SIMULATION OF AUTOGENOUS AND DRYING SHRINKAGE BASED ON MICRO-PORE STRUCTURE AND INTERNAL MOISTURE STATE

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1 SIMULATION OF AUTOGENOUS AND DRYING SHRINKAGE BASED ON MICRO-PORE STRUCTURE AND INTERNAL MOISTURE STATE Y. Luan (1), T. Ishida (1) (1) Depatment of Civil Engineeing, the Univesity of Tokyo, Japan Abstact A multi-scale constitutive model which simulates shinkage of concete is impoved based on ealy age hydation and moistue state in mico-poes. Fist it is found that the oiginal model undeestimates autogenous shinkage at initial days. Theefoe the contibution of chemical shinkage to autogenous shinkage at ealy age is discussed and its influence is added. Besides, long time shinkage is also modified, consideing diving foce acting on poes with diffeent adii espectively. Diving foce is assumed dominated by disjoining pessue in nanoscale poes and capillay tension acting in lage poes. With the impoved model, autogenous shinkage and dying shinkage with vaious w/c atios and elative humidity can be simulated easonably. 1. INTRODUCTION Nowadays concete made with low wate-cement (w/c) atio is being widely used in pactice. Although low w/c atio bings high stength and low pemeability, autogenous shinkage becomes much moe significant. Tensile stess would be induced in concete, esulting in ealy-age cacking and geatly deceasing the long tem pefomance of stuctues. The mechanism of autogenous shinkage is usually egaded as simila to dying shinkage in some extent. Due to wate loss by self-desiccation o exposue, the intenal elative humidity (RH) deceases and diving foce occus. Seveal theoies have been poposed to descibe the diving foce causing the shinkage, such as capillay tension, disjoining pessue, suface enegy and so on. Chemical eaction and coesponding volume change known as chemical shinkage is also consideed as a eason fo autogenous shinkage. With espect to simulation on time-dependent behavious of concete, a multi-scale constitutive model was developed in Concete Lab, the Univesity of Tokyo 1. In this model, the diving foce which pincipally consists of capillay tension is simulated based on the satuation in mico-poes. Combining with a two-phase stiffness model, dying shinkage could be calculated. The analytical esults have been veified by tests. In this study, in ode to expand its application to autogenous shinkage, the authos cay out necessay impovements based on discussion on mechanism of shinkage. Befoe that, the existing model would be intoduced biefly as follows.

2 2. MULTI-SCALE CONSTITUTIVE MODEL 2.1 Simulation on poe-stuctue and moistue state The basis of this model is the poe-stuctue and intenal moistue, which ae simulated by a multi-scale system called DuCOM 1. In this system, hydation, poe-stuctue and moistue tansfe in hadened cement paste (heeafte efeed as hcp) ae coupled and simulated synthetically. Capillay and gel poosities ae calculated. Thei distibutions by size ae descibed by a Raleigh-Ritz distibution function. Moistue tanspot and equilibium in poe stuctue concens not only wate capacity and moistue gadient, but also wate consumption and poosity development. Wate movement is descibed by the following fomula Si P i div( K P t t P) i Si Wp t t chem Whee, i epesents capillay o gel poes, and ρ is the density of wate. S i is poe satuation degee. P is poe pessue. K t is the wate tanspot coefficients consideing both liquid and vapo. β chem is chemically bound wate atio and W p is the weight of cement. Accodingly, the moistue states in hcp including intenal RH, poe pessue, wate content and satuation can be calculated. As to wate capacity in mico-poes, fist the poe pessue diffeence due to capillay meniscus acoss the liquid-vapo inteface is descibed by Kelvin s equation as shown RT P Pl ln h M (2) Whee, ρ is the density of wate, R is the gas constant and T is the absolute tempeatue, M is the molecula mass of wate and h is the elative humidity. Besides, the pessue diffeence can also be obtained by the Laplace equation following the equation 2 Pl c Whee γ is suface tension of liquid and c is the poe