of reinforced concrete elements under the conditions of reinforcement corrosion given in the literature e.g. [4], modified here in order to include the very important changes in the contact zone of steel and

concrete): where: t is the reinforcement corrosion initiation time, t is the propagation time, t is Materials 2020, 13, 5375 2 of 17 the time of activation of mechanical impact by corrosion products, t is the critical time, t is the cracking time, t is the accelerated degradation of the structure time and t is the time of degradation. corrosion initiation (in) and the degradation phase (D), which comprises the work of the corrosion cell and the propagation of mechanical damage in the cover. Schematically the process of degradation can be simplified according to the scheme shown in Figure1. This figure describes the characteristic stages of the corrosion process causing degradation of reinforced concrete elements under the conditions of reinforcement corrosion given in the literature e.g. [4], modified here in order to include the very important changes in the contact zone of steel and concrete): where: tin is the reinforcement corrosion initiation time, tp is the propagation time, t0 is the time of activation of mechanical impact by corrosion products, tcr is the critical time, tcrack is the cracking time, tad is the accelerated degradation of the structure time and tD is the time of degradation.

Figure 1. Scheme of degradation of the structure over time. Figure 1. Scheme of degradation of the structure over time. Commented [M1]: Please add spaces before and after equal sign and plus sign. Is done! The initiation phase of the reinforcement corrosion process has been described in detail in the The initiationliterature phase [5–7]. It of is theassumed reinforcement that exceeding corrosionthe critical concentration process has of chloride been ions described or the critical in detail in the literature [5–7pH]. Itvalue is assumed in concrete leads that to exceeding the development the criticalof an electrochemical concentration cell [8,9] of and chloride consequently, ions to or the critical the flow of the corrosion current and to the development of corrosion products that tightly fill up the pH value in concretefree pore spaces leads around to the the development reinforcing bar ofin the an so electrochemical-called transition layer cell [10]. [8, 9Another] and consequently,effect that to the flow of the corrosionhas an impact current on the and evolution to the of development damage in the cover of corrosion is the propagation products of micro that-cracks tightly in thefill up the free pore spaces aroundstructure the of the reinforcing contact layerbar steel in-concrete the so-called [11–13]. transition layer [10]. Another effect that has an The literature offers a lot of models which enable the description of the incremental process of impact on thevolumetric evolution strains of damage effected by in corrosion the cover products is the [14,15]. propagation The vast majority of micro-cracks of these models in ignore the structure of the contact layerthe influence steel-concrete of the porosity [11– of13 the]. transition layer and the propagation of micro-cracks in the cover, The literatureor they osignificaffers antly lot simplify of models this process which [16]. enable The problem the descriptioncovering the accumulation of the incremental of corrosion process of products in the transition layer has been analyzed, among others, in the paper [17], where the volumetric strainspropagation effected of corrosion by corrosion products products in the pore [14 structure,15]. The surrounding vast majority the rebar of was these analy modelszed. A ignore the influence of the porosity of the transition layer and the propagation of micro-cracks in the cover, or they significantly simplify this process [16]. The problem covering the accumulation of corrosion products in the transition layer has been analyzed, among others, in the paper [17], where the propagation of corrosion products in the pore structure surrounding the rebar was analyzed. A different approach is proposed in [12,13], where an a priori concept of the volume of corrosion accommodation region (car) is proposed. The model for the corrosion degradation of the cover, presented in these papers, erroneously assumes that the corrosion process of the reinforcement proceeds in a uniform manner around the circumference of the rebar. In the general situation this assumption is incorrect and is also not consistent with the method for carrying out the research formulated in [13]. A significant problem in the approach formulated in [13] is also the lack of a clear method for adopting the key parameters of the model calculations, such as the cumulative volume of the corrosion products. This undoubtedly influences the method of describing the effect of corrosion products on the structure of the cover. An additional problem is the assumption adopted in many papers, which says that the entire mass of corrosion products produced spreads in the plane perpendicular to the rebar axis, which, according to the authors of this paper, is not true at the initial stage of the process. The present work clarifies the theoretical concept involving the description of the developmental process of corrosive damages in the cover, using the tensor of corrosive volumetric strain rate [18]. Moreover, this research analyzes the model, along with its numerical verification, which enables the recognition of the deposition of corrosion products in the pores of the transition layer and in micro-cracks developed in the initial phase of the corrosion process. The paper formulates the equations Materials 2020, 13, 5375 3 of 17 Materials 2020, 13, x 3 of 18 for the evolutionThe present of the work corrosive clarifies the volumetric theoretical concept strain ratesinvolving as a the function description of theof the parameter developmentalβ (this parameter process of corrosive damages in the cover, using the tensor of corrosive volumetric strain rate [18]. dependsMoreover, on the this activation research analyzes time thet0 andmodel, the along critical with its time numericaltcr and verification, determines which enables the progress the of cover degradation).recognition The of the paper deposition presents of corrosion a method products of experimental in the pores of estimation the transition of layer the and critical in micro- time tcr, which is a key parametercracks developed for model in the initial description phase of the at thecorrosion initial process. stage The of cover paper degradation.formulates the equations The method for of analysis of the reinforcementthe evolution of the corrosion corrosive volumetric processes strain proposed rates as a in function the paper of theallows parameter the β (this inclusion parameter of the problems depends on the activation time t and the critical time t and determines the progress of cover of non-regular distribution of corrosion products on the circumference of the rebar as well as for the degradation). The paper presents a method of experimental estimation of the critical time t, which changesis a in key the parameter nature offor the model volumetric description strain at the velocityinitial stage field. of cover The degradation. presented The theoretical method of considerations have beenanalysis supported of the reinforcement by experimental corrosion research processes aimed proposed at determining in the paper allows many the key inclusion parameters of the of the model, criticalproblems time t ofand non-regular the eff ectivedistribution electrochemical of corrosion products equivalent on the circumference of reinforcing of the steel rebar kas well. The work also as for thecr changes in the nature of the volumetric strain velocity field. The presented theoreticaleff includesconsiderations an evaluation have ofbeen compliance supported ofby theexperimental obtained research results aimed of experimental at determining research many key with theoretical calculations.parameters The of experimentalthe model, critical research time t and on reinforcementthe effective electrochemical corrosion equivalent processes of reinforcing was carried out using the so-calledsteel k accelerated. The work also corrosion includes an test evaluation [19] and of comp opticalliance strain of the obtained measurement results of techniques experimental[ 20]. research with theoretical calculations. The experimental research on reinforcement corrosion 2. Calculationprocesses Modelwas carried out using the so-called accelerated corrosion test [19] and optical strain measurement techniques [20]. In reinforced concrete elements in which the reinforcement corrosion process is initiated, 2. Calculation Model mechanical effects are not initiated at the same time when the reinforcement corrosion process is activated.In reinforced This e ffconcreteect is fundamentallyelements in which connectedthe reinforcement with corrosion the microstructure process is initiated, of the transition mechanical effects are not initiated at the same time when the reinforcement corrosion process is layer [16activated.,17] and This with effect the is fundamentally propagation connected of micro-damages with the microstructure in the initiation of the transition phase layer of the [16], reinforcement corrosion[17] and process with the [13 propagation]. The mechanisms of micro-damages of in this the initiation process phase are presentedof the reinforcement in a simplified corrosion manner in Figure2process, where [13]. Arabic The mechanisms numerals of denote:this process (1) are cement presented grout, in a simplified (2) products manner from in Figure reinforcement 2, where corrosion, (3) poreArabic voids numerals (with airdenote: and (1) pore cement solution), grout, (2) (4)prod steelucts from rebar, reinforcement (5) corrosion corrosion, pit filled (3) pore with voids products from (with air and pore solution), (4) steel rebar, (5) corrosion pit filled with products from reinforcement reinforcementcorrosion. corrosion.

