Analysis of bifurcation buckling and imperfections effect on the microbuckling of viscoelastic composites by HFGMC micromechanics

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Abstract

Two different approaches are presented for the prediction of the microbuckling of various types of viscoelastic composites under compression. In the first one, an incremental procedure in time is employed to establish, in conjunction with the Laplace’s transform and its inversion, a determinant whose first complex root indicates the occurrence of the failure stress and strain of the composite (with no imperfections) at the critical time. In the second approach, the viscoelastic composite is assumed to possess, due to faulty manufacturing, imperfections. A perturbation expansion of the field in terms of a small parameter establishes a series of problems of various order. It is shown that the solutions of the zero and first-order problems yield, in conjunction with the Laplace’s transform and its inversion, the imperfection growth with applied loading, which asymptotically approaches the bifurcation buckling stress of the viscoelastic composite. In both approaches a repeated application of the high-fidelity generalized method of cells (HFGMC) micromechanics is employed to obtain the solutions. The offered two analyses are verified and applied on bi-layered, continuous and short fiber viscoelastic composites, as well as on viscoelastic woven composites and lattice blocks. The latter two applications necessitate the employment of multiscale HFGMC micromechanical analyses since the yarns in the weaves and the elements of the lattices are themselves unidirectional viscoelastic composites.

Introduction

When a fibrous composite material is subjected to an increasing compressive loading it fails at a certain stage, commonly associated with fiber buckling. For perfect composites, the point of failure is referred to as bifurcation buckling. If however, the composite possesses imperfections such as wavy fiber–matrix interfaces or fiber misalignment, microbuckling (local buckling) occurs through imperfection growth upon the load increase while approaching the bifurcation value. Microbuckling in layered and fiber-reinforced composite materials has been extensively studied over the years. Associating this phenomena with the limit of composites compression carrying capacity, the classical simplified analysis of Rosen (1964) and the following ones, Greszczuk, 1975, Guz et al., 2016, Parnes and Chiskis, 2002, Waas et al., 1990 just to mention a few, established the elastic buckling stress of layered and fiber reinforced materials. Generally, buckling in bending mode with short wavelength (with respect to the thickness of the stiffer phase) and buckling in shear mode with long wavelength were found to be typical to composites of small and high volume fractions (of the stiff material), respectively. However, experimental works revealed a discrepancy between their results and the elastic microbuckling predictions. Consequently, an intensive effort was made to include in the analysis various effects which reduce the elastic theoretical buckling stress, such as combined loading, geometric imperfections and material nonlinearity (Argon, 1972, Budiansky and Fleck, 1993, Gilat and Aboudi, 2008, Kyriakides et al., 1995).

While in most of the works material nonlinearity was represented by plasticity, a few studies considered time-dependent viscous material behavior. The motivation for the pioneering studies of viscoelastic buckling of layered media was the desire to shed light on geologic folding phenomena (Biot, 1961). However, the effect of viscoelasticity has an impact also in structural application of fiber-reinforced polymer matrix composites. This was demonstrated by Schapery (1992) who analyzed the microbuckling of linear viscoelastic layered composites in both shear mode (kinking) and bending mode. The latter was modeled as the buckling of a plate supported laterally by homogeneous linear viscoelastic continuum with the simplifying approximation of negligible shear traction in the fiber–matrix interface. A similar model with perfect-slip condition at the matrix-fiber interface, was employed by Bhalerao and Moon (1996a) in their investigation of time-dependent microbuckling in multi-layered viscoelastic media. Identifying microbuckling occurring during manufacturing as the driving cause of waviness, which is observed in layered (Akbarov et al., 1997) and fiber reinforced composites (Bhalerao & Moon, 1996b), they examined the bifurcation of initially straight layers as well as the growth of initial layer waviness, focusing on the corresponding characteristic dominating buckling wavelength.

An experimental evidence of the effect of matrix viscoelasticity on the long-term compressive strength in fibers direction of polymer matrix composite was provided by Violette and Schapery (2002). The investigation of unidirectional carbon/epoxy composite revealed the influence of the matrix stiffness degradation due to temperature and over time.

The interest in viscoelastic microbuckling in layered and fiber reinforced composites is ongoing in the context of rock folding, Schmalholz and Schmid (2012), as well as in the context of mechanical and biomechanical applications, see Lakes (2020) where the viscoelasticity of cellular solids, bones, tendons and ligaments is discussed. It is further enhanced upon the understanding that viscoelastic microbuckling can occur also under loading conditions which do not involve direct compression along the layers (Makke et al., 2011), is engaged with the behavior of biological systems (Su et al., 2012), and may be utilized for fabrication of material systems with tailored properties enabling for new applications (Alur et al., 2016, Sain et al., 2013). This interest in viscoelastic microbuckling in composites can benefit from the presently suggested systematic and robust micromechanical analysis approach.

In the present investigation, two methods for the prediction of bifurcation buckling of viscoelastic composites are presented. In addition, for viscoelastic composites with internal imperfections caused by faulty manufacturing, the effect of these imperfections on the composite response and their growth is tracked with the increase of applied loading. To that end, two different approaches are implemented to carry out this investigation. Both approaches employ the HFGMC micromechanical analysis which is based on the homogenization technique for composites which possess a periodic microstructure such that a repeating unit cell (RUC) can be identified. This RUC is discretized into several subcells and the governing equations, interfacial and periodic conditions are imposed in the average sense, see Aboudi et al., 2013, Aboudi et al., 2021 for details and computer programs.

