Continuous Retardation Spectrum for Solidification Theory of Concrete Creep

The basic creep of concrete is the time-dependent strain caused by a sustained stress in absence of moisture movements. It is the strain observed on sealed specimens. Similar to other properties of concrete, it is dependent on the age of concrete. as a consequence of long-time chemical reactions associated with the hydration of cement. This paper formulates the solidification theory with a continuous retardation spectrum. and shows how this spectrum can be readily and unambiguously identified from arbitrary measured creep curves and how it then can be easily converted to a discrete spectrum for numerical purposes. The identification of the continuous spectrum is based on Tschoegl's work on viscoelasticity of polymers. Attention is limited to basic creep. The basic creep of concrete is the time-dependent strain caused by a sustained stress in absence of moisture movements. It is the strain observed on sealed specimens. Similar to other properties of concrete, it is dependent on the age of concrete, as a consequence of long-time chemical reactions associated with the hydration of cement. The aging aspect of basic creep of concrete can be mathematically handled by two different approaches (Mathematical 1988): (1) The classical, direct approach which treats the material parameters involved in the creep model as empirical functions of age (Bazant and Wu 1974a.b); (2) the recently proposed approach called the solidification theory (Bazant and Prasannan 1989a,b), in which the material parameters for creep are considered to be age-independent but the volume fraction of the age-independent material increases with age. Only the latter approach has a solid foundation from the viewpoint of chemical thermodynamics. It also has an important practical advantage, namely, the characterization of creep by a nonaging model, which is much simpler. The aging in the solidification theory is introduced separately by means of a variation of the volume fraction of the solidifying viscoelastic material constituent. In the formulation of solidification theory. there are two separate problems. The first is how to describe the variation of the volume fraction of the solidified nonaging material constituent. The second is how to characterize non aging creep for the purposes of large-scale numerical analysis and correlate this characterization to the physics of the problem. Such a correlation is possible only if the creep of the non aging constituent is described in a rate-type form consisting of first-order differential equations. Such a rate-type form can be based on the Kelvin chain or the Maxwell chain. The Kelvin chain is a rheologic model composed of a series coupling of many Kelvin units. each of which consists of a parallel coupling of a spring and a dash pot. The Maxwell chain is a rheologic model composed of a parallel coupling of many Maxwell units, each of which consists of a series coupling of a spring and a dashpot. Roscoe (1950) proved that the Kelvin and Maxwell chains can each describe any given linear viscoelastic behavior with any desired accuracy. This means that there is no need for considering other more complicated rheologic models. and provides a justification for assuming one of the two basic rheologic models. Kelvin chain is more convenient because its parameters can be more easily identified from creep tests. The Maxwell chain parameters can be more easily identified from relaxation tests, but these are harder to carry out. In the original formulation of the solidification theory (Bazant and Prasannan 1989a.b). the creep of the nonaging constituent is described by a Kelvin chain with a finite number N of Kelvin units. Each Kelvin unit number fL is characterized by its spring modulus E". and retardation time T". = T]".IE"., where T]". = dashpot viscosity. The plot of liE". versus T". (fL = I. .... N), which is in viscoelasticity called the retardation spectrum, fully characterizes the material creep properties. For a finite number N of Kelvin units, as used in all the previous studies of concrete creep, the spectrum is discrete (because the T".-values are distributed along the time axis discretely). However, as is well known from classical (nonaging) viscoelasticity. identification of 'Walter P. Murphy. Prof. Civ. Engrg .. Tech. Inst.. Northwestern Univ .. 2145 Sheridan Rd .. Evanston. IL 60208-3109. °Asst. Prof.. Drexel Univ .• Philadelphia. PA 19104; formerly. Postdoctoral Res. Assoc .. N<irthwestern Univ .. 2145 Sheridan Rd .. Evanston. IL. Note. Discussion open until July I. 1995. To extend the closing date one month. a written request must he filed with the ASCE Manager of Journals. The manuscript for this paper was suhmitted for review and pm,sihle puhlication on January 27. 1993. This paper is part of the Journal of Engineering Mechanics. Vol. 12\. No.2. Fehruary. 