International Journal of Solids and Structures
A micromechanical damage model for effective elastoplastic behavior of partially debonded ductile matrix composites
Introduction
Considerable work has been published in the literature on damage in particle-reinforced ductile matrix composites (PRDMCs). We refer to Dvorak (1991), Levy and Papazian (1991), Soboyejo et al. (1994), and Zhao and Weng, 1995, Zhao and Weng, 1996, Zhao and Weng, 1997 for a literature review. Several failure mechanisms have been observed, such as the interfacial debonding between the matrix and inclusions (Lewis et al., 1993; Whitehouse and Clyne, 1993, Whitehouse and Clyne, 1995), the particle cracking (Lloyd, 1991), and the ductile plastic failure in the matrix (Lewandowski and Liu, 1989; Llorca et al., 1991). The mechanism of failure apparently depends on many factors, such as the interfacial strength, the strength of reinforcements, the manufacturing process, and the matrix aging condition (Lewandowski and Liu, 1989).
In recent years, Ju and Chen (1994a) and Ju and Tseng, 1996, Ju and Tseng, 1997 developed micromechanical formulations to predict effective elastoplastic behavior of two-phase metal matrix composites with random particle locations and under general loading histories. They considered the first-order and the second-order stress perturbations of elastic particles on the ductile matrix, and the second-order relationship between the far-field stress and the ensemble-volume averaged stress on the basis of Ju and Chen (1994b, 1994c). On the other hand, Tohgo and Weng (1994) and Zhao and Weng (1995) proposed progressive interfacial complete debonding models for PRDMCs under triaxial tension. They used Weibull (1951) probability distribution function to describe the probability of complete interfacial particle debonding. It was postulated that the debonding of particles was controlled by the internal stresses of particles and the interfacial strength parameter. Further, Zhao and Weng (1996, 1997) derived effective elastic moduli and elastoplastic responses of partially debonded composites using fictitious, perfectly bonded transversely isotropic “equivalent” particles. Very recently, Ju and Lee (2000) proposed an elastoplastic-damage formulation based on a micromechanical framework and the ensemble-volume averaging approach for PRDMCs considering complete interfacial particle debonding. In particular, the authors predicted the overall elastoplastic behavior and damage evolution in three-phase PRDMCs based on the mechanical properties of constituent phases, particle volume fractions, random spatial inclusion distributions, micro-geometry of particles and probabilistic micromechanics.
The primary objective of the present paper is to extend the framework of Ju and Lee (2000) to assess the effects of partially debonded particle interfaces on the overall elastoplastic-damage behavior. The partial interfacial debonding mechanism in a PRDMC is displayed in Fig. 1. Specifically, when a PRDMC is subjected to a uniaxial tensile loading, the particles may partially debond on the top and bottom interfaces normal to the applied loading direction. The resulting partially debonded particles will lose their load-carrying capacity along the loading direction, but will still be able to transmit stresses to the matrix in the transverse direction through the bonded portion of the interfaces. In addition, the particles are assumed to be elastic spheres randomly dispersed in the matrix, and the ductile matrix behaves elastoplastically under uniaxial loading/unloading histories. All particles are assumed to be nonintersecting and initially embedded firmly in the matrix with perfect interfaces. It is further assumed that the partial interfacial debonding is governed by the average internal stress of a particle and the probabilistic Weibull's parameter of the particle–matrix interfacial strength.
This paper is organized as follows. In Section 2, we consider a transversely isotropic and perfectly bonded fictitious particle which is “equivalent” to a partially debonded isotropic particle. The effective elastic moduli of three-phase composites with perfectly bonded and partially debonded particles are micromechanically derived. In Section 3, the effective yield criterion and overall elastoplastic-damage characterization of three-phase composites are micromechanically constructed according to the ensemble-volume averaging procedure and the first-order eigenstrain effects owing to the randomly dispersed, perfectly bonded or partially debonded spherical particles. An evolutionary probabilistic interfacial partial particle debonding model is presented in Section 4 in accordance with the Weibull's function. The proposed probabilistic and progressive elastoplastic-damage formulation is applied to the uniaxial tensile loading in Section 5. Finally, to illustrate the potential applicability of the proposed method, the present predictions are compared with Ju and Lee's (2000) and Zhao and Weng's (1996) analytical predictions, and available experimental data in Section 6.
Section snippets
Effective elastic moduli of three-phase composites considering partial particle debonding
When a two-phase ductile matrix composite containing randomly dispersed, perfectly bonded spherical particles (see Fig. 1(a)) is subjected to remote uniaxial tensile loading, some particles may experience partial debonding on the “top” and “bottom” of the interfaces between the matrix and particles as deformations proceed (see Fig. 1(b)). A partially debonded particle will lose its load-carrying capacity along the debonded direction. Therefore, an initially two-phase composite would become a
The stress norm and eigenstrain formulation
Let us now consider the effective elastoplastic responses of progressively and partially debonded particle composites. That is, an original two-phase composite may gradually become a three-phase composite consisting of the matrix, perfectly bonded particles and partially debonded particles. For simplicity, the von Mises yield criterion with isotropic hardening law is employed in the following. Extension of the present framework to general yield criterion and general hardening law is
Evolutionary probabilistic interfacial debonding
The progressive, partial interfacial debonding may occur under increasing deformations and affect the overall stress–strain behavior of PRDMCs. After the interfacial debonding between particles and the matrix, the debonded particles lose the load-carrying capacity along the debonded direction only and are regarded as partially debonded particles. Within the framework of the first-order (noninteracting) approximation, the stresses inside particles should be uniform. Following Tohgo and Weng
Elastoplastic stress–strain relationship for partially debonded three-phase PRDMCs
In order to illustrate the proposed micromechanics-based elastoplastic damage model for PRDMCs, let us consider the example of uniaxial tensile loading here.
The applied macroscopic stress can be written asWith the simple isotropic hardening law described by Eq. (71), the overall yield function readsSubstituting Eq. (74) into Eq. (75), the effective yield function of partially debonded three-phase PRDMCs under the uniaxial loading
Numerical simulations and experimental comparison
In order to illustrate the influence of partially debonded particles on the behavior of ductile matrix composites, the present micromechanics-based predictions with varying S0 values are compared with the predictions of Ju and Lee's (2000) complete particle debonding model (cf. Fig. 3, Fig. 4) and Zhao and Weng's (1996) damage model (cf. Fig. 5). Specifically, Fig. 4 exhibits the evolution of volume fractions of partially debonded particles versus the uniaxial strains, which is corresponding to
Conclusion
A micromechanical elastoplastic-damage model considering progressive partial particle debonding is proposed to predict the overall stress–strain response and damage evolution of three-phase PRDMCs. To meet the characteristics of partially debonded interfaces, a partially debonded particle is replaced by an “equivalent”, perfectly bonded transversely isotropic particle. The effective elastic moduli of the composite are then micromechanically derived. To estimate the overall elastoplastic
Acknowledgements
This work was sponsored in part by the National Science Foundation, Mechanics and Materials Program, under the PYI Grant MSS-9157238; and in part by the Academic Senate of the Los Angeles Division of the University of California, under Fund Number 4-592565-19914-7. These supports are gratefully acknowledged.
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