Crack Propagation Analysis Due to Rebar Corrosion

Home , Rebar

Fracture Mechanics of Concrete and Concrete Structures - Assessment, Durability, Monitoring and Retrofitting of Concrete Structures- B. H. Oh, et al. (eds) ⓒ 2010 Korea Concrete Institute, Seoul, ISBN 978-89-5708-181-5

Crack propagation analysis due to rebar corrosion

H. Nakamura, K.K.Tran, K.Kawamura & M. Kunieda Department of Civil Engineering, Nagoya University, Japan.

ABSTRACT: Cracking behavior due to rebar corrosion for concrete specimen with single rebar was evalu- ated experimentally and analytically. In the experiment, the propagation of crack was observed by the electric corrosion test. Three-dimensional Rigid- Body-Spring-Method (RBSM) with three phase material corrosion expansion model was applied to simulate internal cracking patterns, surface crack width propagation and internal crack propagation. The effect of corrosion products properties and the penetration of corrosion products into cracks during the corrosion process were investigated and cracking behavior due to rebar corrosion was simulated reasonably. As the results, the mechanism of surface crack width propagation with internal crack propagation was clarified.

1 INTRODUCTION cracks, and the effect of internal cracks to the surface crack width and so on are clarified. Cracking of concrete due to rebar corrosion is one of the major deterioration behaviors and it causes the spalling of concrete cover or acceleration of deterio- 2 EXPERIMENTAL STUDY ration. It is necessary to predict the internal damage from the observable surface condition during main- 2.1 Testing method tenance process. Therefore, it is desirable to estab- Dimensions of the specimen with single rebar are lish a prediction method to quantitatively assess the shown in Figure 1. Two types of specimen are car- internal crack propagation behavior and rebar corro- ried out. Type C30 has concrete cover thickness of sion rate from surface cracks. The internal crack pat- 30mm. Type C10 has 10mm cover thickness. There terns due to rebar corrosion considering the differ- are six specimens in type C30 and one specimen in ence of cover thickness or diameter of reinforcing type C10. bar were clarified (Andrade et al., 1993, Cabrera 1996). However, the relationships between propagation of surface and internal cracks as well as pro- Crack width measurement gress of surface crack width have not been clear. C=30 Cutting section 85 80 85 In this study, the crack propagation behavior is C=10 30 80 85 investigated both analytically and experimentally. In the experiment, the propagation of crack is observed by the electric corrosion test. Then, surface crack 150 D19 D19 widths are measured and internal crack patterns are 250 observed at several rebar corrosion rates. On the oth- 300 er hand, the crack propagation behavior is simulated 150 Unit=mm using Rigid-Body-Spring-Method with three- Figure 1. Specimen setup. dimensional Voronoi particles. In the analysis, the three phase material model which consists of rebar, In order to accelerate the corrosion process, an corrosion products layer and concrete is proposed. external direct electric current density 900µA/cm2 is For the corrosion products layer, the stiffness of cor- applied after 32 days from the casting. Schematic of rosion products is considered by initial strain prob- electric corrosion test is shown in Figure 2. lem due to corrosion expansion. Moreover, the effect At the testing time, concrete compressive strength of penetration of corrosion products into cracks dur- and tensile strength are 18.5 MPa and 1.53 MPa re- ing corrosion process is considered. The applicabil- spectively. During the test, surface crack widths are ity of the model is verified by comparing with the recorded using crack gauges and crack widths are test results. As the results, the progress of surface logged in computer files. Testing time is varied for crack width, the propagation behavior of internal six specimens of type C30 to investigate the effect of corrosionJ = −D(h, Trate)∇h to crack patterns and crack width (1) theirexplicitly lengths accounts increase. for With the furtherevolution amount of hydrationof corro- propagation. When the test has been finished, spe- sionreaction products, and SFinside content. crack initiatesThis sorption and propagates isotherm cimensThe areproportionality cut off along coefficient the positions D(h,T) shown is incalled Fig- insidereads the specimen from the rebar. uremoisture 1 to observe permeability internal and crack it is patterns a nonlinear and measurefunction internalof the relative crack widthshumidity and h lengths. and temperature T (Bažant ⎡ ⎤ Vertical⎢ crack1 ⎥ & Najjar 1972). The moisture mass balance requires w ()()h,α ,α = G α ,α 1 − + that the variation in time of the water mass per unit e c s 1 c s ⎢ Lateral crack∞ ⎥ - ⎢ 10(g α − α )h ⎥ volume of concrete (water+ content w) be equal to the ⎣ e 1 c c ⎦ Inside crack (4) divergence of the moisture flux J ⎡ 10(g α ∞ − α )h ⎤ K ()α ,α ⎢e 1 c c − 1⎥ ∂w reinforcement 1 c s ⎢ ⎥ − = ∇ • J reinforcement (2) ⎣ ⎦ ∂t concrete Figure 4. Internal crack types.

where the first term (gel isotherm) represents the The water content w can be expressed as the sum physically bound (adsorbed) water and the second of the evaporableWooden packer waterCopper w plate (capillaryNaCl 3% water, water e term (capillary isotherm) represents the capillary Figurevapor, 2. and Schematic adsorbed of corrosion water) test. and the non-evaporable water. This expression is valid only for low content (chemically bound) water wn (Mills 1966, 2 of SF. The coefficient G represents the amount of PantazopouloCorrosion rate& WMills(mg/cm 1995)). duringIt is reasonablethe testing tois 1 r water per unit volume held in the gel pores at 100% calculatedassume that by thedividing evaporable mass losswat erW(mg) is a byfunction wet sur- of 2 2 2 2 relative108mg/cm humidity, and435mg/cm it can be expressed749mg/cm (Norling facerelative of rebar humidity, (cm ). h , degree of hydration, α , and c Mjornell 1997) as degree of silica fume reaction, α , i.e. w =w (h,α ,α ) Surface crack width Surface crack width Surface crack width W s e e c s 0.35mm 0.78mm 1.11mm W= r =age-dependent sorption/desorp tion isotherm (1) (NorlingπD ⋅MjonellL 1997). Under this assumption and Figure() 5. Type C30c crack patterns.s (5) G α ,α = k α c + k α s by substituting Equation 1 into Equation 2 one 1 c s vg c vg s where D and L are bar diameter (cm) and bar length In the case of 10mm cover thickness (type C10), obtains (cm) respectively. two cracksc appears on concrete surface and propagate Mass loss W(g) is calculated by using the follow- where k vg and k vg are material parameters. From the tomaximum the rebar. amount Then, lateralof water cracks per unit occur volume when thatamount can ing∂ wempirical∂h formula based∂w on test∂w results as shown of corrosion products increase as shown in Figure 6. in− Figuree +3:∇ • (D ∇h) = e α& + e α& + w& (3) fill all pores (both capillary pores and gel pores), one ∂h ∂t h ∂α c ∂α s n c s can calculate K1 as one obtains W = 0.235I ⋅T (I.T < 66) (2) W = 0.617I ⋅T - 25.305 (I⋅T > 66) ⎡ ⎛ ∞ ⎞ ⎤ where ∂we/∂h is the slope of the sorption/desorption 10⎜ g α −α ⎟h ⎢ ⎝ 1 c c ⎠ ⎥ isotherm (also called moisture capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ where I and T are current intensity (A) and testing 0 c s 1⎢ ⎥ governing equation (Equation 3) must be completed ⎣ ⎦ (6) time (hour) respectively. K ()α ,α = by appropriate boundary and initial conditions. 1 c s ⎛ ∞ ⎞ 10⎜ g α −α ⎟h The relation between the amount of evaporable e ⎝ 1 c c ⎠ −1 water and120 relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity The material parameters kc and ks and g can 100 610mg/cmvg 2 vg 1 humidity and ‘‘desorptionW=0.617× isotherm”IT-25.305 in the opposite be calibrated by fitting experimental data relevant to case. Neglecting80 their difference (Xi et al. 1994), in free (evaporable) water content in concrete at Figure 6. Type C10 crack pattern. the following,60 ‘‘sorption isotherm” will be used with various ages (Di Luzio & Cusatis 2009b). reference to both sorption and desorption conditions. By the way,40 if the hysteresis of the moisture 2.3 Surface crack width propagation Mass loss (g) loss Mass Experiment 2.2 Temperature evolution isotherm 20would be taken into account, two different relation, evaporable W=0.235water vs× relativeIT humidity, must InNote the that, case at ofearly 30mm age, coversince thethickness chemical (type reactions C30), be used according0 50 to the100 sign150 of the200 variation250 of the propagationassociated with of surface cement crack hydration width againstand SF corrosion reaction rates at different rebar mass loss levels is shown in relativity humidity.Accumulated The current shape amount of (A.hr)the sorption are exothermic, the temperature field is not uniform Figureisotherm 3. Rebar for HPCmass lossis infl computation.uenced by many parameters, Figurefor non-adiabatic 7. It can be systemsseen that even opening if the of environmental surface crack especially those that influence extent and rate of the initiatestemperature after is aconstant. significant Heat amount conduction of corrosion can be products are formed. It can be explained that a criti- 2.2chemical Experimental reactions crack and, patterns in turn, determine pore described in concrete, at least for temperature not structure and pore size distribution (water-to-cement calexceeding corrosion 100°C amount (Bažant of corrosion & Kaplan products 1996 is), re-by Figureratio, cement4 indicates chemical the defined composition, internal SFcrack content, types quiredFourier’s to law,build which up enough reads expansion stress and to showncuring intime crack and patterns. method, temperature, mix additives, cause cracking in concrete as B.H.Oh discussed Internal crack patterns at several corrosion rates about the value (Oh et al. 2009). After the initiation, etc.). In the literature various formulations can be q = −λ∇T (7) of type C30 are shown in Figure 5. surface crack width propagates rapidly up to the val- found to describe the sorption isotherm of normal concreteInitiation (Xi of et visible al. 1994). crack However, occurs on inthe the surface present of ue of 0.4mm. After that, the speed of propagation concrete cover (vertical crack). When corrosion reduceswhere qwith is occurrencethe heat offlux, lateral T cracksis the and absolute effect paper the semi-empirical expression proposed by temperature, and λ is the heat conductivity; in this productsNorling increase,Mjornell the(1997) crack is propagatesadopted becauseinside the it of penetration of corrosion products into cracks concrete cover. After that, lateral cracks appear and which is discussed in the later part of the present pa-

Proceedings of FraMCoS-7, May 23-28, 2010 per. Moreover, the speed of propagation increases with the tension Jand= − Dcompression(h,T )∇h caps. The shear (1) explicitly accounts for the evolution of hydration again with the propagation of the inside crack. fracture criterion is expressed as follows (Saito. reaction and SF content. This sorption isotherm reads 1.6 1999): The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function 1.4 τ 2 ≥ 1 of the relative humidity h and temperature(3) T (Bažant ⎡ ⎤ 2 ⎢ 1 ⎥ 1.2 τ f & Najjar 1972). The moisture mass balance requires w ()()h,α ,α = G α ,α 1 − + that the variation in time of the water mass per unit e c s 1 c s ⎢ ∞ ⎥ 1 ⎢ 10(g α − α )h ⎥ where volume of concrete (water content w) be equal to the ⎣ e 1 c c ⎦ (4) 0.8 divergence of the moisture flux J ⎧ ' ⎡ 10(g α ∞ − α )h ⎤ 0.6 ⎪ c −σ tanφ, for σ ≥ 0.5 f c ⎢ 1 c c ⎥ τ f = ⎨ ' (4) K ()α ,α e − 1 ⎪c − 0.5 f tanφ, ∂forw σ < 0.5 f c 1 c s ⎢ ⎥ 0.4 ⎩ c − = ∇ • J (2) ⎣ ⎦ ∂t Surface crack width (mm) width crack Surface 0.2 Rebar is modeled as linear elastic. where the first term (gel isotherm) represents the The water content w can be expressed as the sum physically bound (adsorbed) water and the second 0 200 400 600 800 Zeroof thesprings evaporable water we (capillary water, water 2 term (capillary isotherm) represents the capillary Corrosion rate (mg/cm ) vapor, and adsorbed water) and the non-evaporable water. This expression is valid only for low content Figure 7. Type C30 surface crack width propagation. (chemically bound) water w (Mills 1966, n of SF. The coefficient G represents the amount of Pantazopoulo & Mills 1995). It is reasonable to 1 water per unit volume held in the gel pores at 100% assume that the evaporable water is a function of relative humidity, and it can be expressed (Norling 3 ANALYTICAL MODEL relative humidity, h, degree of hydration, α , and c Mjornell 1997) as degree of silica fume reaction, α , i.e. w =w (h,α ,α ) s e e c s 3.1 Three- dimensional RBSM Tria ngle center point= Nuclearage-dependentVoronoi sorption/desorp particles tion isotherm c s (Norling Mjonell 1997). Under this assumption and G ()α ,α = k α c + k α s (5) Rigid- Body-Spring-Method (RBSM) is one of the Figure 8. RBSM Voronoiby substitutingpolygons. Equation 1 into Equation 2 one 1 c s vg c vg s discrete approaches, which are used as structural obtains σ c s analysis, since it is easy to deal with crack propaga- σ where k vg and k vg are material parameters. From the f f t f = 0.25 f tion of concrete directly. The method represents a c’ tl t maximum amount of water per unit volume that can ∂w ε =∂0.w75G / f h ∂w continuum material as an assemblage of rigid parti- e ∂h tl eF t e ε = 5.0G / f &h & & fill all pores (both capillary pores and gel pores), one − + ∇ • (D ∇h) =tu F αt c + α s + wn (3) cle elements interconnected by zero springs along ∂h ∂t hE ∂α ∂α can calculate K as one obtains G / h 1 fc GFt/h c s their boundaries (Fig. 8). In this study, three- Ec ft1 Ec ε dimensional RBSM is applied. Each element has six ε ε ε εt0 εt1 εtu ε 0 / 2 0 tu ⎡ ⎛ ∞ ⎞ ⎤ where ∂we/∂h is the slope of the sorption/desorption 10⎜ g α −α ⎟h degrees of freedom at center points. Boundary be- ⎢ ⎝ 1 c c ⎠ ⎥ Concrete compressiveisotherm model (alsoConcrete called tensile modelmoisture capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ tween two elements is divided into triangles formed 0 c s 1⎢ ⎥ governing equation (Equation 3) must be completed ⎣ ⎦ (6) by the center and vertices of the boundary. At each c= 0.138f’c K ()α ,α = by appropriate boundaryo and initial conditions. 1 c s ⎛ ∞ ⎞ center point of triangle, three springs, one normal φ= 37 10⎜ g α −α ⎟h The relation between the amount of evaporable ⎝ 1 c c ⎠ and two shear springs are set. The analytical model e −1 water and relative humidity is called ‘‘adsorption is divided into elements using Voronoi random isotherm” if measured with increasing relativity The material parameters kc and ks and g can polygons. In RBSM model, crack widths can be ’’ vg vg 1 humidity and ‘‘desorption isotherm” in the opposite be calibrated by fitting experimental data relevant to automatically measured during the analysis. The Concrete shearcase. model- Neglecting Morh-Coulomb their criterion difference (Xi et al. 1994), in free (evaporable) water content in concrete at with tension and compression caps three-dimensional model is possible to simulate the following, ‘‘sorption isotherm” will be used with various ages (Di Luzio & Cusatis 2009b). complicated problems. Figure 9. Concrete materialreference model. to both sorption and desorption conditions. By the way, if the hysteresis of the moisture 2.2 Temperature evolution 3.2 Material model 3.3 Corrosion expansionisotherm model would be taken into account, two different relation, evaporable water vs relative humidity, must Note that, at early age, since the chemical reactions Modeling expansion of corrosion products is shown Figure 9 shows concrete material models which are be used according to the sign of the variation of the associated with cement hydration and SF reaction in Figure 10. Three phase material model including used in analysis. relativity humidity. The shape of the sorption are exothermic, the temperature field is not uniform rebar, corrosion products and concrete is applied. In the compressive model, f’c is compressive isotherm for HPC is influenced by many parameters, for non-adiabatic systems even if the environmental The merit of the model is that the properties of cor- strength of concrete, Gfc compressive fracture en- especially those that influence extent and rate of the temperature is constant. Heat conduction can be rosion products such as thickness (H) and elastic ergy, Ec is Young modulus of concrete. chemical reactions and, in turn, determine pore described in concrete, at least for temperature not modulus (E ) are assumed directly. The model is ef- The tensile behavior of concrete up to the r structure and pore size distribution (water-to-cement exceeding 100°C (Bažant & Kaplan 1996), by ficient to investigate the effect of corrosion products. strength is modeled by using linear elastic. While bi- ratio, cement chemical composition, SF content, Fourier’s law, which reads linear softening branch of 1/4 model is assumed af- In the model, initial strain is gradually increased in curing time and method, temperature, mix additives, ter cracking as shown, in which f is tensile strength, normal springs located on boundary of the corrosion t etc.). In the literature various formulations can be q = −λ∇T Gft is tensile fracture energy and h is distance be- products layer based on initial strain problem. (7) found to describe the sorption isotherm of normal tween centers of Voronoi elements. Ludgren K.(2002),concrete Oh (Xiet al. et (2009)al. 1994). suggested However, a in the present Tangential springs represent the shear transfer- model of deformation around rebar due to corrosion where q is the heat flux, T is the absolute paper the semi-empirical expression proposed by temperature, and λ is the heat conductivity; in this ring mechanism of concrete. The shear strength is products which canNorling be described Mjornell as shown(1997) in isFigure adopted because it assumed to follow the Morh- Coulomb type criterion 11, in which r, x, U and Ucor are initial radius of re-

Proceedings of FraMCoS-7, May 23-28, 2010 bar,J = −corrosionD(h,T )∇h penetration depth, free increase of (1)the crackexplicitly patterns accounts and surfacefor the crack evolution width ofpropagation hydration radius and the real increase of the radius respec- inreaction comparison and withSF content.the experimental This sorption results. isotherm tively.The The proportionality strain in the corrosioncoefficient products D(h,T) is is called reads moisture permeability and it is a nonlinear function U cor −U ofε cor the= relative= humidityε real − ε free h and temperature T (Bažant (5) 4 ANALYTICAL RESULTS⎡ ⎤ x +U ⎢ 1 ⎥ & Najjar 1972). The moisture mass balance requires w ()()h,α ,α = G α ,α 1 − + thatIn the RBSM variation model, in time initial of thethickness water massof corrosion per unit e c s 1 c s ⎢ ∞ ⎥ In the analysis, mesh sizes⎢ of10 Voronoi(g α − α particles)h ⎥ are productsvolume of layer concrete is modeled (water constantly content w )as be H equal so to the 5mm in the cover area and⎣ 10mme 1 inc the cothers.⎦ An-(4) divergence of the moisture flux J U other arrangement of Voronoi⎡ 10(g αparticles∞ − α ) his also⎤ tried ε free = (6) 1 c c H to confirm the similarity K ()α ,α of⎢e analytical results.− 1⎥ ∂w 1 c s ⎢ ⎥ − This= ∇ parameter• J is input data in the analytical pro-(2) ⎣ ⎦ ∂t gram through increasing of corrosion rate. U is com- 4.1 Simulation of internal crack patterns where the first term (gel isotherm) represents the putedThe from water the content corrosion w can rate be as expressed the following as the equa- sum Analytical crack patterns at several values of the sur- physically bound (adsorbed) water and the second tionof the (Matsuo evaporable et al. 1997)water w (capillary water, water face crack widths are obtained for type C30 speci- e term (capillary isotherm) represents the capillary vapor,W (anddV − 1adsorbed) water) and the non-evaporable mens. The analysis is done with two values of the li- U = r (7) water. This expression is valid only for low content (chemically bound) water wn (Mills 1966, near elastic modulus of corrosion products 500MPa ρs of SF. The coefficient G represents the amount of Pantazopoulo & Mills 1995). It is reasonable to 1 andwater 100MPa per unit to volume compare held the ineffect the gelof this pores property at 100% to assume that the evaporable water 2is a function of where Wr is corrosion rate (mg/cm ),dV is volume internalrelative crackhumidity, patterns. and itFigure can be12 expressedshows the (Norlinginternal relative humidity, h, degree of hydration, α , and expansion ratio of corrosion products (=2.5, inc this crackMjornell patterns 1997) whenas Er is 500MPa and Figure 13 degree of silica fume reaction, α , i.e.3 w =w 3(h,α ,α ) study), ρs is rebar density (=7.85x10s mg/cme e ). c s shows the patterns when Er is 100MPa. = age-dependent sorption/desorption isotherm c s (NorlingConcrete Mjonell 1997). Under this assumption and G ()α ,α = k α c + k α s (5) by substituting Equation 1 into Equation 2 one 1 c s vg c vg s obtainsCorrosion where kc and ks are material parameters. From the products vg vg maximum amount of water per unit volume that can ∂w ∂h ∂w ∂w − e + ∇ • (D ∇h) = e α& + e α& + w& (3) fill all pores (both capillary pores and gel pores), one ∂h ∂Rebart h ∂α c ∂α s n can calculate K as one obtains c s 1

