The influence of micromechanical parameters considering the interfacial phase on effective elastic properties of microencapsulated self-healing composite

Abstract

The microencapsulated self-healing composite (SHC) has received much attention for their ability to automatically detect and repair cracks. However, the general micromechanical models relating to SHC did not adequately consider the complex interaction between components and the interfacial transition zone. To address these problems, the four-phase micromechanical model containing core-wall-interface-matrix is proposed based on the composite-sphere theory and the Mori–Tanaka method in this study, and the effective properties of SHC are investigated by regarding the parameter of microscopic component as variables. The results showed that the predictive effective properties of SHC lie between the upper and lower limits obtained from the investigations of Walpole, indicating that the proposed micromechanical model is feasible, and the effect of interfacial strength and interfacial thickness on the properties of SHC is significant, indicating that it is necessary to build the micromechanical model consisting of the interfacial phase in this study, the core-to-wall ratio, size, and the strength of capsule wall have high sensitivity to the effective properties of SHC. These results can help investigate the influence of individual components on the properties of SHC, which are useful to guide the selection of individual components for functional materials.

Appendices A

Equation (3) can be solved after making use of the equilibrium equations and the continuity conditions at the interfaces between the different regions; the solution can be written formally as

$$\begin{aligned} \frac{\mu _{\mathrm {e}}}{\mu _{\mathrm {m}}}=\frac{-B \pm \sqrt{B^2-A D}}{A}, \end{aligned}$$

(A1)

where

$$\begin{aligned} A= & {} 8\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 4-v_{\mathrm {m}}\right) \eta _{1} \mathrm {v}^{\frac{10}{3}}-2\left[ 63\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right) \eta _{2}+2 \eta _{2} \eta _{3}\right] \mathrm {v}^{\frac{7}{3}} \nonumber \\&+252\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \eta _{2} \mathrm {v}^{\frac{5}{3}} -50\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 70-12 v_{\mathrm {m}} +8 v_{\mathrm {m}}{ }^{2}\right) \eta _{2} \mathrm {v}+4\left( 7-10 v_{\mathrm {m}}\right) \eta _{2} \eta _{3}, \end{aligned}$$

(A2)

$$\begin{aligned} B= & {} -2\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 1-5 v_{\mathrm {m}}\right) \eta _{1} \mathrm {v}^{\frac{10}{3}}+4\left[ 63\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right) \eta _{2} +2 \eta _{2} \eta _{3}\right] \mathrm {v}^{\frac{7}{3}}\nonumber \\&-504\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \eta _{2} \mathrm {v}^{\frac{5}{3}} +150\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 3-v_{\mathrm {m}}\right) v_{\mathrm {m}} \eta _{2} \mathrm {~V}+3\left( 15 v_{\mathrm {m}}-7\right) \eta _{2} \eta _{3}, \end{aligned}$$

(A3)

$$\begin{aligned} D= & {} 4\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 5 v_{\mathrm {m}}-7\right) \eta _{1} \mathrm {v}^{\frac{10}{3}}+2\left[ 63\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right) \eta _{2} +2 \eta _{2} \eta _{3}\right] \mathrm {v}^{\frac{7}{3}}\nonumber \\&+252\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \eta _{2} \mathrm {v}^{\frac{5}{3}} +25\left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( v_{\mathrm {m}}{ }^{2}-7\right) \eta _{2} \mathrm {v}-\left( 7+5 v_{\mathrm {m}}\right) \eta _{2} \eta _{3}, \end{aligned}$$

(A4)

with

$$\begin{aligned} \eta _{1}= & {} \left[ \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right] \left( 49-50 v_{\mathrm {m}}\right) +35\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-1\right) \left( v_{\mathrm {i}}-2 v_{\mathrm {m}}\right) +35\left( 2 v_{\mathrm {i}}-v_{\mathrm {m}}\right) , \end{aligned}$$

(A5)

$$\begin{aligned} \eta _{2}= & {} 5 v_{\mathrm {i}}\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-8\right) +7\left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}+4\right) , \end{aligned}$$

(A6)

$$\begin{aligned} \eta _{3}= & {} \left( \frac{\mu _{\mathrm {i}}}{\mu _{\mathrm {m}}}-8\right) \left( 8-10 v_{\mathrm {m}}\right) +\left( 7-5 v_{\mathrm {m}}\right) . \end{aligned}$$

(A7)