A micro/macro constitutive model for the small-deformation behavior of polyethylene S. Nikolov, I. Doghri * Universite ´ catholique de Louvain, Centre for Systems Engineering and Applied Mechanics (CESAME), 4 Av. G. Lemaitre, B-1348 Louvain-la-Neuve, Belgium Received 3 November 1998; received in revised form 29 March 1999; accepted 5 May 1999 Abstract A micromechanically-based constitutive model for high density polyethylene (HDPE) in small deformations is presented. The micro- structure of HDPE consists of closely packed crystalline lamellae separated by layers of amorphous polymer. Here a semi-crystalline polymer is modeled as an aggregate of randomly oriented composite inclusions, each consisting of a stack of parallel lamellae with their adjacent amorphous layers. For the amorphous phase, the viscoelastic constitutive behavior is modeled, assuming a polydomain liquid- crystal-like structure and micromechanical parameters such as the elastic constant of distortion and the persistent length of polymer molecules are used. The viscoplastic behavior at yield is incorporated through the constitutive modeling of the crystalline lamellae. Constitutive equations for the composite inclusions are proposed and different homogenization schemes for the overall properties discussed. The intermediate phase linking the lamellae and the amorphous layers is assumed to form a surface layer around each lamella and its role in the yield behavior of HDPE is discussed.  1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Polyethylene; Micromechanics; Homogenization 1. Introduction Semi-crystalline polymers are now widely used as structural materials. At small strains they show strongly non-linear stress–strain behavior which depends on such characteristics as the overall crystallinity, the molecular weight, the molecular branch content, etc. Recently, some micro/macro constitutive models for semi-crystalline polymers have been elaborated [1,2]. These are the first successful steps towards understanding how the microstruc- tural evolution during deformation influences the observed overall behavior of these materials. However, the modeling in Refs. [1,2] is restricted to finite strains, and important processes at microlevel (e.g. lamellar-to-fibrillar transition) are ignored. In small deformations, it is very important to know how the microstructure determines the initial Young’s modulus and the yield stress. An important feature of polymers is the dependence of the initial Young’s modulus on the yield stress [3] which, to our knowledge, has not been explained yet from a micromechanical point of view. Viscoelasticity of semi-crystalline polymers is due to the amorphous phase behavior [4], which has not been explicitly modeled either. Rheological models do exist for the small deformation regime (e.g. Refs. [5,6]), but they cannot provide a clear link between the micro and macro scales. In this paper, we develop a small-strain micro/macro model, which enables to simulate the macroscopic behavior of PE from physically based micromechanical modeling. In Section 2 we develop constitutive equations for a composite inclusion consisting of a stack of crystalline lamellae and their adjacent amorphous layers. The viscoelastic behavior of the amorphous phase is explained once the elastic distor- tion of the polymer molecules is assumed to be important. Viscoplasticity of the lamellae is modeled as in rate-depen- dent polycrystalline materials. It is assumed that the intermediate phase (linking the lamellae and the amorphous layers) forms a surface layer around each lamella. The influ- ence of the intermediate phase on the yield behavior is also discussed. The overall behavior of an aggregate consisting of lamellae stacks is considered in Section 3. Results and conclusions are presented in Sections 4 and 5. The notation used is as follows: scalars are given in italics (g ), tensors are designated by bold-face symbols (n, M), the order of which is determined by the context. Tensor contrac- tions over one or two indices are denoted by (·) and (:), respectively. Tensor (dyadic) products are indicated by (  ). Polymer 41 (2000) 1883–1891 0032-3861/00/$ - see front matter  1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0032-3861(99)00330-4 * Corresponding author.