Effective elastoplastic behavior of ductile matrix composites containing randomly located aligned circular fibers

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Abstract

Based on the micromechanical framework and the second-order transverse effective elastic moduli of fiber-reinforced composites derived by Ju and Zhang [Int. J. Solids Struct. 35 (1998) 941], effective elasto-(visco-)plastic behavior of two-phase unidirectional fiber-reinforced ductile matrix composites (FRDMCs) is studied in detail. The circular fibers are assumed to be elastic and unidirectionally aligned while the matrix phase can be either elastic or plastic, depending on the local stress state and effective yield criteria. Furthermore, the ensemble-averaged stress norm is constructed based on the probabilistic distribution of circular fibers, pairwise fiber interactions and the ensemble-area averaging procedure. Together with the plastic flow rule and hardening law postulated in continuum plasticity, the aforementioned stress norm is employed to characterize the overall yield criteria which determine the elastoplastic behavior of FRDMCs under general transverse plane-strain loading and unloading histories. As a special case, the initial effective yield criteria for incompressible ductile matrix containing many unidirectionally aligned cylindrical voids are also derived. In addition, the overall elasto-viscoplastic behavior of FRDMCs is investigated based on the Duvaut–Lions viscoplasticity. Finally, comparison between our theoretical predictions and the available experimental data for FRDMCs is performed to illustrate the capability of the proposed framework.

Introduction

The improvement in processing and manufacturing technology during recent years facilitates the production of advanced composite materials, including the fiber-reinforced ductile matrix composites (FRDMCs). The matrix materials of the FRDMCs are made of ductile metals or alloys (such as aluminum, steel or titanium) with high strain capability, whereas the fibers behave elastically (such as the carbon, boron or glass fibers). Once FRDMCs are loaded beyond the effective yield points, the overall elastoplastic response will follow. The optimal microstructural design of FRDMCs enables the stiffness enhancement within the elastic range and the ductility and strength control within the plastic range. Therefore, it is desirable to characterize and predict the elastoplastic behavior of FRDMCs. Although engineers can control the manufacturing of periodic fiber array in FRDMCs with large-diameter fibers, it is difficult and expensive to achieve periodic fiber distribution with small-diameter fibers (e.g., carbon or glass fibers). The present study investigates the overall elastoplastic behavior of FRDMCs with unidirectionally aligned yet randomly located circular fibers. FRDMCs offer highly directional properties such as high specific stiffnesses along the fiber direction. However, the elastic and elastoplastic properties along the transverse direction are also an important research topic.

Extensive theoretical methods exist in the literature to predict overall elastic moduli of fiber reinforced composites. See, for example, Hill, 1963, Hill, 1964a, Hashin, 1972, Hashin, 1983, Willis, 1981, Willis, 1982, Mura (1987), Zhao and Weng (1990a), Nemat-Nasser and Hori (1993), and Ju and Zhang (1998) for more details. By adopting the overall elastic moduli of fiber reinforced composites with transversely isotropic phases (Hill, 1964a, Hill, 1964b) first incorporated the plastic flow theory to evaluate the elastoplastic incremental moduli of FRDMCs. Adams (1970) utilized a finite element analysis together with classical Prandtl–Reuss flow rule to predict the inelastic behavior of unidirectional composite under transverse normal loading. Lin et al. (1972a) studied the initial yield surfaces of unidirectional B/Al composites subjected to combined longitudinal normal, transverse normal, and in-plane longitudinal shear loadings. Subsequent study (Lin et al., 1972b) indicated that the yielding initiated at opposite corners of the fiber–matrix interface closest to adjacent fibers, and that the plastic zone expanded very fast in the matrix with increasing applied loads.

Hashin (1980) proposed three-dimensional failure criteria for unidirectional fiber composites by considering four distinct failure modes, resulting in a piecewise smooth failure surface. Dvorak and Bahei-El-Din (1987) predicted the shape and position of yield surface with the bimodal plasticity theory. Further, experimental study and comparison with the bimodal theory were presented by Dvorak et al. (1988) and Dvorak (1991). Based on the Mori–Tanaka method (Mori and Tanaka, 1973) and the framework by Tandon and Weng (1988), Zhao and Weng (1990b) derived a multiaxial theory of plasticity for a class of composites containing unidirectionally aligned spheroidal inclusions, including unidirectional fibers as a special case. DeBotton and Ponte Castañeda (1993) employed a procedure proposed by Ponte Castañeda, 1991, Ponte Castañeda, 1992 to obtain both estimates and rigorous bounds for the effective energy functions of fiber-reinforced composites with general ductile behavior. More recently, Ju and Chen (1994a) proposed a micromechanical framework to predict the effective elastoplastic behavior of two-phase metal matrix composites under general loading/unloading histories by considering the first-order stress perturbations of elastic particles to the ductile matrix. Ju and Tseng, 1996, Ju and Tseng, 1997 further improved the foregoing work by incorporating the second-order stress perturbations due to pairwise particle interactions, following the work of Ju and Chen, 1994b, Ju and Chen, 1994c.

In accordance with the plasticity theory, every local matrix point has its own plastic field quantities (such as plastic strains and plastic hardening variables). For a statistically homogeneous ductile matrix composite containing randomly located yet unidirectionally aligned fibers, the Monte Carlo method would require hundreds of simulations in order to obtain pointwise elastoplastic response under specified loading history. Further, statistical averaging of Monte Carlo simulations would need to be performed to render a statistically homogenized (overall) elastoplastic response. This method is impractical owing to the complexity of random microstructure and the tremendous computational effort. An attractive alternative is to employ the ensemble-volume averaging method at the micromechanical level. In the present study, the “local stress norm” is analytically derived for the matrix plasticity formulation by a micromechanical approach which considers complete second-order pairwise inter-fiber interactions for both the elastic and plastic sub-problems. Probabilistic ensemble average is subsequently applied to obtain a homogenized plastic yield function. In addition, the plastic flow rule and hardening law are then postulated at the composite level based on continuum plasticity. Therefore, complete second-order macroscopic effective elastoplastic constitutive models are constructed for FRDMCs.