FigureFigure 2. The 2. The model model of of corrosion corrosion product product deposition deposition in concrete in concrete pores: (a pores:) interface (a steel-concrete) interface steel-concrete transitiontransition zone, zone, (b )(b corrosion) corrosion product product mass mass flows ﬂows in the ininte therface interfacetransition zone transition steel-concrete zone – steel-concrete description in the text. – description in the text. The effective mass balance of the corrosion products that mechanically impacts on the cover Thestructure eﬀective of the mass rebar can balance be expres ofsed the with corrosion the following products relationship: that mechanically impacts on the cover structure of the rebar can be expressed with the following relationship:

...... m = m mpor mtran, m = m m 2+ , (1) eff ekw − − ekw R − R,Fe . . where meff is the effective mass change rate of corrosion products, mR is the source of the mass of . corrosion products—the accumulation rate of corrosion products in the investigated area, mpor is the mass change rate of corrosion products resulting from their being carried away to the micro-cracks . and micro-slits formed in the interface transition layer as a result of mechanical impact, mtran is the rate of mass change of corrosion products due to the transport process of the products into deeper Materials 2020, 13, 5375 4 of 17

. concrete layers, mR,Fe2+ is the rate of mass change of corrosion products due to their deposition in the corrosion pitting of the rebar, and this is proportional to the rate of the development of this loss, in line with Faraday’s law . + ( ) mFe2 = kcompIcorr, kcomp = kcomp k , (2) where Icorr is the total intensity of corrosive current flowing in the microcells, kcomp is the computational electrochemical equivalent which comprises all deviations from ideal conditions, and which will be developed in Section3 of the work, k is the electrochemical equivalent of iron determined in line with the equation M k = (3) zF In Equation (3) M is the molar mass of iron equal to 56 g/mol, F is the Faraday constant equal to F = 9.6485 104 C/mol, and z is the reaction charge number equal to 2. × Assuming a small thickness of the interface transition layer, and assuming a constant density of the corrosion products ρR, the Equation (1) can be formulated in the form [17,18,21] ...... V = V Vpor Vtran, V = V V 2+ (4) eff ekw − − ekw R − Fe . . Where Veff is the rate of change of effective volume of corrosion products with a mass of meff, VR is the . rate of change in mass of products from reinforcement corrosion with a mass of mR, VFe2+ is the rate of change in volume of corrosion loss with a mass of m 2+ tightly filled with corrosion products with . Fe volume VFe2+ = VR,Fe2+ and mass mR,Fe2+ , Vpor is the rate of change in volume of corrosion products . transferred into micro-cracks formed in the interface transition layer, while Vtran is the rate of change in the volume of corrosion products transferred outside the interface transition layer deeper into the . . concrete cover. The determination of volume change rates Vpor and Vtran is very complex from the theoretical point of view, so it was assumed that these quantities change in a manner proportional to . the rate of the change of the equivalent volume Vekw [18]: ...... V = (1 β)V , V = ψI, β V = Vpor + Vtran (5) eff − ekw ekw ekw 1 α− ϑ 1 k 1 1 ψ − α + + = , = mFe2 mR− , ϑ = %Fe2 %R− (6) %Fe2+

ψ + where is a constant depending on the composition of corrosion products, %Fe2 is the density of the mass of iron ions, %R is the density of corrosion products, α and ϑ are parameters adopted depending on the chemical composition of corrosion products [14]. The effect of change over time involving the impact of corrosion products on the concrete of the cover was defined using the parameter β. It was assumed in the research that β can be any function, but to simplify the considerations it was assumed that the said relation is linear [13] (the linear form of the β function was shown in this paper for the first time) [18]:

 0  1, t < t0 Vekw < V (t0) = Vpor  ekw β =  ( ) ( ) 0 ( ) ( ) ( ) (7)  tcr t / tcr t0 , t0 t tcr Vekw t0 Vekw t Vekw tcr  − − ≤ ≤ ≤ ≤  0, t > tcr ( Vekw(t) > Vekw(tcr)).

The function β described by the Equation (7) comprises three main phases that occur during the course of reinforcement corrosion processes. The times t0 and tcr deﬁne the characteristic work phases of the corrosion cell. Time t0 is the time needed for corrosion products to ﬁll up free pore spaces located 0 ( ) = in the transition layer Vekw < Vekw t0 Vpor. The next stage is the time interval from t0 to the critical time tcr. This stage of the corrosion process is connected with a gradual increase of mechanical impacts which result from the increase of pressure exerted by corrosion products on the concrete of the cover. Materials 2020, 13, x 5 of 18

V̇ = (1 − β)V̇, V̇ = ψI, βV̇ = V̇ + V̇ (5)

( ) α ϑ − 1 k ψ = , α = mm , ϑ = ϱϱ (6) ϱ

where ψ is a constant depending on the composition of corrosion products, ϱ is the density of the mass of iron ions, ϱ is the density of corrosion products, α and ϑ are parameters adopted depending on the chemical composition of corrosion products [14]. The effect of change over time involving the impact of corrosion products on the concrete of the cover was defined using the parameter . It was assumed in the research that β can be any function, but to simplify the considerations it was assumed that the said relation is linear [13] (the linear form of the  function was shown in this paper for the first time) [18]:

1 , t < t V < V(t) = V β = (t − t)⁄(t − t), t ≤ t ≤ t (V(t) ≤ V(t) ≤ V(t)) (7)

0, t > t V(t) > V(t). The function β described by the Equation (7) comprises three main phases that occur during the course of reinforcement corrosion processes. The times t and t define the characteristic work Materials 2020, 13, 5375 5 of 17 phases of the corrosion cell. Time t is the time needed for corrosion products to fill up free pore spaces located in the transition layer V < V(t) = V. The next stage is the time interval from t to the critical time t. This stage of the corrosion process is connected with a gradual increase of At that stage,mechanical apart from impacts the which compaction result from of the the increase layer of of pressure corrosion exerted products, by corrosion there produ iscts also on the a propagation of micro-cracksconcrete taking of the place cover. in At the that cover stage, andapart infrom the the transition compaction layer,of the layer into of which corrosion the products, corrosion products there is also a propagation of micro-cracks taking place in the cover and in the transition layer, into penetrate, graduallywhich the blocking corrosion products the possibility penetrate, of gradually carrying blocking away the the possibility corrosion of carryingproducts away into the deeper layers of the cover. Thecorrosion critical products time into described deeper layers in the of literature the cover. The involving critical time such described problems in the can literature be identiﬁed with the concept ofinvolving volume such and problems the evolution can be identified of cumulative with the accommodation concept of volume region and the [ 13 evolution]. Yet, the of application cumulative accommodation region [13]. Yet, the application of this concept is rather complicated of this conceptsince is it rather is difficult complicated to experimentally since assess it is this di ﬃvolumecult of to the experimentally developed corrosion assess products this and volume of the developed corrosionassuming a products predetermined and value assuming of the critical a predetermined time is a burden of value error. of By the using critical the notion time of is a burden critical time t and measurements of displacements in the test this quantity can be relatively easily of error. By using the notion of critical time tcr and measurements of displacements in the test this approximately determined in an experimental way, (compare Figure 3), where L is the basis for quantity can bemeasurement relatively of easilythe test element, approximately ΔL is the elongation determined of this in length. an experimental way, (compare Figure3), where L is the basis for measurement of the test element, ∆L is the elongation of this length.

