Continuum Microviscoelasticity Model for Aging Basic Creep of Early-Age Concrete
Publication: Journal of Engineering Mechanics
Volume 135, Issue 4
Abstract
We propose a micromechanics model for aging basic creep of early-age concrete. Therefore, we formulate viscoelastic boundary value problems on two representative volume elements, one related to cement paste (composed of cement, water, hydrates, and air), and one related to concrete (composed of cement paste and aggregates). Homogenization of the “nonaging” elastic and viscoelastic properties of the material’s contituents involves the transformation of the aforementioned viscoelastic boundary value problems to the Laplace-Carson (LC) domain. There, formally elastic, classical self-consistent and Mori-Tanaka solutions are employed, leading to pointwisely defined LC-transformed tensorial creep and relaxation functions. Subsequently, the latter are back-transformed, by means of the Gaver-Wynn-Rho algorithm, into the time domain. Temporal derivatives of corresponding homogenized creep and relaxation tensors, evaluated for the current maturation state of the material (in terms of current volume fractions of cement, water, air, hydrates, and aggregates; being dependent on the hydration degree, as well as on the water-cement and aggregate-cement ratios) and for the current time period since loading of the hydrating composite material, allow for micromechanical prediction of the aging basic creep properties of early-age concrete.
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Acknowledgments
Financial support by “TUNCONSTRUCT—Technology Innovation in Underground Construction” (Project No. UNSPECIFIEDIP011817-2), sponsored by the European Commission, is gratefully acknowledged.
References
Abate, J., and Valkó, P. (2004). “Multi-precision Laplace transform inversion.” Int. J. Numer. Methods Eng., 60(5), 979–993.
Abramowitz, M., and Stegun, I. (1973). Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th Ed., Dover, New York.
Acker, P. (1988). “Comportement mécanique du béton: Apports de l’approche physico-chimique (Mechanical behavior of concrete: A physico-chemical approach).” Ph.D. thesis, Ecole Nationale des Ponts et Chaussées, Paris, France (in French).
Acker, P. (2001). “Micromechanical analysis of creep and shrinkage mechanism.” Concreep 6: Creep, shrinkage and durability of concrete and concrete structures, F.-J. Ulm, Z. Bažant, and F. Wittmann, eds., Elsevier Science, Cambridge, Mass., 15–25.
Acker, P. (2005). “Ultrahigh-performance concrete: Material engineering by nanoindentation testing and nanomechanical analysis.” Proc., 2nd Int. Symp. on Nanotechnology in Construction—NICOM 2, Y. de Miguel, A. Porro, and P. Bartos, eds., RILEM, Publications S.A.R.L., Cachan, France, 257–267.
Acker, P., and Ulm, F.-J. (2001). “Creep and shrinkage of concrete: Physical origins and practical measurements.” Nucl. Eng. Des., 203(2–3), 143–158.
Aïtcin, P.-C. (1999). “Densifying autogeneous shrinkage.” Concr. Int., 21(11), 54–56.
Alfrey, T., Jr. (1948). Mechanical behavior of high polymers, Interscience, New York.
Atrushi, D. (2003). “Tensile and compressive creep of early age concrete: testing and modelling.” Ph.D. thesis, Norwegian Univ. of Science and Technology, Trondheim, Norway.
Baweja, S., Dvorak, G., and Bažant, Z. (1998). “Triaxial composite model for basic creep of concrete.” J. Eng. Mech., 124(9), 959–965.
Bažant, Z. (2001). “Prediction of concrete creep and shrinkage: Past, present and future.” Nucl. Eng. Des., 203(1), 27–38.
Bažant, Z., Hauggard, A., and Baweja, S. (1997a). “Microprestress-solidification theory for concrete creep. II: Algorithm and verification.” J. Eng. Mech., 123(11), 1195–1201.
Bažant, Z., Hauggard, A., Baweja, S., and Ulm, F.-J. (1997b). “Microprestress-solidification theory for concrete creep. I: Aging and drying effects.” J. Eng. Mech., 123(11), 1188–1194.
