A new three-phase model to estimate the effective elastic properties of semi-crystalline polymers: Application to PET O. Gueguen a,b , S. Ahzi a, * , A. Makradi a , S. Belouettar b a Université de Strasbourg, IMFS, 2 Rue Boussingault, 67000 Strasbourg, France b LTI, Research Center Henry Tudor, 70 Rue de Luxembourg, L-4221 Esch-sur-Alzette, Luxembourg article info Article history: Received 21 May 2008 Received in revised form 28 April 2009 Keywords: Micromechanical modeling Semi-crystalline polymers Effective elastic properties Threephase model Interphase PET abstract In this paper, the effective elastic properties of semi-crystalline polymers are computed through a new three-phase model. The formulation of this approach is based on recent thermal analyses which revealed the existence of an intermediate phase during cooling from the melt in semi-crystalline polymers. At the microscopic level, a three-phase com- posite inclusion constituted by three adjacent layers (a crystalline lamella, a rigid amor- phous interphase and a mobile amorphous phase) is considered to estimate, via homogenization methods, the effective elastic properties of the material. Our model is applied to poly(ethylene terephthalate) and a good agreement is obtained, for different crystallinities, between our predicted results and the experimental ones found in the liter- ature. The model is also compared to the N-phase inclusion model of Hori and Nemat-Nas- ser [Hori, M., Nemat-Nasser, S., 1993. Double-inclusion and overall moduli of multi-phase composites, Mechanics of Materials 14, 189–206] by considering an extension of the dou- ble-inclusion model to a three-phase inclusion model. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction A significant advance is made in the modeling of the effective physical properties of heterogeneous materials due to their large engineering applications. For instance, the statistical continuum approach based on the use of probability functions to characterize the media (Adams et al., 1987; Beran et al., 1996; Lin and Garmestani, 2000) and also the continuum mechanics approaches which are based on the Eshelby’s theory of an inclusion embedded in an equivalent continuum medium or in a matrix (see for instance: Eshelby, 1957; Benveniste, 1987; Mori and Tanaka, 1973; Hori and Nemat-Nasser, 1993; Ahzi et al., 1995, 2007; Nemat-Nasser and Hori, 1999; Aboutajeddine and Neale, 2005; Lipinski et al., 2006; Gueguen et al., 2008; El Mouden and Molinari, 1996; Molinari and El Mouden 1996; Hu and Weng, 2000; Ponte Castaneda and Willis, 1995). Application of these approaches to compute the effective mechanical properties of semi-crystalline poly- mers is currently limited. Based on the Eshelby’s approach (Eshelby, 1957) of the inclusion embedded in an infinite homogenous matrix, dif- ferent models are derived to predict the effective mechan- ical properties of heterogeneous materials. This modeling takes into account the microstructure and shape of the composite constituents through the shape and the constituents of the inclusions. For instance, in the case of semi-crystalline polymers, the crystalline phase can be considered as an ensemble of ellipsoidal heterogeneities (inclusions embedded in an amorphous matrix) when using the Mori–Tanaka approaches (Benveniste, 1987; Mori and Tanaka, 1973), while in the double inclusion ap- proaches (Hori and Nemat-Nasser, 1993; Nemat-Nasser and Hori, 1999; Aboutajeddine and Neale, 2005; Lipinski et al., 2006), the ellipsoidal heterogeneity is assumed to be embedded in an intermediate homogeneous phase which in turn is embedded in an infinite homogeneous domain (the amorphous matrix). Ahzi et al. (1995, 2007) considered the semi-crystalline polymer as a composite 0167-6636/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2009.04.012 * Corresponding author. Tel.: +33 390 24 29 52. E-mail address: ahzi@unistra.fr (S. Ahzi). Mechanics of Materials 42 (2010) 1–10 Contents lists available at ScienceDirect Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat