Micromechanics analysis for thermal expansion coefficients of three-phase particle composites

Abstract

In the present investigation, a new equivalent micromechanics method is proposed, and then, an analysis model has been developed to estimate the nonlinear coefficients of thermal expansion (CTEs) of three-phase composites. As Compared with previous analytical models, the innovative point of this paper is that the influence of the debonding surface thickness is investigated. It is noted that the parameters of thickness of debonding surface have a significant effect on both the longitudinal CTEs and transverse CTEs. The CTEs of composites are also very sensitive to the different inclusion aspect ratios. The constitutive equation curves for different variable parameters can describe the influence of debonding damage on thermal expansion coefficients (CTEs) of the composites. The new model provides a direct prediction of CTEs and can account for the effects of inclusion aspect ratio, volume fractions and thickness of debonding surface.

Appendix

The components of Eshelby’s $S_{mnpq}$ tensor for different shapes of inclusions are provided as follows.

For the fiber-liked particles ( $a_1 >a_2 =a_3 )$,

$$\begin{aligned} S_{1111}= & {} \frac{1}{\left( {2-2\upsilon _m } \right) }\left[ {1-2\upsilon _m +\frac{3\lambda ^{2}-1}{\lambda ^{2}-1}-\left( {1-2\upsilon _m +\frac{3\lambda ^{2}}{\lambda ^{2}-1}} \right) \varphi } \right] \\ S_{1122}= & {} S_{1133} =\frac{-1}{\left( {2-2\upsilon _m } \right) }\left[ {1-2\upsilon _m +\frac{1}{\lambda ^{2}-1}} \right] +\frac{1}{\left( {2-2\upsilon _m } \right) }\left[ {1-2\upsilon _m +\frac{3}{2\left( {\lambda ^{2}-1} \right) }} \right] \varphi \\ S_{1212}= & {} S_{1313} =\frac{1}{\left( {4-4\upsilon _m } \right) }\left\{ {1-2\upsilon _m -\frac{\lambda ^{2}+1}{\lambda ^{2}-1}-\frac{1}{2}\left[ {1-2\upsilon _m +\frac{3\lambda ^{2}+3}{\lambda ^{2}-1}} \right] \varphi } \right\} \\ S_{2222}= & {} S_{3333} =\frac{3}{\left( {8-8\upsilon _m } \right) }\frac{\lambda ^{2}}{\lambda ^{2}-1}+\frac{1}{\left( {4-4\upsilon _m } \right) }\left[ {1-2\upsilon _m -\frac{9}{4\left( {\lambda ^{2}-1} \right) }} \right] \varphi \\ S_{2211}= & {} S_{3311} =\frac{1}{\left( {2-2\upsilon _m } \right) }\frac{\lambda ^{2}}{\lambda ^{2}-1}-\frac{1}{\left( {4-4\upsilon _m } \right) }\left[ {\frac{3\lambda ^{2}}{\lambda ^{2}-1}-\left( {1-2\upsilon _m } \right) } \right] \varphi \\ S_{2233}= & {} S_{3322} =\frac{1}{\left( {4-4\upsilon _m } \right) }\left[ {\frac{\lambda ^{2}}{2\lambda ^{2}-2}-\left( {1-2\upsilon _m +\frac{3}{4\lambda ^{2}-4}} \right) \varphi } \right] \\ S_{2323}= & {} S_{3232} =\frac{1}{\left( {4-4\upsilon _m } \right) }\left[ {\frac{\lambda ^{2}}{2\lambda ^{2}-2}+\left( {1-2\upsilon _m -\frac{3}{4\lambda ^{2}-4}} \right) \varphi } \right] \end{aligned}$$

where $\upsilon _m$ is the Poisson’s ratio of the matrix, $\lambda$ is the aspect ratio of the particles ( $\lambda =a_1 /a_3)$, and $\varphi$ is given as

$$\begin{aligned} \varphi= & {} \frac{\lambda }{\left( {\lambda ^{2}-1} \right) ^{3/2}}\left[ {\lambda \left( {\lambda ^{2}-1} \right) ^{1/2}-\hbox {cos}h^{-1}\lambda } \right] , \quad \left( {\lambda >1} \right) \\ \varphi= & {} \frac{\lambda }{\left( {1-\lambda ^{2}} \right) ^{3/2}}\left[ {\hbox {cos}h^{-1}\lambda -\lambda \left( {1-\lambda ^{2}} \right) ^{1/2}} \right] , \quad \left( {\lambda <1} \right) \end{aligned}$$

For the spherical particles $( a_1 =a_2 =a_3 )$,

$$\begin{aligned} S_{1111}= & {} S_{2222} =S_{3333} =\frac{7-5\upsilon _m }{15-15\upsilon _m} \\ S_{1122}= & {} S_{2233} =S_{3311} =\frac{5\upsilon _m -1}{15-15\upsilon _m} \\ S_{1212}= & {} S_{2323} =S_{3131} =\frac{4-5\upsilon _m }{15-15\upsilon _m} \end{aligned}$$

For the penny-liked inclusions ( ${a_1 }/{a_3 }\ll 0)$,

$$\begin{aligned} S_{1111}= & {} 1-\frac{1-2\upsilon _m }{\left( {4-4\upsilon _m } \right) }\pi \lambda , \qquad \quad S_{2222} =S_{3333} =\frac{8\upsilon _m -13}{\left( {32-\upsilon _m } \right) }\pi \lambda , \\ S_{2233}= & {} S_{3322} =\frac{8\upsilon _m -1}{\left( {32-\upsilon _m } \right) }\pi \lambda , \quad S_{2211} =S_{3311} =\frac{2\upsilon _m -1}{\left( {8-8\upsilon _m } \right) }\pi \lambda , \\ S_{1122}= & {} S_{1133} =\frac{\upsilon _m }{1-\upsilon _m }\left( {1-\frac{1+4\upsilon _m }{8\upsilon _m }\pi \lambda } \right) , \\ S_{2323}= & {} \frac{7-8\upsilon _m }{\left( {32-32\upsilon _m } \right) }\pi \lambda , \quad S_{1212} =S_{1313} =\frac{1}{2}\left[ {1-\frac{2-\upsilon _m }{4\left( {1-\upsilon _m } \right) }\pi \lambda } \right] \end{aligned}$$