Benboudjema, F., Meftah, F., and Torrenti, J. (2005). “Interaction between drying, shrinkage, creep and cracking phenomena in concrete.” Eng. Struct., 27(2), 239–250.
Benveniste, Y. (1987). “A new approach to the application of Mori-Tanakas theory in composite materials.” Mech. Mater., 6, 147–157.
Bernard, O., Ulm, F.-J., and Germaine, J. (2003a). “Volume and deviator creep of calcium-leached cement-based materials.” Cem. Concr. Res., 33(8), 1127–1136.
Bernard, O., Ulm, F.-J., and Lemarchand, E. (2003b). “A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials.” Cem. Concr. Res., 33(9), 1293–1309.
Beurthey, S., and Zaoui, A. (2000). “Structural morphology and relaxation spectra of viscoelastic heterogeneous materials.” Eur. J. Mech. A/Solids, 19(1), 1–16.
Bissonnette, B., Pierre, P., and Pigeon, M. (1999). “Influence of key parameters on drying shrinkage of cementitious materials.” Cem. Concr. Res., 29(10), 1655–1662.
Boltzmann, L. (1874). “Zur Theorie der elastischen Nachwirkung (Concerning the theory of the elastic aftereffect).” Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, 70(2), 275–306 (in German).
Bronstein, I., Semendjajew, K., Musiol, G., and Mühlig, H. (2005). Taschenbuch der Mathematik (Handbook of mathematics), 6th Ed., Verlag Harri Deutsch (in German).
Burgers, J. (1935). “First report on viscosity and plasticity.” Rep., Amsterdam.
Byfors, J. (1980). “Plain concrete at early ages.” Technical Rep., Swedish Cement and Concrete Reserach Institute, Stockholm, Sweden.
CEB-FIP. (1993). Model code 1990, Comité Euro-International du Béton (CEB), and the Fédération International de la Précontrainte (FIP).
Constantinides, G., and Ulm, F.-J. (2004). “The effect of two types of C-S-H on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling.” Cem. Concr. Res., 34(1), 67–80.
Davies, R., Davies, H., and Hamilton, J. (1934). “Plastic flow of concrete under sustained stress.” Proc., ASTM, 34(2), 354–386.
Donolato, C. (2002). “Analytical and numerical inversion of the Laplace-Carson transform by a differential method.” Comput. Phys. Commun., 145(2), 298–309.
Eshelby, J. (1957). “The determination of the elastic field of an ellipsoidal inclusion, and related problems.” Proc. R. Soc. London, Ser. A, 241, 376–396.
Fung, Y. (1965). Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, N.J.
Garrault, S., Finot, E., Lesniewska, E., and Nonat, A. (2005). “Study of C-S-H growth on surface during its early hydration.” Mater. Struct., 38(4), 435–442.
Gaver, D., Jr. (1966). “Observing stochastic processes and approximate transform inversion.” Oper. Res., 14, 444–459.
Gawin, D., Pesavento, F., and Schrefler, B. (2006). “Hygro-thermo-chemomechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete.” Int. J. Numer. Methods Eng., 67(3), 332–363.
Glanville, W. (1933). “Creep of concrete under load.” Struct. Eng., 11(2), 54–73.
González, J., and Abascal, R. (2004). “Linear viscoelastic boundary element formulation for steady state moving loads.” Eng. Anal. Boundary Elem., 28(7), 815–823.
Graham, A., and Toll, D. (2007). “World wide web pages for dam design.” ⟨www.dur.ac.uk/~des0www4/cal/dams/geol/mod.htm⟩.
Granger, L., and Bažant, Z. (1995). “Effect of composition on basic creep of concrete and cement paste.” J. Eng. Mech., 121(11), 1261–1270.
Guénot, I., Torrenti, J.-M., and Laplante, P. (1996). “Stresses in early-age concrete: Comparison of different creep models.” ACI Mater. J., 93(3), 254–259.
