Table 1 Area of surface, and volume and porosity of particles by outermost radius \( R_{t} \) (Maruyama 2003).
From: Modeling of Mechanical Properties of Concrete Mixed with Expansive Additive
Limitless part: \( R_{t} < 0.5 \) | |
Area of a surface | \( S_{1} = 4\pi R_{t}^{2} \) |
Volume | \( V_{1} = \frac{4}{3}\pi R_{t}^{3} \) |
Porosity | \( P_{1} = (1 - V_{1} )\cdot p_{EX} \) |
First period of contact: \( 0.5 \le R_{t} < \sqrt 2 /2 \) | |
Area of a surface | \( S_{2} = 4\pi R_{t}^{2} - 12\pi \left( {1 - \frac{0.5}{{R_{t} }}} \right) \) |
Volume | \( V_{2} = \frac{4}{3}\pi R_{t}^{3} - 6\pi \left( {\frac{2}{3}R_{t}^{3} - \frac{1}{2}R_{t}^{2} + \frac{1}{24}} \right) \) |
Porosity | \( P_{2} = (1 - V_{2} )\cdot p_{EX} \) |
Later period of contact: \( \sqrt 2 /2 \le R_{t} < \sqrt 3 /2 \) | |
Area of a surface | \( S_{3} = \mathop \int \limits_{{\sqrt {R_{t}^{2} - 1/2} }}^{1/2} \mathop \int \limits_{{(R_{t}^{2} - 1/2)/(4 - x^{2} )}}^{1/2} \frac{{R_{t} }}{{\sqrt {R_{t}^{2} - x^{2} - y^{2} } }}dxdy \) |
Volume | \( \begin{aligned} V_{3} = \,&2\sqrt {R_{t} ^{2} - 1/2} + 16\int\limits_{{\sqrt {R_{t} ^{2} - 1/2} }}^{{1/2}} \left( {\frac{1}{2}\cdot 0.5 \cdot \sqrt {R_{t} ^{2} - x^{2} - 1/4} } \right)\\ &+\, \frac{{R_t ^2}-{x^2}}{2} \times \left[ \frac{\pi}{4}-Arc\,cos \left (\frac{0.5}{\sqrt{{R_t ^2}-{x^2}}} \right )\right] {\rm d}x \end{aligned}\) |
Porosity | \( P_{3} = (1 - V_{3} )\cdot p_{EX} \) |