Abstract
In this contribution an analysis of static properties of transversely isotropic, porous and nano-composites is considered. Present work features explicit formulas for effective coefficient in these types of composites. The reinforcements of the composites are a set of spheroidal inclusions with identical size and shape. The center is randomly distributed and the inclusions are embedded in an homogeneous infinite medium (matrix). An study of theoretical predictions obtained by Maxwell approach using two different density distribution functions, which describe the alignment inclusions is done. The method allows to report the static effective elastic coefficients in composites ensemble with inclusions of different geometrical shapes and configurations embedded into a matrix. The effective properties of composites are computed using the Maxwell homogenization method in Matlab software. Another novelty of this contribution is the calculation of new explicit analytical formulas for the control of the alignment tensors N* and Ns∗ which is in charged of the alignment distribution of inclusions within matrix through disorder parameters λ and s, respectively. The alignment tensors N∗ and Ns∗ are obtained as average of all possible alignments of the inclusions inside the composite. Numerical results are obtained and compared with some other theoretical approaches reported in the literature.
Highlights
The Maxwell method is extended in Levin and Kanaun (2012) to an homogeneous anisotropic medium containing an arbitrary set of homogeneous anisotropic ellipsoidal inclusions, where it is shown that the explicit equations obtained by the Maxwell method for isotropic materials coincide in the case of spherical inclusions with the method such as (MT), and for spheroidal inclusions oriented in a parallel direction, the equations for the effective elastic modulus tensor are the same, which are obtained by Maxwell and MT methods
The Maxwell approach allows us to replace tensor N∗ of Eq (9.25) by tensor Ns∗ given by formulae (9.65)-(9.76) of the Appendix for obtaining two Maxwell homogenization approaches that differ one from the other in the density distribution function chosen for modeling the alignment of inclusions within matrix materials
9.5.2 Study of Composites Constituted by Isotropic Matrix and Isotropic Inhomogeneities In Table 9.4 it is shown comparisons between present model through two different density distribution functions (DDF) and the results obtained by Christensen, FEM reported in Dong et al (2005) for effective axial bulk K1∗2 and Poisson ratio ν3∗1 in a two
Summary
It was the scientist James Clerck Maxwell, who proposed a method to calculate the effective conductivity of a homogeneous spherical material that contained a finite amount of inclusions of spherical type in Maxwell (1954). Explicit formula for choosing the aspect ratio of fictitious building is given when it presents spheroidal shape, and the advantages of the Maxwell method are proposed It shows that the reformulation of its scheme as a function of the tensor of contribution of flexibility, is equivalent to the reformulation as a function of the tensor of contribution of rigidity. The novelty of the present contribution is the derivation of explicit analytical formulae for the control of the alignment tensors N∗ and Ns∗ These functions distribute the alignment of inclusions inside the matrix material through disorder parameters λ and s, respectively, obtained as an average of all possible alignments of the inclusions within the nano transversely isotropic composite. Comparisons with other theoretical approaches, such as, closed relations reported by Christensen model and FEM (Dong et al, 2005), Locally-exact homogenization theory (LEHT) (Wang and Pindera, 2016), among others are given
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