Temperature Effect on Concrete Creep Modeled by Microprestress-Solidification Theory

The previously developed microprestress-solidification theory for concrete creep and shrinkage is generalized for the effect of temperature ~not exceeding 100°C!. The solidification model separates the viscoelasticity of the solid constituent, the cement gel, from the chemical aging of material caused by solidification of cement and characterized by the growth of volume fraction of hydration products. This permits considering the viscoelastic constituent as non-aging. The temperature dependence of the rates of creep and of volume growth is characterized by two transformed time variables based on the activation energies of hydration and creep. The concept of microprestress achieves a grand unification of theory in which the long-term aging and all transient hygrothermal effects simply become different consequences of one and the same physical phenomenon. The microprestress, which is independent of the applied load, is initially produced by incompatible volume changes in the microstructure during hydration, and later builds up when changes of moisture content and temperature create a thermodynamic imbalance between the chemical potentials of vapor and adsorbed water in the nanopores of cement gel. As recently shown, this simultaneously captures two basic effects: First, the creep decreases with increasing age at loading after the growth of the volume fraction of hydrated cement has ceased; and, second, the drying creep, i.e., the transient creep increases due to drying ~Pickett effect! which overpowers the effect of steady-state moisture content ~i.e., less moisture—less creep!. Now it is demonstrated that the microprestress buildup and relaxation also captures a third effect: The transitional thermal creep, i.e., the transient creep increase due to temperature change. For computations, an efficient ~exponential-type! integration algorithm is developed. Finite element simulations, in which the apparent creep due to microcracking is taken into account separately, are used to identify the constitutive parameters and a satisfactory agreement with typical test data is achieved. DOI: 10.1061/~ASCE!0733-9399~2004!130:6~691! CE Database subject headings: Temperature effects; Concrete; Creep; Shrinkage; Solidification; Aging, Viscoelasticity; Microstructure; Thermodynamics; Adsorption. Introduction and Overview of Physical Mechanisms The arduous protracted quest for a realistic physically based creep and shrinkage model for Portland cement concrete has been confounded during the last several decades by three complex phenomena: 1. The aging of concrete, which is manifested by a significant decrease of creep with the age at loading and is of two types: a. Shorter-term chemical aging, which ceases at room temperature after about a year and is caused by the fact that new solids are produced by the slowly advancing chemical reactions of cement hydration and deposited ~in an McCormick School Professor and W. P. Murphy Professor of Civil Engineering and Materials Science, Northwestern Univ., Evanston, IL 60208. Research Associate, Dept. of Structural Engineering, Technical Univ. ~Politecnico! of Milan, Milan 20133, Italy. Professor, Dept. of Structural Engineering, Technical Univ. ~Politecnico! of Milan, Milan 20133, Italy; formerly, Visiting Fellow, Northwestern Univ., Evanston, IL 60208. Note. Associate Editor: A. Rajah Anandarajah. Discussion open until November 1, 2004. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November 19, 2002; approved on March 4, 2003. This paper is part of the Journal of Engineering Mechanics, Vol. 130, No. 6, June 1, 2004. ©ASCE, ISSN 0733-9399/2004/6-691–699/$18.00. JO essentially stress-free state! on the walls of capillary pores; and b. Long-term non-chemical aging, manifested by the fact that the decrease of creep with the age at loading continues unabated even for many years after the degree of hydration of cement ceased to grow. Recently, this phenomenon was explained by the relaxation of microprestress ~Bažant et al. 1997a!, although a long-term increase of bonding due to ‘‘polymerization’’ in calcium silicate hydrates might also play a role ~Bažant and Prasannan 1989a!. 2. The drying creep effect, also called the stress-induced shrinkage ~or Pickett effect!, which is a transient effect consisting in the fact that the apparent creep during drying is much larger than the basic creep ~i.e., creep at moisture saturation! while the creep after drying ~i.e., after reaching thermodynamic equilibrium with a reduced environmental humidity! is much smaller than the basic creep. The physical source of drying creep is now known ~Bažant and Xi 1994! to involve two different mechanisms: a. The apparent mechanism