Figure 3. Methodology for determining critical time t and function β: (a) method of determination Figure 3. Methodology for determining critical time t and function β:(a) method of determinationCommented of [M2]: Please add spaces before and after plus of critical time as the intersection point of regressioncr lines, (b) course of changes of function β sign. Is done! critical time astogether the intersectionwith the description point of the of impact regression of corrosion lines, products (b) course on concrete. of changes of function β together with the description of the impact of corrosion products on concrete.

This operation can be made by determining the intersection point of two approximated straight lines characterizing the increments of displacement in the analyzed sample before and after exceeding the critical time (it is necessary to continuously measure the test element to have the necessary displacement data to find approximated straight lines). It is assumed in the analysis that the corrosion products are not pushed away deep into the developing crack at the initial stage of the corrosion process). By allowing for the definition of the time derivative of the first invariant of volumetric corrosion strain tensor . . Veff . V . V . V . V I1εV ε11 + ε22 + ε33 ηε (8) V0 . V where εii are the components of volumetric strain tensor rate in the tangent direction to the reinforcement . V . V . V axis ε33 and in the plane perpendicular to the reinforcement axis respectively ε11 and ε22. Depending on the phase of the analyzed corrosion process, three situations can occur. The first is for t < t0, the parameter η in Equation (8) takes the value η = 3, tensor of volumetric strain rates is . V . V . V . V an isotropic tensor with non-zero coordinates equal to ε11 = ε22 = ε33 = ε . The next situation takes . V place for time t > tcr the coordinate of tensor of volumetric strain rate ε33 in the direction tangent to . V the axis of the reinforcing bar ε33 = 0, the parameter η takes the value η = 2, non-zero coordinates of . V . V . V tensor of volumetric strain rates in the perpendicular plane to the bar axis are equal to ε11 = ε22 = ε . The last considered situation is the intermediate situation, which takes place in the time interval t0 t tcr, the parameter η takes variable values from the interval η (2, 3), η = 3β + 2(1 β), ≤ ≤ . V ∈ − the coordinate of tensor of volumetric strain rate ε33 in the direction tangent to the bar axis is described . V . V . V . V . V . V by the function ε33 = ε33(β), ε33 0, ε = ε11 = ε22 , while the non-zero coordinates of the tensor of ∈ . V . V . V volumetric strain rates in the plane perpendicular to the bar axis take the values ε11 = ε22 = ε . The state of strain depends here on the impact stage of the corrosion process on the concrete of the cover. With the rise of pressure, which goes along with the filling up process of free pore spaces by corrosion products, the character of the strain state changes to a plain strain state. The phases of the Materials 2020, 13, 5375 6 of 17 reinforcement corrosion process and their corresponding forms of volumetric strain tensor corrosion rates were determined by the Equations (9) and (10)

Spatial state of strain, t tcr, • ≤

1 (1 β) α ϑ 1 kcompIcorr . V . V − − − . V . V εαβ = ε δαβ = δαβ, η = 3β + 2(1 β), ε33 = β ε . (9) ηρ + Fe2 V0 −

Flat state of strain, t > tcr, •

1 α ϑ 1 kcompIcorr . V . V − − . V εαβ = ε δαβ = δαβ, ε33 = 0, η = 2, (10) ηρ + Fe2 V0

where V0 is initial volume of the region of the body being subject to volumetric strains and α, β are the indexes, α, β = 1, 2.

3. Experimental Research In order to assess the compliance of the proposed theoretical model with the experimental tests, an accelerated corrosion test was carried out on reinforcing bars 20 mm in diameter, made from St3SX steel (Poland – designation from the production date of the bar and the chemical composition of steel was given in the Table1) and placed in four C50 /60 concrete samples P1, P2, P3, P4 measuring 100 100 80 mm. The concrete mix was made oﬀ the river aggregate of CEM I cement (Góra˙zd˙ze × × Cement SA, Chorula, Poland – chemical composition of cement is presented in Table2) without admixtures and additions at w/c = 0.4. After concreting, samples were demolded after 24 h and then stored in a chamber (Feutron GmbH, Langenwetzendorf, Germany) at 20 ◦C and 90% humidity for 3 months. The axonometry of the sample and the location of the rebar 1 is presented in Figure4a. The front surfaces of the bar 1 were protected with polyester resin (Fenedur S.A., Montevideo, Uruguay) 2 before the development of corrosion processes in that area of the bar.

Table 1. Chemical composition of steel St3SX (S235JRG1) (%).

C Si Mn P S 0.18 0.30 0.32 0.04 0.04

Table 2. Chemical compositions of cement CEM I (%).

CaO SiO2 Al2O3 Fe2O3 SO3 MgO Na2OK2O Cl− 64.5 19.3 5.97 2.71 2.29 0.82 0.37 0.36 0.049

The accelerated corrosion test was carried out with the use of an electrolyser (Silesian University of Technology, Gliwice, Poland) (Figure4b), which forced corrosion of the reinforcement uniformly around the circumference of the rebar. The reinforcing bar 1 was the anode, while the cathode was made of weather-resistant perforated steel sheet 3, which surrounded the concrete sample on all sides. Both electrodes were connected to power supply 4 with insulated wires. The sample was immersed in tap water 5 ensuring electrical contact of all electrodes of the system. The applied power supply unit 4 enabled a simultaneous connection and independent measurement of four samples (Figure4c). The generated voltage for all channels was equal and had the value of 20 V throughout the entire test period. The power supply unit enabled automatic recording of basic electrical parameters of the system, i.e., current I, voltage U and electrical resistance of the system R with the preset frequency of 60 s. The results of the measurement were presented as graphs illustrating the course of changes in Materials 2020, 13, 5375 7 of 17 Materials 2020, 13, x 7 of 18 currentFigure and 4a. electrical The front resistance surfaces in of the the analyzed bar 1 were samples, protected which with for polyester all four samples resin (Fenedur are presented S.A., in Montevideo, Uruguay) 2 before the development of corrosion processes in that area of the bar. Figure5 out during the test: (a) sample P1, (b) sample P2, (c) sample P3, (d) sample P4.