The main objective of this paper is to predict effective elastoplastic behavior of two-phase FRDMCs based on mechanical properties of the constituent phases, volume fractions, random spatial distributions and micro-geometries of the fibers. Furthermore, the fibers are assumed to be elastic circular cylinders (randomly located yet unidirectionally aligned), and the ductile matrix behaves elastoplastically under general loading histories. All fibers are assumed to be nonintersecting and embedded firmly in the matrix with perfect interfaces.

This paper is organized as follows. In Section 2, effective elastic moduli of two-phase FRDMCs are summarized based on Ju and Zhang (1998). A second-order formulation is presented in Section 3 to account for fiber interaction effects. A unified formulation is proposed to derive the overall yield functions for both fibrous and porous composites. In addition, the plastic flow rule and hardening law are postulated according to continuum plasticity to characterize the plastic behavior under general one-dimensional loading and unloading histories (in contrast to monotonic and proportional loadings assumed by many existing works in the micromechanics literature). Initial effective yield criteria for incompressible ductile matrix containing many unidirectionally aligned cylindrical voids are presented in Section 4. Furthermore, in Section 5, plane-strain transverse elastoplastic stress–strain behavior of FRDMCs is studied for both uniaxial and biaxial loading conditions. Comparison between our analytical prediction and available experimental data is also illustrated. Finally, the initial yield surfaces of FRDMCs and viscoplastic extension are derived in 6 Initial yield surfaces of fiber-reinforced ductile matrix composites, 7 Elasto-viscoplastic behavior of fiber-reinforced ductile matrix composites, respectively.

Section snippets

Effective elastic moduli of two-phase composites containing randomly located aligned circular fibers

Following the notation in Ju and Zhang (1998), a two-phase composite consists of an elastic matrix (phase 0) and unidirectionally aligned, infinitely long and randomly located elastic circular fibers (phase 1) with distinct material properties. The two phases are assumed to be perfectly bonded at interfaces. Furthermore, the composite is assumed to be in a plane-strain state throughout this paper. The relation between the stress σ and strain ϵ at any point x in the α-phase (α=0 or 1) are

Basic consideration

In this section, we consider two-phase composites consisting of elastic cylindrical fibers (with bulk and shear moduli κ1 and μ1, respectively) unidirectionally aligned in an elastoplastic matrix (with elastic bulk and shear moduli κ0 and μ0, respectively). We employ the commonly used von Mises yield criterion with an isotropic hardening law for simplicity. Extension to general yield criterion and general hardening law can be derived similarly. Therefore, the stress σ and the equivalent plastic

Initial yield criteria for incompressible ductile matrix containing randomly located yet aligned identical cylindrical voids

Let us consider a special problem in this section – the prediction of the initial yield stresses for an elastically incompressible and perfectly plastic J2-type ductile matrix containing many randomly located yet aligned identical cylindrical voids at various volume fractions. Clearly, there is nothing inside the voids and therefore the bulk and shear moduli are zero for voids. Moreover, we have φf=0. Consequently, the yield criterion becomes (Eq. (60))F̄=σ̄:T̄:σ̄23σyin which the averaged

Transverse elastoplastic behavior for fiber-reinforced ductile matrix composites

To illustrate the proposed micromechanics-based elastoplastic constitutive model for FRDMCs, let us consider two special plane-strain transverse loading conditions.

Initial yield surfaces of fiber-reinforced ductile matrix composites

For general loading conditions, the initial yield surfaces of FRDMCs can be obtained by a procedure similar to that employed in Section 4. In view of the existence of unidirectionally aligned yet randomly located fibers, the initial yield criterion reads (cf. Eq. (60))F̄=(1−φ)σ̄:T̄:σ̄23σy,where the averaged initial yield radius is K=2/3σy. After carrying out the tensor contraction in Eq. (97), we arrive at(T̄1+2T̄2)σ̄112+σ̄222+2T̄1σ̄11σ̄22+4T̄2σ̄122=y231−φ2.

In the case of transverse biaxial

Elasto-viscoplastic behavior of fiber-reinforced ductile matrix composites

Following Simo et al. (1988), Ju (1990) and Ju and Tseng (1997), the rate constitutive equations of the generalized Duvaut–Lions viscoplasticity (Duvaut and Lions, 1972) can be formulated asϵ̄̇vp=1ηC*−1:[σ̄σ̄̄],ē̇vp=−1η[ēvpē̄p],where η is the viscosity coefficient with the unit of time. Moreover, σ̄̄ and ē̄p denote the overall stress tensor and the hardening parameter, respectively, of the inviscid elastoplastic solution. σ̄ and ēvp define the total averaged stress and the hardening

Conclusion

A micromechanical framework is developed in this paper to predict effective elastoplastic behavior of two-phase FRDMCs containing many unidirectionally aligned yet randomly located elastic cylindrical fibers. We have presented a complete second-order formulation using the uniform radial distribution function of the aligned fibers, explicit pairwise fiber interactions (for both the elastic and plastic sub-problems), and the ensemble-area averaging procedure. Subsequently, a unified formulation

Acknowledgements

This work was sponsored by the National Science Foundation, Mechanics and Materials Program under PYI Grant MSS-9157238, and the CUREe-Kajima Phase II Research Project under Grant D950217. These supports are gratefully acknowledged.

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