Expansion ⎡ ⎛ ∞ ⎞ ⎤ where ∂we/∂h is the slope of the sorption/desorption Figure 12. Type C30 crack patterns (Er=500Mpa10⎜ g α ).− α ⎟h strain ⎢ ⎝ 1 c c ⎠ ⎥ isotherm (also called moisture capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ 0 c s 1⎢ ⎥ governing equation (Equation 3) must be completed ⎣ ⎦ (6) Figure 10. RBSM corrosion expansion model. K ()α ,α = by appropriate boundary and initial conditions. 1 c s ⎛ ∞ ⎞ 10⎜ g α −α ⎟h The relation between the amount of evaporable e ⎝ 1 c c ⎠ −1 water and relative humidity is called ‘‘adsorption c s isotherm” if measured Uwith increasing relativity The material parameters k vg and k vg and g1 can humidity and ‘‘desorption isotherm” in the opposite be calibrated by fitting experimental data relevant to case. Neglecting their difference (Xi et al. 1994), in free (evaporable) water content in concrete at Ucor the following, ‘‘sorption isotherm”r will be used with various ages (Di Luzio & Cusatis 2009b). referenceInitia to l rebaboth r sorption and desorption conditions. Figure 13. Type C30 crack patterns (Er=100Mpa). By theradius way, if the hysteresis of the moisture 2.2 Temperature evolution isotherm would be takenx into account, two different relation, evaporable water vs relative humidity, must NoteWith that, the at different early age, values since ofthe the chemical elastic reactionsmodulus be used according to the sign of the variation of the Eassociatedr, the crack with patterns cement are hydration different and but SFthe reaction differ- relativity humidity. The shape of the sorption encesare exothermic, are small. theIn comparisontemperature with field the is notexperimen- uniform isotherm for HPC is influenced by many parameters, talfor resultsnon-adiabatic in Figure systems 5, the valueeven if500 the MPa environmental can simu- especially those that influence extent and rate of the latetemperature better crack is constant. patterns ofHeat the conductioninside crack can when be Figurechemical 11.ifidbdi Deformation reactions aroundand, rebarduein turn, to corrosiondetermine products. pore thedescribed surface incrack concrete, width isat 0.78mmleast for and temperature 1.11mm. Thenot structure and pore size distribution (water-to-cement crackexceeding patterns 100°C of vertical (Bažant crack & and Kaplan lateral 1996cracks), areby ratio,It may cement not be chemical easy to measure composition, properties SF ofcontent, corro- similar.Fourier’s law, which reads sioncuring products time and such method, as thickness temperature, and mixlinear additives, elastic In the case of type C10, we also analyze speci- modulus by experiment. We have tried to analyze men with the two values 500MPa and 100MPa of etc.). In the literature various formulations can be q = −λ∇T (7) specimens with various values of the initial thick- the elastic modulus Er. Analytical crack patterns at found to describe the sorption isotherm of normal nessconcrete and the(Xi linearet al. elastic1994). modulusHowever, of inthe the corrosion present surface crack width 0.5mm are shown in Figure 14. products layer. As the results, the initial thickness Aswhere well qas isin thethe caseheat offlux, type TC30 is specimen,the absolute the paper the semi-empirical expression proposed by temperature, and λ is the heat conductivity; in this (H)Norling 1.0mm Mjornell and the elastic(1997) modulus is adopted (Er) 500MPabecause can it value 500MPa of elastic modulus of corrosion prod- simulate reasonable cracking behaviors in terms of ucts can simulate the crack pattern more closely to