Figure 4. Stand for testing accelerated corrosion of the reinforcement using the potentiostatic method: Figure 4. Stand for testing accelerated corrosion of the reinforcement using the potentiostatic Commented [M3]: Please change dash into minus sign. Is (a) schematicmethod: (a of) schematic the sample of withthe sample the location with the of location reinforcing of reinforcing bar, (b) measuring bar, (b) measuring system, system, (c) view (c of) the done! standview with of powerthe stand supply with power unit and supply four unit samples–description and four samples – indescription the text. in the text.

The use of direct currentTable 1. Chemical significantly composition accelerates of steel St3SX the (S235JRG1) course of (% the) experiment, ensuring homogeneityC of corrosion andSi enables the collectionMn of precise dataP necessary for modellingS the phenomenon. On the other hand, the acceleration of the process causes that the decomposition 0.18 0.30 0.32 0.04 0.04 of corrosion products in concrete is different than in the case of natural corrosion, which is of key importance when comparingTable the results2. Chemical to corroded compositions real of structures. cement CEM In I addition,(%) monitoring the rate of naturally occurring corrosion by electrochemical methods is very difficult and can often be burdened with a large error. 64.5 19.3 5.97 2.71 2.29 0.82 0.37 0.36 0.049 The applied voltage brought about a significant shift of the potential of the steel electrode, where ironThe digestion accelerated was corrosion the dominant test was process carried out with the use of an electrolyser (Silesian University of Technology, Gliwice, Poland) (Figure 4b), which forced corrosion of the reinforcement Fe 2e Fe2+ (11) uniformly around the circumference of the −rebar.→ The reinforcing bar 1 was the anode, while the cathode was made of weather-resistant perforated steel sheet 3, which surrounded the concrete sampleTaking on into all account sides. Both the electrode above dominants were connected process, to the power mass supply of oxidizing 4 with ironinsulated ions waswires. calculated The fromsample Faraday’s was law,immersed in tap water 5 ensuring electrical contact of all electrodes of the system. The applied power supply unit 4 enabled a simultaneousZt connection and independent measurement of

four samples (Figure 4c). The generated mvoltageFe2+ = for allkI channelsextdt was equal and had the value of 20 V (12) throughout the entire test period. The power supply0 unit enabled automatic recording of basic electrical parameters of the system, i.e., current I, voltage U and electrical resistance of the system R whereaswith compared the preset to frequency the formula of 60 (2), s.m TheFe2+ resultsis the mass of the of measurement iron ions carried were away presented from theas graphs steel rebar to theillustrating concrete the microstructure course of changes after in the current duration and electrical time t ofresistance the electrolysis, in the analy kz ised the samples, electrochemical which equivalentfor all four determined samples inare compliance presented in with Figure the 5 Equation out during (3), the while test: the (a) currentsample intensityP1, (b) sample is not P2, related (c) to the ﬂowsample of chargeP3, (d) insample the corrosion P4. microcells, but it is the total intensity of the external current ﬂowing in the electric circuit Iext.

Materials 2020, 13, x 7 of 18

Figure 4. Stand for testing accelerated corrosion of the reinforcement using the potentiostatic method: (a) schematic of the sample with the location of reinforcing bar, (b) measuring system, (c) view of the stand with power supply unit and four samples–description in the text.

Table 1. Chemical composition of steel St3SX (S235JRG1) (%)

C Si Mn P S 0.18 0.30 0.32 0.04 0.04

Table 2. Chemical compositions of cement CEM I (%)

𝐂𝐚𝐎 𝐒𝐢𝐎𝟐 𝐀𝐥𝟐𝐎𝟑 𝐅𝐞𝟐𝐎𝟑 𝐒𝐎𝟑 𝐌𝐠𝐎 𝐍𝐚𝟐𝐎 𝐊𝟐𝐎 𝐂𝐥 64.5 19.3 5.97 2.71 2.29 0.82 0.37 0.36 0.049

The accelerated corrosion test was carried out with the use of an electrolyser (Silesian University of Technology, Gliwice, Poland) (Figure 4b), which forced corrosion of the reinforcement uniformly around the circumference of the rebar. The reinforcing bar 1 was the anode, while the cathode was made of weather-resistant perforated steel sheet 3, which surrounded the concrete sample on all sides. Both electrodes were connected to power supply 4 with insulated wires. The sample was immersed in tap water 5 ensuring electrical contact of all electrodes of the system. The applied power supply unit 4 enabled a simultaneous connection and independent measurement of four samples (Figure 4c). The generated voltage for all channels was equal and had the value of 20 V throughout the entire test period. The power supply unit enabled automatic recording of basic electrical parameters of the system, i.e., current I, voltage U and electrical resistance of the system R with the preset frequency of 60 s. The results of the measurement were presented as graphs illustrating the course of changes in Materialscurrent2020 and, 13 electrical, 5375 resistance in the analyzed samples, which for all four samples are presented8 of 17in Figure 5 out during the test: (a) sample P1, (b) sample P2, (c) sample P3, (d) sample P4.

FigureFigure 5.5. ResultsResults ofof thethe measurement measurement of of current currentI andI and electrical electrical resistance resistance of theof the system systemR carried R carried out duringout during the test:the test: (a) P1, (a) (P1,b) P2,(b) (P2,c) P3, (c) (P3,d) P4.(d) P4.

TheThe indiscriminateuse of direct calculation current ofsignificantly the coefficient accele k fromrates the formulathe course (3) equal of the to k experiment,= 2.894 10 4ensuringg/C = × − 0.00912homogeneityg/µAYear of corrosionand including and it enables in formula the (12) collectio may leadn of to precise large discrepancies data necessary between for themodelling actual mass the ofphenomenon. iron ions transferred On the toother concrete hand, microstructure the acceleration measured of the gravimetricallyprocess causesand that that the calculated decomposition from the of formula (12). Numerous research studies with the application of an electrolyser used to accelerate the corrosion process of reinforcement in concrete have unequivocally proved that there are differences between the gravimetrically measured loss of steel mg and that calculated from the Faraday’s law mFe2+ based on Equation (12). The acquisition of consistent results requires the adoption in the tests of the effective electrochemical equivalent of the process of accelerated corrosion of the reinforcement keff = keff(λg F) − where λg F is the relative difference in mass determined by the formula −

mFe2+ mg λg F = − , keﬀ = 1 λg F k (13) − mFe2+ − −

When analyzing the literature in the field of accelerated corrosion tests on reinforcement in concrete, a large discrepancy in the obtained results can be observed. The discrepancies of λg-F, e.g., in the paper [22], did not exceed 5%, while in [23] the relative difference was larger and ranged from 5% to 25% depending on the applied intensity of the electric current. Much higher values of λg-F, from 2% even up to 200%, were obtained in [24], but here the discrepancies involved both the overestimation and underestimation of the mass of iron ions in relation to gravimetric measurements. The studies presented in [25,26] an electrolyzer were also used to accelerate the corrosion of reinforcement, but the ranges of the abovementioned relative differences were not defined. Yet, it was pointed out that the reaction (11) is not the only one that takes place at the anode, and the calibration of results is needed after the termination of the experiment. The above conclusions and research results prompted us to conduct our own research, in which the impact of cement type (cements CEM I, CEM II (Góra˙zd˙zeCement SA, Chorula, Poland) and CEM III (Góra˙zd˙zeCement SA, Chorula, Poland) were taken into account), and the impact of the method of concrete curing on the λg-F values were estimated. The research demonstrated that the greatest relative differences of up to 65% were obtained for the cement CEM I, and the smallest about 21% for the cement CEM III. And good curing of concrete, i.e., storing the Materials 2020, 13, 5375 9 of 17

