## 2 Prediction of the CTE of Resin Concrete Made by Adding ZrW\({}_{2}\)Os Powders
### First Mixing  
ZrW\({}_{2}\)O\({}_{8}\) powders and resin binder are mixed. Denoting \(c_{1}\) and \(\left(1-c_{1}\right)\) are the volume fractions of ZrW\({}_{2}\)O\({}_{8}\) and resin binder in concrete, respectively. Let \(E_{0},\mu_{0},\alpha_{0}\) be the Young's modulus, [PERSON]'s ratio and CTE of the resin binder as matrix and \(E_{1},\mu_{1},\alpha_{1}\) those of the ZrW\({}_{2}\)Os powders as fillers. Model-A and Model-B are two different computing methods of CTE for determining the interactions between components of resin concrete. Filler strain is approximately equal to the CTE \(\overline{\alpha_{1}}\) of isolated filler embedded in an infinite matrix under the constraint of matrix without considering the interactions between ZrW\({}_{2}\)O\({}_{8}\) powders and between matrix and ZrW\({}_{2}\)O\({}_{8}\) powders. It is named Model-A of \"first mixing\". Thus, the CTE \(\alpha_{1}^{{}^{*}}\) of the mixtures can be given by [18]:  
\[\alpha_{1}^{{}^{*}}{=}c_{1}\overline{\alpha_{1}}+\left(1-c_{1}\right)\alpha_{0} \tag{1}\]  
where  
\[\overline{\alpha_{1}}{=}\frac{\frac{2E_{0}}{\left(1+\mu_{0}\right)}\alpha_{ 0}+\frac{E_{1}}{\left(1-2\mu_{1}\right)}\alpha_{1}}{\left(1+\mu_{0}\right)} \frac{E_{1}}{\left(1-2\mu_{1}\right)} \tag{2}\]  
Due to the dense distribution of fillers, the correlations and interactions between fillers and between the matrix and the filler are under consideration. It is named Model-B of \"first mixing\". \(\overline{G}\) represents the macro effective shear modulus of the mixture. Thus, the CTE \(\alpha_{2}^{{}^{*}}\) of the mixtures can be given by [18]:\[\alpha_{2}^{*} = \frac{\left(4\overline{G}+\frac{E_{1}}{1-2\,\mu_{1}}\right)\left(1- c_{1}\right)\frac{E_{0}\alpha_{0}}{3\left(1-2\,\mu_{0}\right)}+\left(4 \overline{G}+\frac{E_{0}}{1-2\,\mu_{0}\right)}\frac{E_{1}\alpha_{1}}{3\left(1- 2\,\mu_{1}\right)}\,c}{\left(4\overline{G}+\frac{E_{1}}{1-2\,\mu_{1}}\right) \frac{E_{0}}{3\left(1-2\,\mu_{0}\right)}+4\overline{G}c_{1}\left(\frac{E_{1}} {3\left(1-2\,\mu_{1}\right)}-\frac{E_{0}}{3\left(1-2\,\mu_{0}\right)}\right)} \tag{3}\]  
where  
\[\overline{G} = \frac{E_{0}}{2\left(1+\mu_{0}\right)}\left[1+\frac{c_{1}\left( \frac{E_{0}E_{1}}{\left(1+\mu_{0}\right)\left(1+\mu_{1}\right)}-1\right)}{1+ \left(1-c_{1}\right)\frac{2\left(4-5\,\mu_{0}\right)}{15\left(1-\mu_{0} \right)}\left(\frac{E_{0}E_{1}}{\left(1+\mu_{0}\right)\left(1+\mu_{1}\right) }-1\right)}\right] \tag{4}\]  
As pointed out in [19], a three-phase model is presented for the prediction of [PERSON]'s modulus of concrete. The three phases are inclusion, Interfacial Transition Zone (ITZ) and matrix, as shown in Figure 1. The Poisson's ratio of concrete experiences a smaller change than the shear modulus. Hence, in order to simplify the calculation process and be convenient for engineering applications, the transverse series model is used [20]. Thus, the [PERSON]'s ratio \(\mu_{t}^{*}\) of concrete is approximated by:  
\[\mu_{t}^{*}=c_{a}\mu_{a}+c_{b}\mu_{b}+c_{c}\mu_{c} \tag{5}\]  
Where \(E_{a},\mu_{a},c_{a}\) are the Young's modulus, [PERSON]'s ratio and CTE of matrix phase and \(E_{b},\mu_{b},c_{b}\) those of ITZ layer and \(E_{c},\mu_{c},c_{c}\) those of inclusion. According to [20], \(E_{b}=0.3E_{a}\), \(\mu_{b}=0.30\).