Gurtin, M., and Sternberg, E. (1962). “On the linear theory of viscoelasticity.” Arch. Ration. Mech. Anal., 11, 291–356.
Hashin, Z. (1983). “Analysis of composite materials—A survey.” ASME Trans. J. Appl. Mech., 50(3), 481–505.
Hatt, W. (1907). “Notes on effect of time element in loading reinforced concrete beams.” Proc., ASTM, 7, 421–422.
Hauggard, A., Damkilde, L., and Hansen, P. (1999). “Transitional thermal creep of early age concrete.” J. Eng. Mech., 125(4), 458–465.
Hellmich, C., and Mang, H. (2005). “Shotcrete elasticity revisited in the framework of continuum micromechanics: From submicron to meter level.” J. Mater. Civ. Eng., 17(3), 246–256.
Hellmich, C., Mang, H., and Ulm, F.-J. (2001). “Hybrid method for quantification of stress states in shotcrete tunnel shells: Combination of 3D in situ displacement measurements and thermochemoplastic material law.” Comput. Struct., 79(22–25), 2103–2115.
Hellmich, C., Ulm, F.-J., and Mang, H. (1999). “Multisurface chemoplasticity. I: Material model for shotcrete.” J. Eng. Mech., 125(6), 692–701.
Hershey, A. (1954). “The elasticity of an isotropic aggregate of anisotropic cubic crystals.” ASME Trans. J. Appl. Mech., 21, 236–240.
Hill, R. (1963). “Elastic properties of reinforced solids: Some theoretical principles.” J. Mech. Phys. Solids, 11(5), 357–372.
Hill, R. (1965). “Continuum micro-mechanics of elastoplastic polycrystals.” J. Mech. Phys. Solids, 13(2), 89–101.
Hua, C., Ehrlacher, A., and Acker, P. (1997). “Analyses and models of the autogeneous shrinkage of hardening cement paste—II: Modelling at scale of hydrating grains.” Cem. Concr. Res., 27(2), 245–258.
Jennings, H. (2004). “Colloid model of C-S-H and implications to the problem of creep and shrinkage.” Concr. Sci. Eng., 37(265), 59–69.
Jordaan, I., and Illston, J. (1969). “The creep of sealed concrete under multiaxial compressive stresses.” Mag. Concrete Res., 21(69), 195–204.
Knauss, W. (2003). “Viscoelasticity and the time-dependent fracture of polymers.” Comprehensive Structural Integrity, 2(7), 383–428.
Kröner, E. (1978). “Self-consistent scheme and graded disorder in polycrystal elasticity.” J. Phys. F: Met. Phys., 8, 2261–2267.
Lakes, R. (1999). Viscoelastic solids, CRC, Boca Raton, Fla.
Laplante, P. (1993). “Propriétés mécaniques de bétons durcissants: Analyse comparée des bétons classique et à trés hautes performances (Mechanical properties of hardening concrete: A comparative analysis of ordinary and high performance concretes).” Ph.D. thesis, Ecole Nationale des Ponts et Chaussées, Paris (in French).
Larson, M., and Jonasson, J.-E. (2003). “Linear logarithmic model for concrete creep. I: Formulation and evaluation.” J. Adv. Concr. Technol., 1(2), 172–187.
Laws, N. (1977). “The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material.” J. Elast., 7(1), 91–97.
Laws, N., and McLaughlin, R. (1978). “Self-consistent estimates for the viscoelastic creep compliances of composite materials.” Proc. R. Soc. London, Ser. A, 359, 251–273.
Le, Q., Le Pape, Y., Meftah, F., and He, Q. (2005). “Upscaling techniques for modeling the basic creep of cement paste.” Concreep 7: Creep, shrinkage and durability of concrete and concrete structures, G. Pijaudier-Cabot, B. Gérard, and P. Acker, eds., Hermes Science Publishing, London, 113–118.
