Abstract
This paper is concerned with the image force in cubic piezoelectric quasicrystal semi-infinite space and infinite space containing two dissimilar quasicrystal half-spaces. On the basis of the Stroh formalism, the expressions of the Green’s function for generalized displacement and stress under multi-physical loading conditions are obtained exactly. Also, the image force applied to a generalized line dislocation with different boundary conditions is taken into account. The image force for the traction-free and electrically open surface always attracts dislocation to the boundary, while that for clamped and electrically closed surface shows the opposite tendency. Illustrative examples, such as generalized line force and line charge, are given to present the mechanical behaviors of quasicrystals under different loading conditions and investigate the influences of material parameters and dislocation scheme on image force. The results show that generalized line force has little effect on atomic configurations, and image force F is the strongest when phonon and phason dislocations interact together.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12272402, 11972365, and 12102458) and China Agricultural University Education Foundation (No. 1101-2412001).
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Appendix
Appendix
By using Eqs. (1), (2), and (3), we have
By Eq. (29), we introduced a new function
For Eq. (A2), first, the right side is replaced by its complex conjugate, then the \(\alpha\) and \(\beta\) are interchanged, and finally, matrix transpose is applied on the right. Based on these operations, we can obtain that Eq. (A3) is equal to Eq. (A2).
Using Eq. (15), we get
Since H−1S is antisymmetric, Eq. (A4) can be re-expressed as follows:
In the above, H−1 and L are positive definite, while S is singular [40]. We can conclude that H−1- L is positive semi-definite.
Substituting Eqs. (37) and (38) into Eq. (36) to yield
Similar manipulation of Eq. (21), we have
By Eqs. (A7), (A6)1, and Eq. (22)1, we derive
hence, Eq. (A8) has the expression
With the aid of Eq. (A6)2, the following equation is obtained due to similarity
Equation (A9) has an alternative expression
Making use of Eqs. (A10) and (A11) give
or
where
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Mu, X., Xu, W., Zhu, Z. et al. Image force in cubic piezoelectric quasicrystal half-space and bi-material composite space. Acta Mech 234, 5331–5347 (2023). https://doi.org/10.1007/s00707-023-03651-x
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DOI: https://doi.org/10.1007/s00707-023-03651-x