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Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method

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Abstract

A model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal is discussed. The atomic arrangements of the matrix and the inclusion are both periodic along the \(x_{1}\)-direction and quasiperiodic along the \(x_{2}\)-direction in the \(ox_{1}x_{2}\)-coordinate system. Using the hypersingular integral equation method, the inclusion problem is reduced to solving a set of hypersingular integral equations. Based on the exact analytical solution of the singular phonon and phason stresses near the inclusion front, a numerical method of the hypersingular integral equation is proposed using the finite-part integral method. Finally, the numerical solutions for the phonon and phason stress intensity factors of some examples are given.

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Acknowledgements

The authors would like to express their special thanks to the National Natural Science Foundation of China (Project No. 11172320 and No. 11272341).

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, China

    Fei Lou, Taiyan Qin & Chunhui Xu

  2. College of Engineering, China Agricultural University, Beijing, 100083, China

    Ting Cao

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  1. Fei Lou
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  2. Ting Cao
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  3. Taiyan Qin
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Corresponding author

Correspondence to Taiyan Qin.

Appendix A: Material-Related Constants

Appendix A: Material-Related Constants

$$\begin{aligned} A_i= & {} s_1^2 s_2^2 s_3^2 (C_{44} K_2 -R_3^2 -C_{33} K_2 s_i^2 -C_{44} K_1 s_i^2 +2R_3 R_2 s_i^2 +C_{33} K_1 s_i^4 -R_2^2 s_i^4 )/s_i \\ B_i= & {} s_1^2 s_2^2 s_3^2 (-C_{13} K_2 -C_{44} K_2 +R_3^2 +R_3 R_1 +C_{13} K_1 s_i^2 +C_{44} K_1 s_i^2 -R_3 R_2 s_i^2 -R_1 R_2 s_i^2 )\\ C_i= & {} s_1^2 s_2^2 s_3^2 (C_{13} R_3 -C_{44} R_1 +C_{33} R_3 s_i^2 +C_{33} R_1 s_i^2 -C_{13} R_2 s_i^2 -C_{44} R_2 s_i^2 ) \\ D_i= & {} s_1^2 s_2^2 s_3^2 (C_{11} K_2 -C_{44} K_2 s_i^2 -C_{11} K_1 s_i^2 +R_3^2 s_i^2 +2R_3 R_1 s_i^2 +R_1^2 s_i^4 +C_{44} K_1 s_i^4)/s_i \\ E_i= & {} s_1^2 s_2^2 s_3^2 (-C_{11} R_3 -C_{13} R_3 s_i^2 -C_{13} R_1 s_i^2 -C_{44} R_1 s_i^2 +C_{11} R_2 s_i^2 -C_{44} R_2 s_i^4 )/s_i \\ G_i= & {} s_1^2 s_2^2 s_3^2 (C_{11} C_{44} +C_{13}^2 s_i^2 -C_{11} C_{13} s_i^2 +2C_{13} C_{44} s_i^2 +C_{33} C_{44} s_i^4 )/s_i \\ b_1= & {} 2\pi C_{11} (C_{44} K_2 -R_3^2 )(s_1^2 -s_2^2 )(s_3^2 -s_1^2 ) \\ b_2= & {} 2\pi C_{11} (C_{44} K_2 -R_3^2 )(s_2^2 -s_1^2 )(s_3^2 -s_2^2 ) \\ b_3= & {} 2\pi C_{11} (C_{44} K_2 -R_3^2 )(s_1^2 -s_3^2 )(s_3^2 -s_2^2 ) \\ a= & {} C_{33} C_{44} K_1 -C_{44} R_2^2 ,\ d=C_{11} C_{44} K_2 -C_{11} R_3^2 \\ b= & {} C_{33} \left[ {(R_1 +R_3 )^{2}-C_{44} K_2 )} \right] -K_1 \left[ {C_{11} C_{33} +C_{44}^2 -(C_{13} +C_{44} )^{2}} \right] \\&+R_2 \left[ {2R_3 C_{44}+R_2 C_{11}-2(C_{13}+C_{44})(R_1 +R_3)}\right] \\ c= & {} C_{11} (2R_2 R_3 -C_{33} K_2 -C_{44} K_1 )-C_{13} (2R_1 R_2 +2R_3^2 -C_{13} K_2 -2C_{44} K_2 )+C_{44} R_1^2 \\ \gamma _{11}= & {} C_{11} \frac{A_i }{b_i }+C_{13} \frac{B_i s_i }{b_i }+R_1 \frac{C_i s_i }{b_i },\quad \gamma _{12} =C_{11} \frac{B_i s_i }{b_i }+C_{13} \frac{D_i s_i^2 }{b_i }+R_1 \frac{E_i s_i^2 }{b_i }\\ \gamma _{21}= & {} C_{13} \frac{A_i }{b_i }-C_{33} \frac{B_i s_i }{b_i }-R_2 \frac{C_i s_i }{b_i },\quad \gamma _{22} =C_{13} \frac{B_i s_i }{b_i }+C_{33} \frac{D_i s_i^2 }{b_i }+R_2 \frac{E_i s_i^2 }{b_i }\\ \gamma _{31}= & {} C_{44} \frac{A_i s_i^2 }{b_i }+C_{44} \frac{B_i s_i }{b_i }+R_3 \frac{C_i s_i }{b_i },\quad \gamma _{32} =-C_{44} \frac{B_i s_i }{b_i }+C_{44} \frac{D_i }{b_i }+R_3 \frac{E_i }{b_i } \\ \gamma _{41}= & {} R_1 \frac{A_i }{b_i }-R_2 \frac{B_i s_i }{b_i }-K_1 \frac{C_i s_i }{b_i },\quad \gamma _{42} =R_1 \frac{B_i s_i }{b_i }+R_2 \frac{D_i s_i^2 }{b_i }+K_1 \frac{E_i s_i^2 }{b_i } \\ \gamma _{51}= & {} R_3 \frac{A_i s_i^2 }{b_i }+R_3 \frac{B_i s_i }{b_i }+K_2 \frac{C_i s_i }{b_i },\quad \gamma _{52} =-R_3 \frac{B_i s_i }{b_i }+R_3 \frac{D_i }{b_i }+K_2 \frac{E_i }{b_i} \end{aligned}$$

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Lou, F., Cao, T., Qin, T. et al. Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method. Acta Mech. Solida Sin. 32, 249–260 (2019). https://doi.org/10.1007/s10338-018-0072-0

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