Proceedings of FraMCoS-7, May 23-28, 2010 the experimental result (Fig. 6) with two cracks ap- (Val et al. 2009,J = −ToongoenthongD(h,T )∇h & Maekawa. (1) explicitly accounts for the evolution of hydration pearing in the concrete cover. 2004). reaction and SF content. This sorption isotherm The proportionality coefficient D(h,T) is called reads moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant ⎡ ⎤ Vertical crack ⎢ 1 ⎥ & Najjar 1972). The moisture mass balance requires w ()()h,α ,α = G α ,α 1 − + that the variation in time of the water mass per unit e c s 1 c s ⎢ ∞ ⎥ ⎢ 10(g α − α )h ⎥ volume of concrete (water content w) be equal to the ⎣ e 1 c c ⎦ (4) divergenceInitia l of the moisture flux J ⎡ ∞ ⎤ rebar 10(g α − α )h K ()α ,α ⎢e 1 c c − 1⎥ ∂wradius 1 c s ⎢ ⎥ Er=100MPa Er=500MPa − = ∇ • J Lateral crack (2) ⎣ ⎦ ∂t Figure 14. Type C10 crack pattern (surface crack width 0.5mm). Figure 16. Penetration of corrosion products into cracks. where the first term (gel isotherm) represents the The water content w can be expressed as the sum physically bound (adsorbed) water and the second This effect hasof been the simulatedevaporable in waterthe RBSM w (capillary ana- water, water 4.2 Simulation of surface crack width propagation e term (capillary isotherm) represents the capillary lytical program. vapor,One of and the adsorbed advantages water) of RBSMand the non-evaporable Analytical results in the case of type C30 are com- water. This expression is valid only for low content model is that crack(chemically widths and volumebound) of watercracks canw (Mills 1966, pared with experimental results as shown in Figure n of SF. The coefficient G represents the amount of be calculated directlyPantazopoulo during the &analysis Mills (Fig.1995 17).). It It is reasonable to 1 15. The analytical results appear reasonably agree- water per unit volume held in the gel pores at 100% is therefore convenientassume to thatcalculate the evaporable reduction ofwat vol-er is a function of ment with the experimental results, i.e. initiation relative humidity, and it can be expressed (Norling ume of corrosionrelative products humidity, which h,penetrate degree ofinto hydration, α , and c Mjornell 1997) as crack widths occurring with a certain amount of cor- cracks. The reductiondegree of of corrosion silica fume expansion reaction, pres-α , i.e. w =w (h,α ,α ) s e e c s rosion products and then crack opening propagating sure due to the penetration= age-dependent of corrosion sorption/desorp products into tion isotherm speedily up. However, analytical values show larger c s cracks is calculated(Norling by reducing Mjonell the 1997 free) . increaseUnder this U assumption and G ()α ,α = k α c + k α s (5) surface crack widths than the experimental results, in the corrosion expansionby substituting model asEquation shown in1 Figureinto Equation 2 one 1 c s vg c vg s especially when corrosion rates increase. 18. obtains c s We have assumed the following when consider- where k vg and k vg are material parameters. From the 1.6 ing this effect in the analysis: maximum amount of water per unit volume that can ∂w ∂h ∂w ∂w 1.4 Er= 500MPa Corrosion −productse + ∇can• ( Donly∇h ) penetrate= e α& +intoe α& + w& (3) fill all pores (both capillary pores and gel pores), one Er=100MPa ∂h ∂t h ∂α c ∂α s n can calculate K as one obtains 1.2 Experiment cracks if crack widths exceed the cthresholds 1 value of crack width. 1 ⎡ ⎛ ∞ ⎞ ⎤ Corrosion whereproducts ∂w fullye/∂h isfill the in slopethe cracks of the sorption/desorption 10⎜ g α −α ⎟h ⎢ ⎝ 1 c c ⎠ ⎥ 0.8 The free isothermincrease U(also is uniformlycalled moisture reduced capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ 0 c s 1⎢ ⎥ 0.6 around thegoverning rebar. equation (Equation 3) must be completed ⎣ ⎦ (6) K ()α ,α = With a free increaseby appropriate of U, the boundary corresponding and ini cor-tial conditions. 1 c s ⎛ ∞ ⎞ 0.4 10⎜ g α −α ⎟h rosion product volume Vcor is (Figure 18a): ⎝ 1 c c ⎠ Surface crack width (mm) width crack Surface The relation between the amount of evaporable e −1 0.2 water and relative humidity is called ‘‘adsorption c s 2 isotherm”2 if measured with increasing relativity The material parameters k vg and k vg and g1 can 0 200 400 600 800 Vcor = [π (r +U ) −πr ]L = πU (2r +U )L (8) Corrosion rate (mg/cm 2) humidity and ‘‘desorption isotherm” in the opposite be calibrated by fitting experimental data relevant to 2 case. Neglecting their difference (Xi et al. 1994), in i li l f kidh since U ≈0, equation (7) can be approximated as free (evaporable) water content in concrete at Figure 15. Analytical surface crack width propagation (Type the following, ‘‘sorption isotherm” will be used with various ages (Di Luzio & Cusatis 2009b). C30). V = 2πr ⋅U ⋅ L (9) cor reference to both sorption and desorption conditions. With a higher value of the linear elastic modulus where L is the lengthBy ofthe rebar. way, if the hysteresis of the moisture isotherm would be taken into account, two different 2.2 Temperature evolution of the corrosion products, Er=500MPa, corrosion When the penetration of corrosion products into expansion pressure induced by corrosion products is cracks is considered,relation, the evaporableeffective corrosion water vs relativeprod- humidity, must Note that, at early age, since the chemical reactions be used according to the sign of the variation of the associated with cement hydration and SF reaction larger and it causes initiation of surface crack width ucts volume Vcor,eff is: quicker than the smaller case, E =100 MPa. In the relativity humidity. The shape of the sorption are exothermic, the temperature field is not uniform r V = V −V early stage of corrosion, the surface crack width in cor,eff cor isothermcrk for HPC is influenced by many parameters, for non-adiabatic systems even if the environmental the case of E =500 MPa is closer to the experimen- especially those that influence extent and rate of the temperature is constant. Heat conduction can be r (10) chemical reactions and, in turn, determine pore described in concrete, at least for temperature not tal results than the one in the case of Er=100MPa. where Vcrk is volumestructure cracks and as pore computed size distribution in Figure (water-to-cement exceeding 100°C (Bažant & Kaplan 1996), by 17. ratio, cement chemical composition, SF content, Fourier’s law, which reads 4.3 Penetration of corrosion products into cracks If the effectivecuring free increase time and is method, Ueff, the temperature, effective mix additives, During the corrosion process, it is known that corro- volume of corrosionetc.). products In the literaturein Figure various18b will formulations be can be q = −λ∇T (7) sion products can penetrate into cracks in concrete (similar to (9)) found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present so this effect may reduce corrosion expansion pres- Vcor,eff = 2π.r.Ueff .L where q is the heat flux, T is the absolute sure on concrete accordingly as shown in Figure 16 paper the semi-empirical expression proposed by temperature, and λ is the heat conductivity; in this (11) Norling Mjornell (1997) is adopted because it

Proceedings of FraMCoS-7, May 23-28, 2010 fromJ = − (9),(10)D(h,T )∇ hand (11): (1) 0.40mmexplicitly because accounts lateral for thecracks evolution have occurredof hydration and thereaction total volumeand SF of content. cracks becomesThis sorption large whichisotherm in- The proportionality coefficient D(h,T) is called ducesreads a significant reduction on the corrosion ex- 2π.r.U .L = 2π.r.U.L −V (12) moistureeff permeability andcrk it is a nonlinear function pansion pressure on concrete. of the relative humidity h and temperature T (Bažant For the case of 0.2mm⎡ threshold, the changing⎤ V ⎢ 1 ⎥ & Najjar 1972crk). The moisture mass balance requires pointw ()()h, αof ,αthe =propagationG α ,α 1 is− at the value 0.8mm+ of Uthateff the= U variation− in time of the water mass per(13) unit surfacee c cracks width.1 c Thes ⎢ tendency∞ of crack⎥ width 2π.r.L ⎢ 10(g α − α )h ⎥ volume of concrete (water content w) be equal to the propagation is similar with⎣ thee case1 c of 0.1mmc ⎦ thre-(4) divergence of the moisture flux J shold. ⎡ 10(g α ∞ − α )h ⎤ K ()α ,α ⎢e 1 c c − 1⎥ ∂Normalw spring i 1.6 1 c s ⎢ ⎥ − = ∇ • J (2) ⎣ ⎦ εn,i : strain of normal spring i ∂t 1.4

