PSI - Issue 33

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

1112 10

0.1 and v f = 0.1. Such result is related to the fact that, for the investigate nacre-like composite materials, the critical instability mode shapes are mostly global and thus, to capture a global instability mode shapes, a large assembly of unit cells in the loading direction is needed. Contrarily to the results obtained with w = 0.1 and w = 1, regardless of the values of v f and k , the nacre-like composite with w = 10, shows percentage changes of the instability critical loads lower than 3% highlighting that the microscopic stability analysis for high values of platelets aspect ratio leads to low underestimates of the primary instability even considering small unit cell assemblies. In both Tables 1 and 2, before reaching the convergence, negative percentage changes have been observed ( c  approach to c  from below) except for the case depicted in Fig. 4 for which the percentage changes is positive due to the fact that the primary instability mode shape is characterized by short wavelength (microscopic instability). In fact, the microscopic critical mode shape is characterized by a finite wavelength with periodicity coinciding with a 2 × 1 unit cell assembly. Such result highlights that for nacre-like composite materials with k = 20, v f = 0.1 and w = 10 the microscopic instability precedes the macroscopic one, but the related percentage variation is not significant. This means that, also in this case, the macroscopic stability analysis provides good estimates of the primary instability, but it is not excluded that, for different combinations of geometrical and material parameters (which have not been investigated in this work), the macroscopic stability analysis could lead to strong unconservative predictions of the primary instability stretch ratio compared with microscopic stability analysis. In Table 2 the microscopic critical stretch ratios for nacre-like composite materials with k = 1000 clearly show that the microscopic critical stretch ratios are not strongly underestimated by the microscopic stability analysis compared with the macroscopic one. These results highlight that the percentage change, between the critical load factor associated with the onset of primary instability evaluated by means of a microscopic instability analysis and a macroscopic one, remarkably decreases as w and k increase. In fact, for nacre-like composite materials with k = 1000 significantly smaller percentage changes have been obtained compared with those reported in Table 1 (no more than 4%). 6. Conclusions In this study, the instabilities at the macroscopic and microscopic scales have been investigated in 2D periodic nacre-like composite materials subjected to uniaxial loading processes. The onset of instability has been investigated for different combinations of geometrical and material parameters (i.e. platelets volume fraction v f , aspect ratio w and shear modulus contrast k between platelets and matrix phase). The macroscopic stability analysis was performed by monitoring the strong ellipticity condition of the homogenized tangent moduli tensor related to the unit cell and the microscopic one was performed by superimposing an additional displacement field on the current equilibrium configuration of the RVE, that is defined by a single unit cell or by a unit cell assembly. The results obtained from the macroscopic stability analyses highlight that the macroscopic critical stretch ratios and the critical mode shapes are strongly influenced by the geometrical and material parameters. The macroscopic critical stretch ratios show, for low volume fractions, an increasing trend and, once the peak is reached, it shows a decreasing trend. The shear modulus contrast k strongly influences the macroscopic stability phenomena since it acts as stabilizing factor for low values while for high values of shear modulus contrast it acts as an unstabilizing factor leading to macroscopic instabilities with lower critical stretch ratios. Has been also observed that in nacre-like composite materials with high platelets volume fractions a shear modulus contrast change has a greater effect on the critical stretch ratio compared with the nacre-like composite materials with low platelets volume fractions. The results obtained from the microscopic stability analyses highlight that, by adopting small unit cell assemblies with prevalent dimensions in the load direction, strong underestimates of the critical stretch ratios are obtained, while results comparable with those obtained with the macroscopic stability analyses were obtained by adopting large unit cell assemblies. The investigated nacre-like composite materials have shown that, under uniaxial deformation processes along the X 1 direction, the onset of primary instability is characterized by instabilities with long wavelength (i.e. macroscopic instabilities) with the exception of one combination of geometrical and material parameters in which the microscopic instability precedes the macroscopic one.