adius whee the inteface is pesent. By combining Eq.2 and Eq.3, the elationship between h and c is established by 2 M 1 ln h RT c It is assumed that poes with adii smalle than c ae totally filled by wate. Fo lage poes only some ae wate-filled due to ink-bottle effect. The volume and suface aea of inkbottle poes ae calculated based on an entapment paamete f. In the poes which ae not filled by condensed wate, physically adsobed wate exists and is also included in the model. The satuation of gel o capillay poes can be calculated by the thee potions of wate by S S c S ink S ads c max dv fdv Sads (5) min c Whee, S c and S ink ae satuation factions in the poes smalle and lage than c espectively. S ads is satuation faction fo adsobed wate. min and max the minimum and maximum poe adius. 2.2 Diving foce fo shinkage in hadened cement paste Capillay tension is assumed the pincipal diving foce. Combining the poe pessue ΔP in Eq.2 with its effective facto β, the capillay tension is given by the following the equation (1) (3) (4)

3 cp Scp gl Sgl cp P cp gl P (6) Whee φ cp and φ gl ae capillay and gel poosities. S cp and S gl ae thei satuations. Theefoe, β equals the aveage satuation. 2.3 Two-phase composite stiffness model Concete is idealized as a two-phase composite which consists of hcp and elastic aggegate suspended inside. Aggegate is assumed igid and shows elastic defomation. Shinkage caused by aggegates is also teated in the model. Suounding the aggegate ae clustes of hcp which shows ceep defomation unde diving foce. The details of this model can be found in the liteatue DISCUSSION AND MODIFICATION ON THE EXISTING MODEL 3.1 The contibution of chemical shinkage to autogenous shinkage The existing model was used to simulate dying shinkage of nomal w/c atio concete. Heein tentatively it is applied to autogenous shinkage with low w/c atio and compaed with test by Tazawa et al 2, shown in Fig.1. Shinkage befoe 6 initial setting is deducted. Obvious diffeence can 4 be obseved. In the tests, shinkage inceases 2 quickly at initial days, and then the slope tends to be gentle. It has become a distinct featue that autogenous shinkage develops apidly at ealy age. Howeve, the analysis shows continuous incease until long time. It seems that geat undeestimation exists at ealy age. Autogenous shinkage (1-6 ) Analysis Test Test by Tazawa et al W/C atio Age (days) Figue 1: Modelling of autogenous shinkage by the existing model Autogenous shinkage at ealy age is egaded as attibuted to chemical shinkage, which is the absolute volume change of hydates fom unhydated cement and wate. When concete is fluid, it is totally conveted into macoscopic volume change. Duing hadening, concete becomes igid and a potion can be esisted, but it still contibutes to autogenous shinkage. Theefoe the ole of chemical shinkage should not be neglected. Howeve in the existing model it is not taken into account The contibution of chemical shinkage is elated to the w/c atio as epoted 3. Fo lowe w/c, both highe chemical and autogenous shinkages ae obseved. While w/c inceases to nomal level, chemical shinkage still occus but autogenous shinkage tends to be zeo. In this study, the authos attempt to explain the influence of w/c based on micoscale hypothesis, as shown in Fig.2. At the beginning, cement paticles ae dispesed into wate and the aveage distance is dominated by w/c atio. A potion of chemical shinkage induces mico-poes while the othe causes autogenous shinkage. Fo high w/c, the aveage distance is lage, so paticles ae independent and little inteaction happens. Fo low w/c, distance deceases and some paticles even contact each othe. They tend to attact and move towads each othe, esulting in moe obvious macoscale volume change, i.e. autogenous shinkage. An equation descibing autogenous shinkage caused by chemical shinkage is poposed as

