Micromechanics Solution for the Elastic Moduli of Fiber-Reinforced Concrete

The overall elastic moduli of fiber-reinforced concrete composite materials are investigated by employing the theory of micromechanics. A method based on the Mori–Tanaka theory and triple inhomogeneities is found to provide a sufficiently accurate evaluation of the average elastic properties of fiber-reinforced concrete composite materials. The inhomogeneities of the materials are divided into three groups: a fine aggregate, a coarse aggregate, and fibers (steel or polymer). The elastic moduli of fiber-reinforced concrete composite materials are determined as functions of the physical properties and volume fraction of sand, gravel, fibers (steel or polymer), and cement paste as a matrix. The theoretical results obtained are compared with published experimental data. The parameters affecting the elastic moduli of fiber-reinforced concrete are discussed in detail.

Appendix

The explicit expression needed to calculate the tensors used in the text are as follows:

$$ \begin{array}{c}\hfill A=\left[\left(1-{f}_1\right) C+{f}_1{C}_1^{\ast}\right]\left({S}_1- I\right)-{C}_1^{\ast }{S}_1,\hfill \\ {}\hfill B=\left[\left(1-{f}_2\right) C+{f}_2{C}_2^{\ast}\right]\left({S}_2- I\right)-{C}_2^{\ast }{S}_2,\hfill \\ {}\hfill D=\left[\left(1-{f}_3\right) C+{f}_3{C}_3^{\ast}\right]\left({S}_3- I\right)-{C}_3^{\ast }{S}_3,\hfill \\ {}\hfill \begin{array}{cc}\hfill M=\left({C}_1^{\ast }- C\right)\left({S}_2- I\right),\hfill & \hfill N=\left({C}_2^{\ast }- C\right)\left({S}_1- I\right),\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill P=\left({C}_3^{\ast }- C\right)\left({S}_1- I\right),\hfill & \hfill Q=\left({C}_1^{\ast }- C\right)\left({S}_3- I\right),\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill R=\left({C}_2^{\ast }- C\right)\left({S}_3- I\right),\hfill & \hfill S=\left({C}_3^{\ast }- C\right)\left({S}_2- I\right),\hfill \end{array}\hfill \\ {}\hfill {\upsigma}^0\left( I-{C}_1^{\ast }{C}^{-1}\right)+ A\left\langle {\varepsilon}_1^{\ast}\right\rangle +{f}_2 M\left\langle {\varepsilon}_2^{\ast}\right\rangle +{f}_3 Q\left\langle {\varepsilon}_3^{\ast}\right\rangle =0,\hfill \\ {}\hfill {\upsigma}^0\left( I-{C}_2^{\ast }{C}^{-1}\right)+ B\left\langle {\varepsilon}_2^{\ast}\right\rangle +{f}_1 N\left\langle {\varepsilon}_1^{\ast}\right\rangle +{f}_3 R\left\langle {\varepsilon}_3^{\ast}\right\rangle =0,\hfill \\ {}\hfill {\upsigma}^0\left( I-{C}_3^{\ast }{C}^{-1}\right)+ D\left\langle {\varepsilon}_3^{\ast}\right\rangle +{f}_1 P\left\langle {\varepsilon}_1^{\ast}\right\rangle +{f}_2 S\left\langle {\varepsilon}_2^{\ast}\right\rangle =0,\hfill \\ {}\hfill \begin{array}{ccc}\hfill \left\langle {\varepsilon}_1^{\ast}\right\rangle =\alpha {\upsigma}^0,\hfill & \hfill \left\langle {\varepsilon}_2^{\ast}\right\rangle =\beta {\upsigma}^0,\hfill & \hfill \left\langle {\varepsilon}_3^{\ast}\right\rangle =\rho {\upsigma}^0.\hfill \end{array}\hfill \end{array} $$