Analytical Modeling: Three Stages Homogenization Method

Analytical Modeling: Three Stages Homogenization Method

DOI: 10.4018/978-1-5225-4837-9.ch005
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The large number of available natural fibers emphasizes the use of a reliable, non-costly and easy to use, predictive tool with short computation time. In order to predict the ultimate strengths and Young's moduli of green composites, an analytical model known as the three Stages Homogenization Model (3SHM) is used. The model relies on three main parts: a geometrical model, a homogenization method and a strength model. Moreover, the last two models consist of four main parts: a micro-mechanical modeling for elastic properties and ultimate strengths for unidirectional (UD) composites, a homogenization method at meso and macro levels to determine the composite stiffness and stress-strain fields throughout the composite, two 3D failure criteria for the matrix and unidirectional composites and a damaged stiffness model. This model enables the prediction of the ultimate strengths and the 3D elastic properties; Young's and Shear moduli, in addition to the in plane and out plane tensile and shear strengths.
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General Analytical Modeling Strategy

The general scheme of the proposed analytical model that provides the 3D elastic properties, ultimate strengths and a stress and strain diagrams for axial and shear loadings is presented in Figure 1. The proposed modeling is composed from a geometrical modeling, a homogenization method, and a strength model. A geometrical modeling relying on a sinusoidal modeling of the undulated yarns is adopted. However, the homogenization and strength model are composed from different sub-models. Chamis micromechanical model (Chamis, 1989; Chamis et al., 2013) is used to determine the elastic properties and ultimate strengths of subdivisions-UD composites. Then, an analytical homogenization method developed by the authors (Hallal, Younes, & Fardoun, 2013) is employed to predict the stiffness matrix of the composite and to evaluate the stress-strain fields throughout the Representative Elementary Volume (REV). Tsai-Wu failure criterion will be adopted for UD composites subdivisions, while Christensen failure criterion is used for the pure matrix part. The last component used is a damaged stiffness model for subdivisions-UD composites.

This model is developed by one of the principle author of this chapter (Hallal & Younes, 2016) and used to predict the elastic and strength properties of different kinds of 2D and 3D textile composites reinforced by synthetic fibers. The model has shown very good agreement compared to experimental results, and thus it will be used as a tool to evaluate the mechanical properties of natural fiber composites.

Figure 1.

General scheme of analytical modeling of the mechanical behavior of composites


The proposed algorithm shown in Figure 2 tries to predict the failure of the composite based on the following conditions:

  • 1.

    Final failure is assumed when all components (all yarns and pure matrix) are considered at failure.

  • 2.

    Undulated yarns are considered at complete failure only if 90% of subdivisions-UD composites are damaged.

  • 3.

    A multi-failure mode is assigned to the failed subdivisions-UD composites.

  • 4.

    The applied stress on the REV is increased only if no new failure is occurred.

Figure 2.

General scheme of the evaluation of the ultimate strength under static loading of textile composites


In order to propose a reliable strength model, our methodology is based on solving the problematic related to the multi scale homogenization method which has been proposed by (Hallal, Younes, & Fardoun, 2013) quite similar to that proposed by (Whitney & Chou, 1989; Byun, Whitney, Du, & Chou, 1991; Scida, Aboura, Benzeggagh, & Bocherens, 1998) and used to evaluate only the elastic properties of textile composites. It shows better predictions of available experimental data compared to other analytical and numerical models. However, in this study, the capabilities of that method in the evaluation of stress and strain fields throughout the REV are crucial in order to yield good predictions of ultimate properties.


Review On Existing Homogenization Method

Many review papers are found in the literature (Ansar, Xinwei, & Chouwei, 2011; Huang & Ramakrishna, 2000; Byun & Chou, 1989; Tan, Tong, & Steven, 1997; Crookston, Long, & Jones, 2005) that deals with modeling of the mechanical behavior and the architectures of textiles. While in this section, an updated review and a discussion of available analytical models found in literature, focusing on the homogenization methods used, are presented. Analytical models are divided into four categories according to the homogenization approaches based on: Classical Laminate Theory “CLT”, iso-strain assumption (iso-strain models), mixed iso-strain/iso-stress assumptions, inclusion methods and the method of cells.

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