An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

Abstract

In Parnell & Abrahams (2008 Proc. R. Soc. A 464, 1461–1482. (doi:10.1098/rspa.2007.025410.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.

Microstructural Topological Sensitivities of the Second-Order Macroscopic Model for Waves in Periodic Media

“…due to insertion of a small inhomogeneity inside the reference unit cell. Note that such an approximation is also relevant to small-volume-fraction composites [12,2], in which context the expansion has been extended to higher orders [18] in order to handle O(1) volume fractions. To our knowledge, however, such computations are normally performed for biphased materials (i.e., for inclusions in a homogeneous matrix) and not for additional inclusions nucleating in an already periodic medium (see also Remark 6).…”

“…More generally, our work extends the previous TS analyses of periodic media-performed in the context of elastostatics and structural shape optimization [14,3,26]-to dynamic, i.e., wave motion problems described via second-order homogenization. Equivalently, this study can be seen as a follow-up to the small-inclusion asymptotic analyses underpinning the (approximate) effective description of low-volume fraction dilutions [12] and two-phase periodic composites; e.g., [2,18]. In principle, the idea of topological perturbation can also be applied to the (leading-order) effective description [11] of higher, i.e., "optical" solution branches for a given periodic medium; the latter topic is, however, beyond the scope of this study.…”

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation

“…In this regime, effective elastic moduli, viscosities and other mechanical properties can be derived by using homogenization methods and micromechanical techniques [11][12][13]. In particular when the microstructure is distributed periodically, closed-form solutions can often be found [14][15][16][17]. The field of metamaterials is related to, but rather distinct from, this homogenization scenario in the sense that a metamaterial can give rise to a frequency-dependent response even in this low-frequency (homogenization) limit, usually being associated with an induced resonance of the microstructure [18,19].…”

Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface

“…On the computational modeling of problems on domains with small holes we mention, for example, the paper by Babuška, Soane, and Suri [7], which presents a computational method combining analytic knowledge of the solution singularities with finite element approximation of its smooth components. Moreover, a scheme for the effective properties of unidirectional fibre-reinforced media can be found in Joyce, Parnell, Assier, and Abrahams [26].…”