consisting of an apparent additional creep due to microcracking ~Wittmann 1974, 1980, 1982; Wittmann and Roelfstra 1980! or strain-softening damage ~Bažant and Wu 1974a,b; Bažant 1975; Bažant and Chern 1985a,b; Bažant and Xi 1994; Granger et al. 1994!, which are equivalent processes from the viewpoint of constitutive modeling. The creep at variable humidity is conventionally defined by the difference in deformation URNAL OF ENGINEERING MECHANICS © ASCE / JUNE 2004 / 691 between a loaded creep specimen and its companion loadfree shrinkage specimen. This difference is increased not because of creep amplification in the loaded specimen ~in which the microcracking is suppressed by compressive load, considered as compressive!, but because microcracking due to nonuniform local shrinkage tends to decrease the deformation of the load-free companion specimen, causing the overall deformation measured on a drying load-free specimen to be less than the ‘‘true’’ material shrinkage. The reason for calling this mechanism ‘‘apparent’’ is twofold: ~1! It does not reside in the constitutive material properties, and ~2! it is not happening in the creep specimen itself, but in its companion. Obviously, this mechanism has little effect in bending creep or torsional creep, and works in the opposite sense for tensile creep ~Bažant and Moschovidis 1973; Bažant and Xi 1994!. b. A true mechanism that resides in the nanostructure and consists in the fact that the rate of shear ~slip! due to breakages and restorations of bonds in the calcium silicate hydrates is reduced ~or amplified! by a decrease ~or increase! in the magnitude of compressive microprestress that is acting across the slip planes, the stress change being produced by a change in the chemical potential ~i.e., the Gibbs free energy per unit mass! of pore water due to drying ~Bažant 1972a,b, 1975!. 3. The transitional thermal creep, which represents a transient increase of creep after a temperature change, both heating and cooling. In the case of cooling, the transient increase is of the opposite sign than the final change in creep rate after a steady-state lower temperature has been regained. Like the drying creep effect, this effect has two analogous mechanisms: a. An apparent macroscopic mechanism, due to thermally induced microcracking and similar to drying creep; and b. A nanoscale mechanism due to a change in the level of microprestress caused by a change of chemical potential of nanopore water with a temperature change. While the apparent mechanisms 2a and 3a operate on the macroscopic level of the whole specimen or structure ~on the scale of centimeters and meters!, mechanism 1a operates on the level of capillary pores ~on the scale of micrometers!, and mechanisms 1b, 2b and 3b operate on the level of nanopores in calcium silicate hydrates ~on the scale of nanometers!. Development of the solidification theory ~Bažant and Prasannan 1989a,b! showed that the chemical aging ~mechanism 1a! can be separated from the viscoelastic constitutive model if that model is formulated not for concrete but for the solidifying constituent—the hardened cement gel ~i.e., the solid, consisting mostly of calcium silicate hydrates, which forms the skeleton of hardened cement paste!, and if the chemical aging is interpreted as a volume growth of the solidifying constituent ~per unit volume of concrete!. Since epitaxial growth must be ruled out because crystal structure is lacking, the model is formulated under the restriction that the newly solidified material is stress free at the moment it solidifies ~certain earlier models purportedly based on thermodynamics violated this restriction; Bažant 1977!. The solidification theory greatly reduced the number of unknowns in the modeling of aging creep and allowed removing the ambiguity ~nonuniqueness! that previously plagued the identification of agedependent moduli of the Kelvin or Maxwell chain model from the test data. The fact that, in the solidification theory, these moduli are constant allows describing the viscoelastic properties of the 692 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JUNE 2004 solidifying constituent with a continuous relaxation ~or retardation! spectrum, the benefit of which is a smooth spectrum that is unambiguous, i.e., uniquely identifiable from the test data ~Bažant and Xi 1995; Zi and Bažant 2001, 2002!. Mechanisms 1a and 1b were initially modeled separately ~Bažant and Prasannan 1989a!. In a later study ~Bažant et al. 1997a,b!, both 1b and 2b were explained by one and the same physical theory resting on the idea of relaxation of microprestress that is created in the solid nanostructure of cement gel either by microscopic chemical volume changes of various chemical species during hydration or by an imbalance