A note on the divergence problem arising in calculation of Green's function for a 2D piezoelectric thin film strip with free boundaries and containing straight line defects: Fourier approach for bodies of cubic symmetry

作者：Bojar Konrad^*

来源：International Journal of Engineering Science, 2010, 48(12): 1842-1855.

DOI：10.1016/j.ijengsci.2010.07.004

• 2D piezoelectricity
• Green's function
• Electro-elastic fields
• Line defect
• Cubic symmetry

Analytic formulas for the Green's function and the coupled electro-elastic fields for a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects have already been found some years ago. These formulas exploit the well-known Stroh formalism and the Fourier approach, so the result is given as the Fourier integral and therefore its numerical implementation should pose no problem. However, in this note we show that for the case of cubic symmetry this form of the Green's function contains strong divergences, excluding possibilities of direct application of well-known numerical schemes. It is also shown that these divergences translate to divergences of the corresponding electro-elastic fields of a single defect. By means of a rigorous analysis it is demonstrated that imposing physical conditions implied by the nature of the problem all of these divergences cancel and the final, physical result exhibits expected, regular behavior at infinity. Unfortunately, although the nature of this problem is purely mathematical, it leads to irremovable numerical infinity - infinity uncertainties which tend to spread over the whole Fourier domain and severely impede engineering applications of the Green's function. This motivates us to compute the exact form of all divergent terms. These novel formulas will serve as a guide when establishing numerically stable algorithms for engineering computations involving the system in question.

• 出版日期2010-12