PSI - Issue 37

Anastopoulos G. Stylianos et al. / Procedia Structural Integrity 37 (2022) 485–491 Anastopoulos G Stylianos et al/ Structural Integrity Procedia 00 (2019) 000 – 000

486

2

1. Introduction Over recent years, analytical methods are the most prevailing methods used to calculate the effective properties of heterogeneous materials where a matrix with reinforcements exists (Zengrui Song, Xianghe Penga, Shan Tang, Tao Fua, 2020), (Wenya Shu, Ilinca Stanciulescu, 2020), (Djebara, Moumen, Kanit, Madani, & Imad, 2016), (Berger, et al., 2007). The earliest attempts to calculate and predict the effective properties of heterogeneous materials were made by Voigt (Voigt, 1889), who worked on models under axial loading, considering a small unary representative sample of the material with predominating homogenous strain field. Subsequently, Reuss took a resembling approach (Reuss, 1929) , but intending to form a method applicable to models under forces vertical to inclusions’ axis, he assumed a homogenous stress field in a representative material sample. Mori-Tanaka (Mori & Tanaka, 1973) considered that every inclusion is detached from the others, and he studied an infinite matrix subjected to distant load, which is either the average matrix strain EM or average matrix stress TM. At Multi-Step method, the composite is separated into “grains”, consisting of one inclusion family and the matrix. At th e beginning the effective properties of each inclusion are evaluated using Mori-Tanaka model and the overall properties of the composite are calculated by performing homogenization using the Voigt formulation (Documentation, Abaqus, 2017). At the present research multi-step method was regarded as the most suitable method, due to the high number of different orientation vectors of the contained Carbon NanoTubes (CNTs) contained in the cement matrix.

Nomenclature CNTs

carbon nanotubes

effective modulus of elasticity of the homogenized material volume fraction of the carbon nanotubes within the composite

E eff

v f

modulus of elasticity of the carbon nanotubes Poisson ’s ratio of the carbon nanotubes modulus of elasticity of the cement

E CNT ν CNT

E cement ν cement

Poisson ’s ratio of the cement

target effective modulus of the homogenized material calculated effective modulus of the homogenized material

target

Ε eff Ε eff

calculated

2. Methodology The steps to be followed for the optimal volume fraction identification using the proposed method, as shown in Fig. 1, are: 1. Set target effective modulus ( Ε eff target ) of the Homogenized Material, set boundaries for volume fraction. 2. Set initial point for the IPOpt. The initial point is the initial value of the CNT volume fraction. 3. Calculate effective modulus ( Ε eff calculated ) of the Homogenized Material (given E CNT , ν CNT , Ε cement , ν cement , initial volume fraction) using multi-Step method combined with Representative Volume Element (RVE) method at ANSA ® Pre-Processor. 4. Evaluate the difference between the calculated effective modulus and the target effective modulus ( Ε eff calculated - Ε eff target ) of the Homogenized Material 5. If the difference is more than limit objective value and the number of the completed iterations is less than the maximum number of iterations move on to the next iteration. 6. Set new volume fraction value. 7. Start the new iteration using the next point for the IPOpt. 8. Calculate of the Homogenized Material ( Ε eff calculated ) of the Homogenized Material (given E CNT , ν CNT , Ε cement , ν cement , new volume fraction) using multi-Step method combined with RVE at ANSA ® Pre-Processor 9. If the difference Ε eff calculated - Ε eff target is less than limit objective value or the number of the completed iterations are or than the maximum number of iterations move on to the next iteration.