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Introduction

The drying shrinkage of mortar and concrete, as with almost any other physical property of these materials, depends on the microstructure over a wide range of length scales [1,2]. The shrinkage of the C-S-H (C = CaO, S = SiO₂, H = H₂O) gel, which controls the shrinkage of cement-based materials, is determined, at the most fundamental level, by the rearrangement of water in the nanometer-scale gel pores as relative humidity is reduced. Cement paste is a complex composite at the micrometer-length scale, being a mixture of a phase that shrinks under drying (C-S-H), and restraining phases that have negligible shrinkage upon drying, like unreacted cement and calcium hydroxide (CH). In portland cement paste, there are aluminate, ferrite, and sulfate phases that can also act as restraining phases. The analysis in this paper is general, but where we consider cement paste microstructure and properties directly, we specialize to C₃S cement paste for simplicity. Mortar and concrete are also complex random composites, but at the millimeter to meter-length scale. Also composed of both non-shrinking aggregate and shrinking cement paste, their overall shrinkage response depends on the relative amounts of these phases and their arrangement in the microstructure.

Models have already been developed to simulate the movement of water in structural models of C-S-H and porous Vycor at the nanometer scale [3,4]. At this level, a model must account for solid, liquid, and air phases. In addition, shrinkage stresses, generated by surface forces, must be modeled at the solid-liquid, liquid-air, and solid-air interfaces. For this paper's analysis, the material is treated as a composite at a larger length scale where each phase is a continuum material with specified properties. For shrinkage, each phase is given a set of elastic moduli and an unrestrained shrinkage parameter. The unrestrained shrinkage (sometimes referred to as the "eigenstrain" of a phase [5] is the shrinkage strain that would occur in an isolated, unrestrained condition. This description makes use of the analogy between length change due to temperature changes and length changes which occur under humidity changes [6].

This paper reports the results of modeling mortars using digital image models where each pixel is 10 micrometers across. Mortar is known to be a three-phase composite: aggregate, bulk cement paste, and interfacial transition zone (ITZ) cement paste. The ITZ is modelled as a phase with a single property, rather than a gradient of properties, which admittedly is a simplification. Each phase is given appropriate elastic moduli and unrestrained shrinkage strains. These models are two-dimensional because of present computer-imposed resolution limitations in three dimensions. In 2-D, large enough arrays (number of pixels per unit length) can be generated to adequately resolve the interfacial zone region as well as represent a statistically meaningful number of sand grains. Carrying out this kind of computation in 3-D will be dependent on computer memory and speed developments in the next few years, although the high end of present-day supercomputers are very close to being able to run 512³ finite element simulations. This would give resolution comparable to the present 2-D simulations. As a result of the restriction to 2-D, qualitative insight into existing experimental results is sought, rather than quantitative agreement.

It is important to note that only elastic, reversible shrinkage is computed, which is another limitation of the model at present. It is, of course, well-known that the shrinkage of cement-based materials has a strong irreversible component, and contributions from creep [1]. However, a model predicting this type of behavior for arbitrary microstructures is currently unavailable, and would be extremely computer intensive if it did exist. Thus, it seems logical to fully explore the simpler case of elastic shrinkage before moving on to more complicated models. There is general agreement in the scientific community that ITZ properties do affect overall moduli and shrinkage. This paper represents the beginning of an attempt to show how the ITZ affects these properties.

Section II describes the details of the models themselves. Section III discusses the effect on the elastic moduli of mortar of varying the interfacial zone and bulk cement paste moduli. Section IV describes differences in unrestrained shrinkage and moduli between the interfacial zone and bulk cement paste and how this affects shrinkage of mortar. Also, an exact analytical equation is introduced and used to analyze real experimental shrinkage data for the shrinkage of mortars containing small concentrations of sand. Section V describes computational results for cement paste shrinkage at the micrometer scale. These results help explain the relationship of shrinkages in the interfacial zone to shrinkages in the bulk cement paste found in the previous section. Section VI summarizes the results of this work.

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