Thermomechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method
Introduction
Composite materials have been widely used in aerospace structures due to their excellent properties as compared to the metallic materials, e.g. high strength-to-weight ratio, good corrosion resistance. The recent development of composite materials aims to obtain better characteristics, such as enhanced damage tolerance and delamination resistance. However, the complexity and heterogeneity of composites often come with costly computational efforts. A modeling approach whereby all composite constituents (fibers and matrix) are explicitly built renders cumbersome meshing techniques, and requires high computational resources. An excellent technique to cope with this problem is to idealize the composite as a unit-cell representing the whole constituents and the corresponding responses, while the equivalent properties and other detailed stresses can be obtained. Representative unit-cell approach has been commonly used in the analysis of 3-D textile composites. Several studies employing this approach in order to predict the failure strength and ballistic impact damage behavior of 3-D orthogonal woven composites were performed by Tan et al. [1] and Sun et al. [2], respectively.
Dealing with the so-called unit-cell, asymptotic expansion homogenization (AEH) method is often employed. This method assumes that the unit-cell can be repeated in specific directions to represent periodicity. AEH regards the composite structure as a homogeneous macrostructure, which is formed by a large number of periodic and heterogeneous microstructure. AEH incorporates asymptotic expansion analysis to couple between microscopic and macroscopic behavior of a system [3]. Generally, AEH is often formulated to obtain thermomechanical properties of heterogeneous structures. Guedes and Kikuchi [4] derived a rigorous formulation of AEH method, and applied the formulation to analyze sandwich honeycomb plate. AEH is also used to analyze textile composites [5], [6], [7], [8] and metal-matrix composites [9]. Francfort [10] developed asymptotic expansion technique for the case of linear thermoelasticity. The incorporation of thermal effects into the heterogeneous unit-cell was performed by several authors to obtain mechanical properties as well as the coefficient of thermal expansions (CTE) [11], [12], [13].
In order to investigate the response of a unit-cell, localization analysis is usually performed in the framework of AEH. The localization analysis can be seen as a reverse scheme of the homogenization method that aims to obtain the local stresses of a unit-cell in a microscopic scale due to the external loads applied onto the homogeneous macrostructure [5], [14]. However, in the studies described in Refs. [5], [14], thermal residual stresses were not considered. In fact, thermal residual stresses may affect the damage behavior of composites [15], [16].
In this paper, a detailed mathematical and finite element formulation of AEH method is proposed to study the thermomechanical response of 3-D textile composite. The textile composite under consideration is 3-D orthogonal interlocked composites. The numerical analysis includes the calculation of equivalent thermomechanical properties, and the localization analysis whereby the critical regions in the constituents of 3-D orthogonal interlock composites are identified based on the maximum stresses. Two load cases, namely uniaxial tension and bending, are studied to determine the critical loading condition by which the damage may initiate in 3-D orthogonal interlock composites. The method developed in this paper may potentially aid the material selection processes, damage analysis of composites, design of textile composites and other engineering steps in various industries (aerospace, automotive, marine, electronic devices [17], medical [18], etc.). Apparently, the composite material selected for present analysis, i.e. 3-D orthogonal interlock composites, has a limited commercial application due to high cost pertaining to the manufacturing processes. This specific type of 3-D composites has been applied only for a few specific areas where 2-D laminates and metallic materials cannot satisfy the desired properties [19]. Nevertheless, the type of 3-D composites selected herein is deemed appropriate to test the capability of homogenization method because of their microstructural complexities.
Section snippets
General concept
The homogenization method is generally employed in a multi-scale analysis, which involves at least two spatial scales, namely microscopic and macroscopic scales. In this regard, homogenization method correlates both scales by considering a periodic and heterogeneous microstructure in the microscopic scale as a homogeneous macrostructure in macroscopic scale. Viewed from macroscopic scale, the periodicity of the microstructure is assumed very small. Two aforementioned spatial scales can be seen
Characteristic displacement vectors
Characteristic displacement vectors of χ and ψ (also known as correctors) already given in Eqs. (27), (28) can be obtained by finite element method. Eqs. (27), (28) re-expressed in the matrices and vectors reflecting their element parameters as followswherewhere {χ}kl denotes the elastic corrector of mode ‘kl’, {ψ} is the thermal corrector, {v} is the virtual displacement vector, [C]
Numerical models
Fig. 3(a) shows the schematic of 3-D orthogonal interlock composites, and the side-view observed from optical microscopy is shown in Fig. 3(b). A unit-cell of 3-D orthogonal interlocked composites was identified, and the idealized geometry is depicted in Fig. 3(c). The unit-cell consists of x-tow, y-tow, z-tow and resin region. In Fig. 3, the subscripts 1, 2 and 3 of x and y coordinate systems denote the fiber direction of x-, y- and z-tows, respectively. This designation also corresponds to
Modes of elastic and thermal correctors
The homogenization analysis produces elastic and thermal correctors. Fig. 6 shows the six modes of elastic correctors, which consist of Mode 11, Mode 22, Mode 33, Mode 23, Mode 31 and Mode 12. The correctors are shown in terms of normalized magnitude with reference to the maximum value of each mode. Figs. 6(a)–(f) show that the deformation of opposite faces of each mode is the same in terms of magnitude and direction. Because temperature difference is included, present analysis also produces
Conclusions
A formulation of homogenization method has been presented in the framework of asymptotic analysis in linear elasticity by including the thermomechanical effect. The formulation is used to obtain homogenized thermomechanical properties of 3-D orthogonal interlock composites, and to perform localization analysis whereby detailed stresses due to mechanical loading and thermal residual stresses are attained. Several conclusions can be summarized from the analysis:
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The homogenization method developed
Acknowledgement
Authors would like to extend their gratitude to the Tokyo Metropolitan Government for the financial support under the project of Asian Network of Major Cities 21 (ANMC-21).
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2020, Mechanics of MaterialsCitation Excerpt :The asymptotic expansion homogenization (AEH) method was developed by Francfort Francfort (1983) for the case of linear thermoelasticity in periodic structures. The AEH method has been employed to calculate the homogenized thermomechanical properties of composite materials (elastic moduli and coefficient of thermal expansion) (Dasgupta et al., 1996; Nasution et al., 2014). The detailed numerical modeling of the mechanical behavior of composite material structures often involves high computational costs.