Micromechanical investigations of polymer matrix composites with shape memory alloy reinforcement

Abstract

In the present work, a micromechanics model was established to predict the effective time-dependent pseudoelastic responses of composites consisting of thermoviscoelastic polymer matrix and shape memory alloy (SMA) reinforcement. The incremental constitutive equations were firstly derived for polymer and SMA, respectively, and then formulated into a unified formulation. The starting point of the proposed model is to construct a variational statement which is a potential energy functional derived from the unified formulation. On account of the nonlinearity of the composites’ behavior, the present model was developed based on an incremental procedure associated with the instantaneous effective tangential matrix of composites’ coefficients. The present model offers an efficient tool for analyzing the SMA polymer matrix composites with arbitrary number and geometry of SMA reinforcement.

Introduction

Shape memory alloys (SMAs) are metallic alloys that are capable of recovering their original shape through the application of temperature or stress fields due to the phase transformation between austenitic and martensitic phases that SMAs undergo. Two main phenomena related to SMAs are: (i) The first one is pseudoelasticity (PE) in which very large nonlinear elastic strain can be generated especially upon loading, but full recovery is achieved in a hysteresis loop upon unloading; (ii) the second one is called shape memory effect (SME), which may be one-way or two-way. Due to these unique properties, SMAs are excellent candidates for sensors, large strain actuators, and other smart structures widely used in various areas of engineering fields such as aerospace, automotive, and biomechanical applications (Jani, Leary, Subic, & Gibson, 2014). During the past several decades, various numerical models have been developed in order to accurately describe the main behaviors of these alloys (Birman, 1997, Paiva and Savi, 2006). Generally speaking, these models are categorized into micro models (Warlimont et al., 1974, Nishiyama, 1978, Achenbach and Müller, 1982, Perkins, 1975) and macro phenomenological models (Paiva et al., 2005, Brinson and Lammering, 1993, Brinson, 1993, Panico and Brinson, 2007, Lagoudas et al., 1996, Lagoudas, 2008).

SMA composites fabricated by embedding SMA fibers, particles, wires, or thin plates into metal matrix or polymer matrix have attracted great interest in the applications of a wide variety of smart materials and structures. Micromechanics models are indispensable tools to analyze the thermo-mechanical behavior of the SMA composite materials and structures. Boyd and Lagoudas (1994) employed Mori–Tanaka micromechanics model to predict the effective properties of polymer matrix composites with embedded SMA fibers, in which the polymer matrix was modeled as elastic materials. There are many other research works developed to understand the thermo-mechanical behavior of SMA composites with elastic matrix. For example, Damanpack, Aghdam, and Shakeri (2015) recently presented a finite element RVE (representative volume element) model to investigate the off-axis thermo-mechanical response of composites consisting of SMA and elastic polymeric matrix, in which the thermo-mechanical behavior of SMA was described using the constitutive model developed by Panico and Brinson (2007). Birman et al., 1996, Saravanos et al., 1995 presented a combined micromechanics approach for the calculation of equivalent properties of the composite system consisting of SMA fibers within an elastic matrix in uniform thermal fields under longitudinal loads. They derived analytical solutions through the extension of Chamis’s multi-cell micro-mechanics approach (Chamis, 1983) to predict the response of SMA composites with the assumption that the rate of transformation strain tensor is proportional to the rate of the martensite volumetric fraction. Gilat and Aboudi (2004) employed generalized method of cells (Aboudi, 1996) to analyze unidirectional composites with SMA fibers embedded in polymeric or metallic matrices subjected to thermal loadings. They adopted the 3D constitutive model of Lagoudas et al. (1996) to simulate the behavior of SMA. Furthermore, the polymeric matrix was assumed to be a linearly elastic material while the metallic matrix was modeled using the unified viscoplasticity theory of Bodner (2002). Based on the framework of HFGMC (Aboudi, 2004), Freed and Aboudi (2009) developed a micromechanics model to determine the effective mechanical properties and the two-way shape memory effect of SMA composites with SMA phase embedded in elastic resin matrix or elasto-viscoplastic matrix. Jarali, Raja, and Upadhya (2008) proposed an analytical micromechanical approach to evaluate the behavior of SMA composites consisting of SMA and elastic polymeric matrix under hygrothermal environment using the equivalent inclusion method proposed by Mura (1982).

The existing literatures show that numerous researchers have worked extensively to model the thermo-mechanical behavior of SMA composites with SMA reinforcements embedded in polymer or resin matrix. However, most of the research works considered the polymer or resin matrix as elastic materials. It is known that polymer or resin materials exhibit strong time-dependent viscoelastic behavior, which in turn causes the macroscopic viscoelastic response of SMA composites. Hence, it is absolutely needed that micromechanics models dealing with the time-dependent viscoelastic behavior of SMA composites with polymer or resin matrix are established for such purpose. In this study, based on the micromechanics framework VAMUCH (Yu & Tang, 2007), a micromechanics model was proposed to determine the time-dependent and nonlinear pseudoelastic behavior of SMA composites composed of SMA reinforcements and viscoelastic polymer matrix under thermo-mechanical loadings. The behavior of SMA phase was predicted using a 3D model extended from Brinson’s one-dimensional model (Brinson and Lammering, 1993, Brinson, 1993), while the linear thermoviscoelastic behavior of the matrix was modeled by hereditary integral constitutive equation derived on the basis of Boltzmann superposition principle (Wineman & Rajagopal, 2000). Numerical examples were used to demonstrate the capability of the proposed model.