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

489

5

3. Data Specimens with short multi walled carbon nanotubes (MWCNTs) were used at the present study. The characteristics of the CNT were : acquired from Cheap Tubes ® (Metaxa, 2012), purity higher than 95 % (Metaxa, 2012), specific surface 110 m 2 /g (Metaxa, 2012) , length of 10 to 30 μm (Metaxa, 2012), diameter of 20 to 40 nm (Metaxa, 2012), aspect ratio was of 700 (Metaxa, 2012), modulus of elasticity 470 GPa (Yu , et al., 2000), and Poisson ’s ratio 0.1. The cement unreinforced matrix material data were modulus of elasticity 4000 MPa, Poisson ’s ratio 0.2. The target effective modulus of elasticity of the homogenized material is 4031 MPa (Anastopoulos, Givannaki, Papanikos, Metaxa, & Alexopoulos, 2020). and the target CNT volume fraction is 12 ‰ (Anastopoulos, Givannaki, Papanikos, Metaxa, & Alexopoulos, 2020). 4. Results Fig. 3 illustrates the results of the optimization case for 20 maximum iterations and maximum objective value 5 MPa. While during the first four iterations the value of the calculated effective modulus is far away from the target effective modulus, at the sixth iteration and on the calculation of the effective modulus from the algorithm is more accurate (Fig. 3a). As shown in Fig. 3a when the convergence was achieved, the value of the calculated effective modulus of elasticity was 4036 MPa. The decrease of the difference between the calculated and the target effective modulus over the last 14 iterations is depicted in Fig. 3b. The volume fraction that was used from the algorithm at the last iteration was ν CNT = 14.8 ‰.

difference between calculated and targeted modulus of elasticity

4100

80

target effective modulus calculated effective modulus of elasticity

4080

60

4060

40

4040

20

Difference [MPa]

4020

0

Effective modulus of elasticity [MPa]

4000

0 2 4 6 8 10 12 14 16 18 20 -20

0 2 4 6 8 10 12 14 16 18 20

(a) (b) Fig. 3 (a) Effective modulus of the optimization case for 20 maximum iterations and maximum objective value 5 MPa.; (b) difference between calculated and target effective modulus of the optimization case for 20 maximum iterations and maximum objective value 5 MPa. Fig. 4 illustrates the results of the optimization case for 200 maximum iterations and maximum objective value 1 MPa. While at the first iteration the calculated effective modulus value was 4101 MPa (Fig. 4a), at the eighteenth iteration the calculated effective modulus value was decreased to 4039 MPa, a value much closer to the target effective modulus. Finally, the calculated modulus at the last cycle was 4031.92 MPa. Consequently, the difference between the calculated and the target effective modulus, which was 71 MPa at the beginning, dropped to 0.92 MPa (Fig. 4b). When convergence was achieved, the CNT volume fraction was ν CNT = 12.8 ‰. Fig. 5 illustrates the results of the optimization case for 1000 maximum iterations and maximum objective value 1e-4 MPa. The value of the calculated effective modulus has shown a decline by 70 MPa approximating the target effective modulus which was 4031 MPa (Fig. 5a). As shown in Fig. 5b at the first 200 iterations the difference of the calculated and target effective modulus experienced a plunge. Therefore, as the difference diminished to values close to zero during the last 200 iterations, the final effective modulus calculated value was 4031.01 MPa. When convergence was achieved, the CNT volume fraction that was used from the algorithm for the final effective modulus calculation was v CNT = 12.05 ‰. Number of iterations [-] Number of iterations [-]