1995. ©ASCE. ISSN 0733-9399/95/0002-0281-0288/$2.00 + $.25 per page. Paper No. 5521. JOURNAL OF ENGINEERING MECHANICS 281 a broad discrete spectrum from test data is an ill-posed problem because different retardation times can give almost equally good fits of the measured creep curves. Thus, the discrete retardation times must be chosen, although with some restrictions (Mathematical 1988). The arbitrariness of the choice of retardation times is disturbing. Moreover, a simple method to determine the retardation spectrum has been available only for creep curves in the form of the power law, log-law, or log-power law (BaZant and Prasannan 1989b). The purpose of the present paper, whose basic results were summarized at a recent conference (Bazant and Xi 1993), is to formulate the solidification theory with a continuous retardation spectrum, and show how this spectrum can be readily and unambiguously identified from arbitrary measured creep curves and how it then can be easily converted to a discrete spectrum for numerical purposes. The identification of the continuous spectrum will be based on Tschoegl's (1971. 1989) work on viscoelasticity of polymers. Attention will be limited to basic creep. The additional creep due to drying may be determined in the manner described elsewhere [e.g. Mathematical (1988)]. GENERALIZED KELVIN CHAIN MODEL FOR NONAGING BASIC CREEP The linear viscoelastic behavior may be completely characterized by the compliance function J(t, t'), representing the strain f at age t of concrete caused by a uniaxial sustained (constant) stress (T = 1 applied at age t'. The response for any stress history (T(t) then follows by the principle of superposition. For three-dimensional behavior one would generally need a similar compliance function for shear strain, but for concrete this is not needed because the Poisson's ratio for basic creep happens to be approximately constant and equal to its instantaneous (elastic) value (about 0.18). Denoting as lIEo = q] = instantaneous elastic strain caused by (T = I, one may write J(t, t') = q, + qt, t') (I) where C(t, t') = creep compliance function. After lengthy studies (Mathematical 1988) it transpired that the creep description is the simplest if q] is taken as the deformation corresponding to extremely fast loading, shorter than about lO-y s, which of course cannot be directly measured but must be obtained by asymptotically extrapolating the measured creep curve in the logarithmic time scale to -YO. The advantage of doing this is that q] (or Eo, called the asymptotic modulus) can be considered as age-independent because in log-time scale the short-time creep curve (stress duration 104 s to 1 hr) happens to be a smooth (leftward) extension of the long-time creep curve (Mathematical 1988). The normal static elastic modulus, which agrees quite well with the value recommended for design purposes, is then obtained as E(t) = lU(t + ~, t), where ~ = 0.1 day. This expression also gives a good description of the age dependence of E(t). The dynamic modulus is obtained for ~ = lO-x s. For the nonaging Kelvin chain model with N Kelvin units, the creep compliance function C(t, t') is given by the Dirichlet series N C(~) = L. AI<[1 e <iT.]; 1-1-0--1 . h 1 Wit A =I" EI< (2) where ~ = t t'; t = time (age of concrete); t' = time (age) at the moment of loading and E". = elastic moduli of Kelvin units. In (1), the retardation times T I< can be chosen but the choice must satisfy certain well-known restrictions [e.g. Mathematical (1988)]. The values of AI<' which characterize the strain increment during the time leg corresponding to T"., have to be determined by optimum fitting of the measured creep curves. In the previous studies, certain semiempirical formulas have been derived to evaluate A". from the available creep response curves (Bazant and Prasannan 1989a,b). Those formulas have been calibrated by nonlinear curve-fitting of computer results, and their validity was restricted to the log-power law. In general, when a slightly different creep law is required, those formulas are not valid. Another problem of those formulas is that, although a certain set of the optimized A values may suffice to accurately describe the given creep behavior, this set is not unique, d:pending on the choice of values T w To deal with general creep laws and to avoid the aforementioned weak points including the nonuniqueness, an effective approach is to introduce the continuous Kelvin chain model with infinitely many Kelvin units and retardation times spaced infinitely closely, for which A". becomes a continuous retardation spectrum. Such a spectrum may be evaluated by a certain method developed in the theory of viscoelasticity (Tschoegl 1989). Then the discrete spectrum A"., which is required for computer analysis, can be obtained simply by discretizing the known continuous retardation spectrum. CONTINUOUS RETARDATION SPECTRUM AND INVERSE TRANSFORMATION METHOD Eq. (1) may be approximated in a continuous form 282 JOURNAL OF ENGINEERING MECHANICS