4 v ch ch f ( m ) Whee, ε ch is autogenous shinkage and ν ch is chemical shinkage. δ m is the equivalent distance of cement paticles, which is calculated accoding to a steeological model 1. f(δ m ) is its influence function. ν ch is calculated by volume change ate due to convesion fom liquid to chemically bound wate. f(δ m ) is given by an empiical equation. They follow the equations vch W ch /( 1/ l 1/ ch ) (8) f ( ).45exp( a m b m ) Whee, W ch is the weight atio of chemically bound wate. ρ l and ρ ch ae densities of liquid wate and chemically bound wate, and the values of ρ ch is assumed kg/m 3. a and b in Eq.9 ae constants with value and 6.. Fo nomal w/c, f(δ m ) is extemely small thus can be neglected. When w/c and coesponding δ m deceases, f(δ m ) inceases and chemical shinkage stats to affect. In the exteme case that δ m is zeo, f(δ m ) has maximum value.45. It does not happen in eality. Howeve, as measued by Tazawa et al 4, when the chemical shinkage is in the ange of 5%~1%, the autogenous shinkage is lowe than 2 1-6, which implies the value is only seveal pecents of chemical shinkage. The maximum value is in this ange so appeas easonable. (a) High w/c atio Befoe hydation Duing hydation (b) Low w/c atio Befoe hydation Duing hydation (7) (9) Cement paticles tend to move Due to lage distance, cement towads each othe due to small paticles hydate independently distance, so lage extenal volume with small extenal volume change. change occus. wate cement hydates Poe volume Figue 2: Influence of w/c atio on extenal volume change 3.2 Simulation of capillay tension and disjoining pessue based on poe size With the contibution of chemical shinkage in Eq.7 added in Fig.1, analytical esults incease apidly at ealy age and agee with test values as Fig.3 shows. Howeve, in the tests shinkage incease afte initial days tends to be gentle o even stagnate. High modulus 6 may be one eason. Besides, since capillay poes ae 4 Analysis Test gadually filled and tend to vanish in low w/c case, sufficient diving foce may not be povided. Howeve,.3.3 in Fig.3 the analytical esults incease with lage slope W/C atio until long time. It is suspected that the diving foce at long time is oveestimated in the model. Figue 3: Modelling with effect of Accoding to Eq.6, the effective facto β is aveage chemical shinkage satuation of capillay and gel poes. Theefoe, in the Autogenous shinkage (1-6 ) Age (days) existing model all the wate is thought to contibute to capillay tension. Howeve, mico-poe size anges fom tens of micometes to less than nanometes. Such a lage distibution may

5 lead to vaied mico-physical inteaction between poes and wate. Since most of capillay poes ae at micomete scale, capillay tension caused by meniscus should be the pincipal mechanism. On the othe hand, a lage potion of gel poes is at the scale of nanometes. Whethe capillay tension o othe mechanism dominates in those poes needs to be discussed. Beltzung et al 5 pointed out that capillay tension which occus within the scale of nanometes is much smalle than pedicted by the Laplace equation. Futhemoe, as the poe size deceases to nanoscale, due to high specific suface aea and thin wate film, disjoining pessue is consideed moe active to induce shinkage 6,7. Theefoe, it seems not appopiate to include all the poe-wate in capillay tension in the model. Diving foce caused by disjoining pessue in nanoscale poes needs to be teated sepaately. Accodingly, the authos popose a scheme to descibe the diving foce as Fig.4 shows. Fist, the adius c in Eq.4 dominates the moistue distibution. Poes smalle than c ae totally wate-filled and othes ae patly filled due to inkbottle effect. Besides, anothe poe adius is defined to distinguish diving foces. Only in poes lage than (poes L), the intenal wate contibutes to capillay tension. As to smalle poes (poes S), disjoining pessue is active. It acts in poes S and epulsive foce occus. is at the scale of nanometes and assumed constant 1nm. At high RH, c is bigge than so poes S ae totally satuated. The epulsive foce sustains equilibium with the skeleton foce. Diving foce consists of only capillay tension in the poes L. When RH deceases and c becomes smalle than, poes S stats to lose wate and some disjoining pessues disappeas. Accodingly diving foce in poes S is poduced due to foce diffeence fom the skeleton and causes shinkage at low RH. Disjoining pessue