Homogenized moduli and local stress fields of unidirectional nano-composites
Introduction
Despite rapid progress in the science and fabrication technology of materials with microstructures characterized by reinforcement or porosities in the nanometer range, much work remains to be done in establishing better understanding of the structure-property relationship for this class of materials. It is known that surface elasticity plays a substantial role in controlling the overall elastic moduli of this class of materials, which may be quantified through computational techniques such as the homogenization theory or classical micromechanics approaches. A significant body of work already exists illustrating the effects of porosity radius and volume fraction on the homogenized moduli and local stress concentrations at pore boundaries of nanoporous materials based on the surface-elasticity model of Gurtin in Murdoch [1] and classical elasticity and micromechanics approaches, as well as finite-element solutions of the unit cell problem and elasticity solutions of the multiple inclusion problem, e.g., Sharma et al. [2], Duan et al. [3], [4], Chen et al. [5], Gao et al. [6], Mogilevskaya et al. [7], [8]. The reader is referred to the recent review articles by Wang et al. [9] and Eremeyev [10] for additional references in this area. Nonetheless, the classical micromechanical models, which continue to be employed in homogenized moduli calculations of nanocomposites, cf., Kim et al. [11], either neglect the interaction of adjacent reinforcement/porosities or account for this interaction only in an average sense. Such interaction is significant at large reinforcement/porosity volume fractions and its neglect may significantly underestimate the local stress fields that affect failure if not the homogenized moduli.
The alternative approaches that yield both accurate homogenized moduli and local stress fields are based on finite-element or finite-volume solutions of the unit cell problem of a periodic composite, cf., Pindera et al. [12], Charalambakis [13], Cavalcante et al. [14]. However, these approaches require substantial effort during the problem definition stage, e.g., unit cell discretization, and may not be easily employed in parametric studies aimed at identifying structure-property relationships. Another complication specifically related to the surface elasticity effects in nano-composites encountered in finite-element solutions are numerical instabilities at small pore radii due to singular-like stress gradients, Gao et al. [6]. Stress discontinuities at very small pore radii are readily observed in the exact elasticity solution of the Kirsch problem with surface elasticity effects.
Thus far, relatively little work has been conducted aimed at incorporating surface elasticity effects into elasticity-based homogenization approaches, despite reviving interest in these techniques within the past 15 years motivated by the seminal work of Nemat-Nasser et al. [15]. These approaches offer a number of advantages relative to the numerical homogenization schemes, including extremely fast input data construction, and ability to investigate composites with very thin coatings, interphases or thin-walled fibers without experiencing convergence issues common to finite-element analyses. The reconstruction of local fields from homogenized-based results within a multi-scale analysis framework is also significantly faster. The equivalent inhomogeneity technique (EIT) with surface-elasticity effects proposed by Mogilevskaya et al. [8] is a promising exception, but has been only employed thus far in the prediction of homogenized moduli of transversely isotropic nano-composites in limited porosity and pore radius ranges, and limited discussion of the impact of these parameters on full-field stress fields, Mogilevskaya et al. [7].
The locally-exact homogenization theory (LEHT) for periodic materials, Drago and Pindera [16] and Wang and Pindera [17], [18], [19], differs from other elasticity-based solutions of the local unit cell problem such as the eigenstrain expansion technique, Caporale et al. [20], the equivalent inhomogeneity method, Mogilevskaya et al. [8], or the eigenfunction expansion technique, Sevostianov et al. [21], in the manner of periodic boundary conditions implementation. While the eigenstrain and eigenfunction expansion techniques employ doubly-periodic displacement field representations that satisfy periodicity conditions a priori, the locally-exact homogenization theory employs a balanced variational principle in enforcing periodicity conditions along the boundary of a unit cell. This variational principle produces rapid convergence of the displacement fields in cylindrical coordinates which satisfy both the Navier's equations and interfacial continuity conditions in the interior of the unit cell representative of rectangular, square or hexagonal arrays of transversely isotropic or cylindrically orthotropic inclusions. As a result, converged homogenized moduli and local stress fields alike are obtained with relatively few terms in the displacement field representation, Wang and Pindera [22].
In this contribution, we incorporate surface-elasticity effects into the locally-exact homogenization theory and employ the extended computational model to investigate both homogenized moduli and local stress fields in a wide range of porosity volume fraction and pore radius, enabled by the theory's numerical stability. Section 2 describes incorporation of surface effects into the theory's framework, transparently illustrating how the resulting structure is modified. The theory's predictive capability is subsequently verified in Section 3 upon comparison with classical elasticity solution in the limiting case of dilute porosity, equivalent inhomogeneity technique, finite-element and finite-volume solutions. Comparison with classical micromechanics models delineates their limits of applicability. Section 4 presents new results that illustrate the combined effects of pore volume fraction and radius on select homogenized moduli of square and hexagonal arrays of nanoscale porosities and the related local fields for high porosity content arrays where pore interaction is important. Comparison with the results of the widely employed composite cylinder assemblage (CCA) model delineates its limits of applicability while highlighting the importance of adjacent pore interaction on the local stress fields that may affect failure. Discussion and Conclusions are given in Section 5.
Section snippets
Locally-exact homogenization theory
The details of different features of the locally-exact homogenization theory have been documented in a sequence of papers. The original derivation of the rectangular version with isotropic phases has been described by Drago and Pindera [16]. Its extension to hexagonal arrays with transversely isotropic phases, coated and hollow as well as circumferentially orthotropic fibers has been given most recently by Wang and Pindera [17], [18], [19]. Below we focus on the implementation of surface-energy
Comparison with published results
We first validate the extended theory by comparison with the results obtained using classical elasticity solutions for the Kirsch problem, classical micromechanics approaches, the elasticity-based equivalent inhomogeneity technique, and the finite-element and finite-volume methods. The validation is conducted for nanoporous aluminum with different pore volume fractions, radii and arrays because of available data on surface elasticity moduli which are widely used in the literature in validating
Effects of array type on homogenized moduli and stress fields
Having established the accuracy of the locally-exact homogenization theory with surface elasticity effects, we first conduct a parametric study to investigate the effect of array type on the homogenized moduli in unidirectional nanoporous aluminum as functions of both pore volume fraction and radius for select loadings. Such parametric studies may be accomplished efficiently owing to the stability of the locally-exact homogenization theory, its accuracy and quick convergence characteristics.
Discussion and conclusions
The elasticity-based, locally-exact homogenization theory developed for periodic materials characterized by hexagonal and rectangular/square unit cells with transversely isotropic phases has been extended by incorporating the surface-elasticity effects based on the Gurtin-Murdoch interface model, and extensively verified. This extension enables rapid parametric studies of unidirectional composites reinforced by fibrils or weakened by cylindrical porosities with diameters in the nanometer range
Acknowledgment
Q.C. gratefully acknowledges the financial support of the China Scholarship Council and the National Natural Science Foundation of China (Grants 51505364 and 51605365).
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