PSI - Issue 33

Andrea Pranno et al. / Procedia Structural Integrity 33 (2021) 1103–1114 Author name / Structural Integrity Procedia 00 (2019) 000–000

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the macroscopic and the microscopic stability analysis. The first subroutine, once the principal path solution is obtained, allows to evaluate the homogenized tangent moduli tensor evaluating also the eigenvalues associated with the acoustic tensor. The second subroutine allows to evaluate the minimum eigenvalue associated with the microscopic stability functional for each loading step of the principal path and for an increasing assembly of unit cells along the X 1 direction.

Fig. 3. Macroscopic critical stretch ratio as a function of the platelets volume fraction v f with different values of platelets aspect ratio w and shear modulus contrast equal to k = 20 (a) and k = 1000 (b).

Firstly, the onset of macroscopic instability in uniaxially compressed nacre-like composite materials (i.e., elastic instabilities characterized by critical mode shapes with long wavelength) was detected by means of the macroscopic stability analysis. In Fig. 3 the macroscopic critical stretch ratios c  corresponding to the onset of macroscopic instability in uniaxially compressed nacre-like composite materials have been reported for different values of k ( 20 and 1000), w ( 0.1, 0.5, 1, 5, and 10) and v f (ranging from 0.1 to 0.9 with an increment of 0.1). Specifically, in Fig. 3 the macroscopic critical stretch ratios have been reported as a function of the platelets volume fraction v f and we observe that the critical curves show, for low volume fractions, an increasing trend and, once the peak is reached, they show a decreasing trend. The critical curves obtained for a low value of k = 20 (see Fig. 3a) show that peaks are mainly reached for intermediate values of volume fraction (i.e., 0.5  v f  0.7 ), while for a high value of k = 1000 (see Fig.4b) the peaks are reached for higher volume fractions (i.e., 0.8  v f  0.9). Generally speaking, the nacre-like composite materials with a high value of k (i.e., with k = 1000) experience macroscopic instabilities at higher critical stretch ratios which correspond to lower uniaxial compressive loads. By comparing the curves reported in Fig. 3a and Fig. 3b, the results highlight that the shear modulus contrast k strongly influences the macroscopic stability phenomena since, it acts as stabilizing factor for low values (i.e., k = 20) while, for high values of shear modulus contrast (i.e., k = 1000) it acts as an unstabilizing factor leading to macroscopic instabilities with lower critical stretch ratios. The highest critical stretch ratios have been obtained for w = 10 (blue curves), demonstrating that the platelets reinforcement with high aspect ratio acts as a destabilizing effect, while lower critical stretch ratios have been obtained for w  1 demonstrating the stabilizing effect induced by low platelets aspect ratios. Secondly, the onset of microscopic instability was detected for increasing assemblies of unit cell along the X 1 direction, with the aim to investigate the interplay between the macroscopic and the microscopic instabilities. In Tables 1 and 2 the critical stretch ratios corresponding to the onset of microscopic instability have been reported for different values of k ( 20 and 1000), w (0.1, 1 and 10) and v f (0.1, 0.5 and 0.9). It is worth noting that, based on results that have not been reported here for sake of brevity, we exclude the onset of microscopic instabilities characterized by out-of phase instability mode shapes, and thus it was sufficient increases the assembly of unit cells only along the compression direction X 1 (i.e., was considered a RVE assembly with n × 1 cells with n representing the number of