where1.2 the first term (gel isotherm) represents the The water content w can be expressedInterface surface as the i sum physically bound (adsorbed) water and the secondD of the evaporable water we (capillary water, water 1 Si is area of surface i term (capillary isotherm) represents the capillary vapor, and adsorbed water) and the non-evaporable water.0.8 This expression is valid only for low content (chemically bound) water w (Mills 1966, C n of SF. The coefficient G represents the amount of Pantazopoulo & Mills 1995). It is reasonable to 0.6 1 water per unit volume held in the gel pores at 100% assume that the evaporable water is a function of 0.4 No penetration relative humidity,B and it can be Thresholdexpressed =0.1mm (Norling relative humidity, h, degree of hydration, α , and Surface crack width (mm) Threshold =0.2mm hi c Mjornell0.2 1997) as degree of silica fume reaction, α , i.e. w =w (h,α ,α ) Experiment Vcrk is total volume of cracks s e e c s A = age-dependentNSPG sorption/desorption isotherm 0 c 200 s 400 600 800 (Norling VMjonell= ε 1997.h.S ). Under this assumption and G ()α ,α = k α c Corrosion+ k α s rate (mg/cm 2 ) (5) by substitutingcrk ∑ nEquation,i i i 1 into Equation 2 one 1 c s vg c vg s 1 obtains Figure 19. Surface crack width propagation with penetration of where NSPG is numbers of springs corrosion cproducts intos cracks. where k vg and k vg are material parameters. From the wi=εn,i .hi is crack width at interface surface i ∂w ∂w ∂w maximum amount of water per unit volume that can e ∂h e e In comparison with the experimental results, if Figure− 17. Computation+ ∇ • (D ∇h )volume= ofα& cracks+ in αRBSM& + w& model. fill all pores (both capillary pores and gel pores), one h c s n (3) the threshold of crack width is set between 0.1mm ∂h ∂t ∂α ∂α can calculate K1 as one obtains c s and 0.2mm, it can reasonably simulate the surface

crack width propagation as shown⎡ in⎛ Figure∞ 19.⎞ ⎤ where ∂we/∂hU is the slope of the sorption/desorption 10⎜ g α −α ⎟h U However, it is noted that in ⎢the experiment,⎝ 1 c c ⎠ ⎥ cor- isotherm (also called moistureeff capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ rosion products0 mayc not fullys 1 ⎢penetrate into cracks⎥ governing equation (Equation 3) must be completed ⎣ ⎦ (6) r andK ()α the,α speed= of penetration also depends on crack by appropriate boundary and initial conditions. 1 c s ⎛ ∞ ⎞ widths and properties 10of⎜ g corrosionα −α ⎟h products. In the Initia l ⎝ 1 c c ⎠ The relation between the amount of evaporable analysis, we have assumede the fastest−1 and biggest water and rebarrelative humidity is called ‘‘adsorption radius penetration which induces thec largests reduction on isotherm” if measured with increasing relativity corrosionThe material expansion. parameters k vg and k vg and g1 can humidity and ‘‘desorption isotherm” in the opposite be Thecalibrated threshold by fitting of penetration experimental greatly data affectsrelevant the to case. Neglecting their difference (Xi et al. 1994), in free (evaporable) water content in concrete at a) No penetration b) Penetration propagation of surface crack width. The above as- the following, ‘‘sorption isotherm” will be used with sumptionsvarious ages are (Di also Luzio simple & Cusatis cases for2009b). the analysis in into cra cks into cra cks reference to both sorption and desorption conditions. this study and it will definitely require further work By the way, if the hysteresis of the moisture Figure 18. Computation reduction of free increaseU due to to2.2 simulate Temperature the effect evolution to the corrosion process. Espe- penetrationisotherm wouldof corrosion be taken products into into account, cracks. two different cially, it will need further experimental results to ve- relation, evaporable water vs relative humidity, must rifyNote the that, simulation at early results.age, since the chemical reactions be Figureused according 19 shows to the the propagation sign of the of variation surface ofcrack the associated with cement hydration and SF reaction widthrelativity when humidity. this effect The is consideredshape of forthe typesorption C30 are exothermic, the temperature field is not uniform specimen.isotherm for Thresholds HPC is infl ofuenced 0.1mm by and many 0.2mm parameters, crack 4.4for non-adiabaticSimulation of internalsystems crack even propagationif the environmental widthespecially are considered.those that influence The value extent of 500MPa and rate is of used the Ittemperature is necessary is to constant. simulate internalHeat conduction crack propagation can be forchemical the elastic reactions modulus and, of thein corrosionturn, determine products. pore todescribed predict ininternal concrete, cracking at least behavior for temperature from the sur-not structureIt can andbe poreseen sizethat distributionthe propagation (water of-to-cement surface faceexceeding cracks. 100°C Analytical (Bažant results & ofKaplan propagations 1996), byof crackratio, widthcement is chemicalreasonably composition, agreement withSF content,the ex- crackFourier’s width law, near which the readsrebar, lateral crack length and perimentalcuring time results and method, when thetemperature, effect of penetrationmix additives, of lateral crack width are compared with the experi- corrosionetc.). In theproducts literature into cracksvarious is formulations considered. can be mentalq = −λ ∇results.T The effect of penetration of corrosion (7) foundWhen to thedescribe threshold the ofsorption 0.1mm isotherm width is setof normalfor the products into cracks where crack width is larger than concrete (Xi et al. 1994). However, in the present penetration of corrosion products into cracks, the 0.10mmwhere qis considered.is the heat flux, T is the absolute propagationpaper the semi-empirical speed of surface expression of crack proposedwidth is re-by temperature, and λ is the heat conductivity; in this ducedNorling when Mjornell the surface (1997) crack is adoptedwidth grows because up toit

Proceedings of FraMCoS-7, May 23-28, 2010 4.4.1 Propagations of crack width of vertical crack J = −D(h,T )∇h (1) explicitly accounts for the evolution of hydration 1 near rebar reaction and SF content. This sorption isotherm The crack width of vertical crack near the rebar as The proportionality coefficient D(h,T) is called reads 0.8 10mm from rebar (Analytical) shown in Figure 20 is smaller than the surface crack 20mmmoisture from rebar (Analytical)permeability and it is a nonlinear function 40mm from rebar (Analytical) width at the same level of corrosion rate as shown in 10mm fromof rebarthe (relative Exp.) humidity h and temperature T (Bažant ⎡ ⎤ 0.6 ⎢ 1 ⎥ Figure 19. 20mm from& rebarNajjar ( Exp.) 1972 ). The moisture mass balance requires w ()()h,α ,α = G α ,α 1 − + e c s 1 c s ⎢ ⎥ The propagation of crack width near rebar also 40mm fromthat rebar the ( Exp.) variation in time of the water mass per unit 10(g α ∞ − α )h Crack width (mm) width Crack 0.4 ⎢ ⎥ appears reasonably agreement with the experimental volume of concrete (water content w) be equal to the ⎣ e 1 c c ⎦ (4) values when the effect of penetration of corrosion divergence of the moisture flux J 0.2 ⎡ 10(g α ∞ − α )h ⎤ into cracks is taken into account K ()α ,α ⎢e 1 c c − 1⎥ ∂w 1 c s ⎢ ⎥ − = ∇ • J (2) ⎣ ⎦ 0 200 400 600 800 1000 1.6 ∂t 2 1.4 Corrosion rate (mg/cm ) where the first term (gel isotherm) represents the The water content w can be expressed as the sum No penetration Figure 22a. Propagation of lateral crack width (no penetration). physically bound (adsorbed) water and the second 1.2 With penetration of the evaporable water w (capillary water, water e term (capillary isotherm) represents the capillary 1 Experiment 1 vapor, and adsorbed water) and the non-evaporable water. This expression is valid only for low content 0.8 (chemically bound) water wn (Mills 1966, 0.8 of SF. The coefficient G1 represents the amount of 0.6 10mmPantazopoulo from rebar ( Exp.) & Mills 1995). It is reasonable to Crack width (mm) width Crack water per unit volume held in the gel pores at 100% assume that the evaporable water is a function of 0.4 20mm from rebar ( Exp.) relative humidity, and it can be expressed (Norling 0.6 40mmrelative from rebar humidity, ( Exp.) h, degree of hydration, α , and 0.2 c Mjornell 1997) as 10mmdegree from rebar of silica(analytical) fume reaction, αs, i.e. we=we(h,αc,αs) 0.4 20mm from rebar (analytical) 0 200 400 600 800 1000 40mm= fromage-dependent rebar (analytical) sorption/desorption isotherm