in Poes S Repulsive foce F caused by disjoining pessue Poe distibution and moistue state c >, high RH Wate empty Capillay tension in Poes L Skeleton Foce T Poes S ae satuated T = F, Equilibium state T c Poe adius c <, low RH Negative pessue F t F t T Poes S stats to lose wate T > F s, Diving foce occus c Poe adius Figue 4: Scheme of diving foce in poes with diffeent size. Theefoe, fo capillay tension, the wate in poes S must be excluded. Besides, compaed with the aveage satuation in Eq.6, the volume faction of wate seems to be a bette facto, since it eflects the influence of both satuation and poosity. The capillay tension is given by P AV P A V V ) A( S S ) (1) cp L ( cp _ L gl _ L cp cp _ L gl gl _ L

6 Whee, V L is wate volume faction in poes L. V cp_l and V gl_l ae the coesponding factions in capillay and gel poes. S cp_l and S gl_l ae satuation factions only including wate in poes L. A is influential constant with value 8.. As to poes S, following the disciption in Fig.4, the diving foce is deived fom foce diffeence between skeleton and wate, witten as follows F F F t (11) Whee, F is the foce fom skeleton, which equals the epulsive foce when poes S ae satuated. F t is the epulsive foce when some wate in poes S is lost. Now the poblem lies in how to calculate the epulsive foce. Accoding to Dejaguin et al 8, the disjoining pessue between two paallel sufaces can be empiically descibed by ( h) K exp( h / ) (12) Whee, K is the stength of disjoining foces, λ is the decay length which is in nanoscale. h is the distance between the sufaces. With the same equation Mauyama 9 deived the value in hcp as 45 MPa by a statistical thickness-based disjoining model and shinkage tests. Based on Eq.12, if poes ae assumed cylinde-shaped, disjoining pessue acting on an individual poe depends on the poe size. The epulsive foce should be a complex of all the disjoining pessues in vaious poes. Theefoe, convesion fom disjoining pessue to epulsive foce has to be caied out. Poe size distibution, suface aea and intenal wate infomation ae necessay, which ae simulated in DuCOM. Based on those infomation, the epulsive foce is given by the following equation ( ) d w (13) min F t s Whee, φ s is volume faction of poes S. χ is the total suface aea of all those poes. dχ w epesents incemental of suface aea due to wate-filled poes with adius. П() is the disjoining pessue in the poes with adius, following Eq.12. As to the paametes K, efeing to Mauyama s value, it is assumed the value as 4MPa. λ is assumed as 1.nm. Hence, in Eq.13 the integal value is the foce on suface aea of all the wate-filled poes smalle than. With the nomalization by suface aea the epulsive foce is obtained. When c is smalle than, poes S ae unsatuated and wate can be divided into two potions. All the poes smalle than c ae full of wate, wheeas in lage ones only some ae wate-filled due to ink-bottle effect. Theefoe, combing the entapment paamete f in Eq.5, the epulsive foce in Eq.13 can be witten as the two potions c ( ) d f ( ) d min (14) c F t s gl Whee, dχ is the incemental of suface aea fo all the poes with adius. Based on Eq.11 and Eq.14, the diving foce in the fom of stess can be finally witten as max ( ) d c ( ) d max f ( ) d max ( ) d min min (15) C c C F Ft s s s max f ( ) d Combining with capillay tension in Eq.1, the total diving foce is obtained as follows:

7 cp gl (16) 4. SIMULATION OF AUTOGENOUS AND DRYING SHRINKAGE With the impovement on the diving foce, autogenous shinkage shown in Fig.3 is simulated again. As Fig.5 shows, it can be found that by the impoved model the analysis well eflects the shinkage development until long time. Duing the initial days, shinkage develops apidly to hundeds of micostains. Besides, the slope afte ealy age is also alleviated, esulting in slow incease fo long time. The impovement lies in the exclusion of nano-scale poes fom capillay tension discussed above. In low w/c case, most of capillay poes ae filled by hydates and fine poes occupy the main potion. Accodingly disjoining pessue athe than capillay tension dominates the diving foce, which is mainly active at elatively