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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of nacre-like composite materials in a large deformation context. Specifically, no works have yet been published regarding the study of the instability phenomena occurring at both the microscopic and the macroscopic scale. Therefore, this study deals with the onset of instabilities at the microscopic and macroscopic scales in uniaxially compressed bioinspired composite materials with periodic microstructure. In particular, the interchange between instabilities occurring at the microscopic and macroscopic scale has been investigated in incompressible nacre-like composite materials. Parametric numerical analyses, by considering different combinations of platelets aspect ratio, platelets volume fraction and shear modulus contrast between platelets and soft matrix have been performed. It has been demonstrated that the critical stretch ratios and the critical mode shapes are highly influenced by both the geometrical and materials parameters. The stability analyses, performed by means of the commercial software COMSOL Multiphysics, highlighted that the microscopic stability analysis provides in most cases strong underestimates of the critical stretch ratios (adopting a small unit cell assembly), and that, by performing a macroscopic stability analysis based on the strong ellipticity condition of the homogenized tangent moduli tensor, a more accurate evaluation of the load associated with the primary instability can be instead obtained with less computational efforts. 2. Macroscopic constitutive response In this section, the theoretical background related to the homogenization problem and the stability conditions at both the macroscopic and microscopic scales is briefly recalled. Consider a representative volume element (RVE) consisting of an arbitrary assembly of unit cells of a periodic nacre-like material which is made of a soft matrix reinforced with stiff platelets, as depicted in Fig.1. In the undeformed configuration, the homogenized material and the RVE are characterized by a volume equal to ( ) i V and ( ) i V , respectively and the surfaces enclosing the volumes are defined as ( ) i V  and ( ) i V  , respectively. The over-signed variables are referred to the macroscopic problem while the others are referred to the microscopic problem, and the subscript (i) is referred to variables in the undeformed configuration. The RVE is associated with an infinitesimal neighborhood of a generic point X in the homogeneous material. The assumed deformation measures are the macroscopic deformation gradient tensor ( ) F X , for the homogenized material, and the microscopic deformation gradient tensor ( ) /    F X x X , for the RVE, where x and X are the position vectors in the deformed and undeformed configuration, respectively. The Jacobean of the transformation det J  F denotes the volume change with respect the undeformed configuration.

Fig. 1. Homogenized nacre-like material on the left and the representative volume element in the undeformed and deformed configurations on the right.

By considering a finitely deformed nacre-like composite material, the nonlinear stress-strain behavior of the microconstituents undergoing large deformations can be predicted by adopting a nearly-incompressible neo-Hookean constitutive model characterized by a strain energy density function ( ) W F . At the microscopic scale, the equilibrium problem is formulated in terms of ( ) F X and ( ) R T X , where R T denotes the conjugated stress measure of the