Crack width (mm) width Crack c s 2 (Norling Mjonell 1997). Under this assumption and G ()α ,α = k α c + k α s (5) Corrosion rate (mg/cm ) 0.2 1 c s vg c vg s Figure 20. Propagation of vertical crack width nearrebar. by substituting Equation 1 into Equation 2 one obtains where kc and ks are material parameters. From the 4.4.2 Propagations of lateral crack length 0 200 400 600 800 1000 vg vg ∂wCorrosion rate (mg/cm 2) ∂w ∂w maximum amount of water per unit volume that can The analytical lateral crack length propagation is e ∂h e e − + ∇ • (D ∇h) = α& + α& + w& (3) fill all pores (both capillary pores and gel pores), one shown in Figure 21. Figure 22b. Propagation of∂h lateral∂t crack widthh (with∂α penetration).c ∂α s n can calculate K as one obtains c s 1 The tendency of analytical result is similar to the testing results. The propagation of lateral crack 4.5 Mechanism of surface crack width propagation ⎡ ⎛ ∞ ⎞ ⎤ where ∂we/∂h is the slope of the sorption/desorption 10⎜ g α −α ⎟h length is closer to the experiment when the penetra- ⎢ ⎝ 1 c c ⎠ ⎥ with internal isothermcrack propagation (also called moisture capacity). The w − 0.188α s + 0.22α s − G ⎢1− e ⎥ tion effect is considered. 0 c s 1⎢ ⎥ governing equation (Equation 3) must be completed ⎣ ⎦ (6) From the surface crack width propagation (Figure K ()α ,α = by appropriate boundary and initial conditions. 1 c s ⎛ ∞ ⎞ 19) and the internal crack propagations (Figure 20, 10⎜ g α −α ⎟h 60 21,22a and 22b), aThe mechanism relation betweenof surface the crackamount of evaporable e ⎝ 1 c c ⎠ −1 No penetration With penetration width propagationwater with and internal relative crack humidity propagation is called ‘‘adsorption 50 c s Experiment can be estimated asisotherm” shown in if Figure measured 23: with increasing relativity The material parameters k vg and k vg and g1 can 40 a) Verticalhumidity crack initiates and ‘‘desorption from concrete isotherm” sur- in the opposite be calibrated by fitting experimental data relevant to face (Figurecase. Neglecting23a.) and propagatestheir difference to the (Xi et al. 1994), in free (evaporable) water content in concrete at 30 rebar correspondingthe following, to ‘‘sorption the stage isotherm”A to B in will be used with various ages (Di Luzio & Cusatis 2009b). 20 Figure reference19. In this to both stage, sorption surface and crackdesorption conditions. By the way, if the hysteresis of the moisture width rapidly increases. 2.2 Temperature evolution 10 b) Then, lateralisotherm cracks would initiate be taken (Figure into 23b.)account, two different Lateral cracklength(mm) Lateral and increaserelation, in evaporabletheir widths water and vs lengths relative humidity, must Note that, at early age, since the chemical reactions 0 200 400 600 800 1000 which causebe used rotation according on the to concretethe sign ofparts the variation of the associated with cement hydration and SF reaction Corrosion rate (mg/cm 2) as shownrelativity in Figure humidity. 23c. Inside The crack shape initi- of the sorption are exothermic, the temperature field is not uniform Figure 21. Propagation of lateral crack length. ates (Figureisotherm 23c.). for HPC Corrosion is influenced products by many parameters, for non-adiabatic systems even if the environmental penetrateespecially into internal those cracks. that influence In this stage, extent and rate of the temperature is constant. Heat conduction can be 4.4.3 Propagations of lateral crack width the surfacechemical crack reactionswidth also and, increases in turn, but determine pore described in concrete, at least for temperature not Analytical crack widths of the lateral crack at some the speedstructure of surface and porecrack size width distribution propaga- (water-to-cement exceeding 100°C (Bažant & Kaplan 1996), by positions from the rebar are compared with the ex- tion reducesratio, (stagecement B tochemical C in Figure composition, 19). SF content, Fourier’s law, which reads perimental results as shown in Figure 22a and 22b. c) With a curingfurther timeamount and method,of corrosion temperature, prod- mix additives, Again, when the effect of corrosion products pene- ucts, theetc.). inside In crackthe literature propagates various from formulationsthe can be q = −λ∇T (7) tration is considered, the analytical results appear rebar andfound causes to describerotation theon thesorption concrete isotherm of normal closely to the experimental results. parts asconcrete shown in (Xi Figure et a 23d.l. 1994). The speedHowever, of in the present where q is the heat flux, T is the absolute surface papercrack thewidth semi-empirical propagation expressionin this proposed by temperature, and λ is the heat conductivity; in this Norling Mjornell (1997) is adopted because it