samples in limewater until the tests were carried out, resulted in the rise of the λg-F value to about 63% for CEM I, 45% for CEM II and 34% for CEM III. On the other hand, the lack of curing (without the samples being in contact with water after demolding) caused the relative mass differences to be much smaller, 35%, 27% and 21%, respectively. Therefore, it can be assumed that the better the curing conditions the tighter the microstructure of the concrete, and the λg-F values are higher. In the above research, a test was also carried out on bars in concrete of the same specifications and using the same curing method as for the samples P1, P2,P3, and P4 marked respectively as P5, P6, and P7. Before concreting, the bars were weighed, and after the completion of the tests, the samples were split and the bars were taken out, and after cleaning them from corrosion products, they were reweighed. The mean value of λg-F determined with the Equation (13) was here about 40%. The results of the calculationsMaterials carried 2020, 13, x out along with the determination of the resulting values9 of 18 of the effective electrochemical equivalent of iron keff and the average value keff,avg of this parameter used in further In the above research, a test was also carried out on bars in concrete of the same specifications calculations areand summarized using the same incuring Table method3. as for the samples P1, P2,P3, and P4 marked respectively as P5, P6, and P7. Before concreting, the bars were weighed, and after the completion of the tests, the Table 3. Comparisonsamples were of split the and results the bars of thewere gravimetric taken out, and analysis after cleaning and them the calculationsfrom corrosion ofproducts, the bar mass loss λ with the usethey of Faraday’swere reweighed. law The for mean the value samples of g-F complementing determined with the the Equation first series(13) was of here tests. about 40%. The results of the calculations carried out along with the determination of the resulting values of the effective electrochemical equivalent of iron k and the average value k of this parameter used , Effective Electrochemical Bar Mass Loss Test Specimen No.in Initialfurther Mass calculations m0, (g) are Finalsummarized Mass m τin(g) Table Bar3. Mass Loss m, (g) Equivalent of Iron mFe2+ , (g) keff, (g/µAYear) Table 3. Comparison of the results of the gravimetric analysis and the calculations of the bar mass P5 239.115 228.485 10.630 16.529 0.005865 loss with the use of Faraday’s law for the samples complementing the first series of tests. P6 236.183 222.040 14.143 18.959 0.006803 P7Test 240.176 Initial 229.675 10.501Effective 15.582 Electrochemical 0.006146 Avg. 238.491Final 226.733 Mass Bar Mass 11.758Bar Mass Loss 17.023 0.006272 Specimen Mass 𝐦𝟎, Equivalent of Iron 𝐦𝛕, (𝐠) Loss 𝐦 𝐠, (𝐠) 𝐦 𝐅𝐞𝟐, (𝐠) No. (𝐠) 𝐤𝐞𝐟𝐟, (g/𝛍𝐀𝐘ear) P5 239.115 228.485 10.630 16.529 0.005865 The experimentalP6 tests236.183 revealed 222.040 that from14.143 the image 18.959 of crack surface 0.006803 after splitting (Figure6a), from a visual inspectionP7 of240.176 the rebar’s229.675 face in10.501 the area of15.582 the applied 0.006146 resin (Figure6b) and from the contaminationAvg. of water238.491 and the226.733 container 11.758 with corrosion 17.023 products (Figure 0.006272 6c) that some part of the corrosion productThe experimental is washed tests out revealed into that the from solution the image and of crack does surface not after accumulate splitting (Figure at 6a), the rebar surface as assumed byfrom the a calculationvisual inspection model. of the rebar’s Failure face in the to area estimate of the applied this resin aspect (Figure would 6b) andlead from the to further errors in the calculationcontamination model. of Towater determine and the container the with mass corrosion of corrosion products (Figure products 6c) that some that part entered of the the solution, corrosion product is washed out into the solution and does not accumulate at the rebar surface as after the completionassumed of by the the calculation tests the model. entire Fail liquidure to estimate was filtered this aspect through would lead a to medium further errors filter in the and the mass of the products wascalculation determined. model. To determine Assuming the mass that of thecorrosion corrosion products productsthat entered the are solution, a mixture after the of Fe(OH) and completion of the tests the entire liquid was filtered through a medium filter and the mass of the 2 ( ) ( ) ( ) Fe OH 3 (the rangeproducts of was values determined. is related Assuming to that the the adoption corrosion products of diff erentare a mixture ratios of ofFe(FeOH)OH and 2 to Fe OH 3), then the mass ofFe iron(OH) ions(the range carried of values away is related to the to the solution adoption is of 35% different to 45%ratios of Fe the(OH total) to Fe weight(OH)), of iron ions in then the mass of iron ions carried away to the solution is 35% to 45% of the total weight of iron ions the corrosion. in the corrosion.

Figure 6. Accumulation of corrosion products of the rebar steel during experimental tests: (a) sample Figure 6. Accumulationbreak in the fracture, of corrosion (b) detachment products of resin offrom the the rebarbar face, steel (c) contaminated duringexperimental water and container tests: (a) sample break in the fracture,with corrosion (b) detachment products. of resin from the bar face, (c) contaminated water and container with corrosion products. Ultimately, the mass of iron ions calculated from Faraday’s law is in fact reduced due to the losses in the electrolysis process, and reduced by the mass of iron ions being part of the corrosion Ultimately,products the mass carried of away iron to ions the solution calculated through fromthe frac Faraday’sture and cracks law in the is resin in fact shell. reduced The said fact due to the losses in the electrolysiswas process,taken into account and reducedin the research by by the introducing mass ofthe ironion mass ions reduction being factor part χ in of the the formulas corrosion products describing computational electrochemical iron equivalent (9), (10) and (12): carried away to the solution through the fracture and cracks in the resin shell. The said fact was taken

Materials 2020, 13, 5375 10 of 17 into account in the research by introducing the ion mass reduction factor χ in the formulas describing computational electrochemical iron equivalent (9), (10) and (12): kcomp = χkeﬀ = χ 1 λg F k. (14) − − Since it was impossible to obtain precise data, the ion mass reduction factor χ was accepted as χ = 0.5, which was adopted in the numerical calculations. The calculations were also made for χ = 0.4 and χ = 0.35. The analysis of optical test results was limited to monitoring the displacement of the surface points of samples A1 and A2 in the plane A and points B1 and B2 in the opposite plane B (Figure7a) in the abovementioned time periods. Figure7b presents the reading of the displacement of the analyzed points in the program GOM Inspect (ZEISS, Oberkochen, Germany) for the test period t = 388 h. The relative elongation of the sample edges A1–A2 and B1–B2 can be identiﬁed as the arithmetic mean of the change of the width of crack openings on these edges of the sample. The measurement results as well as the results of calculations involving the elongation of the sides (widths of crack openings, presented in the further part of the work) at the abovementioned time intervals are presented in Figures 10 and 11. Materials 2020, 13, x 11 of 18

FigureFigure 7. Measurements 7. Measurements of crackof crack opening opening during during experimentalexperimental tests: tests: (a) ( aschematic) schematic of the of thelocation location of of Commented [M4]: Please change dash into minus sign. Is planes and control points, (b) reading of the displacement of points in the program GOM Inspect for planes and control points, (b) reading of the displacement of points in the program GOM Inspect for done! the testing period t = 388 h. the testing period t = 388 h.

