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Thermo-Chemo-Mechanical Model for Concrete. II: Damage and Creep
Abstract
In this work a coupled thermo-chemo-mechanical model for the behavior of concrete at early ages is proposed. This paper presents the formulation and assessment of the mechanical aspects of the model. Short- and long-term mechanical behaviors are modeled via a viscoelastic damage model that accounts for the aging effects. The short-term model is based on the framework of the continuum damage mechanics theory. A novel normalized format of the damage model is proposed, so that the phenomenon of aging is accounted for in a natural fashion. Long-term effects are included by incorporating a creep model inspired in the microprestress-solidification theory.
... Moreover, only linear creep is considered. The possibility of coupling with damage (Bazant 1995;Cervera et al. 1999;Mazzotti and Savoia 2003;Benboudjema et al. 2005;Rossi et al. 2012;Saliba et al. 2012) that could also exist at an early age (Briffaut et al. 2011;Switek-Rey et al. 2016;Han et al. 2017;Torrenti 2018) is not considered here, meaning that only linear creep tests have been selected. After a brief presentation of these tests, the results are compared to a prediction using Eq. ...
Basic creep plays an important role in assessing the risk of early-age cracking in massive structures. In recent decades, several models have been developed to characterize how the hydration process impacts the development of basic creep. This study investigates the basic creep of various concrete mixes across different ages at loading. The analysis focuses on the very early stages (less than 24 hours) and early stages (less than 28 days) of concrete development. It is shown that a logarithmic expression that contains two parameters describing the material can accurately model basic creep from a very early age. One parameter relates to the creep amplitude and depends solely on the composition of the concrete. The other relates to the kinetics of creep and depends on the age of the material at loading and the nature of the concrete mixture. The logarithmic expression corresponds to a rheological model consisting of a single dashpot wherein viscosity exhibits a linear evolution over time. The model offers the advantage of eliminating the need to store the entire stress history for computing the stress resulting from the restriction of the free deformation. This approach significantly reduces computation time. A power-law correlation is also observed between the material aging parameter and the degree of hydration. This relationship depends on the composition. At least two compressive creep tests performed at two different degrees of hydration are needed to calibrate the material parameters and consider the effect of aging on basic creep compliance.
Investigated are thermodynamic restrictions on rate-type creep laws for porous materials which slowly solidify while carrying load (aging of concrete, i. e. , gradual filling of pores in concrete by hydration products) or slowly melt (gradual dehydration of concrete at high temperature). Thermodynamic potentials (Helmholtz free energy and complementary or Gibbs free energy) are determined. The chemical dissipation of elastic energy is calculated and the condition of its positiveness is proposed; this requires that elastic relations be introduced in terms of stress and strain rates for solidifying materials and in terms of stresses and strains for melting materials. Some creep laws used in practice are found to have thermodynamically inadmissible form. Creep laws of thermodynamically correct form are shown. The known forms of such laws often cannot, however, fit available long-time creep test data for concrete very well, unless some material parameters or rates are allowed to have thermodynamically inadmissible negative values for short periods of time.
This paper presents a detailed study of early age compressive stress-strain responses of low-(30 MPa), medium (70 MPa) and high-strength (100 MPa) concretes. In particular, the stress-strain responses during the first 72 hr after casting are closely followed. Tests were carried out on 100×200-mm (4×8-in.) concrete cylinders. The influence of concrete strength on temperature rise due to temperature-matched curing is presented. The effect of three different curing conditions on the development of concrete compressive strength and elastic moduli is described: 1) temperature-matched; 2) sealed: and 3) air-dried curing. It was found that for early ages, the compressive strength and elastic modulus gain of temperature-matched cured concrete cylinders are higher than those of the sealed cylinders, which, in turn, are higher than those of the air-dried cylinders. After an initial retardation period, the 70 and 100 MPa concretes showed a higher rate of elastic modulus gain than the 30 MPa concrete. It was observed that the current ACI Code expression for predicting the elastic modulus overestimates the stiffness for very early age concretes and for concrete strengths above 50 MPa.
The problem of long time strength (evaluation of time to rupture) is of obvious relevance for various machine parts working under high temperature conditions. The existing data on specimens tested under uniaxial tension are insufficient for general loading conditions and inhomogeneous stress states: intensive experimental investigations are being conducted in this direction. Theoretical modelling of long time strength in the framework of continuum mechanics appears to be important. In recently published work of Hoff (1953), the moment of failure of a rod under tension is defined as the one at which the cross-sectional area becomes zero as a result of quasiviscous flow. His result is in satisfactory agreement with experimental data. A similar concept was used by Katz (1957) and by Rosenblum (1957) in their studies of times to rupture of thick-walled pipes and of a rotating disk with a hole. We note, however, that the experimental data points typically lie below the theoretical predictions and that rupture occurs at elongations not exceeding several tens per cent. The concept of Hoff has certain limitations. It predicts, for example, that creep under torsion does not result in rupture, contrary to observations. Also, his scheme does not explain fractures at small strains (`brittle' ruptures) and the change of character of rupture (from `viscous' to `brittle') if the material is not sufficiently stable. Here, we suggest a theoretical model for the time to rupture with the account of embrittlement. We emphasize that the physics of such phenomena is very complex and has not been studied sufficiently. Although we discuss microcracking, the results can be interpreted in a more general way, in terms of development of damage.
In Part I of this work we developed continuum isotropic and anisotropic elastoplastic-damage models, formulated either in strain space on the basis of the effective stress concept, or in stress space and employing the dual notion of effective strain. In Part II we consider in detail the variational formulation and subsequent numerical implementation of these models. The former relies crucially on the notions of maximum plastic and maximum damage dissipation. The latter makes systematic use of the operator splitting methodology to derive unconditionally stable algorithms for the numerical integration of the elastoplastic-damage equations of evolution. Appropriate extensions to treat the proposed rate-dependent (viscous) damage are also presented. For stress-based damage models, our numerical treatment relies on a new three-step operator split. The algorithms developed lead to simple and efficient stress update procedures suitable for large-scale finite element calculations. Application is made to a class of inviscid and rate-dependent cap models with an isotropic strain-based damage mechanism. Remarkably good agreement with existing experimental data that includes complicated stress paths is obtained. Numerical examples are also presented that demonstrate the good performance of the proposed algorithms.
Continuum Damage Mechanics (C. D. M. ) has developed continuously since the early works of Kachanov and Rabotnov. It constitutes a practical tool to take into account the various damaging processes in materials and structures at a macroscopic continuum level. The main basic features of C. D. M. are considered together with its present capabilities, including damage definitions and measures, and its incorporation into a thermodynamic general framework. Practical damage growth equations will be reviewed in the second part of the paper.