PSI - Issue 28

Stylianos Anastopoulos et al. / Procedia Structural Integrity 28 (2020) 2132–2141 S. Anastopoulos et al. / Structural Integrity Procedia 00 (2019) 000–000

2133

1. Introduction Regarding the modelling of composite materials, where a matrix with reinforcements exists, there have been many methods proposed. The semi analytical methods considered for multi scaled modelling in this research in order to achieve homogenization are that of the semi-analytical method and the direct numerical method. Semi analytical methods considered were The Theoretical Bounds of Hill – Reuss – Voigt (1889,1929), the asymptotic Hashin – Shtrickman bounds (1964), the Eshelby Model (1957), The Mori-Tanaka model (1973), the Multi-Step homogenization method, The Self-Consistent Method and the Interpolative Double Inclusion Model (LIELENS’ Model) . The selection was made making use of comparisons of the various analytical methods (Aboudi, 1992), Ghayath Al Kassem, 2009) and judgment of suitability of each method with regard to this specific research but also the grade by which each method is suitable to be applied in a wider range of problems. Thus, a multi-step homogenization method approach making use of the Mori-Tanaka model was deemed appropriate. Composites may also be homogenized using the numerical method where different types of elements may be used for the matrix and the reinforcement (Fibers, sheets, platelets etc.). Using 3-dimentional elements for both the matrix and the inclusion, may demand a higher computational power investment, but provides more accurate results.

Nomenclature RVE

representative volume element crack mouth opening displacement (multi walled) carbon nanotubes modulus of elasticity of CNTs

CMOD

(MW)CNT

E CNT

2. Methodology 2.1. Modelling Composites Using Numerical method

Concerning the Mori-Tanaka Model proposed by Mori and Tanaka in 1973, each inclusion behaves like an isolated inclusion, in an infinite matrix that is remotely loaded by the average matrix strain EM or average matrix stress TM (Fig. 1), respectively, resulting in a concentration tensor of the following form: (1) 1 1 1 ) ] [ : (      A E C C I I M

Fig. 1 Mori – Tanaka Model representation