4. Numerical4. Numerical Analysis Analysis The FEM computer model of the tested samples was made using the software GiD-ATENA The FEM computer model of the tested samples was made using the software GiD-ATENA (GiD, Center for Numerical Methods in Engineering, Barcelona, Spain – Červenka Consulting, Praha, ˇ (GiD,Czech Center Republi for Numericalc) and Matlab Methods (MathWorks, in Engineering, Natick, United Barcelona, States of Spain– America)Cervenka. The model Consulting, was made Praha, Czechby Republic) the discretization and Matlab of the (MathWorks, sample into solid Natick, spatial United finite elements States of with America). the interface The layer model betw waseen made by thesteel discretization and concrete. of The the support sample method into solid of the spatial specimen ﬁnite and elements the finite withelements the interfacemesh is presented layer between in steel andFigure concrete. 8a. The load The that support corresponded method to of subsequent the specimen increments and the in ﬁnite the volume elements of corrosion mesh is products presented in Figurewas8a. The realized load by that applying corresponded the increments to subsequent of strain increments tensor to in the the rei volumenforcing of bar, corrosion in the plane products was realizedperpendicular by applying to the the axis increments of the reinforcing of strain bar. tensor The to volumetric the reinforcing strain bar, rates in along the plane the axis perpendicular of the to thereinforcing axis of the bar reinforcing that develop bar. at the The initial volumetric phase of strainthe reinforcement rates along corrosion the axis process of the in reinforcing line with bar the Equations (9) were ignored. that develop at the initial phase of the reinforcement corrosion process in line with the Equation (9) Commented [M5]: Does this refer to Equations 9 ? If so, can were ignored. we change it to Equations (9) ? Yes you can! Is done!

Figure 8. Static diagram of the sample adopted for numerical calculations: (a) finite element mesh, (b) boundary condition applied to the test sample – description in the text.

The sum of increments of the volumetric strain tensor produced by the accumulation of corrosion products in line with the Equations (9) and (10) was divided into 12 calculation steps in such a way that the increments of subsequent volumetric strains corresponded to the increments of successive time intervals between the measuring points. The computer simulation was carried out

assuming that the average effective value of the electrochemical iron equivalent k, = 0.006271 μA Year and that the corrosion consisted entirely of iron hydroxide (II)-Fe(OH) or

Materials 2020, 13, 5375 11 of 17 Materials 2020, 13, x 11 of 18

Figure 8. StaticFigure diagram 8. Static diagramof the sample of the sample adop adoptedted for for numerical numerical calculations:calculations: (a) finite(a) finite element element mesh, mesh, (b) (b) boundary condition applied to the test sample–description in the text. boundary condition applied to the test sample–description in the text. The sum of increments of the volumetric strain tensor produced by the accumulation of corrosion The sumproducts of increments in line with the of Equations the volumetric (9) and (10) was stra dividedin tensor into 12 produced calculation steps by inthe such accumulation a way of corrosion productsthat the increments in line with of subsequent the Equations volumetric (9) strains and corresponded (10) was divided to the increments into 12 of successivecalculation steps in time intervals between the measuring points. The computer simulation was carried out assuming that 1 1 such a way thethat average the increments effective value of of thesubsequent electrochemical volu ironmetric equivalent strainskeff,avg corresponded= 0.006271 µA− toYear the− and increments of successive timethat the intervals corrosion consistedbetween entirely the measuring of iron hydroxide points. (II)-Fe The(OH )computer2 or iron hydroxide simulation (III)-Fe( OHwas)3 .carried out The parameters α, ϑ, dependent on the composition of the products, and the assumed thickness and assuming that the average effective value of the electrochemical iron equivalent k, = porosity of the interface transition layer are presented in Table4. 0.006271 μA Year and that the corrosion consisted entirely of iron hydroxide (II)-Fe(OH) or Table 4. Required parameters needed to determine the composition of corrosion products and the iron hydroxide (III)-Fe(OH). The parameters α, ϑ, dependent on the composition of the products, microstructure of the transition layer. and the assumed thickness and porosity of the interface transition layer are presented in Table 4. Parameter Value

Fe(OH)2 : parameter, α (1) 0.523 Table 4. Required parameters neededFe(OH to)2 :determine parameter, ϑ (1) the composition 2.09 of corrosion products and the ( ) α microstructure of the transition layer.Fe OH 3 : parameter, (1) 0.622 Fe(OH)3 : parameter, ϑ (1) 2.24 Porosity of transition layer, ε (1) 0.55 3 Width of transition layer, wws 10 (cm) 7.5 Parameter × Value

Fe(OH): parameter, α (1) 0.523 The increments of corrosion volumetric strains calculated with the Equations (9) and (10) are Fe(OH) : ϑ compiled in the form of graphs in Figure parameter,9a–d. (1) 2.09 The critical time, theFe determination(OH): parameter, methodology ofα which(1) was presented in Figure 0.6223, was deﬁned as the intersection point of two straight lines S1 and S2 approximated with the use of linear regression Fe(OH): parameter, ϑ (1) 2.24 having the slopes ai andPorosity intercepts ofb i.transition Calculations layer, were made ε (1) using the measured values 0.55 of time and average values of the elongation of sample sides. Two values were employed for the calculations, one of which correspondedWidth to the of time transition interval t layer,(0, 48 hwfor the×10 straight(cm) line S1 and the other7.5 being the ﬁrst

5 measured values from the time interval t(48, 216 h for the straight line S2. It was assumed that the The incrementsincrements ofof the corrosion length and volumetric relative elongations strains of the ca samplelculated edge with had athe linear dependencies character. The slopes (9) and (10) are of the straight line and the intercepts determined for the calculation data listed in Tables5 and6 were, compiled inrespectively: the form ofa =graphs0.00052 in, b Figure= 0.125 for9a–d. S1 and a = 0.00276, b = 0.1081 for S2. The determined 1 1 2 2 − critical time, corresponding to the position of the abscissa of the intersection point of the straight lines S1 and S2, was tcr = 53.83 h. The values adopted for the calculations are presented in Tables5 and6.

Materials 2020, 13, x 12 of 18

iron hydroxide (III)-Fe(OH). The parameters α, ϑ, dependent on the composition of the products, and the assumed thickness and porosity of the interface transition layer are presented in Table 4.

Table 4. Required parameters needed to determine the composition of corrosion products and the microstructure of the transition layer.

