Homogenization-Based Mechanical Behavior Modeling of Composites Using Mean Green Operators for Infinite Inclusion Patterns or Networks Possibly Co-continuous with a Matrix

Abstract

The Fourier–Green homogenization method for estimating the behavior of composites was first developed for aggregates and inhomogeneity-reinforced (-weakened) matrices, based on Eshelby (Proc R Soc Lond, A 421:379–396, 1957, Proc R Soc Lond, A 252:561–569, 1959) solution of the isolated inclusion problem. The need to address increasingly complex structures opened fruitful development routes, firstly solving the inhomogeneity pair interaction problem and the one of heterogeneous (double or multilayered) inhomogeneities, in order to account for inclusion dense concentrations and patterns. This work reports recent developments from the authors and co-workers which examined in that framework possibly infinite inclusion patterns, possibly arranged into an infinite network possibly co-continuous with the embedding matrix and possibly evolving under strain. The proposed modeling amounts to determining the representative mean Green operator (mGO) for the infinite pattern or network in its current (strain evolving) state. Once the method foundations being summarized, previously solved “elementary” cases are recalled, concerning infinite coaxial alignments of spheres, spheroids or finite cylinders and planar alignments of infinite parallel cylinders or rectangular beams. It is next shown how other complex patterns or networks could be represented in combining such elementary ones. The mGO solution for a new family of inhomogeneous axial inclusion alignments is reported to support the discussion. Potential other application fields are evoked.

Appendix

The form of the elementary operators $\varvec{t}^{e}$ for isotropic elasticity. (from Franciosi and Lormand 2004; Franciosi 2005), to be retrieved in (Spagnuolo et al. 2020). Each $\varvec{t}^{\text{e}} (\varvec{\omega}) = \varvec{t}^{\text{e}} (\theta \text{,}\varphi )$ elementary operator in the GO $\varvec{t}^{V} (\varvec{r})$ or mGO $\overline{{\varvec{t}^{V} }}$ of V is an axisymmetric operator, defined in reference to the parallel planes of $\varvec{\omega}$ normal direction in some reference medium frame. For general elasticity anisotropy, considering the $(0,0)$-oriented $\varvec{t}^{e} (0,0)$ elementary operator for $(\theta \text{,}\varphi ) = (0,0)$, the only nonzero terms correspond to $\varvec{\omega}= (0,0,1)$ what only involves the $C_{m3p3}$ elastic moduli of the infinite medium, for $\varvec{C}$ expressed in the operator axes frame as $\varvec{C}(0,0)$. In this frame, the nonzero terms of the $\varvec{t}^{e} (0,0)$ operator are the $t_{p3j3}^{e} (\text{0,0})$ terms which make a symmetric 3 × 3 matrix, $\varvec{\Delta t}$ say, such that $t_{p3j3}^{e} = \Delta t_{pj}$. For elastic isotropy, since in all frames $C_{3333} = \lambda + \text{2 }\mu = \text{2 }\mu \text{ (1} - \nu \text{)}/\text{(1} - \text{2 }\nu \text{)}$ and $C_{1313} = C_{2323} = \mu$, the $\varvec{C}$ frame identification is made useless. It remains $\Delta t_{11} = 1 /C_{1313}$, $\Delta t_{22} = 1 /C_{2323}$, $\Delta t_{33} = 1 /C_{3333}$ and $t_{{\text{((2,3),(2,3))}}} = t_{{\text{((3,1),(3,1))}}} = 1/\text{4}\;\mu = B /\text{4}$, $t_{{\text{3333}}} = (1/\mu \text{)(1} - 0.5/\text{(1} - \nu \text{)}) = B + A$, $\forall (\theta \text{,}\varphi ) \equiv\varvec{\omega}$. One arrives at $t_{pqjn}^{e} \text{(}\varvec{\omega}\text{)}$ = $A\text{ }\tau_{pqjn}^{A} \text{(}\omega \text{) } + B\text{ }\tau_{pqjn}^{B} \text{(}\varvec{\omega}\text{)}$ with $\uptau_{{\text{pqjn}}}^{\text{A}} \text{(}\varvec{\omega}\text{)} = \omega_{\text{j}} \omega_{\text{p}} \omega_{\text{n}} \omega_{\text{q}}$ and $\tau_{pqjn}^{B} \text{(}\varvec{\omega}\text{)} = \left. {\left( {\delta_{jp} \text{ }\omega_{n} \omega_{q} } \right)} \right|_{(p,q),(j,n)}$ (Table 15.2).

Table 15.2 iijj (top) and ijij (bottom) terms of $\varvec{t}^{\text{e}} (\theta \text{,}\varphi )$, with “cθ”, “sθ” for “$\cos \theta$”, “$\sin \theta$” (resp. $\varphi$)

These terms are defined by even trigonometric functions $\text{cos}^{{\text{2r}}} \varphi \;\text{sin}^{{\text{2s}}} \varphi$, not vanishing upon integration over the unit sphere when multiplied by a positive and even function in $(\theta \text{,}\varphi )$ as the mean shape function $\overline{{\psi_{V} (\theta \text{,}\varphi )}}$ of V. Owing to the 5 $\left( {(1,0),(0,1)} \right)$ and $\left( {(2,0),(1,1),(0,2)} \right)$ possible values taken by both the (l, m) and (r, s) exponent pairs, and owing to the dependency relations between the trigonometric functions, all terms can be expressed in using only two of the five θ-functions, one in each exponent pair set within brackets, and similarly two of the five ϕ-ones, such as for example $(l,m) = (1,0),(2,0)$ and $(r,s) = (1,0),(2,0)$ which, respectively, correspond to the two functions $\cos^{2} (\iota )$ and $\cos^{4} (\iota )$ for each angle $\iota = \theta \text{,}\varphi$. Thus, the mGO terms over some domain V read $\overline{{t_{pqjn}^{V} }} = \int_{\Omega } {t_{pqjn}^{e} } \left(\varvec{\omega}\right)\overline{{\psi_{V} \left(\varvec{\omega}\right)}} {\text{d}}\varvec{\omega}$ and explicating the elementary (isotropic elastic) operator part in it, the integrals which then need to be calculated are two pairs selected among those of the form:

$$\overline{{t_{pqjn}^{V} }} = \int\limits_{\theta = 0}^{\pi } {\int\limits_{\varphi = 0}^{2\pi } {\overline{{\psi_{V} (\theta ,\varphi )}} \left( {\text{cos}^{{\text{2l}}} \theta \;\text{sin}^{{\text{2m}}} \theta \,\text{cos}^{{\text{2r}}} \varphi \;\text{sin}^{{\text{2s}}} \varphi } \right)} } {\kern 1pt} \sin \theta {\text{d}}\theta {\text{d}}\varphi .$$