Proceedings of FraMCoS-7, May 23-28, 2010 J = −D(h,Tstage)∇h (C to D in Figure 19) is slightly (1) REFERENCESexplicitly accounts for the evolution of hydration higher than the previous stage (B to C). reaction and SF content. This sorption isotherm The proportionality coefficient D(h,T) is called Andrade,reads C., Alonso,C.,Molina,F.J.,1993.Cover cracking as a moisture permeability and it is a nonlinear function function of bar corrosion: Part I-Experimental test, Materi- als and Structures, 26: 453-464.⎡ ⎤ of the relative humidity h and temperature T (Bažant Cabrera,J.G.,1996.Deterioration of Concrete Due to Rein- ⎢ 1 ⎥ & Najjar 1972). The moisture mass balance requires w forcement()()h,α ,α steel= G Corrosion,α ,α 1Cement− & Concrete Composites+ that the variation in time of the water mass per unit e18:47-59.c s 1 c s ⎢ ∞ ⎥ ⎢ 10(g α − α )h ⎥ volume of concrete(a) (water content (b) w) be equal to the Kawamura,K.,Nakamura,H.,Kunieda,M.,Ueda,N.,2009.A⎣ e 1 c c ⎦ fun-(4) damental study about the evaluation of crack propagation in divergence of the moisture flux J ⎡ ∞ ⎤ concrete induced by rebar co10rrosion(g α (in− Japanese),α )h Proceed- ing of JCI meeting, K ()α ,αJapan⎢e Concrete1 c Institute,c − 1⎥CD:1075- ∂w 1 c s ⎢ ⎥ − = ∇ • J (2) 1080. ⎣ ⎦ Lundgren,K.,2002, Modeling the effect of corrosion on bond ∂t in reinforced concrete, Magazine of Concrete Re- where the first term (gel isotherm) represents the The water content(c) w can be expressed (d) as the sum search,54,No.3:165-173. Matsuo,physically T., Nishiuchi, bound (adsorbed)T., Matsumura, water T.,1997.Crack and the Propaga-second of the evaporable water w (capillary water, water e termtion (capillaryAnalysis in Concreteisotherm) induced represents by Rebar the Corrosion capillary Ex- Figurevapor, 23. and Internal adsorbed cracking water) mechanism. and the non-evaporable water.pansion This (in expression Japanese), Concreteis valid Journal,only for Japan low Concretecontent (chemically bound) water w (Mills 1966, n of Institute,19:99-104.SF. The coefficient G represents the amount of Pantazopoulo & Mills 1995). It is reasonable to Nguyen,Q.T.,Millard,A.,Care,S1.,L’Hostis,V.,Berthaud,Y.,2006 water per unit volume held in the gel pores at 100% 5assume CONCLUSIONS that the evaporable water is a function of .Fracture of concrete caused by the reinforcement corrosion relative humidity, and it can be expressed (Norling relative humidity, h, degree of hydration, α , and products, J.Phys.IV France 136:109-120. c Oh,B.H.,Mjornell Kim,K.H.,Jang, 1997) as B.S.,2009.Critical corrosion amount Experimentdegree of silica and fume 3D- reaction,RBSM analyticalα , i.e. w =modelw (h,α was,α ) s e e c s to cause cracking of reinforced concrete structures, ACI studied= age-dependent to evaluate crackingsorption/desorp behavior tionof the isothermconcrete Material Journal,V,106,No.4:333-339. c s specimen(Norling Mjonellwith single 1997 rebar.). Under this assumption and Saito,G ()α S.,1999.Fracture,α = k α c + analysesk α s of structural concrete using(5) by Thesubstituting three phase Equation material 1 intocorrosion Equation expansion 2 one 1springc snetworkvg withc randomvg s geometry， Doctoral thesis， modelobtains was applied in the analysis. The values of ini- Kyushu University. Toongoenthong,K.,Maekawa,K.c s 2005.Simulation of coupled tial thickness and linear elastic modulus of corrosion where k vg and k vg are material parameters. From the maximumcorrosive amountproduct offormation, water per migration unit volume into crack that canand products∂w were recommended.∂w ∂w propagation in reinforced sections, Journal of Advanced e ∂h e e − Internal+ ∇crack• (D patterns,∇h) = surfaceα& + crackα& + widthw& prop-(3) fillConcrete all pores Technology, (both capilla Vol.3,ry No.2:253-265.pores and gel pores), one ∂h ∂t h ∂α c ∂α s n can calculate K as one obtains agation and internal crackc propagations were simu- Tran,K.K., Kawamura,K.,Nakamura,H.,Kunieda,M.,2009.1 Pre- lated quantitatively and qualitatively and compared diction Of Cracking Of Concrete Due To Rebar Corrosion Using 3D- RBSM, Proceedings of⎡ the⎛ Eleventh∞ ⎞Interna-⎤ withwhere the ∂w experimentale/∂h is the slope results. of the Moreover, sorption/desorption the effect 10⎜ g α −α ⎟h tional Summer Symposium, JSCE:⎢ 249-252.⎝ 1 c c ⎠ ⎥ ofisotherm corrosion (also products called penetration moisture into capacity). cracks during The Val,V. D., wChernin,L.,− 0.188α s +Stewar0.22α t,G.M.,2009.Experimentals − G ⎢1− e ⎥ and 0 c s 1⎢ ⎥ corrosiongoverning processequation was (Equation investigated 3) must and be it completedwas con- Numerical Investigation of Corrosion-Induced⎣ ⎦Cover(6) K ()α ,α = firmedby appropriate that the boundarysimulation and results initial were conditions. quantitatively 1 Crackingc s in Reinforced ⎛ Concrete∞ ⎞Structures, Journal of Structural Engineering,10 ASCE,⎜ g α 135:376-385.−α ⎟h agreedThe withrelation the experimentalbetween the resultsamount when of evaporable this effect e ⎝ 1 c c ⎠ −1 waswater considered. and relative humidity is called ‘‘adsorption c s isotherm”Mechanism if measuredof surface with crack incr widtheasing propagation relativity The material parameters k vg and k vg and g1 can washumidity clarified and and‘‘desorption it was strongly isotherm” dependent in the opposite on the be calibrated by fitting experimental data relevant to internalcase. Neglecting crack propagation. their difference Therefore, (Xi et al.in 1994),order toin free (evaporable) water content in concrete at evaluatethe following, the surface ‘‘sorption crack isotherm” propagation, will bethe used internal with various ages (Di Luzio & Cusatis 2009b). crackreference propagation to both sorption must be andclarified. desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different 2.2 Temperature evolution relation, evaporable water vs relative humidity, must Note that, at early age, since the chemical reactions be used according to the sign of the variation of the associated with cement hydration and SF reaction relativity humidity. The shape of the sorption are exothermic, the temperature field is not uniform isotherm for HPC is influenced by many parameters, for non-adiabatic systems even if the environmental especially those that influence extent and rate of the temperature is constant. Heat conduction can be chemical reactions and, in turn, determine pore described in concrete, at least for temperature not structure and pore size distribution (water-to-cement exceeding 100°C (Bažant & Kaplan 1996), by ratio, cement chemical composition, SF content, Fourier’s law, which reads curing time and method, temperature, mix additives, etc.). In the literature various formulations can be q = −λ∇T (7) found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present where q is the heat flux, T is the absolute paper the semi-empirical expression proposed by temperature, and λ is the heat conductivity; in this Norling Mjornell (1997) is adopted because it

Proceedings of FraMCoS-7, May 23-28, 2010