Interphase effect on the effective magneto-electro-elastic properties for three-phase fiber-reinforced composites by a semi-analytical approach

Highlights

• Semi-analytical method provides accurate numerical result.

• Good coincides is obtained for comparisons with other method's results.

• Magneto-electro-elastic effective properties are sensitive to the interphase thickness.

• Magneto-electro-elastic moduli can be manipulated through interphase properties.

Abstract

A semi-analytical approach is proposed to determine the effective magneto-electro-elastic moduli of a fiber-reinforced composite. We especially focus on predicting the effective properties of three-phase periodic composite reinforced with unidirectional, infinitely long and concentric cylindrical fibers with square transversal distribution. The semi-analytical method is developed combining asymptotic homogenization and finite element methods. Asymptotic homogenization method allows the statements of local problems that are solved by finite element method and the associated effective coefficients. Finite element method is implemented via the principle of minimum potential energy. The effect of interphase thickness and the fiber material properties on effective moduli is analyzed. Numerical computations were performed, and an exact agreement is obtained by comparing the semi-analytical approach with asymptotic homogenization method linked to the theory of potential functions of a complex variable.

Introduction

Multi-phase magneto-electro-elastic composites have been receiving great attention in the literature due to the wide application field. Y. Cheng et al. report an updated status for magnetoelectric materials applications (Cheng, Peng, Hu, Zhou, & Liu, 2018). As typical cases, it can be mentioned: field sensors (Reis et al., 2017), energy harvester (Naifar, Bradai, Viehweger, Choura, & Kanoun, 2018; Qiu, Chen, Wen, & Li, 2015; Qiu, Tang, Chen, Liu, & Hu, 2017), random access memory (Kosub et al., 2017; Lee et al., 2017), voltage tunable inductors (Geng, Yan, Priya, & Wang, 2017; Lin et al., 2015), band stop filters (Ciomaga et al., 2016) and tunable resonators (Popov, Zavislyak, & Srinivasan, 2018). This picture that involves a high number of applications shows the current demand for improving magnetoelectric (ME) composite designs.

Homogenization techniques have always been a useful tool to describe the structure-properties relationship of composite materials (Bakhvalov & Panasenko, 1989). In literature, different homogenization implementations can be found, which represent an important advantage, because it allows validations between models by comparing them. Essentially, different mathematical approaches describing the same physical phenomena must provide quite close results. This is an important validation step toward effective properties calculation to better study a wider range of composites. J. A. Otero and colleagues developed a semi-analytical method for computing elastic effective properties of composites with imperfect interfaces (Otero et al., 2013). H. Berger et al. proposed a scheme fully based on the finite element method (FEM) subjected to a set of boundary conditions focused on specific stress-strain, or stress-electric field relations (Berger et al., 2003, 2005).

The effect of phase contact quality on composite properties is an active issue that has been gaining attention during the last years because it can be a structural factor with heavy influence. It is necessary to consider this effect to develop more realistic property estimations. D. Guinovart-Sanjuán and colleagues derived a formulation including imperfect contact for a shell laminated composite (Guinovart-Sanjuán et al., 2018). Y. Koutsawa et al. developed a micromechanical approach to study imperfect thermal contact (Koutsawa, Karatrantos, Yu, & Ruch, 2018). F. E. Alvarez-Borges et al. describe a gain-enhancement of effective properties for a laminate with imperfect contact (Álvarez-Borges et al., 2018). N. D. Barulich et al. report the effect of damage at the interphase based on a computational micromechanics scheme (Barulich, Godoy, & Dardati, 2016). The nature of interphase is another issue of great interest. The imperfect contact can be studied as an interface with a jump in the normal component of stress, electric displacement and/or magnetic induction, but it can also be described as a “third phase” or an active interphase (Espinosa-Almeyda et al., 2017). In this sense, F. Lebon et al. developed a careful analysis of the interphase soft and hard anisotropic behavior (Lebon et al., 2016).

In the present work, a semi-analytical method is implemented for computing the effective coefficients for periodic three-phase fiber reinforced composite (FRC). Herein, the piezoelectric and piezomagnetic constituents exhibit transversely isotropic properties. In addition, an interphase is considered between the fiber and the matrix in order to study the effect of the quality of the constituent contacts. The periodic cell cross-section is a square with two concentric circles and the periodicity is the same in two perpendicular directions. Section 2 illustrates the mathematical formalism for magneto-electro-elastic (MEE) heterogeneous media for a three-phase FRC. In Section 3, the formulation of homogenized antiplane and plane local problems and effective coefficients obtained by a two-scale asymptotic homogenization method (AHM) is reported. Besides, the semi-analytical approach based on FEM, namely, semi-analytical finite element method (SAFEM) is developed. Herein, the principle of minimum potential energy and the FEM with quadrilateral of eight boundary nodes are combined to find the MEE effective coefficients over 1/4 periodic cell, see Ref. Otero, Rodríguez-Ramos, and Monsivais (2016). In Section 4, numerical analysis and model validation are reported and discussed. Herein, some comparisons between AHM solved via the theory of complex variable and SAFEM allow checking the accuracy of the semi-analytical model. The available data in Refs. Hashemi (2016), Kuo (2011), Yan Jiang, and Song (2013) is also considered for further SAFEM validation.

The main contributions of the present research are the determination of a semi-analytical method (SAFEM) for computing MEE effective moduli of periodic three-phase FRC and the study the effect of interphase thickness and the constituent materials on a composite via SAFEM. In comparison with previous works (Otero et al., 2013, 2016), which only considers an elastic periodic FRC, SAFEM formulation is extended to describe the MEE behavior. New local problems arise and they are solved via minimum potential energy through FEM, in contrast with Refs. Espinosa-Almeyda et al. (2017, 2014) and Guinovart-Díaz et al. (2013) where local problems are solved analytically using AHM via complex variable method. The objective of developing SAFEM is to have a more versatile tool to estimate composite effective properties although the numerical implementation could be somehow heavier than analytical solved AHM.