INHOMOGENEOUS INTERFACIAL TRANSITION ZONE MODEL FOR THE BULK MODULUS OF MORTAR

doi:10.1016/S0008-8846(97)00086-0

Cement and Concrete Research

Volume 27, Issue 7, July 1997, Pages 1113-1122

https://doi.org/10.1016/S0008-8846(97)00086-0 Get rights and content

Abstract

The macroscopic bulk modulus of mortar and concrete is modeled by assuming that each inclusion (fine or coarse aggregate) is spherical, and is surrounded by an interfacial transition zone (ITZ) in which the elastic moduli vary smoothly as a power-law function of radial distance from the center of the inclusion. The exponent in the power law can be chosen based on the estimated thickness of the ITZ, or by fitting the power law to measured porosity profiles. For this model, an analytical expression has been found by Lutz and Zimmerman (J. Appl. Mech., 1996) for the macroscopic bulk modulus. The macroscopic bulk modulus depends on known properties such as the elastic moduli of the bulk cement paste and the inclusions, the volume fraction of the inclusions, the elastic moduli at the interface between the cement paste and inclusion, and the thickness of the ITZ. In this paper the inhomogeneous ITZ model is used to analyze the data of Wang et al. (Cem. Conc. Res., 1988) on the bulk modulus of mortar containing sand inclusions. By fitting the measured moduli to the model predictions, we can estimate, in a non-destructive manner, the elastic moduli within the ITZ. For Wang's specimens, it is inferred that the elastic moduli at the interface with the inclusions is 30–50% less than in the bulk cement paste. © 1997 Elsevier Science Ltd

Introduction

The earliest and simplest models of the elastic moduli of cementitious materials were based on the assumption that concrete (or mortar) consists of two phases: aggregate (or sand) particles and cement paste. Under this assumption, simple mixing rules, such as a volume-average of the stiffnesses (Voigt model) or a volume-average of the compliances (Reuss model), can be used to estimate the effective elastic moduli. More sophisticated models can be used that account for the fact that the cement paste is the connected phase, whereas the inclusions form a disconnected, dispersed phase. Various mathematical theories have been proposed to predict the effective elastic moduli of this type of particulate composite. Zimmerman et al. [1]used the Kuster-Toksöz theory [2]to study the effect of sand inclusion concentration on the effective moduli of mortar. Yang and Huang [3]used the Mori-Tanaka theory [4]to account for the presence of both sand and aggregate inclusions, each having their own elastic moduli. A review of two-component models for the macroscopic elastic moduli of concrete and mortar has been given by Mehta and Monteiro [5].

Recently, however, it has become recognized that the cement paste should not be considered to be a homogeneous phase. It is now known that the structure of the cement paste in the vicinity of the inclusions differs from that of bulk cement paste 5, 6. The region in which the presence of the inclusions affects the properties of the cement paste is known as the Interfacial Transition Zone (ITZ). In this zone the porosity is greatest near the inclusions, and decreases with increasing distance from the inclusion [7]. Neubauer et al. [8]and Ramesh et al. [9]have developed models in which the ITZ is represented by a thin, shell-like region that surrounds each inclusion. These three-shell (inclusion/ITZ/cement paste) models should be more realistic than two-component models, but still represent an oversimplification, in that they assume that the elastic moduli are uniform within the ITZ. Lutz et al. [10]proposed a model in which the elastic moduli in the cement paste decay according to a power law function of the distance from the center of the inclusion. An analytical solution for the effective bulk modulus of this inhomogeneous ITZ model has been derived by Lutz and Zimmerman [11]. In the present paper we use this inhomogeneous ITZ model to analyze measurements of elastic moduli made by Wang et al. [12]on a suite of water-saturated mortar specimens that contained varying volume fractions of sand inclusions.

Section snippets

Model of the Transition Zone

Lutz et al. [10]proposed the following conceptual model of concrete or mortar (Fig. 1). The aggregate particles are assumed to be spherical, with radius a. The elastic moduli {K_in, G_in within the inclusions are assumed to be uniform. Outside of each inclusion, the elastic moduli are assumed to vary smoothly with radius, according to the following power law equations: $K(r)=K_{cp} +(K_{if} −K_{cp})(r/a)^{−β},$ $G(r)=G_{cp} +(G_{if} =G_{cp})(r/a)^{−β}$ where the subscript cp refers to the bulk cement paste, and the subscript if

Analytical Solution for the Effective Bulk Modulus

The analytical solution developed in [10]for the effective bulk modulus of a material containing spherical inclusions surrounded by a power-law-type interfacial transition zone described by , can be written as $K_{eff} K_{cp} = 1+(4G_{cp} /3K_{cp})fc 1−fc,$ where $f= 3(K_{in} −K_{if}) ∑ n=0 ∞ Γ_{nβ} + K_{if} +(43)G_{if} ∑ n=0 ∞ nβΓ_{nβ} 3(K_{in} −K_{if}) ∑ n=0 ∞ Γ_{nβ+3} + K_{if} +(43)G_{if} ∑ n=0 ∞ (nβ+3)Γ_{nβ+3} .$ and c is the volume fraction of the inclusions. The coefficients Γ are found from the following recursion relation: $Γ_{n+β} = − (K_{if} −K_{cp}) n^{2} +(β−3)n−3β +(43)(G_{if} −G_{cp}) n^{2} +(β−3)n$

Application of Model to Experimental Data

Wang et al. [12]measured compressional and shear velocities on a suite of water-saturated mortar specimens that contained varying amounts of sand inclusions. The bulk moduli of their specimens can be found from the two measured wavespeeds using the relationship $K_{eff} =ρ_{eff} V^{2}_{p} − 43 V^{2}_{s},$ where the effective density is related to the densities of the cement paste and sand inclusions by $ρ_{eff} =(1−c)ρ_{cp} +cρ_{in} .$

The mortar consisted of cement paste with sand inclusions, the diameters of which ranged from 590–840

Summary and Discussion

A conceptual model of concrete has been proposed in which the aggregate particles are treated as spheres surrounded by a radially-inhomogeneous cement paste. The two elastic moduli in the cement paste are each represented by a constant term plus a term that varies with radius according to a power law. The exponent in the power law can be determined either using a visually-estimated “thickness” of the interfacial transition zone, or by fitting a power law to measured porosity profiles. The

Acknowledgements

The work of M. Lutz was supported by a University of California President's Postdoctoral Fellowship. The work of P. Monteiro was supported by grant NSF-8957183.

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