Concrete is a composite material. A typical mixture contains a cement paste (=hydrated cement) matrix and inclusion particles of various sizes, ranging from the smallest sand grains of diameter 100 µm, to the largest rocks of diameter 10 mm to 20 mm. But concrete is not just a simple two-phase composite. Upon closer examination, one finds that the presence of the grains in the paste changes a thin layer of matrix material surrounding each inclusion. The cement paste matrix in this shell is different, usually more porous, than the bulk of the surrounding cement paste matrix. Typical widths of this layer are in the range 10 µm to 30 µm. This layer is often called the interfacial transition zone (ITZ) [1,2,3,4].
Concrete, therefore, consists of at least three distinct types of constituents. If the altered shell of matrix material is associated with the grains, the point of view may be taken (and this will become our point of view) that concrete consists of a matrix material containing composite inclusions. Assigning each of these phases different linear elastic moduli results in a complicated effective elastic moduli problem.
In fact, the problem is still more complicated. The "shell" around each inclusion actually contains a gradient of properties, since the cement paste matrix density is least at the particle surface and increases outwards to the full matrix density [2,5,6]. If the inclusions in concrete are modelled as spheres, which is a good approximation for many concrete mixtures, then the dilute limit, with a single spherical inclusion surrounded by a thin layer containing a spherically symmetric gradient of elastic properties, can be handled exactly [7,8,9,10]. But the non-dilute microstructure of concrete, with a wide particle size distribution of inclusions, each surrounded by overlapping gradients of properties, is very difficult to treat analytically, by numerical methods, or by effective medium theory (EMT). However, it has been shown that a multi-scale model can be used in order to map this complex microstructure into a simpler, but still complicated, microstructure, where the shell layers can be treated theoretically as having uniform properties [7,11,12]. The three phase composite described above then becomes appropriate for the concrete elastic moduli problem, and is the focus of this paper.
Differential effective medium theory (D-EMT) for this three-phase microstructure has been developed previously in several ways [13,14,15,16,17,18]. Predictions for the electrical properties of this microstructure have recently been checked against random walk simulations [19,20]. These simulations are accurate and simple, but time-consuming, hence the use of effective medium theory. The most accurate D-EMT for this application was developed using a new idea in differential effective medium theory (D-EMT) [20]. The key idea was that a spherical inclusion, along with its surrounding thin shell of altered matrix material, was exactly mapped into a new, slightly larger homogenized inclusion, which included the hard but poorly conducting particle and the softer but better conducting shell, and which had a uniform conductivity. The new system of effective particles embedded in the matrix could then be treated easily using a differential effective medium theory (D-EMT).
A rather different approach to three-phase effective medium theory could be based on the self-consistent formulas of Christensen and Lo [21,22], but as we shall show, the method presented here has considerably more flexibility in the range of complex microstructures that can be incorporated into the model.
This new kind of D-EMT is extended to linear elastic properties in this paper. Similar to the idea used for conductivity, a spherical inclusion, with a surrounding shell layer, is mapped onto an effective particle of uniform elastic moduli. The problem then becomes a simple composite composed of spherical particles, of varying sizes and elastic moduli, embedded in a uniform matrix. This composite can then be treated in the usual D-EMT.
Except for some special models [23], the accuracy of most EMTs is often in doubt. From the point of view of tailoring the approximation to the specific material microstructure that we want to model, EMTs of any kind tend to be uncontrolled approximations. Checking the accuracy of an EMT by comparing its predictions to experimental results is inadequate from the theoretical point of view because the microstructure and phase elastic moduli are usually only approximately known experimentally. Observed discrepancies could be from the EMT, or equally well could arise from the approximate knowledge of the experimental material. A more satisfactory way of assessing the accuracy of EMT is to compare to exact analytical results, where microstructure and phase moduli are controlled by the user. But exact elastic results for non-trivial microstructures are rather rare, and only exist for certain microstructures (e.g., dilute limit of inclusions) or special choices of the moduli (e.g., equal shear moduli) [24,25]. In the case of conductivity, the D-EMT results could be checked on model concrete microstructures using accurate random walk simulations [20]. Recently, a special finite element method has been described that can compute the linear elastic moduli of an arbitrary digital image in 2-D or 3-D [26]. This method is used, after proper error analysis, to provide stringent tests of the new D-EMT equations. The results are found to compare favorably with the essentially exact finite element calculations, in both 2-D and 3-D, with a variety of simple inclusion size distributions.