lowe RH than the condition by self-desiccation. Besides, compaed with Eq.6, the new fomula of capillay tension in Eq.1 adopts the wate volume faction as the influential facto athe than satuation. Fo w/c atio it is moe easonable, since little wate volume exist in capillay poes even if the satuation is not low. Finally, the impoved model is used fo dying shinkage to check the applicability. Two goups of tests ae adopted in the veification. In the test by Tazawa et al 2, concete with w/c atio.5 wee exposed unde RH 4%, 6%, 8%, 9% afte 7d sealed and compaed the dying shinkage with sealed case. In the test by Hiyohaa 1 et al, concete with vaied w/c atio fom.25 to.65 wee exposed at RH 6% afte 7d cuing. All the specimens in the two tests ae pisms with 1mm 1mm 4mm. As Fig.6 shows, the impoved model can well simulate the shinkage development, and also eflect the influence of w/c atio and RH. As RH deceases dying shinkage inceases, while highe w/c atio will induce highe shinkage. The veification by dying shinkage indicates that with the combination of both capillay tension and disjoining pessue in the impoved model, sufficient diving foce can be supplied unde vaied dying conditions. Theefoe, it can be concluded that the modification in this study is easonable. 5. CONCLUSION Autogenous shinkage (1-6 ) Analysis Test W/C atio Age (days) Figue 5: Modelling of autogenous shinkage by the impoved model Seveal mechanisms can be used to explain and simulate autogenous and dying shinkage of concete. In the impoved model a combination of chemical shinkage, capillay tension and disjoining pessue is adopted as the diving mechanism. They ae consideed acting at

8 diffeent stages o in diffeent mico-poes. At initial days, chemical shinkage contibutes significantly to autogenous shinkage, while capillay tension and disjoining pessue act pincipally fo dying shinkage and long time autogenous shinkage. The ole of these diving foces can be descibed by the impoved model fom mico-scale. Veification by expeiment also shows the ationality of the impovement. Dying shinkage (1-6 ) Analysis Test Sealed Sealed 9% 9% 8% 8% RH 6% 6% 4% 4% Dying Time (days) Tazawa et al W/C =.5, Dying fom 7d Dying shinkage (1-6 ) Kiyohaa et al Dying fom 7d RH = 6% Analysis Test Dying Time (days) W/C atio REFERENCES (1) With vaied RH (2) With vaied W/C atio Figue 6: Modeling of dying shinkage and compaed with tests [1] Maekawa, K., Ishida, T. and Kishi, T., 'Multi-scale Modeling of Stuctual Concete' (Taylo & Fancis, 28). [2] Tazawa, E. and Miyazawa, S., 'Effect of constituents and cuing condition on autogenous shinkage concete', in 'Autogenous Shinkage of Concete', Poceedings of the Intenational Wokshop, Hioshima, Japan (Japanese Concete Institute, Hioshima, 1998) [3] Holt, E., 'Contibution of mixtue design to chemical and autogenous shinkage of concete at ealy ages', Cement and Concete Reseach, 35 (3) (25) [4] Tazawa, E., Miyazawa, S., and Kasai, T., 'Chemical shinkage and autogenous shinkage of hydating cement paste', Cement and Concete Reseach, 25 (2) (1995) [5] Beltzung, F. and Wittmann, F.H., 'Role of disjoining pessue in cement based mateials', Cement and Concete Reseach, 35 (12) (25) [6] Espinosa, R.M. and Fanke, L., 'Influence of the age and dying pocess on poe stuctue and soption isothems of hadened cement paste', Cement and Concete Reseach, 36 (1) (26) [7] Powes, T.C. 'Mechanism of shinkage and evesible ceep of hadened cement paste', in Poceedings of Intenational Confeence on the Stuctue of Concete, London, 1965 (Cement and Concete Association, London, 1968) [8] Dejaguin, B.V. and Chuaev, N.V., 'Popeties of wate layes adjacent to intefaces', in 'Fluid Intefacial Phenomena' (John Wiley & Sons Ltd, 1985) [9] Mauyama, I., 'Oigin of dying shinkage of hadened cement paste: hydation pessue', Jounal of Advanced Concete Technology, 8 (2) (21) [1] Kiyohaa, C., Yamamichi, H., Yamauchi, T., and et al, 'Equation fo pedicting dying shinkage stain of concete based on the theoy of composite mateial', in Summaies of Technical Papes of Annual Meeting, AIT (Achitectual Institute of Japan, Hokuiku, 22) (In Japanese)