Parameter Value

Fe(OH): parameter, α (1) 0.523 Fe(OH): parameter, ϑ (1) 2.09 Fe(OH): parameter, α (1) 0.622 Fe(OH): parameter, ϑ (1) 2.24 Porosity of transition layer,  (1) 0.55 Width of transition layer, w × 10 (cm) 7.5

The increments of corrosion volumetric strains calculated with the Equations (9) and (10) are Commented [M6]: Does this refer to Equations 9 and 10? If compiled in the form of graphs in Figure 9a–d. so, can we change it to Equations (9) and (10)? Yes you can! Is

done! Materials 2020, 13, 5375 12 of 17

FigureFigure 9. 9.Coordinates Coordinates of theof the tensors tensors of e ﬀ ofective effective increments increments of volumetric of volumetric strains strainsproduced produced by corrosion by productscorrosion in products reinforced in concretereinforced samples: concrete ( asamples:) P1, (b) ( P2,a) P1, (c) ( P3,b) P2, (d) ( P4.c) P3, (d) P4. Commented [TK7]: The figure was changed.

The critical time, the determination methodology of which was presented in Figure 3, was definedTable as 5. theCalculation intersection parameters point of to two determine straight the lines slope S1 and and the S2 intercept approximated in the simple with regressionthe use of S1.linear regression having the slopes a and intercepts b . Calculations were made using the measured Measurement Point i 1 2 values of time and average values of the elongation of sample sides. Two values were employed for the calculations, one of whichMeasurement corresponded time to tithe, (h )time interval tϵ (024, 48⟩ h for 48 the straight line S1 Average elongation of sample side ∆L, (mm) 0.025 0.038 and the other being the first 5 measured values from the time interval tϵ(48, 216⟩ h for the straight line S2. It was assumed that the increments of the length and relative elongations of the sample edge had Table a linear 6. Calculation character. parameters The slopes to determine of the straight the slope line and theand intercept the intercepts in the simple determi regressionned for S2. the calculation data listedMeasurement in Tables Point 5 and i 6 were, respectively: 1 a = 20.00052 3, b = 0 4.125 for 5 S1 and a = 0.00276, b = −0.1081 for S2. The determined critical time, corresponding to the position of Measurement time ti, (h) 125 144 168 192 216 Average elongation of sample side ∆L, (mm) 0.219 0.298 0.360 0.406 0.496

The computer analysis of the concrete sample was performed using the ATENA system (Cervenkaˇ Consulting, Praha, Czech Republic). The elastic-plastic-fracture material model with the interface steel–concrete connection, which makes use of the Rankin material model [27] in the field of tension, and the Menetrey–Willam model [27] in other cases was applied. The nonlinear geometry was turned on in the analysis. The material model defined in the system as Cc3DNonLinCementitious2 was applied. Strength parameters of the concrete were determined in line with the ATENA program manual for average cube strength under compression fc,cube = 65.89 MPa. In addition, to improve the stability of the solution, the possibility of aggregate interlock and the shear factor were taken into account. The steel bar was defined by adopting elasto-plastic Huber-Misses-Hencky material [27]. The material parameters of steel and concrete (after the conversion from cube strength) and the interface layer used in the calculations are presented in Tables7–9, respectively. The main role of the steel–concrete interface was to stabilize the results. Materials 2020, 13, 5375 13 of 17

Table 7. Concrete material data for FEM calculations.

Material Property Value Modulus of elasticity E, (GPa) 38.28 Poisson’s ratio v, (1) 0.2 Medium tensile strength fctm, (MPa) 3.99 Medium compressive strength fcm, (MPa) 56.4 Cracking energy GF, (kN/m) 0.151 Maximum diameter of aggregate grains dmax, (mm) 8 Active shear factor, (1) 20

Table 8. Interface layer material data for FEM calculations.

Material Property Value 3 8 Stiffness in normal direction knn, MN/m 2 10 × 3 8 Stiffness in tangential direction ktt, MN/m 2 10 × Interface tensile strength fint, (MPa) 0.3 Friction coefficient µ, (1) 0.1 Cohesion coefficient c, (MPa) 1

Table 9. Steel material data for FEM calculations.

Material Property Value Modulus of elasticity E, (GPa) 200 Poisson’s ratio v, (1) 0.3 2 Steel yield strength fy, N/mm 235

The assessment of the compliance of computer calculations with experimental tests involved the average value of the elongation of the wall edge perpendicular to the main direction of the sample crack (the calculated values of ∆LC1D1 and ∆LC2D2 specified in Figure8 were compared with the measured values of ∆LA1A2 and ∆LB1B2—Figure7) as well as the average calculated width of the crack at the edges C1D1 and C2D2 of the test element at Gauss points of the finite elements located in the vicinity of points R1 and R2 (Figure8), which were compared with the measured value of average width of the crack at the edges A1A2 and B1B2. The results obtained from the computer simulations and from optical measurements are presented in Figures 10 and 11. When analyzing the results obtained from experimental research and computer simulations, agreement in respect of the quality of results and their quantitative estimation can be seen. The assessment of the compliance of research results with the results of theoretical calculations is complex due to large fluctuations in the research results which map the elongation of the sides and width of the crack of the sample and due to the small sample size from the viewpoint of statistical surveys. In the case of samples with an assumption of 50% reduction in impact intensity (χ = 0.5) caused by the flushing out of corrosion products, it can lead to the overestimation of the value offered by the mathematical model, which may be caused by the underestimation of the amount of corrosion products carried away by the solution. The results involving the elongation of the sides of the sample Materials 2020, 13, x 14 of 18

In order to assess the correlation of results obtained with the use of optical measurements and the computer model, a number of computer simulations were performed. The results obtained from the computer calculations were compared with the results of optical tests, and they are presented in Figure 8. As mentioned in point 3, in the calculations, allowance was made for a 50% share of corrosion products in relation to the value of corrosion strains obtained with the theoretical approach (χ = 0.5). For comparison, due to the discrepancy of results, χ = 0.4 and χ = 0.35 was also analyzed. The assessment of the compliance of computer calculations with experimental tests involved the average value of the elongation of the wall edge perpendicular to the main direction of the sample crack (the calculated values of ΔL and ΔL specified in Figure 8 were compared with the measured values of ΔL and ΔL—Figure 7) as well as the average calculated width of the crackMaterials at the2020 edges, 13, 5375 C1D1 and C2D2 of the test element at Gauss points of the finite elements14 located of 17 in the vicinity of points R1 and R2 (Figure 8), which were compared with the measured value of average width of the crack at the edges A1A2 and B1B2. The results obtained from the computer as well as the width of the crack opening in the test element diﬀer from the results of experimental tests. simulationsYet, the qualitative and from compliance optical measurements in the assessment are presented of the elongation in Figure ands 10 crackingand 11. of sample sides is noticeable both at the activation and local degradation stage of the propagation process of mechanical damage in the cover within the time range of 0–48 h as well as at further stages of the cracking process of the sample due to reinforcement corrosion.

