Locally-exact homogenization of viscoelastic unidirectional composites

Highlights

• An elasticity-based homogenization theory is extended to accommodate viscoelastic reinforcement via the correspondence principle.

• An accurate and efficient inversion scheme is employed to transform the unit cell solution to the time domain.

• Rapid convergence of the displacement-based solution is obtained in the time domain, validated through comparison with published data for both homogenized moduli and local stress fields.

• New results on the effect of fiber array type (hexagonal vs square) reveal the importance of viscoelasticity on the homogenized response.

• New results on the transmission of matrix viscoelastic features to macroscale support consistent construction of homogenized viscoelastic functions from experimental data.

Abstract

The elasticity-based, locally-exact homogenization theory for periodic materials with hexagonal and tetragonal symmetries is extended to accommodate linearly viscoelastic phases via the correspondence principle. The theory employs Fourier series representations for fiber and matrix displacement fields in the cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement and stress field convergence in the presence of linearly viscoelastic phases with relatively few harmonic terms. The solution’s stability and efficiency, with concommitant simplicity of input data construction, facilitate rapid identification of the impact of phase viscoelasticity and array type on homogenized moduli and local fields in wide ranges of fiber volume fraction. We illustrate the theory’s utility by investigating the impact of fiber array type and matrix viscoelastic response (constant Poisson’s ratio vs constant bulk modulus) on the homogenized response and local stress fields, reporting previously undocumented differences. Specifically, we show that initially small differences between hexagonal and square arrays are magnified substantially by viscoelasticity. New results on the transmission of matrix viscoelastic features to the macroscale are also generated in support of construction of homogenized viscoelastic functions from experimental data.

Introduction

The increasing usage of polymeric matrix composites in applications ranging from aerospace, automotive and civil engineering to bioengineering necessitates the development of predictive tools that gauge their long-term behavior. Such knowledge is key to durable and sustainable structural component designs. Polymeric matrix composites exhibit creep and stress relaxation phenomena which need to be understood in order to design durable composite-based components. For instance, time-dependent stress redistribution in a laminated composite plate due to combined stress and relaxation phenomena may lead to local ply-level failure, producing stress transfer leading to local failure at another location, and so on. Characterizing time-dependent response of polymeric matrix composites may be accomplished through experiment. This, however, is time-consuming and costly, and hence typically conducted for a chosen material system with a specific fiber volume fraction.

The alternative to testing is the use of homogenization techniques to characterize the time-dependent response of different fiber/matrix combinations in a wide fiber volume fraction range, validated experimentally against specific material combinations. The simple geometric micromechanics models of unidirectional composites based on a single fiber embedded in the matrix phase, such as the CCA (composite cylinder assemblage), Mori-Tanaka and GSC (generalized self-consistent) models, which may in turn be embedded in the homogenized medium of sought properties, (Christensen, 1979), yield estimates of homogenized moduli but typically do not provide accurate estimates of stress fields that account for adjacent fiber interaction. This may be obtained using numerical or semi-analytical approaches such as the finite-element or finite-volume methods, cf., Pindera et al. (2009) and Charalambakis (2010). These methods provide the means of modeling complex microstructure composites, but demand substantial training on the user’s part as well as time-consuming input data construction. Hence interest in elasticity-based homogenization methods for periodic microstructures, see the seminal work of Nemat-Nasser et al. (1982) which has motivated current developments of the eigenstrain expansion technique, has revived within the past 15 years, cf. Wang et al. (2005), Drago and Pindera (2008), Mogilevskaya et al. (2010), Sevostianov et al. (2012), Guinovart-Díaz et al. (2013), Caporale et al. (2015), Wang, Pindera, 2015, Wang, Pindera, 2016. The construction of input data for use with these techniques is at least an order of magnitude faster relative to numerical methods, and the execution time is comparable if not faster.

Extensive literature exists that addresses finite-element based, and more recently finite-volume based, homogenization of composite materials containing elastic, elastic-plastic and visco-plastic phases. Substantially fewer contributions are found dealing with viscoelastic response of polymeric matrix composites. The approaches employed for this class of composites include the CCA model, (Hashin et al., 1987); Mori-Tanaka method, (Li et al., 2006); spring models, (Yancey, Pindera, 1990, Jeon, Muliana, 2012); Fourier series-based eigenstrain expansion technique, (Luciano, Barbero, 1995, Caporale, Luciano, Penna, 2013); asymptotic homogenization, (Andrianov et al., 2011); and finite-volume technique, (Cavalcante and Marques, 2014). Fewer contributions still may be found that are based on the elasticity approach for periodic composites containing strictly elastic phases within square, hexagonal and tetragonal unit cell architectures, already demonstrated to be an attractive alternative to variational homogenization techniques for this class of problems.

Herein, the elasticity based locally-exact homogenization theory proposed by Drago and Pindera (2008) for rectangular and square periodic microstructures, and Wang and Pindera (2015) for hexagonal arrays with transversely isotropic phases, is further extended to accommodate linearly viscoelastic phase response via the correspondence principle. The theory differs from other elasticity-based solutions of the local unit cell problem such as the eigenstrain expansion technique, (Caporale et al., 2015), the equivalent homogeneity method, (Mogilevskaya et al., 2010), or the eigenfunction expansion technique, (Sevostianov et al., 2012), in the manner of periodic boundary conditions implementation based on a balanced variational principle. This variational principle produces rapid convergence of the displacement field which satisfies exactly the Navier’s equations and interfacial continuity conditions in the interior of the unit cell representative of rectangular, square or hexagonal periodic arrays of transversely isotropic inclusions. As a result, converged homogenized moduli and local stress fields alike are obtained with relatively few terms in the displacement field representation. The extended locally-exact homogenization theory that accommodates linearly viscoelastic phases is demonstrated herein to exhibit convergence of both homogenized relaxation moduli (or creep compliances) and local stress fields which is just as rapid.

Section 2 describes the locally-exact homogenization theory’s extension which is validated in Section 3. In Section 4 we investigate the combined effects of array type and phase relaxation moduli on the homogenized viscoelastic response and local stress fields, reporting new results, as well as the transmissibility of phase response across scales which is useful in the construction of homogenized response functions from experimental data. Specifically, we address the question whether the homogenized creep compliance elements of a unidirectional composite comprised of a viscoelastic matrix that exhibits power-law creep also exhibit power-law creep response in a wide range of fiber volume fractions. Conclusions are presented in Section 5.