A Modulus gradient elasticity model for nano-reinforced composites

Gülaşık, Hasan
In this work, nanocomposites and their numerical simulations are studied. At the beginning of the study, the properties of the polymer nanocomposites are explained based on a specific nano-inclusion and polymer matrix couple, namely, carbon nanotube (CNT) and thermoplastic polyetheretherketone (PEEK) polymer. In a literature comparison study, it is shown that the properties of the constituents, interface properties, manufacturing methods, characterization methods and therefore mechanical properties of the CNT/PEEK nanocomposites can vary significantly among different studies. Classical elasticity formulations may become inadequate for the modeling of the nanostructured materials. They do not contain any information about the size and applicable from nanometer to meter scale. Moreover, they do not properly describe stress/strain singularities and are questionable if wavelength of deformation is comparable to dominant micro-structural length scale. Therefore, some extensions of the classical elasticity formulations have been proposed in literature. Two of the widely-used extensions, the Eringen’s nonlocal elasticity and the Aifantis’s gradient elasticity formulations, are explained. It is seen that nonlocal/gradient formulations include higher order fields and boundary conditions which are not easy to understand intuitively. They also have complex formulations and are computationally expensive. In this work, a new gradient elasticity formulation, the so-called E-grad model, is proposed to overcome some of the difficulties in the nonlocal and the gradient elasticity formulations. In the new formulation, similar to the differential relation between the local strain and the gradient enhanced strain in the classical models of gradient elasticity, a differential relation is proposed for the elastic constants of linear elasticity. Analytical and finite element solutions of the proposed formulation are derived for a one-dimensional inhomogeneous rod. The results of the proposed model are compared with a classical model of gradient elasticity for a one-dimensional model problem. It is seen that the discontinuities in the modulus, displacement, strain and stress fields are removed by the proposed model. Furthermore, there are no additional higher-order fields and boundary conditions and the numerical formulations are simpler than the nonlocal/gradient elasticity models. Then, the E-grad model is extended to more general three-dimensional inhomogeneous materials with isotropic linear elastic constituents. The finite element formulation for axisymmetric problems is derived and a model problem of a soft cylindrical rod with a stiff spherical inclusion is solved. It is seen that discontinuities and/or sharp changes in the modulus, displacement, strain and stress fields that exist in local formulations are smoothed out with the proposed model. The proposed model is compared with a micromechanical model from literature and experiments conducted on polyimide/silica nanocomposites. The results obtained by the proposed approach agree well with the experimentally measured values of the nanocomposite modulus. The model is also extended to obtain anisotropic macroscopic response by choosing different length scale parameters in different directions. At the end, a CNT reinforced polymer nanocomposite problem from literature is reconsidered in which the nanocomposite is assumed to be composed of four distinct phases: CNT, interface, interphase and bulk polymer. Rather than being homogeneous, the interphase is considered to be graded by the E-grad model. By using the E-grad model and the genetic algorithm optimization, homogenized elastic constants of the transversely isotropic effective fiber are calculated. It is seen that, although the effective fiber has higher modulus in the axial direction, it has lower modulus values in transverse and shear directions compared to the polymer matrix. Then, the effect of the orientation distribution of the effective fibers in a nanocomposite is taken into account by using an orientation distribution function. It is seen that, if effective fibers are aligned in a direction, the modulus of the composite increases in that direction as expected. However, it is also seen that, isotropic distribution of the effective fibers makes the composite to have lower modulus than the matrix due to low transverse and shear moduli of the effective fiber.


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Citation Formats
H. Gülaşık, “A Modulus gradient elasticity model for nano-reinforced composites,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.