Le, Q., Le Pape, Y., Meftah, F., and He, Q. (2007). “Analytical derivation of the asymptotic creep behavior of concrete by multiscale homogenization.” MHM 2007: Modelling of heterogeneous materials with applications in construction and biomedical engineering, M. Jirásek, Z. Bittnar, and H. Mang, eds., 320–321.
Le Chatelier, H. (1904). Recherches expérimentales sur la constitution des mortiers hydrauliques, Dunod, Paris.
Lechner, M., Hellmich, C., and Mang, H. (2001). “Short-term creep of shotcrete—Thermochemoplastic material modelling and nonlinear analysis of a laboratory test and of a NATM excavation by the finite element method.” Proc., Int. Symp. on Discontinuous Modelling of Cohesive-Frictional Materials—Lecture Notes in Physics, P. Vermeer, S. Diebels, W. Ehlers, S. Luding, and E. Ramm, eds., Springer, Berlin, 47–62.
Li, K., Gao, X.-L., and Roy, A. (2006). “Micromechanical modeling of viscoelastic properties of carbon nanotube-reinforced polymer composites.” Mech. Advanced Mater. Struct., 13(4), 317–328.
Lokhorst, S., and van Breugel, K. (1997). “Simulation of the effect of geometrical changes of the microstructure on the deformational behaviour of hardening concrete.” Cem. Concr. Res., 27(10), 1465–1470.
Lynam, C. (1934). Growth and movement in portland cement concrete, Oxford University Press, London.
Maple. (2005). “Maple Ver. 10.” Waterloo maple software, Ontario, Canada, ⟨www.maplesoft.com⟩.
McHenry, D. (1943). “A new aspect of creep in concrete and its application to design.” Proc., ASTM, 43, 1069–1084.
Mehlhorn, G. (1996). Der Ingenieurbau: Grundwissen—Werkstoffe, Elastizitätstheorie (Basic knowledge in civil engineering: Materials and elasticity theory), Vol. 4, Ernst and Sohn, Berlin (in German).
Mitzel, A. (1967). “Concrete creep and shrinkage functions.” Build. Sci., 2(3), 259–265.
Mori, T., and Tanaka, K. (1973). “Average stress in matrix and average elastic energy of materials with misfitting inclusions.” Acta Metall., 21(5), 571–574.
Nilsen, A., and Monteiro, P. (1993). “Concrete: A three-phase material.” Cem. Concr. Res., 23(1), 147–151.
Persson, B. (1998). “Quasi-instantaneous and long-term deformations of high performance concrete with sealed curing.” Adv. Cem. Based Mater., 8(1), 1–16.
Pickett, G. (1942). “The effect of change in moisture content of the creep of concrete under a sustained load.” Proc., J. Am. Concr. Inst., 38(4), 333–355.
Read, W., Jr. (1950). “Stress analysis for compressible viscoelastic media.” J. Appl. Phys., 21(7), 671–674.
Roelfstra, P., Sadouki, H., and Wittmann, F. (1985). “Le béton numérique.” Mater. Struct., 18(107), 327–335.
Salençon, J. (1983). Viscoélasticité (Viscoelasticity), Presses de l’Ecole Nationale des Ponts et Chaussées, Paris (in French).
Sanahuja, J., Dormieux, L., and Chanvillard, G. (2007). “Modelling elasticity of a hydrating cement paste.” Cem. Concr. Res., 37(10), 1427–1439.
Schwarzl, F., and Struik, L. (1968). “Analysis of relaxation measurements.” Adv. Mol. Relax. Processes, 1(3), 201–255.
Schwefel, H.-P. (1977). Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie (Numerical optimization of computer models by means of the evolution strategy), Birkhäuser Verlag, Basel und Stuttgart, Germany (in German).
Sercombe, J., Hellmich, C., Ulm, F.-J., and Mang, H. (2000). “Modeling of early-age creep of shotcrete. I: Model and model parameters.” J. Eng. Mech., 126(3), 284–291.
Sips, R. (1951). “General theory of deformation of viscoelastic substances.” J. Polym. Sci., 7(2–3), 191–205.
Solís-Carcaño, R., and Moreno, E. (2007). “Evaluation of concrete made with crushed limestone aggregate based on ultrasonic pulse velocity.” Constr. Build. Mater., ⟨www.sciencedirect.com⟩.
Spiegel, M. (1990). Statistik (Statistics), 2nd Ed., McGraw-Hill Book Company Europe (in German).
Suquet, P. (1997). Continuum micromechanics, Vol. 377 of CISM courses and lectures, Springer, Wien, New York.
Tamtsia, B., and Beaudoin, J. (2000). “Basic creep of hardened cement paste—A re-examination of the role of water.” Cem. Concr. Res., 30(9), 1465–1475.
Taylor, H. (1997). Cement chemistry, 2nd Ed., Thomas Telford, London.
Tennis, P., and Jennings, H. (2000). “A model for two types of calcium silicate hydrate in the microstructure of portland cement pastes.” Cem. Concr. Res., 30(6), 855–863.
Ter Haar, D. (1950). “A phenomenological theory of visco-elastic behaviour.” Physica (Amsterdam), 16(9), 719–737.
Thienel, K.-C., and Rostásy, F. (1996). “Transient creep of concrete under biaxial stress and high temperature.” Cem. Concr. Res., 26(9), 1409–1422.
Ulm, F.-J., and Acker, P. (1998). “Le point sur le fluage et la recouvrance des bétons (Concrete creep and creep recovery: A review).” Bull. Lab. Ponts Chaussees, 20(2), 73–82 (in French).
Ulm, F.-J., Constantinides, G., and Heukamp, F. (2004). “Is concrete a poromechanics material?—A multiscale investigation of poroelastic properties.” Concr. Sci. Eng., 37(265), 43–58.
Ulm, F.-J., and Coussy, O. (1996). “Strength growth as chemo-plastic hardening in early age concrete.” J. Eng. Mech., 122(12), 1123–1132.
Valkó, P., and Abate, J. (2004). “Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion.” Comput. Math. Appl., 48(3–4), 629–636.
van Breugel, K. (1995). “Numerical simulation of hydration and microstructural development in hardening cement-based materials. I: Theory.” Cem. Concr. Res., 25(2), 319–331.
Vandamme, M., and Ulm, F.-J. (2006). “Viscoelastic solutions for conical indentation.” Int. J. Solids Struct., 43(10), 3142–3165.
Velez, K., Maximilien, S., Damidot, D., Fantozzi, G., and Sorrentino, F. (2001). “Determination by nanoindentation of elastic modulus and hardness of pure constituents of portland cement clinker.” Cem. Concr. Res., 31(4), 555–561.
Wesche, K. (1974). Baustoffe für tragende Bauteile (Building materials for structural components), 3rd Ed., Bauverlag, Wiesbaden, Germany (in German).
Widder, D. (1946). The Laplace transform, 2nd Ed., Princeton University Press, Princeton, N.J.
Wittmann, F., Roelfstra, P., and Sadouki, H. (1984). “Simulation and analysis of composite structures.” Mater. Sci. Eng., 68(2), 239–248.
Zaoui, A. (1997). “Chap. 6.” Structural morphology and constitutive behavior of microheterogeneous materials, Springer, Wien, New York, 291–347.
Zaoui, A. (2002). “Continuum micromechanics: Survey.” J. Eng. Mech., 128(8), 808–816.
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Published In
Journal of Engineering Mechanics
Volume 135 • Issue 4 • April 2009
Pages: 307 - 323
Copyright
© 2009 ASCE.
History
Received: Oct 25, 2007
Accepted: Nov 4, 2008
Published online: Apr 1, 2009
Published in print: Apr 2009
Notes
Note. Associate Editor: Dinesh R. Katti
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