Figure 10. Average elongation values of edges ΔL and ΔL of the reinforced concrete sample Figure 10. Average elongation values of edges ∆LA1A2 and ∆LB1B2of the reinforced concrete sample Commented [M8]: Please add spaces before and after equal together with calculation results for the 50%, 60% and 65% reductions of the impact intensity of together with calculation results for the 50%, 60% and 65% reductions of the impact intensity of sign and plus sign. Is done! corrosioncorrosion products products (λ (λ = 0.5,0.5, λ == 0.4, λλ= =0.35): 0.35): ( a(a)) sample sample P1, P1, (b )(b sample) sample P2, P2, (c) sample(c) sample P3, ( dP3,) sample (d) sample P4. P4. Due to the experimental test result deviation in respect of the results of calculation for the 50% share of corrosion products in the degradation of the cover, an attempt was made in the next phase to assess the elongation of the sides of the elements and the width of crack openings in the sample in which 60% and 65% of the corrosion products were carried away to the electrolyte solution (40% and 35% of the load presented in Figure9 is treated as the load of the concrete sample, χ = 0.4 and χ = 0.35). There is a rational premise that some proportion of the corrosion products at the measurement stage was washed out from the sample, as mentioned in Section3 of the work. When analyzing the results of the experimental research and computer simulations obtained for test samples P1, P2, P3, agreement as to the quality of the results and their quantitative estimation can be observed. The results of experimental

tests on the elongation of the sides of the sample, along with the width of the crack, are between the theoretically determined elongations for the assumed extreme compositions of corrosion products in the form of pure iron hydroxides II and III. There is a quantitative and qualitative compatibility of the elongations of the test element sides both at the activation and local degradation stage of the

propagation process of mechanical damage in the cover (time (0, 48 h) as well as at further cracking stages of the corrosion propagation process. The qualitative compliance of the calculations and experimental tests at the initial stage of the corrosion process is caused by the fact that the calculations allowed for the eﬀects connected with the sealing up of free spaces and the propagation of micro-cracks in the transition layer, so-called CAR-cumulative accommodation region through the function β [13]. Materials 2020, 13, 5375 15 of 17

The discrepancies in the results of experimental tests and computer simulations are noticeable in relation to sample P4, although these discrepancies can be explained by more intense corrosion that occurred at the ends of the rebar (intense corrosion that developed under the polyester resin that protects the rebar face). Materials 2020, 13, x 15 of 18

Figure 11. Average values of crack width at points of the elements’ edges on the sections A1A2 and Figure 11. Average values of crack width at points of the elements’ edges on the sections A1A2 and Commented [M9]: Please add spaces before and after equal B1B2B1B2 of the reinforcedreinforced concrete concrete sample sample at pointsat points R1 andR1 and R2 together R2 together with thewith calculation the calculation results results for the for 50%, 60% and 65% reductions of the impact intensity of corrosion products (λ = 0.5, λ = 0.4, λ = 0.35): sign and plus sign. Is done! the 50%, 60% and 65% reductions of the impact intensity of corrosion products (λ = 0.5, λ = 0.4, λ = (a) sample P1, (b) sample P2, (c) sample P3, (d) sample P4. 0.35): (a) sample P1, (b) sample P2, (c) sample P3, (d) sample P4. 5. Conclusions When analyzing the results obtained from experimental research and computer simulations, The paper presents a model that allows for the analysis of concrete cover cracking, taking into agreement in respect of the quality of results and their quantitative estimation can be seen. The account the stages of the corrosion process, the time of tightening the corrosion products in the assessment of the compliance of research results with the results of theoretical calculations is transition layer t , and the eﬀective electrochemical equivalent of the reinforcing steel. The model complex due to largecr fluctuations in the research results which map the elongation of the sides and can capture both the case of uniform and nonuniform corrosion on the circumference of the rebar. width of the crack of the sample and due to the small sample size from the viewpoint of statistical The obtained results of computer calculations were experimentally veriﬁed. χ surveys.Summing In the upcase the of obtained samples research with an results assumption and the of results 50% reduction of computer in impact simulations, intensity a number ( = 0.5) causedof conclusions by the involving flushing outthe usefulness of corrosion of the products, presented it approach can lead in to assessing the overestimation the opening of width the ofvalue offeredcracks inby reinforced the mathematical concrete elementsmodel, which can be may formulated. be caused by the underestimation of the amount of corrosion products carried away by the solution. The results involving the elongation of the sides of the5.1. sample Conclusion as well No. 1as the width of the crack opening in the test element differ from the results of experimentalThe results tests. obtained Yet, the from qualitative numerical compliance tests are qualitatively in the assessment and quantitatively of the elongation similar to theand results cracking ofof sample experimental sides research.is noticeable This complianceboth at thecan activation be observed and forlocal all stagesdegradation characteristic stage of thethe processpropagation of processreinforcing of mechanical bar corrosion damage in concrete in the and cover involves within the lack the oftime mechanical range of impact 0–48 hat as the well initial as stage at further of stagescorrosion of the for cracking the time processt t , the of time the sample interval due until to critical reinforcement concentration corrosion. time t cr is reached, and the ≤ 0 timeDue interval to the in whichexperime fullntal interaction test result is taking deviation place in between respect the of corrosionthe results products of calculation and the concretefor the 50% shareof the of cover corrosiont tcr products. Very important in the degradation for the correctness of the cover, of the an obtained attempt resultswas made is both in the the next correct phase ≥ to assess the elongation of the sides of the elements and the width of crack openings in the sample in which 60% and 65% of the corrosion products were carried away to the electrolyte solution (40% and 35% of the load presented in Figure 9 is treated as the load of the concrete sample, χ = 0.4 and χ = 0.35 ). There is a rational premise that some proportion of the corrosion products at the measurement stage was washed out from the sample, as mentioned in Section 3 of the work. When analyzing the results of the experimental research and computer simulations obtained for test samples P1, P2, P3, agreement as to the quality of the results and their quantitative estimation can be observed. The results of experimental tests on the elongation of the sides of the sample, along with the width of the crack, are between the theoretically determined elongations for the assumed extreme compositions of corrosion products in the form of pure iron hydroxides II and III. There is a

Materials 2020, 13, 5375 16 of 17 estimation of the critical time, which includes the condition of the cover tightening, as well as defining the effective value keff of the electrochemical equivalent of iron. Incorrect assumption of this parameter may lead in accordance with Equation (3) to large computational discrepancies, which was emphasized in Section2 of this paper.

5.2. Conclusion No. 2

The compliance of the obtained test results is particularly important in the area of t(0, tcr). The fact that the function β was ignored in the calculation leads, in the case of theoretical calculations, to a rapid increment of displacements and propagation of damage in the element, which has a very negative impact on the accuracy of the obtained results and leads to qualitative and quantitative incompliance of the obtained computer simulation results. Similarly, the incorrect assessment of the critical time tcr, has a negative impact on the accuracy of the results.

5.3. Conclusion No. 3 The problem involving the washing out of corrosion products was adapted by way of estimation, but this did not aﬀect the qualitative assessment of the obtained results. The quantitative results obtained with the 40% and 35% attempt at eﬀective increments of corrosion strains are viewed as a satisfactory result.

Author Contributions: Conceptualization, T.K. and T.J.; methodology, T.K. and T.J. and F.R.; software, T.K. and F.R.; validation, T.K. and F.R.; formal analysis, T.K. and F.R. and M.K.; investigation, T.J. and F.R. and M.K. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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