of chemical potentials ~Gibbs free energy density per unit mass! among the four phases of pore water ~vapor, capillary, adsorbed, and hindered-adsorbed phases!. This paper shows how the microprestress-solidification theory, proposed and verified in Bažant et al. ~1997a,b! for mechanism 2b ~and suggested only in general terms, without details and verification, for mechanism 3b! can be extended for mechanism 3b in detail, and verifies that the experimental evidence can be matched. This extension ~broadly hinted in Bažant et al. 1997a,b! rests on recognizing that an imbalance among the chemical potentials of various phases of pore water is created not only by a change in the pore vapor pressure but also a change of temperature. The reason is that the chemical potential is a function of both. The present study finally provides the last building block needed to achieve a grand unity of physical modeling of all the creep-influencing phenomena on the nanoscale. The fact that phenomena, as diverse as the long-term aging, drying creep, and transitional thermal creep, can all be explained by one and the same model lends credence to the validity of the microprestress solidification model. The new unified model, which can be easily handled numerically with the powerful computers that exist today, may be expected to provide a more realistic common basis for the analysis many practically important problems—fire resistance of concrete structures, response to various hypothetical extreme nuclear reactor accidents, long-term safety of radioactive waste disposal, behavior of chemical technology vessels, effects of hydration heat in massive structures, and effects of environmental variations on structures. Interpretation of the available test data on the effect of temperature changes is complicated by two counteracting phenomena, namely the fact that, due to an acceleration of bond breakages causing shear slips in the nanostructure, ~1! a temperature rise amplifies the creep rate, while ~2! it also accelerates the aging due to hydration which is responsible for deceleration of creep. This dichotomy of the thermal effect, modeled in a more limited context long ago ~Bažant 1970a,b; Bažant and Wu 1974a,b; Bažant 1975!, will be automatically embedded in the present formulation. The detailed physical arguments justifying the microprestress-solidification theory are beyond the scope of this paper but can be found in an earlier paper ~Bažant et al. 1997a,b!. Note that the present constitutive law is not usable as a relation between the average stress and average strain in a cross section of beam or plate. Such relations ~e.g., model B3, Bažant 2000; Bažant and Baweja 2000!, popular in preliminary design, take into account the nonuniformity of stress over the cross section caused by drying and cracking, but only in a very crude manner. Microprestress-Solidification Theory Under uniaxial stress s ~and in the absence of significant plastic strains that may arise at high confining pressures!, the normal strain « of concrete can be decomposed as follows ~Fig. 1! «5« i1«v1« f1«cr1«sh1«T (1) where « 5instantaneous strain; «v5viscoelastic strain; « f 5purely viscous strain; «5inelastic strain due to cracking; and « sh and «5shrinkage and thermal strains caused by variations of humidity and temperature, respectively. The triaxial generalization of all the strain components except « can be based on the restrictions of material isotropy, and need not be discussed because it is well known ~e.g., Bažant 1975, 1982; RILEM 1988!. The generalization of cracking strain « to triaxial stress conditions breaks isotropy but it, too, need not be discussed as it is the subject of the works on fracture and damage ~e.g., Bažant and Planas 1998!. The instantaneous strain, i.e., the strain appearing immediately after applying uniaxial stress s, may be written as « 5q1s . At room temperature T5T05296 K ~23°C! and for saturation condition h5h051, the coefficient q1 is age independent. Bažant and Osman ~1976! and Bažant and Baweja ~1995, 2000! demonstrated this property by considering the compliances for load durations t ranging from 0.001 s to 10 h. They fitted the compliances with a smooth formula of the type J5q11ct . Then, they obtained q1 by optimizing the fit of data for various ages t8 at loading. The q1 values for various t8 were nearly the same, and the coefficient of variation of errors of the fit did not increase significantly when q1 was forced to be exactly the same for all t8. The viscoelastic strain «v, originating in the solid gel of calcium silicate hydrates, will be described according to the solidification theory ~Bažant and Prasannan 1989a,b!, which achieves separation of «v from the solidification process that causes longtime aging of concrete