Micromechanical modeling of elastic properties in polyolefins F. Be ´doui a , J. Diani a , G. Re ´gnier b, * a Laboratoire de Microstructure et Me ´canique des Mate ´riaux (LM3-CNRS UMR 8006) ENSAM, 151 bd de l’Ho ˆpital 75013, Paris, France b Department of Materials, Laboratoire de Transformation et Vieillissement des Polyme `res (LTVP) ENSAM, 151 bd de l’Ho ˆpital 75013, Paris, France Received 31 October 2003; received in revised form 19 January 2004; accepted 19 January 2004 Abstract The aim of this article is to try to explain why isotactic polypropylene (PP) is stiffer than high density polyethylene (HDPE) despite the fact that this latter is more crystalline and that its crystallites are stiffer than PP ones. Two micromechanical models were chosen for their ability to represent semi-crystalline polymers. The first one is a differential scheme in which ellipsoidal crystallites are randomly dispersed in an amorphous matrix. The second one is a self-consistent scheme where the material is considered as an aggregate of randomly oriented two layered-phase composite inclusions (crystalline – amorphous). Experiment-model comparisons are clearly in favor of the first model. This latter demonstrates the key importance of the crystalline lamellae aspect ratio on the elastic properties of semi-crystalline polymers. q 2004 Elsevier Ltd. All rights reserved. Keywords: Micromechanical modeling; Semi-crystalline polymers; Elasticity 1. Introduction Nowadays, thermoplastic materials are increasingly used in industrial parts. It is especially true for semi-crystalline materials, which are widely used as structural materials. During the part forming, the stretching or the shearing of the polymer melt under strong cooling conditions lead to a flow- induced crystallization, which generates specific crystalline morphologies such as deformed spherulites, shish – kebab or more complex crystalline macrostructure like in polypro- pylene for example [1]. A high anisotropy of molecular orientation in the crystalline phase is resulting [2,3], even for solidifying shear flows [4–6]. Moreover crystallinity variations along the part and in the part depth can be observed [7]. The crystalline orientation is responsible for possible anisotropic behavior, while variations of the amount of crystallinity induce strong variations of the mechanical properties [8]. For structural polymer applications, there is an industrial need for the prediction of these mechanical properties, especially for the small-deformation behavior, to determine for example the strength of blown bottles [9] or the shrinkage and the warpage of injected parts [10]. Now, process simulations including a flow-induced crystallization law coupled to a viscoelastic behavior based on molecular models [11], allow to predict the final crystallinity and the final molecular orientation in the crystalline phase [12,13]. But, from the predicted crystalline morphology of the polymer, little was done to predict the mechanical proper- ties. Although micromechanical modeling is efficiently used to determine the thermomechanical properties of filled polymers [14], only a few researchers applied these micro – macro models to non-filled semi-crystalline polymers. Halpin and Kardos [15] proposed to use Halpin – Tsai model [16] in order to determine the elastic moduli of semi- crystalline polymers. The model requires the assumption that lamellae be regarded as fibers. Phillips and Patel [17] applied this model to PE. An adjustable parameter in this model was linked to crystallite shape ratio. However, this model, which is generally used to calculate the moduli of short-fiber composite, is known to fit only the experimental data at low volume fraction of filler. This is not the case of semi-crystalline materials, for which the crystallinity can often reach 60–70%. Later, several researchers worked on the prediction of the large deformation behavior to predict the texture evolution induced by a plastic deformation [18,19]. Lee et al. [18] developed a specific micromechanical model in which the crystalline lamellae were assumed to be rigid-visco-plastic and the amorphous phase visco-plastic. Nikolov and Doghri 0032-3861/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2004.01.028 Polymer 45 (2004) 2433–2442 www.elsevier.com/locate/polymer * Corresponding author. Tel.: þ 33-1-44246305; fax: þ 33-1-44246382. E-mail address: gilles.regnier@paris.ensam.fr (G. Re ´gnier).