In the first approach, the time-dependent local field distributions and global response are obtained via the HFGMC as a result of the application in a step-wise manner of a compressive loading on the viscoelastic composite which possesses no imperfections. In every increment another HFGMC micromechanical analysis that includes the buckling terms which arise from the linearization of the governing equations is invoked. This HFGMC model provides, in conjunction with the Laplace’s transform and its inversion, a complex determinant whose root is sought. The specific combination of time, stress and strain corresponding to the occurrence of the first root are the critical bifurcation buckling parameters.

The second approach is based on a perturbation expansion of the field variables which is implemented on the nonlinear governing equations and constitutive relations that govern the response and microbuckling of viscoelastic composites which include imperfections. These imperfections are introduced in the form of interfacial waviness between the composite constituents. The resulting interfacial jump conditions at the wavy interfaces are established by implementing a coordinate transformation together with the perturbation expansion. As a result, zero, first and higher-order problems can be established. The solution of these problems can be achieved by employing the HFGMC micromechanics analysis. The solution of the zero-order problem provides at every increment the local field distribution and the global response as a result of the application of a compressive loading. The solution of the first-order problem, which is coupled to the zero-order one, exhibits, in conjunction with the application of the Laplace’s transform and its inversion, the imperfection growth in the viscoelastic composite which asymptotically approaches its bifurcation buckling. The value of this critical stress should of course coincide with the one that was established by the determinant complex root of the first approach.

Verifications of the predictions of these methods are performed by comparisons with exact solutions that can be established in the case of bi-layered viscoelastic composites, and by approximate solutions in the case of continuous fiber composites where in both of which no micromechanics analyses are involved.

Applications are given for bi-layered, continuous and short fiber viscoelastic composites. For viscoelastic woven composites (in which the yarns are unidirectional viscoelastic composites in addition to the unreinforced viscoelastic matrix) and lattice blocks (in which the elements are unidirectional viscoelastic composites) multiscale HFGMC micromechanical analyses must be implemented. Finally, the effect of imperfect bonding between the fibers and matrix on the composite response is shown in the extreme case of loss of bonding is shear.

This article is organized as follows. Section 2 describes the computation of the field variables of the viscoelastic material which is carried out by an incremental procedure. Section 3 provides an overview of the incremental HFGMC micromechanics analysis for viscoelastic composites. This is followed by Section 4 where the HFGMC analysis that includes the buckling terms is implemented for establishing the determinant whose first complex root provides at a critical time the failure stress and strain of the composite. Section 5, describes the analysis of viscoelastic composites with imperfections by a perturbation expansion and the implementation of the HFGMCs that are needed for the solutions. Applications are given in Section 6 followed by a Conclusion section.

Section snippets

The Viscoelastic constitutive equations

The viscoelastic linear behavior of the polymeric matrix of the composite is modeled by the Boltzmann representation which is given at time t as follows: σmn(t)=tCmnpq(tτ)ε̇pq(τ)dτ,m,n,p,q=1,2,3where σmn(t) and εmn(t) the time dependent stress and strain tensors and Cmnpq(t) are the components of the time dependent stiffness tensor. For isotropic viscoelastic material, these relations reduce to σmn(t)=tΛ(tτ)ε̇pp(τ)δmndτ+2tG(tτ)ε̇mn(τ)dτwhere Λ(t) and G(t) are two time-dependent

HFGMC micromechanical analysis of viscoelastic composites

Consider a viscoelastic composite which is composed of isotropic viscoelastic and anisotropic elastic constituents. Fig. 2 shows the initial and buckling configurations of periodically bi-layered and of continuous fiber-reinforced composites which are subjected to a compressive axial stress σ̄11 loading (which is the average of the axial stress components in the phases) acting in the x1-direction. In the initial prebuckled state of perfect composites, the layers and fibers extend along the x1

Bifurcation buckling of viscoelastic composites by HFGMC

The investigation of buckling of composite materials requires the application of a nonlinear analysis (see Gilat & Aboudi, 2008, for example). In the present investigation however the analysis is based on the linearization of the governing equations as follows. The relation between the first T and second σ Piola–Kirchhoff stress tensors should be considered, which is given by, Malvern (1969) Tmn=σmpδnp+unxp=σmn+σmpunxp,m,n,p=1,2,3In addition, the equilibrium equations are: Tpnxp=0,i.e.,x

The effect of imperfections: Perturbation expansion

Thus far, the viscoelastic composites were assumed to be perfect (i.e., without imperfections). It is presently assumed that due to faulty manufacturing imperfections in the form of periodic waviness exist at the fiber–matrix interfaces. For a composite which is loaded in the x1-direction the following general form of the waviness in the x3-direction is assumed (see Fig. 2(c)–(d)): x3=A0exp(2πix1λ)=ελexp(2πix1λ)εf(x1)where ε=A0/λ1 and A0 and λ denote the amplitude of the waviness and its wave

Applications

The derived two theories are applied for the prediction of the bifurcation buckling stresses and the effect of imperfection on the microbuckling of viscoelastic bi-layered, continuous fibers and aligned short fiber composites. In addition, the applications include the analysis of viscoelastic woven composites and lattice blocks which require the implementation of a multiscale HFGMC micromechanical modeling. Finally, the effect of loss of contact in shear between the fibers and the viscoelastic

Conclusions

Two methods are offered for the prediction of bifurcation buckling of viscoelastic composites. The first one, based on the root of an established determinant, analyzes composites without imperfections whereas the second one, based on a perturbation expansion, considers composites with imperfections. The second one provides, in addition, the imperfection growth with the applied compressive loading. The predictions were verified by comparison with exact solutions for bi-layered composites and

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to Prof. Alexander Gelfgat for his assistance in the computation of the determinants of large sparse matrices. They would also like to thank Mr. Rami Eliasy for generating the RUCs of the woven composites. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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