Elliptic functions and lattice sums for effective properties of heterogeneous materials

Abstract

Effective properties of fiber-reinforced composites can be estimated by applying the asymptotic homogenization method. Analytical solutions are possible for infinite long circular fibers based on the elliptic quasi-periodic Weierstrass Zeta function. This process leads to numerical convergences issues related to lattice sums calculations. The lattice sums original series converge slowly, which make the calculation difficult. This problem needs to be addressed because effective properties are highly sensitive to these values. Therefore, a systematic review and analysis for the lattice sums are a necessity. In the present work, the Eisenstein–Rayleigh lattices sums are reviewed and numerically implemented for fiber-reinforced composites with parallelogram unit periodic cell whose fibers are centered, or not, at the coordinate origin. Numerical values are reported and compared with available data in the literature obtaining good agreements. In this work, new Eisenstein–Rayleigh lattice sums are obtained that are easy to implement and a set of tables with numerical values are given.

Appendix A

The numerical sensitivity of the lattice sums $S_{k+p}$ when $\left| \tau \right|$ increases is performed through an analysis of the numerical values of the infinite series $G^{-n}\left( \tau \right) =\sum \limits _{t=1}^{+\infty } {\frac{\mathrm{1}}{\sin ^{n}\left( {it\tau \pi } \right) }} ,\quad n=k+p$, $\left( {k+p=2,4,6,\ldots } \right)$ which are parts of the $S_{k+p}$ expressions.

In Table 8, an analysis of the numerical sensitivity of the $G^{-n}\left( \tau \right)$ is reported for different $\left| \tau \right|$ values and numbers of terms N of the sums. As it is observed, for a fixed $\left| \tau \right|$ only a few terms N are needed to obtain the sum value. Also, as $\left| \tau \right|$ increases, the required number of terms is getting lower and the sums tend to cero. Then, $\left| \tau \right| \rightarrow \infty$ implies $G_{N}^{-n} \left( \tau \right) =\sum \limits _{t=1}^N {\frac{\mathrm{1}}{\sin ^{\mathrm{k+p}}\left( {it\tau \pi } \right) }} \rightarrow 0$ for all values of $N\ge 1$. Therefore, from Eqs. (32)–(34), it is observed that the lattice sums $S_{k+p}$ is a linear combination of the sums $G^{-n}\left( \tau \right)$ to any period ratio$\left| \tau \right|$; thus, when $\left| \tau \right| \rightarrow \infty$ implies that $S_{2} =\frac{2\pi ^{2}}{\omega _{1}^{2} }\left[ {\frac{1}{6}+\sum \limits _{t=1}^\infty {\frac{1}{\mathrm{sin}^{\mathrm{2}}\left( {it\tau \pi } \right) }} } \right] \rightarrow \frac{\pi ^{2}}{3\omega _{1}^{2} }$, $S_{4} =\frac{2\pi ^{\mathrm{4}}}{\omega _{1}^{4} }\left[ {\frac{1}{90}+\sum \limits _{t=\hbox {1}}^\infty {\left( {\frac{1}{{\mathrm{{si}}{\mathrm{{n}}^\mathrm{{4}}}\left( {it\tau \pi } \right) }} - \frac{2}{{3\mathrm{{si}}{\mathrm{{n}}^\mathrm{{2}}}\left( {it\tau \pi } \right) }}} \right) } } \right] \rightarrow \frac{{{\pi ^\mathrm{{4}}}}}{{45\omega _1^4}}$, and $S_{6} =\frac{2\pi ^{\mathrm{6}}}{\omega _{1}^{6} }\left[ {\frac{1}{945}+\sum \limits _{t=\hbox {1}}^\infty {\left( {\frac{2}{{\mathrm{{15si}}{\mathrm{{n}}^\mathrm{{2}}}\left( {it\tau \pi } \right) }} + \sum \limits _{m = 2}^3 {\frac{{{{\left( { - 1} \right) }^{m + 1}}}}{{\mathrm{{si}}{\mathrm{{n}}^{2m}}\left( {it\tau \pi } \right) }}} } \right) } } \right] \rightarrow \frac{{2{\pi ^\mathrm{{6}}}}}{{945\omega _1^6}}$.

For $\omega _{1} =1$, we have that $S_{2} \rightarrow \frac{\pi ^{2}}{3}\approx \hbox {3.289868133696453}$, $S_{4} \rightarrow \frac{\pi ^{\mathrm{4}}}{45}\approx \hbox {2.164646467422276}$, and $S_{6} \rightarrow \frac{2\pi ^{\mathrm{6}}}{945}\approx \hbox {2.034686123968898}$, which are the values reported in Table 2 when $\left| \tau \right| \ge 6$. The remaining values of $S_{k+p}$ can be computed by the recursive formula Eq. (37). For these cases, a numerical precision of $1\times 10^{^{-8}}$ is considered.

Table 8 Numerical values of the series $G^{-n}\left( \tau \right)$ for different period ratios $\left| \tau \right|$ and number of terms N of the sums. The $(\times 10^{n})$ under the $\left| \tau \right|$ is a factor that multiplies the sum value